วีดีโอ

I come from the set they call potato!

Is this not just a bi-variate tautology?

Slight of hand disappoints again

i want to know what Satan think of mathematics

18:17

So, the interesting thing of the Banach-Tarski paradox is not that we can transform "one ball" into "two balls" (since we know that an "ℝ³" ball has the same cardinality as the whole "ℝ³" set), but "how" we transform "one ball" into "two balls". Isn't it? 👀

Very nice video!

Absolutely beautiful!!!

Hold on... you made two because the original 'cube' in your hands was already in manufactured into two pieces that were put together as a single; you simply separated them in an entertaining and somewhat fanciful way. How does this translate to the Barnach-Tarski Paradox? Serious question....

Philosophy of mathematics??? Numerals are names we have invented for counting things. All math is just addition. Subtraction, multiplication, division are all abbreviation for addition. 3 x 3 = 3 +3+3 etc... Don't make a big deal of math. It scares people not to do math. 😂😂😂😂😂

The foundations of numbers were never defined as Wildberger, Cauchy maintain. For a long time mathematicians couldn't define 'infinity'. Until Cantor managed some kind of a two dimensional array of numbers, that is insufficient to philosophers. The cardinals were able to identify various types of infinities, but infinity remained undefined to most. Mathematical logic stemmed out of Cantor, Russel, Hilbert, Godel etc based on recursive functions and relations hiding behind undefined Metamathematics. What resulted is unproven undecidability of mathematics. What mathematics recently proved with the help of Reimann Zeta functions are a string of correspondences between mathematics and physical reality, based on complex numbers. Complex number i is defined as a ratio between effect and cause, proved by Tristan Needham in VISUAL COMPLEX ANALYSIS.

I think mathematicians do not know math at all

Wonderfully illuminating presentation even for this nonspecialist mathematician!

Thank you for all the enlightening lectures in this series!

Fascinating overview

Professor Hamkins, Is an irrational number a number if we cannot identify or write it with a specific and precise notation? Can we even think of it, if we have no way to pinpoint it relative to other numbers which are known? Does the fact that such an irrational number cannot be specified, except as a member of a general abstract class of such "numbers," detract from its fundamental character as a number?

Thank you for opening up these lectures to the general public. It is a privilege being able to watch them.

@1:03:30 How about the definition of a number is the intersection of all possible definitions of a number 😂

The only raminder question here it's. Where may I buy that cube? Please, let me know.

do you explore any type theory w.r.t. foundations of mathematics (hott as they call it)?

This is the shit you have to go through to become an oxford graduate?

Many thanks! I understood about 'adding 1 to 'omega' (tho how you know this is possible I don't know - unless you simply stipulate that any ordinal can have one added, as an axiom. What I was missing was the notion of a supremum as a number without immediate predecessor. What if so do you get by subtracting 1 from 'omega'??! Unless neither the operation of subtraction, or negative numbers are defined in this system.

I only know in the Euclidean plane, two figures that are equidecomposable with respect to the group of Euclidean motions are necessarily of the same area, and therefore, a paradoxical decomposition of a square or disk of Banach-Tarski type that uses only Euclidean congruences is impossible.

….WHAT?!

@1:33:00 isn't a shorter answer that Peano Arithmetic does not admit power sets. So you do not have ℝ .

@1;23:00 this was lovely, had not heard my affinity with platonism expressed so well before. I am definitely a platonist, but I like this plurality idea of different conceptions of "set", and it follows there could be different conceptions of ℝ. Just like your analogy to non-Euclidean geometry. But it is a fascinating analogy. Non-Euclidean geometries have such clear concrete manifestations (any curved surface or hypersurface). What I've never heard from a set theorist is a similar obvious concrete alternative concept of "set", only different but _very abstract _*_in_*_ their difference_ axiom schemas.

@1:14:00 no? Not necessarily? Woodin's V=Ultimate-L conjecture would render this conception immune to Cohen forcing. It seemed pretty simple. It is _only_ Cohen forcing that gives rise to models where CH is false. I think this is a terrific possible resolution for CH. It does not imply there is only one unique set theoretic platonic universe, just that there is this nice rigid one!

Just a quick note, which I got from Hugh Woodin. Given two doors you can choose one to go through, forbidding forever entry to the other. One leads to an Oracle for Set Theory, the other to an Oracle for Number Theory (and only NT). No one in their right mind is ever going to choose the NT door, since the ST Oracle can already tell you if Number Theory is consistent or not. The number theory Oracle cannot tell you the converse. If the ST Oracle tells you ST is inconsistent, then you can continue on wherever you go developing NT with no worries. But your stupid cousin who took the NT door will never know whether the greater richness of ST could have been a thing.

@1:18:00 you got to my last question, but stopped too short! The thing about CH is that it enriches the conception of ℝ. We could not claim to have realized a conception of ℝ at all before. (And philosophically certainly still no one knows what a true continuum means! You can always squeeze in hyperreals or surreals between the reals.) In some sense the metaphysical continuum is the core issue, and that's just way beyond all of axiomatizable mathematics. What am I trying to say? I think that it is fine to add CH to ZFC as a _provisional theory._ Many logicians might balk at that, but this is the way mathematics is inexorably going. If ZFC+CH cannot force any inconsistency not already in ZFC of course, but more mildly could allow some very obviously stupidly wrong statement to be provable, then so much the worse for CH, but the mathematical exploration is the thing worth pursuing, not the final rejection/acceptance of CH. The thing is, what on earth could be "very obviously stupidly wrong" but not disprovable in ZFC? (Do not tell me Banach-Tarski dissections, since they are beautiful "true" results imho. Showing the richness of sets like ℝ .) To find some such statement surely would require a helluva intuition at the ω-Ramanujan level, and so it'd be only one geniuses intuition, to the rest of us it'd be unfathomable, most likely, unless she could explain it to us children.

@1:17:00 one thing I've never been totally clear about is whether "Set Theory" (or any other axiomatization of the notion of arithmetic or the like) is a lot like Euclidean geometry: If you cannot prove a nice conjecture like the CH, and you really want to prove it, not take it as an axiom, then you are looking at maybe an analogue of non-Euclidean geometries which turned out to "be a _thing!"_ (Who knew?!! We were all flat-earthers before Gauss, Lobachevsky and Bolyai, haha.) So the deeper question is what are non-standard set theories? And can we realize them concretely in a similar way to the way a sphere gives elliptical geometry or Poincaré disk gives hyperbolic geometry?, and the real mind-bender --- the set theoretic or number theoretic analogue of Riemann geometry! To my knowledge we are still "Flat Setters" in this respect. I note that Woodin's V=Ultimate- L conjecture, if true, would establish the CH, and the way he speaks makes it more a set theoretic Universe rather than Multiverse. But surely that is only gauged to CH. I imagine there really should be other varieties of set theory that would not destroy the CH, and so "secure" the finite part of Peano arithmetic relative to ZFC+CH, but still allow for different conceptions of the transfinite, analogous to how Riemann geometry warps Euclid. Thoughts?

@1:16:00 and @51:20 --- is it fair (correct?) to say Tarski gave a criterion for a truth predicate, and showed the predicate is not definable in any formal system? The criterion for a predicate is not a definition _of the predicate_ in a formal system, right? So it's not as paradoxical as you make out?

Related to resiliency, @1:23:30 that issue of a root finding function when there is a tangency is critical. The idea to "solve" it is topological. You deform the function to within floating point tolerance, if you get two roots close by in tolerance the machine can either report them both or consider them "the same". Also related: I used to employ a library that defined an UncertainNumber data type. For work in microwave measurement metrology. I no longer work in metrology, but the idea is of fundamental importance in computing, more people should be computing with uncertain numbers, and someone aught to write an efficient library for them, they should be a fundamental data type, as should data types for dimensioned quantities. These low hanging fruit could go a long way towards establishing some minimal resiliency standards. Still further related is the idea of stochastic computation or analogue or noisy computing, which for a lot of everyday tasks is "good enough" and yet saves tremendous fractions of your real electricity bill! (Learned this from Tim Palmer.) Still even further related: fintec industry is a massive waste of human lives and parasitic on the real economy. As are loads of neural net programs. Really, it should be nearly a crime these days to run an inefficient program if it is sucking megawatts of power from the grid for no good public purpose (internet advertising --- needs to be utterly eliminated, which governments can do. Web companies relying on ad revenue need not exist, if they serve a public purpose the government can subsidize them, without currency inflation risk, to do so the public purpose has to be known and those companies books open, so their real resource and wages are transparent, if the wages can always be paid then there is no need for a profit motive.).

@1:28:00 what insanity is that? We certainly do have a word for "tell the truth", _honestly._ If some grammar nerd tells me, no that's an "adverb," I'll honestly slap them in the face, and can _testify_ to that. I'll also _avow_ to it my dude.

@1:25:00 another answer is that historically people did not understand "=" does not mean "the same" or "identically equal". So _x_ + _x_ = 2 _x_ = _x_ 2, is fine for finite numbers "up to isomorphism". But Cantor, yeah, he's the GOAT.

@1:17:00 not having a clear concept of ℕ, and thinking Dedekind-Peano is "murky" is fine. It gives mathematicians things to do. It does not force one to adopt pluralism. Pluralism is fine too, as a framework. But it is fragile, right? If you could get a non-murky definition of ℕ, then that kills pluralism for ℕ (a damn good thing in my opinion). However, Gödelman always flies to the rescue, no? There's always some structure you can validly desire to be pluralist about, like today the CH. Or "The Continuum" in general --- hyperreal continuua, Surreal continuum (the Surreals are non-murky?), or Cantor-Rucker "The Absolute" continuum (cardinality of Ω = the absolute infinite.) However, does not a lot of this boil down to words? With finer grain knowledge of mathematical structure we can name things that were once thought the same differently, provided we do not run out of words. Like 1960's particle physicists. 🤣 (that's a joke). I honestly think this game can go on forever, the question is whether mathematicians will last for _more than forever_ so the game terminates! 😜

@1:14:00 I am pretty sure, gut instinct, that what Ahmed meant was whether there is a problem with mathematical induction when we allow transfinite numbers, like ω +1, or ω·ω. Maybe? The answer is no (I think). The issue he might have had in mind was that if we can "go to" ω+1 then how come the induction over ℕ is valid? The answer is that it is valid because the proof using induction is induction only over ℕ. But the same idea extends to validity of transfinite induction. For these things I find the category theoretic style better than the indexing style. The CT folks tend to write an "index set" and try to never use an index subscript. It is just conceptually clearer that then one is staying in ℕ (or whatever index set one is using). Although I think I am not expressing this well. I got it off Misha Gromov ("I hate subscripts." He is also known to have said, "I don't know what a matrix is, a matrix is stupid." --- which being a Clifford algebra user I fully agree with.... not just for the laughs! Although mostly for the laughs.).

@59:40 yeah, but you need to be very careful about educating economists. They can be a bit nerdy and autistic in a dangerous way. In economics the Pareto principle is often grossly violated. It goes under various fallacies of composition. Such as the paradox of thrift: everyone is at least slightly better off if they net save. But if everyone net saves then everyone starves and dies (in a monetary economy). In a monetary economy with no counterfeit (so strict accounting laws) some agency has to net dissave. That is typically government, and government can _always_ net spend, since they are the monopoly currency issuer. It is called "running a deficit". Most Europeans think this is a shockingly appalling bad thing to not have people begging for the state currency (hence they introduce insane "fiscal rules" in the EMU which cause needless austerity and mass unemployment), but they've got basic accounting 101 all wrong, it is a *_good thing_* for government to run a deficit --- no ones "tax dollars" are paying for this --- although the government deficit need not be targeted, the supply can float in order to accommodate a stable price level at full employment --- by which I mean zero involuntary unemployment, not the NAIRU myth, the reality is a NAIBER not a nairu, or "non accelerating inflation buffer employment ratio". Employment and price level are all about buffer stocks, which mainstream macroeconomists never seem to consider when theorizing about inflation, being the idiots they are, they are pretty evil really. (Bet you didn't think a few words about reflexive and partial orderings would lead to life & death morality in the real world. But this comment is just basic math for state money accounting systems.) I think it can also be viewed as a comment about directed acyclic graphs? So from DAG's you get an understanding about the source of the price level, which no mainstream economist understands (I am willing to bet). Government deficits accommodate non-government savings desires (to the penny). (Inflation has nothing to do with this.) But try telling this to a Neoclassical or New Keynesian "professional" and they'll wet their pants.

@38:50 what I like is that you "construct" all algebraic numbers by "listing" all the polynomials over rationals --- surely countable, and they're bijective with the strictly finite sub-lists of rationals or integers that are the coefficients. But you never _actually_ construct this list. You only "know you could" if you were a god. This presumption of humble god-like powers is what finitists often object to, but the humility is beautiful. It is "well, if _only_ I had infinite paper and infinite time, or could do the proverbial ω many tasks in 2 seconds by working twice as fast each task step. While I am against tying mathematics to "physical tasks" (or even ω −Turing machine tasks) like this, since it leads to seeming pathologies like Banach-Tarski, it does cry out for a better verb than "construct". It is "mental construction". But what's the shorthand for that? It is "conceivability" or "computability" is it not? But _strong conceivability_ --- one must have a means of computation available _as if_ one were a humble god who merely has vast resources (infinite time or time steps). Which brings me to the question: has anyone got some of these proofs (involving the reals that use such god-like constructive arguments) in the form of computations involving tasks that could be completed in finite time and finite time steps? (i.e., non god-like.) Not that I care, I am fine with the nonconstructive proofs, and infinite tasks. But I'd like to have the occasion now and then to throw old Norman Wildberger some scraps.

Thank you

@1:21:00 who is your colleague in New York? Carl ????

excellent. you can also demonstrate the concept of anti-time by putting the cubes back into one. bravo. 👏

@41:15 The Asgaardians did one better than Yuen Ren Chao, they figured out the any amount of beer *_or_* mead is allowable theorem.

So you are saying that math is mechanical?

On continuous induction at 37:20, this can be extended to connected topological spaces, and linear orders are connected only if they are dense orders with the least upper bound property. The more general versions states that if A is non-empty, whenever x is in A there's a neighbourhood of x contained in A (i.e. A is open), and whenever V is an open subset of A then A contains the boundary of V, A is the whole, connected space. The last condition is like saying 'if points arbitrarily close to x are in A, then x is in A.' The second and third properties tell you A is both open and closed, which in connected spaces means A is empty or the whole space.

I am a lawyer too. Its amazing how this lecture reiterates Socrates' parmenidis. A real and genuine platonist view.

0 1 2 Omega Omega +1 Omega +2 Omega x2 Omega x2+1 Omega x2+2 Omega ^2 Omega ^2 +1 Omega ^2 +2 Omega ^2 +Omega Omega ^2 +Omega +1 Omega ^2 +Omega +2 Omega ^2 +Omega x2 Omega ^2 +Omega x2+1 Omega ^2 x2 Omega ^2 x2+1 Omega ^2 x2+2 Omega ^2 x2+Omega Omega ^2 x2+Omega +1 Omega ^2 x2+Omega +2 Omega ^2 x2+Omega x2 Omega ^2 x2+Omega x2+1 Omega ^2 x2+Omega x2+2 Omega ^Omega

Just like politics...depends on pov.

I am listening on my laptop full volume and somehow struggle to hear. May be it's a good idea if you can edit the video and raise the volume up a bit

Man I am drunk and I have no idea what this is about, but I feel so smart listening to this and kind of picking up on stuff. I don't know why TH-cam suggested me this but I am listening for like an hour now, "general comprehension principle" man, you rock!

Many highly educated people can't see the forest for the trees. Too "strong/rigorous" demonstrations destroy any useful applications. They are a bit like ants that spin around. The hard part is not to be “rigorous”-that can be done by robots-but to set from the beginning the context/axioms and to see where you can go with the results and how to use them. The real is so vast and dense that we can start from anywhere, construct logically some model/structure/theory, that it is self-consistent 99% or more. Nevertheless, it could be completely not fitted for practical use, because it is not compatible with our reality. You can't trust maths as a dogma; this is intellectual fanaticism. Or, as Feynman put it ..., i let u find the citation yourself. Mathematics is tools we have being invented and continues to be invented, not a gift from GOD :)