I love this guy. I disagree with him, but i adore watching him take shots at math and then mentally trying countering him

Thank you for doing this debate, and for putting it on KZread. This has helped me to think in a more complete way about this issue.

@njwildberger

8 жыл бұрын

+Lethargo G Peterson You're welcome.

The question 'does infinity exist' depends crucially upon your definition of 'infinity' and your definition of 'exist'. If you are having a discussion about infinity, and whether or not it exists, it is important to be clear exactly what you mean (as best you can), and be aware of how any answer to the question you come up with depends upon your particular choice of meanings. For example one could legitimately say 'infinity exists, and has hex representation 7ff0000000000000'. This notion of 'infinity' and 'exists' does not coincide with usual mathematical usage (rather it is from IEEE 64bit floating point numbers). Given the flexibility inherent in ascribing meanings to words, the simple, naïve, 'is it true' mentality is perhaps not as productive as one might like.

@njwildberger

7 жыл бұрын

What you are saying is so sensible. Somehow the importance of defining terms precisely before engaging in a discussion about whether or not "something" "exists" has escaped modern mathematics. Thanks for the excellent comment.

@maxpercer7119

3 жыл бұрын

This kind of reminds me of the situation we find ourselves in euclidean geometry, specifically we can define parallel lines as "lines in a plane that do not intersect". But using euclid's four common sense axioms, i cannot show there exist any such non intersecting lines in a plane. For this i have to have to add an axiom, that parallel lines exist.

@Newtube_Channel

3 жыл бұрын

There are legitimate qualifications for booleans in algebra and indeed computing. As far as we're able to conceive mathematics, what's true to us and what's 'eternally' true are different things. Things of an eternal nature tend to be true or false without context. OTOH we introduce a lot of bias when we look/study/talk about things. The IEEE standard of infinity is just a cap of a finite value. There are additional designations of infinity including, 'Not a Number' of +/- sign. This isn't the same infinity in mathematics. Come to think of it, infinity is little understood in mathematics and is often used as a dumpster to conclude results.

@luamfernandez6031

2 жыл бұрын

How far does the need of definitions go?

@conelord1984

2 жыл бұрын

​@@luamfernandez6031 That is a good question. One answer the constructivists have for this is that they should be suficient to construct what you are defining, which is impossible if you need an infinite amount of steps in order to do so.

Regarding the "set of all topological spaces" being too big to be of use, I would like to point out that π is a pperfectly well-defined and incredibly tame functor from the category of topological spaces to the category of groups. You may disagree with the foundations of category theory, but this note certainly shows that there is at least utility in talking about sets which are "too big."

really good arguments Prof Wildberger! really great thought experiments and points that I think a lot of mathematicians don't think about/consider

Loved that last ball game!

id love to see more of those "battles" in the future!! 40 minutes were to short

@conoroneill8067

4 жыл бұрын

If he hadn't cut off the video short it would have been longer. Cheap and dirty move that's completely unprofessional of someone who claims to be a serious mathematician.

@apolloniuspergus9295

3 жыл бұрын

@@conoroneill8067 Holy, do you have the rest of the video?

Good to see some debate between peer mathematicians but I’m left confused and unconvinced, I have to say.

Prof. Wildberg, I wish you would be more rigorous about the following question: You seem to slip indiscriminately between three objections: (1) is there a final number "out at infinity" vs, say, the natural numbers simply progressing one by one beyond our limits to count. (2) Does a number "exist" if it cannot be specifically calculated. (3) choice vs algorithm. Regarding (3), you seem to be vague on what the actual distinction between choice and algorithm is. The natural numbers are created by an algorithm: n + 1. That is utterly definable, understandable, and it can be done to arbitrary lengths, far beyond the "ability" of current computers. I don't understand how you're defining algorithm. Does the algorithm need to end in a finite manner? If so, DO the natural numbers exist, or not? You mix in this concept of how much time we have and how much storage we have somewhat arbitrarily. Why not define "finite" as "what a 2019 Mac is capable of doing in 5 minutes"? Why be arbitrary about what the limit is? IS there a limit? Or not? If there is not a limit, why do you keep talking about how there is a limit "out there somewhere"? How do you know? (

@henrikljungstrand2036

3 жыл бұрын

Very good criticism. Although i would say that, computationally speaking, for humans on Earth, yes there exist more prime numbers now than in 1980. Acting in a meaningful way in computational processes in the physical world (like cryptography). Primes are important in certain natural processes independent of human endeavours though (e.g. survival mechanisms of populations of certain species though, such as the circadian cycles), so possibly these large prime numbers existed/exists somewhere else in the universe as well, even prior to 1980. Being a (spiritual) materialist (hylozoist), i do not believe all these unbounded mathematical structures and theorems are actually "all there" already in the present, rather that they are potentially there.

if a number is so large that we cannot manipulate it because of the limitations of the universe, it follows that that number does not exist. if you're some kind of physicist. mathematics is not about this universe. mathematics is built on logic that transcends physical limitations. therefore, that number exists, since given enough time we could easily count up to it. and we could also calculate its prime factorisation. the time taken for both of these would be longer than the life of the universe with even the fastest processing speed permitted by physics, but, nevertheless, it would be /finite/ and an answer would be reached. so how, after that, would you justify its supposed non-existence?

@njwildberger

8 жыл бұрын

Sure. And ghosts exist, because ghost theory is not about this universe. Given the right vantage point on a higher level of cosmic existence, we could commune with ghosts, and perhaps even go bowling with them. Thus ghost theory is an acceptable form of intellectual activity. How would you justify its supposed non-existence?

@SamBrev

8 жыл бұрын

You raise a good point. You're not wrong; in any universe different from ours, ghosts could plausibly exist. The best counter I can give is that a ghost, to exist, has to represent itself in a physical manifold. Numbers, in order to be applied, do, but pure theoretical mathematics for its own benefit does not.

@ranshoham4918

8 жыл бұрын

+njwildberger We assume in theory that irrational numbers and infinite sets exist because we can deduce interesting conclusions from them - some of which helped solve real world problems, calculus for example. If you could use the assumption that ghosts exist to deduce solutions to complex problems, I would say that such a theory is welcome.

@trumblewumblehumble

8 жыл бұрын

You have a flaw in your argument here: If you can well define what a ghost is and show that at least one object fullfills the definition of 'ghost', then it exists. You use a real life example of something not existing in an answer to the question why math should actually care for what exists in real life. If we would live in a universe with much less particles in it, would that mean suddenly numbers stop existing? That makes no sense at all.

@infinitesimotel

8 жыл бұрын

The thing I have trouble with, is how would one represent infinity? How can you represent infinite numbers in one number? Thing is, you can't represent infinity with a finite entity (a symbol), in the same way one cannot represent the Divine in any single form. This would then become simulation and therefore it's own truth, and will never be the truth that it is supposed to represent. So if you use any type of designation, it would then be making infinity, finite by this very motion, therefore it would not be infinite.

Hi Norman. I think that human brain has the capability of imagining of anything such as very very large numbers and even infinity. We don’t need to be capable of writing large numbers somewhere in the universe in order to accept its existence.

@anthonyymm511

Жыл бұрын

Exactly.

@caveman-cp9tq

11 ай бұрын

Arrogance.

@Felipe_Ribeir0

6 ай бұрын

I agree. Our minds have capacities that computers do not.

Thanks for uploading the video. Shouldn't a distinction be made between criticism of set theory as a whole, and criticism of one particular axiom, namely, the Axiom of Infinity? I am not too familiar with Wittgenstein's criticism of Set Theory, but can't one accept it as a foundation for mathematics while rejecting the Axiom of Infinity?

I heard about this debate at uni but I wasn't able to attend. I'm grateful to listen to it here, so thank you! As for infinity - Up to the early stages of university, first year included, infinity was explored not as something that exists or not but rather a placeholder for a very large number. Such that if you had your example of 10^10^10.... + 23, the effect of adding 1 in most contexts tends to get scaled out. Of course, first year mathematics tends to look at the perspective of applying mathematics rather than actually exploring definitions so maybe thats why the first year textbook doesn't try to prove anything at all - it is completely out of the scope of most students unless you were willing to study the philosophy and theory behind it, which, unfortunately, very few people nowadays do. I also found your box game amusing, it does seem to be a contradiction which in my mind shows that you can't really say that infinity exists. The concept of it being a arbitrarily large number holds and is useful in most if not all applications but existence is really another thing. I also found "James Franklin's" point of view that "z is a recipe of a number rather than an actual number itself" is dodging the question. If you attempt to prove that this number exists or be able to represent it, what exactly does it mean, what exactly can you do with it? How can you possibly play with z in the Natural fields with addition and multiplication especially? Unless if I'm misunderstanding something which is likely. All of this is really interesting, but I don't understand much of it. Is there any further reading/videos that I can look at?

@njwildberger

8 жыл бұрын

+hubewa24 Thanks for your interest. This is a very important topic with many ramifications for mathematics. My MathFoundations series is all about issues like this, with now about 150 videos in it and counting. You should also read my paper: Set Theory: Should you believe, and look at the latest entries in my blog at njwildberger.com. Lots to think about! Don't take the status quo for granted. It is not going to last.

@hubewa24

8 жыл бұрын

+njwildberger Thanks! I'll definitely have a look! Based on my comments, do I seem to understand what's going on or am I missing something?

@njwildberger

8 жыл бұрын

+hubewa24 It is not easy to understand what is going on here. Please watch my videos and then you will find out what areas you can learn more about.

cool ..need more of this discussion.

That was exceptionally interesting discussion! We need the continuation. I won't take any particular position, but what is clear is that there's no set of an infinite number of objects, it's just nonsense. In that I agree with you. One can't enclose infinity, because it's not a finished completed object. Infinity is a process. It's constantly being built. As always the truth is somewhere in the middle.

@northernvibe4870

2 жыл бұрын

Great explanation Sir. Thanks for that.

@siddharthsinha6338

2 жыл бұрын

I actually don't feel like what prof. Wildbereger said is particularly contentious to begin with. If I think back to all the theorems learned upto and including measure theory, we don't really need to emply an infinite set to define limits. I for one, never really enjoyed proofs with extended reals that treated the points at infinity. But I am comfortable with the point at infinity in the extended complex numbers, because the point at infinity is somewhat intuitively exhibited because of the Reimann sphere. I'm not sure what his exact stance is on uncountability though.

The discussion was just great! About the ball example: since the infinite series of adding 10 and substracting one does not converge absolutely, we can get in fact any number out of the series by reordering it in a respective way. So yeah, in particular two or whatever. So what you did was actually perfectly consistent with the permutation theorem for not absolutely converging series. And here is the catch: by talking about infinity we leave the realm of what is compatible with intuition. And if you'd like justification from the real world: Let's try to do quantum mechanics without probailistic continuous considerations (i.e. without probabilistic densities, without Schrödinger equation and wave function formalism); try to do mechanics without langrangian and hamiltonian formalism, without differential operators... So, physics - the realest science of all - just as well wonderfully leaves the realm of what is compatible with our every day intuition. The mistake is to assume that our every day intuition is any measure for anything. Quantum world very effectively disproves that we are allowed to have any such aspirations. And the "suspicious things" in physics like law of energy conservation and alike are much, much "worse" than mathematical infinities, because they truly cannot ever have any consistent or provable foundation without a TOE being discovered. Nonetheless this completely unreasonably physics was able to progress that much and actually give us those computers only based on those "funny assumptions" within maths and physics - playing your game, my game is different. That's why the issue lies much deeper. The issue is just that a human being has no real touch with any of what a human being considers real or at least a human being can never verify such touch completely. And all such considerations simply touch the surface of ambiguity that results from it, well justified and very natural ambiguity. However, they mistake the consequences for the core issue respectively, being raised in philosophy, in particular areas and so on. That is a very general and fundamental insufficiency of our way of existence. And what is displayed in mathematics is one of the most harmless parts of it, because there at least we really can define anything as long as it is consistent in total within itself (or as long as it works with sufficient consistency for all the time of our existence even if it was inconsitent in truth and partially inconsistent - indeed, as long as fixes are there everything _is_ fine. Maths is a collection of techniques of thinking. Yours as well as those of others. Only that this collection allows a great wealth of structural interrelations in large parts of it. But in total, it is a collection of techniques of thinking. Why should it abide to anything?... Everything that makes sense can just be a part of it. Even if different approaches might seem contradict each other.).

@vorpal22

19 күн бұрын

This is what I love about math: it doesn't require any adoption of any point of view. Define some axioms and see what you get out of them. Anything that doesn't start off with contradictions in the axioms is a valid field of study. The axioms behind two theories of mathematics can be quite contradictory and yield different results, but one doesn't really invalidate the other.

ahh the notion of a super task. Doing an infinite amount of discrete operations, in a finite amount of time. At each step you perform it in 1/2 the time it took to do the last one. The "limit" of the time taken over this infinite set is finite!

The concept of infinity certainly does exist. I agree wholeheartedly that in the concrete, real, tangible world, infinity can't be manifested because reality is finite. But being able to talk about 'finite' is, itself, implying that there exists something which is not finite (aka, infinite), and that something may just exist as a concept.

@maxpercer7119

2 жыл бұрын

yes, that is a good observation. thats why im confused on why wildberger says infinity doesn't exist. surely it exists as a concept.

@davidmoir908

Жыл бұрын

The prof.does not believe in infinity and sets so non belief in infinite sets. Let's refute this by Hilbert's paradox of the Grand Hotel.Suppose you are an innkeeper at a hotel with an infinite number of rooms.The hotel is full,and then a new guest arrives.It is possible to fit the new guest in by asking the guest who was in room 1 to move to room 2, the guest in room 2 to move to room 3,and so on, leaving room 1 vacant.We can explicitly write a segment of this mapping 1-2 2-3 3-4 n-n+1. In this way we can see that the set {1.2.3} has the same cardinality as the set{2.3.4} since a bijection between the first and the second has been shown.This motivates the definition of an infinite set as being any set which has a proper subset of the same cardinality; in this case{2.3.4} is a proper subset of{1.2.3}.

A set of "all the natural numbers" is as "provable" as the existence of God. It's a leap of faith. God bless :)

The ball game at the end of the debate was an example of infinity treated both imprecise and precise, and if you want to get precise or imprecise results you have to be consistent in the precision you choose. Infinity treated as sets, or numbers, can be done in different ways. If doing it the Cantorian way, you have all sets that can be paired 1 to 1 as equal, and that leads to the rule that an infinite set can be viewed as equal in size to an infinite subset of it. But, this is a choice you have to make; you can also, without introducing a paradox, choose the rule that "the whole is greater than the part" and view an infinite subset as strictly smaller than the original set, for example, that the set of the prime numbers is smaller than the set of natural numbers. This leads to a more precise way to label the sets, where two infinite sets not only can be considered of different size differing in an infinite number of elements, but also be different by a finite number. So, it depends on how precise you want your result to be what representation of infinity to choose (and you can't mix, because that leads to the paradox in the game). For details, see www.math.uni-hamburg.de/home/loewe/HiPhI/Slides/forti.pdf

@jaredcastro8862

Жыл бұрын

Talking about "larger size" when talking about infinite sets, a couple weeks ago I made a little exercise trying to count all the rational numbers. Instead of doing the common pairing that everyone likes to do, I let "N" be the total amount of natural numbers (trying to fit infinity in a variable in a experimental fashion nd being not rigurous at all) and then counted all the fractions that can be formed with each numerator. For the numerator 1 you get N fractions, for the numerator 2 you get N/2 fractions (since 2/4, 2/6 and 2/8 are the same as 1/2, 1/3 and 1/4 and they already appeared), for the denominator 3 you get 2N/3 fractions, for 4 you get N/2 again, for 5 you get 4N/5 and you continue erasing all the already-appeared fractions until you get a expression like these: N(1 + 1/2 + 2/3 + 1/2 + 4/5 + ...) you can let "k" be the inner sum, so your expression would be the same as N*k. I made some computations calculating k with 10^3, 10^4, 10^5 and 10^6 terms and It turns out that although the sum diverges, the ratio between k and the amount of terms of the sum converges to a number greater than 0.68. I looked for "density of coprimes" in wikipedia and there appeared that the number the ratio converges is 6/(pi^2), so k/N = 6/(pi^2), then k = 6N/(pi^2) and finally N*k = (6N^2)/(pi^2). From my initial setup of "not being too rigurous with the concept of infinity and allowing myself to store it in a variable" the amount of positive rational numbers is (6N^2)/(pi^2). Then you add the negatives plus zero and get |Q| = (12N^2)/(pi^2) + 1. Doing something like that might be impossible for the real numbers.

Great professor. Quite interesting topic

Hello Professor Norman, I appreciate your new approach and it certainly makes me re-question many of the things. I had a few things to add to what you say which is not exactly in line with what you believe (if my understanding is correct), but there are some parallels. My argument is that the idea of limits (and real numbers and so on) is not totally useless but the claim that the definition and the whole structure is well established and rigorous is wishful thinking at best. However, that does not stop us from assigning a number ln(2) to the series 1-1/2+1/3-1/4... because one thing I know for certain is that whatever that number maybe, it certainly lies within some range (that I can define) of a particular number that I can come up with (namely, some approximation of ln(2)). So, idea of limit as such and the question of assigning a particular number to this series cannot be simply abandoned, but it is more humble and I think correct, to just accept that there's no really rigorous, well-established, unquestionable and authoritative structure which gives the method an "absolute truth" value. To extend this philosophy further, I argue that we need to go back to the methods of ancient Indians. In particular, a proof which modern mathematician may call rigorous/not is not the essential/important thing, but an understanding of a particular idea is more important. This leaves the question of truth value of the theorem open forever... As long as someone doesn't come up with a particular counter-example or a very convincing argument as to why my understanding is flawed, I can believe a particular statement to be true (or, in other words, I can assign a high confidence value, and that's all I can do). This is more practical approach to philosophy of mathematics, which is parallel to what you're trying to do by abandoning the idea of infinity altogether. The difference between the philosophy I propose and your philosophy is that (I think) you wish to rebuild the entire structure with no concept of infinity and so on, whereas I accept non-rigorousness of the concepts but I retain whatever the understanding is without requiring to build a structure that helps me to assign "absolute truth" value. I hope you will take your time to read this and respond. Thanks.

First, I would like to say that, having watched several of Prof. Wildberger's videos on traditional number theory, I find that they provide clear and interesting approaches to the subject. Thankyou for them! His videos on his philosophy of Infinity seem to be less about teaching, and more about raising awareness of certain foundational issues in Math. These are bound to be controversial. I can not speak for Prof. Wildberger, but his views seem similar to those of the "Constructivist" and "Intuitionist" approaches to the foundations of mathematics, which are supported by some renowned mathematicians, which the reader can learn about on various websites. Both Prof. Wildberger and Prof. Franklin make surprising statements about the example of "(10^...^10)+23". Prof. Franklin claims it is "not a number" but merely a name for a number; I suspect very few mathematicians would agree. On the other hand, Prof. Wildberger claims there is no prime factorization of this number, presumably because it is so large that such a factorization could neither be found nor verified in the lifetime of the universe. I think even the "Constructivists" would find this statement hard to accept. It would be foolish, I believe, to toss out all theorems proved using notions of infinity, because they are clearly useful and accurate as far as we can tell. On the other hand, there are many unanswered questions about the foundations of Math, as Prof. Wildberger points out. These are not "silly questions", but represent the attempts of many smart people to discover "where Math comes from, ultimately". The Constructivists and Intuitionists may offer insights that more traditional approaches have missed, and help us to make better sense of these deep issues. I recommend the youtube video on "Brouwer and the Mathematics of the Continuum", at kzread.info/dash/bejne/iYJ1z5mNeJPPf9o.html which discusses a quite different understanding of the Real numbers that the Intuitionist approach leads to. Most of that presentation is in English, though French is spoken for the first couple of minutes. According to an Intuitionistic argument by Brower, the Continuum is "unsplittable". I won't pretend that I understand this.

@iliyapeshikan7783

5 жыл бұрын

Isn't James Franklin's view the common sense physics view nowdays? Big bang coupled with heat death and limited observational capacities, of course!

@eatfastnoodle

4 жыл бұрын

"(10^...^10)+23" isn't merely a name of a number but what about pi with "(10^...^10)+23 digits? That'd be merely a name of a number cuz we can't calculate the number, can't store the number, can't represent the number, can't reason the number even if we use every quantum of the entire universe, then it doesn't and can't exist, even on conceptual level since we can't even conceptualize it in any kind of mathematic sense.

@henrikljungstrand2036

3 жыл бұрын

Brouwer has done some good work, and so has Heyting, but i would recommend Bishop for a more nuanced and pragmatic stance on intuitionism/constructivism.

@henrikljungstrand2036

3 жыл бұрын

Girard and Blass et al have also done some good work on intuitionism, by making the logic resource sensitive, this is called linear logic, and it largely preserves the dualism of negation (in classical logic) that intuitionism usually breaks (though it splits AND and OR, and TRUE and FALSE into both additive and multiplicative versions, duplicating most logical operators, while explaining implication better), while still faithfully embedding intuitionistic logic into itself using the "bang" or "of course" necessity modal operator (written as "!") to describe unlimited resources we can reuse over and over in a computation (which has a dual "query"/" why not"/"?" possibility modal operator signifying multiple communicating answers to multiple querents, which seems at first very strange, but later natural, for parallell communicating processes).

@elijahr_1998

2 жыл бұрын

​@@eatfastnoodle Of course we can "reason" and "conceptualize" (10^10^10^10^10^10^10^10^10)+23. I just wrote it down for you. I can write it down again. It's reproducible. It has precisely one innate meaning. It is a unique member of the natural numbers. No other natural number equals this one. I can do arithmetic on (10^10^10^10^10^10^10^10^10)+23 by hand and explicitly write down the result. This result is reproducible and will represent precisely another unique member of the set of natural numbers. Computational unbelievability or impossibility should never contradict basic arithmetic, the very thing these computers are built on.

Fascinating discussion. I found that both made valid points and both made invalid points. I'm Buddhist, so the philosophy of mathematics that I use is based on a Buddhist psychology. For example, Franklin began by saying "There is a serous problem because the mind is finite, and so there is a problem grasping infinity, if there is such a thing as infinity." From a Buddhist view (as differentiated from the Aristotelian, Platonic, or nominalist views), mind is NOT finite, so that mean's mind is infinite. Thus the problem with understanding infinity is the problem of *grasping* infinity, because the mind cannot grasp itself, infinity cannot grasp infinity, in the way that the finger tip cannot touch itself and the eye cannot see itself, directly. We are fooled into thinking that the eye can see itself by the use of the mirror, but the eye is only seeing a reflection of itself, not itself. Thus, when mathematics provides a view of infinity, we are not seeing infinity directly and we are only seeing the reflection of infinity as it is reflected in mathematics. From this Buddhist view, Wildberger is correct that infinity does not exist as a separate thing outside and what is called infinity is not the "real thing", but Franklin is also correct that infinity exists; even though our mind does not exist as a separate thing outside, yet our mind exists as unlimited and inconceivable presence, and like infinity, our infinite mind is not defined into existence. I'm a Buddhist psychologist and philosopher, not a mathematician,, so I don't have enough knowledge of mathematics to extend the application of the Buddhist view in contradistinction to the Aristotelian view, but one important distinction is that Aristotle confused everyone with his view of truth being constrained by the idea of dilemma. Infinity is not subject to Aristotle’s construct of the dilemma. Wildberger’s comments about the universe are erroneous. He states “If the universe is finite, which most physicists probably think is true, then its clear that there is only a finite number of possible things that you can write down in the universe. So conceptually there is only a finite number of possible things that could be numbers.” However, the idea of a finite universe if definitely not clear, as it overlooks the factor of time and variations within time. There is no static universe. As time never “stands still,” there is no finitude possible in a universe that is perceived as constantly expanding, and since the horizon of the universe is ever expanding from within each point of the universe, that continuous expansion must itself demonstrate that the universe is infinite, not finite. Using the Aristotelian view, when Franklin says that we perceive numbers by their aspects, that is correct, but that does not establish the existence of numbers as objects outside of perception. The Buddhist view is that all objects of perception are merely objectifications of aspects, features, qualities, etc. and on that basis all objects are said to be empty of any inherent existence. This would apoply to numbers also. Any particular number has no inherent existence, and is perceived only by the concatenation of qualities such as symmetry, order, pattern, etc. While numbers seem to have a quality of endurance far beyond any physical object, that quality of continuity is part of the construction of the number as a mental configuration. The Platonist is right to see the number as having an “ideal” existence, but is wrong to see this mental configuration as external to mind. Thus all finite numbers are the actual representations of the mind’s inherent infinity, which is not a “nether” world, but is this actual world, thus making this aspcct of the Aristotelian view ring true. Thus, both Plato and Aristotle were part correct and part erroneous. I have to say that the "box and balls in the corner" game is very silly and totally irrelevant to any demonstration as a thought experiment, unless it is taken as a demonstration of the absurdity of Wildberger’s position. Because the game is entirely based on the false premise that the infinite number of balls in the corner can be put into a box. Wildberger sets up (pun intended) the question by saying, “Now let’s go to infinity. Let’s take the limit. We’ll go super duper fast and perform all these infinite number of operations, so that there are no more balls over there. How many balls are in the box?” Clearly, if there are indeed an infinite number of balls in the corner, then there will never be a time when there are no more balls in the corner, because there will always be one more ball in the corner as the definition of infinity. So by defining infinity as something finite, Wildberger’s “game” amounts to nothing more than the logical fallacy of assuming the conclusion.

if i devide a triangle into two which one of the 2 takes the end point(the pic of the original)?if we assume they have the same area after the division,did they share that point?i think we have to live with infinity and infinitesimal or give up on math and geometry completely

Would it be possible to find a mathematical representation of the game related to the infinite boxes on the corners and the box? Thanks!

@jpgarciaortiz

9 жыл бұрын

Ok, forget my question... I've finally understood the game and why there are 2 balls in the box... :-)

In rebuttal to the last point, where each operation takes 10 balls out of the original bin, puts 10 in the next one, the removes 1 from that and puts it in the corner (that 1 being the 2+nth ball): at each point, the 2+nth ball is removed, but 9 balls are being added, or if we think of this as a set or list, 9 entries are being appended. In this scenario, we will accomplish an infinite amount of removals and additions. I believe we can describe this as having performed "omega" operations. So we will have the 1st and 2nd ball in the bin as described in the video, but then we'll also have the "omega + 1"th ball, "omega + 2"nd ball, etc, to account for the "missing" balls which were added.

@fleisch19843

9 ай бұрын

I think you can use the idea that the set of all balls taken out has a 1 to 1correspondence with the balls NOT taken out, in that for any ball you said was added, I can give you an n such that it removes that ball. Although you might say, "but by that point, the set of balls NOT taken out has gotten even WORSE!!" I could say "That's ok, I can do this forever. You can't put a ball in that I won't eventually take out. If you can, then tell me which number. I'll tell you when I'll take it out, just to prove my point" So, you don't have to worry about this list that contains all the balls NOT taken out, because they get "cancelled out" eventually by the items of the set of the balls that DO get taken out. Clearly, in my opinion, doing analysis with infinity is stupid, it becomes semantics. I sympathize with the argument that infinity has no business in rigorous mathematics.

Goood day sir!! I asked myself this question and I had this answer::: My opinion is that infinity is like numbers itself, does numbers exist in the universe? no. but we use numbers to explain the universe..but we need infinity to explain numbers...if we try to get rid of infinity, we will have to get rid of irrational numbers, that means we must assign a rational number to every important constant in universe like PI planks constant and all that...

I am curious about how Renormalization in physics would work (looking more at the Quantum Electrodynamics, than the Stat Mech application), since it involves equations that involve the concept of infinity, but are tamed to finite (and physically measured) quantities. Is there a way to do this with finite-ist mathematics?

just because one cannot factorize that big number does not mean it cannot be included in an infinite set

Would it be correct to posit that there in the first 'box' thought experiment, that there are more balls in the near box than the box in the corner?

Guys this is good stuff but the debate is over and infinity demonstrably exists. Everyone listening has infinite patience for your mics

Very nice presentation. Especially in combination with your MF91 and MF92 lectures about real numbers (those lectures were also very helpful). I have lately also found the spell and glamor of the Cantorean way of thinking diminishing in my mind and have been searching out alternatives, like Brouwer, Wittgenstein etc. It seems to me that there is a weird psychology going on that you captured nicely I think in MF91: "we are not gods". Specifically, there is a weird mental quantum jump: We start with a process of adding 1 to keep generating increasing numbers. Obviously the process us unbounded and can go on and on. However, people somehow then make the jump from this unbounded process to the collection of all natural numbers. It is like mistaking a verb for a noun. It seems to be an unmerited quantum jump in people's thinking, like a basic category error. Also, I think it was weird when Prof Franklin said we don't need to name something to think about it. Obviously we need to have a name for anything to be able to think about it. While the relationship between symbols and meanings is generally arbitrary, nonetheless, all meanings we can access have to be clothed in symbols for our mind to operate on them. Further, as you stress in MF92, an algorithm is a perfectly fine substitute for an explicit representation of a number, where the latter is usually taken as the "name", At any rate, I found this to be a rather lame argument and seemed to me to be an appeal to fuzzy nebulous thinking. Finally, if you might comment. Have you read Nicolas of Cusa's "On Learned Ignorance"? It is from 1440 I believe. This work is about infinity, which he calls "The Maximum". I think the things he says there make the most sense I have ever read. The basic gist is that infinity exists, it is God, and there is only one, and everything else derives from this condition. In some sense he is the sane mirror image of Cantor, where Cantor is some weird distorted parody. Anyway, if you've not read "On Learned Igorance", I think you might find it interesting. Thank you for all the work you too. I have learned much from you. Best wishes. Don

@njwildberger

9 жыл бұрын

Thanks Don for the interesting and insightful post. I am also intrigued by the reference to Nicolas of Cusa, and will look this up.

in that tennis ball example, what does it mean to have picked up all the tennis balls from the corner?

When he mentioned data structures, in Haskell (programming language) you can represent infinite lists symbolically, and VIA lazy evaluation, perform computation on such.

Starting at 0 & at each step we add 1 . At each step we have a finite number . So it's indefinitely finite . So the real question is : are indefinitely & infinity the same thing ?

How do you feel about the set of natural numbers from and including 1 to the huge number you wrote on the board?

@njwildberger

9 жыл бұрын

Corey Monsta There is no such `set'.

@coreymonsta7505

9 жыл бұрын

njwildberger Out of curiosity, are you an ultrafinitist? If you don't mind, could you list for me a few precise facts of interest to you regarding the existence of large numbers?

@njwildberger

9 жыл бұрын

+Corey Monsta I am not a member of any particular school of thought, at least not consciously. I do want (pure) mathematics to be completely clear and correct, and I believe abstract theories should be well supported by explicit examples and concrete computations to ensure that we are actually talking about something real. I will be talking more about large numbers. Their `theory' is very very difficult.

@zapazap

4 жыл бұрын

@@njwildberger : If by 'something real' you *mean* something 'explicit' and 'concrete' then your statement sounds like it's dancing near the border of a tautology. It sounds as if what you are saying can be summarized as: "I believe abstract theories should be about something real to insure that we are talking about something real". I would like to steelman your claim but I am not sure how. What some call the 'list of natural numbers less than four' I would model as a generator function. What some might call 'the list of all natural numbers' i would also model as a generator function. It may well be that any real world machine running the latter generator function would eventually overflow the stack, but both generator functions are finite objects. A generator function for [0,1,0,1,0,1,0,1,0...] will use only a bounded memory and will never terminate. Nonetheless it *will* terminate at the heat death of the universe. The generator is an object. What is the most fruitful conception of the object? As something which will terminate because of universal heat death, or as something which never terminates? What conception is most fruitful for what circumstances?

@HDitzzDH

3 жыл бұрын

@@njwildberger By that logic you would also reject complex numbers since there is no explicit, real world example of these numbers. That's the beauty of mathematics, it doesn't have to be restricted to real life applications/computations, the abstraction part of it is what advances new mathematical discoveries today. Infinity is obviously not a number and should not be treated as one, however it's a concept which is essential to mathematics. It is true that there exists an infinite number of integers, rationals etc for example, and if you deny that (which I hope you don't) then there's a major problem here. Out of curiosity, do you dislike real numbers as well? What's your view on transcendental numbers such as "pi" and "e"?

Good stuff. I have to watch early or my brain wont let me sleep/

Great discussion and I agree that a full set of infinity cannot exist because its always growing, I agree with Prof James Franklin - but I'm biased because he's a mate and I love his work.

I'd like to hear you argue for a non-Platonic view of mathematics and how you moved away from your 'closet Platonist' position.

Great place to be as computers take over the world...We have so much to learn!

Doesn't the principle of mathematical induction (MI) depend on the continuum of the infinite set of natural numbers? When MI is used as a method of prove the proposition P for n, isn't the deduction of P(n+1) from P(n) saying that proposition P is true for any natural number n -- i.e., no matter how infinitely large?

(Revised and expanded from an earlier comment I posted, since Norm asked me to repost this as a separate comment thread.) Norm, your Z (that is, 10^10^10^10^10^10^10^10^10^10+23) is divisible by 3, 13, 139, 673, 18301, 400109, 27997373, and 360987373. Any additional prime factors of Z (there's at least one more, but probably many) are all greater than a billion. (That is, I tested all possible factors up to 10^9.) But more interesting to me is that even if we consider a similar expression, let's call it A, with an INFINITELY tall stack of tens in the power tower and still having the +23 part, the new result (now infinite) would still be easily shown to be divisible by 3. See this by considering that all but three of the infinitely many digits of the result are zero, and the digit sum of A is 6. Furthermore, a computer could determine the remainder after dividing A by any reasonably-sized integer, an operation which completes in finite time in spite the power tower being infinitely tall. (Similar to how we know the final digits of Graham's number, in spite our not being able to write down the whole number.) On the other hand, the question of whether A has finitely or infinitely many prime factors, while well-defined, may be unanswerable.  Anyway I calculated the factors using some code I adapted from code I had previously used for calculating some sequences for the Online Encyclopedia of Integer Sequences. See for examples A246491 and A240162, and related sequences linked from those. The later can be interpreted as analysis of a infinite number similar to the A I mentioned above. On the other hand, alternate interpretations just involving finite bounds are probably preferable. The "infinite tower" bit is a simple consequence of expressions of this form modulo a fixed divisor always having a fixed value once the height of the power tower exceeds a certain minimum. See the OEIS entries for more discussion.

Is the paradox you expressed in the end yours?

@CXVIGeriCXVI

7 жыл бұрын

It sounds like the Littlewood-Ross paradox; although I've only read the banker variation :)

Its very simple. I know that we normally claim about natural numbers various properties. To exemplify this we for example use the induction principle and order the numbers accordingly. But to "go" to the limit is something totally different than to "be at the limit". You yourself with your wonderful examples have shown us that actually it is impossible to "go" to the limit. All the problems you mentioned at the end of your two examples (at the end of the lecture) arise from this fact. That you actually do not "reach" the limit with your simple method "to go straight to the limit" does not mean it is not there.... So actually you might speak about something that does not exist as a result of an algorithm. But the fact that it does not exist as a result of an algorithm does not answer the question whether there is or might be a natural number "at infinity". A nice example is the ideal line in projective geometry. By the way i enjoyed your lectures about algebraic topology, in which you actually use these concepts to actually work out some proofs that otherwise would be "impossible". To come back to my example. And i know it might be a crazy one. Lets pretend there is a mathematical problem that gives rise to only natural numbers ( as we understand them ) as a solution and that gives rise to only natural numbers that are bigger than any bound, that you might check them against. I do not know whether such a problem exists, but i am sure, if it exists, it will give you interesting insight into the question you are debating here.

@jonathanmitchell8698

2 жыл бұрын

I'm curious if the difference in perspective is the difference between functions and objects. "Going" to a limit is essentially recursively applying a function, but "at" the limit there is an object that is not defined by the recursive function.

Professor Wildberger, I loved your Math History videos (and look forward to watching more of your excellent videos), but I have a few questions for you: Do geometric objects such as a point, a line, a sphere, etc. exist (as they have no counterpart in the physical universe)? Also, if space/time is "continuous" as opposed to "discrete" (whatever that precisely means), could there be room in our universe to store/display/compute all of those extremely large natural numbers you mentioned? Do we have to rely on our current computational/computer paradigm which is inherently digital/discrete? Also, I don't understand what is meant by "exist" in this debate. What does it mean for the number 5 to exist? For a finite set to exist? I don't have a definition to make a decision as to whether these exist or whether an infinite set exists. I'm not even sure it is important whether an infinite set exists or not, as it is just a way of speaking that is intuitive for some people. If you read texts on the foundations of math, axiomatic set theory, and you learn about the cumulative type hierarchy of sets, then you find out that all of the set theory paradoxes have been resolved in natural ways. Even the distinction between a "set" and a "class" too big to be a set has been adequately described. (For some of this see Frank Drake's "Set Theory, An Introduction to Large Cardinals.") People aren't just waving their hands and playing games. Also, the construction of real numbers has been accomplished (and since it is quite difficult and long, of course it is waved off in introductory calculus texts). Are you sure the universe has only room to store finitely many numbers? Physicists aren't even sure if the universe is finite or infinite, but even if it is finite, it may be "continuous," which in some sense provides "infinity" in an infinitesimal way. Of course, I'm not even convinced it matters whether an infinite set exists or not. I am convinced that they are useful and haven't provided any problems/contradictions in mathematical research (since the early paradoxes have been resolved). I'm not convinced of anything by your game with the balls and the box. First of all, I don't know what it means to repeat a physical task an infinite number of times, so it does sound like nonsense. Secondly, just because a result seems to offend our sense of what "ought to be" which is shaped by our limited human perspective, does that make it wrong? Mathematicians have never arrived at an infinite set by starting out with a finite set and adding elements to it. Unlike your debater's comment, they actually do just postulate the existence of an infinite set. Pick up any textbook on (ZFC) axiomatic set theory and you will see an "axiom of infinity" that says: "there exists an infinite set" (where the word "infinite" has been previously defined). Whether there is a physical counterpart to a mathematician's infinite set, I do not know, but I don't need the answer to that to carry on using infinite sets in mathematics.

@njwildberger

9 жыл бұрын

It is easier in this kind of discussion, i.e. comments to KZread videos, to keep your thoughts focussed. How about asking one question or issue at a time? That way it is easier to respond, thanks.

@lbblackburn

9 жыл бұрын

njwildberger Ok, sorry. I'll start with this: What does it mean for a mathematical object to "exist"? Also, by your definition of "existence" does a geometric object like a point or a line exist?

@njwildberger

9 жыл бұрын

Leonard Blackburn Often questions of existence arise because we do not take care of definitions clearly enough. If we define a point to be an ordered pair of rational numbers of the form A=[a,b], then we can exhibit the existence of such a point by showing A=[3,5]. The problem with the `existence' of `infinite sets' is actually more a problem of definition: what exactly do the words mean??

@lbblackburn

9 жыл бұрын

njwildberger Well, first we would have to agree on a definition of the word "set" (whether finite or infinite). A first failed attempt at this would be to say that a set is any collection of objects taken as a totality, which as you point out leads to contradictions and paradoxes. We then realize that small collections of objects are ok (like finite sets) and set out to describe the cumulative type hierarchy of sets. Then we define a set as any collection of objects that can be found on some level of this hierarchy. Such a definition of set so far has not produced any contradictions or paradoxes. So we accept this as our definition of set. Then we define "finite set" in a suitable manner such as a set whose objects can be put into one-to-one correspondence with {1,2,...,n} for some natural number n. Finally we define "infinite set" as any set that is not finite and we point to examples of infinite sets in the cumulative type hierarchy. This is a non-mathematical definition but it is the best that we can do. Not every mathematical object can be defined mathematically--there must be some mathematically undefined terms unless you want circular definitions or an infinite regress.

@lbblackburn

9 жыл бұрын

Leonard Blackburn Also, I meant whether you believe geometric objects like points or lines exist in their own right, not as representations of other objects like pairs of numbers. For example, when the ancient Greeks like Euclid were doing geometry and proving results about synthetic geometry, they didn't interpret points as pairs of numbers or lines as algebraic equations. They were imagining points as locations without size or infinitesimal specks (whatever this means) and lines as having no thickness (however that is possible). They took real objects (like drawings) and mathematically abstracted them down to their "essence." Now I find it strange that such a thing is possible and I don't fully understand the Platonic point of view in mathematics, but I have to admit that doing geometry in this way gave rise to theorems that we still consider true today. If a line without thickness or a point without size don't exist (because they have no real-world physical counterpart) then how does one explain that working with such purely mental constructs yields real scientific progress? Maybe it is so with infinite sets. They might not have any real world counterpart (then again they might) but they may still be useful mental constructs that yield real scientific results. In fact they have been proven to be wildly useful in mathematics. For example, read about Goodstein's Theorem on Wikipedia. Infinite ordinals can be used to prove that any Goodstein sequence (which is just a sequence of natural numbers) eventually stabilizes at 0. Once you understand exactly how a Goodstein sequence works, you can easily convince yourself of the truth of the theorem without writing down a formal proof. Then when you read a proof using infinite ordinals, you'll see how it mirrors exactly your own intuitive proof. Whether these infinite ordinals really exist or not is immaterial.

I really dont see nwwildberger argument ...the finiteness of (information) in the universe vs infiniteness of numbers. Is he talking about real or potential existence? And what is existence?

At 10:57-11:38 Wildberger talks about the size of the universe and that many numbers between 1 and z can’t be be written down because there’s not enough space in the universe. Therefore, he says, ”their existence is problematic”. Why would the numbers care about the size of the universe? Does the number of existing numbers increase as the universe expand?

C'mon dude! Physical existence is not the same as mathematical existence. Putting aside the fact even physicists do not truly know "what exists" the criteria are just so different. Mathematically we demand relative consistency (or assume it and hope, or prove it eventually). Physically there is no such criteria because physical stuff is so much messier and does not necessarily conform to our definitions, so physically we hypothesis some type of abstract (often mathematical model) and then have to test it, we can also ask for mathematical consistency, if the physical model is mathematical, but mere mathematical consistency is no guarantee for physical reality, however you define that experimentally. So it is a bit unfair to accuse mathematicians of talking about ghosts. No mathematician attempts to define ghost and fairies and unicorns. But if they did, these would merely be names for mathematical structures, and existence then is quite plausible, using the only criteria mathematics ever has, relative consistency. It is just not the same as the more absolute type of experimental existence afforded by physics. So it is extremely unfair - and a form of sophistry - to liken infinities in mathematics to fictions like ghosts. You don't reject the concept of a circle just because no one has ever fabricated a perfect circle physically (which is in any case impossible). Put another way, no perfect physical circle exists, but does that mean circles are on a par with ghosts and unicorns? Clearly not. But, in fact, if I define a "ghost" carefully, then I can claim that ghosts exist, and in fact Faddeev-Popov ghosts are a well-defined notion in quantum field theories (en.wikipedia.org/wiki/Faddeev-Popov_ghost). Ok, they are not the ghosts of fairy stories, but I have defined the term. So these ghosts do exist, even though they are not physical (see the article). This bias of physicalism or materialism which I think lies behind the notions that we should do only finite constructible mathematics, is the far worse disease, compared to the disease of introducing Cantorian (or ZFC or NBG) sets, I don't think sets count as problematic, although they introduce severe challenges. But physicalism is enormously problematic, not just a disease, but an ultimately fatal cancer eating at mathematics. Sure, Cantor did not fix the problems of sets, and arguably ZFC and NBG and other schemes only partially heal the wounds, but they are honest attempts. Introducing biases of physicalism on the other hand is a wound that can never be healed. I think, once physics becomes mature, you could probably split off a branch of mathematics which deals solely with physical intuitions, finitism, constructivism, etc., but to me, that would not be all of mathematics, finitism will never be all of mathematics. You cannot stop people from thinking by forcing them to stay within your biased limits.

Hi ! Interesting video. Maybe we should think that after all Cantor's argument about the uncountability of real numbers shows that the continuum is something of a completely different nature from discrete sets of objects. Maybe the paradoxes come from the fact that somehow we think of the continuum as being made out of "elements" or points. Maybe that is false. I think the philosopher Bergson said interesting things about this.

Stuff like this makes me wish I gave math more of a chance in school. Very interesting stuff. Makes me think of this classic joke, which illustrates it perfectly. An infinite number of mathematicians walk into a bar. The first one orders a beer, the second one orders a half a beer, the third orders a quarter of a beer, and this trend continues on for some time. After a while, the bartender gets fed up and hands them 2 beers, shakes his head and says: - “You mathematicians just don’t know your limits.” We could keep dividing that beer down to the very last beer molecule, at which point it stops being beer. If that's not enough, keep on dividing until the last quark. At some point, there's always a limit. Numbers need to be associated with something real, for it to have any limit other than the capacity of the storage medium that represents it.

@MrKidgavilan

Жыл бұрын

you are an empiricist, not all mathematicians are...

I would like to know the title and author name of that book where this ambiguity is written. Kindly write me back

The question is easily resolved when we do not confine ourselves to "real" numbers. If we realize that we are always dealing with vectors, where the slope of the vector [x y] is y/x, then when y=0 the slope is x/0, and we say the slope is "infinite" but the arctan (x/0) is (pi)/2 radians (or 90 deg). Any scalar multiplication of a slope is still the same slope.

Why do we conflate "Existence" with "Physical expressed"?

You are so right. It is truth.

@12:19: It is true that every natural number has a prime factorisation. It was proven thousands of years ago. As for your number "z", there is nothing complex about it. Given sufficient time, it can be written down. Whether or not computers can store it, is quite frankly irrelevant. z has a prime factorisation and the size of the universe has nothing to do with it. You know, in order to back up some of your false claims, you would essentially have to disprove the prime factorisation theorem. Good luck with that.

nice overview. Yet, still even Cantor would be confused nowadays

If we could pretend infinity was graphable and we graph its location on the numberline then there would only be two kinds of states, either zero or infinity. 100 would be located at the origin, 1 googol plex would be at the origin, even Z and grahams number would be at the origin. Infact infinity would represent the entire number line and all finite numbers are stuck at just one point.

circle: a line which is equidistant from a point. sphere: a surface. ... 1/0, which, for the purposes of trigonometric calculation, equals zero.

How far does the need of definitions go?

Doesn't any intro textbook on Real Analysis explain how to construct R?

@chasebender7473

5 жыл бұрын

Those constructions rely on the axioms that Prof. Wildberger is opposing though.

I would like to know if, whenever we speak about infinity, we "take the limit", "go to infinity", "take the limes". Are there any mathematical objects that evidently are only possible to exist at infinity on a selfevident basis? What i would like to know is, if there are mathematical objects which do not rely on taking a limit to prove them to be at infinity....?

@njwildberger

9 жыл бұрын

A good answer to this kind of question was given by James Franklin: we know what mathematics without infinity would look like, because that's what computers do.

I am not knowledgeable enough on the subject, but it seems like the whole argument against infinity is that it has no analog in our universe. But the beauty of mathematics is that we can "play make believe" and manipulate abstract concepts that can be simplified to represent real world phenomena. Also, I think that whether computers can or cannot emulate certain things isn't relevant in debating the foundations of pure mathematics.

@njwildberger

9 жыл бұрын

That is not the beauty of mathematics as I see it. The beauty resides in so many remarkable patterns and lovely theorems. They are not whimsical or arbitrary in my experience. Furthermore they do not change or shift depending on our moods. If you have a mathematics based on ``play make believe'', I believe most mathematicians would not be particularly interested in hearing about it. Make believe is easy: finding out the way the world really is is much more challenging.

@Wildberger: does Pi exist? @all: the ball example was spot on what if infinity is there but it's just our limitation that we cannot handle it (we are limited to make only a finite amount of steps of anything) the desire of infinity is clear as day: one should always be able add one on the other hand that doesn't mean we can properly tackle infinity with the finite tools we have however, our lack of understanding infinity doesn't imply it can't exist (it's not about our abilities)

The big number discussed at 12:30 is divisible by three.

Overall, I found James Franklin's arguments to be more persuasive, but that may just be my bias towards accepting infinite sets. I found your arguments to be very interesting in clarifying my understanding of the notion of infinity in mathematics and really enjoyed the thought experiment, which I have attempted to make sense of: As for the boxes at the end, let A, B and C be the "Boxes", where C is the box with balls labeled 1,2,3,... Let A_n, B_n, C_n be the boxes at time n. Now, it makes no sense to talk about the limit as n goes to infinity of sets, since we have no notion of "closeness". We, however, can talk about the cardinality of these sets and also the infinite intersection or union of the sets. Let a_n, b_n and c_n be the cardinality of A_n, B_n and C_n respectively. First case: take 10 from C, put 9 in A and 1 in B. Then, clearly, a_n = 9n and b_n = n. Thus, as n goes to infinity they will go to infinity aswell. The sequence c_n is always going to be the same thing (cardinality of natural numbers), but I'm not sure if you can talk about limits of transfinite numbers. Since this was inconclusive, you might want to look at the infinite intersection of the C_n's, which would be the empty set, which has cardinality 0. Second case: take 10 from C, put 10 in A. Then, take 1 from A, put 1 in B. Clearly, the case for the B_n's and C_n's is the same as before. However, now it depends on how exactly we define the "take 1 from A, put 1 in B" as for the answer for the A_n's. Using the strategy proposed in the video, we can make the infinite intersection have any cardinality we want (0,1,2,.. and cardinality of natural numbers) depending on how we remove a ball from A. The cardinalities a_n will however always be 9n, so they tend to infinity. These notions of infinite sets, cardinality and limits are very useful in dealing with this unintuitive thought experiment.

Hi, since your viewpoints are fundamentaly different from mainstream maths, I think you should put more emphasis on the argument from computability. Because like Franklin said, oh I just don't believe it is relavent that you can do this or that only in a finite time and so on, then there is no point in arguing about this.

What about Chaitin omega number, does it exist ?

38:26 I think there's a logic error there. You can't go super duper fast and perform all infinite operations because you'll be doing it forever and never reach and end. On the other hand I think there's also a problem with infinity. It's true, you can always add one but at some point the information content of a number becomes so large you will neither be able to compute it in ANY way nor be able to hold its information content. Adding +1 to 10^10000000000 may be easy to imagine but for numbers like 9999......99999 it will become a hopeless task to change all the digits to zero. So, counting to infinity is - imho - impossible, just like Professor Wildberger says. All you can do is count up to a certain point - limited perhaps by the energy/ information content of the universe and then accept loss of information if you continue to add +1 (it will be like adding +1 und subtracting -1 add the same time)

If we take ZFC, and a sentence P in the language of set theory, and ZFC proves P (in the usual sense), then P can be deduced from a finite subset of the axioms of ZFC, using one of a finite number of rules of deduction. Let this finite subset be called S. If we pick some positive integer N, and let T(N,S) be the set of all sentences in the language of set theory which are provable using only the axioms in our finite set S, using at most N steps (for some sensible definition of number of steps of deduction), then we get a finite set of sentences. All these sentences are provable from the axioms of ZFC. If we then ask about the behaviour of T(N,S) as N gets larger, or S becomes larger and larger subsets of ZFC, where does this take us? Every sentence provable from ZFC is provable from some finite subset of ZFC in a finite number of steps. What the weird infinity stuff allows is for us to ask about the 'asymptotic' behaviour of T(N,S) as N increases without bound and S increases to eventually encompass all of ZFC. Ackermann's result that ZFCfin (ZFC with axiom of infinity negated) is biinterpretable with PA is worth bearing in mind. If you then want to further restrict things, the maths you can do will fit within some fragment of PA. The only slight issue with the above is that 'finite' can include things too big to write down in the physical universe.

Maybe Prof Franklin needs to do his own MathFoundations series. I am still concerned I may be biased towards Prof Wildberger. And, what's bad about multiple mathfoundations series?

Many people say 'this makes no sense' while eliding 'to *whom*?'

Dear Norman, I am totally convinced that you are right. At a next opportunity explain to the people that immensely big numbers are not ”out there” but only “in there” (point at your brain). The numbers we use in our calculations are actually already much bigger than the number you can “experience”. You may find 4 parrots or three eggs, but assuming you count non-stop at the speed of 1 item a second, you can only count up to less then 6x10power17 during the time since the big bang up to now. We can actually name such numbers, but we van never count so many real items. They cannot be “found" (4x10p3 per hour, 1x10p5 per day 4x10p7 per year, 6x10p8 in 15 years, 6x10p17 in 15 billion years) Also realize that the box in which you can store an infinite number of balls must be so big that there is no “outside” so in fact you can’t really put anything into it, nor take anything out. Neither can you go to Hilbert’s “Hotel infinity” as you must necessarily be already there, for that exact same reason.

@barefeg

9 жыл бұрын

He didn't say what size the balls are so in this imaginary game you can think of the balls as infinitely small

@Velzen5

9 жыл бұрын

Alas fysics does not allow for anything smaller than an atom so you still need an infiitely big black hole at the very least! (which by the way has a schwarzschildradius beyond the universe :) )

@barefeg

9 жыл бұрын

Peter Velzen That is incorrect. Have you heard of electrons or quarks? Those are smaller than atoms. The smallest physical scale for that matter would be the plank scale but that is beyond the point. It is just a thought experiment that shows that if you treat infinity as an "ordinary number" then things start making no sense at all.

@Velzen5

9 жыл бұрын

it does not matter how small, infinitely small things do not exists. you still need more then the known universe to hold an infinite number of real things

@jamesfranklin2412

9 жыл бұрын

Peter Velzen ?? The brain is smaller than the "great outdoors" so surely the size of numbers that fit in the brain should be smaller than the ones "out there"?

As a physicist, I know that the foundations of my subject are uncertain and touch very much upon philosophy. I find it interesting to see that the same seems to be true of mathematics. I can see that one has to be a Platonist in some sense to believe that the set of natural number is a real thing.

@devalapar7878

3 жыл бұрын

Numbers are just as invented as infinity. Infinity stands for an infinite process. It has rules. You can work with it and it delivers true results. For example in Achilles paradox Achilles can never reach the turtle. It is an infinite process. But we all know that Achilles is able to reach the turtle in reality. So that means that some infinite processes have solutions and can be calculated. You can calculate the solution in two ways. You either say the position Achilles reaches the turtle is the limit of the process: pos = sum (k=0->infinity) s/10^k = s * 1/(1 - 1/10) = s * 10/9 Or you do a kinematic calculation where you subtract the velocity of both acteurs, find the time Achilles closes the gap and calculate the position by multiplying the time with Achilles velocity: pos = s / (v_A-v_T) * v_A = s / (10 - 1) * 10 = s * 10/9 Both lead to the same result, even the same formula! Both are the same thing, but in different formulations. Edit: mistakes are due to auto correction. I don't know if I found all mistakes.

@ROForeverMan

10 ай бұрын

@@devalapar7878 The reason Achilles reaches the turtle is because time is a quale in consciousness that exists in the eternal present moment, and "Achilles reaching the turtles" is just a story in which such a thing happens, like the story of Santa Claus reaching all the children in one night. It is possible for the simple reason that this is precisely the plot of the story, for Achilles to reach the turtle. And I'm talking about the "physical world". "Physical world" is a story that consciousness tells itself in the eternal present moment. All the movement that you see around is just a story that happens in the eternal present moment.

@devalapar7878

10 ай бұрын

@@ROForeverMan You can say that from a philosophical standpoint but I am talking about very specific things. Mathematicians and physicists know what I mean.

@ROForeverMan

10 ай бұрын

@@devalapar7878 Those "very specific things" appear because improper understanding of the philosophy involved.

@devalapar7878

10 ай бұрын

@@ROForeverMan I don't think you know what i mean. Achilles catches the turtle in every possible world. The math example is often misunderstood. Achilles doesn't only move in space but also in time. And just as his steps in space become smaller, his steps in time become smaller too. People forget that.

Interesting to hear the arguments for maths and numbers from realistic and perceivable point of view and from abstract and beyond this point of view. We can't ignore philosophy and nature of reality and existence in defining maths, it's just impossibly. We don't even able to define time and space yet; the two things "infinity" where "infinity" happens. Here is the catch: - If we think beyond this universe then, who knows, there is no time and space there and no need of maths and numbers! so beyond this universe there is really no infinity and maths and numbers and laws and axioms as we perceive! Its all in this universe and for us to understand these concepts. And really as far as our universe is concern we have not explored it yet and will never able to be with any advance mathematical and scientific system we ever build ... so why bother with this continuum thing as we which creates so much paradoxes, why not take the stance of pragmatisim in cimoitational way and then extend it to the limit we can and then understand whats going on ... we can't think beyond this universe with the tools and technologies we have, we can only go beyond with spirtual ways which migjt not be quantifiable or percievable with our technologies and mathematical tools ...! the whole new realm!

What does he mean by 'there never will be a factorisation' 10^10^10^10^10^10^10^10^10^10+23 has (for instance) the prime factor 3. How does he know that we won't find some systematic way to find all of the prime factors of a number like this?

@exili9743

8 жыл бұрын

+Integral Cube A prime factorization of a number exists when a number has been completely factored. A number is completely factored when it has been written as a product of prime numbers. If you cannot write a number as a product of prime numbers then it cannot be completely factored and therefore the numbers prime factorization doesn't exist. The prime factorization of 10^10^10^10^10^10^10^10^10^10+23 will most likely never exist. Norm states this with total confidence.

@picitnew

8 жыл бұрын

+The Mad Corn Stalk - "Most likely" is not a mathematical proof. - His confidence has no meaning in this case.

@theWebWizrd

8 жыл бұрын

+The Mad Corn Stalk That's nonsensical. Every natural number must have a prime factorization. By the definition of the word prime, if it can';t be written as a product of smaller primes it is itself a prime. One of those two statements are true, because either A is true or (not A) is true. We might not *know* the prime factorization or *know* which one of those two statements are true, but one of them must be, and in either case the number has a prime factorization. You can't question that argument unless you refute the axiom that either A is true or A is false, in which case you're working with a different logical system.

@exili9743

8 жыл бұрын

+theWebWizrd "Prime Factorization" is determining which prime numbers multiply together to construct the original number. Even though we know that z=10^10^10^10^10^10^10^10^10^10+23 can be written as the product of prime numbers it's prime factorization doesn't exist until we determine these primes. Determining these primes is currently impossible due to the size of z; therefore, the prime factorization of z doesn't exist and most likely never will. This is what I think +njwildberger is trying to express to the audience.

@exili9743

8 жыл бұрын

+picitnew When the prime numbers which make up the prime factorization of z have been determined then its prime factorization will exist. Until then, the prime factorization of z doesn't exist.

The way to get around the infinite structure is : (10^.1)^.1^.1.....to nine stacks is the number 23*10^-9. This related to the rich prime,semi-prime number formula , (.75*x^2)+(1.5*x) +23 = prime or semi prime numbers when x is an even number.

This sounds like a great topic. But with the audio i can't sit 30+ min. If possible, reupload

Saying "infinity exists" is not the same as saying "There exists things that are infinite". I don't really know what the former even means. Just because our language allows us to form a noun like 'infinity' out of an adjective like 'infinite' does not mean that the noun refers to anything cogent. Some configurations of Turing machines halt. Others do not.

I really enjoy all of the content on this channel. I think with the ball thought experiment... maybe I'm missing something. If the one box starts with an infinite number of balls... and we take ten away an infite amount of times, the experiment will never complete because infinity minus 10 is still infinity. It may not be useful to even talk about the results "in the end" of this experiment. It may be useful to take a look after a given amount of "turns", but at any state there are still an infinite amount of balls in the first box. The number does not decrease because 10 are removed. Infinity minus 10 is still infinity. You can take 10 balls at a time away forever because Infinity minus 10 balls performed an infinte amount of times still leaves infinity in the box. To play this out over an infinte number of turns, the best estimation would be the second box would always be whatever is in there already plus the infinite amount of balls still to come from the first box, and likewise there would be an infinite number in the corner (what is there plus the infinite amount still to come). Mainly, if something is done an infite number of times the notion of even talking about that process "completeing" is flawed. All boxes will need to accommodate an infinite amount of balls for this process to play out an infinite amount of turns.

2 Question for Norman. 1) How did you exhaust the balls in the first corner? In my view, infinite is assumed to be inexhaustible. 2) How did you remain with two balls in the box? According to what you said, you kept on getting one ball for every ten balls. On the other hand, I was perplexed when I saw infinite multiplying itself 3 times, the 1st corner, box and 2nd corner, making it look like it's possible to have some infinites which are a lot more(bigger) than others.

@maxwang2537

2 жыл бұрын

I agree. According to the classical theories, infinities even have different orders, which mean an indignity divided by another can be zero, a nonzero constant, or another infinity. I don’t see an obvious issue here.

@JT-sv9bi

2 жыл бұрын

2) I guess the idea is that you add 10 but take out number 3, you add another 10 but take out number 4, you add ten more, but take out number 5, etc... After an infinite number of times you will have taken out all the balls from 3 to infinity. So you are left with 2. But of course, you've also added 9 balls each time, an infinite amount of times. So the infinity in the new corner would have to be "bigger" that the infinite number of balls or operations. Which doesn't make sense.

@joaogoncalves-tz2uj

Жыл бұрын

1) that is the whole point of modern math. It takes the infinite sequence of natural numbers and pretend it can be captured in the set of natural numbers. It gives a name to that infinity: aleph zero. When Noah says "now let's go to infinity", he really means let's do what modern math does.

I dunno, seems like this video lasted an infinity. lets take space, remove everything from it and reduce the vacuum to perfect zero. is what remains infinite, finite, or both simultaneously?

I think Mr Wildberger should have defined what is "exist". Because, infinity clearly doesn't exist in real world (even if we say "infinitly big" physics is never "infinitly" big, but huge, exactly the same with "infinitly small") included "actual infinity". But what does it mean "exist" in math ? Something exists if it's not contradictory with logic law, or itself, no? So infinity doesn't exist means infinity is in contradiction with itself or with logic axioms. Why not defined what is "intuitionism" "platonicism" "aristotle school" ... ? I think polemic will be clearest. Just one example of "actual infinity" take a segment, say [AB] take a second, say [CD], not on the line (AB) and as CD trace [AC) and [BD), they intersect in O 1- If actual infinity doesn't exist (in math sense), I could found a point in ]AB[, say P, that (OP) never cross (CD) 2- If actual infinity exists, whatever point I choose, there will always a point in ]CD[ which is the intersection of (CD) and (OP), and it never the same for 2 differents P. In computer science we are clearly in case 1. In mathematic, we can choose.

Dimensional Gauge Symmetry; combinatorial infinity in nature. And you got it Norman. Look at the casio calculator memory error. How long before it can be fixed? $$$$, wrong metric. And the economic Lagrangian too.

I'm increasingly becoming an adherent of NJW. Thanks.

Professor Wildberger knows a bunch more of math that I currently know and that I'll probably ever be aware of, but why is it neccessary that math be limited to the real world? Why can't it be a separete world in which we accept the existence of irrational numbers, infinite sets, imaginary numbers and so on? What's the reason for preventing us from conceiving these fascinating and interesting object provided we don't derive contradictions and everything is consistent and logically justified?

@WildEggmathematicscourses

6 жыл бұрын

@ Juan 12345: Without proper definitions we never really know if everything is consistent and logically justified.

@Achill101

3 жыл бұрын

Why can't we think of a story that didn't happen in the physical world? Asking the question means answering it in the negative. Of course, we can have fictious stories. Similarly, we can think of the infinite and describe it, without it having any correspondence in the physical world.

@Achill101

3 жыл бұрын

@@WildEggmathematicscourses - ZF set theory has been found to be logically consistent. What do you mean with justified? Do we need to justify everything we're creating in our thoughts? Couldn't it be a game? Even if we needed to: math has been been very useful for describing and manipulating the physical world, and that math used the infinite.

You are drawing a distinction between the practical and the conceptual. I can conceptually understand infinity but cannot know what infinity is from a practical standpoint.

@njwildberger

4 жыл бұрын

I think most of us can't even understand infinity "conceptually".

@aa697

4 жыл бұрын

@@njwildberger I understand its meaning. And l can understand that 1/× as x gets larger and larger goes to i.e. approaches zero. Or for example 10 to the x power as x approaches " infinity" is infinitely larger and larger. But I get your point. Great stuff to think about.

Instead of "defining" the set of natural numbers N, we can list some of it's properties: it is a well ordered set that is closed under addition and multiplication and satisfies certain laws like distributivity and associativity. No finite set that we can come up with satisfies this. So you might think that N is just a made up thing; however, it seems kind of strange that when we assume the existence of an N with these properties, we never get a contradiction when applying the laws of logic. That's why I think N exists. So essentially, we can use our finite experience as evidence that an infinite object exists; the fact that we never get a contradiction in any our of finite-length proofs about the set N supports the idea. It's kind of like how we can know what the center of the earth is like without ever having been there; we can have evidence that inifinity exists (or at least is non-contradictory) without ever actually experiencing it. I remember physicists talking about how we can have evidence of 10 or more dimensions even if we'll never be able to observe them with our senses by noticing strange things in 3 dimensions (such as particles bizarrely popping in and out of existence) and then showing that a 10 dimensional theory would provide the simplest and most elegant explanation of them.

@michaelgreenberg782

5 жыл бұрын

Just as we cannot define N but can only list its properties, we cannot define what a set is, but we can list its properties (the ZFC axioms, for example). The fact that they never contradict in our finite length proofs gives us evidence that "sets" exists (or at least are non-contradictory). In fact I'm not sure if you can make a distinction between something existing and it being non-contradictory.

By this logic, the number of subatomic particles in the "known universe" is a well defined natural number n and to add 1 to such number breaks mathematics?

If you define a "number" as as abstract concept that describes the quantity of a group of objects, and the objects can be actual things or abstract concepts themselves, then it seems obvious that infinity is a number. It is the number that tells you that the objects are never ending. That certainly describes the quantity of the objects, does it not? I guess you might also want a definition of "quantity". I would define it like this: Draw a box, put a dot in the box. Draw another box, put a dot in the box, now put another dot in the same box. Do this a few more times and follow the pattern. The quantity is everything that is different about the dots in each box other than their location. We can define a "number" for each box to describe the quantity of the dots in that box. The way I would describe the quantity of a collection of objects is like this. Take a dot for something that belongs to the collects of thing you wish to describe the quantity of. Put the dot in a box, now discard your object. Keep doing this until you are all out of objects, putting the dots in the same box of course. The number that matches that box from the previous paragraph is the quantity of your collection of objects. Now, lets say that this process will never end based on the nature of your collection, we describe that quantity as "infinity". Do you disagree with this definition? Or do you disagree that never ending abstract concepts exist? Is your objection something else all together? Also, the fact that within a never-ending process there will be multiple disjoint never ending processes, doesn't really seem like an issue. The even and odd numbers within the natural number for example. It just means that if you were to count every other natural number, or every other anything infinite, that this process would never end, so we describe its quantity by what we define to be never ending, infinity. I love your videos by the way! Especially the history of mathematics lectures. I studied Chemical Engineering and was 1 class short of a math minor in college and I do a lot of self studying of more advanced math than what I studied in college.

I agree with Wildberger and here's why. To me, an "infinite set" is a bit of a misnomer. It is actually just a finite set with the additional property that you can always include more elements using a predetermined algorithm. The algorithm can be anything your imagination can conjure. It has specific inputs and input criteria, is deterministic, may have additional, internal criteria checks, and outputs elements... even collections of elements at a time if you prefer... even an empty set for certain inputs if you prefer. You don't need this new word "real". Rational numbers already do all this. I can make the set arbitrarily large and have special subroutines in my algorithm that specifically produces PI or root-2 to the desired number of decimal places. *The number of elements and precision of individual elements is limited only by what I can afford.* I think that Wildberger is right that the mathematics community has been lazy and did not create a meaningful framework for generating the real numbers. Maybe everyone is so embarrassed by the fact that real numbers are actually just rational numbers. We did even worse by only saying "It's infinite" in the textbooks. I'm not even a mathematician by profession and I have provided a much better definition stated above. Sure, keep the word "infinite" -- it does, after all, fit with the fact that you can keep generating new elements -- but provide something like the definition I gave above to go along with the word "infinite". So the next time you say "the set of all" of an "infinite set", think of it as starting from the empty set and you populate the elements in based on what you need and can afford (time/money/technology). It is always finite but can grow as large as you want. Doing things this way avoids esoteric ideas and philosophical debates which will probably lead to "What is life?"... sheesh!

@eddielam2875

8 жыл бұрын

I think you are taking mathematics too realistically. Mathematics is not the study of real life objects, it is the theoretical foundations and studies about what will the consequence be if we are given such and such systems and axioms. Of course in a real life sense, it is impossible to have an infinite set, no computer could have generate one. But in pure mathematics, we can define infinite sets by using different rules. Because in the mathematical world a set is just an object that needs not exist in the real world! We allow the set of all natural numbers to exist because while it is impossible to write it down physically, it is completely logical and obvious that 1,2,3,4,5,... are all inside the set. Also I can assure you that real numbers (including irrational numbers) do exist. As in a perfect right angle isosceles triangle with side length 1, the length of the hypotenuse is the square root of 2. To see that it is irrational (which means it cant be written as a quotient of two integers), let sqrt(2)=p/q be an irreducible fraction, then p^2=2q^2 , which means that p must be an even number p=2k, then 2k^2=q^2, then q must also be an even number, contradicting the fact that it can be represented by a fraction, meaning it is irrational. Of course back to real life, irrationality may not be too common because we tend to round things up to the nearest cm,mm etc in our daily measurements. But if I define 1meter= sqrt(2) arbitrary length unit, then in terms of this metric system, all our rational length will become irration! As in the idea of real numbers, it is a fact that they exist, e.g. it is proven that pi, the ratio between the circumference and diameter of a circle has non repeating digits, which is equivalent to irrationality of pi. The problem about maths is that you cant take it like physics or chemistry because it is a field in which theoretical objects are the main interests. What does it exactly mean to be a manifold (geometric objects) in the 7th dimension anyway? It doesn't make sense to talk about it but we do have ways to study about it and it does have theoretical applications in string theory etc.

@graysonsoldahl

8 жыл бұрын

+chin hang lam You say that there's such a thing as a mathematical world, but that's mental experience I'm having. For me it's text. Text is just lines. This reality seems like a game about what you desire. Every world has a limit that I've experienced. Even though I remember meditating in lucid dreams and feeling time stretch for what seemed timeless, it had a time. Hence I woke up when I remembered this life. Sure you can play with axioms that are useless in the world but that's not what I want to do.

@justcrank9088

8 жыл бұрын

+Grayson Soldahl ''Sure you can play with axioms that are useless in the world but that's not what I want to do."" SIR!!! do you realize the allgeations you are making against the most respected INFINITY??? Infinity is a very practical thing. Infinity makes mathematics possible simple and complete why make something needlessly difficult or incomplete...

@justcrank9088

8 жыл бұрын

+Euquila "To me, an "infinite set" is a bit of a misnomer. It is actually just a finite set with the additional property that you can always include more elements using a predetermined algorithm. The algorithm can be anything your imagination can conjure. " No way....you cannot use anything in your imagination....if you could do that you would have a infinite series named after everyone in the world

@okuno54

8 жыл бұрын

The definition you propose sounds closest to the definition of "recursively enumerable set". These sets can be infinite, such as the natural numbers, or the digits of pi. Wildberger is taking the much more extreme stance that even these sets do not exist, even though there are finite algorithms generating them.

This is very interesting. If I am understanding Dr Wildberger's argument early in the video correctly, then it would seem that one consequence of it would be that there should exist a largest natural number, and that the precise value of that number would be an empirical question. Am I on the right track? If so, I cannot help but feel that there has been a change of subject somewhere along the way... Because if there exists a largest natural number n-a number n than which none greater can exist-then it follows that the question "What is n+1?" must be answered with "There is no such thing." But it really seems that whatever we mean to say does not exist when we say something like that simply isn't we generally mean when we make reference to natural numbers... Alternatively, perhaps we could posit that a number exists if and only if (1) we have conceived of it, and (2) it meets certain "pragmatic-mathematical" criteria (e.g., we can in actual practice list all of its prime factors). This would seem to have the interesting consequences that numbers come into existence in dependence upon our thoughts, and that there will exist some number n such that it exists but such that n+1 can be conceived of but does not exist. The second of these seems once again like it might be a change of subject, for the same reason mentioned above (i.e., we're redefining the concept of "number," rather than stating something new about what we were referring to all along). Abandoning the second criterion while holding on to the first, on the other hand, would indeed seem to reduce the natural numbers from the realm of actual infinity to the realm of potential infinity. And it seems to do this not by redefining our definition of "number," but rather by bringing into focus our definition of "existence." Does it sound like I'm on the right track? Update: Having watched the rest of the video, it now feels very clear to me that this debate does not hinge on a single disagreement as to whether infinity exists or not, but hinges rather on two separate disagreements: (1) what does "number" mean? and (2) what does "exist" mean? I think that anyone who accepted one party's answers to both of these questions would, trivially, agree with that party's stance regarding the root thesis of the debate. Incidentally, I don't think the argument about the number of balls left in the container is very effective. It seems to show only that our current theories of infinities lead to results that are profoundly counterintuitive. But this alone does not settle the question of whether the resulting paradox is falsidical or veridical (there does not seem to be any question of its being antinomical). If profound counterintuitiveness were to be accepted as a valid refutation of mathematical theories, then the majority of probability theory would have to be abandoned!

As for prime factors of (10^10^....)+23, what is wrong with 3 ?

But what about the Infinitesimal dX in calculus?

Any one else working on designing new counting systems? All we need to do is be able to represent irrational numbers, they're finite but endless digits, seems impossible but keep trying

There are is a difference between being able to imagine a mathematical object and being able to conceive of that mathematical object. The latter transcends the former. I can conceive of the number of all possible states of a non expanding universe in terms of Plankian limitations. I am unable to imagine it. But I conceive of it. And I can conceive of that number plus one. And I think it is meaningful to say, counterfactually, that of these two numbers only one is even. That is, I can conceive of reducing such a number by two, repeatedly, until I am left with one or zero.

Infinity: does it exist?? Not yet, but we're working on it.

Maybe "unbounded" is a more suitable concept in this context

The infinite sequences have a solid foundation. We have learnt how to work with it and it is extremely useful. Creating Analysis without infinity is just a redefinition of the same. If you want to get rid of infinity, you have to prove that it leads to wrong results.

@devalapar7878

2 жыл бұрын

@alone from society You have no idea what infinity means.

Just because you cannot physically factorize that large number into primes doesn't mean that you literally can't. This doesn't exclude existence in the slightest... Sure, it may be difficult, but that's because you're looking for specific values, which is different from existence. In any case, if you want a specific value, then 3 is one of the primes for that so called 'unfactorizable number'.