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We have stated that the computer can handle tedious low-level details of proof. This isn't just an airy idea; on the contrary the LCF methodology allows the user to create higher-level proof structures which decompose to simple primitives inside the computer. But what grounds do we have for believing that this is practical even for a computer? (Accepting that some kind of definitional principle will be used as detailed above.) It's well-known that computers are amazingly fast, and getting faster, but it's equally well-known that their speed cannot defeat problems that have an inherent combinatorial explosion. The fact that fully automatic theorem provers have not displaced human mathematicians, and that the World Chess Champion is not (quite yet) a computer, are evidence of this. For example, if the length of a fully formal proof were an exponential function of the length of an informal proof, then the proposed approach would probably be doomed.

We've already seen that for any formal system of the kind we need to consider, there are true statements which aren't provable in the system (Gödel's first theorem). At the same time we have argued that such statements are pathologies and we're unlikely to hit them in practice. We will make an exactly parallel claim about the feasibility, rather than possibility in principle, of proofs.

The measure of feasibility will be the size of a proof, which is meant to denote something like the number of symbols in the proof based on a finite alphabet --- this seems the most reasonable practical measure. Certainly, from a theoretical perspective, there must be sentences out there whose proofs are possible but not feasible. Indeed, given any (total) recursive function f (e.g. a large constant function or a huge tower of exponentials), there are provable sentences of size n that have no proofs of size le f(n). For otherwise, provability would be decidable (just check all possible proofs up to size f(n)), contradicting the results of Church and Turing that the Entscheidungsproblem is unsolvable. More explicitly, we can modify Gödel's diagonal argument and exhibit a sentence which says not 'I am unprovable', but rather 'all proofs of me are unacceptably big':39

|- phi iff forallp. Prov(p,qqtelphiqqter) implies size(p) > f(qqtelphiqqter)

Now, whether or not there are proofs of phi whose size is bounded by f(qqtelphiqqter) is decidable, and so phi is provable iff it is true. But phi cannot be false, since that would mean there exists a proof of it, indeed, one within the stated size bounds. Therefore phi must be true, provable, and yet have no proof within the stated bounds.

But now we claim once again that all such theorems are likely to be theoretical pathologies, never to arise in 'normal' mathematics. Of course such a claim is rather bold, and always open to refutation, but we can at least point to some agreeable empirical evidence. First, looking at a modern, rigorous mathematical textbook such as Bourbaki's, it's hard to see how an unbridgeable gap between informal and formal proofs could arise. And in practice some significant pieces of mathematics have been formalized, e.g. in Mizar, without any exponential dependency appearing. On the contrary, there is some empirical evidence so far that the length of a formal proof is a fairly modest linear function of the length of a thorough textbook proof, as explicitly stated by [debruijn-sad]:

A very important thing that can be concluded from all writing experiments is the constancy of the loss factor. The loss factor expresses what we lose in shortness when translating very meticulous 'ordinary' mathematics into AUTOMATH. This factor may be very big, something like 10 or 20 (or 50), but it is constant; it does not increase if we go further in the book. It would not be too hard to push the constant factor down by efficient abbreviations.

Of course, the 'very meticulous' should be noted; Landau's book is a most unusual one, and de Bruijn chose it after great deliberation. Although Mizar has been used for an unrivalled number of mathematical formalizations, we are unaware of any published comparisons between the original textual sources (such as they are) and the resulting formal texts. Perhaps this is because the Mizar formalizations are usually arrived at independently after consulting many sources, rather than by translating existing texts or papers directly. However, more efforts of this latter kind should throw new light on the question we are considering. One attempt to follow a textbook faithfully is detailed by [paulson-rubin]. They translated the early parts of set theory books by [kunen] and [rubin-ac2] into Isabelle, and found that the relationship between the size of the formal and informal texts was much more variable. (Though not to the extent of infeasibility, it should be stressed. They merely had to work rather harder themselves at certain points.)

left right up home John Harrison 96/8/13; HTML by 96/8/16