Infinity: does it exist?? A debate with James Franklin and N J Wildberger
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- เผยแพร่เมื่อ 27 ก.ย. 2014
- Infinity has long been a contentious issue in mathematics, and in philosophy. Does it exist? How can we know? What about our computers, that only work with finite objects and procedures? Doesn't mathematics require infinite sets to establish analysis? What about different approaches to the philosophy of mathematics--can they guide us?
In this friendly debate, Prof James Franklin and A/Prof Norman Wildberger of the School of Mathematics and Statistics, Faculty of Science, UNSW, debate the question of `infinity' in mathematics.
Along the way you'll hear about Jim's new book: `An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure', published this year by Palgrave MacMillan.
Unfortunately, the microphone could not pick up audience questions and responses very well. The correct answer to Norman's question at the end of the game he described was given by Roberto Riedig: `any number you want'! As for this interesting game itself, Norman seems to remember getting the idea from Wolfgang Mueckenheim, who also ventures into heretical waters: see for example his paper "Physical Constraints of Numbers", Proceedings of the First International Symposium of Mathematics and its Connections to the Arts and Sciences, A. Beckmann, C. Michelsen, B. Sriraman (eds.), Franzbecker, Berlin 2005, p. 134 - 141.
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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 TH-cam. This has helped me to think in a more complete way about this issue.
+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.
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.
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.
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.
How far does the need of definitions go?
@@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.
Loved that last ball game!
id love to see more of those "battles" in the future!! 40 minutes were to short
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.
@@conoroneill8067 Holy, do you have the rest of the video?
really good arguments Prof Wildberger!
really great thought experiments and points that I think a lot of mathematicians don't think about/consider
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
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.
cool ..need more of this discussion.
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 th-cam.com/video/WNAm7TH0iOw/w-d-xo.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.
Isn't James Franklin's view the common sense physics view nowdays? Big bang coupled with heat death and limited observational capacities, of course!
"(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.
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.
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).
@@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.
(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.
Great professor. Quite interesting topic
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
You are so right. It is truth.
being and existence (expression) should be distinguished here.
something exists (ex-sists) if it could be _expressed_ in a graspable manner.
the prime factorization of a number has *being*, although there might be no way to express it.
the *set* of natural numbers neither *exists* nor *is*.
for something to constitute a set, it has to form into one.
now the number 1 is a set, here is its one-ness in "action": {0}
the natural numbers do not constitute a set, but they *are* without *being one*.
they are multiplicities whose "totality" exceeds the power of one-ness.
we could certainly talk about their *cardinality* though.
we know for example that the cardinality of N is a real thing, although N itself isn't *a* thing.
the point is simply that, when it comes to infinities, our language (our power of expression) naturally fails. so we say things such as *N* although we know that it is not a set but we say it anyway because we are not robots!
Z is a cursor that is defined recursively
data Z = Z | Succ(Z) | Pred(Z)
it needs a "middle" point for construction
Norman is a genius
I don't think that either one is right - I agree with Norman on certain things, but I think he goes a little bit overboard.
I agree that: 1) Infinity does not exist in reality (physics). The Time-Space continuum that we as a species can physically influence is finite; anything meaningful or coherent (like a large number that is the result of a mathematical problem) necessarily would have to originate from a single point and can only grow at a finite speed. I don't think there is an infinite amount of time ahead of us either, so... And 2) Infinity is not a number, you cannot calculate with it as with finite numbers.
Before I state what I disagree with I have to add that I am convinced that reality, the universe, quantum mechanics, us, everything *is* pure mathematics. We exist because equations exist in mathematics that represent reality - we exist in mathematics. So -- from that you'd think that I'd oppose the idea that mathematics would have anything that can be called infinity, as per point 1 above. But I don't. It is not necessary for things that can exist in mathematics to be part of our reality. We can be the solution of a certain equation that simply does not include all natural numbers, not even a particular finite one like the largest prime factor of 10^10^...^10 + 23. But I DO think that that prime factor exists in mathematics itself.
I think that 3) Every (finite) number that can be described in any way, exists (in mathematics) EVEN if we have no way to actually write it out or even "know" it. For example, 10^10^...^10 + 23 *does* have a unique prime factorization. 4) For the same reason, any irrational number that can be described in any way - exists, even if we can't write it down. Aka, the sqrt(2) exists: it is one of the two solutions to x^2 = 2 (but there is no such thing in reality / physics - anything that links a square to the number of two would be "smeared out" (Heisenberg) and require an infinite number of time to establish itself to infinite precision to sqrt(2)).
Here is what I think 'infinity' is. Long before youtube existed I already thought this. Each problem that needs an infinity in modern mathematics can also be solved by using a finite number N that is so large that it is irrelevant FOR THAT PARTICULAR PROBLEM to make it any larger. So, N is arbitrarily large, but not infinity. I think Norman made a mistake with the example of infinite balls therefore; what I'd say is the following: If you have 'infinity' number of balls in the corner - then per definition - that amount is larger than whatever you can take out of it (you can take out an *arbitrary* large number N, but if that would exhaust the original amount then the original amount wouldn't be "infinite" since infinity means that it is irrelevant for the problem to make it even larger, hence there must be more balls in the corner than you take out. So, let N be the number of times you run to the corner and get 10 balls. Then next you can argue that if you put 10 balls in the box and take one out and put that in the other corner then there will be N balls in that corner, and 9N in the box. Next let N "go to infinity" (that is, make it arbitrary large, so large that it is irrelevant to make it larger - and call that infinity) then the end result is that there are an "infinite" number of balls in the left corner, there are an infinite number of balls in the box, and there are still an infinite number of balls in the corner on the right. This, in my opinion, is the only way to think about infinity; saying that there are only 2 balls left in the box is *nonsense* in every way. Of course, that is what Norman wanted to argue too: he wanted to show that it was nonsense. His mistake is to say that it is possible to continue 'to the end' until there are no more balls in the right corner - nor do I agree with him on that there are only a finite number of natural numbers because you run out of space in the universe.
Applying this idea to, for example, the sqrt(2) - Let x^2 = 2 and x > 0. Then we can think of x to produce a x^2 that is arbitrarily close to 2, so close that it is irrelevant for anything related to whatever we do next with it not to consider it to be actually equal to 2, and CALL it "sqrt(2)". The funny thing is that if you apply this to computer calculations then this irrelevancy is suddenly very imprecise! Aka - if we want to calculate the color of pixel on the screen as the result of a 3D project of some model etc, then you don't need a very precise sqrt(2) to arrive at the quantized value of the (most) correct digital color, and at a certain precision - as used by computers - becomes "infinitely" precise, because it would be irrelevant for that case to make it more precise.
Considering the game that Wildberger played. Having an infinite amount of balls in one corner, putting 9 in a box and throwing 1 in the opposite corner at each move. In order for the original corner to be emptied of balls one has to think of infinity as something finite. I would rather think that the infinite pile of balls are displaced at the rate of 9 balls to the box, 1 ball in the opposite corner at each move. I think that there would be an infinite amount of balls at each place as we go towards the immeasurable. What we can grasp is the concept of growth, which are 9 balls per move for the box, which thus grows faster than the opposite corner which increases by 1 ball per move and finally the original pile is decreasing by 10 balls per move. But no matter what measurable/immeasurable amount of balls are removed from an infinite amount, there would still be an infinite amount left. What is of interest, by my opinion is the ratios between them. For example, what we can say about the natural numbers is that they grow by one unit at each step. So that is a recognisable pattern, as is the the ratios between the two corners and the box, and thus an object of study. Moreover those ratios seem to be constant for each time step, and if we set the rules as such, we may assume that they are. But if we are oblivious to the underlying rules we may only make qualified guesses based on our experience. Mathematics, I have heard is the study of patterns :) Correct me if I'm wrong, thank you!
I'm increasingly becoming an adherent of NJW. Thanks.
Sorry, but that is, by modern standards, a somewhat weak argument of Descartes. It’s much in the direction of earlier arguments in favour of the existence of God by Spinoza and others. It’s important to remember that 17th century and earlier discussions of infinity were thinly disguised arguments about the existence of God.