The specific characteristics of mathematical argumentation all depend on the centrality that writing has in the practice of mathematics, but blindness to this fact is near universal. What follows concerns just one of those characteristics, justification by proof. There is a prevalent view that long proofs pose a problem for the thesis that mathematical knowledge is justified by proof. I argue that there is no such problem: in fact, virtually all the justifications of mathematical knowledge are ‘long proofs’, but because these real justifications are distributed in the written archive of mathematics, proofs remain surveyable, hence good.

Some interesting claims in a footnote here:

Philosophy of mathematics is in poor shape for, as Putnam

(1979) put it long ago, ‘in philosophy of mathematics, nothing works.’ Apart from

some welcome attention to practice, not much has changed. This is because the

only positions taken seriously are variations of those associated with foundational

studies — logicism, intuitionism, formalism — all of which have mortal wounds.

There’s lots of activity in the field, but almost all of it takes for granted the claim

of formal logic to authority over the argumentation in at least the proofs of pure

mathematics (even the occasional deviant). Correlative to that is an obsession with

the ‘objects’ of mathematics and their metaphysical salvation from skepticism, or

subjection to it. The real nature of mathematical knowledge is hardly discussed,

the assumption being that if we get the nature of the objects straight, then what

knowing them consists in will be a simple corollary. Many people deny that there are

any mathematical objects (‘really’) because if there were we couldn’t know anything

about them. This leads to much tortuous nonsense.

The study of argumentation might seem to be just the source for a better understanding

of mathematical knowledge, since it is concerned with real argumentation

rather than the idealised simulacrum that formal logic indicates. There are two

reasons why I doubt this. The first is that the quite proper concern of the study of

argumentation with non-deductive argumentation might lead the discussion toward

philosophically peripheral aspects of mathematical practice. This is evident in uses

which some writers in the mathematical education community have made of the socalled

‘Toulmin model’ (for example Inglis et al., 2007). My point is not that I want to

insist that these elements of practice can be ignored for all serious purposes—on the contrary — but that they distract us from trying to give a better treatment of those

elements traditionally and rightly seen as central for the philosophy of mathematics,

such as proofs as given by Euclid. My other reason for doubting the usefulness of

the extant study of argumentation is that there is no real theory of argumentation

to rival formal logic, just a congeries of interesting and useful but weak ‘approaches’

to the study of argumentation (cf. the survey book of van Eemeren et al 1996).

There are some examples of problematic proofs: the four-colour theorem, Wiles’ proof of Fermat’s last theorem, and The Enormous Theorem.

No serious doubts about

the correctness of this proof have been expressed recently, but there is

some disquiet about how properly we can speak of a proof which is not

available to most mathematicians, let alone most people. (I recently

read a mathematican blogging that he intends to spend two years full-

time getting up to speed on the theories required, in order to read the

proof.)[…]

Finally, P is only a good proof of T if P is interesting — if the

ideas on which it depends are valuable. A real proof justifies asserting

the theorem; but a good proof must also give some kind of insight.

As Lord Rayleigh is reported (for example, by Huntley and by Kline,

though I have been unable to locate the original source) to have said:

‘Some proofs command assent. Others woo and charm the intellect.

They evoke delight and an overpowering desire to say, ‘Amen, Amen’.’[..]

Azzouni suggests that more surveyability makes for better proofs be-

cause they are easier to understand (1994, p. 125). Later he rebuts an

objection to one of his arguments by saying that certain proofs are

not unsurveyable, though long. He recognises that in reality individual

mathematicians have not actually surveyed many of the proofs which

justify their knowledge, but accept them on the authority of others who

have — a point to which I will return in section 10. But he evidently

thinks that this ‘in principle’ surveyability, because it is shown by actual

survey distributed in the community, is actual surveyability.

He runs through Tymoczko, Wittgenstein, Descartes and Azzouni and their arguments for requiring surveyability.

Formalists regard the proofs given in ordinary mathematics as ‘informal’:

they think the real proofs that anchor mathematical truths are

their formal counterparts in some more polished descendent of Prin-

cipia Mathematica or Grundgesetze. But a more convincing suggestion

for the ‘real’ proofs is that they are fully expanded informal proofs.

Unfortunately virtually all such proofs are long.

This is brought up to be dismissed, thankfully.

7. Uses of proofs presuppose written practice

What are proofs really, and what must surveyability be in order to

contribute to them? To answer these questions we need to understand

that real proofs are written texts.[…]

To show that a proof is surveyable it suffices to give a perspicuous

representation of it. Some simple examples have already been given:

− the idea of Euclid’s proof of Pythagoras’ Theorem is that you

divide the big square with a line from the right-angle vertex parallel

to its sides, and then use theorems about triangles between

parallels to show that the two parts are equal to the two smaller

squares;

− the idea of the proof of the irrationality of p2 is to argue by

contradiction that no candidate fraction is in lowest terms;

− the idea of the classical proof (Euclid IX.20) of the infinity of the

primes is to construct a bigger one from any candidate biggest by

forming its factorial plus 1.

His conclusion is that:

Surveyability is the requirement that the proof be capable of supporting the construction of a perspicuous representation of the proof-idea.

…this as separated from checkability. This seems to diverge from others’ definitions, which seem to be many and various.

The moral of the story is that we need to distinguish between proof

and justification. A proof is a kind of conditional justification. The full

justification of virtually any item of mathematical knowledge is a long

and unsurveyable ‘proof’, but this is not a problem because the archives

are so organised that each chunk (proof) is surveyable.[Conclusion:] Long proofs are not a problem, and nor is the understanding of long

proofs. What has been a problem is the understanding of the understanding

of long proofs. The key to that, is to recognise that mathematics

is a written practice which depends on the accumulation and

deployment of an archive.4 Because of that, one does not need to have

all of a proof in one’s head because it is all on record. Understanding

it means you have the main idea of it in your head, and knowing

it requires both that, and the facility with the archive to get at the

details if wanted.