I'm Daniel Rubin, professional mathematician and man of many interests.
I host a conversation show in which I talk to interesting people about the things I care about: math, science, movies (and the arts generally), education, history, philosophy, standup, politics, and how to make it in life.
I also have have math lecture videos and videos with general opinions and advice on math.
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I disagree with your statement that the definition of Cauchy sequence doesn't depend on the axiom of choice; it does. And the axiom of choice is the source of all 'mathematical evil' so to say. If there is a contradiction in the foundations of mathematics, in the sense of Voevodsky (who suspected there was, and therefore he initiated his program of computer proofs), it must be in the AoC.
Mathematicians duelling at 40 paces and Wildberger was left standing. Love it!
Does anyone know a course that uses this book so I can find accompanying problem sets? I've looked everywhere.
Hi ! I want to know does springer textbooks have solutions and answer in COMPLEX ANALYSIS AND WHAT THE BEST TEXTBOOKS FOR MORE PRACTICAL APPLIED COMPLEX ANALYSIS THANK YOU
Richard!!! How fun to see you and hear you in this interview. Grateful for your wisdom and artistry - and for how you helped me at a pivotal point in my life. Thanks, Daniel, for having him on and for asking good questions.
Very well made in lots of ways and I tried for a long time to make time for this video, enjoyed the first ten minutes or so, but then you start making all kinds of insane leaps with absolutely 0 explanation as if they were common sense. After about the 20th time you did this, halfway through the video, I just had to give up. I can't deduce the foundations of modern mathematics independently by hand every 30 seconds. If these details are not important, acknowledge it, and if they are assumed background knowledge, say so. But this was a real let down, it seems common on math youtube but lots of creators also explain things very nicely, even assuming some background.
That is just a waste of time- just words joggling. When people have nothing to say they start to pretend they know what they are talking about. - There is a difference between how people are naming/marking/some points on the infinite continuum and some human psychological problems with some machines called computers. Mathematics has very little to do with computers or their limitations. or human limitations to symbolically represent some particular number. that so called "Proffessor" is mixing "flies with putty".
Anyone here studied Spivak as a first calculus book? I was considering doing that (well, I did some calc at school, 20 years ago, which I've now mostly forgotten). Most other folks recommend Stewart or similar first. Which is actually how I ended up here - I was looking for recommendations for books that might get me up to speed quickly. Sounds like Prof Rubin thinks it's possible to jump straight in to Spivak? I do prefer that idea, just because the idea of re-learning something I already studied at school seems less appealing that learning something new. (Or rather, learning the same thing but in much more depth.) The exercises in Spivak are notoriously difficult.
Norman seems to be completely ignorant of the fact that we have an entire branch of analysis (numerical analysis) that is interested in computing the things that are defined using various notions of infinities in a finite number of steps.
19:20 Joel: “…there’s only one way to put a finite set in order…” This is misleading. I know you mentioned order type but that’s an equivalence relation that’s obvious to mathematicians in certain areas but isn’t very clear for lay readers/viewers/listeners. I think the use of games would help, and for many, you can describe simpler games (than chess) that can help intuitively capture the notion of equivalent well orderings on a finite set of natural numbers. I’m sure that anyone who can imagine infinite chess can also imagine “arbitrarily large” games that are much simpler than chess, in which an important fundamental rule involves a specific well-ordering of nodes of a graph of a specific kind. Calling it a network rather than a graph would perhaps help.
The universe of hereditarily finite sets provides a way to discuss existence of arbitrarily large finite numbers without accepting the axiom of infinity, so I think that’s the clarification you seek at 28:02.
At 24:42 you use the word “silly”. I’m afraid that I must not concur, as I think that that’s as disrespectful to his philosophy as is his claim that acceptance it use of the axiom of infinity is a” religion “. I suggest avoiding that kind of characterization of his position.
Here (@ 17:57), you’re disagreeing with Norman’s disavowal of the “added value” of post-19 th century analysis. I think that your argument is stated as overly subjective - opinion. It wiles benefit from specific examples, and I’m convinced that you can find many. For example, I’m convinced that there are useful algorithms in areas of mathematics to which Norman does not object which were developed specifically because analysts used mainstream mathematics to lead them to understand and create and implement such algorithms. Proving to Norman that an example you find has that property may be more difficult than you’d like, but providing more evidentiary support for your argument among your viewers, including the proverbial “choir” is probably even more important to you than getting the “blessings” of Norman or any of his followers.
Here (@ 17:57), you’re disagreeing with Norman’s disavowal of the “added value” of post-19 th century analysis. I think that your argument is stated as overly subjective - opinion. It wiles benefit from specific examples, and I’m convinced that you can find many. For example, I’m convinced that there are useful algorithms in areas of mathematics to which Norman does not object which were developed specifically because analysts used mainstream mathematics to lead them to understand and create and implement such algorithms. Proving to Norman that an example you find has that property may be more difficult than you’d like, but providing more evidentiary support for your argument among your viewers, including the proverbial “choir” is probably even more important to you than getting the “blessings” of Norman or any of his followers.
It’s important to have a good foundation, so I disagree here with you (16:56). However, one must have some notion of what constitutes “good foundations”, and this is where I differ with the implicit notion of “good foundations” to which Norman ascribes. His view is subject to a kind of “cascade of collapse of the acceptability”, which I mentioned in a comment on your video of the debate itself. The criterion for “goodness” to which I ascribe includes a “sufficiency” portion and a “necessary” portion. I don’t require a metaphysical component but I consider that a sideline worthy of discussion in the philosophy of mathematics and the philosophy of science. The “cascade of collapses of acceptability” that I ascribe to Norman’s philosophy is that his objections led naturally, and I believe, inexorably, to a nihilistic philosophy of numbers, in which we may as well deny the existence of every natural number. This is a very strongly anti-platonic view of mathematics, and one who subscribes to such a view may as well also posit that nothing exists. I digress. Your view - that foundations are almost irrelevant - fails in the same way that the lack of secure foundations failed in the 1800s, in that it allows contradictions in our foundations, and hence leads to mathematics that is devoid of any certainty of validity at all. I think then that it’s clear that I’ve taken a middle road between these, in that we need foundations that at the very least are not provably inconsistent (the “necessary” condition for a philosophical stance in mathematics), but are, from a pragmatic standpoint, at least provide a way to develop the solutions of problems that may be formulated in our mathematical language. Inherently, this allows for at least the existence of arbitrarily large positive integers, if we are to consider the problems of analysis as relevant to either pure or applied mathematics. …
@55:35 or so, Norman objects to adding 3 irrational numbers, but when he points out that it’s analogous to asking a grade 6 student to add three fractions. This leads to the following, which is my “domino theory” approach to Norman’s philosophical objections to irrational numbers: I claim that this philosophical objection reduces to an objection to the validity of the equation 0+0=0, since his objection is metaphysical, and the formalist objection to the Platonic view of the metaphysics of mathematics already objects to “actual” existence of any mathematical objects, including the number zero. If you actually object to analysis as a form of religion, then you might as well object to all of mathematics as a form of religion. If you want to object to mathematical concepts using that metaphysical concern as your foundation of objecting to things, then why do you call yourself a mathematician? Is that not a form of hypocrisy?
@about 53:00, Norman: “…as we try to figure out what’s really going on here…” Norman, you belie your claims (implicit or explicit) that you’re not wanting to go down philosophical rabbit holes with this because you’re here making it clear that you’re really focused on the metaphysical issues around mathematical concepts. …
@about 53:00, Norman: “…as we try to figure out what’s really going on here…” Norman, you belie your claims (implicit or explicit) that you’re not wanting to go down philosophical rabbit holes with this because you’re here making it clear that you’re really focused on the metaphysical issues around mathematical concepts. …
@31:19, Again, for the nth time, where n is larger than 3, Norman has indicated that he’s limiting his philosophy of mathematics to computer science and algorithmic foundations, which didn’t really exist as a philosophical consideration; this is in spite of his frequent claim that he’s wanting to avoid a philosophical discussion. For example, he clearly wants to deny that what he’s calling “choice sequences” of natural numbers exist or that such a restriction is a philosophical issue. Daniel has trouble addressing this, and I think the reason is that he’s not addressed these kinds of questions in holds own mind previously. It helps me to realize that I’m a human being in a world that has many things in it that I’ve never observed. When I do observe one of those things, it’s extremely unlikely that I’ll be able to view it as having been “generated” using an “algorithm” that I know or will either know or understand very soon after observing it. I cannot deny the existence of such a thing at the moment that I first observe it; that may even be dangerous. To live in this actual universe, I am compelled to be able to imagine certain things as being possible by virtue of not having introduced into my system of thought an obvious contradiction. Mathematically, this leads me to accept as potentially existing, arbitrarily large positive integers, and sets of them, and
@about 19:30, Norman states that pure mathematics didn’t develop until the 19th century, but I disagree. He’s ignoring the fact that recreational mathematics is actually a part of pure mathematics, and it has been around almost as long as applied mathematics, notwithstanding the fact that many questions in recreational mathematics have a “flavor “ of being related to or derived from certain specific applications, but with some aspect of entertainment injected, often via a modification of hypotheses to something that is motivated by a philosophical conversation about what we can imagine, instead of what we actually see and touch and use.
@about 9:03, Norman introduces “a slightly more difficult” problem, but it’s actually too difficult to illustrate his concerns. He goes from considering an example that’s a linear equation with integer coefficients, to a cubic equation, x^3=15. A simple example is the following: Let e denote the smallest positive rational number representable in my desktop computer using Mathematica today (2024-06-09), and consider the equation 2x=e. Then no solution to this equation exists in the set of numbers that I can compute today using Mathematica in my desktop computer. This even more directly represents Norman’s viewpoint as he presents it in his videos, because he points out that he actually believes there is a largest positive integer. Here’s an ultra-finitist. Daniel, I would suggest that you re-watch Norman’s videos very carefully if you wish to enjoin him in a new “battle” or debate. …
@about 9:03, Norman introduces “a slightly more difficult” problem, but it’s actually too difficult to illustrate his concerns. He goes from considering an example that’s a linear equation with integer coefficients, to a cubic equation, x^3=15. A simple example is the following: Let e denote the smallest positive rational number representable in my desktop computer using Mathematica today (2024-06-09), and consider the equation 2x=e. Then no solution to this equation exists in the set of numbers that I can compute today using Mathematica in my desktop computer. This even more directly represents Norman’s viewpoint as he presents it in his videos, because he points out that he actually believes there is a largest positive integer. Here’s an ultra-finitist. Daniel, I would suggest that you re-watch Norman’s videos very carefully if you wish to enjoin him in a new “battle” or debate. …
Fun fact: Robert E. Greene taught Russel Crowe to play the violin for the movie Master and Commander.
That's pretty cool! (Great movie, by the way.)
@17:00 misguided young sprite? Foundations are important because we don't want houses built on quicksand. The issue is that most people trust classical mathematics, for good reason, no glaring inconsistencies. So "it's all right Jack" for applied mathematics. But the thing is, we other weirdos tend to look to the future of civilization when we ourselves will long since be dead, and yet still worry. We don't want a Trisolarian invasion to catch our descendants with their pants down when some strange esoteric piece of obscure mathematics turns out to be wrong because of an inconsistency in the foundations. I joke. But I am half serious. I noticed on your channel, too often for comfort, that some of your talk is individualistic. Things don't matter _to _*_you._* That's fine, but other people exist.
@1:16:00 for your philosophical question, I think a good answer is that a Set is a Many that can be thought of as a One. (I read this first in Rudy Rucker's wonderful little book.) This is our ordinary intuitive concept that the mathematical notion of "a set" captures. But like Hamkins would probably say, that's just one of many valid responses. Mathematics is useful in part because it is re-useable.
@4:00 but it is _not_ just "philosophical". If NJW is serious in rejecting a much of mathematics it should only be because it is inconsistent. There is a proof his side is "proper" mathematics if indeed it is the only proper math, which is to show anything more is inconsistent. You don't just assume there is no proof of inconsistency of a system, say of ZFC. That's getting Gödel the wrong way around, Le Chien.
@2:50 yes. In science and mathematics having to ""deal with a problem" is a good thing, not a bad thing. At some points NJW get's close to admitting this, and seems to suggest that the issue is we need a firmer foundation for transfinite sets and defining ℝ in a nicer way, which is a nice open puzzle. He even suggests more use of "algebra" harking back to Gauss et al, but he seems to not recognize Category Theory is a firmer foundation he might seek. At other times he comes across as an ideologue, like Wolfram and others, which I find distasteful. Although _de gustibus non est disputandum_ I still find _such_ ideology to be a sign of knuckle-headedness
Nö number exist
q/p=1/r is the point where I lost you
Hi ! What the best calculus practical and problem solving critical thinking mathematics textbooks Calculus concepts 3 edition by Stewart Calculus 3 or 5 edition, Stewart Calculus 9 edition, Larson Calculus 12 edition, Thomas Thank you
All these books have no solution to the exercises which is so frustrating.
@1:03:00 you can perfomr transfinite arithmetic in some software. That fact you cannot do more is because no one's yet written the general code, but they have not written general code even for a tiny subset of ℕ either, so this "write-downable" is ridiculous. Everytime what we can write down increases he is saying mathematics expands. The mathematical expanding universe. But I doubt FRWL would've approved. It's an impoverished view of the word "mathematics" if you you constrain it by "what we can write down". Moreover, if he is saying "what we *_could_* write down" then it is completely undefined, since he has no idea what in the future is possible.
@57:00 this is where Daniel you failed perhaps?, since you dissed Category theory 🤣. NJW is talking there about a transformation (functor let's say) from the category of real number systems, where π + 𝑒 + √2 exists perfectly fine algebraically, but not as "numbers". You do not know what a _number_ means until you define a system. That means some category. Then you can consider the functors, and one of them is a mapping from ℝ to ℚ, the latter which a computer can represent perfectly. That's NJW's proper notion I think. You can give it to him if you have a categorical perspective. To put it another way, the set ℝ is not really a set of numbers. It is a different beast. We call it a number system or set of numbers only because we use language loosely. But there is no concept in all of formal mathematics as "number". There are only certain systems, and ℕ is one of them. ℝ is totally different. I think what NJW might want to say is that the objects in ℕ are "the numbers" and everything else is something else, not a number, and arithmetic on these other things is very differently defined. For good reason, it is treacherous.
@17:00 the ℂ numbers are not "points in a plane". They are algebraic, as he alludes to a minute earlier. Moreover, the setting for ℂ numbers really is the Clifford algebra, the bivectors or even subalgebra. And the proper setting for _that_ in turn is a Category. Such mathematical objects are not "numbers", they might however be homomorphic to a number system. The whole idea of "number" NJW is trying to motivate here is ill-conceived, or impoverished. You can restrict your notion to "what is computational in finite memory" but that's not all there is to play with in _Cantor's paradise._ When you play, the demand is relative consistency. When you eventually find inconsistency it is perfectly fine, you know you were not doing mathematics, which is a good thing to know, and move on from (or do whatever else you want with, like art). Mathematics, the sociology thereof, has to be such art, for if everyone instantly took to perfection when beginning mathematics it'd be impossible to make mistakes and we'd be gods, that's NJW's "religion". You have to be able to be wrong to be right.
@5:55 nonono... real numbers, like π 𝑒, and all those you'll never know, are not _defined_ by decimal expansion. The representations of concepts are not the concepts themselves. The whole point NJW misses (it seems) is that mathematics is not physics and is not computational, if it were it'd be the lesser. It is _because_ we can entertain nonphysical concepts with mathematics that allows us to refine physics (and a lot more besides). Science becomes feeble if we have no ability to entertain what is nonphysical. That's because so much of science is _negativa_ --- it's often what is _not true_ that you know the best.
@15:00 he admits this when he pivots to "algebraic concept ok". But "point on the number line" or "point in the plane" is not a mathematical concept, it's physical, or invoking physical intuition. There is no one true notion of a "continuum".
The following links are broken fyi: Edwards, Galois Theory Akhiezer, Elements of the Theory of Elliptic Functions
Hola, si hay alguien que esté estudiando, investigando estos temas, de habla hispana, y quiera charlar de estos temas, nos contactemos!
I want to known does springer textbooks have answers in them or its better to get a functional analysis textbooks Thank you
I've said the same thing for years -- I'd rather have learned electrodynamics than calc 3 because the former was unmotivated and thus abstract in the wrong way.
Rewritten with greater clarity.
I guess the most difficult thing to wrap your mind around is about whether something can be infinite or not. On the one hand you can't imagine that you can only go up to a certain point and then it stops. On the other hand you also can't imagine that you can go on for forever (either on a smaller and smaller scale or a bigger and bigger scale). It's a real dilemma. My personal opinion is that we always have to deal with finite obects. For example planets, stars, galaxies only contain a finite number of atoms. Then there are finite / discrete energy levels of electrons in atoms, discrete charge etc. Everywhere we look the universe seems to offer only finite / discrete stuff. Does that give us a hint that in the end there is no continuity in general? What bothers me though is the question of time. It's hard to believe that time has a beginning or an end. Because you can always ask: what was before that? what will come after? ... Unless time is just a figment of our thinking and doesn't really exist - or rather a sort of 4th spatial dimension / space-time.
I am a theoretical physicist, therefore most mathematical disputes usually do not bother me unless they are directly relevant to my field. In this instance, however, I quite firmly side with Prof. Wildberger, for at least one very reason: to me mathematics is fundamentally the study of how to count objects. If this premise is accepted, and it might not be for many of us, then I expect, or rather demand mathematics to give me exact solutions. Not approximations. If I were interested in the latter, I would rely upon physics or engineering. It seems to me that most modern mathematics is more concerned with axiomatic thinking than actual hard proofs. When I was a young student, I was taught that mathematics is synonym of exactness, rigour and hard logic. Well, apparently not really. Cauchy sequences, Dedekind cuts, real numbers are artificial constructs to provide us with a framework that 1) we can assume as correct 2) we can forget and move on to other topics. This approach would not be accepted in Physics for instance, as it would dangerously sound more like a religious belief than Science. A divine knowledge bestowed upon us never to be unveiled or questioned. Either you can prove your theory with an experiment, scalable and repeatable, or your theory is a conjecture and so it will be treated. Galilei taught us this method, Feynman elevated it to the highest intellectual peak humanly possible at this given time in our evolution. I however do appreciate the effort made by this channel to bring forth this important discussion, openly and honestly, regardless of your own stand on the matter. Keep up the good work
Re: "demand mathematics to give me exact solutions. Not approximations". As far as I can tell, real analysis offers both? To even talk about an approximation to a number or a function, we need a well-defined notion of the number or function in the first place. Re: "axiomatic thinking than actual hard proofs": what does this mean? Those two go firmly hand in hand.
I don't have anything against people who argue we should stay within a specific level of "axiom-strength" for mathematics, but I feel like saying that we should unilaterally ignore everything beyond purely computable stuff is throwing out a lot of really interesting stuff. Even if you don't believe that non-computable numbers exist, there's still a lot of really beautiful mathematics involved in understanding the relationship between computability and non-computability (this can be seen with the arithmetic hierarchy or how reverse mathematics shows that the major theorems of analysis are perfectly contained in neat rings of increasingly non-constructiveness).
Bruh, 2:22-3:14, life changing.
Goes well with Bressoud's A Radical Approach to Real Analysis. Bressoud starts with Fourier to motivate the development of real analysis.
Bruh. This is phenomenal. Ty ty
He seemed defensive from the start, with a hostile tone, often going on tangents instead of answering questions. You appeared uncomfortable and baffled. When you raised objections, he cut you off and changed the subject. The interview was difficult to listen to. He tries to present a new view of mathematics but is insulting, using ad hominem attacks and calling things religious. He comes off as hostile and emotionally driven, resembling a modern conspiracy theorist rather than a serious academic.
Take a simple function over the Rational numbers, say y=x². How many members in the codomain? If you don't say 'infinity', you deluded. To deny this would entail that Q doesn't exist. Q is an infinite set. Wildberger denies it exists, or does he? He often says, 'Rational Numbers' without qualification. Finitism is a busted flush.
It is not necessarily a matter of analyzing infinities. It is more so actually a case of potentiating & encompassing rolling infinities, in our factitious systematizations. Its not about analyzing infinities. Its accepting that conducing them is in our best interests. As a true wholistic systematization innately manifests (particularized) infinities.
Thanks for participating in very important and very nuanced discussion. I share many similar views with Norman, and also reject the notion of "completed infinite process" as inherentry contradictory. As we are discussing pure mathematics, arguments from utility don't apply, as they belong to the pragmatic truth theory of applied mathematics. I also reject the theological notion of "timeless being" of Platonia as an arbitrary and unnecessary axiom. So, I'm left with Intuitionism and coherence theory of truth, for which I argue as coherent ontology of mathematics and mathematical truth. The foundational crisis of mathematics has been going on unsolved since the Brouwer-Hilbert controvercy. The crisis was swiped under the rug by Acacemic sociology (Hilbert's cancel culture against Brouwer etc.), not solved by power of argumentation. Greek pure geometry made clear distinction between pure (no neusis) and applied (neusis allowed). We are basically continuing the sama debate of how neusis relates to foundations of mathematics, as was also Berkeleys criticism of infinitesimals in his "Against Analysis". Norman does at least in practice accept many kinds of infinities, including the kind that 1/0 refers to, but of course I can talk only for myself. I agree that the Stern-Brocot construct (SB) is very foundational, and IMHO the best candidate offer a coherent solution to our foundational problems. AFAIK standard analysis is generally committed to field arithmetics, and thus rejects 1/0 and consequently also SB as a form of analysis. A very crucial foundational problem is that we have no coherent theory nor philosophical consensus over what does "number" mean. I take conservative position is that regard, and define numbers as product of tally operations. Instead of numbers, I suggest that we call operators operators and algorithms algorithms. Much confusion can be clarified this way, and instead of "algebraic numbers" we can speak of algebraic algorithms. With the inclusion of origami in the tool pack of classical constructive geometry, the classical meaning of "transcendental number" has become outdated, and we need another term for algorithms which can't be solved by the method of straight edge, compass and origami. Naturally, this definition excludes "real numbers", especially "non-computable real numbers" which are not algebraic algorithms and don't have any closed form demonstration, which can function as in input for computation. What cannot be named, cannot have linguistic existence, and if mathematics is a "language game", as Formalists claim, then playing language games with what can't be named is inconsistent and dishonest wrong playing. If somebody claims that "real numbers form a field", then according to basic syllogism it is necessary to prove that claim by demonstrating that each and every real number can participate in arithmetic operations. If that requirement is not accepted, then we lose the ability to falsify a conjecture by a counterexample, and end up with the situation that mathematics as a whole is a logical Explosion where anything goes. I don't accept that as a foundationally coherent position, but of course heuristically we are free to assume what ever we like. Formalist method of arbitrary language games, assumining "finite infinities" and what ever can be heuristics, but not claim the status of mathematical truth and coherent foundation. *** Holding on the principle that mathematical coherence theory of truth includes also empirical truth conditions, the question arises, what is the temporal ontology of mathematics, if we reject Platonist ontology of timeless being? The bi-directional T-symmetry of quantum is already as mathematical as mathematics gets, so most certainly we are not limited only to classical consecutive unilinear time. The strong both mathematical and empirical proof of the tautochrone-brachistochroen property of the cycloid provides also a very fundamental notion of duration. We can't prove that a mathematical proof is eternal and immutable, but we can conclude that mathematical proofs and truths can have very large duration spreading both to past and future from a proof event, even if foundationally indefinite duration coherently with the Halting problem. Reversible computing in quantum duration might be able to also empirically support computational pure mathematics that purely classical ontology would forbid, and without assuming logical and empirical absurdities such as "completed infinity". Such view would mean that we and our pure mathematics are nested in and reflecting quantum cosmology, and new challenges would arise, such as mathematically more coherent formulation of holistic quantum mechanics. Formally we can present holistic quantum duration as the following nesting algorithm, in which the fundamental operators < and > (cf. arrows of time, relational operators, Bra Ket notation and Dyck language) symbolize continuous directed movement: < > < <> > < <<> <> <>> > etc. When we define that < and > have the numerical value 1/0, we can define that 1/0+1/0=0/1. Thus for the second row < <> > we get the numerical interpretation 1/0 0/1 1/0 which functions as numerical generater for numerical two-sided SB-construct, and we get the same numerical result by defining <, <> and > as countable elements and tallying them in each generated mediant word. The blanks between the mediant words form a binary tree, in which continued fractions can be defined as L/R paths. This gives all that Cauchy sequences does, but in foundationally coherent and fully computable manner (see also Gosper arithmetic.). From this holistic perspective, (mereological) fractions come first, and integers and naturals are mereological decompositions of fractions.