0

By Godel's incompletness theorems we know there are problems that cannot be proven. Is this sufficient to claim mathematics (the set of axioms and theorems) is finite?

As a counter-argument, we might say that for a finite set of axioms we can produce an infinite number of theorems by permutation and combination, and also that Infinity - X = Infinity (X being the number of conjectures Godel's theorems apply to. It could be infinite).

Does the first argument still have some merit? Is there another argument that proves maths is finite?

6

Actually, it's the opposite. Here's a simplistic overview of why: Suppose you do a finite amount of mathematics (prove a finite number of theorems from a finite axiom set). But by the incompleteness theorems, there are some theorems that simply cannot be proved. Thus, no finite amount of mathematics suffices to encompass all of mathematics.

It's instructive to look at a specific case study of what situations like these actually look like in the course mathematical research. The first and still most famous case is the ancient question of whether or not Euclid's parallel postulate (PP) can be derived as a theorem from the first four axioms (the postulate states that There is at most one line that can be drawn parallel to another given one through an external point).

One way geometer's posed this problem was to attempt to re-prove all the old Euclidean results from scratch without invoking the axiom of parallels, and see how far they could get. This system of geometry, Euclidean minus parallels, is called Neutral Geometry because it is neutral about whether or not the parallel axiom is true. The question is then, does Neutral Geometry end up being equivalent to Euclidean Geometry anyway?

And of course, they failed time and again to prove PP within Neutral Geometry, and so they eventually started focusing their research on more indirect approaches, like considering what taking "PP is false" as an axiom would imply and presumably something absurd would result. One way to falsify PP is to say that we can find at least two parallel lines through a point, in contrast to at most one. Counter-intuitive results followed, but it this system was surprisingly proved to be consistent regardless, and it came to be called hyperbolic geometry. Whereas before geometry was a single unified system codified by Euclid, both complete and consistent, was shown to be but a stem of an ever branching family geometr*ies*.

This is what really makes the implications Godel's incompleteness theorems so profound. It guarantees the openendedness of mathematics. In the beginning there was geometry and arithmetic, and then algebra and eventually calculus/analysis. The subject matter of mathematics could be understood in completely taxonomic terms: space, quantity, structure and change. But today we can see that mathematics exceeds any definition in terms of subject matter. On might say that mathematics is more of an art: the art of being creatively logical through the media of abstraction.

  • How would that change if we assume Godel's theorems do not hold? Would it be meaningful to make that assumption? And even so, that only proves there are some theorems that cannot be proven with any amount of available theorems – rath Jul 28 '13 at 4:33
  • @rath Godel's theorems are consequences of mathematical logic, so you're not allowed to assume they don't hold. The only way they can cease to hold in non-classical systems of logic. I'm not qualified to comment on what happens in this scenario. If you're curious, that would be a good question for mathSE. – David H Jul 29 '13 at 16:22
  • Good answer. Godels theorem could be taken as an answer to the epistemology of mathematics. One can also take a more indirect route, and start from the first line of the Tao, that they Way that can be walked is not the Way, and interpret that as saying the truth that can be formalised is not the truth - meaning that truth is always an excess, fuller than any means that we have to capture it. Still this is only my interpretation, one would have to look at the literature round the Tao to see whther this line of thought had been taken up. – Mozibur Ullah Jul 30 '13 at 9:51
  • 1
    Of course where Godels theorem is innovative is that it expresses this excess in a formal manner, and also it opens up new questions and new routes within mathematical logic itself. Goedels theorem holds in at least one non-classical logic - intuitionistic logic. Its a question I've asked on this site, I think. There is another non-classical logic which is dual to intuitionistic logic - a paraconsistent logic, where the law of non-contradiction is given up. It would be interesting to see what form Godels theorem takes there, if it all. – Mozibur Ullah Jul 30 '13 at 9:52
  • @rath: This answer is completely wrong!!! Godel's incompleteness theorems have nothing to do with and says nothing about Euclidean or hyperbolic or spherical geometry! The theorems only apply to formal systems with decidable proof validity that also interpret arithmetic. You will find the proper explanation of this in any decent logic textbook. – user21820 Oct 9 '16 at 4:59
2

From a purely theoretical perspective, mathematics is easily seen to be infinite, in the sense that it contains infinitely many theorems. Why? You can just produce a machine that spits out true mathematical statements without end. For example (if one is lazy): 1 = 1, 1+1 = 1+1, 1+1+1 = 1+1+1,... Of course, these are special cases of a general rule that x = x for any x, whatever this x is. But I don't think there is any mathematically definable line to divide such "trivial statements" from "true mathematical theorems".

If you accept that integer arithmetic speaks of a universal object that just "is", then Godel's Incompleteness Theorem tell you (among other things) that to describe arithmetic you need infinitely many axioms. This is a rather strong statement, and it goes in the direction reverse to your hypothesis: mathematics is very infinite. It isn't just that there are infinitely many theorem, there are infinitely many basic rules that cannot be derived from the rules you start with. If you want to do mathematics in a purely formal manner (just transforming formulas, without reference to their meaning) then you can't even say what integer arithmetic is (in finite time).

Of course, the above remarks are about the purely theoretical (meta)mathematics. If we think about mathematics as something that we do in our particular universe, and we accept that this universe is somehow finite in principle (or that our civilization has finite span, or that human being have a finite aptitude for mathematics, etc...), then mathematics is finite as well, as was pointed out in other answer.

1

There is a much easier way to show that 'maths is finite'. Suppose that the universe has a finite lifetime. Then the human race has only a finite amount of time to prove theorems, hence there can only be a finite number of theorems proven!

Take any mathematics expressed as a set of axioms, and enumerate every possible proof, this process will not halt, but in theory you're proving every possible theorem in this axiomatic system. Of course you haven't exhibited a single significant proof, which is generally the point of forming an axiomatic system and proving a theorem! What this shows is that the set of theorems is countable. Now from the perspective of any large cardinal axiom, countablity is small. So you haven't shown that the mathematical truths enumerated by this set of axioms is finite, but that they are small, for a suitable idea of small.

Of course I'm not being particularly serious above, as this misses the point of mathematics which is to come up with significant new ideas, questions, theorems & proofs.

  • You're assuming that one step in the computation always at least takes some minimal bounded length. But maybe hypercomputation will be possible. :) – Nikolaj-K Jul 29 '13 at 9:28
  • @Kidman:Yes, and there is hyper-hyper-computation; and hyper-cubed computation etc :). – Mozibur Ullah Jul 30 '13 at 9:56
  • Transfinite computation? – Nikolaj-K Jul 30 '13 at 11:14
  • @Kidman: using Oracles, yes. But all of these are speculative models, as far as I understand. – Mozibur Ullah Jul 30 '13 at 13:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.