1

The result of Godel's theorems was that we knew for sure that a formal axiomatic system wasn't capable to derive all of mathematics. The math derived under the system cannot be consistent and complete.

Why isn't the onus of failure put on mathematics as well? Seems to me like if mathematics cannot be derived under any formal axiomatic system, then the problem is with mathematics as well, because you'd certainly need a formal axiomatic system to derive any complete sentence.

How do we even know then for sure that mathematics is complete and consistent?(If these words still apply)

novice
  • 578
  • 4
  • 15
  • 3
    You are missing a large part of what Gödel's (really Rosser's generalization) theorem says. It doesn't say that there is no compete and consistency theory of arithmetic, it says that there is no complete, consistent, and recursively axiomizable theory of arithmetic. Take the theory Th(N) of all first order sentences that are true of the natural numbers. That is a complete and consistent theory of arithmetic. However, it is not recursively axiomaizable. – Not_Here Apr 25 '18 at 18:04
  • @Not_Here Yes. The recursively axiomizable part didn't seem very interesting I guess at first, so got left out when I wrote this. – novice Apr 25 '18 at 18:27
  • And I think that is a major part of the issue you're having on this subject, it's one of the most important parts. Gödel was a platonist and he assumed that there is an answer to every mathematical question, that isn't contradicted by this theorem. What is contradicted is the assumption that it can be recursively generated. – Not_Here Apr 25 '18 at 18:40
  • @Not_Here : I'd be happy to know better. Wait for your answer. Do you mean that the theorem which proves the statement, can not necessarily be recursively generated? – novice Apr 25 '18 at 23:12
  • 1
    You might find Tarski's Undefinability Theorem interesting. It uses similar approaches to show that there are similar issues in a wide range of formal languages. – Cort Ammon Apr 26 '18 at 05:45
  • @novice I will get an answer out as soon as I can. What recursively axiomaizable means is that there is a Turing machine which will accept if and only if it's input is one of the axioms of the theory and reject otherwise. Again, Th(N), which is the true theory of arithmetic (see this also, is complete and consistent, but, as an easy consequence of Gödel's theorems and the halting problem, it is not recursively axiomizable. – Not_Here Apr 26 '18 at 10:24
  • @novice you can imagine Th(N) being a theory where every single true sentence about the natural numbers are it's axioms, all of those sentences. That means for it to be recursively axiomizable (sometimes people say effectively generated but it means the same thing in this context), there would need to be a Turing machine that accepts all true statements of arithmetic and rejects all false, which obviously is impossible because we have undecidability results, like the halting problem, about arithmetical statements. The point is that the theory is still complete and consistent, just not "useful" – Not_Here Apr 26 '18 at 10:26

2 Answers2

2

Godel's theorems are about mathematics. In fact, his theorems say that we cannot have a recursive (very roughly: 'finite'), complete, and consistent axiomatization (foundation) of basic arithmetic, and thus certainly not one of mathematics as a whole. He thus blew a hole in the hope (see the Hilbert Program) that we could know for sure that 'mathematics', as we typically practice it, is consistent. Indeed, as of today we don't know if the ZF axioms for set-theory, which can be used to underpin a lot of mathematics, are consistent.

Bram28
  • 2,709
  • 11
  • 14
  • So we don't know if mathematics is consistent? – novice Apr 25 '18 at 16:58
  • @novice That's right. The hope was that mathematics could somehow prove its own consistency; i.e. somehow 'bootstrap' itself. Godel showed that to be impossible. On the face of it, that seems pretty shocking; it's as if much of mathematics is built on quicksand ... but practicing mathematicians are not too concerned: it's working fine for us ... and will keep working fine us, even if we find an inconsistency. – Bram28 Apr 25 '18 at 17:01
  • 1
    This answer is very naive in terms of the philosophy of mathematics that it expresses. I completely agree with you that working mathematicians don't care much about the result of the foundational crisis, but that is like asking what working physicists think about Popper or the fall of the Vienna Circle. I don't mean to say that your answer is naive in general, or that you are naive; what I mean to say is that in terms of a general exposition of the philosophy of mathematics, which would be expected on phil.se as opposed to math.se, this is a very weak and non nuanced answer. – Not_Here Apr 25 '18 at 18:10
  • 2
    And even if it weren't lacking a nuanced view of the subject matter, it's incredibly short and doesn't explain anything that you're expressing. I highly disagree with this being an accepted answer, especially since the OP who asked the question had to reiterate one of their main questions in the comments after you already answered the question since you didn't fully address it in the body of the question. If another answer shows up (I'll write one if I have time) I implore them to reconsider accepting this as the best answer. – Not_Here Apr 25 '18 at 18:13
  • 1
    @Not_Here I know ... but I didn't want to make things too complicated. From the question, I felt that novice was .... well ... a novice in this area, and so I felt a fairly high-overview answer with the take-home punchline would be appropriate. But, if you feel a different answer is more appropriate, I encourage you to add your own answer. – Bram28 Apr 25 '18 at 18:14
2

The result of Godel's theorems was that we knew for sure that a formal axiomatic system wasn't capable to derive all of mathematics. The math derived under the system cannot be consistent and complete.

What I'm going to write here isn't a widely accepted position, but it has no unanswered criticisms.

Mathematical knowledge, like all other knowledge, is not derived from anything. Any argument uses premises and rules of inference from which a conclusion supposedly follows - if the premises and rules are correct then so is the conclusion. But we have no way of guaranteeing the truth of premises or rules so arguments can't be used to prove conclusions.

Maths isn't guaranteed to be complete and consistent for many reasons, including Godel's theorem. Another reason is that all our reasoning is conducted using physical systems, such as pen and paper and human brains. Those systems make errors, so mathematics can't be guaranteed to be error-free.

Mathematical knowledge is created by guessing mathematical ideas as solutions to problems and criticising the guesses, not by deriving them from conclusions. The same is true for other knowledge.

See the books by Popper listed here for more on the epistemology described above:

http://fallibleideas.com/books#popper

and "The Fabric of Reality" by David Deutsch, especially chapter 10 which is specifically about maths.

alanf
  • 7,748
  • 13
  • 20