I'm definitely in the "math is something we made up" camp. Math is a language and language is something humans made up. It's relatively ordered and so is the universe, so they tend to correlate. But, as a physicist, I can tell you that correlation is /not/ perfect. It just sits firmly in the "close enough" category. It is not the language of the universe. If you want to say the universe even speaks a language, then math is just our best translation of it into "human"... but, like with any other language, some things get lost in translation.
That "close enough" sounds an awful lot like a limit. If you consider the philosophical implications, it seems an awful lot, if the universe isn't explicitly doing so, like the universe just wants us to believe it.
Is Pi infinite because a circle has an infinite number of small straight sides? Infinities exist in math but I'm not too sure about in our observable reality because of quantum limits like the Planck length etc...? So if Pi really is infinite and real that means our universe if infinite in size!? 😲 Or maybe we look at circles wrong and true curvature is possible but can only be described with the correct description, which we don't have yet?
There might be an Einstein quote, forgot it, along those lines...that math doesn't perfectly predict the universe, it's only precise when abstract, etc, or something, forgot the quote...at any rate, I wonder what a universe that didn't somehow follow math rules would even look like, lol...if it's going to follow any rules at all, wouldn't we end up calling those rules "math" anyway?...Well, I suppose one could say that there are plenty of things that follow rules but which aren't as directly "mathematical", like biological beings, etc, one would have to zoom in to find the mathematical rules, etc. But still, lol...how would a universe that doesn't follow mathematical rules even be like?...Is it conceivable, would it be a universe that doesn't follow any rules at all?...
@The Science Asylum I know this is 5 years old but this is more wrong than anything can possibly be wrong. Math isnt *just* the language of the universe but it IS the universe Like... Its baffling how any thinker can even believe that humans calling a thing 'something' actually affects that thing in real life. Thats absurdity and reeks of ego Humans arent an integral part of the universe... Mathematical symmetries exist in nature regardless of what humans call it. Math is simply cause + effect, logic, shapes, and what the gradients of the universe are doing in the present time. Logic on paper man logic on paper Like... Did triangles not exist before Pythagoras "invented" his theorem? Euler "Invented" countess mathematical equations that are fundament to reality itself. So.... Did reality not exist before Euler was born? Were the rules of reality different before Newton "discovered" gravity? Scientist are the most egotistical people on the planet jeeeeeez. These people are simple OBSERVING whats already there not CREATING.
Lol...I don't know...there are infinitely many things we could define, our there, lol...there is a sense in which they only start existing when we define them...a sense...but anyway, lol...
I'm a strong formalist. At its base, Mathematics is simply defining some rules (or axioms) and carrying them out to their conclusion. The reason it describes reality so well is that we tend to pick axioms that we feel define our reality. We could pick completely different axioms and get completely different mathematics, but it just wouldn't describe reality. I believe this is most noticeable when we discover independent axioms, like the continuum hypothesis. Math works no matter what version we choose, but maybe we will see that one version lets us describe our reality better. Okay, long speech over. Just want to say I love the series, and I'm really excited to see more videos! You all are amazing, keep it up!
Would you not agree that the set of conclusions from given axioms forms some kind of reality? If there is only one outcome of the rules and every observer would measure the same outcome, how is the outcome any less real than say, a photon?
Nice! You might be interested in a branch of the philosophy of mathematics called "logicism". I also think the Axiom of Choice would be an even better example for your argument, since it's now (essentially) accepted because it proved so many statements that mathematicians felt were true.
Modus Ponens Now we get into the question of what is really "real". Through the philosophy of skepticism, you could make the argument that our objective reality is created through our minds, since the only way we can truly "prove" something exists is through our senses, and our senses have been wrong before. Therefore, through the eyes of skepticism, math could be made up in or minds, since our reality itself could have been created in our minds. Of course, there are fakes with this theory, but I feel like it's worth noting
+Kevin Merrick But you don't explain the problem posed by Wigner about the effectiveness of math. The thing is that simply claiming that we chose axioms to mimic reality doesn't brush off the idea that these mathematical axioms/objects exist independantly of human minds. If anything it just reinforces the idea because if our axiomatic system is simply "copying something" then by definition it means we didn't really invent them. Also If that "something" we're copying is just an other mathecatical axiomatic system then it would mean that math exist!
Modus Ponens I am a formalist and I have thought before about the question you raise. I think you're absolutely right; the logical deductions do form a objective reality. So, how does this not make me a mathematical platonist? Well, a platonist would accept the axioms as having some objective ontological significance. I still think the axioms are just the "rules of the game" that we defined. I also believe that when we say we've proven something to be true, we're really just saying that if we accept ZFC (or whatever we're using as our foundation), then such and such theorem is necessarily true. Since I don't accept the axioms as ontologically objective entities, but I do accept the logical deductions that follow from a set of axioms as ontologically objective entities, I like to say I'm a "logical platonist" but not a mathematical platonist. Logic is definitely real and objective in my opinion.
except english changes. Its words dont just accumulate, but evolve. There are also other languages that describe other things better. Math is special. Plato was right. Math is more real than us.
English and every known natural language is spectacularly poor at describing reality. We just aren't aware of this because part of what we call our thoughts is expressed internally as words. Since part of what we are describing as 'thinking' is built out of words, well then this is like saying something is itself. But this is an incomplete statement because we have other modes of thinking, that we experience, that are not built out of words. For instance what are dreams built out of? Or what does your brain do when it 'sees' something? For all these other kinds of thought, words are really bad at conveying, identifying, or describing what is going on.
Maths itself is abstract, however we can use it to define and manipulate real things in terms of physics. Physics is applied maths. Our buildings don't fall down, and we used physics to make them.
Mathematics is based on logic. The vocabulary is freely invented, but the rules of grammar for this formal language are the logic. The rules for logical thinking are constructed in a way, that is consistent with the *physical properties of the universe*. It means, the rules of logic to apply mathematics, to proof conjectures, are specifically designed to set the *physical system "human brain"* (or computer or alien brain) into a *restricted state, which always produces the same consistent results independent* from the specific system used or any superficial conditions (like mood). That's why different mathematicians using the same assumptions and axioms obtain the same results and why students can relatively easily follow the way of thinking by reading a proof. If, on the other hand, they would read some complete nonsense, a flawed proof or fail to apply proper rules, they discover an inconsistency between the way their brains produce results and the material at hand. I.e. the proof is logically inconsistent or the applied logical rules were not stringent enough to force consistent results.
I take it you do not subscribe either to the Copenhagen interpretation of quantum mechanics, or to an idea of discrete spacetime units? Also, what happens when large swathes of mathematics are based on things like hyperbolic or higher-dimensional geometries? Sure the rules of logic used are the same, but the axioms are switched up.
Dustin Rodriguez The Copenhagen interpretation doesn't interpret anything. It just states the observed effect in a more formal manner. Still want to know why.
This has already proven itself to be an incredible series. The metaphor of writing the first page of a book was very nice. Thank you for producing it so wonderfully!
@@lEGOBOT2565 that is very strange, as 2 is definitely a number that meets every requirement to be a prime and functions like primes too. Also at another point in the video it was shown with other primes
I still haven't found a better alternative than the one which was proposed by Gottlog Frege in his Foundations of Mathematics: mathematical objects are existent, but abstract: they are 'found' when we are using our language on a higher level. While in natural language there are concepts in which the value of reference is empirically determined, mathematical objects are a second order concepts, they are concepts of concepts. Nevertheless, they are true and objective: their logic, relations and operations are valid for more than one person, and work independently of our opinions or taste about math. In a certain way, Frege was called a platonist, but if we consider that mathematical objects, although true and objective are only found when one is thinking on a higher level of language, his approach was much more kantian and formalist. Math seems to work as a tool and a language to describe nature only because we already perceive nature as mathematically ordered. Math seems to fit so perfectly not because of math itself nor nature itself, but because of how our own mind works.
You have ten fingers, I have ten fingers, bob has ten fingers. Abstract away who has ten fingers and you get the abstraction "ten fingers" Like wise, I have ten apples, you have ten apples, bob has ten apples. Abstract away who has ten apples and you get the abstraction "ten apples" So we have the abstractions "ten apples", "ten fingers", "ten toes" etc. What if we abstract away what we have ten of? We get an abstraction of an abstraction. The number "ten" Every culture developed its mathematics through this abstracting from simple counting of physical things like fingers or apples to these abstractions. Numbers. And then considering that "fingers" and "apples" are already abstractions themselves, numbers are 4th order abstractions. They are so abstract as to be universal. And all of mathematics is just abstraction upon the abstraction of numbers, so no wonder they work so well, because they are so bloody abstract!
@Juliano Bittencourt I know this is 5 years old but you just used a bunch of words to say a whole lot of nothing smh Its not that hard people overcomplicate things because of their massive god complexes. Humans arent gods in hat regard humans didnt create the universe which is founded on mathematical principles. The symmetries of the universe exist regardless of what we call them. Subatomic particles are the physical manifestation of 1+ (-1) = 0 as charges and spin cancel out. Plants grow geometrically regardless if we have a word for geometry The circumference of a circle will ALWAYS be 2πr and always have been even before humans existed.
Let a few students lose in the jungle without a guide of any kind, and they would soon develop a language to learn to live there. The real information comes from the environment and the language is used to sort it out, and adjusts continuously to fit change. Maths is language, and part of us like all tools. Plus, everyone is is allowed to have a hypothesis about what is possible and test number relationships until the rules emerge - and then find the real world correlation and or corrections. It's endlessly interesting.
I just realized I wasn't even subscribed... I've watched so many of the vids cause they appear in suggested and home pages... time to change it I think!
With all these interesting discussions going around, I thought I'd share my own view. I'm a struggling musician with a passion for math, physics and philosophy. I've read the essentials of Platon, Hilbert, Russell etc. I especially enjoyed reading some of Russell's ideas, but in the end I believe that math is like the product of a compiler outside this universe. It's the underline structure, or the predetermined framework on witch our universe exists. It's created by something we simply cannot fathom. I like to think about it like a computer game. All games have rules and more or less the world created inside that game follows basic logic and it's self consistent in that framework. It can also have physics and gravity programmed in it. The player, or the AI will always obey those rules. Now, if the AI was sentient, he would be aware of all these rules and would analyze its world from his perspective, but he will never have a clue that he's the product of some functions and scripts written in C#, outside his world. The very notion of C# is utterly meaningless in the product of C#. Anyway, fruitcake or not, that's just my take. It's not like I'm gonna get better answers this lifetime.
Formalist here. The reason that math works so well in science, engineering and technology is because those subjects have long been strong motivators for mathematics. And because they work so well, those are the math systems we teach and study. There is nothing unreasonable about the fit, any more than it is unreasonable that my door fits into my door frame. (Though I still puzzle why my refrigerator doesn't fit into the space that is purported to have been built for it in my kitchen). Our invented math only appears to describe nature. Oh, it comes close, but when we look closer, it fails. Newtonian physics and Euclid is all neat and tidy and was assumed to explain nature, but it doesn't. We go fast, and we have to find a new geometry to make things work. Or closer to home, the Euclidean stuff doesn't really work measuring things on the surface of the earth because the earth is not a plane, it is a sphere. So all those lovely triangle rules don't really work. So we looked around at some of the new ideas and picked one that seems to fit nicely... so far. We go small and find that there are no circles because matter is quantized and thus inherently fuzzy. The concept of a circle is still useful in approximating reality., but it isn't reality. And that is the crux of the matter. It isn't that there is some "ideal circle" that the real world approximates. It's the other way around. There is a real world with which we interact, and our math approximates it. Well, some of it does. Chess is another form of math, just like the Game of Life. Neither describes reality, but the latter especially gives us a lot of insights into how replicating patterns behave over time. New math. And more complex mathematical models have been built on this insight and do an even better job of describing the real world. But I understand why we are so tempted to think of our math as "real". That is our great human talent, to imagine things and make them real, even when they remain imaginary. Money, countries, laws, corporations,... none of these are real, they are all inventions of our imagination. But because we believe them to be real, because we see these figments of our imagination as real and tangible and are so utterly convinced that we act accordingly, these abstractions act as though they were real. In fact, in a certain sense, we make them real. Just like math. They are abstractions that sprung forth from our imaginations and we endowed with the gift of existence and reality.
Thomas R. Jackson The problem with formalism is that it implies that their are other systems that we could have just as easily have invented that would also be able to describe the universe, but there aren't. Only one mathematics, our mathematics, exists and can described the natural world. You can explain particle physics with matrices or differential equations but it's all the same math.
I'm sorry but 2+2=4 is a constant. Quoting someone really smart doesn't mean that their quote holds any water. No matter what, math is going to be a constant in our reality.
MrDajdawg I think you're missing the point. No one is claiming that at some point 2+2 will equal anything but 4; it follows from the basic axioms. We use the maths we do because it works so well at describing the natural world, but certainly over the past few centuries, and probably millennia, mathematicians have explored maths that doesn't appear to describe the natural world. Of course, in some cases, some of that maths has turned out to be useful. Similarly, I can imagine an alien species on a planet with radically different material circumstances, different chemistry, biology, and society etc developing ways of understanding their world that use different sorts of concepts and maybe miss out lots of what seems basic to us. Part of the problem is that maths is so ingrained into our thought and perception that it's hard to think about understanding the world without it.
You say laws and corporations are not real, but how are these things different from an EM wave or a cell nucleus? Aren't all things just trying to adhere to rules? I think a relatively stable environment respects more rules and we can talk about "time" and "energy" or the realness that is intertwined with causality. But just because there are perceivable layers to the rules of a causal world, it doesn't mean that some layers are less real than others. If anything, I would say the only thing that is not real is an environment that has no "time" or way of measuring "energy", no "causality", no patterns or logical progression. No rules!
I would argue that math is like a game, where we're following rules to their necessary conclusions, but the rules aren't completely arbitrary: they're rooted in nature. We didn't just decide 1 + 1 = 2, it's a natural fact. We didn't decide a circle was the shape where all points are equidistant from the center, the universe did. Mathematicians took the rules nature gave them and ran with them, which is why math, while game-like, describes the real world so well.
Did the universe define the circle though? There exists no circle in the universe with those properties. Or a sphere for that matter. It was probably a mathematician who sat down a while ago and defined what an ideal circle is and then ran with it. :p
M.W. Vaughn But that is what our human brains think is the right shape. Maybe some other sentient being would differ with us on that. :p This can go on and on.
Maybe the key to the puzzle is the origin of mathematics. Just like you said math is a book that write itself but we have to write the first page. And it's not difficult to see why 1 + 1 = 2, just count your fingers. We agree on the basic rules of adding, subtracting, dividing etc. because they make sense in the real world - I still remember learning about division with X apples and Y plates to put them in... So, math isn't real but it's not unreal either. And it's effective because the real world is in itself consistent, just like math is. It's fascinating that our logical brains can extrapolate unknown real world properties by just looking at the math - I'm talking black holes, dark matter, time dilation and so on - all real world objects which were predicted by the math.
Hi, I'm a human from a simpler time. I fish for a living. I keep finding lesser fish in my basket than I caught, I can tell someone is stealing them but I can't keep track of them. one day I decided to give a different name to each amount of fish starting from a fish to many fish and increasing by a single fish each time. And that is how I invented numbers who do not exist on their own whatsoever and the only reason they are so good at describing how many fish I have is that that is exactly what I modeled them after. TLDR: numbers are only so good at describing reality because, in the beginning, they were concepts that were INVENTED to Describe reality, like how many fish someone had caught.
Formalism FTW :) I never understood (and probably never will understand), why some mathematicians think that every concept/idea has some physical object attached to it. At some point in time humans started counting sheep and made up _numbers_. When we started thinking about shapes, humans made up concepts like _distance_ and _angle_ by defining them. There might be big circles floating around in the universe, but the concept of a perfect _circle_ is just a definition a human made up (possibly based on real-world-objects). Integrals were made to calculate the area under a curve, then a definition came by and suddenly we can do much more with integrals. Great video :) PS: A prime number is not defined to be only divisible by itself and one, because this would include one as a prime number (because one is divisible by one and also divisible by one), but one isn't a prime number. There is a similar definition though: Every number greater than one is called _prime_, if and only if it is divisible by itself and one. Another definition (I personally like the most) is this: A positive integer is called _prime_, iff it is divisible by two different positive integers exactly.
+Cubinator73 But was anyone claiming that concepts and ideas have physical objects attached to it? People who think mathematical objects really exist have to accept that they are non-physical abstract objects. Ironically, it's the naive empiricists who want to say all mathematical statements are tied up with physical objects that would want to draw this link between mathematical objects and physical objects (even though that was a failed philosophical project).
Ideal forms of humanly perceived, elementary objects! Simplified by stripping them down to their bare minimum, but still keeping them recognizable (e.g. a circle)! Numbers are tools for counting objects (1,2,3,4,.. etc), AND also NAMES, assignable to objects, even non-sequentially!
"non-physical abstract objects" don't exist in the way I use the word 'exist'. I don't accept that because we think about something that means it has an objective existence. If what you are saying is that things exist because people think about them, then that implies God exists simply because people think about the concept of God, which is an even weaker argument than St Anselm's. Is that your position? While there has been some sloppy thinking on this thread, I think the reason antirealists (certainly in my case) have drawn links between physical & mathematical objects is to show that Wigner's effectiveness is not unreasonable at all. Rather, we've invented the abstract objects in order to explain the phenomena we see in the natural world, and we've got better and better at inventing and refining the abstract tools we use for this over the millenia.
+Donach mc kenna Nope. I don't say things exist, because people think about them. I can think of a perfect circle and yet there is no perfect circle anywhere. But the idea of a perfect circle exists (not as physical object) and is denoted / described with language / remembered with electrons buzzing around in my brain. I can think and talk about God, but that doesn't make him suddenly exist as physical entity. I can even think about perpetuum mobiles, but that doesn't make them possible in reality. Even though, there is no real Harry Potter, he still exists in books, in movies, ... The (probably idealized) idea of some thing is not the thing. The thing *might* exist as physical object. The idea cannot exist as physical object, but it exists as description that might be physical letters in a book, electrons in our head / in a computer...
A phisycist and an engineer were talking. The engineer told the phisycist: You phisycists think everything is easy and quick, you just sit down with your equations and computers and wait for instruments to pop up. And the phisycist replied: We phisycist think of engineers as a magic lamp; you rub it's surface, tell them what you want and telescopes and instruments pop out of it.
That should be astonishing! It would prove that Math is actually the language of the universe, and that instead of us having invented it, it would tell us we understood and written it. Altough I believe it's not the case here, it would be nice...
It could prove that we have discovered only a part of it or that we both have discovered the same part, if, as me, you think math exists besides humans.
I never understood this debate. To me the answer seems obvious: just like you explained, we invent the axioms and discover the emergent properties of the system defined by our axioms. Maybe the real meat of the issue is what is considered math: the axioms, the theorems, or both? Well, in that case are you arguing semantics or philosophy? As for the "unreasonable effectiveness" of mathematics in describing the natural world, neither does this seem to be a good counterargument for formalism, nor does it seem at all mysterious. A mathematician typically works with axioms that were intentionally selected to resemble how we understand our world to work, so of course the emergent properties would likely also resemble our world. Yes, chess does not describe our world but that's because the rules of chess were not devised to resemble our world... Smells like an inverse error to me. Just because a game does not describe reality does this mean that no game can? I simply fail to see any real distinction between the rules of a game and mathematical axioms besides the level of rigor you might typically expect. For instance, I can describe axiomatic systems that do not model our world any better than the chess game you describe that nobody would hesitate to call "mathematics", and suddenly the "unreasonable effectiveness" argument falls flat on its face. Anyway, I enjoyed this video and like things that make me think. I am looking forward to more.
Okay, but that doesn't really solve the problem. Sure, there exist axiomatic systems which do a poor job of describing the universe. We could pick one of those to explore, but we choose rather to pick the one that describes reality. That doesn't explain why there exists an axiomatic system which does describe reality. Why should the universe follow coherent rules at all? And if the universe does follow coherent rules, doesn't that mean that those rules exist independently of a system that we invented?
As for why the universe should follow coherent rules, I'd like to apply the anthropic principle to state that any universe that didn't follow coherent rules almost certainly wouldn't support life intelligent enough to ask the questions we're discussing. Of course, this just changes the discussion to why our universe has all the nice properties that it does so that we are able to be here to observe it.
If multiverse theory applies, wouldn't it be the fact that there has to exist universes in the infinity that do meet those properties and intelligent life could only ever exist in those universes that do have those properties? Thus, our ability to have intelligence by necessity dictates we are in one of said universes with said properties?
I don't think it's even necessary to invoke anthropic principle... we can see our axioms on mathematics change to suit the needs of reality. Newtonian mathematics breaks down on very close orbits like Mercury. But Relativity using new tensor geometry works well for all orbits including Mercury. Axioms of mathematics are infinite, and only those that are useful get any attention. The rigidity of a mathematical systems is what gives the sense of independence. But the fact we have to invent new forms of geometry and probability of which axioms come from the outside world and old mathematics that fail due to their own rigidity shows even the rigidity is an illusion.
jmitterii2 You have ten fingers, I have ten fingers, bob has ten fingers. Abstract away who has ten fingers and you get the abstraction "ten fingers" Like wise, I have ten apples, you have ten apples, bob has ten apples. Abstract away who has ten apples and you get the abstraction "ten apples" So we have the abstractions "ten apples", "ten fingers", "ten toes" etc. What if we abstract away what we have ten of? We get an abstraction of an abstraction. The abstraction ten. If we have ten, 1, 5, 9, etc. and we abstract away the magnitude from each of these things, we get the abstraction numbers. Every culture developed its mathematics through this abstracting from simple counting of physical things like fingers or apples to these abstractions of numbers. And then considering that "fingers" and "apples" are already abstractions themselves, numbers are 4th order abstractions. They are so abstract as to be universal. And all of mathematics is just combunations upon the abstraction of numbers, so no wonder they work so well, because they are so bloody abstract! So much for the unreasonable effectiveness argument. One might as well ask about the unreasonable effectiveness of the laws of logic, or of abstraction itself, to pick the few things that are even more abstract than math. But isn't the rules of chess also an abstract system? So why does it not work to describe reality? The reason is because it is not abstract enough. Chess is a lower order abstraction than mathematics, because mathematics is needed to describe chess. You need the abstraction of numbers to describe how many pieces there are on the board, but you don't need the abstraction of a stalemate to describe any concept of mathematics. This is also why many mathematical fields have no bearing, no connection to anything in the real world, because they are lower order abstractions than those of numbers.
In science, the simplest solution that explains the most amount of things always prevails. Certain types of math are purely platonic, but other types explain reality so simply and generally it would be crazy to say that that math only existed in our heads.
For some reason youre in the vast minority. When i look at the ratios, most idiots think humans made up math. Even so called intellectuals believe such an easily disproveable thing Like BRUH THE UNIVERSE EXISTS! Humans didnt create the universe and math is fundamental to the forming of the fundamental particles that make up reality and then some
What a wonderful video, I'm so glad to see a mathematician taking the Philosophy of Mathematics so thoughtfully. I tend to think of mathematics as "schematically true". That is, it is true for anything adequately described by the premises of the structure. If an object's motion is close enough to f(t) = 2t+1 then you can predict closely enough that after the stop-watch strikes 10 seconds the object will be located at 21 meters away. 2+3 works well for describing trucks and apples up to a point, unless you dump 2 and then 3 apples into a blender. Does 2 exist? Only as a schematic place holder for two objects that will persist over time long enough for the mathematical structure to tell us something useful.
All languages have effectiveness in everything, like the one I'm using right now to tell you this, which I couldn't do nearly as effectively with mathematics.
A lot of people conflate "concrete" (as in concrete vs abstract) with "real". They view concrete things like "matter" and "energy" as "real" while abstract things like "love" and (potentially) "math" as "unreal" or "made up". But there's no sufficient evidence that indicates that being "real" is contingent on being concrete. There's also no sufficient evidence that what we understand to be concrete is necessarily "real". Descartes's famous statement, "je pense, donc je suis" is abstract in nature but it's also tautological; it cannot be untrue. Or, in other words, it absolutely *must* be "real". But this concept is outside the scope of science because it does not yield to empirical analysis. In other words, there are lots of things that Science is equipped to handle, but just because science can't handle it doesn't automatically make it unreal. Hence, even if there is no empirical evidence for the existence of math, no concrete manifestation thereof, it can still be considered "real".
+Darkenedbyshadows Yeah, I definitely have more sympathy with platonism. It's very odd because pure math students that I talk to in real life are always platonists, never formalists, but nearly everyone online seems to be a formalist.
I think that speaks about the environment in which we're using the mathematics more than anything else. I'm a math and sciences tutor and on any given workday I can be using math for physics, chemistry, statistics and levels of math from arithmetic up to the calculus series and sometimes beyond. And if things are exceptionally slow, sometimes computer science. It's more or less impossible to work that range of material and not be forced into formalism. And that's not even considering the instructors at the college I work at who insist upon bringing their pure mathematics into the classroom and the myriad approaches I see to the same basic problems. Formalism has the rather significant advantage that you can focus more of your attention on the mathematical validity of what you're doing while you do the problem and then after you have a result, you can consider whether or not the result is one that applies to the application. You can also consider at that time whether or not the mathematical model even applies as a lot of that information comes from the form of the problem.
Since there's a lack of platonists on this comment section, let me come out of the closet. One argument for platonism that should be taken seriously is the _indispensibility argument_ which says, as a principle, we are ontologically committed to the objects about which our best scientific theories (i.e. physics) make statements about and since our best theories make statements about mathematical objects, we're committed to abstract mathematical objects, as they are _indispensible_. Perhaps if someone someday comes along with physical theories that work just as well and don't make statements about mathematical objects, we can ditch this argument.
Penelope Maddy addresses P1 - we ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories - of Quine's indispensability argument in Indispensability and Practice (1992), Naturalism and Ontology (1995), and Naturalism in Mathematics (1997). Her strategy is to undermine Quine's naturalist and holist justification for P1 of the argument, by showing that the former is incompatible with the latter. Maddy forwards three objections: 1. Naturalism says we ought to respect the attitudes of working scientists. Holism says there shouldn't be significant variations in those attitudes. Yet, working scientists often disagree heavily on existing theories. 2. Maddy suggests mathematics exists not within the true elements of scientific theories, but rather those idealised elements used to model reality. Water is assumed to be infinitely deep in analysing water waves; matter is continuous in fluid dynamics. 3. According to holism, logicians mathematicians should be assessing and revising the foundations of mathematics (the ZFC axioms) in accordance with the needs of physicists and other scientists. Given that such arguments for/against ZFC are almost always intra-mathematical, and we should respect that, this demonstrates that naturalism is favourable compared to holism.
Mathoma Isn't it the other way around? Don't our best scientific theories make statements about the universe oftentimes *using* mathematical objects? If science was making statements about mathematical objects, it would be math.
+Benjamin Przybocki I think it's pretty obvious the theories are making statements most directly about mathematical objects. We then say, secondarily, that the statements have something to do with physical objects. Whether they are directly or indirectly about mathematical objects, we both recognize that mathematical objects are needed in this picture _somewhere_.
+Communisation This is a complex comment so I'm not sure I can address it all at once. Let me just give my reactions to Maddy's objections to P1: the ontological commitment premise. Actually, a note on P1: why do we have to say _all and only_? I was initially under the impression that we should, at a minimum, be committed to such objects, not exclusively the objects needed for the best scientific theories. Perhaps this goes against Quine's aesthetic need to have a sparse ontology, but I don't really care about being profligate for the time being. 1. It seems obviously correct that scientists have major disagreements on existing theories but one thing I've never heard a disagreement on is whether the theories should be mathematically phrased. 2. I don't see what's left of a scientific theory once you've removed the mathematics, so how could it not be a true element of a scientific theory? Furthermore, the scientific theories often use mathematical terms right in their statements. For example, Faraday's law makes a statement about a _derivative_ of a magnetic _flux_, both terms already being mathematical concepts. 3. I don't understand this objection well enough to see how it's actually an objection to P1. Nonetheless, I'll have to check out some of Maddy's work.
I think the mention of the "unreasonable effectiveness of mathematics in the natural sciences" is the most important point of this video. Not the "real vs unreal" debate, which I really don't consider "serious" in any sense - rather just a play with words. On the other hand, the effectiveness of mathematics is something that is stumping me for a long time and I think getting into bottom of this might be one of the most important tasks human kind can (if) ever deal with. Is the physical world a natural consequence of math? If so, why? And can we exploit this "causality" to create another physical realm similar to ours? This is the big question.
Do not forget that our models are seldom, if ever, entirely complete or correct! We typically start with a crude approximation, enough to give us some rudimentary insight and ability to predict physical outcomes! And then we start a tedious model refining process, in order to make the answer fit closer and closer to the outcome of the physical process! Sometimes the necessary math, happily turns out to be wonderfully simple and astonishingly elegant, but for the most part, it is a female dog! And many times, we are forced to adopt entirely new mathematical structures, in order to achieve some success in our efforts! You don't just throw some numbers together, and out comes the perfect model, miraculously, as it is presented here!
Near the end... "knowing the rules of chess doesn't help scientists launch rockets into space" .......K, but a game about launching rockets into space would. Easy semantics, gg close.
i always thought of math as a logical language for describing and considering stuff in the world around us. it works so well because we've been building up the rules that math uses based on useful observations. numbers started as counting things that exist. then measuring things like distance and size.
Except... we explicitly and definitely do NOT do what you said. What you described is science. Mathematics is different. Mathematics absolutely is not built on rules based on useful observations. In fact, no observation of reality could ever disturb a single mathematical fact. Mathematics is a product of pure reason, totally divorced from reality except in its respect of basic logic. (And reality seems to perhaps not respect that logic so much, such as in the Copenhagen interpretation of quantum mechanics.) For instance, there is substantial physical evidence to suggest that spacetime is not continuous in the way the real number line is continuous, there being an infinite number of distinct positions between any two selected points regardless of their distance from one another. This means that the circle mathematics deals with can not exist in our universe. Every circle must have an exact answer for the ratio of its circumference to its diameter expressed in that 'smallest unit'. But, this does not cause any problems whatever for the definition of Pi, the real number line, or the geometry of circles. Those are intact because they were never built upon physical observation or definition to begin with. They were derived purely from a small set of basic axioms, and logical extensions thereof. Nothing more.
Dustin Rodriguez I am not convinced that math is a product of pure reason. Much of our math, including numbers, is rooted in practical problems, such as counting stock, making trades, measuring fields. Sure, abstract logic and constructions were used, but they were inspired by the real world. Sure, an abstract world of reason developed from this, and often took a life of its own. But whenever it bumped into a problem with the real world, new math was developed, or picked up off the shelf. That is the root of the "unreasonable" correspondence of math with science. Science, business, engineering, all inspires a great deal of math, whether mathematicians like it of not. Archimedes, Newton, Noether, all mathematicians that developed their math based on scientific problems.
Thomas R. Jackson Things like the natural numbers (not including 0) historically originated from counting stock and such, but that has nothing to do with mathematics. Mathematics is, by definition, exclusively the abstract arguments built upon reasoning from axioms through use of proofs. And while, yes, mathematics was expanded in ways to address physical problems, they never broke from the pure basis of it. The fact they never needed to is why the correspondance is 'unreasonable'. And in many cases, the math came first. Complex numbers and complex geometry are the usual example used. They were developed with no practical application, simply an outcome of pure reason. Then Einstein came along, and they made explaining relativity simple. Why? There's no good answer for that.
Dustin Rodriguez Yours is an interesting perspective. Tradesmen learn to count and add, and group, but that is not math because... I am at a loss. They may not have published a paper, but they did reason it out. It seems to me that the "pure" basis of math developed over time. But its very "purity", the idea that you can reason these abstract concepts, even invent new ones, perhaps inspired by the physical world, but maybe not, makes my point. These are abstractions, dependent upon our own thoughts to make them out. If they are "real" , then they are real in the same way that we make so many of our abstract thoughts real. Money, nations, laws,... these are all "real" in a sense too. They have measurable effects on our behaviour. That is empiric reality, something the proper subject of science. But if humans disapeared, so would these things because they have no independent reality. So to math, in a sense. The sense is that so many of our concepts in math were founded on our observation of the natural world, as seen through our perceptions. So that will live on beyond us. Will mathematical "truths" remain true when we are gone? I think the question is nonsensical because we invented the whole concept of truth as well. If we destroy a computer and its software, it does not exist. We might rebuild it, and perhaps the new code we construct will develop the same concepts and behaviours, and exist again, perhaps not. But the bottom line is that information has no existence apart from its physical matrix. Remove that, and information is gone. I think the idea of our abstract ideas having an existence apart from us or our creations, or records, is mysticism, pure and simple. It is a story our brains give us, a perceptual illusion if you will, to help us respond to our world. It is like the idea of truth, or self, or soul, or or any of the other things we conceive of as distinct from actual matter. Philosophically you may disagree, and I can not "prove" you wrong. But neither can you prove your spiritual conception right.
It's a matter of historical fact that the first mathematics came from trying to understand and manipulate the natural world. Then we humans, being as intelligent as we are, were able to abstract, refine and expand the knowledge we had so gained. But although parts of maths had followed axiomatic systems as far back as Euclid, it was only in the late 19th century that there was a systematic attempt to derive the whole of maths "from a small set of basic axioms, and logical extensions thereof" So, in seeking to understand Wigner's "unreasonable effectiveness" (the objection to antirealism) the historical roots must be kept in mind. If someone had come up with ZFC set theory purely recreationally,, then its effectiveness would be astounding, but instead ZFC was an attempt to systemise and axiomatise abstract and logical procedures we (humans) had developed primarily (tho not exclusively) for understanding the natural world. Don't get me wrong, I understand we have abstracted that knowledge and then run with it; and the 'running with it' doesn't necessarily have to have anything to do with the natural world any more; and abstractions, pretty much by definition, do not exist in the real cconcrete world.
Math isn't like science, where you gain enough evidence and declare something true, you have to have a precise logical proof. Greatest sentence ever uttered by a human being ever.
Mathematics exists beyond everything else. Humanity discovered and developed mathematical rules and concepts, but other intelligent extraterrestrial lives could understand the same concepts, beginning with the counting of stars.
stardude692001 And that is precisely why, the sum can be 10 (base 5), 11 (base 4), or 12 (base 3). Of course, in the case of base 3, one would have to have the 3, of the addends, expressed differently, to avoid confusion, since the numeral 3 could not be written as is, in that base, or it would be meaningless!
It seems to me that formalism is the best explanation for math, because math is just a logically self-consistent system of precisely describing relationships. Any such system would be equally useful, whether you used numbers or colors or sounds, as long as the definitions are precise.
> Any such system would be equally useful, whether you used numbers or colors or sounds, as long as the definitions are precise That statement actually makes you a platonist, because you agree that the underlying mathematical concepts exist beyond our brains or our selected numbers, symbols and rules. They are "there", waiting to be discovered, no matter what names we give them.
I'm very sorry to say but there is a mistake at 1:32. 2 is a prime number and 9 is NOT! I'm disappointed... The first few prime numbers are 2, 3, 5, 7, 11, 13.
Here’s my guess: Our mind is a pattern recognition machine, it’s how neurons work. We sometimes make simplified analogies of commonly observed patterns and articulate them. Culture accumulates the most useful of these observations and weeds out inconsistency. Five rocks are a common pattern, but coarse sand and very small rock fragments confuse things in nature on the beach, so we grab five roughly the same pebbles to demonstrate “five” to others. We filter, or simplify ideas, out the patterns that interest us. The idea of five can be used to build other simplified analogies of commonly observed patterns, such as five groups of five, so on. The ideas and symbols of commonly observed patterns represent something both in and outside of our minds. We can say very precise things about “five”, that we could never say about a small pile of rocks, sand and dirt.
I fell in love with PBS Space Time, and now I have Infinite Series? Awesome stuff! With regards to realism vs. anti-realism, is it fair to say that considering numbers to be a fundamental property of our universe (like the speed of light, or the planck constant) is a realist viewpoint? To me, saying that numbers aren't real is to say that nothing is discrete. Particles exist, and they are discrete. So numbers just arise from that fundamental truth. Once you have 1+1=2 the rest of mathematics just reveals itself through logic.
Although there are many non-platonists in this comment section, can anyone offer an actual philosophical argument for why they hold their position? It seems everyone is giving their opinion, e.g. _math is invented_, _math is just a tool_, ... without giving an argument for their philosophical position.
Lovely! Is this your personal discovery, or is this one of those arcane facts known only to a small circle of intimates, or is this a commonly known fact that I am naively ignorant of? Thanks for pointing it out, you have made my day.
0:30: 1 is divisible by one and itself and we don't assume that 1 is a prime number. Obviously is the matter of definition, however in order to formulate in the elegant way the theorem about the factorization of any natural number into primes, we have to exclude 1 as the prime number (if we want to have uniqueness statement).
Mathematics is the study of patterns. The symbols and notation of mathematics is simply our language for describing different patterns. Pondering whether numbers are real is like pondering whether words are real. The word "banana" is used to communicate about something real. The word "unicorn" is used to communicate about something not real. Mathematics can be used to describe patterns we observe in the universe and patterns we don't observe in the universe. When we study mathematics we are essentially exploring our ability to describe all different kinds of patterns. Prime numbers are interesting because they are constructs of our language of patterns, yet their distribution is so difficult to characterize with that language.
we are not talking about the representation of numbers we are talking about their meaning. you may represent 1,2,3 by a,s,d or any other symbol but their relationship won't change.number are highly consistent with natural phenomena even if we don't intend them to be which is weird and suggests the independent existence of numbers
Aashray Narang I'm not talking about the representation of numbers either. I'm talking about their meaning. We intend math to describe patterns. The reason it's so effective at describing the universe is that there are patterns in the universe.
Abe Dillon there are patterns in universe only when you see them through maths. there are no patterns if there is no maths. we didn't create maths to see patterns because we notice patterns when we have maths (don't consider abstract patterns like constellations) and this application of maths in physical world is weird.
"there are only patterns in the universe only when you see them through maths" That's absurd. Plants adapted to seasonal patterns long before humans invented math. Pulsars still pulse periodically even without math. Atoms would still bind to form molecules in specific geometric structures even if pattern language didn't exist. "We didn't create maths to see patterns because we dnotice patterns when we have math." My claim is that we created math to *study* and *describe* patterns. It's ridiculous to say that we need math to "notice" patterns. The brain is an exceptionally good at pattern recognition. It has been long before we came up with math. Simply knowing the complex behavioral pattern required to reproduce is evidence enough for patterns existing before we developed a system for studying and describing them. The universe is clearly not 100% chaotic. You don't see TV static every time you open your eyes. You see a universe with lots of predictable, repeating phenomenon.
Abe Dillon i want to clarify that i am not talking about patterns like plant seasonal pattern or abstract patterns and i am also not saying that objects in universe know maths and then they show patterns , i am saying that we can predict the behavior of objects in universe by using maths take for example hydrogen, energy of electron in hydrogen atom is given as E= (2π^2×m×e^4)/(n^2×h^2) put n =1 and you get the ionisation energy of electron that is 13.6eV per atom provide electron this much energy and ya it leaves hydrogen atom, provide energy less than this the electron do not leaves hydrogen atom this predictability in the behaviour of electron is weird (don't say its chemistry and not maths) and there are thousands of other examples which predict the behaviour of physical objects using maths and i am talking about such patterns
there is a real sense in which something like a number is "made up", because if humans did not exist, the universe would be indifferent to its inner workings, the ratio of the circumference of a circle to its diameter, etc. But the fact that we can communicate at all using abstract concepts like circle means that there is something that we can call objective reality, but not a Platonic reality in which perfect circles exist
That's kinda a false dichotomy isn't it? We defined math, for certain. But we defined aspects of math based on natural observations. We did just THINK up numbers, we thought up numbers to label counts of actual objects. Newton didn't invent calculus and then realize it magically mimicked the motion of the planets, he specifically defined calculus to represent the motion of planets. The fact that calculus can be used to describe other natural phenomenon is more unsurprising than it is surprising, many things in nature have a relationship between states similar to the relationships that cause planetary motion. Math exists in natural form only in the sense that things can interact in physics and have causal relationships that can be described with math, that doesn't mean there is some universal calculator that exists independent of the human (or alien) mind. (unless we're in a simulation). It's quite the opposite, the particle interaction of nature that allows for causal relationships is what allows computation of the human brain and computers. It does kinda turn into semantics, semantics of definitions. Is a tree a house? Depends on how we define house, does it keep off the rain? Does it keep out the wind? A house without a windows or doors (but holes for them) doesn't keep out the wind, is it still a house? Is a cave a house? If we define the relationship of natural phenomenon to be mathematical, it is, because we defined it. But the similarities between abstract numbers and their relationships of mathematical functions are distinct from the natural relationships. The similarities are not surprising, because we've defined the rules of math to mimic natural phenomenon and as such, the fact that it mimics natural phenomenon in unexpected ways, isn't surprising at all. It's like if you make a life simulator and it creates lifelike behaviors that seem to defy the original definition, it might be surprising emotionally, but it's not unexpected, because it's what you defined it to do.
This subject is the biggest battel I ever fought (and still fight) within my little brain. And this video is by far the best summary of it and introduction to it. Thanks a lot! This makes it easier to communicate with other people on fascinates me and keeps me busy.
I'm definitely a formalist, I've drifted more and more that direction over the years because such a large portion of the problems I see are ones that I wasn't taught and I don't usually have the time necessary to do any research. Ignoring the reality of it is pretty much the only way to get any work done, not to mention that it's nigh impossible to comprehend what you're doing until you have something on the paper to understand. Trying to understand prematurely, just leads to the possibility that you hedge out a productive direction of attack whereas ignoring that and focusing on the form puts you in the position where the worst case scenarios are either not getting a result at all or checking the result against the application and finding that it doesn't check. As opposed to not getting anything at all. For me personally, I advocate knowing what you're working with and what you're looking to accomplish, then focusing on what the form of the situation is talking about and working from there. Once you've got a possible solution, that's the time to look at the work and try to understand whether it applies and if so how it applies and worry about whether or not the solution makes any sense. Trying to do that prematurely has never worked out well for me.
bloodyadaku in Z/4 for example 2+3=1. In Z/5, 2+3=0. In fact, in every Z/n the sum 2+3 where "n" is not equal to 1 and is less or equal than 5, the result would be a different value than 5. Magic right? haha
+bloodyadaku Yes, you are right, Z is the integers. Z/n would be the algebraic space where the number is the remainder of dividing the result, calculated like we normally do, by n. This are other spaces or classes within algebra. If you ask me, it changes the standard definition of what the operators "+", "-", etc, do. And this is a far as I understand it. I am sure that a mathematician could give a much better and clear definition than what I did. I am an engineer, but my wife is a mathematician. Z/4 I think is the notation that is used in english, in Argentina (spanish language) she always used the notation Z_4, or more generic Z_n. But that is just a notation, although I would have to say that is much more clear the english than the spanish one. When many afternoons in the bar are filled with researchers in math this sort of things come up lol. In the end it is just another definition that is useful for them in some way that goes beyond my ability to understand what they say. It is like when they ask "what is the sum of all possitive integers", I answer "infinity" and they say "that is one possible answer, but it depends of what you are talking about". Standard maths will yield the result infinity, but if you change only one definition (how you calculate the limit of an infinte series), you get the increidible answer "-1/12". This last result is greatly explained by mathologer (just type "mathologer ramanujan" in youtube and you will find it). To be honest, I was really unconfortable with all of this at first (and was really grumpy about it). For my engineer head it was extremelly annoying to say the least hahaha. But I guess math is all about logical consistency. It is not that they wanted to make "2+3=0" or "2+3=1" or "1+2+3+... = -1/12", it is just a result that is the logical conclussion when you just change a definition. I guess math is much more amazing and even much more complicated (and maybe much more annoying) than we all non mathematicians realize at first glance.
7 років тому+1
"If a tree falls in a forest and no one is around to hear it, does it make a sound?"
Maci Sordi Math is a language, sure, but this language is a tool humans invented to solve problems in reality, which is why it has the ability to model reality.
Maci Sordi I don't think I am making conjectures. For example, there are tribes whose adults cannot understand numbers, because their language never invented them. We can try all sorts of visual comparison methods, but they cannot tell the difference in quantity between a group of 5 sticks, or a group of 6 sticks. They nevwr needed to invent a number system, brcause numbers didn't serve any purpose for them, and didn't solve any problems for them. The way we think is based on our evolutionary psychology, and societal development. We know humans create languages, and we know humans create tools to solve problems. Our species didn't have language or civilization until primitive agriculture began to make civilization possible. Mathematics is a creative part of modern languages, and was invented to solve specific problems by simplifying reality into a useful fiction, allowing for trade, taxation, and engineering to be done more consistently. There's no magical universal form that humans are psychicly reading from.
This doesn't mean that their ears cannot do Fourier Transform to listen... Our brain is made by things that follow the rules of our universe, mathematical rules. We discover(ed) our "special mathematic", but it is only a limited part of the total mathematic that exists apart from us. There is no magic in what i say.
There is a total false dichotomy here. Math is either a made up game or about imaginary objects? Those are the only two possibilities worth mentioning? Why did you not consider the idea that math is about _real_ objects _viewed from a certian perspective?_ Take the example of a "perfect" circle. If you look around you, you can find countless examples of circularity in many forms. If you take those examples and _abstract away_ their differences, you are left with the concept of a circle (with each example being a variant on the same theme). There clearly isn't a "perfect" circle somewhere out there waiting for us to discover it. It's also clear that the concept of a circle wasn't just made up out of nowhere (how would you even come up with the idea without first looking at reality?).
1+1. i know 1 is not prime (nor a composite...), but that's a rule we made up. Technically 1 is devidable only by 1 and itself, just like all other primes. but this is only speculation...
Formalist. A platonic interpretation cultivated my love of mathematics, but was also a HUGE OBSTACLE to clearly thinking about the deep problems in mathematics, like foundations.
The answer is that we didn't simply made up rules that mathematics it built upon. We discovered them while observing natural world. Yes, math is like rules of a game, but those are rules embedded in the universe we live in. Mathematical rules govern all natural phenomena, including very first moments of our life. We see perfect mathematical division in the development of human zygote, which after four divisions consists of exactly 16 blastomeres. Golden ratio determines proportions between individual bones in our hands. We see logarithmic spirals in shells of snails and arms of the galaxies. There's really nothing surprising in the fact that mathematical rules we uncover from the world around us help us accomplish scientific miracles.
You could make much of these same arguments about any language or even abstractions in general. Mathematics is simply the rules that come from abstracting certain observable aspects of reality.
Cool poem, I'm sure Kelsey's looks as well as her intellect are what landed her this hosting gig. Both are on display and I'll comment on both. If she doesn't like it she should do voice overs.
"If she doesn't like it she should do voice overs." "My name's Dylan and instead of respecting women I think they should just not show their face to prevent me from disrespecting them."
Yea you're totally right but your commenting on her body is completely irrelevant and off topic and women have been undermined in the STEM fields since they originated and you making it about her body rather than the subject she's talking about is pretty disrespectful and gross. Not to mention the fact that what you just did happens to women of high esteem constantly. Female actors, singers, business owners, and just in general women are asked about their physical appearance rather than their accomplishments and your comment is the exact same. Plus your idea about how we should be able to comment on each other is more for friends telling each other if they're being dicks or if they should stop doing something. Not for a stranger online to discuss whether or not an intelligent woman has a tattoo under her dress or not.
This is how I felt when I studied Grafting Numbers. That I was discovering something that was already there and the connections that I found to Catalan Numbers were completely unexpected!
Formalist here. To address your point, the reason math tends to describe things in reality is because we tailor the rules to model objects we see in reality. Math's usefulness in physics relies heavily on the accuracy of this model. Also, there is plenty of math that doesn't describe reality; it just doesn't get as much press.
Is Pi infinite because a circle has an infinite number of small straight sides? Infinities exist in math but I'm not too sure about in our observable reality because of quantum limits like the Planck length etc...? So if Pi really is infinite and real that means our universe if infinite in size!? 😲 Or maybe we look at circles wrong and true curvature is possible but can only be described with the correct description, which we don't have yet?
1:37 "Mathematics can feel like you wrote the first page of a book and then, you're figuring out the rest." Kinda reminds me of programming: You write a piece of source code and you don't immediately anticipate what happens next. But what happens next is a logical consequence of what you wrote into the source code, meaning that the same source code will always result in the same result, even though it only happens after you actually run the program!
Goldback conjecture for 2n: list [odds in] 1 to n in top row and [odds in] n - 1 down to n in bottom row, at least one prime is above another. For 32: 01, [03], 05, 07, 09, 11, [13], 15 31, [29], 27, 25, 23, 21, [19], 17
Not a mathematician here as such but love the sciences without any formal education of the subjects! Eg. Frequency - sound and light, heat and cold are all related and our understanding of them relies on our abilities to calculate. Heat and cold, sound and light are all observable. Understanding what they are relies on our use of maths! One of my favorite math concepts is Infinity - a line an inch long technically only has 1 dimension - length and that length of 1 inch is infinitely divisible. A line a million miles long also has only 1 dimension - length, and that length of million miles is also infinitely divisible, so is it just a perception that makes the difference between the two or is our perception limiting us to consider distance as a factor which may turn out to be an illusion? Then you have the axis of a rotating body - this axis also only has one dimension - length. The further out from the center of the axis, the more observable the rotation is but as you get closer to the center of the axis - is there a point where the rotation ceases to exist or maybe becomes or exists purely as a void like a nano-sized black hole!? Whichever is the case, the axis may be infinite or infinitesimal in length and have only that one dimension yet hold almost unimaginable power as it seems that is lays at the heart of almost everything we can observe! As I said, I am not a mathematician or a scientist, I am more of a philosopher and so I love these questions which often keep me awake at night. Thanks for your channel, I have subscribed and look forward to future episodes. Cheers Jeff
I feel formalist. Yes, we made it up, we made the axioms that we liked. They might happen to be "Unreasonably Effective" because we made them that way. We can see that others accept the theorems, so other thinking people may make up the same things. At 2:03, may I suggest that you use scientific notation? Who do you think you are talking to here? Do you want us to count up zeros, or do you want to tell us the magnitude of the number?
I miss Kelsey and this series. This is so fun and open my mind widely about things I thought to be hard and boring. So sad that it is cancelled now.
yeah
my mind: crys
me: oh my god that annoying brain that keeps crying
I agree.
I'm definitely in the "math is something we made up" camp. Math is a language and language is something humans made up. It's relatively ordered and so is the universe, so they tend to correlate. But, as a physicist, I can tell you that correlation is /not/ perfect. It just sits firmly in the "close enough" category. It is not the language of the universe. If you want to say the universe even speaks a language, then math is just our best translation of it into "human"... but, like with any other language, some things get lost in translation.
That "close enough" sounds an awful lot like a limit. If you consider the philosophical implications, it seems an awful lot, if the universe isn't explicitly doing so, like the universe just wants us to believe it.
Is Pi infinite because a circle has an infinite number of small straight sides? Infinities exist in math but I'm not too sure about in our observable reality because of quantum limits like the Planck length etc...? So if Pi really is infinite and real that means our universe if infinite in size!? 😲
Or maybe we look at circles wrong and true curvature is possible but can only be described with the correct description, which we don't have yet?
There might be an Einstein quote, forgot it, along those lines...that math doesn't perfectly predict the universe, it's only precise when abstract, etc, or something, forgot the quote...at any rate, I wonder what a universe that didn't somehow follow math rules would even look like, lol...if it's going to follow any rules at all, wouldn't we end up calling those rules "math" anyway?...Well, I suppose one could say that there are plenty of things that follow rules but which aren't as directly "mathematical", like biological beings, etc, one would have to zoom in to find the mathematical rules, etc. But still, lol...how would a universe that doesn't follow mathematical rules even be like?...Is it conceivable, would it be a universe that doesn't follow any rules at all?...
@The Science Asylum
I know this is 5 years old but this is more wrong than anything can possibly be wrong. Math isnt *just* the language of the universe but it IS the universe
Like... Its baffling how any thinker can even believe that humans calling a thing 'something' actually affects that thing in real life. Thats absurdity and reeks of ego
Humans arent an integral part of the universe... Mathematical symmetries exist in nature regardless of what humans call it. Math is simply cause + effect, logic, shapes, and what the gradients of the universe are doing in the present time. Logic on paper man logic on paper
Like... Did triangles not exist before Pythagoras "invented" his theorem?
Euler "Invented" countess mathematical equations that are fundament to reality itself. So.... Did reality not exist before Euler was born? Were the rules of reality different before Newton "discovered" gravity?
Scientist are the most egotistical people on the planet jeeeeeez. These people are simple OBSERVING whats already there not CREATING.
Lol...I don't know...there are infinitely many things we could define, our there, lol...there is a sense in which they only start existing when we define them...a sense...but anyway, lol...
I'm a strong formalist. At its base, Mathematics is simply defining some rules (or axioms) and carrying them out to their conclusion. The reason it describes reality so well is that we tend to pick axioms that we feel define our reality. We could pick completely different axioms and get completely different mathematics, but it just wouldn't describe reality. I believe this is most noticeable when we discover independent axioms, like the continuum hypothesis. Math works no matter what version we choose, but maybe we will see that one version lets us describe our reality better.
Okay, long speech over. Just want to say I love the series, and I'm really excited to see more videos! You all are amazing, keep it up!
Would you not agree that the set of conclusions from given axioms forms some kind of reality? If there is only one outcome of the rules and every observer would measure the same outcome, how is the outcome any less real than say, a photon?
Nice! You might be interested in a branch of the philosophy of mathematics called "logicism".
I also think the Axiom of Choice would be an even better example for your argument, since it's now (essentially) accepted because it proved so many statements that mathematicians felt were true.
Modus Ponens Now we get into the question of what is really "real". Through the philosophy of skepticism, you could make the argument that our objective reality is created through our minds, since the only way we can truly "prove" something exists is through our senses, and our senses have been wrong before. Therefore, through the eyes of skepticism, math could be made up in or minds, since our reality itself could have been created in our minds. Of course, there are fakes with this theory, but I feel like it's worth noting
+Kevin Merrick
But you don't explain the problem posed by Wigner about the effectiveness of math.
The thing is that simply claiming that we chose axioms to mimic reality doesn't brush off the idea that these mathematical axioms/objects exist independantly of human minds.
If anything it just reinforces the idea because if our axiomatic system is simply "copying something" then by definition it means we didn't really invent them. Also If that "something" we're copying is just an other mathecatical axiomatic system then it would mean that math exist!
Modus Ponens I am a formalist and I have thought before about the question you raise. I think you're absolutely right; the logical deductions do form a objective reality. So, how does this not make me a mathematical platonist? Well, a platonist would accept the axioms as having some objective ontological significance. I still think the axioms are just the "rules of the game" that we defined. I also believe that when we say we've proven something to be true, we're really just saying that if we accept ZFC (or whatever we're using as our foundation), then such and such theorem is necessarily true. Since I don't accept the axioms as ontologically objective entities, but I do accept the logical deductions that follow from a set of axioms as ontologically objective entities, I like to say I'm a "logical platonist" but not a mathematical platonist. Logic is definitely real and objective in my opinion.
I'm more of a procrastinist when it comes to math
Is English real ? It seems to have an unreasonable effectiveness describing my thoughts.
so its real
except english changes. Its words dont just accumulate, but evolve. There are also other languages that describe other things better.
Math is special. Plato was right. Math is more real than us.
English and every known natural language is spectacularly poor at describing reality. We just aren't aware of this because part of what we call our thoughts is expressed internally as words.
Since part of what we are describing as 'thinking' is built out of words, well then this is like saying something is itself.
But this is an incomplete statement because we have other modes of thinking, that we experience, that are not built out of words. For instance what are dreams built out of? Or what does your brain do when it 'sees' something? For all these other kinds of thought, words are really bad at conveying, identifying, or describing what is going on.
Maths itself is abstract, however we can use it to define and manipulate real things in terms of physics. Physics is applied maths. Our buildings don't fall down, and we used physics to make them.
Circular argument. English is defining your thoughts, therefore it should be capable of describing your thoughts.
Mathematics is based on logic. The vocabulary is freely invented, but the rules of grammar for this formal language are the logic. The rules for logical thinking are constructed in a way, that is consistent with the *physical properties of the universe*. It means, the rules of logic to apply mathematics, to proof conjectures, are specifically designed to set the *physical system "human brain"* (or computer or alien brain) into a *restricted state, which always produces the same consistent results independent* from the specific system used or any superficial conditions (like mood).
That's why different mathematicians using the same assumptions and axioms obtain the same results and why students can relatively easily follow the way of thinking by reading a proof.
If, on the other hand, they would read some complete nonsense, a flawed proof or fail to apply proper rules, they discover an inconsistency between the way their brains produce results and the material at hand. I.e. the proof is logically inconsistent or the applied logical rules were not stringent enough to force consistent results.
I take it you do not subscribe either to the Copenhagen interpretation of quantum mechanics, or to an idea of discrete spacetime units? Also, what happens when large swathes of mathematics are based on things like hyperbolic or higher-dimensional geometries? Sure the rules of logic used are the same, but the axioms are switched up.
Dustin Rodriguez The Copenhagen interpretation doesn't interpret anything. It just states the observed effect in a more formal manner. Still want to know why.
This has already proven itself to be an incredible series. The metaphor of writing the first page of a book was very nice. Thank you for producing it so wonderfully!
at 1:32 why is 9 there, 9 is NOT a prime (it's 3*3 btw)
and why is 2 not there (assuming they wanted to show the first primes)
@@pietervannes4476 2 is generally regarded as "not prime". Don't know why
@@lEGOBOT2565 no its not
@@pietervannes4476 not what I heard from doctorate level mathematicians that I have met at a nerd convention
@@lEGOBOT2565 that is very strange, as 2 is definitely a number that meets every requirement to be a prime and functions like primes too. Also at another point in the video it was shown with other primes
I still haven't found a better alternative than the one which was proposed by Gottlog Frege in his Foundations of Mathematics: mathematical objects are existent, but abstract: they are 'found' when we are using our language on a higher level. While in natural language there are concepts in which the value of reference is empirically determined, mathematical objects are a second order concepts, they are concepts of concepts. Nevertheless, they are true and objective: their logic, relations and operations are valid for more than one person, and work independently of our opinions or taste about math.
In a certain way, Frege was called a platonist, but if we consider that mathematical objects, although true and objective are only found when one is thinking on a higher level of language, his approach was much more kantian and formalist. Math seems to work as a tool and a language to describe nature only because we already perceive nature as mathematically ordered. Math seems to fit so perfectly not because of math itself nor nature itself, but because of how our own mind works.
You have ten fingers, I have ten fingers, bob has ten fingers. Abstract away who has ten fingers and you get the abstraction "ten fingers"
Like wise, I have ten apples, you have ten apples, bob has ten apples. Abstract away who has ten apples and you get the abstraction "ten apples"
So we have the abstractions "ten apples", "ten fingers", "ten toes" etc. What if we abstract away what we have ten of? We get an abstraction of an abstraction. The number "ten"
Every culture developed its mathematics through this abstracting from simple counting of physical things like fingers or apples to these abstractions. Numbers. And then considering that "fingers" and "apples" are already abstractions themselves, numbers are 4th order abstractions. They are so abstract as to be universal. And all of mathematics is just abstraction upon the abstraction of numbers, so no wonder they work so well, because they are so bloody abstract!
@Juliano Bittencourt
I know this is 5 years old but you just used a bunch of words to say a whole lot of nothing smh
Its not that hard people overcomplicate things because of their massive god complexes. Humans arent gods in hat regard humans didnt create the universe which is founded on mathematical principles.
The symmetries of the universe exist regardless of what we call them. Subatomic particles are the physical manifestation of 1+ (-1) = 0 as charges and spin cancel out. Plants grow geometrically regardless if we have a word for geometry
The circumference of a circle will ALWAYS be 2πr and always have been even before humans existed.
Let a few students lose in the jungle without a guide of any kind, and they would soon develop a language to learn to live there. The real information comes from the environment and the language is used to sort it out, and adjusts continuously to fit change. Maths is language, and part of us like all tools.
Plus, everyone is is allowed to have a hypothesis about what is possible and test number relationships until the rules emerge - and then find the real world correlation and or corrections. It's endlessly interesting.
I'm already hooked on this series! Totally looking forward to more episodes :D
Glad to hear it! We're looking forward to delivering more episodes. :)
PBS Infinite Series please do one on Complex Functions and their Graphs!
I just realized I wasn't even subscribed... I've watched so many of the vids cause they appear in suggested and home pages... time to change it I think!
With all these interesting discussions going around, I thought I'd share my own view. I'm a struggling musician with a passion for math, physics and philosophy. I've read the essentials of Platon, Hilbert, Russell etc. I especially enjoyed reading some of Russell's ideas, but in the end I believe that math is like the product of a compiler outside this universe. It's the underline structure, or the predetermined framework on witch our universe exists. It's created by something we simply cannot fathom. I like to think about it like a computer game. All games have rules and more or less the world created inside that game follows basic logic and it's self consistent in that framework. It can also have physics and gravity programmed in it. The player, or the AI will always obey those rules. Now, if the AI was sentient, he would be aware of all these rules and would analyze its world from his perspective, but he will never have a clue that he's the product of some functions and scripts written in C#, outside his world. The very notion of C# is utterly meaningless in the product of C#.
Anyway, fruitcake or not, that's just my take. It's not like I'm gonna get better answers this lifetime.
*If A Perfect Circle exists, how do you interact with it?*
_With a Tool_
Found the engineer.
Formalist here. The reason that math works so well in science, engineering and technology is because those subjects have long been strong motivators for mathematics. And because they work so well, those are the math systems we teach and study. There is nothing unreasonable about the fit, any more than it is unreasonable that my door fits into my door frame. (Though I still puzzle why my refrigerator doesn't fit into the space that is purported to have been built for it in my kitchen).
Our invented math only appears to describe nature. Oh, it comes close, but when we look closer, it fails. Newtonian physics and Euclid is all neat and tidy and was assumed to explain nature, but it doesn't. We go fast, and we have to find a new geometry to make things work. Or closer to home, the Euclidean stuff doesn't really work measuring things on the surface of the earth because the earth is not a plane, it is a sphere. So all those lovely triangle rules don't really work. So we looked around at some of the new ideas and picked one that seems to fit nicely... so far. We go small and find that there are no circles because matter is quantized and thus inherently fuzzy. The concept of a circle is still useful in approximating reality., but it isn't reality.
And that is the crux of the matter. It isn't that there is some "ideal circle" that the real world approximates. It's the other way around. There is a real world with which we interact, and our math approximates it. Well, some of it does. Chess is another form of math, just like the Game of Life. Neither describes reality, but the latter especially gives us a lot of insights into how replicating patterns behave over time. New math. And more complex mathematical models have been built on this insight and do an even better job of describing the real world.
But I understand why we are so tempted to think of our math as "real". That is our great human talent, to imagine things and make them real, even when they remain imaginary. Money, countries, laws, corporations,... none of these are real, they are all inventions of our imagination. But because we believe them to be real, because we see these figments of our imagination as real and tangible and are so utterly convinced that we act accordingly, these abstractions act as though they were real. In fact, in a certain sense, we make them real. Just like math. They are abstractions that sprung forth from our imaginations and we endowed with the gift of existence and reality.
Thomas R. Jackson The problem with formalism is that it implies that their are other systems that we could have just as easily have invented that would also be able to describe the universe, but there aren't. Only one mathematics, our mathematics, exists and can described the natural world. You can explain particle physics with matrices or differential equations but it's all the same math.
I'm sorry but 2+2=4 is a constant. Quoting someone really smart doesn't mean that their quote holds any water. No matter what, math is going to be a constant in our reality.
That was a great post by youtube comment section standards.
Thanks for writing this.
MrDajdawg I think you're missing the point. No one is claiming that at some point 2+2 will equal anything but 4; it follows from the basic axioms. We use the maths we do because it works so well at describing the natural world, but certainly over the past few centuries, and probably millennia, mathematicians have explored maths that doesn't appear to describe the natural world. Of course, in some cases, some of that maths has turned out to be useful.
Similarly, I can imagine an alien species on a planet with radically different material circumstances, different chemistry, biology, and society etc developing ways of understanding their world that use different sorts of concepts and maybe miss out lots of what seems basic to us. Part of the problem is that maths is so ingrained into our thought and perception that it's hard to think about understanding the world without it.
You say laws and corporations are not real, but how are these things different from an EM wave or a cell nucleus? Aren't all things just trying to adhere to rules? I think a relatively stable environment respects more rules and we can talk about "time" and "energy" or the realness that is intertwined with causality. But just because there are perceivable layers to the rules of a causal world, it doesn't mean that some layers are less real than others. If anything, I would say the only thing that is not real is an environment that has no "time" or way of measuring "energy", no "causality", no patterns or logical progression. No rules!
I would argue that math is like a game, where we're following rules to their necessary conclusions, but the rules aren't completely arbitrary: they're rooted in nature. We didn't just decide 1 + 1 = 2, it's a natural fact. We didn't decide a circle was the shape where all points are equidistant from the center, the universe did. Mathematicians took the rules nature gave them and ran with them, which is why math, while game-like, describes the real world so well.
Did the universe define the circle though? There exists no circle in the universe with those properties. Or a sphere for that matter. It was probably a mathematician who sat down a while ago and defined what an ideal circle is and then ran with it. :p
+Danial Haseeb Maybe the universe didn't define the perfect circle, but the way space is shaped suggested that a circle would be the right shape :p
M.W. Vaughn But that is what our human brains think is the right shape. Maybe some other sentient being would differ with us on that. :p
This can go on and on.
Man, I love Devil's Advocate :p
You're absolutely right, it can. Maybe there's no good answer
M.W. Vaughn humans tend to think of circles as perfect, but imagine trying convince a sentient bee that circles are better than hexagons.
infinite series, what is your limit?
darksid007 I tend to converge to pi but occasionally break loose and diverge
+
It converges to infinity
@@98bransonfun it does indeed, but if you watch an episode of Khan academy at the same time you can use la'hopitals rule to show it converges to 4
"la'hopitals"
Maybe the key to the puzzle is the origin of mathematics. Just like you said math is a book that write itself but we have to write the first page. And it's not difficult to see why 1 + 1 = 2, just count your fingers. We agree on the basic rules of adding, subtracting, dividing etc. because they make sense in the real world - I still remember learning about division with X apples and Y plates to put them in... So, math isn't real but it's not unreal either. And it's effective because the real world is in itself consistent, just like math is. It's fascinating that our logical brains can extrapolate unknown real world properties by just looking at the math - I'm talking black holes, dark matter, time dilation and so on - all real world objects which were predicted by the math.
***** yeah, I guess I should have mentioned I don't know shit about maths and I'm talking out of my ass... well, mentioning it now.
well this has been the nicest youtube argument I've ever had :D
Hi, I'm a human from a simpler time. I fish for a living. I keep finding lesser fish in my basket than I caught, I can tell someone is stealing them but I can't keep track of them. one day I decided to give a different name to each amount of fish starting from a fish to many fish and increasing by a single fish each time. And that is how I invented numbers who do not exist on their own whatsoever and the only reason they are so good at describing how many fish I have is that that is exactly what I modeled them after.
TLDR: numbers are only so good at describing reality because, in the beginning, they were concepts that were INVENTED to Describe reality, like how many fish someone had caught.
Formalism FTW :)
I never understood (and probably never will understand), why some mathematicians think that every concept/idea has some physical object attached to it. At some point in time humans started counting sheep and made up _numbers_. When we started thinking about shapes, humans made up concepts like _distance_ and _angle_ by defining them. There might be big circles floating around in the universe, but the concept of a perfect _circle_ is just a definition a human made up (possibly based on real-world-objects). Integrals were made to calculate the area under a curve, then a definition came by and suddenly we can do much more with integrals.
Great video :)
PS: A prime number is not defined to be only divisible by itself and one, because this would include one as a prime number (because one is divisible by one and also divisible by one), but one isn't a prime number. There is a similar definition though: Every number greater than one is called _prime_, if and only if it is divisible by itself and one. Another definition (I personally like the most) is this: A positive integer is called _prime_, iff it is divisible by two different positive integers exactly.
+Cubinator73
But was anyone claiming that concepts and ideas have physical objects attached to it? People who think mathematical objects really exist have to accept that they are non-physical abstract objects. Ironically, it's the naive empiricists who want to say all mathematical statements are tied up with physical objects that would want to draw this link between mathematical objects and physical objects (even though that was a failed philosophical project).
Ideal forms of humanly perceived, elementary objects! Simplified by stripping them down to their bare minimum, but still keeping them recognizable (e.g. a circle)!
Numbers are tools for counting objects (1,2,3,4,.. etc), AND also NAMES, assignable to objects, even non-sequentially!
"non-physical abstract objects" don't exist in the way I use the word 'exist'. I don't accept that because we think about something that means it has an objective existence. If what you are saying is that things exist because people think about them, then that implies God exists simply because people think about the concept of God, which is an even weaker argument than St Anselm's. Is that your position?
While there has been some sloppy thinking on this thread, I think the reason antirealists (certainly in my case) have drawn links between physical & mathematical objects is to show that Wigner's effectiveness is not unreasonable at all. Rather, we've invented the abstract objects in order to explain the phenomena we see in the natural world, and we've got better and better at inventing and refining the abstract tools we use for this over the millenia.
+Donach mc kenna Nope. I don't say things exist, because people think about them. I can think of a perfect circle and yet there is no perfect circle anywhere. But the idea of a perfect circle exists (not as physical object) and is denoted / described with language / remembered with electrons buzzing around in my brain. I can think and talk about God, but that doesn't make him suddenly exist as physical entity. I can even think about perpetuum mobiles, but that doesn't make them possible in reality. Even though, there is no real Harry Potter, he still exists in books, in movies, ...
The (probably idealized) idea of some thing is not the thing. The thing *might* exist as physical object. The idea cannot exist as physical object, but it exists as description that might be physical letters in a book, electrons in our head / in a computer...
Cubinator73
My comment was directed to Mathoma. I'm not sure why the +Mathoma disappeared from it. I largely agree with you on this.
Brain, body and beauty all in one.
Sex appeal, as well!
A phisycist and an engineer were talking.
The engineer told the phisycist: You phisycists think everything is easy and quick, you just sit down with your equations and computers and wait for instruments to pop up.
And the phisycist replied: We phisycist think of engineers as a magic lamp; you rub it's surface, tell them what you want and telescopes and instruments pop out of it.
at 6:52 , did you guys just find that graphic somewhere or did one of you animate it?
As a physicist, I don't care whether math is "real." It works in the world we live in. That's good enough for me.
Ah, the good old physicist nihilism approach.
wow such interest in the topic
Classic physicist.
"it works in the world we live in."
I put on my set theory glasses and discovered that not one of the words in that statement is well founded...
I used to think physics was the coolest subject, but now I feel like math is closest to the deepest truths.
It is as simple as this: Math is ideas extracted from the real world. What is so difficult to comprehend?
the proof that math is real will be when we manage to contact aliens and see if they math the way we do.
Do plants count?
That should be astonishing! It would prove that Math is actually the language of the universe, and that instead of us having invented it, it would tell us we understood and written it. Altough I believe it's not the case here, it would be nice...
It could prove that we have discovered only a part of it or that we both have discovered the same part, if, as me, you think math exists besides humans.
Not prove, but strongly indicate! What if there is some alien race that doesn't math? What if 50% "maths", while the rest doesn't?
It will be thrilling to know how they think. You use math even if you don't know it!
I never understood this debate. To me the answer seems obvious: just like you explained, we invent the axioms and discover the emergent properties of the system defined by our axioms. Maybe the real meat of the issue is what is considered math: the axioms, the theorems, or both? Well, in that case are you arguing semantics or philosophy?
As for the "unreasonable effectiveness" of mathematics in describing the natural world, neither does this seem to be a good counterargument for formalism, nor does it seem at all mysterious. A mathematician typically works with axioms that were intentionally selected to resemble how we understand our world to work, so of course the emergent properties would likely also resemble our world. Yes, chess does not describe our world but that's because the rules of chess were not devised to resemble our world... Smells like an inverse error to me. Just because a game does not describe reality does this mean that no game can? I simply fail to see any real distinction between the rules of a game and mathematical axioms besides the level of rigor you might typically expect. For instance, I can describe axiomatic systems that do not model our world any better than the chess game you describe that nobody would hesitate to call "mathematics", and suddenly the "unreasonable effectiveness" argument falls flat on its face.
Anyway, I enjoyed this video and like things that make me think. I am looking forward to more.
Okay, but that doesn't really solve the problem. Sure, there exist axiomatic systems which do a poor job of describing the universe. We could pick one of those to explore, but we choose rather to pick the one that describes reality. That doesn't explain why there exists an axiomatic system which does describe reality. Why should the universe follow coherent rules at all? And if the universe does follow coherent rules, doesn't that mean that those rules exist independently of a system that we invented?
As for why the universe should follow coherent rules, I'd like to apply the anthropic principle to state that any universe that didn't follow coherent rules almost certainly wouldn't support life intelligent enough to ask the questions we're discussing.
Of course, this just changes the discussion to why our universe has all the nice properties that it does so that we are able to be here to observe it.
If multiverse theory applies, wouldn't it be the fact that there has to exist universes in the infinity that do meet those properties and intelligent life could only ever exist in those universes that do have those properties? Thus, our ability to have intelligence by necessity dictates we are in one of said universes with said properties?
I don't think it's even necessary to invoke anthropic principle... we can see our axioms on mathematics change to suit the needs of reality.
Newtonian mathematics breaks down on very close orbits like Mercury.
But Relativity using new tensor geometry works well for all orbits including Mercury. Axioms of mathematics are infinite, and only those that are useful get any attention. The rigidity of a mathematical systems is what gives the sense of independence. But the fact we have to invent new forms of geometry and probability of which axioms come from the outside world and old mathematics that fail due to their own rigidity shows even the rigidity is an illusion.
jmitterii2 You have ten fingers, I have ten fingers, bob has ten fingers. Abstract away who has ten fingers and you get the abstraction "ten fingers"
Like wise, I have ten apples, you have ten apples, bob has ten apples. Abstract away who has ten apples and you get the abstraction "ten apples"
So we have the abstractions "ten apples", "ten fingers", "ten toes" etc. What if we abstract away what we have ten of? We get an abstraction of an abstraction. The abstraction ten.
If we have ten, 1, 5, 9, etc. and we abstract away the magnitude from each of these things, we get the abstraction numbers.
Every culture developed its mathematics through this abstracting from simple counting of physical things like fingers or apples to these abstractions of numbers. And then considering that "fingers" and "apples" are already abstractions themselves, numbers are 4th order abstractions. They are so abstract as to be universal. And all of mathematics is just combunations upon the abstraction of numbers, so no wonder they work so well, because they are so bloody abstract!
So much for the unreasonable effectiveness argument. One might as well ask about the unreasonable effectiveness of the laws of logic, or of abstraction itself, to pick the few things that are even more abstract than math.
But isn't the rules of chess also an abstract system? So why does it not work to describe reality?
The reason is because it is not abstract enough. Chess is a lower order abstraction than mathematics, because mathematics is needed to describe chess. You need the abstraction of numbers to describe how many pieces there are on the board, but you don't need the abstraction of a stalemate to describe any concept of mathematics.
This is also why many mathematical fields have no bearing, no connection to anything in the real world, because they are lower order abstractions than those of numbers.
YAAY coem from the PBS Space Time, now another PBS show to subscribe. very happy
In science, the simplest solution that explains the most amount of things always prevails. Certain types of math are purely platonic, but other types explain reality so simply and generally it would be crazy to say that that math only existed in our heads.
For some reason youre in the vast minority.
When i look at the ratios, most idiots think humans made up math. Even so called intellectuals believe such an easily disproveable thing
Like BRUH THE UNIVERSE EXISTS! Humans didnt create the universe and math is fundamental to the forming of the fundamental particles that make up reality and then some
What a wonderful video, I'm so glad to see a mathematician taking the Philosophy of Mathematics so thoughtfully. I tend to think of mathematics as "schematically true". That is, it is true for anything adequately described by the premises of the structure. If an object's motion is close enough to f(t) = 2t+1 then you can predict closely enough that after the stop-watch strikes 10 seconds the object will be located at 21 meters away. 2+3 works well for describing trucks and apples up to a point, unless you dump 2 and then 3 apples into a blender. Does 2 exist? Only as a schematic place holder for two objects that will persist over time long enough for the mathematical structure to tell us something useful.
This channel is insane! Subscribed!
All languages have effectiveness in everything, like the one I'm using right now to tell you this, which I couldn't do nearly as effectively with mathematics.
arent mathematical axioms based on our real world intuitions though?
A lot of people conflate "concrete" (as in concrete vs abstract) with "real". They view concrete things like "matter" and "energy" as "real" while abstract things like "love" and (potentially) "math" as "unreal" or "made up". But there's no sufficient evidence that indicates that being "real" is contingent on being concrete. There's also no sufficient evidence that what we understand to be concrete is necessarily "real". Descartes's famous statement, "je pense, donc je suis" is abstract in nature but it's also tautological; it cannot be untrue. Or, in other words, it absolutely *must* be "real". But this concept is outside the scope of science because it does not yield to empirical analysis. In other words, there are lots of things that Science is equipped to handle, but just because science can't handle it doesn't automatically make it unreal. Hence, even if there is no empirical evidence for the existence of math, no concrete manifestation thereof, it can still be considered "real".
Great content. Who produced the music that is heard in the background during the explanation ?
So many Formalists here!
I would say I'm a Platonist, but I mean it kinda goes with the territory when commits to studying Number Theory. :D
+Darkenedbyshadows
Yeah, I definitely have more sympathy with platonism. It's very odd because pure math students that I talk to in real life are always platonists, never formalists, but nearly everyone online seems to be a formalist.
I think that speaks about the environment in which we're using the mathematics more than anything else.
I'm a math and sciences tutor and on any given workday I can be using math for physics, chemistry, statistics and levels of math from arithmetic up to the calculus series and sometimes beyond. And if things are exceptionally slow, sometimes computer science.
It's more or less impossible to work that range of material and not be forced into formalism. And that's not even considering the instructors at the college I work at who insist upon bringing their pure mathematics into the classroom and the myriad approaches I see to the same basic problems.
Formalism has the rather significant advantage that you can focus more of your attention on the mathematical validity of what you're doing while you do the problem and then after you have a result, you can consider whether or not the result is one that applies to the application. You can also consider at that time whether or not the mathematical model even applies as a lot of that information comes from the form of the problem.
first video i see from these channel, i love it :) congratulations
Since there's a lack of platonists on this comment section, let me come out of the closet. One argument for platonism that should be taken seriously is the _indispensibility argument_ which says, as a principle, we are ontologically committed to the objects about which our best scientific theories (i.e. physics) make statements about and since our best theories make statements about mathematical objects, we're committed to abstract mathematical objects, as they are _indispensible_. Perhaps if someone someday comes along with physical theories that work just as well and don't make statements about mathematical objects, we can ditch this argument.
+Keating Allen
I don't understand how this _gridline objection_ touches any of the premises of the argument.
Penelope Maddy addresses P1 - we ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories - of Quine's indispensability argument in Indispensability and Practice (1992), Naturalism and Ontology (1995), and Naturalism in Mathematics (1997). Her strategy is to undermine Quine's naturalist and holist justification for P1 of the argument, by showing that the former is incompatible with the latter.
Maddy forwards three objections:
1. Naturalism says we ought to respect the attitudes of working scientists. Holism says there shouldn't be significant variations in those attitudes. Yet, working scientists often disagree heavily on existing theories.
2. Maddy suggests mathematics exists not within the true elements of scientific theories, but rather those idealised elements used to model reality. Water is assumed to be infinitely deep in analysing water waves; matter is continuous in fluid dynamics.
3. According to holism, logicians mathematicians should be assessing and revising the foundations of mathematics (the ZFC axioms) in accordance with the needs of physicists and other scientists. Given that such arguments for/against ZFC are almost always intra-mathematical, and we should respect that, this demonstrates that naturalism is favourable compared to holism.
Mathoma Isn't it the other way around? Don't our best scientific theories make statements about the universe oftentimes *using* mathematical objects? If science was making statements about mathematical objects, it would be math.
+Benjamin Przybocki I think it's pretty obvious the theories are making statements most directly about mathematical objects. We then say, secondarily, that the statements have something to do with physical objects. Whether they are directly or indirectly about mathematical objects, we both recognize that mathematical objects are needed in this picture _somewhere_.
+Communisation
This is a complex comment so I'm not sure I can address it all at once. Let me just give my reactions to Maddy's objections to P1: the ontological commitment premise. Actually, a note on P1: why do we have to say _all and only_? I was initially under the impression that we should, at a minimum, be committed to such objects, not exclusively the objects needed for the best scientific theories. Perhaps this goes against Quine's aesthetic need to have a sparse ontology, but I don't really care about being profligate for the time being.
1. It seems obviously correct that scientists have major disagreements on existing theories but one thing I've never heard a disagreement on is whether the theories should be mathematically phrased.
2. I don't see what's left of a scientific theory once you've removed the mathematics, so how could it not be a true element of a scientific theory? Furthermore, the scientific theories often use mathematical terms right in their statements. For example, Faraday's law makes a statement about a _derivative_ of a magnetic _flux_, both terms already being mathematical concepts.
3. I don't understand this objection well enough to see how it's actually an objection to P1.
Nonetheless, I'll have to check out some of Maddy's work.
I think the mention of the "unreasonable effectiveness of mathematics in the natural sciences" is the most important point of this video. Not the "real vs unreal" debate, which I really don't consider "serious" in any sense - rather just a play with words. On the other hand, the effectiveness of mathematics is something that is stumping me for a long time and I think getting into bottom of this might be one of the most important tasks human kind can (if) ever deal with. Is the physical world a natural consequence of math? If so, why? And can we exploit this "causality" to create another physical realm similar to ours? This is the big question.
Do not forget that our models are seldom, if ever, entirely complete or correct!
We typically start with a crude approximation, enough to give us some rudimentary insight and ability to predict physical outcomes!
And then we start a tedious model refining process, in order to make the answer fit closer and closer to the outcome of the physical process!
Sometimes the necessary math, happily turns out to be wonderfully simple and astonishingly elegant, but for the most part, it is a female dog! And many times, we are forced to adopt entirely new mathematical structures, in order to achieve some success in our efforts!
You don't just throw some numbers together, and out comes the perfect model, miraculously, as it is presented here!
Near the end... "knowing the rules of chess doesn't help scientists launch rockets into space" .......K, but a game about launching rockets into space would. Easy semantics, gg close.
Agreed
Kerbal
Not sure which camp I fall into, I am 37 and have just begun to find math interesting for the first time in my life. Great series!
i am already in love with this series.
I want to play that mathematical chess game.
i always thought of math as a logical language for describing and considering stuff in the world around us. it works so well because we've been building up the rules that math uses based on useful observations. numbers started as counting things that exist. then measuring things like distance and size.
Except... we explicitly and definitely do NOT do what you said. What you described is science. Mathematics is different. Mathematics absolutely is not built on rules based on useful observations. In fact, no observation of reality could ever disturb a single mathematical fact. Mathematics is a product of pure reason, totally divorced from reality except in its respect of basic logic. (And reality seems to perhaps not respect that logic so much, such as in the Copenhagen interpretation of quantum mechanics.) For instance, there is substantial physical evidence to suggest that spacetime is not continuous in the way the real number line is continuous, there being an infinite number of distinct positions between any two selected points regardless of their distance from one another.
This means that the circle mathematics deals with can not exist in our universe. Every circle must have an exact answer for the ratio of its circumference to its diameter expressed in that 'smallest unit'. But, this does not cause any problems whatever for the definition of Pi, the real number line, or the geometry of circles. Those are intact because they were never built upon physical observation or definition to begin with. They were derived purely from a small set of basic axioms, and logical extensions thereof. Nothing more.
Dustin Rodriguez I am not convinced that math is a product of pure reason. Much of our math, including numbers, is rooted in practical problems, such as counting stock, making trades, measuring fields. Sure, abstract logic and constructions were used, but they were inspired by the real world. Sure, an abstract world of reason developed from this, and often took a life of its own. But whenever it bumped into a problem with the real world, new math was developed, or picked up off the shelf. That is the root of the "unreasonable" correspondence of math with science. Science, business, engineering, all inspires a great deal of math, whether mathematicians like it of not. Archimedes, Newton, Noether, all mathematicians that developed their math based on scientific problems.
Thomas R. Jackson Things like the natural numbers (not including 0) historically originated from counting stock and such, but that has nothing to do with mathematics. Mathematics is, by definition, exclusively the abstract arguments built upon reasoning from axioms through use of proofs.
And while, yes, mathematics was expanded in ways to address physical problems, they never broke from the pure basis of it. The fact they never needed to is why the correspondance is 'unreasonable'. And in many cases, the math came first. Complex numbers and complex geometry are the usual example used. They were developed with no practical application, simply an outcome of pure reason. Then Einstein came along, and they made explaining relativity simple. Why? There's no good answer for that.
Dustin Rodriguez Yours is an interesting perspective. Tradesmen learn to count and add, and group, but that is not math because... I am at a loss. They may not have published a paper, but they did reason it out.
It seems to me that the "pure" basis of math developed over time. But its very "purity", the idea that you can reason these abstract concepts, even invent new ones, perhaps inspired by the physical world, but maybe not, makes my point. These are abstractions, dependent upon our own thoughts to make them out. If they are "real" , then they are real in the same way that we make so many of our abstract thoughts real. Money, nations, laws,... these are all "real" in a sense too. They have measurable effects on our behaviour. That is empiric reality, something the proper subject of science. But if humans disapeared, so would these things because they have no independent reality. So to math, in a sense.
The sense is that so many of our concepts in math were founded on our observation of the natural world, as seen through our perceptions. So that will live on beyond us. Will mathematical "truths" remain true when we are gone? I think the question is nonsensical because we invented the whole concept of truth as well. If we destroy a computer and its software, it does not exist. We might rebuild it, and perhaps the new code we construct will develop the same concepts and behaviours, and exist again, perhaps not. But the bottom line is that information has no existence apart from its physical matrix. Remove that, and information is gone.
I think the idea of our abstract ideas having an existence apart from us or our creations, or records, is mysticism, pure and simple. It is a story our brains give us, a perceptual illusion if you will, to help us respond to our world. It is like the idea of truth, or self, or soul, or or any of the other things we conceive of as distinct from actual matter. Philosophically you may disagree, and I can not "prove" you wrong. But neither can you prove your spiritual conception right.
It's a matter of historical fact that the first mathematics came from trying to understand and manipulate the natural world. Then we humans, being as intelligent as we are, were able to abstract, refine and expand the knowledge we had so gained. But although parts of maths had followed axiomatic systems as far back as Euclid, it was only in the late 19th century that there was a systematic attempt to derive the whole of maths "from a small set of basic axioms, and logical extensions thereof"
So, in seeking to understand Wigner's "unreasonable effectiveness" (the objection to antirealism) the historical roots must be kept in mind. If someone had come up with ZFC set theory purely recreationally,, then its effectiveness would be astounding, but instead ZFC was an attempt to systemise and axiomatise abstract and logical procedures we (humans) had developed primarily (tho not exclusively) for understanding the natural world.
Don't get me wrong, I understand we have abstracted that knowledge and then run with it; and the 'running with it' doesn't necessarily have to have anything to do with the natural world any more; and abstractions, pretty much by definition, do not exist in the real cconcrete world.
Math isn't like science, where you gain enough evidence and declare something true, you have to have a precise logical proof. Greatest sentence ever uttered by a human being ever.
I'd love to see a response video on this from Hank Green over at CrashCourse: Philosophy.
what is the intro music? is that custom or part of a song?
Team descriptivism, standing by.
Ready to describe reality and take names.
Mathematics exists beyond everything else. Humanity discovered and developed mathematical rules and concepts, but other intelligent extraterrestrial lives could understand the same concepts, beginning with the counting of stars.
Math is essentially logic
2+3 is always gonna be 5? Tell that to George Orwell
2+3 could be 10 or 11, and that's without even changing the definitions of what the numbers are.
The base, is part of the definition!
2 represents the same thing in base 4 as base 10. I would say only 2 digit numbers have their base as part of their definition.
stardude692001
And that is precisely why, the sum can be 10 (base 5), 11 (base 4), or 12 (base 3).
Of course, in the case of base 3, one would have to have the 3, of the addends, expressed differently, to avoid confusion, since the numeral 3 could not be written as is, in that base, or it would be meaningless!
Ζήνων Ελεάτης opps, you got me there. I'm a bit rusty on the old maths.
This has been one of the most satisfying playlist of math videos, second only to Grant.
Thank you.
Thank you very much.
When 1+1=2 I'm a formalist. When 1+1=10 I'm just a realist(ic) old code slinger.
1+1=11
When 1+1=3?
It seems to me that formalism is the best explanation for math, because math is just a logically self-consistent system of precisely describing relationships. Any such system would be equally useful, whether you used numbers or colors or sounds, as long as the definitions are precise.
> Any such system would be equally useful, whether you used numbers or colors or sounds, as long as the definitions are precise
That statement actually makes you a platonist, because you agree that the underlying mathematical concepts exist beyond our brains or our selected numbers, symbols and rules. They are "there", waiting to be discovered, no matter what names we give them.
I'm very sorry to say but there is a mistake at 1:32.
2 is a prime number and 9 is NOT! I'm disappointed... The first few prime numbers are 2, 3, 5, 7, 11, 13.
Here’s my guess: Our mind is a pattern recognition machine, it’s how neurons work. We sometimes make simplified analogies of commonly observed patterns and articulate them. Culture accumulates the most useful of these observations and weeds out inconsistency. Five rocks are a common pattern, but coarse sand and very small rock fragments confuse things in nature on the beach, so we grab five roughly the same pebbles to demonstrate “five” to others. We filter, or simplify ideas, out the patterns that interest us. The idea of five can be used to build other simplified analogies of commonly observed patterns, such as five groups of five, so on. The ideas and symbols of commonly observed patterns represent something both in and outside of our minds. We can say very precise things about “five”, that we could never say about a small pile of rocks, sand and dirt.
3:03 I have it on good authority that 2 + 2 = 5, for extremely large values of 2. ;-)
Yes, if your value for 2 is 2.499, then 2+2 = 4.998 or 5.
out of all of the videos on youtube, this one made me think about what math is, the most.
This is an important video.
I fell in love with PBS Space Time, and now I have Infinite Series? Awesome stuff!
With regards to realism vs. anti-realism, is it fair to say that considering numbers to be a fundamental property of our universe (like the speed of light, or the planck constant) is a realist viewpoint? To me, saying that numbers aren't real is to say that nothing is discrete. Particles exist, and they are discrete. So numbers just arise from that fundamental truth. Once you have 1+1=2 the rest of mathematics just reveals itself through logic.
Although there are many non-platonists in this comment section, can anyone offer an actual philosophical argument for why they hold their position? It seems everyone is giving their opinion, e.g. _math is invented_, _math is just a tool_, ... without giving an argument for their philosophical position.
Lovely! Is this your personal discovery, or is this one of those arcane facts known only to a small circle of intimates, or is this a commonly known fact that I am naively ignorant of? Thanks for pointing it out, you have made my day.
@@wsmith49 It is just the solution of the equation x^2 = x + 13
@@wsmith49 which is, by definition, defining a number 'x' with the said property.
0:30: 1 is divisible by one and itself and we don't assume that 1 is a prime number. Obviously is the matter of definition, however in order to formulate in the elegant way the theorem about the factorization of any natural number into primes, we have to exclude 1 as the prime number (if we want to have uniqueness statement).
love your hair as much as the concepts you explain 😍
I just discovered this channel and love it so much!
Keep it up!
I'm so distracted by whatever is on her right shoulder.
I'm distracted by her atrocious vocal fry.
Scy I'm distracted by whatever is her
Nah, it's just a part of her bra or skirt.
It is not behaving like a piece of clothing. It might be a tattoo or a wire to a worn microphone.
It's a microphone wire I believe
I'm a formalist. The effectiveness of math is not unreasonable as long as perfect knowledge can exist.
Mathematics is the study of patterns. The symbols and notation of mathematics is simply our language for describing different patterns. Pondering whether numbers are real is like pondering whether words are real. The word "banana" is used to communicate about something real. The word "unicorn" is used to communicate about something not real. Mathematics can be used to describe patterns we observe in the universe and patterns we don't observe in the universe.
When we study mathematics we are essentially exploring our ability to describe all different kinds of patterns. Prime numbers are interesting because they are constructs of our language of patterns, yet their distribution is so difficult to characterize with that language.
we are not talking about the representation of numbers we are talking about their meaning. you may represent 1,2,3 by a,s,d or any other symbol but their relationship won't change.number are highly consistent with natural phenomena even if we don't intend them to be which is weird and suggests the independent existence of numbers
Aashray Narang
I'm not talking about the representation of numbers either. I'm talking about their meaning.
We intend math to describe patterns. The reason it's so effective at describing the universe is that there are patterns in the universe.
Abe Dillon there are patterns in universe only when you see them through maths. there are no patterns if there is no maths. we didn't create maths to see patterns because we notice patterns when we have maths (don't consider abstract patterns like constellations) and this application of maths in physical world is weird.
"there are only patterns in the universe only when you see them through maths"
That's absurd. Plants adapted to seasonal patterns long before humans invented math. Pulsars still pulse periodically even without math. Atoms would still bind to form molecules in specific geometric structures even if pattern language didn't exist.
"We didn't create maths to see patterns because we dnotice patterns when we have math."
My claim is that we created math to *study* and *describe* patterns. It's ridiculous to say that we need math to "notice" patterns. The brain is an exceptionally good at pattern recognition. It has been long before we came up with math. Simply knowing the complex behavioral pattern required to reproduce is evidence enough for patterns existing before we developed a system for studying and describing them.
The universe is clearly not 100% chaotic. You don't see TV static every time you open your eyes. You see a universe with lots of predictable, repeating phenomenon.
Abe Dillon i want to clarify that i am not talking about patterns like plant seasonal pattern or abstract patterns and i am also not saying that objects in universe know maths and then they show patterns , i am saying that we can predict the behavior of objects in universe by using maths take for example hydrogen, energy of electron in hydrogen atom is given as E= (2π^2×m×e^4)/(n^2×h^2) put n =1 and you get the ionisation energy of electron that is 13.6eV per atom provide electron this much energy and ya it leaves hydrogen atom, provide energy less than this the electron do not leaves hydrogen atom this predictability in the behaviour of electron is weird (don't say its chemistry and not maths) and there are thousands of other examples which predict the behaviour of physical objects using maths and i am talking about such patterns
there is a real sense in which something like a number is "made up", because if humans did not exist, the universe would be indifferent to its inner workings, the ratio of the circumference of a circle to its diameter, etc. But the fact that we can communicate at all using abstract concepts like circle means that there is something that we can call objective reality, but not a Platonic reality in which perfect circles exist
That's kinda a false dichotomy isn't it? We defined math, for certain. But we defined aspects of math based on natural observations. We did just THINK up numbers, we thought up numbers to label counts of actual objects. Newton didn't invent calculus and then realize it magically mimicked the motion of the planets, he specifically defined calculus to represent the motion of planets. The fact that calculus can be used to describe other natural phenomenon is more unsurprising than it is surprising, many things in nature have a relationship between states similar to the relationships that cause planetary motion. Math exists in natural form only in the sense that things can interact in physics and have causal relationships that can be described with math, that doesn't mean there is some universal calculator that exists independent of the human (or alien) mind. (unless we're in a simulation). It's quite the opposite, the particle interaction of nature that allows for causal relationships is what allows computation of the human brain and computers.
It does kinda turn into semantics, semantics of definitions. Is a tree a house? Depends on how we define house, does it keep off the rain? Does it keep out the wind? A house without a windows or doors (but holes for them) doesn't keep out the wind, is it still a house? Is a cave a house?
If we define the relationship of natural phenomenon to be mathematical, it is, because we defined it. But the similarities between abstract numbers and their relationships of mathematical functions are distinct from the natural relationships. The similarities are not surprising, because we've defined the rules of math to mimic natural phenomenon and as such, the fact that it mimics natural phenomenon in unexpected ways, isn't surprising at all. It's like if you make a life simulator and it creates lifelike behaviors that seem to defy the original definition, it might be surprising emotionally, but it's not unexpected, because it's what you defined it to do.
This subject is the biggest battel I ever fought (and still fight) within my little brain. And this video is by far the best summary of it and introduction to it. Thanks a lot! This makes it easier to communicate with other people on fascinates me and keeps me busy.
Why not a perfect superposition of both camps?
I'm definitely a formalist, I've drifted more and more that direction over the years because such a large portion of the problems I see are ones that I wasn't taught and I don't usually have the time necessary to do any research. Ignoring the reality of it is pretty much the only way to get any work done, not to mention that it's nigh impossible to comprehend what you're doing until you have something on the paper to understand. Trying to understand prematurely, just leads to the possibility that you hedge out a productive direction of attack whereas ignoring that and focusing on the form puts you in the position where the worst case scenarios are either not getting a result at all or checking the result against the application and finding that it doesn't check. As opposed to not getting anything at all.
For me personally, I advocate knowing what you're working with and what you're looking to accomplish, then focusing on what the form of the situation is talking about and working from there. Once you've got a possible solution, that's the time to look at the work and try to understand whether it applies and if so how it applies and worry about whether or not the solution makes any sense. Trying to do that prematurely has never worked out well for me.
Well... 2+3 is not always 5.. you are a mathematician so you know that.
she quickly made the point... maybe too quickly for some of us
Please show me an instance where 2+3 is not equal to 5.
bloodyadaku in Z/4 for example 2+3=1. In Z/5, 2+3=0. In fact, in every Z/n the sum 2+3 where "n" is not equal to 1 and is less or equal than 5, the result would be a different value than 5. Magic right? haha
what is Z/4? What are you defining as Z? ℤ In mathematics represents all integers. Not sure if you're referring to that.
+bloodyadaku Yes, you are right, Z is the integers. Z/n would be the algebraic space where the number is the remainder of dividing the result, calculated like we normally do, by n. This are other spaces or classes within algebra. If you ask me, it changes the standard definition of what the operators "+", "-", etc, do. And this is a far as I understand it. I am sure that a mathematician could give a much better and clear definition than what I did.
I am an engineer, but my wife is a mathematician. Z/4 I think is the notation that is used in english, in Argentina (spanish language) she always used the notation Z_4, or more generic Z_n. But that is just a notation, although I would have to say that is much more clear the english than the spanish one. When many afternoons in the bar are filled with researchers in math this sort of things come up lol. In the end it is just another definition that is useful for them in some way that goes beyond my ability to understand what they say.
It is like when they ask "what is the sum of all possitive integers", I answer "infinity" and they say "that is one possible answer, but it depends of what you are talking about". Standard maths will yield the result infinity, but if you change only one definition (how you calculate the limit of an infinte series), you get the increidible answer "-1/12". This last result is greatly explained by mathologer (just type "mathologer ramanujan" in youtube and you will find it).
To be honest, I was really unconfortable with all of this at first (and was really grumpy about it). For my engineer head it was extremelly annoying to say the least hahaha. But I guess math is all about logical consistency. It is not that they wanted to make "2+3=0" or "2+3=1" or "1+2+3+... = -1/12", it is just a result that is the logical conclussion when you just change a definition.
I guess math is much more amazing and even much more complicated (and maybe much more annoying) than we all non mathematicians realize at first glance.
"If a tree falls in a forest and no one is around to hear it, does it make a sound?"
Saying that mathematics exists independent of human minds is like saying the rules of chess exist independent of human minds.
Math is the language in which it's written reality.
Maci Sordi Math is a language, sure, but this language is a tool humans invented to solve problems in reality, which is why it has the ability to model reality.
I don't think so, we are part of this reality therefore ( a fortiori) we think in a mathematical way. But this are just conjectures...
Maci Sordi I don't think I am making conjectures. For example, there are tribes whose adults cannot understand numbers, because their language never invented them. We can try all sorts of visual comparison methods, but they cannot tell the difference in quantity between a group of 5 sticks, or a group of 6 sticks. They nevwr needed to invent a number system, brcause numbers didn't serve any purpose for them, and didn't solve any problems for them. The way we think is based on our evolutionary psychology, and societal development. We know humans create languages, and we know humans create tools to solve problems. Our species didn't have language or civilization until primitive agriculture began to make civilization possible. Mathematics is a creative part of modern languages, and was invented to solve specific problems by simplifying reality into a useful fiction, allowing for trade, taxation, and engineering to be done more consistently. There's no magical universal form that humans are psychicly reading from.
This doesn't mean that their ears cannot do Fourier Transform to listen... Our brain is made by things that follow the rules of our universe, mathematical rules. We discover(ed) our "special mathematic", but it is only a limited part of the total mathematic that exists apart from us. There is no magic in what i say.
Very captivating and well-put-together video series. Can't wait for more... keep it up!!! :)
There is a total false dichotomy here. Math is either a made up game or about imaginary objects? Those are the only two possibilities worth mentioning?
Why did you not consider the idea that math is about _real_ objects _viewed from a certian perspective?_
Take the example of a "perfect" circle. If you look around you, you can find countless examples of circularity in many forms. If you take those examples and _abstract away_ their differences, you are left with the concept of a circle (with each example being a variant on the same theme).
There clearly isn't a "perfect" circle somewhere out there waiting for us to discover it. It's also clear that the concept of a circle wasn't just made up out of nowhere (how would you even come up with the idea without first looking at reality?).
Very well stated!
Very well stated!
does the goldbach conjecture make an exception for 2?
Yes, of course. Otherwise it would be obviously false.
Erik S At 0:25 she puts "(greater than 2)" on the screen to specify this.
1+1. i know 1 is not prime (nor a composite...), but that's a rule we made up. Technically 1 is devidable only by 1 and itself, just like all other primes.
but this is only speculation...
The intro sound track and the host herself are so classy 😀
At 3:30 saw her shoes which don't at all go with her dress. You broke my opinion 🙈
Check that dress strap. It bugged me all episode.
Formalist. A platonic interpretation cultivated my love of mathematics, but was also a HUGE OBSTACLE to clearly thinking about the deep problems in mathematics, like foundations.
The answer is that we didn't simply made up rules that mathematics it built upon. We discovered them while observing natural world. Yes, math is like rules of a game, but those are rules embedded in the universe we live in. Mathematical rules govern all natural phenomena, including very first moments of our life. We see perfect mathematical division in the development of human zygote, which after four divisions consists of exactly 16 blastomeres. Golden ratio determines proportions between individual bones in our hands. We see logarithmic spirals in shells of snails and arms of the galaxies. There's really nothing surprising in the fact that mathematical rules we uncover from the world around us help us accomplish scientific miracles.
Why have I only just found this channel? Subscribed
I love her hair!
Just discovered this channel and absolutely love it!
one of the hottest mathematicians I've ever seen.
smart and sexy
Yesterday I read your article about this issue on Scientific American Japanese Edition. It was a very interesting and clear explanation. Thanks!
Talk about fruitless over analysis.
You could make much of these same arguments about any language or even abstractions in general. Mathematics is simply the rules that come from abstracting certain observable aspects of reality.
Do I see a hint of a hot tattoo under that dress?
Cool poem, I'm sure Kelsey's looks as well as her intellect are what landed her this hosting gig. Both are on display and I'll comment on both. If she doesn't like it she should do voice overs.
Dylan T Watch her left forearm, Sherlock! 🔍🤔😱😳😅
"If she doesn't like it she should do voice overs."
"My name's Dylan and instead of respecting women I think they should just not show their face to prevent me from disrespecting them."
We dont know what might offend someone. We can't live in a world where no one comments on others for fear they might possibly be offended.
Yea you're totally right but your commenting on her body is completely irrelevant and off topic and women have been undermined in the STEM fields since they originated and you making it about her body rather than the subject she's talking about is pretty disrespectful and gross.
Not to mention the fact that what you just did happens to women of high esteem constantly. Female actors, singers, business owners, and just in general women are asked about their physical appearance rather than their accomplishments and your comment is the exact same.
Plus your idea about how we should be able to comment on each other is more for friends telling each other if they're being dicks or if they should stop doing something. Not for a stranger online to discuss whether or not an intelligent woman has a tattoo under her dress or not.
This is how I felt when I studied Grafting Numbers. That I was discovering something that was already there and the connections that I found to Catalan Numbers were completely unexpected!
This is absolutely fascinating! Have you listened to John Lennox at all?
Formalist here. To address your point, the reason math tends to describe things in reality is because we tailor the rules to model objects we see in reality. Math's usefulness in physics relies heavily on the accuracy of this model. Also, there is plenty of math that doesn't describe reality; it just doesn't get as much press.
How have I never seen this channel before. Came here from PBS spacetime
Actually it's probably because there's only seven videos and not a countable infinite number such as the name suggests
Is Pi infinite because a circle has an infinite number of small straight sides? Infinities exist in math but I'm not too sure about in our observable reality because of quantum limits like the Planck length etc...? So if Pi really is infinite and real that means our universe if infinite in size!? 😲
Or maybe we look at circles wrong and true curvature is possible but can only be described with the correct description, which we don't have yet?
Goldback conjecture for 2n: for any positive integer n, there is at least on k, k >= 1, k
Does the conjecture work under base 11 or 12?
It has nothing to do with base whatsoever.
numbers are properties of matter and maths are the interactions. numbers are like happy, you can't touch happy
Math is like writing a paragraph and finding a book that starts with it.
1:37 "Mathematics can feel like you wrote the first page of a book and then, you're figuring out the rest."
Kinda reminds me of programming: You write a piece of source code and you don't immediately anticipate what happens next. But what happens next is a logical consequence of what you wrote into the source code, meaning that the same source code will always result in the same result, even though it only happens after you actually run the program!
Goldback conjecture for 2n: list [odds in] 1 to n in top row and [odds in] n - 1 down to n in bottom row, at least one prime is above another. For 32:
01, [03], 05, 07, 09, 11, [13], 15
31, [29], 27, 25, 23, 21, [19], 17
Not a mathematician here as such but love the sciences without any formal education of the subjects! Eg. Frequency - sound and light, heat and cold are all related and our understanding of them relies on our abilities to calculate. Heat and cold, sound and light are all observable. Understanding what they are relies on our use of maths! One of my favorite math concepts is Infinity - a line an inch long technically only has 1 dimension - length and that length of 1 inch is infinitely divisible. A line a million miles long also has only 1 dimension - length, and that length of million miles is also infinitely divisible, so is it just a perception that makes the difference between the two or is our perception limiting us to consider distance as a factor which may turn out to be an illusion? Then you have the axis of a rotating body - this axis also only has one dimension - length. The further out from the center of the axis, the more observable the rotation is but as you get closer to the center of the axis - is there a point where the rotation ceases to exist or maybe becomes or exists purely as a void like a nano-sized black hole!? Whichever is the case, the axis may be infinite or infinitesimal in length and have only that one dimension yet hold almost unimaginable power as it seems that is lays at the heart of almost everything we can observe! As I said, I am not a mathematician or a scientist, I am more of a philosopher and so I love these questions which often keep me awake at night. Thanks for your channel, I have subscribed and look forward to future episodes. Cheers
Jeff
I feel formalist. Yes, we made it up, we made the axioms that we liked. They might happen to be "Unreasonably Effective" because we made them that way. We can see that others accept the theorems, so other thinking people may make up the same things. At 2:03, may I suggest that you use scientific notation? Who do you think you are talking to here? Do you want us to count up zeros, or do you want to tell us the magnitude of the number?