The Banach Tarski paradox - is it nonsense? | Sociology and Pure Mathematics | N J Wildberger

[@PeterHarremoes]

If we abandon the axiom of choice (in its usual formulation) then we may assume that all subsets of real numbers are measurable and if this is the case Banach-Tarski is wrong. As student at the university I had a course called "measure and integration". Although I scored the highest marks at the exam, after the course I was not able to integrate a single function that I was not able to integrate before the course! All we learned was how to avoid some theoretical problems that could only emerge due to the axiom of choice.

[@andrewsheehy2441]

It is great that you have the intellectual self-confidence to write those words. I myself - as an electronics student scored 100% on the paper on electromagnetism in my final year exams. This was a 3-hour exam and, having finished the actual questions in about 1-hr I spent the rest of the time deriving all the equations given on the first page - including Maxwell’s Equations - from first principles. Except that now, I realise I did not understand what I was doing or how many arbitrary assumptions I was making along the way - like simply assuming you can move from a static system to a time-variant system which carries with it an assumption that light travels infinitely fast. The common notion that a changing E-field causes a coupled changing B-field is parroted everywhere but it is nonsense. Science - not just math - needs more people like Mr. Wildeberger - to inspire and encourage a new generation of scientists to ask the hard questions and respectfully challenge foundational assumptions that cannot be justified.

[@PeterHarremoes]

@@andrewsheehy2441 Yes, much teaching is based on misleading explanatinóns that only few dare to challange.

[@20-sideddice13]

This is wrong. The Lebesgue integral is much stronger than Riemann's. You should be able now to calculate the integral of a function defined as such : f(x)=0 if x is irrationnal f(x)=1 otherwise On the domain [0;1]. Because there are much more positive measurable functions than positive continuous ones. Plus how do you do probabilities rigourously without theory measure ?

[@PeterHarremoes]

@@20-sideddice13The indicator function on the rational numbers is measurable, so the question is how to define the integral. You are right that it is Lebesgue integrable but not Riemann integrable. The Lebesque integral is better than the riemann integral, if the goal is to integrate as many functions as possible, but then the Kurzweil-Henstock integral is even better than the Lebesgue integral. The Kurzweil-Henstock integral also has the advantage that it can be defined without reference to measurability. I agree that measure theory is highly relevant for probability theory.

[@PeterHarremoes]

@Greg LeJacques The theory of fractals rely heavily on measure theory. The coloring of figures that illustrate fractal sets can have high aesthetic value, but the coloring has little to do with mathematical theory.

[@freed.man.1]

You can't subtly rearrange a dollar to get two. Federal Reserve Bank: Hold my printer.

[@antizephdaniel7868]

Are you that stupid? The dollar would be fake because it would be larger.Stupidity is now top comment?

[@exodus_from_babylon]

Now talking political science in this sort of discussion is normally unacceptable. Your comment bucks the trend. Ha ha ha

[@Amine-gz7gq]

and that is why when a leader wants to sell oil in another currency than the dollar, he gets killed (Kadafi, Saddam, etc...).

[@zackyezek3760]

I think his point is essentially this: Logic only really works for computable things. Therefore when you try to define or manipulate uncomputable or even 'ineffable' things with it, logic as a tool becomes unreliable. Because logic itself is a kind of algorithm, a computation. Therefore using such things in mathematics "voids it's warranty"- you can sometimes do it, and even sometimes with great practical benefit, but at the cost of compromising the field's overall logical integrity. By allowing serious logical paradoxes and other flaws right in parts of the foundations, all the math above them risks being built on quicksand. Proofs only are reliable when everything below them is logically sound, and you can no longer guarantee that's true. So the dispute really is: Does Logic work reliably for things well beyond the current formal notions of computability (e.g. Turing machines)? Current pure math implicitly claims it does, but NJW and others say no. His "Law of logical honesty" really equates to "Do not treat non-computable things AS IF they were". Formal math claims things like large cardinals and unmeasurable infinite sets ARE computable in the limited sense that the algorithms of pure logic (e.g deduction, excluded middle) still work when given such things as inputs. NJW says that this is lunacy, because things like Banach-Tarski are EXPLICIT EXAMPLES of that very machinery breaking down & throwing errors when you do that.

[@TheFeatInk]

Do you suppose one could successfully demonstrate, or prove, that the laws of logic are not absolute? Of course one could say they are only temporary or convention, but that would not make them so. The laws of logic have a far greater status than just a computation or algorithm. Do you not assume their validity when even proposing an argument or demonstration?

[@ROForeverMan]

All these happen because mathematics is a creation of consciousness, and consciousness is beyond logic.

[@RandyH524]

You're an inspiration to me. Thank you for all of your wisdom you've shared through this youtube channel. 🙏🏼

[@yinsound]

Great example. As a computer guy, I suspect any 'finding' that is defined as not computable. Although I am not a mathematician, I challenged one on Quora on the definition of math itself. I claimed that useful, provable math must, at least, be computable and congruent with physical reality. Regardless of the branch of math, I still think it must be representable, albeit in reduced slices in some cases, in a physical model. For example, all topological entities must have some graphical equivalent. Or, any claim for number theory must have an algorithm. To abandon even the possibility of computability or representation is to make an inherently nonsensical claim. Thanks for bringing math to non-mathematicians!

[@Kindiakan]

My math professor joked that this was how Jesus multiplied the loaves and the fish! XD

[@zaclanderos1624]

Hahaha I thought the same

[@zapazap]

I find it fascinating that combining certain sets of strings (ZF) with another string (C), in certain interesting systems of string rewriting, yields the fascinating string (Banach-Tarski). Call it mathematics or don't call it mathematics. I don't care. I find it fascinating. Many others find it fascinating. Perhaps an certain *interpretation* of these strings does not make sense, but that does not make it less fascinating. It may make it even more fascinating. Questions of the *expressive* and *generative* power of such systems can be treated in a finite way, and to be certainly _looks_ like mathematics to me. Whether it looks like mathematics to another, or whether another wishes to _call_ it mathematics, does not change this. Is this a worthy area of research that warrants funding? That might be disputed. But it seems to me to be a noble activity. We fund music in the academy. I see no reason not to fund this. Where in the academy would such research be most at home in? Departments of mathematics would seem the most obvious place. And I say this while wholeheartedly supporting an algebraic approach to all of mathematics, a la Dr. Wildberger's work. Thoughts?

[@tylerchambers6246]

By that logic why don't we just deliberately go about putting together axiomatic systems, alternative mathematics with no mind to their real life application or inherent meaning,- alternative systems that produce the most illogical, mind boggling paradoxes that can be conceived, and then spend time thinking about those? Through axioms you can construct an infinite number of alternative mathematical systems and then you can create paradoxes within them; so what?

[@jonadabtheunsightly]

Banach Tarski relies on the assumption that an object is made up of mathematical points. That's true for abstract mathematical objects, such as lines, planes, and spheres; but it is NOT true of physical objects. Physical objects are made up of matter; and matter is made up not of points, but of atoms. This is an important distinction: the number of atoms in an object may be extraordinarily large by the standards of non-mathematicians, but it is nonetheless always finite, no matter how large the object is. Banach Tarski absolutely relies on the number of points in a mathematical object being infinite, so it does not apply to physical objects. It can't be generalized to apply to physical objects, because they don't have the required properties. You can get similarly weird results if you try to imagine that subatomic phenomena (electrons, photons, etc.) are like macroscopic objects, and expect them to behave as such. Or vice versa. Macroscopic objects have properties that subatomic phenomena don't have, such as shape, size, texture, and orientation. Thus rules that apply to one, don't always apply to the other as you might expect. This allows subatomic phenomena to do things that seem very counterintuitive to people who are used to thinking in terms of macroscopic objects: see for example the double-slit experiment, which weirds people out every bit as much as Banach Tarski but is nonetheless objectively verifiable. Of course, it only works with photons: you can't demonstrate the double-slit experiment with tennis balls. And you can't demonstrate Banach Tarski with a tennis ball either. It won't work, because a tennis ball isn't a mathematical sphere any more than it's a photon. More generally, you can't take a theorem that's been proven about one type of thing, and expect it to necessarily hold true for a completely different kind of thing with a completely different definition and completely different properties. That's very imprecise thinking, and anyone who has had high school math should know better.

[@santerisatama5409]

If quantum "matter" is a property of Hilbert space, it's made of points (and point is an undefined primitive notion, according to Hilbert). :P

[@MartinPitti]

Exactly. That whole conjurer trick ("a line is an is infinite set of 0-sized points") is nothing more than multiplying by zero or dividing by infinity. n * 0 == 2n * 0. As such, it's entirely uninteresting and unsurprising.

[@santerisatama5409]

@@MartinPitti Well, not entirely uninteresting, the whole foundational crisis has been revolving around that issue since Newton, Leibnitz and Berkeley, and the debate has had it's creative aspect in both good and bad. Intuitionist logic and bridges between classical and intuitionist - and how deeply those relate to quantum, classical and decoherence between are also products of the foundational debate and investigation.

[@Hyperparoxysm]

​ @Jonadab the Unsightly One The problem is that this kind of mathematical idealism is fundamentally dishonest sophistry, and is infesting not only pure math but physics as well, as per your example. Subatomic particles are defined not as real physical objects but mathematical objects that by definition have no correspondence to the real world. Starting from the Copenhagen interpretation in 1926 Bohr and others basically decreed that physics at the quantum level does not make sense, and it is futile and ill-advised to try to diagram it or model it mechanically. All you can do is write math equations that describe it. Since physics has stopped being physical (that is, mechanical and corporeal) and been subverted by pure math, we find all sorts of paradoxes. There actually is a mechanical explanation of the double-slit experiment that resolves the paradox, which you can read here: milesmathis.com/double.html Paradoxes are a result of incorrect underlying assumptions or flawed logic, and once that is resolved, they evaporate. Paradoxes should never be pedestalized and worshiped.

[@MartinPitti]

@@santerisatama5409 True -- it's certainly interesting in the sense of deciding "does that make sense for how we want to describe the world". So one conclusion would be "describing a line as an uncountable infinite set of nothings" makes the description both worthless (as it loses the "essence" of what we mean by a line), as well as infinitely (hah) more complicated. I meant that once you divide by infinity, then particular results like 1 == 2 are uninteresting -- not the whole process of thinking about it.

[@MisterrLi]

Sure, this is a real paradox, there is even a real contradiction involved here. But in what lies the cause of this contradiction? To double a volume just by re-arranging its parts seems just totally impossible. But the interesting question is what is meant by re-arranging in this case. There is a method in math to define geometric size as sets of points, and if you don't care about doing this in a very precise way, you can easily come to the conclusion that since a unit length has an equal (infinite) number of points as a double unit length, they must be equal in length, which is obviously nonsense. But if you don't constrain this use of set-to-size measurement, you will be able to stretch volumes to the double because you still have equally many points as you had when you started, there is a one-to-one mapping between the points in the two objects. Despite the technical complexities in the Banach Tarski paradox, this can be "the" main explanation of the result. The methods used to manipulate volumes defined as sets of points are simply not constrained sufficiently, causing volume to change in those cases of manipulations. The relation between the infinity of points in a unit length of a line segment is simply not precisely defined, it has to be defined precisely to set it apart from a length of two, and even a length of infinity, if we want length, area and volume to be unchanging units when using point sets to measure them.

[@yipengguo2732]

In introduction analysis course, you will learn how dangerous the rearrangement is. Even the limit for the series can be changed if you rearrange it. Only absolutely converged series can be rearranged. That’s one of the most basic idea in mathematics.

[@antizephdaniel7868]

“The study of numbers, said Socrates, should be followed by, in successively lesser importance, the study of geometry, then astronomy-as far as it concerns the laws of motion-then perhaps harmony, the study of sound. Only through the study of abstractions of the mind-as he viewed these disciplines-could one release oneself from the chains that bind us all to the rigid world of our senses.”

[@michael2974]

Where did you find that in Plato's work? I came across the same thing in the Pythagorean Sourcebook. It was given as: 1. Mathematics is the study of numbers. 2. Geometry is the study of numbers in space. 3. Music is the study of numbers in time. 4. Astronomy is the study of numbers in space and time. Anyway, pretty cool. Take care

[@Anders01]

I think the Banach Tarski paradox can be resolved by using what Aristotle called potential infinities instead Cantor's 'completed' infinities (transfinite numbers). Then for example Pi = 3.14159... has an endless number of decimals but not an infinite number of decimals as with standard real numbers.

[@1DJDUST]

Super interesting. Can we experiment this?

[@JUUULiiAAAN]

i just starting writing a thesis on the subject of amenable groups and the Banach-Tarski-Paradox and you really have me questioning my choice right now ^^

[@njwildberger]

Hi Julian, I hope you at least consider that the orthodoxy is not all it is cracked up to be. Please watch more of my Maths Foundations series for example related to real numbers. There are serious issues that people are not wanting to address. Be careful you don't end up choosing a dead end topic --there are lots of really exciting directions for modern mathematics. How to identify them? They usually have a strong computational aspect. Have a look at some of the Advice lectures which are in the Members section on my other channel --Wild Egg Maths : for example kzbin.info/www/bejne/r6q4XoOohb-ohsk

[@benhbr]

The gist of BT is that real numbers are not suited to describe the continuum of physical space.

[@thomaskember4628]

If they're not able to, what numbers can?

[@JoelSjogren0]

Another possible culprit, which is more general, is the 'analytic' view of bodies in space as composed merely of points. Should this view make it into the definition of a geometric object, or should it be secondary (however powerful)? For instance, a limited kind of equations could be primary instead. Only in the eyes of a set theorist (or someone else adopting this 'analytic' view) would a geometric object be precisely "the set of all points satisfying said equation".

[@MartinPitti]

It is not at all a given whether physical space is a continuum. We have absolutely no idea what the nature of space (and time) really is, and in fact the current standard physics teaching is that space is *not* infinitely subdivisible ("quantum foam", "Planck length", etc.)

[@MartinPitti]

@Quantum Bubbles Of course, I didn't mean to say that GR doesn't work; but we know that it's not complete either: there is consensus in the physics world that actual infinities are "unphysical", i.e. just point out situations where the theory breaks down (black holes, infinite speed/energy etc.). I am not a professional physicist, but it seems to me that GR is formulated in the standard "real number" analysis framework mostly because that just happened to be (and still is) what's being taught at schools, not because it *actually* needs the "god-like" properties of real numbers. There is no such thing as a "real number" in physics, simply because there is not an infinite amount of information/energy/space in any finite portion of our universe -- such singularities are precisely the above case where the theory breaks down. So in a sloppy sense, GR works not *because* of the real-number analysis formulation, but *despite* it :-) And that's what I meant with "discrete" -- there is only so much precision any measurement or description of particles/space etc. can have. The mathematics finesses over these, but reality doesn't. So IMHO it's not the physics that "gets in the way" (as you say), it's the assumption of continuous and infinitely divisible space that does. The latter is just a approximation to make the formulation of the maths (apparently) easier, but one needs to be aware of that.

[@MartinPitti]

Hi @Quantum Bubbles ! I don't think one actually needs to (deeply) invoke QM for this. As you say, my issue is exactly these "intensive" infinities (and I agree we can ignore the "extensive" ones for this). If you would need an actual-infinite amount of information for any theory/model (like, an infinitely precise measurement, or any "real" number), then the entirety of physics goes down the drain. The idea of a physical model is to explain a large/larger set of data/measurements/observations/information with a smaller set of rules/assumptions/information. So requiring an infinite amount of information as input of any model would break that -- a good model should "scale" the quality of its output with the quality of its input in some reasonable manner. *That* is what I meant with "formulation of GRT in a real-number framework is merely coincidental", as obviously you can get good approximations as output when you put in good approximations as input. With that, I still reject the idea of an actual continuous space as unphysical, and merely as an approximation, or artifact of formulation of current mathematics. It's fine in most cases, but shows its limits exactly when you drill down to ever-smaller scales.

[@santerisatama5409]

The philosophical argument, if such is made, that pure mathematics should be restricted by classical physics, and concretely, what can be written down in terms of classical physics, would be very problematic as postulating metaphysical materialism restricted to classical physics. One would think that is the natural position of formalism, which approaches mathematics as arbitrary language games based on arbitrary axiomatics, but as we know historically formalism goes with the Cantor's paradise. According to intuitionist philosophy of math what can be written down etc. linguistically constructed, is only a limited reflection of mental ontology of pure mathematics in some version of idealist metaphysics. Main difference with Platonistic idealism is that intuitionism does not subscribe to eternalism, as it is open and evolutionary and hence closely connected with philosophy of time. In intuitionist ontology of pure math, all linguistic constructions as such are already in a sence applied mathematics, hence the esthetic and ethical norms are historically more strict than in formalist language games, and intuitionism is historically in the "Cantor = joke" camp. Should pure mathematics be founded on metaphysical axiom of either materialism or idealism in some version? Not in my view, as philosophy of math is the critical open question in the debate between materialism and idealism, and intuitionism concerns also the empirical inquiry about the phenomena of mathematical intuitions and mathematical cognition, and is open towards also metaphysics of process philosophy, dynamic instead of static view of mathematics. Perhaps what we can agree on, is that mathematics simply cannot methodologically subscribe to the amputation of introspection, as is the norm of post-Cartesian Western science, and hence math can't be fitted in the confines of contemporary physicalism. Physicalist claim that mathematical cognition and mathematical theories are caused by e.g. quantum fields or some other physicalist fundament as described by mathematical physics, would be some kind of category error. We can't make much sense of sociology of pure mathematics if we don't honestly and openly discuss this wider philosophical and metaphysical context, and try to make some sense of it.

[@qswaefrdthzg]

I wish Dr. Wildberger could honestly talk with somebody like you instead of propagandistically misrepresenting mathematics, though I don't argue that he makes interesting points.

[@santerisatama5409]

@@qswaefrdthzg I don't think Wildberger is misrepresenting mathematics, and I agree with his criticism of infinite sets and axiomatic set theories in general.

[@ThePallidor]

The whole notion of "isms" is part of the problem. Math, like all rigorous communication, is a mode of getting across what someone visualizes in their mind. If someone's message doesn't conjure up a clear and unambiguous visual, the only thing to do is ask for clarification. If the person refuses to clarify, no matter the reason, but especially if they claim not to be even conveying a visualizable concept in the first place, they're done. It's then a social game, not an intellectual one.

[@santerisatama5409]

@@ThePallidor Yes. In that sense, mathematicians job is rather similar to that of an translator. In my own experience with various intuitions, only small part of them in translatable into clearly communicable language, and the translation process can take a long time to mature. Deep and creative mathematical intuitions are close kin to psychadelic experiences, of which only very small part is communicable in language. I very much agree and admire Wildbergers insistence on clear and communicable language as ethical and esthetic fundaments of pure mathematics. Not least because we organize so much of our social reality based on mathematics, and being ruled by esoteric monk Latin is far from ideal.

[@antizephdaniel7868]

@@ThePallidor That is why we have proofs.But it is time we start accepting proofs by programming.

[@brendanward2991]

This theorem could be regarded as a reductio ad absurdum for the existence of real numbers.

[@acortis]

Exactly what I was thinking! you beat me to it

[@twwc960]

Not really, because you don't need the Axiom of Choice to construct the real numbers, whereas you do to derive the Banach-Tarski paradox. If anything I see the Banach-Tarski paradox as an argument against accepting the Axiom of Choice. Most of real analysis will survive without Choice. Much of topology will too, but some proofs are definitely more awkward without it. And of course, some theorems such as the Tychonoff theorem (the product of a possibly infinite number of compact spaces is compact) will fail without Choice, as will some function extension theorems. But most of working real analysis (that used by engineers and scientists) will work just fine without the Axiom of Choice.

[@tantzer6113]

It is possible to do math with the Axiom of Choice or without it, with infinities or without it, etc. Each axiomatic system generates its own delightful body of mathematics (set of theorems). All these versions of mathematics are TRUE and free of contradiction relative to the axioms on which they are based. One might argue that a particular axiom or set of axioms reflects or does not reflect the physical world in which we live, and that may or may not be true, but that makes no difference to mathematics. Mathematics is not the study of the physical world; it is the study of abstractions. Some abstractions might have physical analogues, and some might not.

No system equal or greater than arithmetic can be proved to be free of contradictions.

[@ostihpem]

First of all: All your versions of mathematics _could_ be true and free of contradiction relative to the used axioms. But they _could_ be also false due to Goedel's 2nd theorem or due to the fact that axioms could turn out just false despite being assumed as true (for instance it could be the case that our world is only finite which would mean that our concept of infinity is only about a very big finiteness which would mean that we'd contradict us when talking about infinity as non-finiteness but always only being able to mean finiteness due to what the world is all about). Wildbergers subtle point seems this: if we assume things beyond our control it's basically 50/50 that we are wrong and we may never find a contradiction since the whole mess is beyond our capabilities. So we may work with false concepts, we may even trust them because we cannot prove contradictions or we even can prove (false) concepts to be true (superficially within our only limited realm). Wildberger warns us about it - and rightly so - and with his kind of method we'd avoid this risk, making math safer (but also leaner).

I believe his point is that mathematical definitions should be more restrictive, disallowing inconstructible concepts that can only be defined/proved by vacuity/lack of formal impossibility. Having said that, you are essentially correct: whatever is done with the Axiom of Choice and with infinities can still be explored by logic, whether or not you want to call it Mathematics.

[@tantzer6113]

@ Yes, and why would humans NOT want to know what mathematics WITH infinities and Axiom of Choice looks like? I personally love infinities and Axiom of Choice (and the Banach Tarski theorem, which I don’t find illogical), and yet I also want to know what math looks like without them. Both versions of math seem equally valid to me, even though I am emotionally attached to the standard version, because it feels mind expanding to investigate conceptual universes that may be different from our own.

[@tantzer6113]

@@ostihpem There is a fundamental misconception in what you write. If it turns out that our universe is finite and discrete, it does not follow that axioms that entail infinities are false. Why not? Because the point of axioms is not to describe physical reality. Math is about abstraction and inference, not about the physical reality. For physical reality, go to physics.

[@antizephdaniel7868]

I just solved this problem with some C++ programming. If you can turn a sphere inside out topologically(which has been proved) then you can make it hold 2 times more data than one would think it can hold at first sight.

[@rogerstone4948]

Elaborate on how this is possible.

[@antizephdaniel7868]

@@rogerstone4948 If the data is on the surface area of the sphere. A concise explanation is to imagine that the the surface area is a holographic sheet that generates the volume.Turning it inside out will give the opportunity of generating a new volume.This also correlates with the -1 and +1 oscillations (Majorana) I explained earlier.

[@QuestforaMeaningfulLife]

You've been pointing out for years how there's something off with the modern math treatment of "infinity". No better example than this.

[@brendawilliams8062]

Yeah. Some one blind sighted me with it. It was an awakening.

[@andrewsheehy2441]

Norman Wildberger is a unique intellectual talent who should be cherished and encouraged. His ability to explain complex mathematical ideas so clearly is extraordinary - and likely rather scary to those in academia who either lack that ability, or who prefer to hide behind a facade composed of obscure symbolism, very subtle logical contortions and intellectual abstraction that 'lay' people fine impenetrable. So well done Mr. Wilderberger and please do keep going. I plan to watch all of your videos and reference them as widely as I can.

[@andrewsheehy2441]

@Quantum Bubbles Interesting. I can see that I've struck a nerve here. Apologies - you clearly found my remarks irritating and that was not the intention. The specific point I was making by my 'facade' comment was that when it comes to tricky subjects then there are a lot of mathematicians who simply parrot stock explanations and derivations without having ever properly understanding the underlying point themselves. Maybe they do understand it, but just can't communicate clearly. Maybe they do understand it, but their understanding of the point is based purely on what can be done with mathematical machinery, tools and techniques - at which they might have become very adept, but they have lost any intuition for the point and can't see the wood for the trees, as the saying goes. But maybe they never understood it - but just say they do. Of course, there are plenty of mathematicians who really do very deeply understand the topic and are capable of meta-analysing the whole field and its axiomatic foundations. Norman Wildberger is one such person. Sadly, the ‘herd’ mentality results in subtle topics being persistently mis-explained and mis-communicated - and this happens in all intellectual domains, not just math. But I'll give you two examples in math which illustrate my point: === Bessel’s Correction === This is the one where the standard line is "If you want to estimate the population variance then you should divide by (n-1), not n. Dividing by (n-1) gives an 'unbiased' estimate of the sample variance." This is completely wrong: most statistics textbooks and math professors have never deeply understood this themselves. If one takes the time to think carefully then it is very clear that, even for small sample sizes where you'd think that this is important, dividing by (n-1) is more likely going to give you a WORSE estimate of the population variance than simply dividing by n. And the so-called derivations of this supposed mathematical fact are plain wrong. The reason is that dividing by (n-1) ONLY works if you are drawing samples with replacement and you can take the mean of the variances for all possible samples. Under these conditions then, yes, using (n-1) rather than n to calculate an ‘unbiased’ sample variance will produce the exact population variance when the ‘unbiased’ sample variances are averaged. But dividing by (n-1) when you have just one sample which was drawn without replacement is mathematically meaningless and has zero justification. And yet, that is what we are told. === Hilbert's Hotel === In this case the stock explanation focusses a lot of what happens at the 'beginning' but the detail of what happens at the end is, in effect, left to the reader’s imagination. I think it is right that most people are left feeling that this is a bit of a 'fudge', or just that they never really understood the argument. Here's a summary... So we have a very clear picture of a hotel with a countably infinite number of occupied rooms (Room 1, Room 2, Room 3...). There is also a hotel manager and a new quest who wants to a room for the night. So far so good - the experiment makes sense. Let's call the state of the hotel at this stage STATE 1. The hotel manager then executes his guest reassignment algorithm which results in all existing guests moving to room number (R+1), where R is their existing room number. This then frees up Room 1, which the new guest can use. After the new guest has moved into Room 1, let's call the state of the hotel STATE 2. If we now revist our definition of STATE 1 then we see it is the same as STATE 2. So that's the stock explanation, which I'm happy to parrot. And I can also do the one where a countably infinite number of busses show up, each containing a countably infinite number of new quests... Very good. But there is something wrong here. What's wrong is that we assumed that the 'guest reassignment algorithm' can be completed. Or, in other words, all guests in the hotel in STATE 1 can be moved to a new room. But this is nonsense as - given the hotel has an infinite number of rooms - the guest reassignment algorithm can never end. No matter how many guests are reallocated there would always be more guests to reallocate. I realise that this is an intellectual exercise and so we do not have to imagine that there is a finite time needed to reassign a given guest. But the rather neat set-up of this thought experiment grates with a glaring logical problem - which nobody seems to mention, even though we can all see it. To fix this, we should fess up and say explicitly that 'we will assume the presence of an imaginary process that can complete a countably infinite number of operations - of any complexity - instantly. I'd be OK with that - because we have now completed the explanation and it would make logical sense. But, as it is, we are comparing STATE 1 (a stable, static state) with STATE 2 (an unstable, dynamic state) and calling them the same. In STATE 2 there will always be a room with two guests and there will always be a reassignment algorithim running. How can this in any way be considered the same as STATE 1? This is plainly nonsense. And yet people consistently fudge this. I have a long list of similar examples - across multiple intellectual domains, and the deeper and wider I look I find more and more. So in closing, I’m a very strong supporter of math (and am what you might call an ‘advanced learner’) but I can see room for improvement. It’s not all roses.

[@andrewsheehy2441]

@Quantum Bubbles Your verbose replies are both flattering, and mildly irritating - in broadly equal measure. I imagine that was your intention. I'd first like to thank you for the reference to the paper that claims to expose a range of failed attempts to refute Cantor's diagonal argument. I will study that and come back to you by replying to that specific comment. However, I note that you have not responded to the specific point I made which is - as far as I can see - not explicitly included in the Hodges paper. I did quickly scan it and saw one argument based on "Well, I can't actually draw this diagonal map therefore the argument makes no sense" >> This is clearly a silly argument, and one that I'd not make: the person advancing this has not studied the history of Cantor and Hilbert at the relevant time and clearly did not understand that these arguments are playing out in a intellectual realm that is separate to physical reality. I get that, and I’m OK with it. On the statistics stuff, I'm no statistician (although I have completed Joe Blitzstein’s outstanding course ‘Stat 110’ - which is not exactly a walk in the park. For the record I have a huge amount of respect for Joe who is a tremendous ambassador for his subject). But I can categorially assure you that the n vs. n-1 topic is very widely mis-explained. Including in maths textbooks. I did not say that any of the results or formulae quoted are wrong - but it is certainly the case that they are widely misunderstood with scant regard for the back story. The result is that generations of students do not understand what they are doing when they divide by (n-1). I'll come back to you with more detail when I've taken a close look your detailed comment. More generally, and if I may say so, I find your style of writing and choice of words rather insulting and patronising. You insinuate that I have some sort of cognitive disability and am one step from donning a tin hat. I find your classification of Norman Wildberger as a ‘novice’ to be unbecoming, ungentlemanly and a poor way to present your colleagues within the academic establishment to the outside world. As to what my intellectual level is - then if you’re interested then I’m sure you can find me on LinkedIn. For another, your narrative presents a clear picture that "Everything is fine in academia. And how dare outsiders criticise what we're doing." Given the well-known problems with peer-reviewed research (reproducibility, statistical validity, spinning, conflict of interest, dysfunctional reward mechanisms etc.) I think that a bit more humility at your end would be prudent. In closing, thanks for your replies and I’ll come back more fully later.

[@andrewsheehy2441]

@Quantum Bubbles Hello, So I've reviewed the Hodges paper and was unable t find anything here that was relevant to the specific point I made which, I see, you are unwilling to critically analyse. Specifically, the conclusion that "There is nothing wrong with Cantor's argument" is one that I'd agree with. It is very easy to imagine an infinite countable set and then, beneath that, a much larger non-countable. That is not the point I was making. What I was saying is that this topic sits on the boundary between two intellectual domains - one that includes ideas that we can imagine using visual imagery, and one that includes ideas that are not possible for human minds to comprehend visual analogs. But nobody has the guts to fess up and say this. Without wanting to nit-pick, I would actually challenge the verbatim on p. 2 where, in Point (3) we see the sentence: "Write 0 . an1 an2 an3 ..." # For me, this is a total fail because of the use of the word 'write'. Plainly, one cannot 'write' an infinite series! Assuming we agree that the number of elementary particles in the human mind is finite then we cannot even imagine a countably infinite set, let alone an uncountable infinite set (the reals). So why use the word 'write'?? If I was the editor I’ve have sent this script back for change. I'd have insisted on communicating the actual assumed scenario which, broadly, is "Imagine that within an intellectual realm - which is not fully accessible by human minds - a construct exists that contains a finite set of sets as follows: [0 . an1 an2 an3 ...]" (or something like that). This would have been ballsy enough to make it clear what we're actually talking about. And it would have immediately nullified all manner of spurious objections like "Oh, I can't actually write that down so the claim cannot be right". My point was - once again - that you cannot simply consider the first state of the hotel (countably infinite rooms, all occupied) with the second state (countably infinite rooms, one of which contains two occupants and where a reassignment algorithm is running without ever being able to end. This, in Turing speak, is analogous to an infinite loop. Surely, you can see that these two hotel states are not equivalent? To fix the mess all we have to do is assume the presence of an algorithm that can execute any countably infinite set of instructions instantaneously. This is quite similar to the ‘Intellect’ imagined by Pierre Simon Laplace - the source of the famed ‘Laplace’s Demon’ which now - reimagined as ‘determinism’ - forms the bedrock of all scientific thinking.

[@andrewsheehy2441]

@Quantum Bubbles Thank you. Well, I can see from that that we are very different. I am very respectful of those who are operating multiple levels above me. But I am also - over 30+ years professional experience in electronics and software engineering - able to spot problems in foundational thinking, gaps in thinking, inconsistent thinking and BS, You seem to be a consensus guy. I’m not. I subscribe to what I consider to be quality thinking - and I don’t care where it comes from. I suggest you consider researching the field of metacognition - the most advanced form of human thinking. You seem to be a pretty articulate and smart person. But are you just a ‘yes man’ - or can you really think for yourself?

[@andrewsheehy2441]

@Quantum Bubbles I must say that I'm rather enjoying our exchanges, although we must soon wrap things up. To be as brief as I can: 1. I made a specific point (see: page. 2 of the Hodges paper) where, in Point (3) of the argument, we see the sentence: "Write 0 . an1 an2 an3 ...". It is Hodges - NOT ME - who has chosen to use the word 'write.' It is Hodges, NOT ME who has introduced a 'foreign element' into the argument - as you put it. If this is a formal mathematical proof then, frankly, I'm not impressed. What does 'write' mean? The answer is that we don't know, because it is not defined. Maybe it literally means 'write three dots’ - we don’t know. Maybe it means 'write in your mind'. We don't know - because it is not defined. Fyi...I build large-scale software products and direct team sizes of up to about 100 engineers. Software, as you will know, is based on logic and if I saw something as sloppy as this - analogous to a variable that is undefined (no data type, no data structure etc.) - then I'd clip someone's ear. Surely you can see that I have a valid point, even though it was not the specific one I made at the beginning?? The specific point I made at the beginning - which, and I'm sorry to press you on this, you're persistently refusing to specifically address - was how can we consider the initial state of the hotel (countably infinite number of occupied rooms) to be the same as the final state of the hotel (countably infinite number of occupied rooms with one room always contains two occupants) as the same UNLESS we are rigorous and explicit about how the reassignment algorithm is assumed to work? You seem to be referring to some level of understanding of the concept of ‘algorithm’ that is very advanced. Fine. Please define it or refer to it or use a technical term that people can then research. What's wrong with saying that we're just assuming that the algo can - in a purely intellectual sense only - perform the room reassignment on a countably infinite set in a bounded way? If we cannot say that then the algorithm can never end and the states of the two systems are not the same. This is extremely clear. Surely you can sense that this is a valid point? Look, I just genuinely want to understand this - but cannot do that unlessthe argument is rigrourous and crystal clear. You cannot say "Well, this is a advanced concept and only very experienced mathematicians have the intellectual capacity to understand it" That is total BS. And you cannot say "Yes, alright so it doesn't make much sense - but nor does countable infinity so get used to it, laddie" Either the argument is sound as it stands, or it isn’t. And I think any reasonable, intelligent person who looked at the above two points would agree that the argument as presented by Hodges is not fully formed and needs some edits. As it stands it the Hodges argfument is sloppy and the Hilbert Hotel is illogical (see above two specific points, respectively) 2. On my agreement with the last sentence in the Hodges paper, I was simply referring to Cantor’s philosophical position - which was essentially Platonic (world of forms idea): he saw infinity as a mathematical construct that exists within in a purely intellectual realm that is separate to reality. Being a Platonist myself, I’m on board with this. Even Hilbert thought that infinity had nothing to do with the real world. He wrote in 1924: “The infinity is nowhere to be found in reality. It neither exists in nature nor provides legitimate basis for rational thought. The role that remains for the infinite is solely that of an idea.” All I’m asking - to return to the point - is that we also invent another idea which is that within this intellectual realm that contains the idea of ‘infinity’ there is also another idea that can perform mathematical operations - like the room reassignment algorithm - on a countably infinite set in a bounded way. If this was included then I’d be 100% on board with the Hilbert Hotel, but until it is then it is nonsense.

[@robinbrowne5419]

Prof Wildberger, I wonder if you could do an episode on Computer Numbers. For instance: - Made of bits 8, 16, 32, 64 - 2 ^ 8 = 256 possible numbers. - Types of numbers Int, Long, Signed, Float. - Operations and Symbols + - × / ^ - Functions sqrt(x) etc. - Constants pi etc. - Accuracy increases with number of bits. - Rounding errors. - All computer numbers are Rational Numbers. Just a thought. Would be interesting for Sociology of Math.

[@rhaq426]

What do you think about the axiom of determinacy then?

[@antizephdaniel7868]

Bob and Alice can play games of infinity and "forget" their true form. Charlie will always know his finite form and has nothing to do with that game lol.

[@rhaq426]

@@antizephdaniel7868 How about the axiom of REAL determinacy

[@antizephdaniel7868]

@@rhaq426The axiom of real determinacy involves polish spaces and a good example is the real line which as you know has a paradox because there are infinitely many "Real" numbers between every interval of real numbers.When I work with set theory I do not use infinity but I prefer to use supremum and infimum.Infinity is misleading. Non-trackable and extremely long processes are what give the perception of infinity.The Cantor space is the result of a long/non-trackable process for example.This is also a polish space.

[@rhaq426]

@@antizephdaniel7868 ah i see - nice

[@AllothTian]

During the middle ages, scholastic philosophers would argue about questions like "How many angels can dance on the head of a pin?" It didn't matter whether such questions make sense to begin with, all that mattered was the ability to embed them into a logical framework. I was exposed to the history of philosophy before I embarked on my study of pure mathematics, so I could never let go of the impression that a lot of modern mathematics is just modern scholasticism in disguise. For example, let's take Cantor's diagonal. We're asked to assume that such a diagonal can be constructed and we proceed from there to establish the theory of Aleph numbers. Was our initial assumption about the constructability of such a diagonal reasonable? Turing set out to answer this question in his famous thesis, and in this quest he settled the issue of the Halting problem. How many CS or maths students are taught that the Halting problem was just a stepping stone towards a proof of the fact that Cantor's diagonal requires one to produce an algorithm with an infinite number of steps - which would be impossible even if we tried to brute force enumerate all possible algorithms (due to the aforementioned Halting problem)? I'd venture to guess that very few, if any, are taught this fact. And in light of Turing's proof, should we assume that the constructability of Cantor's diagonal is relevant to any human that exists in our universe? I'd say it's about as relevant as a discussion of the logical consistency of Tolkien's fictional universe - interesting to prospective writers, perhaps, but hardly relevant outside questions of aesthetics.

[@qswaefrdthzg]

Well yes. Mathematics is just a logical game. With the added bonus that it seems to be able to describe reality rather well and that we can properly truncate abstract so that they are useful in applications. E.g. the fundamental theorem of algebra tells you that in fact a polynomial has a root, and further theorems tell you that you can compute explicitly arbitrarily closely and that you are approaching "something".

[@AllothTian]

@@qswaefrdthzg Oh, I'm not disputing that. In fact, analysis, to me, feels a lot closer to the field of formal verification in computer science where you can reason about the behaviour of a particular algorithm / system without actually running it. We can determine whether algorithms will terminate with an error under certain restrictions (general case is obviously undecidable). We can even reason about whether they'll satisfy certain properties. However, the emphasis here is on algorithm analysis, not Cauchy sequences or Dedekind Cuts. Algorithms are far more useful because they tell you how something is constructed and how it changes. I was somewhat surprised when I first learned that the algorithmic point of view is dismissed as "limiting" in higher mathematics... To me, this is similar to saying that the scientific method is limiting because it rejects the possibility of supernatural phenomena if they can't be subjected to empirical testing. I guess the mindset of wanting to spend your life playing logic games (I know not all mathematicians fall in this camp) is quite foreign to me, hence why I chose not to pursue pure mathematics as a career.

[@ThePallidor]

@@AllothTian I would call much of the math since Cantor semantic games rather than logic games. The goal is to see how far you can advance your career via wordplay and loose definitions. A fun game for some but not my cup of tea.

[@tantzer6113]

Why limit math to what is physically possible?

[@qswaefrdthzg]

@@AllothTian I guess my point of view concerning for example the real numbers and Cauchy sequences is a fundamentally geometric one (I have to admit I have always hated doing actual calculations). It gives meaning and logical structure to our intuitive notion of continuity: when do a circle and line cross? When it looks like they do, not when some to me esoteric Diophantine equation has a solution. Watching Dr. Wildberger's videos I had to think through a lot of his points, but I am fundamentally ok with the non-constructive aspects of the real numbers. It simply reflects the fact that when you bisect an interval you can always bisect the other two intervals and thus for example get something like an infinite random binary sequence describing a point on the line. Otherwise you are forced to work with intervals of uncertainty, Wildberger does for his "angles", the real numbers just allow you to do both. Also if you prove something within the real numbers constructively, you can always truncate and get approximate rational results, showing that nothing fundamentally new or inconsistent with what you new before happens.

[@rufismalan]

Kudos to Mr.Widlerberger, this topic and godel second IT is what hinder people like me (some kind of engineer) to really delve into mathematics and appreciate it more yet manage to continually keep me intrigued over time and at the same time continuously keep making me questioning my reasons (cannot be an overstatement enough 😂). The pov of non computational is maybe philosophical and sociological, but whether the indexing system - or 'choice framework' - that Mr Wildberger mention briefly here is sufficient to outline the problematic nature of BTP is one thing i still not understand clearly. The way i see it is if i am (and imperative at certain level) being empathetic, giving the general mathematician consensus a benefit of a doubt then what the choice framework real problem is not only the unknown oracle thing as i think there maybe other factors in play, i perceive indexing problem here like the problem of infinite set, cardinality, aleph 0 vs aleph 1 etc. Meanwhile based on scant reading of some materials of BTP i suspect that there any other underlying concepts in BTP that suggest finitism as Mr.Wildberger talk here, this is not a problem of arbitrary unknown infinite indexing but maybe about other concept like 'existence of immeasurable finite set(cmiiw)... etc' that make this intriguing and counterintuitive that i not know about??? Back again if it is within 'finite' context I belief then it is visualisable but i also suspect that it is not, even though i think should and please Mr.Wildberger if you can (even not fully accurate) please make attempt on this and relief me from this insanity 😂. There should exist communicable resolution on general BTP understanding that is accessible for layman and not dismiss it as pure nonsense, i am really twisting my brain trying to write this comment 😂😂😂.

[@winsomehax]

I'm not a mathematician... I mean REALLY not one. With that disclaimer: so much of the stuff that confuses me involves a sneaky infinity somewhere in the argument.

[@ThePallidor]

If nonsense doesn't confuse you, there's something wrong with your thought process. The proper response to nonsense is to say you don't understand what the person who said it is getting at, but in academia that loophole is used to pretend at higher knowledge.

[@mahanvazsilva112]

There is no problem with this theorem. It only says that the way the continuum behaves is counter intuitive.

[@mahanvazsilva112]

@Gennady Arshad Notowidigdo To be a paradox, it should lead to a contradiction, which it does not, given that it can be proved. So it is, strictly speaking, not a paradox - although it is debatable what is exactly a paradox anyway. That's why I prefer calling it a theorem, since it can be proved from axioms (ZFC) or other theorems that Banach-Tarski holds. As for the "underpinnings of the entire paradox" it is underpinned in measure theory, more specifically, the non-Lebesgue measurability of the reals. You can check "The Banach-Tarski Paradox" by Tomkowicz and Wagon to understand it better. Also, Hrbacek and Jech have a good introduction in Set Theory which mentions the weird behavior of the reals. Cheers.

[@thesmallestatom]

I agree OP. This is the reason why I always struggled with traditional presentations of calculus.

[@pmcate2]

Just curious, what are your thoughts on John Gabriel?

[@StevenSiew2]

The Banach Tarski paradox is NOT nonsense. But it does not work in the real world because real matter is NOT infinitely divisible. Pure Mathematical sphere is infinitely divisible. Matter in the real world is made out of atoms and once you get down to the level of atoms, you cannot divide them anymore.

[@High_Monk]

Of course you can divide atoms, what are you talking about?

[@matthewcory4733]

As Stan Wagon points out, analysis is not even helped by eliminating the Axiom of Choice: "It turns out that ZF + LM implies that a certain paradox exists that is just as troubling as the BTP, if not more so. To set the stage, consider the National Football League, which has 32 teams and 53 · 32 = 1696 players. If the players were assigned to teams in some other way, subject only to the conditions that a team cannot have zero players and each player can be on only one team, then there can easily be more than 32 teams. But could there be more than 1696 teams? Of course not. The idea of grouping players into nonempty teams so that there are more teams than players is ludicrous. Yet this is the essence of the phenomenon that Blass told me about: this sort of thing can arise in a world without the Axiom of Choice." Long live ultrafinitism.

[@Pygmygerbil88]

thanks for your kindness professor

[@markthiele770]

where did he get the values for the first points ie .742,.214,.380. I must have missed something

[@baze3541]

Is ur comment a joke that im too stupid to understand?

[@ivand8393]

If a ball with R=1 consists of infinite number of points then a ball with any real Radius can be "build" with the same "number" of points , right ?

[@antizephdaniel7868]

All is DREAM and all is REAL, because dreams are part of reality .Set theory makes it possible to create any Axiom that suits your fancy.I call it virtualization, creating a simulation within a simulation. Access to all possible dreams is what is Real and that is Mathematics.

[@ThePallidor]

The set theory axiom "There exists an infinite set" is not even possible to visualize in a dream. It is just meaningless words.

[@antizephdaniel7868]

@@ThePallidorIn set theory there is the notion of Infimum and Supremum and it applies to topology.In the realm of a topological quantum computer where my dream will be taking place, I think you will find an infinite set as elements in the ordered set T, greater than the supremum of the Subset S.

[@davidste60]

Dreams are PART of reality, but they do not CORRESPOND to reality.

[@antizephdaniel7868]

@@davidste60In terms of dream as an imagination, many things that you see in your reality were once dreams of the people who created them.Secondly I believe in Plato's Universe which states that Space is in some sense absolute.Hence when you sleep at night the space you find yourself in your dream state actually exists some billions of lightyears away!

[@martinflygar7699]

Maybe You could make a video of the paradox of Gabriels horn as well?

[@christopherellis2663]

Numberphile have done one

[@thomassynths]

@@christopherellis2663 Numberphile is very hit and miss. Their horn one was ok, but it also prescribed to standard-taught mathematics of today.

[@iosephusgranicae8769]

So I wonder what would it be from a finitistic perspective for one to view the Gabriel's Horn...

[@stephenmorton9789]

Well the real numbers come from ZF axioms and the Axiom of choice and then the Banach- Tarski theorem is derived. I don’t think you need to give lecture after lecture when basically you just don’t like the axiom of choice.

[@cariboubearmalachy1174]

I dunno. B-T to me is just the same as saying theres the same number of even numbers as there are natural numbers bc you can make a 1-1 correspindence between them. Yeah, we can match points 1-1 between spheres of different size, but we already knew that all spheres, no matter what size, had the same number of points, that is, infinite points.

[@BrynSCat]

If you treat 3d space as a SU3(Hopf Fibration ?) and you split into the six pieces as the paradox states you can no longer do the rotations as stated in the paradox if you want the rotations the half the volume. You can probably prove this with a quaternion ?.This positively closed topology limits infinities and infinitesimals ?.

[@luisrosano3510]

You can put in bijection a segment lengh one with another segment lengh two because both have a infinite numbers of points. If is it correct, then the unit sphere can put in bijection with the sphere radio two. I am right???? That is what they teach us on the college.

[@alberthadonlyone]

You can create a bijection from one set to another if they have the same number of elements (so both being infinite is not enough) but yes, you can create a bijection from a sphere (in R^3) to one with a radius twice as big (the bijection in this case is simply multiplying by 2, assuming the center of the sphere is at (0,0,0)) but that's not really what happens in the Banach Tarski paradox.

[@thexen3120]

Drumstick pointer....make a video of you playing some polyrhythms!

[@exodus_from_babylon]

I watched this in my local. I'm now barred

[@jakeb.2990]

sets defined that way have indeterminate cardinality, I really wonder why do they talk about their size at all

[@lultopkek]

I really enjoy your videos. Thank you so much. My brain can really use some destraction from the insanity going on atm.

[@thomassynths]

I initially misheard you saying "weird mathematics" instead of "pure mathematics", which I think sums it all up nicely.

[@dsm5d723]

Well I'll be (?). This is essentially a thought experiment of mine, but in physics. No pure math involved. Just the LANGUAGE of 3D unpacking of it through the middle voice displacement and case (decline/incline) switching of ergativity. How would you weight the Earth? Not use an abstract method for computing a mass, but actually put the bitch on a scale. To be brief, Universal Scaling Metrics are an analogy for the Euclidean plane. So is phase change. Solid, liquid and gas; plasma is "special." To weigh something, you must align its force vector down. By inverse, you can get a helicopter or postal scale up and do it backwards. Or, you must be standing on a curved 3D surface. So, by the scaling laws of binary, you would need an Earth Squared, and you would have to STOP the dynamical friction value of energy in the system, in the same way that a battery is slightly heavier when charged. Or, YOU could compute the infinitesimal values from the IIT full description of the system. This is a great advancement, because it replaces the Hilbert Space needed for quantum wave functions. Finite and discrete calculus, that is your lane. Try to charm quantum idiots into looking at your work.

[@MsAerokiss]

Please make a video about model theory (math logic)

[@pn8646]

The simplest way to phrase this, is that something finite (i.e. a sphere made of matter) cannot be made into the infinite. If the hypothetical sphere can be made into an identical sphere, then the original sphere, and therefore all subsequent spheres, were made of an infinite matter. While hypothetically an infinite matter substance can exist, It does not exist in the world of humans or human observable science, including advanced sound theory math. This theory postulates that, essentially, something can be made from nothing. And further, it assumes that a sphere, a finite object, has infinite points on it, which it does not; any quantifiable shape, even at the most analytical level, is just that....quantifiable if you spend a finite amount of time (no matter how long it may be) quantifying it. Math gone wrong. Simple.

[@romnickbuenaflor4703]

No. You simply assumed Infinity does not obey rules applying to the other numbers. The paradox appears only because of that assumption. When a concept such as infinity assumes many “values”, then there is no real or physical equivalence for it. Only in the thought process… MATH has mislabeled infinity as a concept applied to numbers. What MATH actually requires, is a concept of undefined number, or unknown value, which can take on only one “value”. Then there will be no paradox…

[@alexandartheserb7861]

Σ & = 1, i.e. in phyosophy/religion: we are all one

[@oriongurtner7293]

I see it this way: you can represent doubling a sphere on paper like you can represent the volume of a 3-dimensional space on paper, as in it’s not real at that point. A sphere is only a true sphere in 3d space, as volume is only truly volume in 3d space Essentially: there is no volume in 2d space, and there are no spheres in 2d space, these exist in higher dimensions and their representations in 2d space are more imaginary than ‘imaginary numbers’

[@theoremus]

This is above my head. Is it similar to the proof that 1 = 2? There are KZbin videos on that.

[@alessandrodellacorte1737]

No it isn’t. Those are just fake proofs.

[@gausssto570]

Those are joke videos for fun. They aren't serious.

[@Adam-rt2ir]

Banach-Tarski "paradox" is completely serious math

[@calebcurry4458]

Those "proofs" usually involve dividing by zero somewhere, so they aren’t legit

[@G0ldbl4e]

The entire paradox is nothing more than a very fancy way to say 1 / aleph(1) = 0 0 * aleph(1) = 2 1 = 2! By "defining" (dubious word) an object as an infinite number of zero-size points, you are giving that object properties it didn't have before. If you only look at this as "infinite array of points" and cut out the smoke and mirrors of pretending that these arrays "are" the singular objects at the beginning and end, then there's nothing unusual or paradoxical as far as math surrounding infinities goes. It's not a paradox of "infinities", it's a problem entirely derived from being sloppy with how you can define something.

[@pacajalbert9018]

V škole ostatné čísla pre mňa nemali význam pre to že hneď som zistil že počítam to samé a existujú len dva čísla po ľavé strane a práve strane preto som hľadal Einstana pro to že som mal v podobný obraz kvantovtove mechaniky teraz si predstavte si že som mal asi len 7 rokov

[@joecotter6803]

Nobody knows how many fallen leaves there are in the world ar time t but we do know that it is a finite number. The fact we can't write it down is irrelevant. Similarly withe the digits of root 2. Norman's arguments against real numbers apply to finite sets. If we denote the number of leaves fallen by N, we can then manipulate N. Similarly we can manipulate root 2. This finitism project is interesting but excludes people from the richness of Pure Mathematics. Pure mathematics does not contain a time dimension. Hence all infinite or finite 'choices' occur instantaneously. Which is impossible in the real world, which has a necessary time dimension. This does not invalidate Pure Mathematics. For readers who want an account of numbers, find John Conway's videos on the Surreal numbers and Donald Knuth's book on the subject. Conway and Knuth are giants in their fields.

[@santerisatama5409]

The main criticism is to my understanding against the deep dishonesty of the purely axiomatic and arbitrary ad hoc claim that real numbers satisfy field axioms. Norman is not a strict finitist, AFAIK, there's much to be explored and said about unbounded / transfinite phenomena. The social and communicable aspect of mathematics has it's own requirements which we need to take very seriously, and that's what Norman does with passion. The relation of mathematics and time is an open question much less discussed than it deserves; the fact that most work so far has been static system building does not mean it has to be so. Computation theory naturally involves temporality, and undecidability of Halting problem is temporal issue with foundational level meaning. I've read only Knuth's book. I think that in general, formal language games evolving into game theory is a positive step, but I'm not quite sure what to think of the fact that in Surreal numbers you need infinite game to find 1/3. I think that in that respect the game(s) of Stern-Brocot trees (yep, there are many) are much richer in deep structure with much wonder waiting to be found. The most important key finding of Surreal Numbers is the revelation that equivalence relation reduces to relations of relational operators. Ie. if A is neither more nor less than B, then A = B (in a given context). No more need for presupposition of Law of Identity.

[@joecotter6803]

@@santerisatama5409 It depends what number base you use. In base 10 1/3 =0.33333333 recurring. In base 3 it is 0.1.

[@santerisatama5409]

@@joecotter6803 The comment was about Surreal numbers. Number bases have nothing to do with the issue.

[@joecotter6803]

@@santerisatama5409 errrr... You can define surreal numbers in any base you like. Finite representations can be made of any rational number. Norman has to get over his obsessions. Is the whole of the world of Pure Mathematics wrong? Does y=x² - 2 intersect the x axis at a point? Or at a hole in the line? We must accept that root 2 + root 3 is a number. Norman doesn't.

[@santerisatama5409]

@@joecotter6803 The distance of the objects that the game of Surreal Numbers generates from the start of the game is independent of base. The distance of generating surreal 1/3 in any base is infinite, because the game is binary. In other words, in Surreal numbers 1/3 is not anymore a rational number, it becomes analogous to irrational real number. A problem in English is that it has only one word, 'number', which leads to shallow debates about what is a number and what is not. In Finnish we have both the loan word (numero) and our own word (luku). In our language 'numero' refers mainly to the signs, ie. what can be written, but 'number theory' is called lukuteoria. So in Finnish we don't need to accept that sqrt2+sqrt3 is a number, which it obviously is not in our meaning, and we can think and discuss what kind of luku it is, if any kind, perhaps more freely and philosophically. Wildberger has good insight of this issue, in his discussion of pi he says it is not a number, it's a landscape. There was an era when Hungarian mathematicians were exceptionally prominent, and that might have something to do with the linguistic and cultural background and how it orients to intuit and think about math.

[@rationalagent6927]

Not sure where else to ask this but if we reject the axiom of choice does godels incompleteness theorem hold?

[@santerisatama5409]

Yes. Gödel's results don't depend from axiom of choice or axiomatic set theories in any form.

[@talonward2494]

Came here to see if N J Wildberger were still batshit crazy. Found his two solid balls.

[@talonward2494]

@Quantum Bubbles he may hide it better, and he may make fewer mistakes, but his ultimate argument is essentially the same as the guy who says, "There are only finitely many natural numbers because only God is infinite." He has a fundamental misunderstanding of what Mathematics is, right from the start. And, in the true spirit of Dunning-Kruger, he has a snarky attitude and no doubt whatsoever that nearly every other Mathematician in the world is wrong -- but he is right -- even though he can't even state how. Like the anti-vaxxers, the COVID conspiracists, the climate change deniers, and the flat earthers, he uses his inability to comprehend or accept something as evidence that he is right and others are wrong. That is basically the definition of batshit crazy.

[@talonward2494]

@Quantum Bubbles oh, certainly, I expect his work to be accurate. When I say "Dunning-Kruger," I don't mean in regard to all of Mathematics, and I certainly don't mean to address Mathematics as a monolith; it is an enormous subject. Rather, I mean specifically to this area in particular: grasping the concept of infinity. And of course infinity is hard for humans to understand. Even basic Probability is notoriously difficult for humans. Our brains were not built for this; we're stretching our limits. And certainly you can have Mathematics without infinite sets (just remove the Axiom of Infinity, haha), and, if he prefers to work without infinite sets, that's fine. If he reworks trigonometry so that you can avoid irrational numbers, that's borderline interesting. But the idea of rejecting "infinity" because you've never seen (or could never see) infinitely many things is so off-base, so ridiculous, so infantile, and so laughably, obviously false. "Hello, I would like √(-1) apples, please." Do imaginary numbers not exist because he can't count them? (Wait, does he have a video on that?) I suspect he would say that an imaginary number is an ordered pair (x, y) of (rational! haha) numbers and believe in them because he can write them down. But, of course, that's not what an imaginary number is. Just like a "real number" isn't an equivalence class of Cauchy sequences or a Dedekind cut. These are constructions that prove the existence of something that obeys the axioms. Real numbers are not literally Dedekind cuts any more than the number two is literally two apples. Saying real numbers don't exist because you can't finish writing out the decimal expansion of them is such a gross misunderstanding of what a number even is. Just because we only have access to finite classical information, and there are uncountably infinitely many real numbers (almost every) containing infinite classical information doesn't mean those numbers don't exist. It just means we are limited in what we can say about them. I may never know what the 10^10^10^10^10^10th prime number is, and I may never be able to write it out, but I know it exists, and I know it's odd. This is the very basic, foundational essence of Mathematics -- a core concept seemingly completely lost on him. At that point, how could one even call themselves a Mathematician?

[@andrewsheehy2441]

​@@talonward2494 I think you haven’t understand Wildberger's point. He is not saying that very large numbers (meaning numbers so large that they cannot be represented in our physical reality) are not needed to make our reality what it is. He is specifically saying that the 'idea' of infinity occupies a different class to numbers that are needed to make our reality what it is. We simply do not know whether the architect of base reality (I'm using the term 'architect' in a metaphorical sense) needed to use of the idea of infinity - at all. I mean, let’s imagine that the simulation hypothesis is true then it would be reasonable to assume that everything is running within a computational realm that is not only quantised by which is defined by finite numbers: one would expect that the 'computational structure' being used is limited to a finite number of decimal points (say). Sticking with the simulation idea, we would then be talking about an engineering problem and the way things work there - is that people (engineers) - do integrals by counting discrete chunks, because the closed form expressions are either too hard to work out or cannot be expressed mathematically. As you will know mathematics cannot handle the things when complexity increases beyond a modest threshold. This is a very well know problem. Remember that both Cantor and Hilbert never thought that their concept of infinity was part of our reality. Cantor observed that while there was (and is) no evidence that shows how infinity is expressed in our reality this does not invalidate its use as a legitimate mathematical object. The same goes for the idea of a sphere. Thus, both ideas - those of infinity and a sphere - are legitimate mathematical objects. Hilbert wrote in 1924: "The infinity is nowhere to be found in reality. It neither exists in nature nor provides legitimate basis for rational thought … The role that remains for the infinite is solely that of an idea." I think Wildberger's point is that while we can see examples of objects that resemble the idea of a sphere in our reality (e.g. a black hole) - which one can imagine are somehow ‘convergent to' their ideal mathematical form - there is no such evidence of such an example when it comes to infinity, especially when the idea of infinity comes in different flavours (naturals, reals). Once can conceptually see how the surface of a black hole is an close approximation to that of a sphere. The number you mentioned - the 10^10^10^10^10^10th prime number - and the number of electron transitions in all the atoms that have ever existed in the universe both 'exist' in a clearly bounded way. I don’t think Wildberger has a problem with this: we cannot write these numbers down but it is possible to imagine their existence in a mathematical realm. For the reasons I’ve just given, Wildberger's claim is that the idea of infinity occupies a different CLASS and it should therefore not be lumped into the same class as ideas that we can easily see correlate with our reality in a bounded way, even when those ideas cannot be explicitlly expressed within it (meaning written down or represented in computer memory device). I think this is a very sensible and legitimate point. I frankly doubt that his effort to redefine ZFC axioms will get anywhere (at all) - but he is asking good questions. I'm sure you know that many leading mathematicians at the time were strongly opposed to 'infinity' being regarded as a legitimate mathematical idea. It was only Hilbert's force of character, intimidating style and an inner belief that he was some type of ‘saviour’ whose sole purpose was to 'save' mathematics - his life's work - that resulted in infinity being what it is today. Things could well have turned out very differently. You might consider being a little less dismissive and binary in your assessment.

[@talonward2494]

@@andrewsheehy2441 I understood what he thinks his point is; you didn't understand mine. The example of the 10^10^10^10^10^10th prime number was precisely something he would accept to illustrate that there is no difference. Reality has no concept of "two." Nowhere in the universe is "two" used. There are, however, instances of two objects. Reality has no concept of "infinity." Nowhere in the universe is "infinity" used. There are, however, infinitely many natural numbers. To say that a real number "doesn't exist" because you could never have the infinite classical information required to store an algorithm that could generate its digits is to completely misunderstand what a number even is in the first place.

[@andrewsheehy2441]

@@talonward2494 "The example of the 10^10^10^10^10^10th prime number was precisely something he would accept to illustrate that there is no difference." What? How can you possibly say that the idea of a bounded, discrete natural number is in any way the same as the idea of a countably infinite set. These are two radiacally different concepts. If your position is to argue that they are the same then I'm completely lost.

[@ja524309]

You don't even need Banach Tarski to see things like this happening in mathematics. Take for example the entire set of natural numbers. This set can be put into bijection with a strict subset of itself, a seeming contradiction akin to Banach Tarski and his two balls arising from one. Except this is extraordinarily uncontroversial and doesn't even require Real Numbers, which Wildberger incorrectly seems to think is what is causing all of this trouble here, instead of the obvious fact that infinite sets just behave differently than finite ones.

[@robharwood3538]

You're already assuming that infinite sets exist, and appealing to authority (and/or popularity) in that it is (supposedly) extraordinarily uncontroversial. But that is just begging the question. If it's so extraordinarily uncontroversial, then it should be extraordinarily easy to demonstrate the existence of such an -- or indeed *_any_* -- infinite set. Yet the only thing you can fall back to is an unfounded *_assumption_* that such sets exist. That is the very definition of begging the question: Presuming exactly what you're trying to prove in the first place.

[@ja524309]

@@robharwood3538 I'm not really sure what you mean by "exist". Isn't it much more arbitrary to stop at some natural number N and say "this is our biggest set".

[@rynin8019]

@@robharwood3538 Talking about existence in mathematics has always seemed strange to me. What is the issue with simply defining a set of elements x such that x is in S if x is a natural number?

[@Kraflyn]

^this. BT paradox is nothing other than blowing dust over the simple fact that the Cantor theory is inconsistent: there is no bijection between even integers and all integers. Now, the entire modern mathematics is built upon the Cantor's theory of infinity...

[@rynin8019]

@@Kraflyn What do you mean, there's no bijection between the even integers and all integers? There's an incredibly easy one: f(x) = x/2.

[@Runnin_Wit_Scizzahs]

May be the thing to look into to help explain some of the Quantum observations. Perhaps the rotations required to reproduce the second are not rotations but something else. Even smart people like yourself are limited to a marginal perception. If you can prove its impossibility, I would love to watch the youtube video you make for it.

[@alexandartheserb7861]

1mother + 1father = 3babymotherfather. From where comes third spirit if 1+1= just 2?

[@carly09et]

This is just points : it is the same as the Hilbert's Hotel paradox. :)

[@santerisatama5409]

I once tried to ask Hilbert what is point, what is his definition of the concept. I should not have done that, his axioms of geometry responded.

[@Nick08352]

But doesnt the Paradox just show that we need something like measure theory and not that wie can make 2 Balls out of 1? Thats why its a paradox right?

[@BrunnerNathan]

I agree, this paradox is a huge red flag.

[@spiveeforever7093]

Another big problem that choice interpretations neglect is the halting problem!

[@santerisatama5409]

That's very interesting observation. For intuitionist philosophy of math, there's time before and after the proof that Halting problem is undecidable. Before it was thought that a proof event spreads immediately to eternity, both past and future. Halting problem together with Curry-Howard correspondence and Church-Turing thesis implicates that we can't no more consistently speak of eternity of mathematical proofs, only indefinite temporal durations. Which is closely connected with Bergson's philosophy of time. Halting problem is truly a foundational event. Imposing LNC (which the proof of undecidability presupposes) over temporal processes in not unproblematic, but any case Halting problem offers fairly precise and foundational concept for transfinite / open-ended processes. Reinterpreting relational operators < and > as open ended processes is a very interesting possibility.

[@reubenhaynes]

Completely wrong. - The paradox is real and I experience it every day. Yes I’m writing a new paper to disprove that anybody who believe that isn’t true. It is! You’re not think out-of-the-box or at least out of your own reality because your reality, it’s not three dimensional. It’s eight dimensional.

[@TheDavidlloydjones]

No, Norman, it's not "First" at 3:34. The flim-flam has gotten underway well before your quite correctly mocked "free groups." We veered off into pure fiction earlier with the apparently realistic specification of "three-dimensional Euclidean space." The very basic problem here is one of the normal magicians' misdirection. Saying the soothing words "three-dimensional Euclidean space," (yes, yes, we all know what that is) lulls us into imagining there might be some other kind of space. There isn't. Three-dimensional, omnidirectional, space -- floors, ceilings, planets going "around" in their ellipses, and all that -- is the only kind of space there is. Other uses of the word "space" are other meanings of the word. An eleven-dimensional space-time construct of banana-theory* or whatever physicists are putting in grant applications for this month is not space. It is a construct, so called because physicists imagine they have constructed, i.e. made, something when in fact they have merely dreamed it up. They're very good at dreaming stuff up because the big money tossed around in the field since atom bombs became dangerous attracts a lot of talent. Physics is kinda like Hollywood without the attractive women so of course a lot of imaginative men turn up to sample the wares. *"Banana theory." Oops. I remember now. It's string "theory." This is the discovery that if you got an inconvenient infinity, (these days renamed an N.A.N. for "Not A Number") by dividing something by zero, a no-no, you can get rid of it easily: you just replace the zero with a little itsy-bitsy. We can't call our little itsy-bitsies "points" because physics is already full of fictional point-masses which are inconveniently zero in height, length, and breadth. Let's call them strings. We'll make them too small to see. They'll be much smaller than the wavelengths of light, so they'll be permanently either or both unseeable because invisible or invisible because you can't see 'em. X-rays? Gamma-rays? Nope. By the time the wave-length of your illuminator becomes short enough to see *these* strings, it, too, will have turned into strings. So there. Take that, audit committee. Now, strings vibrate. Harmonically. Harmonic vibration has been studied by Legitimate Academics for at least five thousand years now. Wonderful: we have a notion that will appeal to funding committees. And thus we have all the physics since Congress stopped being willing to fund a big hole in the ground in Waxahachie, Texas.

[@TheRosyCodex]

Music to my ears

[@TheRosyCodex]

I'm working on a video series to re examine this whole notion of dimension

[@qswaefrdthzg]

Ok first things first I think string is as bogus as you do, however, you are completely mischaracterising the problems with it. We can "see" what are to the best of our knowledge point particles of the standard model just about as well as we could potential strings. Also what point particle means is not that they are in fact "points" but that they lack any further internal structure (if you are interested take a look at form factors and those turn out to be point like Delta functions for the elementary particles as far as we can measure). Also saying that we know that 3D Euclidean space exists, also strains what we have experimental evidence for. Anyway, mathematics fundamentally is just a game of setting up some logical premises and looking where they might lead. The fact that it turns out we can describe reality to some degree is a nice bonus but completely independent of mathematics. Hence saying that we take a d-dimensional space is simply a logical premise, and we would like to study its properties. But yes it's just a game. There is never a claim to the physical existence anywhere.

[@robharwood3538]

"The very basic problem here is one of the normal magicians' misdirection. Saying the soothing words "three-dimensional Euclidean space," (yes, yes, we all know what that is) lulls us into imagining there might be some other kind of space. There isn't. Three-dimensional, omnidirectional, space -- floors, ceilings, planets going "around" in their ellipses, and all that -- is the only kind of space there is. " Except: We already know that the space (spacetime) we live in isn't Euclidean. Black holes for the most extreme counter-example. But just Einsteinian relativity in general as a more thorough counter-example.

[@antizephdaniel7868]

@@robharwood3538 Why does it have to be either Euclidean space or hyperbolic space? Why not both.Mathematics is different than other subjects.We think as not to limit ourselves .This might sound stupid but ask yourself if the puppeteer interprets the puppets time as "time"?Minkowski space is too time dependent but if you understand Euclidean space then you hear the music.

[@reintsh]

Finally stumbled upon this video that's already 3 years old, so my comment is a bit late. I have always rejected the B-T paradox. It is not a paradox. It yields an impossibility, i.e. a contradiction, thus falsifying at least one of the premises. Ex falso sequitur quodlibet, from falsehood follows anything you like. My simple reasoning is that pretending to touch infinity and then come back home invalidates ANY proof. Infinity is uncome-at-able and NOT a number. It easily beats Graham's number plus one... It is like "proving" the echo of an infinitely deep well, not by shouting and waiting until you hear it, but by mere reasoning. A bit like those morons saying infinity minus infinity equals pi. Yes, pi plus infinity is infinite, but you cannot simply move infinity to the other side of the equals sign and swap its sign. Albert Einstein: Propositions obtained by pure logic are completely empty with regard to reality. And yes, he DID say this (at the Herbert Spencer lecture, Oxford, June 10, 1933).

[@TheOneMaddin]

What is the contradiction? Is it an axiom of your personal set theory that doubling a ball is not possible, or what are the axioms you derived it from?

[@reintsh]

@@TheOneMaddin The physical impossibility is the contradiction. It is definitely against all known laws of nature. I am a realistic physicist who does not automatically accept things because maths says so.

[@calebcurry4458]

Thank you wonderful teacher! Another win for finitism!

[@brendawilliams8062]

Love from your biggest fan. 🌻

[@TheFeatInk]

Great explanation 🙂

[@Adam-rt2ir]

That's a good pronouncuation of those Polish surnames

[@rhaq426]

Wheres the sociology??

[@antizephdaniel7868]

Sociology in terms of saying you can do something when there is no proof that you can. It is like a salesman's pitch.

[@rhaq426]

@@antizephdaniel7868 I meant sociology the academic discipline - the title says ' Sociology and Pure Mathematics '

[@rhaq426]

@@antizephdaniel7868 "Sociology in terms of saying you can do something when there is no proof that you can" ->sociology still uses evidence although it doesn't use deductive proofs

[@antizephdaniel7868]

@@rhaq426 The problem is that the Benarch Tarski paradox occurs in science but are members of the society like you willing to accept it? The schools will never teach you what will lead you to the truth.Magic is not separate from math.Is this not a social problem?. Btw I have proved the paradox in 2 different ways.

[@rhaq426]

@@antizephdaniel7868 Yes I understand that this paradox has its place in science. But at the same time, you have to think that if I taught group theory in video series called 'Maths and Biology', without explaining the non-obvious link and how it would be useful I'd be making a mistake. Is there any 'episode' in this lecture series that actually explains the link between sociological concepts and research and the math concepts described in that video (excluding statistics). Of course, you can find a link but not a very good one. If I ask why this series is not 'Math and Philosophy' or just math instead of 'Maths and sociology, could you give me a good answer? I don't want you to ponder into mathematical beauty. BTW good job on proving the paradox in 2 ways.

[@christopherellis2663]

A century of waffling!

[@chrissegroves9241]

Seems like a similar issue is in science. They say a lot of stuff that can never be tested or observed over the course of the scientists lifetime, or even the time since we've even had the scientific method as we know it.

[@stretch8390]

I'm not sure what specifically you may be referring to but if it can't be tested or observed its not going to be in a journal and its certainly not science

[@chrissegroves9241]

@@stretch8390 Science journals are full of untested, unobservable things. Example: People came from an ape like ancestor. All they got is some bones and people saying they were an ancestor. There is no test. No one observed it happen to record it happening over time. It's just a story they made up based on some bones they found.

[@ambeshsinghable]

You are best

[@outofbox000]

I love you professor😚

[@YawnGod]

+1