My congratulations to numberphile for making a philosophy of mathematics video. In my opinion, this is a very interesting side of math that gets almost completely overlooked and it should get more attention
This guy (who I'm assuming is a professor) is really knowledgeable about the subject. This video is the only place on the entire internet that explains the Platonist/nominalist/fictionalist battle concisely and well. Thanks, Numberphile!
Mathematics is just the study of abstract ideas, the properties and facts of abstract objects that obey axioms. The subject emerged out of studying physical objects, and in modern times, has now been generalised to study abstract problems. Mathematics exists in the realm of thoughts.
Alq No, I'm studying to be an engineer. Applied mathematics is the only thing I've studied. Did you read the paper? They derived Wallis' pi formulation from a hydrogen atom... It shows a relationship between pure mathematics and the physical world. Maybe your abstract world is a representation of a real mathematical property of the Universe.
irving c It is certainly true that Mathematics used to be inspired by Science, where many of the insights in Mathematics came from observations in Science. However, in modern age, Mathematics has been largely independent of Science. Take for example, the space R^k where k is an arbitrary positive integer. Is there any physical space in the real world where k is, say, 50 million? Another example is a ball or other geometrical objects defined in that very R^50,000,000 space! Can you find such objects that are tangible in the real world? The answer is obvious. Mathematics is abstraction, a huge part of which is a study of the properties of Mathematical objects. It is difficult to explain what "abstraction" is (or, rather informally what "generalization" is) to someone who hasn't been exposed to such topics as Abstract Algebra, Analysis, Topology, etc.
I can render a rather subtle Socratic argument as follows: The standard way this is cashed out in analytic philosophy goes back to Frege (although it has its roots much deeper than that). The idea is that thoughts are not public. You cannot have my thought; thoughts are metaphysically private. However, the contents of thoughts are not private. Although you cannot have one and the same thought as me, you can have a thought with the same content as me. We can both think about the proposition 2+2=4. So thoughts essentially belong to the minds that think them, and are private to those minds; but the contents of thoughts (propositions, i.e. "ideas") do not belong essentially to minds and are not private. The proposition 2+2=4 doesn't need me or you (or anyone) to be thinking it for it to be true, or for it to exist. If minds suddenly disappeared from the world, so to would all thoughts. But the things that thoughts are about (propositions) didn't depend on any particular mind, and it doesn't follow that if minds disappeared then propositions would too.
I really like the talks about the philosophy of numbers. I feel they could go in depth about each one and more about where it's troubles are, and maybe even how they've gone alone historically (with more resolution). It's an interesting topic.
I used to be a nominalist before I entered university. Then linear algebra started, and this is when nominalism totally collapsed for me. It was a momentary aha moment. All it took was to reflex on the axiomatic definition of what a vector is. Vectors aren't unique concepts in this manner, but it was the thing that striked me. Vectors and vector space create a world completely detached from reality. You don't need anything real to define vector space, and you actually can't use anything real to define a vector. Some vectors can be discribed using numbers and some basis, but vectors aren't those sequences of numbers. And there are vectors that can't be fully described using numbers, for example vectors in spaces with infinite dimensionality. And everything can be a vector if operations are defined. Vectors are literally "anything, for which operations of addition and scaling is defined and well behaved". Look, I say, that I define operation multiply(my bed, any number) = my bed; add(my bed, my bed) = my bed, vector space = set, containing only my bed. Now my bed (a literal physical object that I am sitting on right now) is a vector, because it satisfies all the axioms of vector space! Not a description of the bed, not a representation of the bed, but the bed itself is a vector. And no, it's not like some vector is used to represent my bed, but my bed IS a vector in this context.
Is it not plain to you that universal can only possibly be imaginary or images in the dreaming or associative apparatus - they clearly cannot be directly immediately personally experienced, or do you suggest otherwise? You could rephrase the question two numbers exist as do descriptions exist - numbers are a form of description are they not not?
Seemingly a " nominalist" embraces those like me and your former self that take the view or merely observe that all universals are imaginary in the sense that they cannot be directly immediately personally experienced other than as images or symbols in the associative or dreaming apparatus or mind. If we rephrase the question into a form that makes some sense, might we not ask can numbers be directly immediately personally experience other than as proxies or symbols conjured up by the dreaming/associative apparatus or mind. Probly best just avoid blaubs like exist, or just recognise that they are blaubs-pronounced blorbs
My stance on this is, and I can't believe it hasn't been brought up - that Mathematics describe the relationships and laws of things in our universe. In some way. We can say anything, for example "let there be i=sqrt(-1)". However, simply by ensuring that the implications of that claim are consistent and without contradictions to itself and anything earlier, we derived a new and true relationship of our world. We might not understand, how exactly the physical world works in its most fundamental way, there are many ways to formalize things. But eventually - by ensuring those rules - we see, that left and right claims of Mathematics appear to be particularly useful in the physical world, and make claims that hold true in observation. That's why I don't think, all those things are existing, abstract objects in their own. I think, that all those concepts we have in Mathematics relate to physical relationships in one way or another, we just have no way to discover what they actually relate to fundamentally. We only observe at the very end, that all the logical conclusions we've made line up with the observed reality of our world, and so all the logical steps must somehow be true, how the world works. It's like closing your eyes and driving for 30 minutes, only by exact data where your start position is (assumed perfection) and logical deduction. Somehow those deductions tell you all the actions you have to perform. Then you open your eyes, and you are actually where you worked out you must be theoretically. In the end, observational evidence shows it's all true. But we have no idea, what exactly happened in between, fundamentally, with all the details. It's a model of the world. And the more accurately we can model it, the more accurate our predictions are. But the relationships and laws in mathematics themselves seem to be universally true, since they don't change, they are always the same for every calculation. We found a way to work with the most fundamental laws of how our universe works, without having to understand it. We just worked out the logics, what happens in between, and it works out in the end - and in my opinion - because it actually relates in one way or another, to how the physical world works.
@@IroAppe this is an approach from reality to math. In the approach you describe, math seem to be rooted in reality and is checked by it. This is intuitive. But the most fascinating thing about math for me is the fact that it can be based entirely on itself, without any connection to reality. It reverses the conclusions, not "math is correct, because it passes the reality check", but "reality is correct, because it passes the math check". This empowers me, because it gives to me an even stronger instrument, than reality check.
Numbers, letters, and symbols are all abstract units we have devised over the ages to better communicate and rationalize about the world as we experience it. Language is one of the earliest attributes we developed, but it took a while before we started writing down what we thought our language sounded like. How we decided to represent the sounds of our language in symbols draws from our creative faculty, which is linked to our senses and perception, which differs from person to person. Words are special concepts we recognize and associate with objects, ideas, emotions, etc. Numbers are a lot like words in that they allow us great freedom in describing the world; on the other hand, whereas numbers are infinite, our ability to communicate ideas, and especially to convey an experience, is limited. Our use of language as a primary means of communication is still far from perfect. Language is only our best attempt at conveying our thoughts and emotions. For example, you cannot absolutely convey your current state of being using language. Likewise, you cannot concretely define 'happy'. Granted, there are some experiences that a majority of us can agree we would say tend to make us 'happy', but 'happy' is just a string of letters. That's it. It's an identifier, and just like any other word we readily recognize, it points us to number of other things stored in our memory banks that we associate with the term. But what about when we see a word or symbol we don't recognize? It has little or no meaning at all to us, and no switches in our mind are tripped because we don't have any images or experiences to attach to the word. On the other hand, we aren't restricted by the inherent obstacles of language when engaging in mathematics. Numbers can completely and absolutely describe any physical quantity we would possibly want to record to any degree of precision we want, and then some. Numbers are only as ugly as we need them to be, but the more beautiful side of mathematics lies in an ideal world where we can say that we have a circle of radius 4 or an infinite sum of integers or even such abstract concepts as dividing zero objects among zero recipients. However, in the real world, it means very little to me if someone says they measured a sandbox to be 6 feet by 8 feet with a depth of 6 inches, so they know they can fit exactly 24 cubic feet of sand in it. The molecular make-up of the sandbox tells me that the width varies microscopically due to the uneven surfaces, and no matter how precise a measurement you give me, I can always go one decimal place further, and we can play that game on and on. Though really we're limited by the tools we use to measure said quantities, but I think we can all agree that there comes a point where a measurement is "close enough" for our purposes. We could instead say that there's a certain maximum amount of sand particles we could fit in the sandbox. It turns out that units of measure are ideal, fixed quantities we can envision, and we apply them to the best of our ability, but they really don't "fit" in the real world though they've worked wonders for us throughout many fields of study. Likewise, numbers exist in theory and are at the core of science, but the way I see it is that number is a property that describes either the ordinal ranking of something, or how many of something there are. That is, . Anything beyond that and you're treading in the hypothetical land of mathematics, where solids have faces that are perfectly flat, and we can say we know a quantity exactly. There's only so far you can go applying mathematical principles to the world we live in. As far as numbers existing, they all exist in theory: the imaginary unit, i, Euler's number, e, pi, the golden ratio, even the Fibonacci sequence, prime numbers, perfect numbers, real numbers, infinity, one, zero, rational numbers, irrational numbers, etc. They are all at the root of many natural phenomena and play major roles in many applications, but I truly believe that only natural numbers can be seen as tangible, existing concepts in the real world. I can hold three bricks, sixty thumbtacks, one shovel, and I can be number 1 in a list, number 5, number 1000000, etc. I cannot hold zero staplers or be number zero on a list. Neither of those statements makes sense. Using a number outside the natural number set, or using natural numbers for anything other than ordinal ranking or counting is to apply abstract concepts from an ideal world to a world that's not ideal, and there are plenty implications to consider when making such a move.
The problem with nominalsim is, even for simple concepts like the number 3, if you have 3 apples on one hand and 3 oranges on another, what do these two sets of fruits have in common? Would we say they have the same quantity, but that there's no such thing as quantity?
Quantity is a characteristic about sets of objects. That characteristic exists. However, the number three does not always indicate quantity! The same number three could be ordinal, and describe the order an object or a person is in a line. What is the "threeness" in common in these two uses of the word "three"? It's an abstract concept that has no real counterpart in the world. We have created that concept, it's an idea, and ideas do not exist in the same sense as material objects or entities do.
This was a terrific video with a terrific interviewee. Thank you! I want to point out that (being something closer to a mathematical fictionalist myself) that part of the problem in explaining these things has to do with confusion between the words being used, namely "true," "exist," and (barely used here at all) "real." These all stand to be delineated from each other to prevent probably equivocation (a logical fallacy) due to their colloquial synonymy. true ≠ exist ≠ real It's worth taking the time to parse these out. A fictionalist indeed won't agree that numbers offer "true" descriptions of things, but that's in part because of one of two things (depending on the sort of fictionalism you belong to): (1) nothing is real anyway (a.k.a. "anti-realism," which I do not endorse), or (2) what is real is impossible to access directly by any means available to humans and non-humans alike (a.k.a. speculative realism). As a speculative realist, the problem around numbers is that they exist (notice by saying they exist, I'm not necessarily saying they're true nor that they're real) as a type of intentional objects (which has a specific meaning derived from medieval philosophy and, more recently now, phenomenology) which give literal descriptions of real *qualities* of objects (real ones), without actually being real objects themselves nor able to "tap into the truth" of real objects. It would take a whole other video or essay to really get into this, but I think it is inaccurate to say that fictionalism is necessarily extreme. What's extreme to me would be denying that numbers exist at all (which as I've indicated, isn't what all fictionalists would argue) or that they are "merely useful" as opposed to being both useful and aesthetically interesting in their own right.
@@davejacob5208 that word implies a quantity. No matter how you turn it around its a word that describes a quantity. He's referring to the quantity of ideas he will talk about. So I knew I would get one more than 2 different ideas to listen to. With his words in a non mathematical sentence he conveyed mathematical info to me. So yea still numbers. In case you want to argue further, the dictionary states the word is a number. So yea still numbers.
@@user-el4np5xt8c he also states that he's essentially a nominalist. So perfectly in line with the argument, he hasn't used the number in a platonically mathematically sense. Either way, does it really matter? 🤷♂️
I think the (truncated) platonist idea is that numbers have properties-like primeness-independently of how you conceive of them-hence you can misconceive of a number-and therefore they have mind independent properties. As such, they exist.
I'd rather see numbers as the linguistic filter through which humans interpret their physical reality. If that makes sense to anyone else? Doesn't make mathematics "absolutely true" or "absolutely false". Since they occur to us in our thoughts, numbers are conceptually true, but since they are physically intangible, they can just as easily be called "false", if you wish to see it that way.
It makes sense to me! I was surprised when he said something like fictionalists say that mathematics is false. I know nothing about "mathematical fictionalists" (perhaps they just make themselves up ;) ", but writers of fiction tend to speak of fictional truth, and rarely about falsehood. And I'm not happy about him mixing up the concept of mathematics with the concept of number.
Maxime Boileau If I had to invent a name for this stance, I would call it idealism, and this is definitely where I come from. I would however say _a_ rather than _the_ linguistic filter. Mathematics sets itself apart from natural languages like English in that mathematical languages are carefully defined to be able to express with great specificity, and mathematicians all agree on the definitions. Finally, since I'm a physicalist, I'd say our thoughts (being patterns of neuron-firings) actually are physical entities, and just as "real" as eddies or storms. If you aren't a physicalist, then that might not follow.
I believe you just described nominalism. For example, in the case of the square root of negative one, that can be explained by breaking it down into its fundamental parts: one, square root, and subtraction, and those are all things that can be related to the real world. Even in the case of higher dimensions in mathematics, they can be used to relate to physical properties that have more than three variables, such as elements.
Well isn't mathematics sort of a language? A language can't be true of false, its type isn't a boolean far as I know. It can be used in any environment to "describe" a problem and build up frameworks to generalize and simplify the system we interact with (which could be real life, a computer program, observations basically anything we can study and break down), this is how I interpret mathematics, many toolsets within mathematics lets us express ourselves better than we could do orally. Neither of these philosophical statements proposed on the video make any sense.
Math is not 'just a language'. Languages are arbitrary, and math is logic based. Can you give me a logical reason we call cups "cups"? Probably not, it's just convention. You can, however, PROVE mathematical concepts
Aeroscience we call cups "cups" because that's the notation, then we can use further observation and define what a cup is mathematically and other tools/languages such as physics, chemistry even biology. Etc. the geometry of the cup and the elements it consists of. Basically every observable entity that defines a cup we can write in a "language". Or brain has this infrastructure to generally define the geometry of a cup and then recognize it with the notation or word "cup". If you've ever studied very basic programming it would be suitable to say that "cup" is a map pair with the geometry. Is that so difficult to comprehend? I wouldn't say that the properties of a cup is equal to any specific "concept" as you yourself phrased it. Our entire written and spoken language is just a huge dictionary mapped to structures. The KEYS can be meaningless (sometimes not, if they are based upon another key such as different suffixes or prefixes). But the structure or pattern the keys is mapped to are certainly not arbitrary, in most languages the keys even have history thus they are not as random as you might think.
Aeroscience You need to provide me with examples, sorry if I was to harsh in my preceding comment (my digital tongue is a bit demeaning). No one has been able to define mathematics clearly it's a collection of different subjects. But it is still a unified universal language. A language that is mainly used to express your logical conclusions and/or observations. I kind of get it though you can't go about saying mathematics is a language that's the same as saying programming is a language (which is clearly is not). Mathematics, Programming are the same as "talking" in a "specific manner : a language" to describe logically drawn observations to a specific demographic or interpreter, to explain how or why something works or is or should be constructed. Maybe a bit to explicit as it is more generic than that of course but I hope you understand why I could see this video as and its propositions as complete rubbish. It feels as his logics are applied to mere arithmetics which is just a very small part of of a much larger scheme.
But making divide by 0=1 does not make logical sense. You can't prove it with conventional mathematics, so you would have to add it as an axiom. The problem is, there is nothing in nature to support that axiom, so it not valid.
@@larsthomasdenstad9082 I told my bank I'm a mathematical fictionalist . . . they told me that they told me they had known that for quite a while, but didn't want to tell me and risk interrupting the cash flow from my overdraft fees.
You joke, but the idea of economic wealth is a fiction used to leverage power. Think of the Great Depression. Did natural resources and labor suddenly disappear? No, it was all there. In a very real sense, people starved to preserve the status quo of a clever lie.
What strikes me, having just watched this again, is that all these theories are based on a correspondence theory of truth. They are true, and they are criticised, based on how well they refer to the world. Even the fictionalist's argument is based on a concept of true= referring to 'the world'. I've started to consider maths as being an example of a coherence theory, where something is true because it is consistent with the other statements in the set. Of course this comes to problem of 'if there is an infinite number of numbers, can we ever check that every number is coherent with any other'. The answer, of course, is no. But in broad terms, maths seems to work because statements are consistent with each other. Disproving a mathematical theory involves finding it to be inconsistent with other, accepted, theories. It's is probably true that nominalisation brought maths about, but for these abstract terms it seems more prudent to focus on it's consistency.
I agree on all points. It's natural enough to understand the origins of mathematics in correspondence with objects or features of the physical world, and likewise our initial sense of utility in mathematics lies in its application to the physical world. But it doesn't take long before we begin to see places where that correspondence and utility break down, in a direct sense of "√2 cows" or "π/4 wheel rotations." And that breakdown is where mathematics becomes interesting in its own right, because it starts to tell us about itself. It turns out that this self is remarkably rich. Despite having no physical existence, it does have its objects and measures and properties and so on. Some of what we can develop within mathematics indeed tells us about the physical world, and that's nice, but not strictly required. But what seems most fundamental is, as you say, the principle of coherence. It would be hard to work with an axiomatic formalism that allowed incoherence. As a practical matter, it would be hard to get anything out of it. It would also be a limiting feature of a formalism if it produced incoherence. It's not hard to devise trivial examples of incoherent formalisms. It can't be used for much. But then what's the first thing we do? Try to develop a formal account of how incoherence arises, in such a way that the account itself is coherent. Then we've got something. Gödel undecidability is such an account, not of outright incoherence - a true statement that a given system is incoherent - but of proving the existence of systems the truth of whose coherence can't be decided. Isn't that interesting? I think it's interesting. And it's interesting long past the point of counting sheep, in other words concerns over what "existence" means, philosophically, with respect to these abstractions. Existence, as defined by the abstraction, is sufficient to make it interesting.
I would refer you to the story of the man in Ireland who asked how you get from X to Y and his interlocutor replied: "if I were you sir, I wouldn't go there from here. Perhaps if you asked yourself what do I mean by numbers and just remained with that question without answering it or killing it.
You know, I went to community college and there are, most certainly, "real" numbers. Haha. Anyways, there's a subtle difference between 'real' (as in reality) and existing (existence) -- although these words are used interchangeably in every day contexts. Reality refers to perceptions, senses, and even beliefs. It's actually a very loaded word that holds human interpretations of existence. You have your reality; I have mine. Someone might say "these are the realities children in (insert third world country here) face everyday." In America, middle class families live in different economic and social realities. People come from 'different' worlds, so to speak. Existence, on the other hand, tends to be more concrete. For instance, France -- its people, geographical land marks, etc-- exist. You and I can not only feel and sense objects in France, these things exist even if we can't sense them. Deaf, blind, and physically insensitive people aren't evidence the Eiffel Tower doesn't exist. They exist because objective standards allow us to verify their existence in physical reality. We can measure (use standards) the Eiffel Tower, a thing that exists, but we can't measure (at the moment) people's differing perception on artwork, love, or lifestyles. Are numbers real? Yes. Do they exist? hmmm. Standards like meters, grams, etc, which comprise of numbers help us verify if objects are actually in existence. After all, if everyone objectively measured the Eiffel Tower in meters, we'd all agree upon some accepted value in meters. However, what standards do we use to ensure that numbers actually exist?
qwerty222999 Well it's possible that, since logic is a concept that living beings create trough pattern recognition in their brain, maths is based on a human concept.Our recognition of patterns makes us want to name and symbolize those patterns that we see.So if you had the time you can pretty much find the logic behind every equation and prove that that equation is based on human understandings of the world.To summarize my opinion is that maths is the language of the universe from the living beings's perspective (mostly humans or only humans), since there is a lot of evidence that caters towards that, although I don't really think that this could be the most correct point of view describing maths.
qwerty222999 Of course languaged don't exist, in any objective or concrete sense. Its an emergent property of minds, and if you remove the minds, your remove the language. Languages are a way of communicating ideas, but they along with math are no more real than chess or mancala...
In defence of mathematical Platonism: the reason that people themselves can interact with the abstractions in mathematics is simply because consciousness is "the same type of thing" as those abstractions, which I call _information_ (i.e. consciousness and abstractions are both forms of information). The mind and the body are a fantastic link between the material world (i.e. matter) and the immaterial world (i.e. abstractions and thought).
This is extremely interesting from a self examination standpoint. With my very limited mathematical background (I watch these UA-cams out of interests, not application), I strongly relate to Nominalism (how I learned math) yet Platonism seems impossible to grasp. This helps explain why I can't warp my mind around there being 'imaginary' numbers or even something life infinity. For me, a nominalist, I need a real number to exist in my mind; for a Platonist, that number does exist regardless. VERY interesting indeed. PS I'm a soccer coach... math plays very little in my profession. I just enjoy Numberfile and Sixty Symbol videos a lot! thank you.
may i redirect you to a youtuber named "another roof"? they have an *amazing* series about how the numbers are defined, which without getting too deep into the philosophy of anything, is really interesting in it's own right
Do squares exist? Do they have properties? What are they made of… Are they made of only matter in squares, and were there no matter there cannot be squares, or do they still exist then too, ‘outside’ space and time
I seem to be somewhere between Platonism and nominalism. I do like the notion that numbers are mathematical objects, which did start off as interpretations of real phenomena but aren't restricted to it. The fact that numbers can only interact with each other in a finite number of ways, to me, seems eerily similar to how physical objects too can only interact with each other in finite ways. That doesn't mean that numbers exist like physical objects. But they're different from simple abstractions because they're not subjective. Does that make sense?
I don't agree with any of those completely, but I see pieces of each. Math doesn't really exactly describe objects. If you have 2 apples, those 2 apples are different, therefore you don't have an exact two apples, you have two sets of mass that are separate from each other that are relatively the same. And therefore an approximation of what you actually have. There is no physical depiction of a circle that has a surface, volume, or area that can be accurately calculated by pi. In fact, a circle itself is just another abstract object. I don't have a circle for a mirror, I have a mirror shaped approximately like a circle. Mathematical objects are just tools, like a hammer. The hammer strikes a nail, but the full surface of the hammer doesn't meet the full surface of the nail, there are grooves that hide surface away from each other, but at the end of the day, the job is done just as well.
But the idea that two dissimilar things count as two of something, is completely abstract. There is nothing in reality that can be measured from those two things, and equal two.
Lee Louviere i like the way you put it! interesting follow up question: do you think ideal cubes absolutely cannot exist in nature, or do you think that the reason we dont see them is because of the nature of measurement (i.e. uncertainty)? those are (very arguably) two different things.
i'm a biologist, but i know a little bit of physics and i think of it in this way: sinusoidal waves depend on circles and pi and all that jazz. a wave (which we can observe in nature in the form of light or sound or vibrations or oscillations) has a wavelength/frequency, which are related to energy (energy = planck's constant * frequency (a property directly related to wavelength). however, the uncertainty principle in physics dictates that there is a time-energy uncertainty relation, so the only way we can know a wavelength with absolute precision is to observe it for an infinitely long duration of time (roughly speaking) which is impossible; this is why measuring a perfect circle by extension is impossible. but the nature 'behind' the uncertainty follows mathematical laws where waves interfere to cause the uncertainty. this means that these waves really could be governed by what you might call the platonic circle. perhaps someone could correct me or explain if i'm reasoning incorrectly. cool things to think about, huh :-)
It all depends on your definition of exist. To me exist means has observable, measurable physical properties. This definition excludes such things as love, hate, ideas, numbers etc.
Platonism: mathmaticians are so reliable because unlike all of us in different realms, they have thoughts that are pretty clear in each step, there is no use of the subconscious mind, you might intuitively get a correct math answer but not always and the same is for all kinds of thinking (math or others) but we still use intuition in others.
yea, yet intuition can do complex calculations like 'where to place my hand to catch a moving object in space.'. it just can't deal with the abstraction of numbers. it is better at dealing with the world of mathematics directly without going through the proxy of abstraction.
Numbers are similar to language, intellectual property, thoughts, ideas, memories, and other brain states. In this regard, they are as real as the concepts of a chair, a bed, or a house. These are different from the objects (chair, bed, and house), but the concepts (i.e. brain states) are needed for humans to construct chairs, beds, and houses from real resources (wood, plastic, cotton, etc.). This does not imply that people could not sit on rocks, sleep on the ground, or live in caves without these concepts, but we would not be as comfortable doing so. We can extend this logic to another level. Wood, for example, results from Nature's DNA memory of how to grow and to reproduce trees. Rocks result from Nature's memory that SiO2, Al2O3, etc. can form crystalline grains and that gravity and rain can pack these grains together into dense solids. There is a theory, in fact, that the fundamental physical particles are merely the knowledge of how to form these superstring states from space-time fabric. Given these facts (I know the theory is not a fact..) it is reasonable to ask which is more real? Is the chair more real or the knowledge of how to build a chair and the knowledge of how it will make us more comfortable?
I think the fictionalism is the most accurate, since Mathematics are simply models of the things we consider true… Mathematics uses true logics, but isn't always using accurate representation of the true world… Mathematic is a platonic/nominalist science. Real sciences are connected to the real world, not to fictional models.
From my understanding of fictionalism 1 and 1 is 2 is just a piece of our construct of the universe that makes sense to us. But math is made of axioms that are described by other axioms within that same construct so there isn't an absolute truth to nail that construct down to. Personally I think fictionalism lacks the rigor of realist views of mathematics and if you can't stay grounded within the axioms we have (at least until they're disproven) then you just get stuck in solipsistic loops where you can't really refine anything with thinking. But I'm no expert and I've heard wiser people than me argue it both ways.
Marcus Grant Well it doesn't change if you call the numbers anything different. If you make 1 tau and 2 Mao then tau+tau is still equal to Mao and Mao is still Mao times tau. In this universe that remains constant.
SymbolX Our sciences are dependent on three things: mathematics, the scientific method, and logical reasoning. If you take aways any of those three we will end up with much less scientific knowledge.
_>is 1+1=2 fiction?_ By that same token: "Elves have pointed ears," "Leprechauns wear green," "Unicorns have horns." So it basically depends on whether you believe "true" statements can be made about fictional things. Perhaps "canon" is a better word to describe it.
The deeper problem about fictionalism is that it can't account for WHY math beliefs are so useful if they are fiction. F=MA consistently works to launch rockets, build homes, etc., whereas using some other equation like F=MA^2 fails miserably. That can only be the case if F=MA is a real part of the structure of the world, that it governs the world of objects that are moving at sub-relativistic speeds. A fictionalist can't account for why one fiction works and another does not if both are equally made up. The one that works must match something in the world.
Numbers exist, the very act of knowing about it maintains it's existence so it really doesn't matter if they're a metaphysical object or just an abstract fiction it doesn't change the fact that the idea of them exists. So I guess I'm a Platonist?
He just seems to lack an understanding of it. It's not about calling mathematics false. It's about calling mathematics a human construct. It's a system we created to govern how we do things. No, numbers do not exist. Not as abstract objects, and not as properties of objects. They only "exist" as concepts, that humans have created. I can say 1+1 = 2. That's not something determined by nature, that's something determined by humans. *We* created numbers, and yes, they are helpful. No, they don't physically exist.
Zebo12345678 They have a referent; the structural configuration of physical objects. Which is not irrelevant. Depending on the arrangement of physical substances, for instance, distinct forms of matter result. What exists is not only about material composition; it's also about structural 'shape' or form _of_ said material.
But what does that have to do with numbers? Yes, we use human-made numbers to reference natural phenomenon. When two hydrogen atoms and an oxygen atom join to make a water molecule, that isn't because numbers said 2H + 1O = H2O, it's because that's just how atoms work. Humans are the ones who decided to label the components from their number variable. It's humans who created the numbers to do that.
The question about Platonism comes from someone who has not studied Plato, and is really quite easy to address. In a Platonist standpoint: Anybody can interact with the Forms, or "abstract objects", and in fact the world would not exist as it does without this interaction. The difference is that one interacts with a chair with the senses - you see, touch, smell, etc. a chair - while one interacts with the Form of "Chairhood" through the mind. Mathematicians can interact with numbers in just the same way a person interacts with Justice, or Freedom, or Goodness, or Chairhood; through the mind. Through Reason.
Well, we interact with the physical world with our bodies. Our senses inform our minds about interactions. And we use language to describe and communicate things.
Can't we just say that "mathematical truth" is derived from the set of axioms what we've defined which are the rules of the game? It's really no different to chess it seems like. No one would seriously claim that chess moves exist outside of space and time, there are a merely a set of rules which govern the game. So the case of irrational or imaginary numbers... they just emerge from the rules of the game. The epistemology here is Mathematics and the truth or any mathematical claim is then determined by verifying it against the axioms of the epistemology... no?
Yes you are on the right track. The rule of the imaginary number is simply declared "it is a number which you can multiply by itself to produce negative one". It's simply true by definition, just like saying "all bachelors are unmarried" is simply true by definition.
bpansky Such gives rise to the practicality of language. It's kind of like how many times we will know a word and exactly how to use it and the meaning it's suppose to convey, and yet when called upon to define it we just can't find the words. Sometimes, it's the word itself being used that is it's definition.
bpansky I mean this in a certain way and within a certain context I know that I, in the past, knew how to use a word and what it implied within a context, but when asked what it meant I didn't know how to describe it but only how to use it. This is what I mean by "the word itself is the definition"
How about nothing but numbers exist? Basically, every particle has its values, its location, its properties etc. All the numbers we make up are ways to connect those values. Numbers don't exist in the sense of 1, 2, 3 but as relations. So this is twice that, and this is 3 times that and so on. The complex numbers like phi are simply infinitely long connections. In the end, if we'd take away the human interpretation of the universe, it's all just trillions times trillions times trillions times trillions of data points with interaction with one another. Just my view on the topic, be welcome to share your thought.
Pseudo Arch Because theorizing is just that, theorizing. I might be fundamentally wrong, but it is just a theory. Unless you have strong reasons/proofs of me being wrong, it will be a theory. Also, just because we are humans doesnt mean we cant think or theorize about things beyond the human ways, even though we cant fully comprehend it. If I get data on a piece of paper and I know enough formula's, I could do massive calculations on everything without ever needing to see what exactly I'm calculating. So up to a point, the universe is just raw data that changes itself (in a way, just not sure what to think of quantum physics in this)
I actually agree with you. My own personal theory is that all of reality is in fact a quantum bit. And by that I mean a piece of information that equals both zero and one at the same time (not that I consider time to even exist at this point, I just used it for lack of a better word). So ultimately this means that all combinations of the two must exist making all of reality just a product of infinity. First thing people ask me when I try to explain this is usually "How do you get from numbers to physical things?" and I tell them to look at a 3D game. You can see that you are playing within a world of seemingly solid objects but at the end of the day you could take the machine code, convert it to decimal and what your left with is a number. A number that contains all the information needed to create a 3D world. If you think of just enormous infinity actually is would it be any surprise that somewhere in there is the code for Earth and the rest of the universe?
Devilofdoom Since the observable universe is finite there is a number encoding all of it. The unobservable universe, if any? It may be infinite in which case no finite code exists.
i (as a student of math) agree most with Fictionalist but what annoys me is that existing and being true or false is getting mixed up here. for me they are totally independent...
Human cognition: recursive operation #1: objects represent other objects. #2 objects represent the relation between objects representing other objects #3 objects represent the relation between objects representing the relation between objects representing other objects etc.
I think the 'discovery' of mathematics is like the creation of a language, albeit one that's a lot more rigorous (i.e. logically deductive) in its creation than that of a spoken language. And it's a language for describing the patterns/ structures we can see or imagine. I don't think we know enough about the conscious mind to understand what it means for a structure/pattern to be mathematically describable by it (and not just by our mind, but one that's maximally 'intelligent').
The numbers in our maths are logical concepts which at first best described the physical world around us and the abstract understanding is developed and refined over time. However, those numbers have original ties to the physical world and then became abstract describing reality that may or may not exist. At the end of day, they are just representations and not the actual thing being represented. Also, human concept of numbers might be totally alien to another creature and that creature might say, ""Wtf are you are guys talking about, there's a more logical and easier way to represent these things" Like numbers, truths are concepts -- something that we've agreed on over time.
I personally adopt a stance of mathematical nominalism and only dabble in mathematical Platonism when it aids me in learning new concepts/ideas. It is much easier to treat a new thing with old rules than learn new rules and all that entails.
I agree. Physics tells us that all atoms are connected together, so there's really no such thing as 1 thing. We created that according to our senses. If our brains evolved, mathematics would change
Numbers exist. Every application they have, every property, has always existed. From the time the universe was created, the concept of a number, say, "7" was real. Base 10 also did. We gave 7 that shape and took Base 10 as our main guide. Even the newest theorem, has been true all along. Or was it something none existant at some time? You think we humans could've invented this vast, perplexing, ever expanding world of mathematics by ourselves? You see, we don't invent numbers and properties, we discover them .
I definitely see where you are going with that, but I still disagree. Because mathematics is used solely to apply to the real world, that is why we are "discovering" new mathematical concepts all the time. It's a concept we made, but we use it to apply to the real world. We were the ones who figured it would help a lot if we had a system of numbers to count our apples and oranges. Addition, Subtraction, Multiplication, and Division stemmed from that purely because we used numbers to apply to the real world, which is consistent. No matter what, 1 apple is 1 apple. We had 2 pairs of apples, and discovered that if we put them together, we get 4 apples. It doesn't matter which 2 pairs of apples we used, we would always get 4 apples. That doesn't mean that addition already existed, it just means that when we invented numbers, we also managed to invent addition, even if we didn't know it at first. Take Conway's Game of Life, for instance. He invented it, and inadvertently discovered patterns that existed from it. It definitely didn't exist before he created it, but plenty of scientists discovered all sorts of constructs that did peculiar things. Conway defined a set of rules, and the rest created itself. The same thing happened with math. We defined numbers, and because numbers are consistent - every 1 is the same, every 2 is the same - patterns and rules can always be found. That's why I believe numbers are still a completely human construct, and they do not "exist" as long as we don't *let* them exist.
Zebo12345 That's like saying that the ocean doesn't exist, because it's something we gave a name, and just used for our purposes. Giving names or using something doesn't imply it's not there. You can pretend that the number pi is not there, but it is, and is part of every circle. You can choose to see it or not, but it won't change its reality. All we did with numbers was just to give them names and symbols, just like we named our trees and oceans.
I'm pretty sure mathematics is a language- a system used to describe the relationship between different elements. The number one is no more or less real than the word "one", it's just more flexible as a tag. Irrational numbers are just hypothetical relationships that are not logical, and do not exist in our world.
I've always wondered whether it was a universal language: if there was another way people could have come up with it, or if there was no other way to describe those relationships. We've sent prime numbers into space thinking that math would be universal but it's interesting to wonder if our math it would be as different to aliens as speaking another language.
If the aliens had 8 fingers, then they may be using a different base, such as octal. We have only 10 words to describe our numbers because we had 10 fingers to count and had to repeat after 10 counts. Other than all our mathematical relationships describe the behaviour of the universe, and there is full reason to beleive that they might also have discovered maths the same way as us
Sinclairelim "Irrational numbers are just hypothetical relationships that are not logical, and do not exist in our world." Quantum theory relies on imaginary numbers. They are perfectly logical and if any numbers exist, imaginary numbers exist too.
No, the numbers describe something, that something is real, the numbers are not, they are just a description of the relationship between different elements of a given system.
I think the nominal way to define pi is to call it the ratio of the circumference of a circle to its diameter. In exactly the same manner we define “two” to be “a singularity plus a singularity”. We use an approximation for pi, sure. But the concept of its existence/definition as a ratio is as real as any other “number” which can be called “true” by definition under the rules of mathematics.
Thanks for this channel. I'm very interested in Maths, so this channel is the best opportunity. But my english isn't so good at all so I'm training with your videos my english to. The prounciation of all Numberphile members is very good/perfect. So I have no problems with searching for a translation if there is a vocab which I don't know. Thanks for all :*
0:29 "so [numbers] don't exist in the same way that tables and chairs do" 1:37 "so in the same way a table exists, a chair exists, a number exists" Talking about Platonism both times -_-
"Ontology is mathematics." - Alan Badiou "Dasein already exists." - Martin Heidegger "Our external physical reality is a mathematical structure." - Max Tegmark (Mathematical Monism) "The only kind of verification of which ontological claims are capable is phenomenological." - Paul Gorner "The Universe is the complete spatio-temporal logic of the self." - Robert Lanza (7th Principle of Biocentrism)
Is 'blue' real? One can define a range of frequencies as being blue - but that definition would be arbitrary; that is to say that if some children were raised without being told that frequency range, and then asked to define blue, they likely would not come up with the same range. Blue is just a concept we use to encapsulate a commonly perceived phenomenon. Are our perceptions 'real'? Let the ontological debate begin!
Heather Spoonheim True, if an experiment could be conducted on that I'd be very interested to see the results. You could even go as far to say the colour I have assigned to my perception of the wavelength that corresponds to "blue" might be different to yours. We both recognise that it is blue, but what do we really see? Is it the same perception?
NoriMori Yeah, I have encountered it - and it was sort of what I was trying to introduce here. To me, numbers are just adjectives - the way we describe quantity to each other, or even to ourselves. If no one is counting, numbers do not exist.
Heather Spoonheim It makes me think about things like, how different cultures and languages have interpreted and defined colours over the millennia. There are languages that don't distinguish between blue and green. And when you think about it, you could draw the lines between colours anywhere. We consider a wide range of different shades to be variants of "brown", but if we wanted we could consider some of those variants to be distinct colours, in the same way we consider pink to be distinct from red even though pink is just red with white added. In English, at least, this is not symmetrical with blue, which we still consider to be "blue" (even if we can call it something else, like "sky blue") if we add white to it. We could demarcate colours however we want!
NoriMori Every culture distinguishes between green and blue - you might be thinking of Vietnamese where they are in one group that is further delineated by a prefix - because leaves and the sky have always been delineated by humans. Red is a big one as well because blood - but, yeah, pink, purple, orange and yellow can get rather squiggly in many languages.
If I draw the blueprints for a house that hasn't been built, are those blueprints "false"? Certainly the house doesn't exist, but nor is it an abatract name for houses. If I were to build a house, I could use those blueprints I carefully planned out and thought about, but the blueprints do not become "true" if I build a house closely resembling the abstraction I put on paper. So my position is that numbers are not real, they are a mental tool, but that doesn't make math somehow "false."
+MountainBlade100 ... no. It's just true. 1+1=2... PROBABLY?! lol only the symbols are probable here, not the numbers nor their relations, which is math.
Seeing a lot of passionately written walls of text, but not seeing anything that does not fall into one of the three philosophies described in the video.
What do you mean by "fictionalists" think math is false? That seems to imply that they think it's incorrect in some way. By false do you mean not real? or do you mean incorrect? I'm leaning to think that numbers aren't real. They're just a concept we humans use to explain things. And with things like Pie and square of -1 we are beginning to see our concept, aligned with proper logic, has the power to show us things we don't understand and will never be able to. I think the very fact that imaginary numbers exist and we can prove it with math proves that there are certain un-explainable things that exist somewhere outside of our reality.
Mike Y That's what I wanna know. he said they say that math's false however very useful / helpful, however never specified in what way math is "false", as if they knew what "correct" were :p
Models aren’t real. They’re a tool to help us mentally understand or see something that might be unreal. They’re a medium that describes something like a surrealism painting but they don’t define what they’re trying to describe,
What is mathematics? That's a deep question. Here's what I think: Sets are collections of elements. Set theory deals with operations on sets of elements. Sets are the most fundamental of all mathematical objects. Sets are very general, and thus, it's only really meaningful to think of isomorphisms* between sets. *an isomorphism ("iso" meaning same, "morph" meaning shape)is a type of relation that basically says that "two sets are similar" Numbers describe one kind of commonality between sets. They describe a relation between the elements of one set and the elements of another. This set has 3 elements. Numbers are, in some sense, a manifestation of an isomorphism between sets. Number Theory describes certain properties that seem to arise in these numbers. Primeness, perfectness, etc. Geometry describes sets with elements ordered in a specific way (a shape), and the properties of these sets. Geometry makes use of a more general type of number, one that does not necessarily indicate "value", but rather one that indicates the existence of a relation. Numbers like pi, sqrt(2), etc. tell us something about the way these shapes are ordered, and similarities in their ordering from one shapes to another. Trigonometry studies triangles, the first 2 dimensional instance of a shape, and their properties. Trigonometry is extremely important. Algebra is the study of holes in a certain kind of relation: This is where variables come in. For example, x-2=0, what does x equal? Algebra introduces new kinds of ordered relations, equations and polynomials, and gives us insight into the nature of them. Algebra studies functions of the set of Real Numbers on to the set of Real Numbers. An equation is a specific instance of a polynomial. Algebra started as a study of specific instances, but was generalized to deal with polynomials over the Reals as a whole. Polynomials do not represent specific numbers, but what these instances look like over all the Real Numbers. Graphing, which comes about in algebra, allows us to study how a function changes the geometry of the Real line. Abstract Algebra studies generalizations of the Reals, as well as generalizations of polynomials and functions. Abstract Algebra leads to results such as the Fundamental Theorem of Algebra, saying that "every polynomial with complex coefficients has complex roots." Fields, Rings, Groups, Morphisms, Modules, everything. Abstract Algebra is fundamental. Topology is like geometry, except dealing with the geometry of sets themselves rather than shapes, which are more specific than just sets. Calculus is the study of smooth polynomials. Calculus is as fundamental as algebra. Calculus, in some sense, allows us to study the algebraic properties of smooth functions. The derivative, the integral, etc. these are all things that are easy to find in linear functions, but hard to find on continuous functions. Differential equations- The polynomials of calculus. Linear Algebra- Points are generalized into lines Differential Geometry- Linear Algebra, Topology, Calculus, and Abstract Algebra had a baby Mathematics is the study of relations in systems. Simple as that. As long as there exist nonhomogeneous systems, there will exist mathematics.
Agreed, Mathematics is the study of the relationships in systems, that is all. The language used to write math has been invented, the same way we invented the turbine to harness the energy of a waterfall. It definitely exists.
I think I'm a nominalist. I don't think that i, π, and the square root of two really debunk the relationship of numbers with things. i is a number that can accurately describe the third dimension in euclidean space. This becomes useful in polynomial algebra, where functions can be imagined as forms in three dimensions, as can be demonstrated by the calculation and graphing of impossible roots. Pi appears practically everywhere in nature and is exactly the 'ratio' or relation of the diameter and circumference of any circle that is properly constructed. The square root of two is the number by which a geometric square's side lengths can be multiplied to double its area. There are numbers which can be generated by stopping a computer at a random point in time that don't have any relation to objects and cannot be reproduced. Some mathematics probably does lie outside of numerical logic, but I think that arithmetic, geometry, and calculus are undeniably true.
I've discovered that I was fundamentally a mathematical nominalist, in the sense that I always thought numbers were descriptions of concrete objects. Explains why I've been so baffled by imaginary numbers and irrational numbers. Thanks, Numberphile!
Would it help if you understood imaginary to convey cannot be directly immediately personal experienced other than as an image or symbol/proxy in the dreaming/associative apparatus or mind? If you can directly immediately personally experience numbers, what exactly do you experience if you do directly immediately perorally experience when you do experience numbers if you do indeed experience numbers as directly immediately and personally as you experience or *Know, pain? So simple yes or no question:*Do* you or can you directly immediately personally experience numbers as directly immediately and personally as you experience pain or the thing on the end of your left leg as your left foot? Yes or no, and if yes, what exactly are you experiencing? Numbers or just multiplicity? Numbers are a form of description are they not?Probly (sic-because it pleases me) best avoid the question : "Do descriptions exist?" Can a mirror reflect itself? While we are at it: Where is the where the where the where?
why didn't he mention Rorty and pragmatism when he was talking about fictionalism. which reminds me of bentham idea of truth as a fiction or as an agreement.
Thank for the video and response. I guess I left it brief as well. I will just quote the Stanford Enclyopedia of Phil. : "Bentham's analytical and empirical method is especially obvious when one looks at some of his main criticisms of the law and of moral and political discourse in general. His principal target was the presence of "fictions"-in particular, legal fictions. On his view, to consider any part or aspect of a thing in abstraction from that thing is to run the risk of confusion or to cause positive deceit. While, in some cases, such "fictional" terms as "relation," "right," "power," and "possession" were of some use, in many cases their original warrant had been forgotten, so that they survived as the product of either prejudice or inattention. In those cases where the terms could be "cashed out" in terms of the properties of real things, they could continue to be used, but otherwise, they were to be abandoned." However, considering Rorty, where the possibility of change becomes politically impossible, or not possible even in the spectrum of reasonable argument, where we only refer to concrete-tangible things, Rorty concludes, since we cannot reach agreement in principle with platonists, but that it maybe be possible to simply validate their truth on a term or a proposition's effective usefulness (pragmatism.) Anyways, feel free to make any corrections here in my description, if you have the time. I am also being far too brief here. Thanks for taking the time to do the video, and to respond to my comment. It was interesting to someone, like myself, who has studied philosophy on my own and is working towards an Economics-Mathematics degree.
As a nominalist, pi is easy: it's a relative value to circular calculations. Just like the square root of -1, yeah it's more complex, but ultimately it has relational value to other numbers.
It is a work of fiction, but some people just can't handle that. They cherry pick for the good parts, and ignore the evils. Also, numbers are abstract. They don't exist physically in the universe, without us making physical number objects, so not naturally, anyway. We created numbers as a means of counting, measuring, and labeling.
Rumford Chimpenstein You can't say natural science is fiction. Natural science is literally based on observations on nature. But would you mind elaborating on your view of science?
Fantastic video! I think I’m on the end of Fictionalism in general but never considered it before now. Thanks for putting in the effort of making great videos 😁
Numbers are concepts used to define quantity. Quantity is a conceptual product of language. Nothing really exists without understanding. Existence is a conceptual product of understanding. The questions are: Is understanding a product of consciousness or sensory information?
Jeroen Verschaeve For what you said to mean anything you actually have to provide evidence and not just say that they exist as i can change one word in your sentence from "numbers" to "flying spaghetti monster".
+Jeroen Verschaeve The fourth dimension does exist in physics, but it is considered spacetime and can be shown to curve using general relativity (which, for the most part, has been demonstrated to be correct). Existing close to a gravity source will cause time to move slower than further away; ie: clocks on orbiting satellites have to be adjusted in order for GPS to work. Another extension of three dimensions into 4 dimensions could also just be time. You can create a 4-dimensional "graph" of an object's position, and watch it change throughout time. Even objects which pop into and out of reality (ie: virtual photons) would have a graph, albeit not a continuous one. In a mathematical sense, the fourth dimension does exist, as it requires 4 "dimensions" in order to perform linear algebra transformations from one three dimensional vector space into the same three dimensional vector space. Within just three dimensions, you can get rotation and scaling, but not translation. If you want to talk about the fourth dimension as some additional physical dimension, then you would need some pretty amazing evidence to back that up.
The point about pi just seems misinformed. Pi absolutely has a precise value, and you can perfectly visualize it using a circle. I think that nominalism is true in a historic sense. It's how maths was born. However, when concepts that are internally consistent take a foothold, humans are able to start treating them as a seperate world on its own, and words like 'one' and 'two' start linguistically referring to a real abstract thing. At that point, platonism becomes true. But I guess it's an on and off thing.
Pimp Trizkit The point is that things don't have to exist physically to be real, and for the reasons that I then stated. The world 'real' even sometimes has this specific meaning when used in contrast with the world 'actual'. Pi is simply the precise value of the ratio of any circle's circumference to its diameter. In turn, a circle is the collection of all points equidistant from a centre point. We don't need an ending decimal expansion to do maths with it, and one certainly doesn't need to exist to make it precise. Just because we can't express its value using a ratio of integers doesn't mean it's suddenly not within our understanding of the universe. People throw around this 'plank length' as a "unit out of which all lengths are built", but it's not like the universe is divided in an evenly spaced grid of planck lengths...
I think that the "objects" that math is derived from is a coordinate plane/space. We exist in a three dimensional space and points definitely exist through out our world. Infinitely many in fact. We use numbers to denote that space and can do intricate calculations about that space.
Isn't existing outside space and outside time a bit of a contradiction? As what you are really saying is that they exist at no place and at no time. Which is the same as they do not exist at any place and do not exist at any point in time. Which start to sound an awful lot like the definition of not existing.
Tethloach Kingofreason Yet nothing is a flawed concept. It can't exist because if it does then it is something. I've always thought its that very paradox that created reality.
It kind of makes sense to make use of the tools we are given. Math is a tool. It would not make sense to throw it away in favour of some incoherent mysticism. So in a sense it exists to us and us only. It is our own creation and ours to use. It is not a creation of some mystical being, it is a creation of logical thinking. It is a tool for logical reasoning. And sometimes, logic is the least logical thing you can imagine. At least by the definition of "logical" being that it makes sense. Hence why logic itself has led to many paradoxes. But when you realize that you are wrong, perhaps you will learn a better way. And every time you encounter a paradox, it might be a moment to learn something. For example: "I know that I know nothing" is just a way for Socrates to brag about his knowledge. Ironic, isn't it?
Boril Boyanov I would say thats impossible to know. It also depends on how you define "universe". If you mean all that is, then i guess the value is infinity. But once you define a finite amount of space-time, such as the observable universe then it will have a finite value.
Numbers are there for explaining things and complicated numbers are there because we created those numbers, because we needed them. We needed the pi number for explaining something so we created it. Let's say you want to understand a chemical reaction, and to understand it you need to form an equation and there is a power in that reaction that effects the equation but you cannot symbolize it with a simple number like 5, 7, 10. So here, you'll calculate what effect does the power has on the reaction and you'll create a complicated number to explain that effect, to understand that power, and effect of that power to the equation. So I think you can explain and understand complicated numbers in nominalism.
Unfortunately it's not that easy. If they were only communicative abstractions, why should a chinese '3' should be the same entity as an english '3'? Or does the 3 have some attributes on it's own?
The numerical symbols are part of a language. But the numbers themselves ARE the quantities. For example number 3 is the "three-ness" of something, and is different from for example "two-ness" and "four-ness", etc. I think that the platonist view has something to it.
Asking "Do numbers exist?" is like asking "Does red exist?" "Does red exist on the color wheel?" Yes! "Does red exist as a physical object?" No! "Does Harry Potter exist in the story of the book series?" Yes! "Does Harry Potter exist as a person on our planet?" No! "Do numbers exist in number theory?" Yes! "Do numbers exist as objects in the physical world?" No! Now, the fictionalist says "the discourse of math is false" and he does so because he says maths isn't a real thing. Sure, if you take certain definitions, you will end up with this conclusion. But are they useful definitions? I do think it's useful to use sentences like "1 + 1 equal 2 in maths" and while denying this might be a valid consequence of a valid definition, it's ultimately not a very useful view. It's not useful because the language of logic and mathematics you use operators such as "for all" and "exists". Sure it's all made up by humans but in that made up environment, it makes sense to talk about existence. Now, the nominalist says "numbers represent real things" and I will tell you that matrices doesn't represent real things. A pegasus (a horse with wings) doesn't represent a real being. And they're not approximations. But I can still think of them. They can be useful concepts in certain contexts, so it isn't very useful to deny this. So I think this view is actually the most problematic because we can only discuss real things. Theoretical physics would almost be impossible. You can make predictions but you don't have a representation for everything (i.e. a concrete value in a wave function before you calculate the probability of it by squaring it). Lastly, the platonist. He distinguishes between physical existence and abstract existence, physical objects and abstract objects. You can simply think of "abstract objects" as being concepts or ideas. And yes, it's useful to say concepts or ideas exist in the minds of people. They might not represent real things but thinking of "existence" in the right context is useful if you are precise about what you mean. Regarding the causal interaction of numbers, it's the same thing and I could ask "Does Harry Potter interact with me in the story of the book series?" and the answer is no. But the concept entertains me and gets the author a lot of money. Same with maths. Bottom line, you just have to be precise in what you ask. If you ask a ill-defined question or one that is open for interpretation, you might just get 42.
Through my "career" as a elementary school kid, to A-levels and now university I've went through all three stages of maths and now at the end of my bachelors degree/soon starting masters degree, I'm switching through all three every few weeks xD
What? only three options? False tricotomy? (disclaimer: "To me it seems..." should be applied to every sentence written here) Numbers are arbitrary chosen symbols to describe different systems of logic. Some logic system is based on 2 choices - 0 and 1. Sometimes it's based on 10 variations. Numbers are real to us inside our heads... but they are completely made up. Numbers have real use, but are nothing more than logic rules used inside our heads to appraise situations. Some real world problems can't be "truly" described by a value system based of a 10 variation, for instance describing the relation between a circles diameter to it's circumference. Like every logic system it's not applicable on all problems. The decimal system can't describe "1/3" absolutely accurate or Pi -it has limits in it's descriptive properties. The classical statement that "2 + 2 always equal 4" is actually only true in our base 5 or higher number system where "+" is an addition operator. So 2+2=4 does not make sense in bi-, ter- or quatenary number systems (with zeros) I think other higher primates/animals, especially those who attack or defend in groups, also think in atleast few numbers to appraise the situation but maybe in a way strange to us -just as I cannot really know at all how others percieve the color yellow in their mind.
If you asked me, "What is the most absurd combination of three words you can make in the English language?" I would have to say it'd be, "Math is false."
Does Rassel's teapot sound familiar to you? If yes, than you should proof that math is true before you state that. Let me guess, you don't have a single proof.
I feel more like math is a big concept that we follow, that is nor true nor false. Like, say, what is circle for example? I don't mean that i don't know what that is, but in "real" world does it even exist? What's a line? Does that exist? A square? It may not exist but it has it's aplications. And math certainly doesn't lie. It's just the world that we exist in isn't the same as we think it is.
In my opinion, numbers are simply lovely. Mathematics is a language, a way to communicate with nature. You understand nature through physics but with the help of mathematics. Each number signifies a physical concept. Pi is a ratio that is the property of every circle. Numbers exists if languages exist. They exist more as a concept than as objects .
***** By "theoretical math", I presume you mean pure maths, the study and development of maths itself. Once you develop the natural numbers you can then go on to define the rest of the number systems which make up the number systems I come at it from a historical perspective, maths was developed out of necessity, and whilst we can have a physical representation of what means `1' or `2' you can then make up a set of abstract rules. Then you come on to geometry where the abstraction of the physical representation is accurate to a certain number of decimal places say.
Lets say I build a missile. And I want to hit a target on the other side of the world with that missile with accuracy. I would need the missile's guidance system to interact with GPS satellites in order to hit that target. The computers in the missile and the satellite are programmed using our understanding of "math". You could pin point any location on earth with coordinates. If I were to invent a coordinate system that mapped out the earth, that doesn't use numbers, but random symbols. Then I program a computer that understands those symbols, which in turn build a satellite and a missile that runs on that computer. Could I hit a target anywhere on earth with that missile? If so, what does that say about numbers. If numbers are truly real, would that missile not work?
DongerLady ToTen I just realized, though the analogy is interesting, it fails to evade the ubiquitous nature of numbers. By comparing numbers to random symbols (what is a random symbol?). It is basically asking: if we can imagine a universe that has different laws than our own, do laws exist? It is a question we can never answer
No, your missile, using only symbols, would completely miss the target. Because the relations between numbers, and the continuity of space, are needed in order to chart the path the missile takes from the launcher to the target. The end coordinates are not enough. Renaming only the end coordinates doesn't allow you to rebuild the entire path. Even if you have all the coordinates along the paths, you also need the direction, in order to control the missile's navigations, since a missile doesn't just move arbitrarily, it moves with momentum and inertia. If you rename every point along the path, and all the directions, then you've essentially constructed the real line, and given it relations roughly equivalent to continuity and derivation. In other words, you've simply renamed numbers. You're back to numbers, with all of their original mathematical meaning.
Shiro 史郎 Wretched Egg So the same fundamental concepts of math, just with different symbology... Still math. Much like how English and French are still languages.
This is all philosophical semantic nonsense. Numbers are an abstract concept by definition, we DEFINE them to be what we want. We define everything in math, because math is simply an abstract concept that we use to derive truth through self-consistency. To say something "exists" but has no physical component is a pointless discussion. You may as well argue other abstract concepts like "love" or "music" exist. It's completely pointless. These things are labels we assign to things. They are labels and nothing more. Math is compilation of labels.
Just so you know anything you say will make you both a Nominalist , Fictionalist and Platonist , bickering which has the better view does really make you a child .
+Katie Katie , i'll use this natation: "cat" - label. cat - object. Cat exists as a physical object. You can prove that by touching it. "Cat" definetly does not exjst as an physical object, it is either an abstract object or nothing. "1" definetly does not exist as an object. The question is, does 1 exist? I don't think so, because you can't touch it or see it. in that sense, "1" is more real than 1, because we can see "1" written on paper, for example.
Katie Katie. Ammmm no. When you say "cat" you know what kind of physical object you are refering to. When you say "one" you don't know what kind of physical object you are talking about.
+Honest Tech Reviews That's exactly what I told my surgeon after the operation! Or that's what I *would* have told him, but since the tracheotomy didn't work I couldn't talk and I suffocated.
as a nominalist you would probably understand pi as the curve of a circle imperfectly represented as a number, or symbol - a fictionalist could say that the representations are applicable and repeatable but that it doesnt prove that that method of representation is the only possible way to represent something applicable and repeatable, think of an alien intelligence using alien methodology (inherently something we cannot envision or imagine) accurately and repeatedly solving or working, or that mathematics is not THE truth but part of the truth or an effective application that holds true i think all three platonist nominalist and fictionalist are accurate at the same time
You can think of 42 as the product of 6 and 7. Pi is simply the quotient of the circumference and diameter. We may not know its exact value, but it does have one that can be represented in mathematics.
Nominalism is the one I agree the most. We start mathematics describing simple things that exist in the concrete world and the more complex numbers, like the irrational or the complex themselves are just descriptions of things we cant measure (like Pi), things in more than 2 or 3 dimensions, or just descriptions about abstract concepts that are not concrete, but depend their existence from beings that think or believe is such concepts.
Unfortunately, the idealist thinking has been criticized for so long that people forgot to take what it really meant into account. Idealists never claimed numbers exist. What we propose (from before Plato) is that numbers reflect notions that are aprioristic to reality, perception and language. Scientists are idiotically aprioristic about matter, but cannot stand someone stating that underneath nature laws there is structure. As much as today we have strong evidences that grammar precedes language, we state that there is structure underneath reality. That is not to say numbers exist, per se, but they are language elements we use to conceive and communicate meaning. And this ability to see and transmit meaning is the reason we call numbers objects. Those are the very terms in Plato's books.
I've been playing with this for a while. Say you have two whole apples on a table. One is 200g and the other is 300g. If you asked anyone how many apples there were, they would say that there are two apple. "Two" represents the number of apples and yet it doesn't because they aren't the same things. In this case "two" is merely an approximation but it is a true approximation because no other number could be used to describe the number of apples on the table. How much apple "matter" is on the table is 500g which is a better approximation but still just an idea that is useful. I would take the position that numbers are ideas and are real as long as they represent reality.
for a nominalist, sqrt(-1) is the solution to the equation x^2+1=0, which is equivalent to the question: if you have x^2 objects arranged in a square with lengths x, and you add 1, what length is needed so that the resulting number of objects is equal to zero?
I think an argument a nominalist can make with complicated numbers is that they describe a property of a real object, for example six lots of seven being the size of the set of objects. It seems a trivial distinction, but it opens the door to say the square root of negative 1 exists, because it is number which allows the schrödinger equation of particle properties to be represented in exponential form (for example, of which there are many) and pi, although we dont know its exact value, saying that it is the ratio of circumference to diameter for any circular object is sufficient to define its existence.
Coming back on this, I think I combine platonist and nominalists. Numbers do exist as abstract objects. But we can't interact with them directly. What we can do, however, is what we can do with other platonic ideals: we can interact with the idea of them. And, in so doing, we learn more about them. So, in other words, we interact with the names for the abstract concept. So still nominalism, but of the platonic ideals rather than the physical objects. That's not to say we don't treat numbers as a property of an object. But that object is a shade of the platonic ideal. And what we manipulate in our minds is the platonic ideal, then convert it back to the physical world. That said, there are times where we take math further, and work in a realm where axioms are neither true nor false, or even explicitly false. But I would argue that we're then just interacting with different ideas, which means a different platonic ideal of that idea. That's what platonic ideals are: the abstract concept of a thing.
A non-dualistic alternative to Mathematical Nominalism: Mathematics describes in precise detail the infinitesimally thin boundary that separates reality from fantasy. For an object to exist in the real world, it must strictly conform to the mathematical laws that govern physical reality. One step outside those laws lies an endless realm of fantasy. One step outside the classically presumed Law of the Excluded Middle lies the realization that a mathematically defined boundary is not necessarily a member of either of the sets that it separates. As for Imaginary Numbers (at 4:39 in the video), it's easy to show how they are useful in measuring physical properties of things that exist in the real world. Electricity and magnetism, for example, are physical forces that exist natively in a Complex Number domain.
Numbers are abstractions we use to describe real life phenomena - they're expressly created for this purpose which is why it's no surprise that they do it so well. In other words, the nominalists have got it mainly right, but since numbers in fact abstract it's not necessary that they always have to be describing real things (in the same way that my thoughts about a chair don't need to correspond to any real life chair and can be detached from any practical purpose). I can conceive of √-1 in the same manner as I can conceive of multiple earths - imagination.
You know how there’s machine code for computers (binary) then there’s abstracted languages to access machine code then more abstracted languages to those languages and so on, until you have something simple like your computer (how it functions). I think math and the universe are like that. There’s an underlying base language for mathematics within the universe, but the one we’ve developed is incredibly abstracted and simplified, or rather it’s a subset of this larger, more inclusive base language.
My congratulations to numberphile for making a philosophy of mathematics video. In my opinion, this is a very interesting side of math that gets almost completely overlooked and it should get more attention
gijijijijijijijijijijji so efin true!
Ya I'm also work on mathematics related to philosphy
@@AshokKumar-os6vo what work you do?
so true..
!!!
This guy (who I'm assuming is a professor) is really knowledgeable about the subject. This video is the only place on the entire internet that explains the Platonist/nominalist/fictionalist battle concisely and well. Thanks, Numberphile!
This was one of the most fascinating videos I've watched in a long time. Thank you so much @Numberphile.
Easter Egg:
In the thumbnail: RIIDIICIIIPO
R2-D2 and C-3P0
Canal do Sky what do they mean
Ahmed Elhori, R2-D2 and C3PO are android autonoton or 'Droid' characters in George Lucas's Star Wars film franchise
Wut? They must have changed the thumbnail then, i don't see anything.
@@FunkyEspelhoCat they must have, I remember seeing those numbers there 8 years ago.
Mathematics is just the study of abstract ideas, the properties and facts of abstract objects that obey axioms. The subject emerged out of studying physical objects, and in modern times, has now been generalised to study abstract problems. Mathematics exists in the realm of thoughts.
+Alq I consider it to be the study of _relationships of quantity_.
frtard Relationship is just one of many Mathematical objects.
irving c I don't think you've ever studied pure maths.
Alq
No, I'm studying to be an engineer. Applied mathematics is the only thing I've studied. Did you read the paper? They derived Wallis' pi formulation from a hydrogen atom... It shows a relationship between pure mathematics and the physical world.
Maybe your abstract world is a representation of a real mathematical property of the Universe.
irving c It is certainly true that Mathematics used to be inspired by Science, where many of the insights in Mathematics came from observations in Science. However, in modern age, Mathematics has been largely independent of Science. Take for example, the space R^k where k is an arbitrary positive integer. Is there any physical space in the real world where k is, say, 50 million? Another example is a ball or other geometrical objects defined in that very R^50,000,000 space! Can you find such objects that are tangible in the real world? The answer is obvious.
Mathematics is abstraction, a huge part of which is a study of the properties of Mathematical objects. It is difficult to explain what "abstraction" is (or, rather informally what "generalization" is) to someone who hasn't been exposed to such topics as Abstract Algebra, Analysis, Topology, etc.
Do ideas exist? Is existence dependent on materiality? What defines existence?
I define existence.
@Orson Cart You might've outsmarted me this day, but who defines defining? That should keep you busy for a while.
@Orson Cart thats who defines who not who defines defining
@Orson Cart Don't you guys have anything better to do with your brain other than causing useless argues that noone can win?
I can render a rather subtle Socratic argument as follows:
The standard way this is cashed out in analytic philosophy goes back to Frege (although it has its roots much deeper than that). The idea is that thoughts are not public. You cannot have my thought; thoughts are metaphysically private. However, the contents of thoughts are not private. Although you cannot have one and the same thought as me, you can have a thought with the same content as me. We can both think about the proposition 2+2=4.
So thoughts essentially belong to the minds that think them, and are private to those minds; but the contents of thoughts (propositions, i.e. "ideas") do not belong essentially to minds and are not private. The proposition 2+2=4 doesn't need me or you (or anyone) to be thinking it for it to be true, or for it to exist.
If minds suddenly disappeared from the world, so to would all thoughts. But the things that thoughts are about (propositions) didn't depend on any particular mind, and it doesn't follow that if minds disappeared then propositions would too.
I really like the talks about the philosophy of numbers. I feel they could go in depth about each one and more about where it's troubles are, and maybe even how they've gone alone historically (with more resolution). It's an interesting topic.
"I feel .... "
Feel meaning what or what do you seek to convey by the word "feel"?
I used to be a nominalist before I entered university. Then linear algebra started, and this is when nominalism totally collapsed for me. It was a momentary aha moment. All it took was to reflex on the axiomatic definition of what a vector is. Vectors aren't unique concepts in this manner, but it was the thing that striked me. Vectors and vector space create a world completely detached from reality. You don't need anything real to define vector space, and you actually can't use anything real to define a vector. Some vectors can be discribed using numbers and some basis, but vectors aren't those sequences of numbers. And there are vectors that can't be fully described using numbers, for example vectors in spaces with infinite dimensionality. And everything can be a vector if operations are defined. Vectors are literally "anything, for which operations of addition and scaling is defined and well behaved". Look, I say, that I define operation multiply(my bed, any number) = my bed; add(my bed, my bed) = my bed, vector space = set, containing only my bed. Now my bed (a literal physical object that I am sitting on right now) is a vector, because it satisfies all the axioms of vector space! Not a description of the bed, not a representation of the bed, but the bed itself is a vector. And no, it's not like some vector is used to represent my bed, but my bed IS a vector in this context.
Is it not plain to you that universal can only possibly be imaginary or images in the dreaming or associative apparatus - they clearly cannot be directly immediately personally experienced, or do you suggest otherwise? You could rephrase the question two numbers exist as do descriptions exist - numbers are a form of description are they not not?
Seemingly a " nominalist" embraces those like me and your former self that take the view or merely observe that all universals are imaginary in the sense that they cannot be directly immediately personally experienced other than as images or symbols in the associative or dreaming apparatus or mind.
If we rephrase the question into a form that makes some sense, might we not ask can numbers be directly immediately personally experience other than as proxies or symbols conjured up by the dreaming/associative apparatus or mind.
Probly best just avoid blaubs like exist, or just recognise that they are blaubs-pronounced blorbs
This is similar to what is sometimes called "structuralism", look up Stewart Shapiro.
My stance on this is, and I can't believe it hasn't been brought up - that Mathematics describe the relationships and laws of things in our universe. In some way. We can say anything, for example "let there be i=sqrt(-1)". However, simply by ensuring that the implications of that claim are consistent and without contradictions to itself and anything earlier, we derived a new and true relationship of our world. We might not understand, how exactly the physical world works in its most fundamental way, there are many ways to formalize things. But eventually - by ensuring those rules - we see, that left and right claims of Mathematics appear to be particularly useful in the physical world, and make claims that hold true in observation.
That's why I don't think, all those things are existing, abstract objects in their own. I think, that all those concepts we have in Mathematics relate to physical relationships in one way or another, we just have no way to discover what they actually relate to fundamentally. We only observe at the very end, that all the logical conclusions we've made line up with the observed reality of our world, and so all the logical steps must somehow be true, how the world works.
It's like closing your eyes and driving for 30 minutes, only by exact data where your start position is (assumed perfection) and logical deduction. Somehow those deductions tell you all the actions you have to perform. Then you open your eyes, and you are actually where you worked out you must be theoretically. In the end, observational evidence shows it's all true. But we have no idea, what exactly happened in between, fundamentally, with all the details.
It's a model of the world. And the more accurately we can model it, the more accurate our predictions are. But the relationships and laws in mathematics themselves seem to be universally true, since they don't change, they are always the same for every calculation. We found a way to work with the most fundamental laws of how our universe works, without having to understand it. We just worked out the logics, what happens in between, and it works out in the end - and in my opinion - because it actually relates in one way or another, to how the physical world works.
@@IroAppe this is an approach from reality to math. In the approach you describe, math seem to be rooted in reality and is checked by it. This is intuitive. But the most fascinating thing about math for me is the fact that it can be based entirely on itself, without any connection to reality. It reverses the conclusions, not "math is correct, because it passes the reality check", but "reality is correct, because it passes the math check". This empowers me, because it gives to me an even stronger instrument, than reality check.
Numbers, letters, and symbols are all abstract units we have devised over the ages to better communicate and rationalize about the world as we experience it. Language is one of the earliest attributes we developed, but it took a while before we started writing down what we thought our language sounded like. How we decided to represent the sounds of our language in symbols draws from our creative faculty, which is linked to our senses and perception, which differs from person to person. Words are special concepts we recognize and associate with objects, ideas, emotions, etc. Numbers are a lot like words in that they allow us great freedom in describing the world; on the other hand, whereas numbers are infinite, our ability to communicate ideas, and especially to convey an experience, is limited. Our use of language as a primary means of communication is still far from perfect. Language is only our best attempt at conveying our thoughts and emotions. For example, you cannot absolutely convey your current state of being using language. Likewise, you cannot concretely define 'happy'. Granted, there are some experiences that a majority of us can agree we would say tend to make us 'happy', but 'happy' is just a string of letters. That's it. It's an identifier, and just like any other word we readily recognize, it points us to number of other things stored in our memory banks that we associate with the term. But what about when we see a word or symbol we don't recognize? It has little or no meaning at all to us, and no switches in our mind are tripped because we don't have any images or experiences to attach to the word. On the other hand, we aren't restricted by the inherent obstacles of language when engaging in mathematics. Numbers can completely and absolutely describe any physical quantity we would possibly want to record to any degree of precision we want, and then some. Numbers are only as ugly as we need them to be, but the more beautiful side of mathematics lies in an ideal world where we can say that we have a circle of radius 4 or an infinite sum of integers or even such abstract concepts as dividing zero objects among zero recipients. However, in the real world, it means very little to me if someone says they measured a sandbox to be 6 feet by 8 feet with a depth of 6 inches, so they know they can fit exactly 24 cubic feet of sand in it. The molecular make-up of the sandbox tells me that the width varies microscopically due to the uneven surfaces, and no matter how precise a measurement you give me, I can always go one decimal place further, and we can play that game on and on. Though really we're limited by the tools we use to measure said quantities, but I think we can all agree that there comes a point where a measurement is "close enough" for our purposes. We could instead say that there's a certain maximum amount of sand particles we could fit in the sandbox. It turns out that units of measure are ideal, fixed quantities we can envision, and we apply them to the best of our ability, but they really don't "fit" in the real world though they've worked wonders for us throughout many fields of study. Likewise, numbers exist in theory and are at the core of science, but the way I see it is that number is a property that describes either the ordinal ranking of something, or how many of something there are. That is, . Anything beyond that and you're treading in the hypothetical land of mathematics, where solids have faces that are perfectly flat, and we can say we know a quantity exactly. There's only so far you can go applying mathematical principles to the world we live in. As far as numbers existing, they all exist in theory: the imaginary unit, i, Euler's number, e, pi, the golden ratio, even the Fibonacci sequence, prime numbers, perfect numbers, real numbers, infinity, one, zero, rational numbers, irrational numbers, etc. They are all at the root of many natural phenomena and play major roles in many applications, but I truly believe that only natural numbers can be seen as tangible, existing concepts in the real world. I can hold three bricks, sixty thumbtacks, one shovel, and I can be number 1 in a list, number 5, number 1000000, etc. I cannot hold zero staplers or be number zero on a list. Neither of those statements makes sense. Using a number outside the natural number set, or using natural numbers for anything other than ordinal ranking or counting is to apply abstract concepts from an ideal world to a world that's not ideal, and there are plenty implications to consider when making such a move.
Mucho texto
The problem with nominalsim is, even for simple concepts like the number 3, if you have 3 apples on one hand and 3 oranges on another, what do these two sets of fruits have in common? Would we say they have the same quantity, but that there's no such thing as quantity?
Quantity is a characteristic about sets of objects. That characteristic exists. However, the number three does not always indicate quantity! The same number three could be ordinal, and describe the order an object or a person is in a line. What is the "threeness" in common in these two uses of the word "three"? It's an abstract concept that has no real counterpart in the world. We have created that concept, it's an idea, and ideas do not exist in the same sense as material objects or entities do.
This was a terrific video with a terrific interviewee. Thank you!
I want to point out that (being something closer to a mathematical fictionalist myself) that part of the problem in explaining these things has to do with confusion between the words being used, namely "true," "exist," and (barely used here at all) "real." These all stand to be delineated from each other to prevent probably equivocation (a logical fallacy) due to their colloquial synonymy.
true ≠ exist ≠ real
It's worth taking the time to parse these out. A fictionalist indeed won't agree that numbers offer "true" descriptions of things, but that's in part because of one of two things (depending on the sort of fictionalism you belong to): (1) nothing is real anyway (a.k.a. "anti-realism," which I do not endorse), or (2) what is real is impossible to access directly by any means available to humans and non-humans alike (a.k.a. speculative realism). As a speculative realist, the problem around numbers is that they exist (notice by saying they exist, I'm not necessarily saying they're true nor that they're real) as a type of intentional objects (which has a specific meaning derived from medieval philosophy and, more recently now, phenomenology) which give literal descriptions of real *qualities* of objects (real ones), without actually being real objects themselves nor able to "tap into the truth" of real objects.
It would take a whole other video or essay to really get into this, but I think it is inaccurate to say that fictionalism is necessarily extreme. What's extreme to me would be denying that numbers exist at all (which as I've indicated, isn't what all fictionalists would argue) or that they are "merely useful" as opposed to being both useful and aesthetically interesting in their own right.
yeah, I think I am believe what you believe
He starts with "we are gonna be thinking about three different schools of thoughts on this"! So he believes that the number 3 exists!
Nice! 🔥 Which means theres a 2 and 1
no, he just uses the word. also, he uses it like an adjective for a thing that is not part of mathematical language ("different schools of thoughts")
@@davejacob5208 that word implies a quantity. No matter how you turn it around its a word that describes a quantity. He's referring to the quantity of ideas he will talk about. So I knew I would get one more than 2 different ideas to listen to.
With his words in a non mathematical sentence he conveyed mathematical info to me. So yea still numbers.
In case you want to argue further, the dictionary states the word is a number. So yea still numbers.
@@user-el4np5xt8c he also states that he's essentially a nominalist. So perfectly in line with the argument, he hasn't used the number in a platonically mathematically sense. Either way, does it really matter? 🤷♂️
@@davejacob5208 Someone here no humor
I think the (truncated) platonist idea is that numbers have properties-like primeness-independently of how you conceive of them-hence you can misconceive of a number-and therefore they have mind independent properties. As such, they exist.
I ended up from A4 paper sizes to the square root of 2 to whether numbers even exist..i love youtube
I just made this same journey today, so I guess UA-cam is using the same algorithm as they were three years ago...?
4:40
* camera awkwardly stares at square root of minus one *
"Wtf is dis ".
@@RazorM97 i
I'd rather see numbers as the linguistic filter through which humans interpret their physical reality.
If that makes sense to anyone else?
Doesn't make mathematics "absolutely true" or "absolutely false". Since they occur to us in our thoughts, numbers are conceptually true, but since they are physically intangible, they can just as easily be called "false", if you wish to see it that way.
It makes sense to me! I was surprised when he said something like fictionalists say that mathematics is false. I know nothing about "mathematical fictionalists" (perhaps they just make themselves up ;) ", but writers of fiction tend to speak of fictional truth, and rarely about falsehood. And I'm not happy about him mixing up the concept of mathematics with the concept of number.
@Zousteen From Holland
how about this one man
Maxime Boileau If I had to invent a name for this stance, I would call it idealism, and this is definitely where I come from.
I would however say _a_ rather than _the_ linguistic filter. Mathematics sets itself apart from natural languages like English in that mathematical languages are carefully defined to be able to express with great specificity, and mathematicians all agree on the definitions.
Finally, since I'm a physicalist, I'd say our thoughts (being patterns of neuron-firings) actually are physical entities, and just as "real" as eddies or storms. If you aren't a physicalist, then that might not follow.
I believe you just described nominalism. For example, in the case of the square root of negative one, that can be explained by breaking it down into its fundamental parts: one, square root, and subtraction, and those are all things that can be related to the real world. Even in the case of higher dimensions in mathematics, they can be used to relate to physical properties that have more than three variables, such as elements.
Your identity (your memories, emotions, feelings) aren't physical either. Should I assume that you're also 'not real'? How will that impact my ethics?
Well isn't mathematics sort of a language? A language can't be true of false, its type isn't a boolean far as I know. It can be used in any environment to "describe" a problem and build up frameworks to generalize and simplify the system we interact with (which could be real life, a computer program, observations basically anything we can study and break down), this is how I interpret mathematics, many toolsets within mathematics lets us express ourselves better than we could do orally. Neither of these philosophical statements proposed on the video make any sense.
Math is not 'just a language'. Languages are arbitrary, and math is logic based. Can you give me a logical reason we call cups "cups"? Probably not, it's just convention. You can, however, PROVE mathematical concepts
Aeroscience we call cups "cups" because that's the notation, then we can use further observation and define what a cup is mathematically and other tools/languages such as physics, chemistry even biology. Etc. the geometry of the cup and the elements it consists of. Basically every observable entity that defines a cup we can write in a "language". Or brain has this infrastructure to generally define the geometry of a cup and then recognize it with the notation or word "cup". If you've ever studied very basic programming it would be suitable to say that "cup" is a map pair with the geometry. Is that so difficult to comprehend? I wouldn't say that the properties of a cup is equal to any specific "concept" as you yourself phrased it. Our entire written and spoken language is just a huge dictionary mapped to structures. The KEYS can be meaningless (sometimes not, if they are based upon another key such as different suffixes or prefixes). But the structure or pattern the keys is mapped to are certainly not arbitrary, in most languages the keys even have history thus they are not as random as you might think.
What I'm saying is, you can't prove anything in regards to language, as it is not logic based, unlike math.
Aeroscience You need to provide me with examples, sorry if I was to harsh in my preceding comment (my digital tongue is a bit demeaning). No one has been able to define mathematics clearly it's a collection of different subjects. But it is still a unified universal language. A language that is mainly used to express your logical conclusions and/or observations. I kind of get it though you can't go about saying mathematics is a language that's the same as saying programming is a language (which is clearly is not). Mathematics, Programming are the same as "talking" in a "specific manner : a language" to describe logically drawn observations to a specific demographic or interpreter, to explain how or why something works or is or should be constructed. Maybe a bit to explicit as it is more generic than that of course but I hope you understand why I could see this video as and its propositions as complete rubbish. It feels as his logics are applied to mere arithmetics which is just a very small part of of a much larger scheme.
But making divide by 0=1 does not make logical sense. You can't prove it with conventional mathematics, so you would have to add it as an axiom. The problem is, there is nothing in nature to support that axiom, so it not valid.
I've told my bank I'm a mathematical fictionalist... they looked at my account and agreed, but still expect my bills to be paid on time.
Banks are assholes.
@@larsthomasdenstad9082 I told my bank I'm a mathematical fictionalist . . . they told me that they told me they had known that for quite a while, but didn't want to tell me and risk interrupting the cash flow from my overdraft fees.
I told my bank I was a mathematical Platonist. In an abstract realm my wealth is numerically limitless.
You joke, but the idea of economic wealth is a fiction used to leverage power. Think of the Great Depression. Did natural resources and labor suddenly disappear? No, it was all there. In a very real sense, people starved to preserve the status quo of a clever lie.
What strikes me, having just watched this again, is that all these theories are based on a correspondence theory of truth. They are true, and they are criticised, based on how well they refer to the world. Even the fictionalist's argument is based on a concept of true= referring to 'the world'. I've started to consider maths as being an example of a coherence theory, where something is true because it is consistent with the other statements in the set. Of course this comes to problem of 'if there is an infinite number of numbers, can we ever check that every number is coherent with any other'. The answer, of course, is no. But in broad terms, maths seems to work because statements are consistent with each other. Disproving a mathematical theory involves finding it to be inconsistent with other, accepted, theories. It's is probably true that nominalisation brought maths about, but for these abstract terms it seems more prudent to focus on it's consistency.
I agree on all points. It's natural enough to understand the origins of mathematics in correspondence with objects or features of the physical world, and likewise our initial sense of utility in mathematics lies in its application to the physical world.
But it doesn't take long before we begin to see places where that correspondence and utility break down, in a direct sense of "√2 cows" or "π/4 wheel rotations." And that breakdown is where mathematics becomes interesting in its own right, because it starts to tell us about itself.
It turns out that this self is remarkably rich. Despite having no physical existence, it does have its objects and measures and properties and so on. Some of what we can develop within mathematics indeed tells us about the physical world, and that's nice, but not strictly required.
But what seems most fundamental is, as you say, the principle of coherence. It would be hard to work with an axiomatic formalism that allowed incoherence. As a practical matter, it would be hard to get anything out of it.
It would also be a limiting feature of a formalism if it produced incoherence. It's not hard to devise trivial examples of incoherent formalisms. It can't be used for much. But then what's the first thing we do? Try to develop a formal account of how incoherence arises, in such a way that the account itself is coherent. Then we've got something. Gödel undecidability is such an account, not of outright incoherence - a true statement that a given system is incoherent - but of proving the existence of systems the truth of whose coherence can't be decided.
Isn't that interesting? I think it's interesting. And it's interesting long past the point of counting sheep, in other words concerns over what "existence" means, philosophically, with respect to these abstractions. Existence, as defined by the abstraction, is sufficient to make it interesting.
I would refer you to the story of the man in Ireland who asked how you get from X to Y and his interlocutor replied: "if I were you sir, I wouldn't go there from here. Perhaps if you asked yourself what do I mean by numbers and just remained with that question without answering it or killing it.
Numbers is just a language used to define logic.
Is language non-existent too?
*****
Grunting? When I type, I don't make any noise at all. When I read, the words don't grunt in my head.
You know, I went to community college and there are, most certainly, "real" numbers. Haha. Anyways, there's a subtle difference between 'real' (as in reality) and existing (existence) -- although these words are used interchangeably in every day contexts. Reality refers to perceptions, senses, and even beliefs. It's actually a very loaded word that holds human interpretations of existence. You have your reality; I have mine. Someone might say "these are the realities children in (insert third world country here) face everyday." In America, middle class families live in different economic and social realities. People come from 'different' worlds, so to speak. Existence, on the other hand, tends to be more concrete. For instance, France -- its people, geographical land marks, etc-- exist. You and I can not only feel and sense objects in France, these things exist even if we can't sense them. Deaf, blind, and physically insensitive people aren't evidence the Eiffel Tower doesn't exist. They exist because objective standards allow us to verify their existence in physical reality. We can measure (use standards) the Eiffel Tower, a thing that exists, but we can't measure (at the moment) people's differing perception on artwork, love, or lifestyles.
Are numbers real? Yes. Do they exist? hmmm. Standards like meters, grams, etc, which comprise of numbers help us verify if objects are actually in existence. After all, if everyone objectively measured the Eiffel Tower in meters, we'd all agree upon some accepted value in meters. However, what standards do we use to ensure that numbers actually exist?
qwerty222999 Well it's possible that, since logic is a concept that living beings create trough pattern recognition in their brain, maths is based on a human concept.Our recognition of patterns makes us want to name and symbolize those patterns that we see.So if you had the time you can pretty much find the logic behind every equation and prove that that equation is based on human understandings of the world.To summarize my opinion is that maths is the language of the universe from the living beings's perspective (mostly humans or only humans), since there is a lot of evidence that caters towards that, although I don't really think that this could be the most correct point of view describing maths.
qwerty222999 Of course languaged don't exist, in any objective or concrete sense. Its an emergent property of minds, and if you remove the minds, your remove the language. Languages are a way of communicating ideas, but they along with math are no more real than chess or mancala...
qwerty222999 whoa dude
In defence of mathematical Platonism: the reason that people themselves can interact with the abstractions in mathematics is simply because consciousness is "the same type of thing" as those abstractions, which I call _information_ (i.e. consciousness and abstractions are both forms of information).
The mind and the body are a fantastic link between the material world (i.e. matter) and the immaterial world (i.e. abstractions and thought).
This is extremely interesting from a self examination standpoint. With my very limited mathematical background (I watch these UA-cams out of interests, not application), I strongly relate to Nominalism (how I learned math) yet Platonism seems impossible to grasp. This helps explain why I can't warp my mind around there being 'imaginary' numbers or even something life infinity. For me, a nominalist, I need a real number to exist in my mind; for a Platonist, that number does exist regardless. VERY interesting indeed.
PS I'm a soccer coach... math plays very little in my profession. I just enjoy Numberfile and Sixty Symbol videos a lot!
thank you.
may i redirect you to a youtuber named "another roof"? they have an *amazing* series about how the numbers are defined, which without getting too deep into the philosophy of anything, is really interesting in it's own right
Do squares exist? Do they have properties? What are they made of…
Are they made of only matter in squares, and were there no matter there cannot be squares, or do they still exist then too, ‘outside’ space and time
I seem to be somewhere between Platonism and nominalism. I do like the notion that numbers are mathematical objects, which did start off as interpretations of real phenomena but aren't restricted to it. The fact that numbers can only interact with each other in a finite number of ways, to me, seems eerily similar to how physical objects too can only interact with each other in finite ways. That doesn't mean that numbers exist like physical objects. But they're different from simple abstractions because they're not subjective. Does that make sense?
I don't agree with any of those completely, but I see pieces of each.
Math doesn't really exactly describe objects. If you have 2 apples, those 2 apples are different, therefore you don't have an exact two apples, you have two sets of mass that are separate from each other that are relatively the same. And therefore an approximation of what you actually have.
There is no physical depiction of a circle that has a surface, volume, or area that can be accurately calculated by pi. In fact, a circle itself is just another abstract object. I don't have a circle for a mirror, I have a mirror shaped approximately like a circle.
Mathematical objects are just tools, like a hammer. The hammer strikes a nail, but the full surface of the hammer doesn't meet the full surface of the nail, there are grooves that hide surface away from each other, but at the end of the day, the job is done just as well.
Lee Louviere Either way it's still two apples, much like a truck and a van are still two vehicles.
But the idea that two dissimilar things count as two of something, is completely abstract.
There is nothing in reality that can be measured from those two things, and equal two.
Lee Louviere i like the way you put it! interesting follow up question: do you think ideal cubes absolutely cannot exist in nature, or do you think that the reason we dont see them is because of the nature of measurement (i.e. uncertainty)? those are (very arguably) two different things.
i'm a biologist, but i know a little bit of physics and i think of it in this way: sinusoidal waves depend on circles and pi and all that jazz. a wave (which we can observe in nature in the form of light or sound or vibrations or oscillations) has a wavelength/frequency, which are related to energy (energy = planck's constant * frequency (a property directly related to wavelength). however, the uncertainty principle in physics dictates that there is a time-energy uncertainty relation, so the only way we can know a wavelength with absolute precision is to observe it for an infinitely long duration of time (roughly speaking) which is impossible; this is why measuring a perfect circle by extension is impossible. but the nature 'behind' the uncertainty follows mathematical laws where waves interfere to cause the uncertainty. this means that these waves really could be governed by what you might call the platonic circle. perhaps someone could correct me or explain if i'm reasoning incorrectly.
cool things to think about, huh :-)
Lee Louviere Pieces of fruit.
It all depends on your definition of exist. To me exist means has observable, measurable physical properties. This definition excludes such things as love, hate, ideas, numbers etc.
Platonism: mathmaticians are so reliable because unlike all of us in different realms, they have thoughts that are pretty clear in each step, there is no use of the subconscious mind, you might intuitively get a correct math answer but not always and the same is for all kinds of thinking (math or others) but we still use intuition in others.
yea, yet intuition can do complex calculations like 'where to place my hand to catch a moving object in space.'. it just can't deal with the abstraction of numbers. it is better at dealing with the world of mathematics directly without going through the proxy of abstraction.
Numbers are similar to language, intellectual property, thoughts, ideas, memories, and other brain states. In this regard, they are as real as the concepts of a chair, a bed, or a house. These are different from the objects (chair, bed, and house), but the concepts (i.e. brain states) are needed for humans to construct chairs, beds, and houses from real resources (wood, plastic, cotton, etc.). This does not imply that people could not sit on rocks, sleep on the ground, or live in caves without these concepts, but we would not be as comfortable doing so. We can extend this logic to another level. Wood, for example, results from Nature's DNA memory of how to grow and to reproduce trees. Rocks result from Nature's memory that SiO2, Al2O3, etc. can form crystalline grains and that gravity and rain can pack these grains together into dense solids. There is a theory, in fact, that the fundamental physical particles are merely the knowledge of how to form these superstring states from space-time fabric. Given these facts (I know the theory is not a fact..) it is reasonable to ask which is more real? Is the chair more real or the knowledge of how to build a chair and the knowledge of how it will make us more comfortable?
I think the fictionalism is the most accurate, since Mathematics are simply models of the things we consider true… Mathematics uses true logics, but isn't always using accurate representation of the true world… Mathematic is a platonic/nominalist science. Real sciences are connected to the real world, not to fictional models.
im confused. is 1+1=2 fiction?
From my understanding of fictionalism 1 and 1 is 2 is just a piece of our construct of the universe that makes sense to us. But math is made of axioms that are described by other axioms within that same construct so there isn't an absolute truth to nail that construct down to. Personally I think fictionalism lacks the rigor of realist views of mathematics and if you can't stay grounded within the axioms we have (at least until they're disproven) then you just get stuck in solipsistic loops where you can't really refine anything with thinking. But I'm no expert and I've heard wiser people than me argue it both ways.
Marcus Grant Well it doesn't change if you call the numbers anything different. If you make 1 tau and 2 Mao then tau+tau is still equal to Mao and Mao is still Mao times tau. In this universe that remains constant.
SymbolX Our sciences are dependent on three things: mathematics, the scientific method, and logical reasoning. If you take aways any of those three we will
end up with much less scientific knowledge.
_>is 1+1=2 fiction?_
By that same token: "Elves have pointed ears," "Leprechauns wear green," "Unicorns have horns."
So it basically depends on whether you believe "true" statements can be made about fictional things.
Perhaps "canon" is a better word to describe it.
The deeper problem about fictionalism is that it can't account for WHY math beliefs are so useful if they are fiction. F=MA consistently works to launch rockets, build homes, etc., whereas using some other equation like F=MA^2 fails miserably. That can only be the case if F=MA is a real part of the structure of the world, that it governs the world of objects that are moving at sub-relativistic speeds. A fictionalist can't account for why one fiction works and another does not if both are equally made up. The one that works must match something in the world.
Numbers exist, the very act of knowing about it maintains it's existence so it really doesn't matter if they're a metaphysical object or just an abstract fiction it doesn't change the fact that the idea of them exists.
So I guess I'm a Platonist?
That portrayal of mathematical fictionalism was very far from neutral.
Agreed.
It's more neutral than I could have ever managed.
He just seems to lack an understanding of it. It's not about calling mathematics false. It's about calling mathematics a human construct. It's a system we created to govern how we do things. No, numbers do not exist. Not as abstract objects, and not as properties of objects. They only "exist" as concepts, that humans have created. I can say 1+1 = 2. That's not something determined by nature, that's something determined by humans. *We* created numbers, and yes, they are helpful. No, they don't physically exist.
Zebo12345678 They have a referent; the structural configuration of physical objects. Which is not irrelevant. Depending on the arrangement of physical substances, for instance, distinct forms of matter result. What exists is not only about material composition; it's also about structural 'shape' or form _of_ said material.
But what does that have to do with numbers? Yes, we use human-made numbers to reference natural phenomenon. When two hydrogen atoms and an oxygen atom join to make a water molecule, that isn't because numbers said 2H + 1O = H2O, it's because that's just how atoms work. Humans are the ones who decided to label the components from their number variable. It's humans who created the numbers to do that.
0:13 plato's theory of forms
The question about Platonism comes from someone who has not studied Plato, and is really quite easy to address. In a Platonist standpoint: Anybody can interact with the Forms, or "abstract objects", and in fact the world would not exist as it does without this interaction. The difference is that one interacts with a chair with the senses - you see, touch, smell, etc. a chair - while one interacts with the Form of "Chairhood" through the mind. Mathematicians can interact with numbers in just the same way a person interacts with Justice, or Freedom, or Goodness, or Chairhood; through the mind. Through Reason.
Well, we interact with the physical world with our bodies. Our senses inform our minds about interactions. And we use language to describe and communicate things.
Can't we just say that "mathematical truth" is derived from the set of axioms what we've defined which are the rules of the game? It's really no different to chess it seems like. No one would seriously claim that chess moves exist outside of space and time, there are a merely a set of rules which govern the game.
So the case of irrational or imaginary numbers... they just emerge from the rules of the game. The epistemology here is Mathematics and the truth or any mathematical claim is then determined by verifying it against the axioms of the epistemology... no?
exactly, it was kind of weird that they were conflating, truth as in correctness through a set of axioms, and truth as in a universal law in reality.
Yes you are on the right track. The rule of the imaginary number is simply declared "it is a number which you can multiply by itself to produce negative one". It's simply true by definition, just like saying "all bachelors are unmarried" is simply true by definition.
bpansky
Such gives rise to the practicality of language.
It's kind of like how many times we will know a word and exactly how to use it and the meaning it's suppose to convey, and yet when called upon to define it we just can't find the words.
Sometimes, it's the word itself being used that is it's definition.
"Sometimes, it's the word itself being used that is it's definition."
That sounds like complete nonsense. Do you have an example?
bpansky
I mean this in a certain way and within a certain context
I know that I, in the past, knew how to use a word and what it implied within a context, but when asked what it meant I didn't know how to describe it but only how to use it.
This is what I mean by "the word itself is the definition"
How about nothing but numbers exist? Basically, every particle has its values, its location, its properties etc. All the numbers we make up are ways to connect those values. Numbers don't exist in the sense of 1, 2, 3 but as relations. So this is twice that, and this is 3 times that and so on. The complex numbers like phi are simply infinitely long connections.
In the end, if we'd take away the human interpretation of the universe, it's all just trillions times trillions times trillions times trillions of data points with interaction with one another.
Just my view on the topic, be welcome to share your thought.
How can you, a human, theorize what the universe is like in the absence of yourself?
Pseudo Arch Because theorizing is just that, theorizing. I might be fundamentally wrong, but it is just a theory. Unless you have strong reasons/proofs of me being wrong, it will be a theory.
Also, just because we are humans doesnt mean we cant think or theorize about things beyond the human ways, even though we cant fully comprehend it. If I get data on a piece of paper and I know enough formula's, I could do massive calculations on everything without ever needing to see what exactly I'm calculating. So up to a point, the universe is just raw data that changes itself (in a way, just not sure what to think of quantum physics in this)
Thats a bit like nominalism, which I support!!!!
I actually agree with you. My own personal theory is that all of reality is in fact a quantum bit. And by that I mean a piece of information that equals both zero and one at the same time (not that I consider time to even exist at this point, I just used it for lack of a better word). So ultimately this means that all combinations of the two must exist making all of reality just a product of infinity.
First thing people ask me when I try to explain this is usually "How do you get from numbers to physical things?" and I tell them to look at a 3D game. You can see that you are playing within a world of seemingly solid objects but at the end of the day you could take the machine code, convert it to decimal and what your left with is a number. A number that contains all the information needed to create a 3D world. If you think of just enormous infinity actually is would it be any surprise that somewhere in there is the code for Earth and the rest of the universe?
Devilofdoom Since the observable universe is finite there is a number encoding all of it. The unobservable universe, if any? It may be infinite in which case no finite code exists.
i (as a student of math) agree most with Fictionalist but what annoys me is that existing and being true or false is getting mixed up here. for me they are totally independent...
Human cognition: recursive operation #1: objects represent other objects. #2 objects represent the relation between objects representing other objects #3 objects represent the relation between objects representing the relation between objects representing other objects etc.
It's fascinating as well as depressing to see something so much powerful and magical can't even define it's existence.
I think the 'discovery' of mathematics is like the creation of a language, albeit one that's a lot more rigorous (i.e. logically deductive) in its creation than that of a spoken language. And it's a language for describing the patterns/ structures we can see or imagine.
I don't think we know enough about the conscious mind to understand what it means for a structure/pattern to be mathematically describable by it (and not just by our mind, but one that's maximally 'intelligent').
The numbers in our maths are logical concepts which at first best described the physical world around us and the abstract understanding is developed and refined over time. However, those numbers have original ties to the physical world and then became abstract describing reality that may or may not exist. At the end of day, they are just representations and not the actual thing being represented. Also, human concept of numbers might be totally alien to another creature and that creature might say, ""Wtf are you are guys talking about, there's a more logical and easier way to represent these things"
Like numbers, truths are concepts -- something that we've agreed on over time.
I personally adopt a stance of mathematical nominalism and only dabble in mathematical Platonism when it aids me in learning new concepts/ideas. It is much easier to treat a new thing with old rules than learn new rules and all that entails.
For me, mathematics is an idea. Just like any other idea. It exist in the minds. Also, it's a form of communication.
I agree. Physics tells us that all atoms are connected together, so there's really no such thing as 1 thing. We created that according to our senses. If our brains evolved, mathematics would change
Numbers exist. Every application they have, every property, has always existed. From the time the universe was created, the concept of a number, say, "7" was real. Base 10 also did. We gave 7 that shape and took Base 10 as our main guide. Even the newest theorem, has been true all along. Or was it something none existant at some time? You think we humans could've invented this vast, perplexing, ever expanding world of mathematics by ourselves? You see, we don't invent numbers and properties, we discover them .
I definitely see where you are going with that, but I still disagree. Because mathematics is used solely to apply to the real world, that is why we are "discovering" new mathematical concepts all the time. It's a concept we made, but we use it to apply to the real world.
We were the ones who figured it would help a lot if we had a system of numbers to count our apples and oranges. Addition, Subtraction, Multiplication, and Division stemmed from that purely because we used numbers to apply to the real world, which is consistent. No matter what, 1 apple is 1 apple.
We had 2 pairs of apples, and discovered that if we put them together, we get 4 apples. It doesn't matter which 2 pairs of apples we used, we would always get 4 apples. That doesn't mean that addition already existed, it just means that when we invented numbers, we also managed to invent addition, even if we didn't know it at first.
Take Conway's Game of Life, for instance. He invented it, and inadvertently discovered patterns that existed from it. It definitely didn't exist before he created it, but plenty of scientists discovered all sorts of constructs that did peculiar things. Conway defined a set of rules, and the rest created itself.
The same thing happened with math. We defined numbers, and because numbers are consistent - every 1 is the same, every 2 is the same - patterns and rules can always be found. That's why I believe numbers are still a completely human construct, and they do not "exist" as long as we don't *let* them exist.
Zebo12345 That's like saying that the ocean doesn't exist, because it's something we gave a name, and just used for our purposes.
Giving names or using something doesn't imply it's not there. You can pretend that the number pi is not there, but it is, and is part of every circle.
You can choose to see it or not, but it won't change its reality. All we did with numbers was just to give them names and symbols, just like we named our trees and oceans.
I'm pretty sure mathematics is a language- a system used to describe the relationship between different elements. The number one is no more or less real than the word "one", it's just more flexible as a tag.
Irrational numbers are just hypothetical relationships that are not logical, and do not exist in our world.
I've always wondered whether it was a universal language: if there was another way people could have come up with it, or if there was no other way to describe those relationships.
We've sent prime numbers into space thinking that math would be universal but it's interesting to wonder if our math it would be as different to aliens as speaking another language.
If the aliens had 8 fingers, then they may be using a different base, such as octal. We have only 10 words to describe our numbers because we had 10 fingers to count and had to repeat after 10 counts. Other than all our mathematical relationships describe the behaviour of the universe, and there is full reason to beleive that they might also have discovered maths the same way as us
Sinclairelim "Irrational numbers are just hypothetical relationships that are not logical, and do not exist in our world."
Quantum theory relies on imaginary numbers. They are perfectly logical and if any numbers exist, imaginary numbers exist too.
No, the numbers describe something, that something is real, the numbers are not, they are just a description of the relationship between different elements of a given system.
Almost all measurements in the real world are irrational, we just approximate them.
I think the nominal way to define pi is to call it the ratio of the circumference of a circle to its diameter.
In exactly the same manner we define “two” to be “a singularity plus a singularity”.
We use an approximation for pi, sure. But the concept of its existence/definition as a ratio is as real as any other “number” which can be called “true” by definition under the rules of mathematics.
Thanks for this channel. I'm very interested in Maths, so this channel is the best opportunity. But my english isn't so good at all so I'm training with your videos my english to. The prounciation of all Numberphile members is very good/perfect. So I have no problems with searching for a translation if there is a vocab which I don't know.
Thanks for all :*
I like Seven Up, more than I like sprite
+Miguel Almeida If Sprite was called 8up I may like it better as well. So I like Mountain Dew. Caffeine.
best comment
CERTAINLY.
7 likes for you
Passionate Blow with a Right Jab to the abdomen of (what is analytical in relation to logical).
0:29 "so [numbers] don't exist in the same way that tables and chairs do"
1:37 "so in the same way a table exists, a chair exists, a number exists"
Talking about Platonism both times -_-
there's a difference between just existing and existing in the *same* way 😂
@@aditichegu1951 „so numbers don't exist in the SAME way a table exists“
„so im the SAME way a table extists, numbers exist“
"Ontology is mathematics." - Alan Badiou
"Dasein already exists." - Martin Heidegger
"Our external physical reality is a mathematical structure." - Max Tegmark (Mathematical Monism)
"The only kind of verification of which ontological claims are capable is phenomenological." - Paul Gorner
"The Universe is the complete spatio-temporal logic of the self." - Robert Lanza (7th Principle of Biocentrism)
Is 'blue' real? One can define a range of frequencies as being blue - but that definition would be arbitrary; that is to say that if some children were raised without being told that frequency range, and then asked to define blue, they likely would not come up with the same range.
Blue is just a concept we use to encapsulate a commonly perceived phenomenon. Are our perceptions 'real'? Let the ontological debate begin!
Heather Spoonheim True, if an experiment could be conducted on that I'd be very interested to see the results. You could even go as far to say the colour I have assigned to my perception of the wavelength that corresponds to "blue" might be different to yours. We both recognise that it is blue, but what do we really see? Is it the same perception?
Stark Clark That question is the root of the concept of _qualia_. It's an awesome concept, if you haven't already you should read all about it. :D
NoriMori Yeah, I have encountered it - and it was sort of what I was trying to introduce here. To me, numbers are just adjectives - the way we describe quantity to each other, or even to ourselves. If no one is counting, numbers do not exist.
Heather Spoonheim It makes me think about things like, how different cultures and languages have interpreted and defined colours over the millennia. There are languages that don't distinguish between blue and green. And when you think about it, you could draw the lines between colours anywhere. We consider a wide range of different shades to be variants of "brown", but if we wanted we could consider some of those variants to be distinct colours, in the same way we consider pink to be distinct from red even though pink is just red with white added. In English, at least, this is not symmetrical with blue, which we still consider to be "blue" (even if we can call it something else, like "sky blue") if we add white to it. We could demarcate colours however we want!
NoriMori Every culture distinguishes between green and blue - you might be thinking of Vietnamese where they are in one group that is further delineated by a prefix - because leaves and the sky have always been delineated by humans. Red is a big one as well because blood - but, yeah, pink, purple, orange and yellow can get rather squiggly in many languages.
If I draw the blueprints for a house that hasn't been built, are those blueprints "false"? Certainly the house doesn't exist, but nor is it an abatract name for houses. If I were to build a house, I could use those blueprints I carefully planned out and thought about, but the blueprints do not become "true" if I build a house closely resembling the abstraction I put on paper. So my position is that numbers are not real, they are a mental tool, but that doesn't make math somehow "false."
+Ryan N Yea , it only makes it "probable" .
+MountainBlade100 ... no. It's just true. 1+1=2... PROBABLY?! lol only the symbols are probable here, not the numbers nor their relations, which is math.
Seeing a lot of passionately written walls of text, but not seeing anything that does not fall into one of the three philosophies described in the video.
Booga .Boo What is your position on the subject?
This video made me think about this more than I thought I would.
What do you mean by "fictionalists" think math is false? That seems to imply that they think it's incorrect in some way. By false do you mean not real? or do you mean incorrect?
I'm leaning to think that numbers aren't real. They're just a concept we humans use to explain things. And with things like Pie and square of -1 we are beginning to see our concept, aligned with proper logic, has the power to show us things we don't understand and will never be able to.
I think the very fact that imaginary numbers exist and we can prove it with math proves that there are certain un-explainable things that exist somewhere outside of our reality.
Mike Y That's what I wanna know. he said they say that math's false however very useful / helpful, however never specified in what way math is "false", as if they knew what "correct" were :p
Mike Y "Pie" LOL
It's spelled "Pi".
:p
“All models are wrong, but some are useful” - George E. P.
Models aren’t real. They’re a tool to help us mentally understand or see something that might be unreal. They’re a medium that describes something like a surrealism painting but they don’t define what they’re trying to describe,
I just quoted that :)
I have a pencil. I have another pencil. Oh, two pencils!
What is mathematics?
That's a deep question. Here's what I think:
Sets are collections of elements. Set theory deals with operations on sets of elements. Sets are the most fundamental of all mathematical objects. Sets are very general, and thus, it's only really meaningful to think of isomorphisms* between sets.
*an isomorphism ("iso" meaning same, "morph" meaning shape)is a type of relation that basically says that "two sets are similar"
Numbers describe one kind of commonality between sets. They describe a relation between the elements of one set and the elements of another. This set has 3 elements. Numbers are, in some sense, a manifestation of an isomorphism between sets.
Number Theory describes certain properties that seem to arise in these numbers. Primeness, perfectness, etc.
Geometry describes sets with elements ordered in a specific way (a shape), and the properties of these sets. Geometry makes use of a more general type of number, one that does not necessarily indicate "value", but rather one that indicates the existence of a relation. Numbers like pi, sqrt(2), etc. tell us something about the way these shapes are ordered, and similarities in their ordering from one shapes to another.
Trigonometry studies triangles, the first 2 dimensional instance of a shape, and their properties. Trigonometry is extremely important.
Algebra is the study of holes in a certain kind of relation: This is where variables come in. For example, x-2=0, what does x equal? Algebra introduces new kinds of ordered relations, equations and polynomials, and gives us insight into the nature of them.
Algebra studies functions of the set of Real Numbers on to the set of Real Numbers. An equation is a specific instance of a polynomial. Algebra started as a study of specific instances, but was generalized to deal with polynomials over the Reals as a whole. Polynomials do not represent specific numbers, but what these instances look like over all the Real Numbers. Graphing, which comes about in algebra, allows us to study how a function changes the geometry of the Real line.
Abstract Algebra studies generalizations of the Reals, as well as generalizations of polynomials and functions. Abstract Algebra leads to results such as the Fundamental Theorem of Algebra, saying that "every polynomial with complex coefficients has complex roots." Fields, Rings, Groups, Morphisms, Modules, everything. Abstract Algebra is fundamental.
Topology is like geometry, except dealing with the geometry of sets themselves rather than shapes, which are more specific than just sets.
Calculus is the study of smooth polynomials. Calculus is as fundamental as algebra. Calculus, in some sense, allows us to study the algebraic properties of smooth functions. The derivative, the integral, etc. these are all things that are easy to find in linear functions, but hard to find on continuous functions.
Differential equations- The polynomials of calculus.
Linear Algebra- Points are generalized into lines
Differential Geometry- Linear Algebra, Topology, Calculus, and Abstract Algebra had a baby
Mathematics is the study of relations in systems. Simple as that. As long as there exist nonhomogeneous systems, there will exist mathematics.
Agreed, Mathematics is the study of the relationships in systems, that is all. The language used to write math has been invented, the same way we invented the turbine to harness the energy of a waterfall. It definitely exists.
loved, loved, loved! just thanks!!!
I'm with plato on this one
Two plus two is four minus one that's three quick maths
I think I'm a nominalist. I don't think that i, π, and the square root of two really debunk the relationship of numbers with things. i is a number that can accurately describe the third dimension in euclidean space. This becomes useful in polynomial algebra, where functions can be imagined as forms in three dimensions, as can be demonstrated by the calculation and graphing of impossible roots. Pi appears practically everywhere in nature and is exactly the 'ratio' or relation of the diameter and circumference of any circle that is properly constructed. The square root of two is the number by which a geometric square's side lengths can be multiplied to double its area. There are numbers which can be generated by stopping a computer at a random point in time that don't have any relation to objects and cannot be reproduced. Some mathematics probably does lie outside of numerical logic, but I think that arithmetic, geometry, and calculus are undeniably true.
I've discovered that I was fundamentally a mathematical nominalist, in the sense that I always thought numbers were descriptions of concrete objects. Explains why I've been so baffled by imaginary numbers and irrational numbers. Thanks, Numberphile!
Would it help if you understood imaginary to convey cannot be directly immediately personal experienced other than as an image or symbol/proxy in the dreaming/associative apparatus or mind?
If you can directly immediately personally experience numbers, what exactly do you experience if you do directly immediately perorally experience when you do experience numbers if you do indeed experience numbers as directly immediately and personally as you experience or *Know, pain?
So simple yes or no question:*Do* you or can you directly immediately personally experience numbers as directly immediately and personally as you experience pain or the thing on the end of your left leg as your left foot?
Yes or no, and if yes, what exactly are you experiencing? Numbers or just multiplicity?
Numbers are a form of description are they not?Probly (sic-because it pleases me) best avoid the question : "Do descriptions exist?"
Can a mirror reflect itself?
While we are at it: Where is the where the where the where?
why didn't he mention Rorty and pragmatism when he was talking about fictionalism. which reminds me of bentham idea of truth as a fiction or as an agreement.
Mostly because I didn't have very long. Stacks of stuff I'd like to have talked about that I didn't, but only so much time. :)
Thank for the video and response. I guess I left it brief as well.
I will just quote the Stanford Enclyopedia of Phil. :
"Bentham's analytical and empirical method is especially obvious when one looks at some of his main criticisms of the law and of moral and political discourse in general. His principal target was the presence of "fictions"-in particular, legal fictions. On his view, to consider any part or aspect of a thing in abstraction from that thing is to run the risk of confusion or to cause positive deceit. While, in some cases, such "fictional" terms as "relation," "right," "power," and "possession" were of some use, in many cases their original warrant had been forgotten, so that they survived as the product of either prejudice or inattention. In those cases where the terms could be "cashed out" in terms of the properties of real things, they could continue to be used, but otherwise, they were to be abandoned."
However, considering Rorty, where the possibility of change becomes politically impossible, or not possible even in the spectrum of reasonable argument, where we only refer to concrete-tangible things, Rorty concludes, since we cannot reach agreement in principle with platonists, but that it maybe be possible to simply validate their truth on a term or a proposition's effective usefulness (pragmatism.)
Anyways, feel free to make any corrections here in my description, if you have the time. I am also being far too brief here. Thanks for taking the time to do the video, and to respond to my comment. It was interesting to someone, like myself, who has studied philosophy on my own and is working towards an Economics-Mathematics degree.
Plato to the core!
As a nominalist, pi is easy: it's a relative value to circular calculations. Just like the square root of -1, yeah it's more complex, but ultimately it has relational value to other numbers.
"What use....."? Was my question. I know the answer but I get the feeling you are maybe missing the point buddy?
It is a work of fiction, but some people just can't handle that. They cherry pick for the good parts, and ignore the evils. Also, numbers are abstract. They don't exist physically in the universe, without us making physical number objects, so not naturally, anyway. We created numbers as a means of counting, measuring, and labeling.
Numbers are a concept
+burpie a human concept
True
science is completely fiction, finally someone is willing to say it!
Rumford Chimpenstein You can't say natural science is fiction. Natural science is literally based on observations on nature. But would you mind elaborating on your view of science?
Fantastic video!
I think I’m on the end of Fictionalism in general but never considered it before now.
Thanks for putting in the effort of making great videos 😁
Numbers are concepts used to define quantity. Quantity is a conceptual product of language. Nothing really exists without understanding. Existence is a conceptual product of understanding. The questions are: Is understanding a product of consciousness or sensory information?
For something to exist outside space and time seems to be like saying something exists no where and never.
+Kelbie The fourth dimension exists, except we just can't imagine it since we look in three dimensions. Surely numbers can exist in that was as well.
Jeroen Verschaeve For what you said to mean anything you actually have to provide evidence and not just say that they exist as i can change one word in your sentence from "numbers" to "flying spaghetti monster".
***** Thats what i said isn't it?
***** Haha its okay:)
+Jeroen Verschaeve The fourth dimension does exist in physics, but it is considered spacetime and can be shown to curve using general relativity (which, for the most part, has been demonstrated to be correct). Existing close to a gravity source will cause time to move slower than further away; ie: clocks on orbiting satellites have to be adjusted in order for GPS to work.
Another extension of three dimensions into 4 dimensions could also just be time. You can create a 4-dimensional "graph" of an object's position, and watch it change throughout time. Even objects which pop into and out of reality (ie: virtual photons) would have a graph, albeit not a continuous one.
In a mathematical sense, the fourth dimension does exist, as it requires 4 "dimensions" in order to perform linear algebra transformations from one three dimensional vector space into the same three dimensional vector space. Within just three dimensions, you can get rotation and scaling, but not translation.
If you want to talk about the fourth dimension as some additional physical dimension, then you would need some pretty amazing evidence to back that up.
The point about pi just seems misinformed. Pi absolutely has a precise value, and you can perfectly visualize it using a circle.
I think that nominalism is true in a historic sense. It's how maths was born. However, when concepts that are internally consistent take a foothold, humans are able to start treating them as a seperate world on its own, and words like 'one' and 'two' start linguistically referring to a real abstract thing. At that point, platonism becomes true.
But I guess it's an on and off thing.
I agree with you
Pimp Trizkit The point is that things don't have to exist physically to be real, and for the reasons that I then stated. The world 'real' even sometimes has this specific meaning when used in contrast with the world 'actual'.
Pi is simply the precise value of the ratio of any circle's circumference to its diameter. In turn, a circle is the collection of all points equidistant from a centre point. We don't need an ending decimal expansion to do maths with it, and one certainly doesn't need to exist to make it precise.
Just because we can't express its value using a ratio of integers doesn't mean it's suddenly not within our understanding of the universe. People throw around this 'plank length' as a "unit out of which all lengths are built", but it's not like the universe is divided in an evenly spaced grid of planck lengths...
I think that the "objects" that math is derived from is a coordinate plane/space. We exist in a three dimensional space and points definitely exist through out our world. Infinitely many in fact. We use numbers to denote that space and can do intricate calculations about that space.
Bolino I love the internet. Where else could a high school kid tell an expert that he's misinformed?
They can also do that in high school. hehe
Isn't existing outside space and outside time a bit of a contradiction? As what you are really saying is that they exist at no place and at no time. Which is the same as they do not exist at any place and do not exist at any point in time. Which start to sound an awful lot like the definition of not existing.
But what its the other way round? Lets say numbers don't exist within space and time, rather space and time exist within numbers.
Tethloach Kingofreason
Yet nothing is a flawed concept. It can't exist because if it does then it is something. I've always thought its that very paradox that created reality.
It kind of makes sense to make use of the tools we are given. Math is a tool. It would not make sense to throw it away in favour of some incoherent mysticism. So in a sense it exists to us and us only. It is our own creation and ours to use. It is not a creation of some mystical being, it is a creation of logical thinking. It is a tool for logical reasoning. And sometimes, logic is the least logical thing you can imagine. At least by the definition of "logical" being that it makes sense. Hence why logic itself has led to many paradoxes. But when you realize that you are wrong, perhaps you will learn a better way. And every time you encounter a paradox, it might be a moment to learn something. For example: "I know that I know nothing" is just a way for Socrates to brag about his knowledge. Ironic, isn't it?
Devilofdoom OK. Universe "has" time-space so it could be said Universe "is like" a Number... What's the value of the Universe then? :)
Boril Boyanov
I would say thats impossible to know. It also depends on how you define "universe". If you mean all that is, then i guess the value is infinity. But once you define a finite amount of space-time, such as the observable universe then it will have a finite value.
Numbers are there for explaining things and complicated numbers are there because we created those numbers, because we needed them. We needed the pi number for explaining something so we created it. Let's say you want to understand a chemical reaction, and to understand it you need to form an equation and there is a power in that reaction that effects the equation but you cannot symbolize it with a simple number like 5, 7, 10. So here, you'll calculate what effect does the power has on the reaction and you'll create a complicated number to explain that effect, to understand that power, and effect of that power to the equation. So I think you can explain and understand complicated numbers in nominalism.
Numbers are a language for communicating quantities.
YES !!! Thank you!
Unfortunately it's not that easy. If they were only communicative abstractions, why should a chinese '3' should be the same entity as an english '3'? Or does the 3 have some attributes on it's own?
The numerical symbols are part of a language. But the numbers themselves ARE the quantities. For example number 3 is the "three-ness" of something, and is different from for example "two-ness" and "four-ness", etc. I think that the platonist view has something to it.
Soo what is the quantities of square root of minus one
@@arifnurhikam that is more easy than u think/imply. It has the quantity of one in the complex plane.
Asking "Do numbers exist?" is like asking "Does red exist?"
"Does red exist on the color wheel?" Yes!
"Does red exist as a physical object?" No!
"Does Harry Potter exist in the story of the book series?" Yes!
"Does Harry Potter exist as a person on our planet?" No!
"Do numbers exist in number theory?" Yes!
"Do numbers exist as objects in the physical world?" No!
Now, the fictionalist says "the discourse of math is false" and he does so because he says maths isn't a real thing. Sure, if you take certain definitions, you will end up with this conclusion. But are they useful definitions? I do think it's useful to use sentences like "1 + 1 equal 2 in maths" and while denying this might be a valid consequence of a valid definition, it's ultimately not a very useful view. It's not useful because the language of logic and mathematics you use operators such as "for all" and "exists". Sure it's all made up by humans but in that made up environment, it makes sense to talk about existence.
Now, the nominalist says "numbers represent real things" and I will tell you that matrices doesn't represent real things. A pegasus (a horse with wings) doesn't represent a real being. And they're not approximations. But I can still think of them. They can be useful concepts in certain contexts, so it isn't very useful to deny this. So I think this view is actually the most problematic because we can only discuss real things. Theoretical physics would almost be impossible. You can make predictions but you don't have a representation for everything (i.e. a concrete value in a wave function before you calculate the probability of it by squaring it).
Lastly, the platonist. He distinguishes between physical existence and abstract existence, physical objects and abstract objects. You can simply think of "abstract objects" as being concepts or ideas. And yes, it's useful to say concepts or ideas exist in the minds of people. They might not represent real things but thinking of "existence" in the right context is useful if you are precise about what you mean.
Regarding the causal interaction of numbers, it's the same thing and I could ask "Does Harry Potter interact with me in the story of the book series?" and the answer is no. But the concept entertains me and gets the author a lot of money. Same with maths.
Bottom line, you just have to be precise in what you ask. If you ask a ill-defined question or one that is open for interpretation, you might just get 42.
@@ScootalooInMyShoe27That would be the cause of the perception of red though, not red.
DEEEEP.
give this man a cookie :D
Through my "career" as a elementary school kid, to A-levels and now university I've went through all three stages of maths and now at the end of my bachelors degree/soon starting masters degree, I'm switching through all three every few weeks xD
So many pretentious 13 year olds in the comments...
If only you-tube would make it so that only those who are smart enough to get drunk and watch numberphile are eligible to form accounts on you-tube~
Sadly there are 13-year-olds who are more intelligent than certain adults...
That's EXACTLY what a pretentious 13 year old would say!
What? only three options? False tricotomy?
(disclaimer: "To me it seems..." should be applied to every sentence written here)
Numbers are arbitrary chosen symbols to describe different systems of logic. Some logic system is based on 2 choices - 0 and 1. Sometimes it's based on 10 variations. Numbers are real to us inside our heads... but they are completely made up.
Numbers have real use, but are nothing more than logic rules used inside our heads to appraise situations. Some real world problems can't be "truly" described by a value system based of a 10 variation, for instance describing the relation between a circles diameter to it's circumference.
Like every logic system it's not applicable on all problems. The decimal system can't describe "1/3" absolutely accurate or Pi -it has limits in it's descriptive properties.
The classical statement that "2 + 2 always equal 4" is actually only true in our base 5 or higher number system where "+" is an addition operator.
So 2+2=4 does not make sense in bi-, ter- or quatenary number systems (with zeros)
I think other higher primates/animals, especially those who attack or defend in groups, also think in atleast few numbers to appraise the situation but maybe in a way strange to us -just as I cannot really know at all how others percieve the color yellow in their mind.
If you asked me, "What is the most absurd combination of three words you can make in the English language?" I would have to say it'd be, "Math is false."
Does Rassel's teapot sound familiar to you? If yes, than you should proof that math is true before you state that. Let me guess, you don't have a single proof.
I feel more like math is a big concept that we follow, that is nor true nor false. Like, say, what is circle for example? I don't mean that i don't know what that is, but in "real" world does it even exist? What's a line? Does that exist? A square? It may not exist but it has it's aplications. And math certainly doesn't lie. It's just the world that we exist in isn't the same as we think it is.
Александр Бантьев Read Gödels Theorem
In my opinion, numbers are simply lovely. Mathematics is a language, a way to communicate with nature. You understand nature through physics but with the help of mathematics. Each number signifies a physical concept. Pi is a ratio that is the property of every circle. Numbers exists if languages exist. They exist more as a concept than as objects .
philosophy of maths????
***** Philosophers do tend to make an utter mash of things don't they?
Mat Hunt Some bits of philosophy have some use, but the rest of it is no more use than ornament.
***** I have to agree, I think philosophy is the correct way of thinking about morality and law for example.
Mat Hunt Exactly.
***** By "theoretical math", I presume you mean pure maths, the study and development of maths itself. Once you develop the natural numbers you can then go on to define the rest of the number systems which make up the number systems
I come at it from a historical perspective, maths was developed out of necessity, and whilst we can have a physical representation of what means `1' or `2' you can then make up a set of abstract rules.
Then you come on to geometry where the abstraction of the physical representation is accurate to a certain number of decimal places say.
Lets say I build a missile. And I want to hit a target on the other side of the world with that missile with accuracy. I would need the missile's guidance system to interact with GPS satellites in order to hit that target. The computers in the missile and the satellite are programmed using our understanding of "math". You could pin point any location on earth with coordinates. If I were to invent a coordinate system that mapped out the earth, that doesn't use numbers, but random symbols. Then I program a computer that understands those symbols, which in turn build a satellite and a missile that runs on that computer. Could I hit a target anywhere on earth with that missile? If so, what does that say about numbers. If numbers are truly real, would that missile not work?
But, you could say that numbers are the coordinate system that maps out our universe, at least the universe as we see it.
DongerLady ToTen I just realized, though the analogy is interesting, it fails to evade the ubiquitous nature of numbers. By comparing numbers to random symbols (what is a random symbol?). It is basically asking: if we can imagine a universe that has different laws than our own, do laws exist? It is a question we can never answer
No, your missile, using only symbols, would completely miss the target. Because the relations between numbers, and the continuity of space, are needed in order to chart the path the missile takes from the launcher to the target. The end coordinates are not enough. Renaming only the end coordinates doesn't allow you to rebuild the entire path.
Even if you have all the coordinates along the paths, you also need the direction, in order to control the missile's navigations, since a missile doesn't just move arbitrarily, it moves with momentum and inertia.
If you rename every point along the path, and all the directions, then you've essentially constructed the real line, and given it relations roughly equivalent to continuity and derivation. In other words, you've simply renamed numbers. You're back to numbers, with all of their original mathematical meaning.
Okay. That makes sense.
Shiro 史郎 Wretched Egg So the same fundamental concepts of math, just with different symbology... Still math. Much like how English and French are still languages.
This is all philosophical semantic nonsense. Numbers are an abstract concept by definition, we DEFINE them to be what we want. We define everything in math, because math is simply an abstract concept that we use to derive truth through self-consistency.
To say something "exists" but has no physical component is a pointless discussion. You may as well argue other abstract concepts like "love" or "music" exist. It's completely pointless. These things are labels we assign to things. They are labels and nothing more. Math is compilation of labels.
+bob smith So you're a nominalist ?
MountainBlade100
I'm a touchsmallchildren'sgenitaliaist
bob smith
So you always have the urge of touching yourself ?
Just so you know anything you say will make you both a Nominalist , Fictionalist and Platonist , bickering which has the better view does really make you a child .
I think this is similar to asking:
Does beauty exist?
Does justice exist?
Numbers do not exist. Numbers are created by humans.
The measure of a quantity exists, we chose what to name it.
The things you are measuring or quantifying exist. The quantity or measure does not. It's an abstraction.
+Katie Katie ,
i'll use this natation:
"cat" - label.
cat - object.
Cat exists as a physical object. You can prove that by touching it.
"Cat" definetly does not exjst as an physical object, it is either an abstract object or nothing.
"1" definetly does not exist as an object.
The question is, does 1 exist? I don't think so, because you can't touch it or see it.
in that sense, "1" is more real than 1, because we can see "1" written on paper, for example.
Katie Katie. Ammmm no. When you say "cat" you know what kind of physical object you are refering to. When you say "one" you don't know what kind of physical object you are talking about.
Katie Katie. I have seen many "cats". But I have never seen a "one" in my life. Numbers do not exist per se, as well as words do not exist per se.
you are confusing numbers with the symbols you use to denote those numbers
this is a false trichotomy
+Honest Tech Reviews They never claimed that these are the only ways to think about this.
+Jyrik Gauldurson true
+Honest Tech Reviews That's exactly what I told my surgeon after the operation! Or that's what I *would* have told him, but since the tracheotomy didn't work I couldn't talk and I suffocated.
So, what is a fourth perspective?
You don't need a universe for pi to exist, but you need pi for a universe to exist. :-)
as a nominalist you would probably understand pi as the curve of a circle imperfectly represented as a number, or symbol - a fictionalist could say that the representations are applicable and repeatable but that it doesnt prove that that method of representation is the only possible way to represent something applicable and repeatable, think of an alien intelligence using alien methodology (inherently something we cannot envision or imagine) accurately and repeatedly solving or working, or that mathematics is not THE truth but part of the truth or an effective application that holds true
i think all three platonist nominalist and fictionalist are accurate at the same time
apparently numbers and gods exist in the same place
maybe because Numbers are the gods?
No shortage of wannabe scholars here I see lol.
Plenty of those
You can think of 42 as the product of 6 and 7. Pi is simply the quotient of the circumference and diameter. We may not know its exact value, but it does have one that can be represented in mathematics.
Nominalism is the one I agree the most. We start mathematics describing simple things that exist in the concrete world and the more complex numbers, like the irrational or the complex themselves are just descriptions of things we cant measure (like Pi), things in more than 2 or 3 dimensions, or just descriptions about abstract concepts that are not concrete, but depend their existence from beings that think or believe is such concepts.
Unfortunately, the idealist thinking has been criticized for so long that people forgot to take what it really meant into account. Idealists never claimed numbers exist. What we propose (from before Plato) is that numbers reflect notions that are aprioristic to reality, perception and language. Scientists are idiotically aprioristic about matter, but cannot stand someone stating that underneath nature laws there is structure. As much as today we have strong evidences that grammar precedes language, we state that there is structure underneath reality. That is not to say numbers exist, per se, but they are language elements we use to conceive and communicate meaning. And this ability to see and transmit meaning is the reason we call numbers objects. Those are the very terms in Plato's books.
I've been playing with this for a while. Say you have two whole apples on a table. One is 200g and the other is 300g. If you asked anyone how many apples there were, they would say that there are two apple. "Two" represents the number of apples and yet it doesn't because they aren't the same things. In this case "two" is merely an approximation but it is a true approximation because no other number could be used to describe the number of apples on the table. How much apple "matter" is on the table is 500g which is a better approximation but still just an idea that is useful. I would take the position that numbers are ideas and are real as long as they represent reality.
for a nominalist, sqrt(-1) is the solution to the equation x^2+1=0, which is equivalent to the question: if you have x^2 objects arranged in a square with lengths x, and you add 1, what length is needed so that the resulting number of objects is equal to zero?
I think an argument a nominalist can make with complicated numbers is that they describe a property of a real object, for example six lots of seven being the size of the set of objects. It seems a trivial distinction, but it opens the door to say the square root of negative 1 exists, because it is number which allows the schrödinger equation of particle properties to be represented in exponential form (for example, of which there are many) and pi, although we dont know its exact value, saying that it is the ratio of circumference to diameter for any circular object is sufficient to define its existence.
Coming back on this, I think I combine platonist and nominalists. Numbers do exist as abstract objects. But we can't interact with them directly. What we can do, however, is what we can do with other platonic ideals: we can interact with the idea of them. And, in so doing, we learn more about them.
So, in other words, we interact with the names for the abstract concept. So still nominalism, but of the platonic ideals rather than the physical objects.
That's not to say we don't treat numbers as a property of an object. But that object is a shade of the platonic ideal. And what we manipulate in our minds is the platonic ideal, then convert it back to the physical world.
That said, there are times where we take math further, and work in a realm where axioms are neither true nor false, or even explicitly false. But I would argue that we're then just interacting with different ideas, which means a different platonic ideal of that idea.
That's what platonic ideals are: the abstract concept of a thing.
A non-dualistic alternative to Mathematical Nominalism:
Mathematics describes in precise detail the infinitesimally thin boundary that separates reality from fantasy. For an object to exist in the real world, it must strictly conform to the mathematical laws that govern physical reality. One step outside those laws lies an endless realm of fantasy. One step outside the classically presumed Law of the Excluded Middle lies the realization that a mathematically defined boundary is not necessarily a member of either of the sets that it separates.
As for Imaginary Numbers (at 4:39 in the video), it's easy to show how they are useful in measuring physical properties of things that exist in the real world. Electricity and magnetism, for example, are physical forces that exist natively in a Complex Number domain.
Numbers are abstractions we use to describe real life phenomena - they're expressly created for this purpose which is why it's no surprise that they do it so well. In other words, the nominalists have got it mainly right, but since numbers in fact abstract it's not necessary that they always have to be describing real things (in the same way that my thoughts about a chair don't need to correspond to any real life chair and can be detached from any practical purpose). I can conceive of √-1 in the same manner as I can conceive of multiple earths - imagination.
You know how there’s machine code for computers (binary) then there’s abstracted languages to access machine code then more abstracted languages to those languages and so on, until you have something simple like your computer (how it functions). I think math and the universe are like that. There’s an underlying base language for mathematics within the universe, but the one we’ve developed is incredibly abstracted and simplified, or rather it’s a subset of this larger, more inclusive base language.