Philosophy of Math | Harry Binswanger
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- Опубліковано 5 лют 2025
- A continuation of Dr. Harry Binswanger’s “ • Saving Math from Plato ” (OCON 2023): how the Aristotelian, perception-based approach to mathematics refutes or re-interprets wrong ideas advanced by figures such as Russell and Cantor. Topics will include number, infinity, limits, and the axiomatic concepts of mathematics. (The lecture assumes no knowledge of mathematics beyond beginning algebra.)
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This is a fascinating, well articulated lecture and its thesis is correct. The detractors criticizing it may be more familiar with higher mathematics than the lecturer, but are poor philosophers. They simply don't get Binswanger's argument, that infinite objects rest on a ridiculous form of realism about mathematical concepts. They probably think that somehow in higher maths everything makes sense and one can wiggle away from apparent absurdities (which makes me wonder whether they understand the higher math to which they allude.) No. The realist premises are actually stronger in formal constructions of the reals than in Russell's example of the natural numbers covered in the lecture. Kudos to the lecturer for identifying a profound mistake by carefully analyzing basic math. I am looking forward to his book, which I believe will be very valuable.
No, you are wrong. I am the best philosopher my video debunking this one proves that. Cute attack though. You said nothing
This was an awesome talk. Having studied engineering physics and real number mathematics i wish this had been around in my university days.
He's criticisms are absolutely naive it is motivated by his false ideology of that woman doctrine of "objectivism". He not only misunderstood Bertrand Russell theory of descriptions or overall his significance in mathematical logic he also misconstrued Cantor which he never had grasp of since he never took higher mathematics than pre algebra.
@@SeanAnthony-j7f That's your opinion. Having personally studied Real Number analysis (Cauchy sequences, Cantor sets) and Complex number analysis (Integration in the complex plane) in university as part of an Engineering Physics degree I found many illogical ideas in mathematics that professors glossed over and NEVER tried to tie back to reality. I think that a proper grounding of mathematics in perceptual reality can do nothing but good for the field and for engineering and physics. In engineering, for example we NEVER need infinite precision, we always manufacture to tolerances and his presentation acknowledges that fact right in the beginning where he calculates the length of the hypotenuse.
@@zardozcys2912 how does challenging the most basic assumptions in mathematics affects how engineers or natural scientists calculate or perform their tasks when the study of mathematical analysis (which you are acquainted of) can be generalized in advanced calculus like integration and differentials in order to solve and even predict behaviors on how materials should be constructed or planning how to build 4 floor building. Mostly cantor sets and all other set theories after him that solve all of the paradoxes naturally emerged like the Zermelo-Fraenkel set theory, homotopy theory, type theory etc. are important in proofs across mostly other fields of mathematics even the most practical ones but not necessarily useful in measuring since it doesn't deal with quantity but rather sets and counting transfinite cardinals or perhaps in philosophy of mathematics in different schools of thought unlike the realist which introduced subjectivism (our mind constructing across space and time with the use of intuition) like intuitionism or neo-Kantian in nature which is not typically necessary in practical affairs except if you wanted to pursue the most abstract truth about the nature of mathematics itself or its relationship to reality from the very bottom of its iceberg.
@@zardozcys2912how does challenging the most basic assumptions in mathematics affects how engineers or natural scientists calculate or perform their tasks when the study of mathematical analysis (which you are acquainted of) can be generalized in advanced calculus like integration and differentials in order to solve and even predict behaviors on how materials should be constructed or planning how to build 4 floor building. Mostly cantor sets and all other set theories after him that solve all of the paradoxes naturally emerged like the Zermelo-Fraenkel set theory, homotopy theory, type theory etc. are important in proofs across mostly other fields of mathematics even the most practical ones but not necessarily useful in measuring since it doesn't deal with quantity but rather sets and counting transfinite cardinals or perhaps in philosophy of mathematics in different schools of thought unlike the realist which introduced subjectivism (our mind constructing across space and time with the use of intuition) like intuitionism or neo-Kantian in nature which is not typically necessary in practical affairs except if you wanted to pursue the most abstract truth about the nature of mathematics itself or its relationship to reality from the very bottom of its iceberg.
how does challenging the most basic assumptions in mathematics affects how engineers or natural scientists calculate or perform their tasks when the study of mathematical analysis (which you are acquainted of) can be generalized in advanced calculus like integration and differentials in order to solve and even predict behaviors on how materials should be constructed or planning how to build 4 floor building. Mostly cantor sets and all other set theories after him that solve all of the paradoxes naturally emerged like the Zermelo-Fraenkel set theory, homotopy theory, type theory etc. are important in proofs across mostly other fields of mathematics even the most practical ones but not necessarily useful in measuring since it doesn't deal with quantity but rather sets and counting transfinite cardinals or perhaps in philosophy of mathematics in different schools of thought unlike the realist which introduced subjectivism (our mind constructing across space and time with the use of intuition) like intuitionism or neo-Kantian in nature which is not typically necessary in practical affairs except if you wanted to pursue the most abstract truth about the nature of mathematics itself or its relationship to reality from the very bottom of its iceberg.
I had someone tell me that you could divide a circle indefinitely.
I told him if I bought a pizza pie could he feed humanity forever?
Of course the circle he was talking about existed purely in one's imagination. When presented with reality, his rationalization crumbled... Infinitely.
😂
With infinitemaly small slices, of course.
I had someone tell me that you could make a pizza.
I told him if I got wheat grains crushed into a powder and smashed tomato's, do I have a pizza?
Of course the pizza he was talking about exists purely in his imagination. When presented in reality, his imagination crumbles... like wheat grains
@@ExistenceUniversity "I had someone tell me..."
If you *felt* that was analogous to the original comment, you were wrong.
Next time, actually watch the video to which you post comments.
@bleigh3369 I did, and it was. You are just grumpy
There is nothing absurd about the concept of infinity as a limit. It is a celebration of Cantor's mind to have both comprehended and articulated that there Exists an Order of Infinities. The method of diagonalization is not some parlor trick; rather, it is the application of a well-defined process to prove the premise of the inequality of infinities and to advance the understanding of Cardinality. Sad, that Harry is not able to comprehend this. Then again, Harry is a philosopher and not a mathematician.
Should be entitled Philosophy of Numbers
why
@@TeaParty1776Because that is more accurate
@@ExistenceUniversity for what purpose, by what standard. knowledge is contextual, starting w/senses
No. This fits well with synthetic geometry. It still contains measure but there are no numbers.
@@lamalamalex Ordinal math?
Harry's awesome!
0:55 what is the hypotenuse? In what units do you need it?
Length lol
This was immediately apparent when Harry said it. One side has length of some unit, and the Pythagorean theorem is about the ratio of lengths of one corner is a right angle. The ‘one’ right now is arbitrary and Harry is angry that there is a ratio between lengths and that ratio is not a perfectly whole number. How does this show math as a field is bad?!?
18:51 You read this Wikipedia posting like Diego reads anything. You have to finish the sentence before you insert your own imagination to force a disagreement.
A 1-to-1 correspondence between infinite sets and well-ordered sets.
AKA, more than just counting. He GENERALIZED the process of counting.
19:26 Like seriously dude, that's a comma for clarification of what sets are being corresponded to, not an Oxford comma to separate his multiple achieves.
Infinite sets and well-ordered sets are the two sets he put together. Like this is a basic reading comprehension problem
19:40 Not even higher math, it's literally a reading comprehension problem
19:55 YOU ADDED THE WORD DEFINED!!!
This is absurd cherry picking and quote mining
20:07 Harry 3 minutes ago:
"Infinity was invented by the ancient mythics that's why I reject it."
3 minutes later.
"Cantor defined and invented Infinity and that's why I reject it"
Ok dude
31:12 is Harry angry that the fraction 1/5 can be cleanly expressed in decimal notation as 0.2 while the fraction 1/3 cannot be since the 0.3333 would need to repeat forever? Again these are all unitless quantities, and one could speak about having six balls and wanting to divide them into three groups and doing so. You could also imagine a whole pizza that is cut into three pieces. These won’t be perfect thirds at the atomic level, but will usually be good enough for the people eating the pizza.
Thanks for Your Presentations
Truthfully it's The Sophistication of MATTER
Assumptions create a whole new constructed world, realities start from inherent properties to figure out what's true.
No
Earth and Moon are at different places. Their volumes compared to the distance are nill. Since there is only one distance, there is niether close nor far.
Sun is at a different place too. Contrasting the three places we have similarity along the lesser distance.
Instantaniousness is inability to resolve cause and effect. How close do two events need to occur to be considered simultaneous? Simultaneity similarly requires more than two events to grasped.
Simultaneity is when two events reach an observer at the same time. Simultaneity is relative.
Why is it necessary to multiply before adding in the same formula? The closest thing to an explanation I found was that it is like this because it was established in a convention, and/or because of a supposed greater complexity of multiplication. Anyway, it doesn't seem like an explanation to me, at least not a complete explanation.
It's quite simple.
Multiplication is addition but by a set value, and so when you multiple, then add, the addition is the remainder of the work.
For example:
Say we are selling apples, but we need to move them to the store first! So we buy apple cartons which hold 10 apples long and 5 apples wide. (Fits nice in the back of the truck).
We can also stack the cartons on top of each other.
Now, say we have a big pile of apples, and we start loading them into the cartons. We fill 3 cartons. And there are apples left over, 3 to be exact.
So how many apples are we taking to the store to sell?
1 carton = 5×10
3 cartons = 3×5×10
Plus 3 remaining apples from the pile = 3×5×10 + 3
(3×50)+3
(150)+3
153
If you added first, then it would become 3×5×13, which is physically not true!!
@ExistenceUniversity Now that I can call a explanation. Thank you! 😄
@@ANascente You're welcome!
I found it all rather interesting and self-explanatory/self-evident only I don't think he is talking about what is usually called pure math and its philosophy for the most part, he is talking about natural philosophy, about how the world is supposed to be. Math as a discipline of its ows wants to be independent of nature(it can't be, completely but that's another story) and reality. The talk is about how math is used as a tool for physics and in that respect I think he gets it right. All the criticism I've seen here is basically what he says about Russell nd numbers, but he is approaching it from the practical applied side so my criticism is why he calls this philosophy of math when it is his take(or Ayn Rand's take) about how the mathematical tool should be applied to reality. Of course I'm also of the opinion that mathematics should come back a little to its applied to nature roots because it's become thru its platonism too mystical.
Added 10 years to the scourge😅😅😅
I am very curious how Dr. Binswanger would deal with imaginary numbers😂
47:46
@@ggefrygHmm yes I think he’s right about it being an operator. When you use phasor notation for fields and manipulate them in the end you always take only the real part.
Ayn Rand answers a question about them in Intro to Objectivist Epistemology.
@@thebestofallworlds187 Thanks I will have to read that for sure now
@@ab_c4429 It is certainly an operation, given by the equation: z = reⁱᶿ it is useful NOTATION for working with any oscillatory system. It simplifies calculation but you have to remember that IN REALITY it represents something that is oscillationg, or has wave behaviour. Without the convenient notation you would be writing out multiple simultaneout equations.
Harry is wrong, of course. Math is valuable apart of units it can measure. Math is not a tool of calculation! Private keys in cryptography are numbers, yet they don't measure anything. Math is all about isomorphisms. You can use the same Lagrange polinomial in both Shamir cryptography and digital processing. Another example is Fourier transform, which can be re-used in very different settings like video compression and mathematical proofs of function's properties. Fourier himself never imagined how his series will be used in the future. Math operate objects that cannot represent measurements or calculations, e.g. lattices, finite fields, and topologies.
Cantor's bijection isn't about counting infinities, it is about PROOFS! :)
Very good speech…..And we have zealots that believe that math is racist….
21:06 how many numbers are there? In whose mind?
No matter how many numbers there are, you can always add 1 to the last. Or substrace one from negative numbers.
This video is why I am worried about American Education...
Why
@@Felapa999 - don’t feed the trolls (it’s a GenZ specialty these days because of the toxic Gynocracy).
@@ExistenceUniversity
I'd have to agree with Felapa here. Even if he were an normal American in upbringing, which I guess he is, his current views on mathematics _aren't _*_remotely_* representative of general mathematics in the states or otherwise. And I don't see why you think they are either, considering how obviously fringe objectivism is _inside_ of philosophy and economics, let alone outside of it...
@ivoryas1696 Math skills are not apart of philosophy. He is clearly using American Education system math he learned 50 years ago
@ExistenceUniversity
So you're saying you're worried about math education based off of its past? Fair enough, I suppose, but he's still not terribly representative of math students from the 70's either, considering that people who aren't unique in their mathematic skill (as in outliers) don't generally make it to or through MIT. 🤷🏾♂️
Science is rigor. Logic is rigorously discovered relationships that always replicate, a sub-set of science. Math is relationships of quantity, a sub-set of logic. Quantity is fungibility - dividing something into equivalent parts.
30:14 No they are expressions...
1 divided into 5. That's a whole sentence in that symbol
The point I am actually making, as I just realized no one here will understand unless I hold their hands, is that if 2/3 is a number then so to is 0.6666666666666... as it is literally the exact same subunit.
30:35 if you did 12ths then you'd get to (4/12 = 1/3 = 0.333...)
I knew I would regret clicking on this video. 14:08 is where I'm starting to lose patience. I doubt that the lecturer understands Russel's point.
Reinventing the foundations of mathematics is a noble and fun pursuit, I'm all for it, but one has to do way better.
I'm losing patience on this comment being unable to point out exactly where the lecturer is supposedly wrong and tell us that obvious answer that had escaped us.
@@Valmoorer Not unable, just unwilling.
Since you asked, I'll say something about the ballroom example. The lecturer says: "suppose Ayn and Frank come walking into a ballroom. 2 has walked into the ballroom. The number 2 has walked into the ballroom. That's what he [Russel] says".
That's not what Russel says at all.
If Ayn and Frank have walked into the ballroom, it doesn't follow that {Ayn, Frank} has walked into the ballroom, and it definitely doesn't follow that the number 2 has walked into the ballroom. 2 isn't the same as {Ann, Frank}. 2 is the class of all classes similar to {Ann, Frank}.
These distinctions are important, and one has to take them seriously if one wants to fairly assess whatever Russel may have said.
You don't understand the objectivist point of basing all concepts on actually existing percepts. Anything not based on that is an invalid concept even if mathematicians pretend they are not. They are inventing a world that does not exist so that their notation can work.
@@danshved What is the class of {Ayn, Frank}? Is it different than Ayn and Frank themselves?
@@RashadSaleh92 I'm sorry, I can't answer this unambiguously without talking to you properly, and with a whiteboard. There's too much potential for misunderstanding.
The short answer is, in math generally even x, {x}, and {{x}} are three different things. Between "Ayn" and "2" there are two layers of "nestedness", so to say. Ayn is a member in class {Ayn, Frank}, and that class *itself* is a member in class 2. So when Ayn does something and Frank does something, it doesn't mean that 2 does something.
For example, if I say "Ayn is healthy" and "Frank is healthy", it doesn't follow that "{Ayn, Frank} is healthy", because {Ayn, Frank} isn't even a human. It's a class containing humans as members.
I think the best you can do is to read Russell himself.
Or you can get a modern (serious) text on set theory. That would be different from Russell's work but would still give you the gist of the kind of mathematical rigor that is required.
14:00 Not what he is saying, he said, if you please learn how to read math, a set of two people have walked in. He is doing exactly what Binswanger is doing. Harry just can't do basic research.
16:27 Most Rationalistic garbage argument I have ever heard
18:50 CAN TeR
20:47 Yes!! Just as there are an infinite amount of words!
This is embarrassing to watch
21:42 Literally arguing that words are invalid because we were not born with them
22:11 Your answer "How many you need" and "infinity" is literally the exact same answer. I want an infinite amount. Every time you give me one, I'll divide it into 100 parts. And we can imagine do this forever until a quantitative measure pops out!
I think we can still precisely place a moving thing. It would use the same interval language as, say, 1 + õ. Where is Dr. Binswanger's plane? Flying somewhere on Earth. We have precisely placed the plane on Earth. Okay. Travis, can you get closer to that? Yes, his plane is flying over the state of Florida. Or we can say the plane is between two particular longitudes and two particular latitudes. Or something like, "Travis is on Highway X between the 45th and 46th mile markers."
The thing is that math assumes instant points are universally and exactly well defined, not just the interval, that's the limit at infinity. Binswanger quoting Rand is arguing that reality doesn't necessarily works like that, even if math gives good approximations of local reality within the precision of measurements(since he refers to mathematical tools as ways to relate certain measuremnets with other ones), that can never be infinitesimal.
Of course the lecture assumes no knowledge of mathematics. If you do have the knowledge, it is ridiculous.
Justify.
Pretty funny.
These 3 comments in response are emotional trash.
If you understand math, Harry is making a fool of himself and Objectivism. You three and your inability to do math have no argument by claiming he has none lol
Assumptions create a whole new constructed world, realities start from inherent properties to figure out what's true.
57:56 “math is a tool” - i would say that math 🧮 is a language. therefore everything harry said is incorrect 🎊 🎉
Like languages can't be tools, duh!
27:08 But where did the obsessed man get the next three from? It wasn't created by him, he discovered it! If you discovered the next number, not created it, then wasn't it always there?
Very very radical. As someone who has proved many of the fundamental theories of calculus in analysis I am suspicious that these ideas break calculus since calculus employs many things that are being dismissed here. I am mildly irritated by these claims, as an emotional reaction.
Calculus identifies rates of change, one of the identities of existence. Its a valid tool but it needs to be identified relative to it hierarchical abstraction from the perception of concretes. You suggest that it is defended in some other, invalid, way. Mans methods ofm identifying reality must be validated relative to metaphysical and epistemological fundamentals or the subjectivists and mystics will destroy any rational view of it. The culture is regressing to primitivism under the influence of modern philosophys metaphysical primacy of consciousness and the disintegration of the mind. Math was established as a theoretical study when the mind was recognized as a method of knowing reality. These values are under attack in very fundamental ways.
@@TeaParty1776 Not sure you're following what I said. Have you had analysis? Have you seen the underlying concepts behind calculus, such as: between any two points on the Real number line, there is another point?
@@drstrangelove09 Youre defining by non-esentials. Philosophy of math is about essentials, ie, the widest causes and explanations. Your example, real numbers, has a context that connects it to a view of the universe a whole. Thats how philosophy guides the mind to unify mans knowledge. You take facts and methods out of context . Knowledge is contextual starting w/perceived concretes and widening and unifying until the universe as a whole is identrified. Major alternatives are floating abstractions (rationalism, mysticism) and concrete-boundedness (empiricism, Pragmatism). Mans life and knowledge requires common human experience and unity. Thats how science started w/aristotle and became disintegrated w/Kant. The mind is a tool for living, not a toy for evading life. Look out at reality, not inward. Focus your mind.
@@drstrangelove09 >Have you had analysis?
Freudian or Jungian? You comments are wildly out of context, as if you started a book in the middle. Knowledge starts w/the evidence of the senses which is absolute. Your "knowledge" rationalizes subjective or mystical ideals in your unfocused mind. Look OUT at reality, not INWARD. Focus your mind. Youve been trying to get high without having to pay.
You must realize he is not really talking about the philosophy of math, he is referring to how math should be used as applied to natural science. Anyway your emotional reaction is just an automatic reflex, common to most mathematicians(more acute if platonist like 95% of them are), so don`t worry, it'll pass like a cough.
29:06 I am sorry, but this is ... insane. Pretty much a waste of time.
What do you mean? He's simply stating the results of Cantor's transfinite numbers. Most Mathematicians accept this.
@samueldeandrade8535 Sorry you feel that way. Would you like to make an argument to add to your statement?
the more that subjectivists and mystics talk, the more their unfocused mind is revealed
@@RashadSaleh92 Emotional declaratives - NOT rational arguments - seem to be the only thing most people who don't like Dr. B's arguments have here.
We should not assume what are the identity of numbers they could be straight or bi. Just saying...
7:34 What gives you this impression? Your mind! How do you know that anything has any shape, form, quality, or quantity? You perceive it.
Quantity exists outside the mind like that of quality... what does that mean?
You perceive quantity. It can metaphysically be the case that quantity doesn't exist, that Parmenides is correct, it is all just 1 stuff. Yes, there is your quantity then, but outside of Parmenides axiom that SOMETHING must exist, how can you claim quantities exist outside the mind? What if your mind, like it does with color, isolates differences of singular objects, and its all just one object that we think has quantities because of how we interact with the quantum soup we swim in?
Everything exists., the products of the focused and unfocused mind. Man must choose, a terrifying fact for the unfocused mind
@TeaParty1776 You are saying nothing at all
@@ExistenceUniversity Agreed, in your unfocused mind
Can somebody tell me a practical use for knowing all the stuff (rubbish) thats being spouted here?
Zero
keeps mind based in reality
@@TeaParty1776 how so?
@@ExistenceUniversity You evade common human experience/
@@TeaParty1776 Speak in full sentences like an adult
Just from the headline: I see philosophy as a study of how man should live. Why would math be considered a philosophy? Math is a product of thinking, and the philosophies most responsible for human thinking, (which would have to be individual liberty, as it frees the mind), allowed math to be created.
*"I see philosophy as a study of how man should live."*
No. The "study of how man should live" is but a single BRANCH of philosophy, called ETHICS (or MORALITY). Philosophy also includes Metaphysics (the study of WHAT exists), and Epistemology (the study of HOW man knows WHAT exists). Ethics is merely the APPLICATION of these other two branches TO human action. In other words, not only is Ethics NOT the entirety of philosophy, it isn't even one of the PRIMARY branches of philosophy. It is instead a COROLLARY branch of a philosophy's Metaphysical and Epistemological branches.
5:07 No! "The science of measurement-calculation" implies that no child can ever form a unit, because he would first have to calculate the singularity as 1.
Harry did this YEARS AGO back in the day, he made reason require reason to start. Today he fixed that, but he has moved that issue to math. Now he is saying, to get started in math, one needs to calculate the unknown into math. Calculate how?
try focusing first, then read HB
How to sound clever for 57.56 minutes while saying absolutely nothing.
My education was very basic, I don't pretend to have any knowledge of science or maths but many years ago I worked for a University.Physics department in the UK
Nonetheless I decided to try to read a book by Russel (no idea what title was) I remember reading about 10 pages of it.
At the early stages of his book Russel made a the following statement (more or less) when a Physic's professor looks at a chair he does not see a chair but atoms ?
I remember asking several of the departments senior academic staff what was the object they were sitting on ?
After giving me a look they all answered I'm sitting on a chair.
So much for Russell.
Luckily for me my lack of a well founded education has not stopped me from having a fantastic life
Any body like a pile of atoms?
I could be me just a telling a silly story.
Or it could be me making the point that most Philosophers talk bollocks.but use $64 words saying it.
"It is a tale told by an idiot, full of sound and fury, signifying nothing”
It's Emperors new cloths
I'm glad I had a poor quality education at least the world I live in is real,
He should be old enough and educated enough to know what he says has no practical purpose at all.