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Can all math results, that have been correctly proven so far, be formalized and checked by a computer? If so, what type of logic would need to be used there?

I've heard that the first-order logic is not expressive enough: certain facts can not be expressed in it, and sometimes proofs can be given, but they have asymptotically large size (not just merely syntactically inconvenient).

MWB
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    By "all results", we can assume you mean "all results found by humans that aren't wrong", rather than the more philosophically challenging "all correct results that could ever be found" ? – DanielV May 12 '14 at 18:28
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    Does this question count as a "math result"? – Brian Fitzpatrick May 12 '14 at 18:38
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    IMHO, if it can't be formalized, then it isn't mathematics. (Not to say that if it isn't formalized, then it isn't mathematics.) And any formal proof can be verified by a suitably programmed computer. That is the nature of formal systems and computers. I am not aware of any mathematical proofs or statements that cannot be expressed in the language of FOL with notation extended as required, e.g the $\in$ symbol for set theory. (Speaking as a developer of such software. See my website: http://www.dcproof.com ) – Dan Christensen May 12 '14 at 20:54
  • @DanielV Yes; edited – MWB May 12 '14 at 21:05
  • @DanChristensen First-order logic under usual formations has all of the connectives and functions as constants. So, unless I've missed something variable functions cannot get expressed in first-order logic, nor can quantified variable functions. One can't prove, for example, that C$\delta$pC$\delta$Np$\delta$q is an axiom of a propositional calculus which has all of the theorems of more common propositional calculi, and allows us to deduce all C-$\delta$-N tautologies. Nor does $\forall$p$\forall$q$\forall$$\delta$ C$\delta$pC$\delta$Np$\delta$q make sense in first-order logic. – Doug Spoonwood May 13 '14 at 03:14
  • @DanChristensen Also, if I understand correctly, you can't do proofs like the one I did here http://math.stackexchange.com/questions/516912/how-do-i-prove-the-theorems-needed-for-the-deduction-meta-theorem-from-ccpqcp/780476#780476 in first-order logic. – Doug Spoonwood May 13 '14 at 03:16
  • @DougSpoonwood A function $f: A \to B$ can be defined in FOL with $\in$ and functions as $\forall x:[x\in A \implies f(x)\in B]$. Some (most?) authors will say you can't quantify over sets or functions in FOL, but it seems to work in my version of it. I don't quantify over logical predicates or propositions. Your notation is not one I am familiar with. – Dan Christensen May 13 '14 at 05:08
  • @DanChristensen I don't think that your definition will work for variable functions. You'd need something like $\forall$f∀x:[x∈A⟹f(x)∈B]. Also, quantification over propositions comes as something also investigated to some degree in the study of propositional calculi. No doubt there exists a shortest axiom(s) of propositional calculus where quantification over propositions comes as allowed. Additionally, that some authors say that you can't quantify over sets or functions indicates that in some systems of first-order logic you can't express all mathematics. – Doug Spoonwood May 13 '14 at 13:04
  • @DougSpoonwood Based on several years of my own use writing hundreds of proofs, my definition of functions seems to work quite well. Your definition doesn't make sense to me. Why the $\forall f$? – Dan Christensen May 13 '14 at 16:50
  • @DanChristensen Because when you make a statement about variable functions for a given domain, you can make a statement about all such functions or at least one such function. They differ significantly, just as a claim about all numbers of a given domain differs from a claim about the exist of a number in a given domain. For say binary operations of a domain of two elements one can say that $\exists$f such that f it can represent all other operations. However, to say that $\forall$f f can represent all other operations is false. When I wrote ∀δ it has the same meaning as $\forall$f. – Doug Spoonwood May 13 '14 at 17:09
  • @DanChristensen This "∀p∀q∀δ CδpCδNpδq " can get read "for all p, for all q, for all delta, if delta of p, then if delta of not p, then delta of q." The meaning of it could get interpreted as "for any proposition p, for any proposition q, for any truth-function delta of one argument, if delta of the first proposition p, then if delta of the negation of the first proposition p, then delta of the second proposition q." – Doug Spoonwood May 13 '14 at 17:37

1 Answers1

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First, the good news:

There is a strong belief among mathematicians, particularly among mathematical logicians, that nearly all (correct) proofs in the literature could be formalized to the point that a computer could completely verify the proof.

There are many automated theorem verifiers ("proof assistants") that have been developed for this purpose. Recent developments include completely verified proofs of the Prime Number Theorem (two independent formalizations), the Jordan Curve Theorem, and the Four Color Theorem. It is certainly possible to completely formalize nontrivial mathematics.

The bad news is that

The process of taking a proof in a journal and converting it to a proof that can be verified by computer is tedious, arduous, and non-trivial.

Even though a proof assistant program takes proofs in a high-level format as input, the format is still much more detailed and verbose than a natural language proof would ever be. Moreover, the standard foundational systems (such as ZFC) are particularly difficult to work with to actually write formalized proofs.

Avigad (2007) reported about the proof of the Prime Number Theorem, which was written in the much more convenient system Isabelle, that:

"The five-line derivation of the Mobius inversion formula in Section 3.1 translates to about 40 lines, and the proof of the form of the Selberg symmetry formula discussed there, carried out in about two-and-a-half pages in Shapiro’s book, takes up about 600 lines, or 13 pages. These ratios are more typical."

He also reported

"This process stabilized, however, and towards the end [I] found that [I] could formalize about a page of Shapiro’s text per day."

So, although we all believe it is possible to fully formalize a proof, the effort required with current technology makes it impractical. Most top-tier research mathematicians can assimilate much more than one page of a textbook in a full working day. Perhaps technology will improve.

There is also a question of purpose. Some (perhaps influenced by technology) view computer verified proofs as a goal to work towards. But others, including Bourbaki, have viewed it as a distraction. Many of the latter kind feel that the purpose of proofs is to communicate ideas with other mathematicians. While computer verified proofs have some additional certainty, the form in which they are written is not at all suitable for conveying ideas to other mathematicians. After all, what mathematicians are usually interested to see is not only the "proof", but also the "idea" or "method" that underlies the proof, because that is what can be used to prove other theorems.

Carl Mummert
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