Pure maths has painted itself into a corner | Sociology and Pure Maths | N J Wildberger
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- čas přidán 22. 04. 2024
- It is long past time that pure mathematicians as a community address the serious foundational weaknesses that beset almost all areas of the discipline outside of combinatorics and some adjacent areas. This is also hugely important for students of pure mathematics and those wishing to embark on a career either as a maths teacher or a researcher in mathematics.
Our AI machine friends/competitors will soon be breathing down our mathematical throats. Let's correct our errors and misunderstandings ourselves, before we are humiliated into doing so by our computational devices.
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I love you Norman. Don't quit Brother. Your message has made it to South Carolina, USA. Many of us out here are sharing the facts with others and liberating folks from the grip of living in a mathematical fantasy world.
Theoretical computer science is exactly the kind of math where most of the objects of study are finite. Indeed, the description of an algorithm and the number of steps it takes to compute a language are all finite, and in some sense, it is nicely aligned with the finite nature of reality. This is why fundamental problems such as P vs NP or P vs BPP are not just of theoretical interest, but point to a deeper understanding of our reality.
I would shoot the moon to travel along with Wildberger on whatever. The typed lambda calculus is an amazing thing.
An algorithm is a routine that terminates according to Alexander Stepanov.
Especially 'Functional programming' (bad name, sound idea) teaches that computing is ultimately about temporal processes, not about static objects. A duration of a computation is not any fixed magnitude, the available theories of computation durations concern comparative inequivalence relations (computation x is faster than y etc.); on the other hand mereology of duration is very important (routines, their nested subroutines, etc.).
Pure mathematics desperately needs clearly defined and communicable notion of "lazy algorithm", by which from ongoing process some values can be given. The current notions terminates / doesn't terminate are insufficient for lazy evaluations, which are somewhere between.
and that's because computer science is forced to remain grounded in concrete computation as a justification for all the abstractions. one can't leave the universe so to speak.
*which is a good thing
You've given me a lot to think about, Professor Wildberger! I'm not a mathematician, just a high school math teacher, but you have me questioning whether I am helping to perpetuate a theory of "real numbers" that does violence to my students' natural, down-to-earth logic--are they right, after all, to object to the idea of viewing the result of an infinite process as a real number? More disturbingly, am I robbing them of sound, solid grounding in computational reality that would actually be useful to them from a career perspective? Is the fact that students continue to struggle working with fractions due to the fact that their teachers do not actually value the rational number system, thinking of it as an uninteresting subset of the "real numbers"?
My problem is that I am undecided--I am not sure that I agree with you philosophically that only what is finitely constructible can exist. To me, that seems to impoverish our world. The idea that numbers are a great, dark ocean, and that the numbers we can know weigh as nothing to that ocean does not surprise or repel me. As a theist, I am at peace with the idea of a reality outside of myself that does not depend on my ability to construct it. Combinatorics was a large part of what drew me to mathematics as a young person, and it remains my favorite area of mathematics, but the other strong draw was the fascination with numbers like pi, e, and the transcendental functions like sin and cos.
I've watched enough of your videos now to know that you are not interested in throwing out geometry, just that you don't believe that there is one-to-one correspondence between numbers and points on line. You would prefer to approach measurement from a computational perspective, and I can see the sense in that. What gives me pause is that I wonder if this would give students' minds a better training.
I'm not sure! You have put out an enormous amount of content, and I won't be able to go through it all any time soon. But you certainly have me thinking!
What a great comment Mark! I feel instinctively that I should make a video just to address your questions and thoughts. Not to put you down in any way whatsoever, indeed just the opposite. Would you have any objection to me doing that? I would let you listen to the video first and approve it.
@@njwildberger I would be completely OK with that! I'm not sure I would call it a professional crisis, but I can see clearly that at some point it is necessary for me to definitively come down on one side of the fence or the other. These views of the number system are not compatible with each other, and they have very real ramifications for how we teach students at every level. I was just describing to my wife how your videos have made me question whether I am doing my students a disservice by forcing on them a "pi in the sky" view of numbers that they can't actually use and don't want to use! Would that time be better spent in helping them understand the computational processes employed by the technology they use? At the same time, I am still in love with the beauty of calculus--though not on an extremely educated level, only an undergraduate one.
Today I do not see mathematicians who are problem conscious,curios and think but are engaged in computing.This professor makes a difference,for he is curious and is thinking.There is a problem to sociological research to be undertaken why do professors lack thinking in clear way why do they confuse themselves and their students using sophism.
In what way is this different to Brouwer's prorgramme from a hundred years ago?
Keep going Norman! As a software engineer I love your takes, and I think it's very sensible to re-examine math in terms of computation, as you are doing, and I'm using some of your ideas in my own CAS program. My test for correctness is usefulness., and your ideas are useable, set theory is mostly not.
Finite group theory and combinatorics really make me happy.
Any math that actually works is finite. Even limits.
"actually works" lol
You adduce the cosine of 7 as an example of something which cannot be written down explicitly in finite time, and so casts doubt on the idea of functions whose domain is a real interval. You might as easily have used the square root of 2 as an example, or any other irrational number. For apparently related reasons you acclaim combinatorics (you obviously intend finite combinatorics) as a field which deals with genuinely specifiable (because finite) numbers. But does eg Graham's number, which issues from results in finite combinatorics, exist? It cannot be written down in full specificity either, certainly not in this universe. If your arguments about the inadequate specifiability of real numbers as such are thought to undermine our confidence in these, should we not give up on sufficiently large finite ones too? But in that case the obvious question arises: where exactly (or even approximately) does the sequence of integers come to an end? Or should we instead conceive of various levels of reality, the smaller more tractable numbers being more real than the bigger unwieldy ones; or perhaps integers gradually fade out of existence (or reality) as they get larger and larger? These are the sorts of bizarre speculations that one seems driven to, once one refuses to countenance anything as real which is not finitely computable.
Prof Wildberger has argued in a previous video that the integers actually do gradually fade out of existence as they get larger: The sporadic nature of big numbers | Data Structures in Mathematics Math Foundations 176, czcams.com/video/Z8I68E7yZeY/video.html
He has a series on really big numbers and basically argues that they are nearly as dubious as the concept of "infinity."
However, I think that approach barks up the wrong tree. It's not about what can be written down or computed, but what is even possible to conceive of so as to serve as the referent for an act of communication. It comes down to application, and there we have no use for gigantic numbers anyway.
@@Rudi_F_Vienna It's not entirely clear whether he thinks that the (so to speak) "level of reality" of numbers decreases monotonically with the size of the number concerned, or whether their level of reality is rather a function of how easy it is for us to represent them in feasible notations. As far as I can see, it appears to be the latter - hence the word "sporadic": only the feasibly nameable ones exist, seems to be the idea, feasibility is not a black and white matter, and feasibly specifiable numbers, even very large ones like Graham's number or TREE(3), exist (to whatever degree they do) on isolated islands amid an unimaginable ocean of number-line "here be dragons" indiscernibility. If that is what he thinks, then it is not that a number's reality is inversely proportional to its size; perhaps it is inversely proportional to the size of its definition in a notation system, together with (the size of) the definition of that notation system, and a proof that that system is free of contradiction. But postulating such levels of existence seem a long way from the advertised desire to keep mathematics concrete and computational and free from metaphysical speculations. It looks like a reductio ad absurdum of his entire project to me.
@@ThePallidor You say we have no use for gigantic numbers, but that rather depends on who 'we' are. Wildberger says he has no problem with (finite) combinatorics, and seems to think that this field is some sort of shining of example of how pure mathematics should be done and the sort of questions that pure mathematicians should concern themselves with; yet Graham's number, and similarly utterly unimaginable - I'm not sure about 'inconceivable' - numbers emerge naturally from finite combinatorics all the time.
We should return to foundations and start cautiously fixing errors.In my opinion the foundation on which calculus is built is illogical and vague.But it is not beyond the ability to make corrections.
It can be built on rigorous foundation.The weak point of the teachers and learners is that all learners learn ready-made information from formal educational institutions.Karl Marx seems to be stunningly more correct when he observed that man's consciousness is determined by the knowledge system to which he is introduced.This knowledge is forwarded continuously from past to present. and future.A learners learns this knowledge and this knowledge formulates his consciousness.He acts,speaks and thinks in this line.ONLY EXCEPTIONALLY GENIUS THINK ON THEIR OWN.
I have seen the students learning sy/DX without conceiving the real reflective concept nor the teacher is able to.In differential calculus DX is an increment to a variable quantity x,and astonishingly Leibnitz uses this DX equivalent to zero and finite quantity at the same time.Marquis l'hospital takes the postulatum that a quantity increased or diminished by this infinitely small quantity may be considered neither increased nor decreased.I.e x+DX=x.If this is true,then DX is zero.But these incorrigible mathematicians who gained this knowledge from their elders go on holding in their mind the differential DX is not zero.What an absurd postulate.
Is it agreeable?.
This is great, Professor Wildberger! I'm excited to see the rest!🌞
I see two immediate benefits in Professor Wildberger's initiative to question the basics. The first is that we might succeed and find the appropriate scaffolding. The second is even more important. Posing the "trivial" question about the very "foundations" encourages the wider audience like myself to believe there is merit in coming back to studying math even at an older age because it is neither too late nor everything is set in concrete, far from it. It may just attract the right crowd of the citizen scientists into the field of Mathematics. This latter sociological aspect might prove to be the most poignant in Norman Wildberger's initiative and who knows, this might even become one of his greatest legacies for Humanity. Thanks a million Professor for your past, present and future contributions! ❤
I always had issues with continuous math. For discrete things like graphs, it's easy to imagine all the details. Once you convince yourself something is true, you firmly believe it. But when it comes to analysis, there are lots of assumptions. I constantly ask questions about the nature of the thing but people seem to ignore most of them because they are "trivial".
I heard a lot of advice like "Don't understand, just accept". I need to do this all the time for continuous math, but I never fail to understand something for combinatorics, even if I don't know something for now, I'm confident I will eventually get it. I think there is a big cultural difference between them and it's pretty artificial.
I must be a weirdo, because for me graph theory seems obviously continuous math. The vertices are connected by continuous edges. I see graph theory just as a version of continuous geometry. Some people seem to think that the numerical aspects of the graph theory matter more than what the numbers count, so they think that it's discrete in that sense?
Our confused and confusing notions of "continuous" vs. "discrete" demonstrate well the current semantic bog.
@@santerisatama5409 A Graph is probably the most discrete structure you can get.
I would advise you to look up Konig's lemma. It states that in a graph with infinitely many edges, if all edges have finitely many vertices, then there exists an infinitely long path.
Seems obvious ? Well too bad you need the axiom of choice to prove it. And it is discrete maths
@@ailrky4765 Please explain. What definitions are we using, to get to the main issue of the current topic?
@@20-sideddice13 Sounds close to CT object with infinitely many self-arrows. And all they say is "completed infinity", which is completely boring.
Ongoing participatory mereological processes of rotations is much less boring - sounds like life on Earth! Squaring the circle by taxicab norm is infinitely more interesting application of graph theory/continuous geometry.
A say-so doeth not make a make-so. Still waiting for sound arguments why I should agree that graph theory is FOUNDATIONALLY discrete.
Last year I left my PhD program after a year. I had entered expecting to study differential geometry or something adjacent to it, but that didn't end up happening obviously. Upon reflecting over my time in graduate school, it became very apparent to me how naïve I had been about the nature of mathematics. "We accept the axiom of infinity because modern mathematics would not be possible without it" would be something you could have easily heard exiting my mouth. I've since become much more jaded after realizing how very little pure math has advanced in modern times and how much overhead cost there is to reaching a point where you can perform research on modern problems. Pure math is reaching a place of death-via stagnation, and that stagnation is caused at least in part by an explosion of hyper-abstraction as a result of the axiom of choice and the axiom of infinity. That you must rely on these to talk about the existence of foundation objects now seems to me as folly, damning the field to an eternity of gnostic idealization. Is there power in the abstraction these axioms provide? Yes. But there is also immense danger in relying on them too much
Quantum mechanics depends heavily on infinite-dimensional Hilbert spaces to work. Yet this theory is necessary to construct the modern computers whose calculations you cherish so much
Quantum mechanics is complete hokum.
It is possible to work with Hilbert spaces in some fashion without assuming that we can do an infinite number of things.
@@njwildberger could you please refer to peer-reviewed literature ?
I don't think QM depends from Hilbert space. We need equilateral triangle etc. symmetries on plane, and we don't need inner product space for that, continuous geometry is sufficient. We need straight edge, compass, origami and two hands (chirality). And mereology.
@@santerisatama5409 of course it does ^^
How do you decompose an operator along its eigenspaces without a Hilbert space ?
as a CS student I really loved Norman's videos/books. He showed me some of the elegence of math.
Symbolic vs Semantic awareness - conflict in our current modern existence, We seem to confuse those two, and that creates cognitive failure in a whole system. So this idea of going thru the basic definitions is awesome.
Looking forward to the series. What would you call the branches, computational number theory vs pure number theory? I’ve just learned to accept limits and infinities as a tool to get an answer that works, especially after seeing your videos years ago.
I deeply admire anyone who is willing to challenge what others take for granted. You sir embody that sentiment and with all my being I congratulate you and encourage you to carry it on.
My own experience falls in line as well. Most of my calculus loving friends are unaware you only need the quadratic discriminant to create a tangent line and thus the "derivative". You can create tangent functions that are not lines just lesser in degree and carry on until you reach that line.
I look forward to a real grounding of math, and I salute you sir!
Calculus really has nothing to do with tangent lines. Secant lines are all that is needed. No ongoing processes.
Epsilon delta basically uses "potential infinity", or for limits to a point a, doesn't even use infinity so I don't know what you're on about.
@@ThePallidor the derivative is moreso what my comment centered on. You don't need limits and calculus to find tangents to curves if you use algebra
@@lox7182 that makes two of us. Why bring up epsilon delta ? My main idea was that we can use algebra to calculate derivatives . No need for limits, no epsilon nor delta
@@mikeh283 How do you prove that the wierstrass function has no derivative without the concept of the limit derivative?
My biggest gripe is the claim uncountable infinities exist without an axiom. Uncountable sets require undefinable elements. How does one construct the reals if it contains numbers which cannot be defined and without an axiom explicitly stated to justify it? I can live with countable infinity rather comfortably since I think most of that can be rephrased back and forth with unbounded inductive reasoning.
Counting ceaselessly is a movie than never ends as long as one cares to keep watching, rather than a static image. These are separate concepts. There is no noun "infinity" or adjective "infinite" but a verb "count" and an adverb "endlessly."
You just need the axiom of power sets to go from countable to uncountable infinities
Im right on board w/ you nj i think more extreme, we havent actually thought enough into as you put it "pure mathematics", or the whole system. Mathematics the language. Math as an idea of it being both scientific, artistry and computation. I see what you mean about it describing the imperical. I think of it even a step further, as its explored artisticly, that is to say the processes that might require looking at math in new ways that would apply to the math itself.
Also historically, im looking into those ideas as im still fairly new to where these ideas came from, and the world that they emerged out of. Because it seems to be quite different depending on at what time period, and what was the influential things. Thats quite interesting. In that way, theres a very evolving thing that we're trying to get ahold of and work with.
I think it could be a huge help in advancing computational applied mathematics and computation in general if the notion of floating point numbers somehow being a slightly deficient representation of real numbers could be dispensed with in a world where pure mathematics dispensed with real numbers, and in its place we put computing unashamedly into the realm of only rational numbers, with floating point numbers being one powerful way of representing rational numbers (all floating point numbers are themselves actually rational numbers).
While rational numbers are great theoretically, they are often not practical for real life applications. This was already understood by Simon Stevin 500 years ago. There is something really powerful about a decimal (or the Babylonian sexagesimal) floating point system in which the amount of accuracy you have available is reflected in the numbers you use.
But certainly "real numbers" are completely off the scale in terms of disconnect with the actual real world in which we live. They are almost 100% a fantasy that allow us pure mathematicians to keep on pretending that the world is a lot simpler than it really is. (The "Fundamental Theorem of Algebra is an excellent example which I will be discussing in detail here shortly.)
It can suffice to say, "There are no objects called 'real numbers,' only processes."
@@njwildberger Can't say that "integer divided by integer" is theoretically really that great, and IMO there are better definitions of rationals. The intuitive notion of fraction, "how many equal size parts of a whole" is of vital real life importance when dividing a cake for the kids. :)
More generally, I've been very impressed by Lou Kauffman's argument that math is really about studying distinctions, and in that sense all algorithms that generate distinctions and uniqueness are very important. Also Gosper noted that most practical tasks of computing concern finding a satisfying inequivalence relation (abc-conjecture being a prime example); in that sense Gosper had similar approach as Babylonians, but instead of any fixed word length system he went straight for arbitrary word length systems. Gosper argued that continued fraction arithmetics are in many ways not only theoretically but also pragmatically superior to Stevin's decimals.
I don't know enough of the fine details to offer a firm opinion. I'm hoping that some day soonish enough you are ready to offer your insights and opinion on this issue, for which I'm most curious. :)
So what is Pi? What is cos(x)? These objects can be realised geometrically
There's a lot of interesting work going on in formalization with interactive proof assistants at the moment. With great difficulty they are able to prove things about real numbers, right down to the axioms. I think this is the right direction for pure maths going forward.
I feel we should be skeptical about “proof assistants” that have been pre-programmed to believe in real numbers and infinite sets
@@njwildberger I'm not an expert on this, but my understanding is that they just use dependent type theory. Real numbers have to be defined as cauchy sequences of rational numbers.
I think we need to think about the foundations for applied mathematics first perhaps. For example what is the multiply sign "x" to signify. The most general algorithm for multiplying two numbers. But can the algorithm give a result for pi "x" e different from pi "x" e. So we either say it also represents unrealizable algorithms in finite time (or algorithms which never terminate) within the generality of the definition, or we accept pi and e are not numbers. They are perhaps more like real physical quantities (like the height of a tree) which can be studied in ratio with a similar physical quantity to become a number and enter the realm of "applied" mathematics.
Numbers and magnitudes are two different things. Some magnitudes are incommensurable. The ancient Greeks had it right.
@@ThePallidorso your understanding of mathematics hasn't advanced further than 200BC 😂
Fields Medalist Voevodsky has a brilliant lecture called "What if Foundations of Mathematics are Inconsistent" at the Institute for Advanced Study in Princeton, and to paraphrase, he states that The Real Number system is an over-idealized (albeit false) notion of continuity, suitable for the limited human brain, which necessarily leads to contradictions in physics (including Quantum Mechanics). He predicts that a new Foundation of Mathematics, built systematically with upcoming AI technology, will play a major role in remedying both the shambles that modern-day mathematics is in, as well as the mess that physics is in. Don't listen to the Naysayers, Norman. You've got a Fields Medalist who indirectly is saying the same thing (although he is slightly more diplomatic and gentle in his assertions)...Edward Witten, also a Fields Medalist and a superstar in the theoretical physics community, asks a great question at the end, suggesting that the Real Number system in Analysis would have to be different if the foundations of Mathematics (i.e. Set Theory) were inconsistent. So, these two great pioneers and mathematicians have addressed critically two of your main gripes with mathematics, namely Real Numbers and Set Theory.
So Keep up the good work mate! You're on the right track!
Thanks NJW for making this crystal clear to a non-mathematician.
I think it is worth mentioning Aristotle who said that one should distinguish between potential infinity and actual infinity. Finite sequences that can be extended to a longer finite sequence is an example of potential infinity. Considering an infinite sequence as a whole is an example of actual infinity. Potential infinity is no conceptual problem but actual infinity will lead to all kind of problems.
Retired applied linguist in Japan here, but.with a lot of background in philosophy of math and science ... I climbed Wittgenstein's Ladder without a parachute back in undergrad days.
Just thought I'd weigh in with my observation of a fundamental difference between two mostly mutually exclusive mind-sets ... those who believe that mathematics is THE language of nature/universe and it is explanatory ... and those who believe that mathematics is merely one of many descriptive languages of nature/universe. What the two mind-sets have in common is that they are both necessarily grounded in unprovable beliefs (thank you, Gödel).
Bridging domains between linguistics, the social sciences, and math ... I would say that practical, predictive mathematical models are ultimately parochial (locally limited such as building a bridge), provisional (as in definitions of standards for quantifying such as the metric system) social constructs. A meter or kilogram is not fundamental to nature. The metric system, or any mathematical model positing an ideal abstraction as a proxy for a number (what does "1" mean?), is a variable mesh through which we filter and share experience. Increasingly granularized measurements and manipulations can lead to dangerous amounts of power (such as nuclear fusion), but when approaching plank-unit "infinities", I suspect the law of diminishing returns becomes increasingly salient.
When talking about those infinities, in linguistics, I take the stand of what would be called the strong version of the Sapir Whorf hypothesis of linguistic relativity. In the social sciences, maybe Joseph Campbell's Jungian inspired insight into the metaphorical nature and agreed-upon assumptions of trying to share experience through language. In philosophy and religion, Schopenhauer's appropriation of the Hindu "Veil of Maya" ... the limits of our language and our senses as producing sometimes sophisticated epiphenomenon in our attempt at expressing something about fundamental nature.
Having resigned from a tenured position in protest, my experience of the petty politics of academia ... its own rituals, gate keeping, competition, and personal ambitions ... is merely anecdotal. But those who have survived the publish or perish paradigm long enough to sit in on academic assembly meetings that are merely a cloak for what has already been decided behind closed doors, can not help but to nod in recognition.
For those who are looking from the outside in, one influential book is T.S. Kuhn's "The Structure of Scientific Revolutions" ... a much easier and understandable read than its title suggests. Another less known is Robert Friedman's "The Politics of Excellence: Behind the Nobel Prize in Science". Also an easy read, that book shows that collectively, regardless of domain, we behave not so differently from observations of power struggles among chimpanzees. Frans de Waal is the man for that ... and funny too. His TED talk was a hoot.
I also agree with Wildberger in that the power of A.I., in the hands of bad actors, poses an existential threat to homo sapiens. But I may be more pessimistic about the capacity for our collective wisdom to control our collective capacity for cleverness. In the words of Stephen Hawking ... "Greed and stupidity will mark the end of the human race."
I generally agree with the skepticism about arbitrary arguments from infinity, but what about well defined infinite series like those of Ramanujan?
Ongoingness isn't a problem. It's the confusion of a process with an object that is a problem. Modern math essentially confuses verbs with nouns.
I'm far from expert on all the wonders that Ramanujan delivered, but my hunch is that he/his Goddess intuitively computed continued fraction arithmetics with great ease and generality. Gosper's computable algorithms of continued fraction (and continued logarithm) arithmetics came much later, and remains rather poorly known even today. Wildberger gave a glimpse of a matrix form of Gosper arithmetics in his lectures about Stern-Brocot tree.
Don’t refer to a man as goddess, that’s weird dude. He was brilliant but still just a man
@@avinashreji60 Ramanujan said that he received math from the goddess Namagiri.
@@avinashreji60He's talking about the Goddess that Ramanujan worshipped.
You reminded me of Ludwig Wittgenstein 🤧
Those Al toys can only make subsets of the power set of phenomena. They cannot see beyond Immanuel Kant.
Mr. Norman, You are a precious person. Stay healthy!
I think the problem is not unique for pure mathematics, other applied mathematics such as mathematical physics, probability theory, statistics and information theory also share the same problem.
I don't know if Prof.Wildberger has a deep understanding of probability theory, but I think probability theory suffers more than pure mathematics of those logical difficulties regarding "infinity" and "limits", or even more difficulties such as "randomness". I think in the future you may discuss about the usage of those ill-defined concepts in probability theory.
What do you think is going wrong in pure mathematics? Are people disagreeing on mathematical results, are students repulsed by the field? I doubt demanding better definitions in human language for mathematical objects will lead to any positive change, but I'm willing to listen.
. . . It's Ok to reject actual infinities that Plato proposed and only accept potential infinities as Aristotle, but your work in math gets much harder: are the results worth taking this philosophical stance?
The problem is that the whole concept of infinity could be false eventually. What if the world is inherently finite which would imply our brain‘s mental concepts are finite? Then we could not refer to an infinity, we‘d fool ourselves by calling a somewhat large entity „infinity“ and givin it different properties. It may have been a warning sign „from god“ when Russell discovered his paradoxical set or Goedel his incompleteness theorems - both do not happen with finite sets. Finitism is just more robust, but poorer in its implications.
um... under that argument can we refer to graham's number?
I have an idea that aligns with your sentiments in this video. The scenario I propose bears a striking resemblance to the scenario outlined by the French mathematician Jules Richard, known as "Richard's paradox." Richard introduced his paradox in 1905, predating the existence of electronic computers or programming languages. Had these tools been available at the time, Richard might have employed them similarly to my proposal.
In "Richard's paradox," he represents real numbers using English language descriptions, while I suggest code segments. He organises his descriptions by length and then alphabetically for strings of equal length. My proposed code segments, or 'Turing machines' if preferred, could be arranged in the same manner.
For instance, consider the code segment for √2. It might encompass the code for a general square root function along with a line of code that could invoke that function, such as 'result=SQRT(number=2,base=10,significant_digits=0)'. These parameters would enable the function to execute for a specified number of significant digits (to correspond with the mathematical definition for a 'computable number') or to run unrestrictedly by setting the third parameter 'significant_digits' to zero. If, hypothetically, the most concise way to do this in a given programming language took up 300 characters (including the line of code containing the function call), then we could conclude that the code segment length corresponding to √2 is 300 characters long.
Considering that there is a finite alphabet for any given programming language, there will be a finite number of code segments that are 300 characters long. Therefore, there must be a finite number of 'real numbers' that can be encoded (and thus considered well-defined or in a closed form) into a code segment length of 300 characters.
Richard then defines another real number 'r' as follows: "The integer part of r is 0, the nth decimal place of r is 1 if the nth decimal place of rn is not 1, and the nth decimal place of r is 2 if the nth decimal place of rn is 1." This 'r' value is akin to Cantor's anti-diagonal value, except that Richard is only applying it to definable/specifiable real numbers. And so with respect to Richard's paradox, we don’t need to consider whether or not any non-specifiable numbers can be said to exist.
Richard argues that this is an English expression that unequivocally defines a real number 'r'. Thus, he presumes 'r' must be one of the rn numbers. However, he deems this paradoxical since 'r' was constructed to avoid being any of the rn numbers.
In both cases, whether 'real number' is specified in English or via a piece of code, we describe an ongoing process. We start with shorter lengths and gradually increase to longer ones. Upon close examination, at any stage, we will have specified a finite number of 'real numbers' with self-contained code. However, the specification of Richard's 'r' number requires further consideration.
Initially, I believed it would not be possible to specify a self-contained code segment to calculate Richard's 'r' number. However, upon further contemplation, I began to question whether I was correct in making that assertion. Its construction would necessitate a formulaic approach to creating the other number specifications. Then, conceivably, we could produce a single self-contained piece of code that would emulate the creation of the other numbers, calculate the nth digit of each of them, and output the altered digit as required.
If such a thing were possible, then I would have to agree with Richard that his specification of 'r' could be described as unambiguously defined. However, as I delved deeper into this idea, it became somewhat mind-boggling to consider whether we could continue creating more of these 'r' values or anti-diagonals ('r1', 'r2', and so on). That is, could we proceed to create further code segments, each of which differs from all previous 'r' values as well as all specifiable numbers?
Yet, while pondering this perplexing scenario, I stumbled upon a more fundamental issue that I had overlooked in my analysis of the original problem. If we assume that 'r' can be encoded into a piece of code, then what transpires during its processing when it has to deal with code segment lengths equal to its own?
It appears that the code segment for 'r' would need to emulate or execute its own functionality and then apply further functionality to change the nth digit. This seems contradictory as it would require all of "its own functionality" plus "some more functionality" to be contained within "its own functionality," which is evidently impossible. Also it would need to represent not just its own real number, but its own real number with one digit altered, which is also impossible. Consequently, I reverted to my original belief that Richard's 'r' value is not well-defined as it cannot be constructed as a self-contained code segment.
Richard concludes that his 'r' statement refers to the construction of an infinite set of real numbers, of which 'r' itself is a part, and so it does not meet the criteria of being unambiguously defined. Contemporary mathematicians agree that the definition of 'r' is invalid, but they claim it is because there is no well-defined notion of when an English phrase defines a real number. My proposed code segment approach would seem to negate this objection.
Note that should mathematicians concur with Richard that the diagonal is not well-defined, it would suggest that Cantor's diagonal could not be defined, thus rendering the diagonal argument invalid.
If all specifiable real numbers were said to already exist, all infinitely many of them, then Richard's description of 'r' would seem to be a valid specification (not only is the concept of infinite repetition readily accepted in formal definitions of real numbers, but the concept of Cantor's infinite anti-diagonal is also widely accepted by the mainstream). However, as such, the value it describes would need to already exist in the static set of all specifiable numbers. Hence, it would have to describe a value that is different from its own value, forming a trivial contradiction.
Therefore, after much thought, I still maintain that the most reasonable resolution of Richard's paradox is that the concept of 'infinitely many' is incoherent. No other proposed solution can avoid contradiction to my mind (for the reasons explained above). It also renders all infinite diagonal arguments invalid.
Around 2010 there was a guy who posted a series of blog posts and maybe newsgroup posts arguing that Cantor's diagonalization argument was incorrect. One was called, "Cantor's Theory Visualized." He got raked over the coals for it, of course, by mainstream-defender blogs about "bad mathematics" "mathematical cranks" and now the whole thing seems to have been scrubbed from the Internet.
@@ThePallidor Moderators ought to eliminate abusive, racist, or violence-promoting content. However, they also often remove well-thought-out ideas presented in a fair, unbiased, and professional manner.
They wield ultimate power over their particular corner of the internet and misuse it by eliminating any argument they personally detest. This misuse of power is particularly common in mathematical discussions, where there's a prevalent belief among mathematicians that mathematical proofs are unquestionably true. This leads them to believe that removing all opposing viewpoints is entirely justified.
Sigh.... Of course the r constructed of r_n that are computable isn't computable, but the whole point is that not all of the reals are computable. There is a bijection from the natural numbers to the computable reals, though that bijection itself isn't computable, but not from the natural numbers to all the reals.
The reason why r isn't computable is because if it was computable it would mean that there was a way to solve the halting problem computably, which there isn't, but your argument for why r isn't computable works too I guess
Also what you're saying is literally the reason why cantor's diagonal argument works. Cantor's diagonal argument assumes that there does exist a bijective function from N to R. Then it shows that this is contradictory. The whole point of cantor's diagonal argument is "you can construct a new real number even though the function was supposed to cover every real, meaning that function doesn't actually exist in the first place"
@@ThePallidor Moderators wield ultimate power over their particular corner of the internet and misuse it by eliminating any argument they personally detest. This misuse of power is particularly common in mathematical discussions, where there's a prevalent belief among mathematicians that mathematical proofs are unquestionably true. This leads them to believe that removing all opposing viewpoints is entirely justified.
what do you think of type theory as a foundation instead?
I’ve tried to understand that a little bit but have been unsuccessful
@@njwildberger HOTT indeed ... The trick is to keep track of the amount of information contained in those infinite objects or processes. Isomorphisms sometimes are not enough. Univalence is some shape of hierarchical form of equivalence classes.
But of course we can do an infinite number of things.
This was exhibited in the simplest of Zeno's Paradoxes. Movement is impossible, the argument goes ... because to move from A to B, we first have to move from A to the midpoint, say M; and from there to the midpoint of the remaining distance, say MM; and from there to MMM, and so on without end. To move from A to B we must complete an infinite number of ever-decreasing steps.
But we do it all the time. And this shows that a finite quantity can be the limit of an infinite series.
Now this has got me interested in the range of the Professor's topics. I too am concerned with foundations, from an amateur standpoint. But I don't think it helps to take a detour into sociology. The crucial thing for me is to bring out the fundamental arguments within a given topic.
The objection that "we cannot do an infinite number of things" is not a very clean one. A cleaner way to dispense with it is to say we cannot even coherently speak of "an infinite number of things."
I think the Dr is talking about only being able to do a finite number of things within the context of computation as described around 2:45
It is impossible for someone to sit down and sum up all of the numbers in the sequence representing Zeno’s paradox using arithmetic computation.
cos(7) exists in potency but not in act as with all irrational numbers.
Thanks for the talk. I deeply agree that pura mathematics is an empirical science. That doesn't mean that we can especape philosophy - on the contrary, since Socrates and Plato, finding coherent definitions has been a main task of philosophy. In my view, pure mathematics is ultimately search for Truth and Beauty. Which means that the love of wisdom and pure mathematics serve the same purpose in complementary manner.
What is mathematics without mathematical truth? Applied math serves pragmatic truth theory. What is the truth theory of pure mathematics? Correspondence with "timeless being" (as Gödel expressed it) of Platonia? There are no empirical grounds for such postulate, timeless being would not be coherent with our empirical condition of constructibility. Intuition is traditionally very important empirical phenomenon and method for pure mathematics. Can we leave out e.g. Ramanujan's intuition out of the discussion, when trying to comprehend and communicate what we mean by empirically defined pure mathematics?
Not in my opinion. To say something meaningful and important about the nature of mathematical truth, I think we can say that the empirical truth conditions of mathematics are constructibility (cf. computability) and intuition. Intuition is phenomenally a some kind family resemblance of of empirical experiences, but seems that intuition is undefinable, because phenomenally it largely occurs in the borderzone between pre-linguistic and linguistic. We can be intuitively inspired and possessed by some mathematical idea or question which is seeking a way of a crisp and clear linguistic expression.
AI can in many ways do syntactic manipulaition of formal languages better than us, they are strong allies on the side of the constructive aspect of empirism. Can AI intuit? Dear teatcher, I don't think we can answer the challenge you posed without a profound discussion of intuition, whether we call such discussion "philosophy" or something else. I hope you help us raise to the challenge with the attention you have earned and deserve.
If I understood you correctly your conditions for empirical truth in mathematics are constructivity and intuition. From my perspective there is one or two other conditions that are important. Maybe they are about something else than truth or maybe they can be subsumed into either your condition of constructivity or intuition.
Here I try to formulate my conditions: Whatever we present as an important result in mathematics ((who cares about the unimportant results?)), should be somehow *surprising*, or *unifying*, or *operationalizing*. (Maybe there is a better word than "operationalizing". Here it is meant for example finding an algorithm for answering a kind of query.) Unification is inherently surprising, so maybe it is redundant as a condition. Operationalization is closely related to conditions of falsifiability, coherence, objectivity. Maybe it could also be subsumed under the condition of being "suprising". But on the other hand the word "surprising" is pretty vague and maybe strictly speaking not so helpful.
@@JoelSjogren0 Yes! The element of surprise is very close to experiencing mathematical beauty.
I consider that part of the intuitive aspect. The constructive aspect is in essence the basic method of proof by demonstration.
I find it hard to dismiss real numbers and the continuum without going even further than you're suggesting though...
By that logic I would even call into question the foundations of concepts such as addition and multiplication, as it's hard to justify that any truly commutative operations exist.
At the end of the day it's all just a model, and you should choose the model that motivates you to explore.
(Although, I'm not saying I'd pick real numbers personally either)
Time to orient ourselves in the right direction, What would that direction be? And where do you suggest we start?
See his Math Foundations playlist.
Please start watching the videos on this channel and on wild egg maths
Mathematics is what it is. It doesn't care what you want it to be. I don't know, does mathematics have its own existence separate from our physical realm?
Computational mathematics on a finite rational machine is interesting, but it's just an approximation no matter how powerful these classical machines become.
Yes, Godel has shown us how flawed all of the mathematical frameworks are that's probably why he was a Platonist.
I'm not holding my breath for results from AI. 🙂
The other problem with AI and mathematics is symbolism. What if the AI manipulates symbols that aren't printable. They could be as unintelligible to us as the bong tones of an old 300 baud telephone modem negotiating a connection with another modem. One AI talks to another over the internet and all we hear is noise. Are they playing GO or are they designing a machine to remove that corrosive oxygen from the atmosphere. They don't need it to breathe like we do, but since the ozone layer shields ground dwellers from solar radiation, the machines need it for shielding. But, shielding can be done by other means while we need oxygen to exist.
You should check out Type Thoery and constructive mathematics
I agree with your points. Surprisingly many people have an intuition that computation is way too limited to explain what mathematicians are really doing, maybe because the kinds of computations they can *imagine* seem too simple. But when you look beyond terms and what (most) mathematicians claim to be doing and just observe, it will come down to computation every time, essentially "playing" formal language games with axioms.
Definitely. Once you have a taste for it, every magicaly axiomatic math concept just comes out very obviously as some computational concept, just with all your 'exact' solutions revealed to be approximate, all your 'set theories' revealed to be formal languages (with no referent other than subsets of itself), and all your 'hierarchies of infinities' revealed to be variations on the Halting problem.
I find intuitive the notion that as x gets bigger and bigger, y gets closer and closer to some value L-- and it's usually clear from a graph, and it seems reasonable to me in that case to infer that the ultimate value is exactly L. I can see that the exact L could be considered impure because there is no rigorous proof, but ...
The problem is that the lesson of pure maths has often been that our intuitions deceive us. See: Galois theory and the absence of a quintic formula
Also consider a fundamental theoretical question in quantum mechanics that is an applied mathematical problem carrying the baggage of pure maths for which we have no intuition: does a quantum object, say an electron in its orbital around a proton, have a finite or infinite number of states that it could hypothetically occupy? The answer has profound physical implications.
As 𝑥 gets bigger and bigger, 𝑦 gets closer and closer to ... π? 𝑒? The problem is there is no "value" π or 𝑒. There is no exactly π or exactly 𝑒.
Yes 𝑦 gets more and more tightly bounded, but as far as exactness it remains no closer to "infinite" exactness than it ever was.
I left set theory type math 50 years ago and never looked back. There are so many important problems in Medical, Defense and Nature that needs to be solved. Don't waste your talent on the basement of Math.
Bro if you're the type who wants to work on something practical then why are you working in pure maths in the first place.
Like I have nothing wrong with wanting to work on something that affects the real world but I'm just wondering
Infinity is where 2 parallel lined intersect at right angles.
Dear Professor Wildberger. It's a pleasure to see you back in great shape.
"The machines are coming."
Just as soon as they can figure out how many fingers are on our hands. No, I don't think the machines are coming anytime soon. That's just a wet dream of the managerial class. There's something fundamentally different with the way they learn, but also broken. You've never seen anyone draw an absolutely picture perfect representation except for the three joints in their elbows, or shutdown at the phrase "solid gold magikarp."
Is AI impressive? Yes, in a way. Is it coming? No. Not yet. But every time someone says so, an investor gets their wings... at least until the bubble bursts and we all see that the check doesn't clear.
"Wet dream of the managerial class"? Well maybe. But another equally apt description might be "Nightmare of the working people".
As student of Philosphy and lover of math, I consider power of abstraction in pure mathematics is like theoritical structure available to be applied in new emerging realities.
Consider the "proof" that there are infinitely many prime numbers. We know we can't start with an infinite set of prime numbers because if you have the thing you're trying to prove, then you don't need the proof. So we start with a finite set of known prime numbers. We multiply them all together and add one. The result must have a prime factor (which may be itself) that is not in our set because it's larger than the product of all the ones we have, and the fundamental theorem of arithmetic tells us it has a prime factorization. We find that new prime factor and add it to our set, which is now one larger than when we started. Rinse and repeat. The problem is obvious. You start with a finite set of prime numbers and end with a finite set of prime numbers. No matter how many times you literate, EVEN FOREVER, you will always have a finite set. At no point can a finite set ever transition to an infinite set. It is, in fact, a proof that there is no largest finite set of prime numbers (assuming you have the ability to keep adding more). Any conclusion that results in an infinite set of prime numbers must assume the thing into existence without the proof.
Yes I believe you have stated that well. Thanks
But you could see the proof as showing there is no largest prime number. You could take that as what „infinitely many prime numbers“ means. In such an approach you would not need to assume infinity as a separate concept. You could do finite mathematics but still (indirectly) talk about infinity.
@@ostihpem Infinite vs. unbounded. Yes the natural numbers, for example, proceed without end but that is not that same as saying "the set of natural numbers of infinite." The former is a process (one that carries on as long as one cares to) and the latter purports to be an object ("infinite set").
@@ThePallidor But you can also define an infinite set := not a finite set. So you can work with the set of infinite prime numbers as the pure complement of all sets of finitely many prime numbers. That is different to defining an infinite set like in the infinity axiom of ZFC but still it is infinity in the full sense. The difference is the perspective: one is a god-like-from-above-perspective, the other is just our inner-perspective. Like in astrophysics they just describe the universe from within, if they‘d adopt modern math they‘d construct some meta-realm on which our universe would sit. That shows that maybe my proposal is better because more lean.
@@ostihpem This becomes a bit tricky because there is a largest set of prime numbers currently known to humanity, and a largest prime number known to humanity. That list is expanding, but there will come a point in time when humans have found the largest prime number they will ever find. Personally, I don't see any reason why the process wouldn't continue to work given more resources, so I am ok with saying that it's a proof that there is no largest prime number, in the abstract. Practically speaking there is a largest prime number, and there will eventually be a largest prime number for all eternity. Regardless, the proof itself never produces anything but finite sets of prime numbers. What does infinity even mean in a context like this? If you imagine an object that can continue to grow in size forever, that doesn't mean that it is infinite in size. It will always take up a finite amount of space. Actual/Completed/Axiom of infinity is probably the biggest mistake in the history of math.
The diagonal argument in non-classical logic. 0.00011010b101001... is a real number. The 'b' is the 3rd digit, which means "both zero and one". Alternatively you can write 'n' which means neither true nor false, neither 0 nor 1. So you can make a list of all the reals. Just some of the digits in those numbers are unknowable. It's more like quantum mechanics (which is reality, or so I've been told)
Quantum mechanics is unfortunately complete fiction.
I agree on some points but essentially what you are saying is that abstraction in pure mathes can be cancerous in a sense ? Well it had proved to be very useful (if you think about modern AG ) but maybe now abstraction is done for the sake of abstraction and it might not lead to interesting concrete results but still there a lot of young mathematicians who like maths for its abstraction and for some it is the main reason they got into maths field. But I agree there must be at least some kind of balance between concrete and abstract
I don’t have any problem with abstraction. It is faulty logic that I object to.
I am curious on where to start. Why not an encyclopedia with computable constructions for mathematical objects? Like OEIS but for computables. Might encounter challenges.
I think the end is going to happen and an Alphazero-like mathematician(s) will ultimately rebuild it all. Refactor the entire bloated codebase.
There are an infinite number of reasons why this argument fails, but this margin is too short to contain it, and they wouldn't be accepted by Dr. Wildberger either way since it's a neverending list :^)
One totally correct and clear argument, would probably suffice
If math can define zero into existence (and why we use + and - operators in physics) all heck got loose.
Why don't just accept infinity like any other unreal thing in math like a "number" for ex. We just plaing game(games) here...
Category Theory #RM3 --- there is certainly an implicit bias in modern maths towards binary logic. Quite a lot of paradoxes (The Liar, relevance fallacies in general) are solved when you stop forcing your decisions to be binary. Truth must be replaced with Validity
What? Transform the mathematics bais in less than 10 years? Did i get it wrong?
Anyway, if some AI applications develop some mathematical rules, it won't be in human comprehensive language and they will be confined in their own application. But, maybe applications will be able to share their knowledge. Anyway this won't stop me sleeping.
Start with 1+1. It only equals 2 if you are talking about organizing these 1's into a set. You are not adding them together. You are adding them into one set.
Take one vector and add it to another vector. You get one vector.
Look down at your hands and figure out why humans use a base 10 number system.
There is no 2. It's a made up branch of mathematics to organize like concepts when in reality it is far more difficult to find exact objects and organize them together because of atomic variation. But for some reason, people ignore this fact and play make believe.
What does it mean to add one thing regardless of it's self to another? Crickets....
Vector calculus gives us a way to do this. 1 vector plus 1 vector equals 1 vector. This does not have to be a set. Notice that the 1's are vector notation rotated 90°.
In any case, 1+1 is taught to everyone to be equal to 2 when we do not know what the 1's are and how exactly they were added together.
So, I agree at a fundamental level. And you can not spell fundamental without fun and mental.😊
First of all, opposition is uncalled for. I agree that the foundations are, not only shaky, but often pure fluff and fantasy. But most of what has been built shows every sign of true discovery. And besides, people are very happy with it. Inspired and motivated by it.
So, all that is needed is more people being enthusiastic about re-working the foundation, regardless of the modern house of cards spiralling off. Perhaps such two branches hook up one day. And perhaps much of what is now shaky may be salvageable. Or perhaps not ... Either way, modern analysis and its branches shouldn't be seen as a threat. Firm foundations should be its own forrest fire.
Secondly, Aİ is no threat, except as a monkey-type imposter. It is dumb as f. It only imitates. And it has no real concept of logic or concern, only imitated and empty versions of them.
I'll leave your rant unperturbed aside from the note that trigonometric functions are not restricted to 'pure math' and neither is the concept of infinity. Both are pivotal in the history of influence of applied math. Attempts to shrug off their usefulness is ignorance to science itself. You can argue about the philosophy of those functions and concepts and extension of study past what has already been established, but not about their usefulness.
Norman wildberger probably hasn't studied the theory of modular forms deeply enough. It's beautiful! Lots of combinatorial ideas need analysis to prove them!
I have studied it well enough to know that the notion of a modular form is highly questionable.
@@njwildberger Modular forms have led to a lot of good combinatorial fruit, like formulas for the number of ways a natural number can be written as the sum of 2n squares, or congruences in the partition function. Let me know if you have a better way of proving those things.
There are a lot of really interesting aspects to the theory of modular forms. Just because a theory develops in interesting directions does not mean that it is a correct theory.
As a natural scientist I agree. Modern pure math is mostly done for the purpose of itself. That is way different from the time of, say the 19th century, where real problems were of the highest importance to be studied and solved. The quest for the foundations, set theory, Hilbert, Goedel, Bourbaki, Grothendieck, to name just a few, set the stage for the change to the extreme abstraction. But where is this abstraction really needed or helpful? Why are people proud to demand and enforce it, even when it hinders progress? Where is the pragmatic middle ground?
7:13 is an incredibly sloppy definition of a function.
That might be the case, but the point wasn't to define what a function is, but to give a meaning to it's value at a particular point, not to speak of it's value at an arbitrary point.
Hello Mr. Wildberger- I know of a certain philosopher who can validate the existence of infinity, and is willing to debate the subject, via a livestream. Would you be interested in this opportunity? I can get in contact with him, if the answer is yes.
yo bro mabye try his email next.
Who?
@@MrSlugmuffin If you have his email address, I would definitely send him one. (I'm not using n.wildberger@unsw.edu.a because he retired from that university and I assume he no longer uses that address.)
Sorry I am too busy at the moment
@@njwildberger Noted. The offer remains open, if you wish to take it up at some point in the future.
Sociology is just good English
Analysis does not require “performing an infinite number of tasks”. That’s a strawman argument.
Then please explain how many arithmetical operations or steps you or your computer require to compute e=1+1+1/2!+1/3!+…
@@njwildberger no one claims that the series can be summed up by performing an infinite number of addition operations. Nonetheless, we can coherently say that the number e, defined as the limit of the partial sums of that series, exists. And whenever we need approximations, we can perform a finite number of computations to get them. What’s the problem?
@3:00 Computation cannot give you _meaning._ If you seriously want meaning from your mathematics you are talking about meta-mathematics and the whole thorny problem of subjective consciousness, but that quest defeats your purpose of avoiding philosophy. So I am afraid you are not going to succeed. Taking a Finitist approach is fine for what it does, giving simpler concrete structure, but it is not going to provide _meaning._ There is no sentient meaning in material structure, since there is no mind there to be found.
It's slightly quixotic to complain that 'pure' mathematics has somehow become estranged from the reality of everyday life. "Why is this job taking so long?" "We're waiting for a full evaluation of pi" "Oh, right - get back to me". Pure Maths is 'Ivory Tower' because it has always wanted to be - towering edifices of infinitely many ever 'more infinite' infinities (What Quine described as 'Recreational Mathematics') - these have nothing whatsoever to do with everyday life, and precious little to do with working mathematicians either. A great deal of the subject seems to be pretty 'incestuous' - theorems whose only value is facilitating the generation of even more abstruse theorems, each iteration narrowing the audience to a smaller and smaller set (see what I did there?) of practitioners. As usual, as is quite rightly pointed out in this video, Infinity is the fly in the ointment.
I guess on the upside it still gives some people practice at hard-core reasoning and creative proofs. But it's a shame they won't just apply all that ingenuity to something grounded in computational theory
I completely disagree. The notion of infinite iterability and absolute, ideal symmetry is the very essence of mathematics. Finding the solution of some mundane sum that we can simply ask to a computer about is not mathematics, that is mere calculation.
Expressions like Euler's formula, the sin and cosin, infinite sequences approximating pi... These are the heart of what maths is about.
Mathematics _is_ calculation, in one way or the other. And "infinite iterability" is not a thing which can be done in reality. Wildberger ist right, as far as I can see.
We don't study infinite sets to actually calculate the values we're talking about. We need to talk about the theoretical limits of mathematics, because we don't know the exact practical limits. It's easier to work with the function that all approximations of cosine will converge to, than with many different approximations of cosine. We need the concept of cosine in order to know we are all talking about the same thing even though we calculated slightly different values.
That's why there is no problem with uncountably infinite sets for themselves.
Pure mathematics is usually not about the specific things inside the sets they talk about, it's about the sets.
8:35 so since cos7 is not rational then cosx is not a function, basically that’s the argument 🥸
Most mathematicians are instrumentalists - realists on weekdays and nominalists on Sundays. They "believe" in the reality of the abstract objects their theories depend upon, but when pressed they have no actual ontological commitment to the existence of abstract entities, even numbers. Wildberger is an instrumentalist who wants to reduce mathematics to an ordinary practical science or discipline, like chemistry or geology. This is nothing new. The solution proposed by philosophers (most notably, Quine) has been to adopt an instrumentalist approach to any successful scientific theory attempting to describe the nature and action of quasi-observables. The instrumentalist regards this unverifiable portion of the overall theory as a useful fiction which satisfies the need for explanation and prediction without making any ontological commitment to the objects the theory describes. By modeling the cognitive processes of knowledge and belief in terms of biological mechanisms, which are themselves grounded in physical laws, epistemology itself reduces to a study of the effects of biological determinism upon these human cognitive processes. Since, under this model, there is no knowable truth superior to what human beings presently know, the vexing questions of epistemology become irrelevant and simply evaporate. Rejecting the a priori /a posteriori Kantian distinction, Quine thus advocates an “ends oriented” approach to scientific advancement. The idea is that we already know what the ends of science are because these are biologically determined for our species through evolutionary processes. These ends pertain to the fulfillment of mankind through the mastery, domination, and control of nature, and therefore, the singular purpose of science is to bring about that end.
Therefore, what constitutes a “true” scientific theory is simply one that advances this purpose most efficiently. Abandoning its pristine, enlightened image as a search for truth, “science” thus becomes a very specialized, pragmatic metaphysics consisting of a vacuous epistemology and an ontology committed to nothing beyond the objects of the physical senses. Interestingly, this is simply a restatement of the maxim “man is the measure of all things,” uttered 2500 years ago by Protagoras. What a dull, dreary, lifeless and pointless thing Mathematics would become if Wildberger and his ilk had their way.
The problem with "useful fictions" is they limit the range of applicability. The reason for getting the foundations right (epistemology) is because eventually a house built on sand can be built no further.
Instrumentalism doesn't require ignoring the foundations, but rather presents the possibility of tempting shortcuts via fudged foundations or a more sustainable and ultimately powerful field by taking the time to get the foundation right at the start.
I don't perceive Wildberger as an instrumentalist. I think he's ultimately a (closet) Intuitionist, driven by Truth and Beauty for the sake of Truth and Beauty. That could be also just my own projection, but I think the projection is supported by Wildbergers Law of Logical Honesty, and the book title "Divine proportions". Constructivism at large is committed to constructive methods because of commitment to truth seeking, not because of pragmatic concerns of instrumentalism. If experiences of beauty and wonder were not present in these lectures, I would not be following.
I see no empirical evidence for mathematical intuition being grounded in the generally rather bad math of "physical laws" and "biological mechanisms". On the contrary, there are many examples of intuitively reached a priori mathematical verbalizations, which some a posteriori day become instrumentalized by theoretical physics and then pragmatic engineering motivations of showing of their practical wonders of math magic. Not always with great wisdom (Atom bombs etc.) but anyhow.
But we really don’t know what would happen if we started thinking clearly. Except that the subject would make more sense and consequently be much easier to learn and appreciate. And making logical sense is not something to be sneezed at.
The fundamental ideas in mathematics are motivated by real life contexts, such as numbers, infinity (something/a process that never ends) and so on. We prove things based off made-up definitions. Why is -1x-1=1? Take a look at the definitions... The definitions are not "fixed" they can be whatever we want, and usually we only care about ones that come out to be useful in a setting.
Nobody is claiming the axioms (assumptions) are "right", they just seem to be right. It's like anything, how do you know your not in a dream, and everything is your subconscious? You don't, but you just assume that to be false because it seems right. Take a look at the Munchauseen Trilemma.
Much, probably most, of modern pure mathematics is essentially a new religion.
Unfortunately, it isn't the only new religion haunting us.
Critical as I am of mathematicians using supposed words, like "infinite", much like older religionists use "blessed", that is without much regard for meaning, I have to acknowledge that as mere monks and nuns they are harmless outside their own discipline.
My first maths teacher at grammar school was an old finitist presbyter, who greatly valued computation. He retired and was succeeded by a young presbyter in love with structures and formalism and little interest in computationnal results. The subject became boring, but never worse than that. Mathematics, even when it goes wrong, is wonderfully civilized.
Let us be glad that this discipline has only spawned abbots, abbesses and their followers, at least keeping out of politics.
I don't agree at all. Even if irrational numbers, the real line, topological vector space, manifolds, quaternions or complex numbers don't "exist" as infinitely precide and detailed objects and distances in the real physical world, doesn't mean that they aren't of TREMENDOUS use in physics. Knowledge and precise definitions are an end in and of itself, not just a mean, though they are certainly a means to an ever-shifting goal, that being more knowledge and more understanding of how the world works, whether it would be the physical world, that is real in the measurable sense, or whether it would be the abstract, etherial world, that is nevertheless "true" and consistent. You cannot do the applied math only. You need analysis, topology and measure theory in order to understand dynamical systems and statistics well and you need those things to understand the real world, even if most real numbers, as well as topologies and sigma algebras "don't exist" in the physical world.
I would agree that analysis, topology and measure theory, for example, are highly useful in the real applied world. The point I am making is that the current versions of these theories that pure mathematicians study are not connected directly to the real world of the applications that the applied mathematicians require. Is it logically necessary to have real numbers just to be able to compute trajectories of a space probe around Jupiter? No it is not: those calculations are always done with finite decimals and approximations and basically the arithmetic that our computers are capable of. There is a big divide between the practical world of computational reality that drives modern technology and the wishful dreaming of us pure mathematicians with our heads in the clouds about "infinite this" and "infinite that".
The situation in physics is like if it were said that there are these ghosts that pull the car toward the earth with a certain number of newtons of force.
Then someone comes along and says, "The whole ghosts and demons thing isn't necessary, adds baggage, and actually ends up limiting how far we can build out these useful tools. Let's start again with a non-superstitious foundation, instead a rigorous one with explicit definitions for all key concepts."
The ghost-and-demon system was very useful but that doesn't mean it was the optimal tool for the job; and really, practical applications never actual make any direct use of or reference to ghosts or demons anyway, so the most that can be said is that these are useful fictions. Or more like useful nonsense. But it's better to have foundations that are at least not nonsense.
Irrational numbers ARE grounded in physicality, it's just that the current mathematical model is wrong. Infinity becomes necessary when you have a solid line with a small gap at the end that never gets filled. That gap is the empty space inside all of matter. A better model is having the empty space dispersed throughout the line. Then, it becomes clear that √2 and Pi are actually ratios of physical matter to empty space. This could also be described as the ratio of a unit of energy to the space around it. The problem for everybody is that this ratio is a base-12 ratio. The big wrong assumption is that there is no difference between base-10 and base-12 math. This is the big discovery everybody is looking for/waiting for, so I'll explain it right here - the difference lies in the size of the decimals. There is a geometric pattern that exists on the base-12 Cartesian Plane that can be created just by connecting the dots, or the lattice points, and is therefore very precise, and yet it is invisible on the base-10 Cartesian Plane. This pattern has to do with the geometry of the circle, and it enables the creation of a diagram which perfectly divides the circle into 360 degrees, something I was surprised to discover doesn't exist in base-10...and so much more. Anyway, you heard it here first.
I appreciate where you're coming from. Pure maths is becoming esoteric. But applied maths still has not broken through the nonlinear space, and nature is overwhelmingly nonlinear. Pure maths can flourish by integrating itself with other fields where non-exact features abound, such as biology, economics, cognitive sciences etc. AI is no threat here. It's imagination that counts.