Talk given in Paris 20/10/23

1. The concept of
mathematical proof: how
much trouble are we in?
Brendan Larvor
FPMW15 : 15ème French PhilMath Workshop 1

2. Abstract
Philosophers of mathematical practice have been discussing the nature of
mathematical proof for a quarter of a century, if we take Yehuda Rav's (1999) paper
as a point of origin. Recent developments in computer proof assistants have added
urgency to this discussion as it is now possible to create fully formalised proofs of
significant theorems in systems available to undergraduates (such as Lean). One
natural approach is to insist on the diversity (BUNTES Gemisch) of mathematical
proofs and argue that 'mathematical proof' is a Wittgensteinian family resemblance
concept or a cluster concept in the sense of Lakoff.
FPMW15 : 15ème French PhilMath Workshop 2

3. Abstract (cont.)
I will argue that neither of these approaches succeeds. We do not have a diversity
of discrete uses of the term 'proof', each unproblematic within its proper language-
game, because mathematics is not a constellation of discrete language-games, all
sealed off from each other (so the Wittgensteinian route is blocked). Nor do we
have a cluster of proof-like properties that combine to form a consistent
paradigmatic proof-concept (so the Lakoff route is blocked). Rather, we have a
single, rich, incoherent concept of mathematical proof. This is not a problem for
research mathematicians, but it does present difficulties for anyone committed to
making ideological use of the concept of proof and for everyone employed to teach
it.
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4. We’ve been talking about proof 4

5. We’ve been talking about proof 5

6. How do proofs relate to formal proofs?
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7. Statements of the ‘Standard View’
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In practice, a proof is a sketch [esquisse], in
sufficient detail to make possible a routine
translation of this sketch into a formal proof…
...the test for the correctness of a proposed
proof is by formal criteria and not by reference
to the subject matter at issue.
(Saunders Mac Lane 1986 pp. 377-8).
The ultimate standard of proof is a formal
proof, which is nothing other than an unbroken
chain of logical inferences from an explicit set
of axioms. While this may be the
mathematical ideal of proof, actual
mathematical practice generally deviates
significantly from the ideal.
(Thomas Hales Dense Sphere Packings : a
Blueprint [photocalque] for Formal Proofs
2012, p. x)

8. Statements of the ‘Standard View’
FPMW15 : 15ème French PhilMath Workshop 8
Proof and truth… lie on opposite sides of the
syntax-semantics divide, for at bottom, a proof is
a kind of argument, a collection of assertations
structured syntactically in some way, while the
truth of an assertion is grounded in deeply
semantic issues concerning the way things are…
Proof… lies solidly on the syntactic side, since
ideally one can verify and analyze a proof as a
purely syntactic object, without a concept of
meaning and without ever interpreting the
language in any model.
(Joel D. Hamkins 2020, pp. 157-8)
Resistance:
Most contemporary mathematical
proofs are written in prose, essay-
style… In mathematical practice, a
proof is any sufficiently detailed
convincing mathematical argument
that logically establishes the truth
of a theorem from its premises.
(Op. cit. p. 160)

9. Statements of the ‘Standard View’
FPMW15 : 15ème French PhilMath Workshop 9
...the Formalizability Thesis should be given a very strict
reading, namely that
(i) every good proof has an underlying logical structure,
(ii) that structure is completely analyzed in the derivation
that formalizes the proof, and, finally
(iii) that derivation assures the correctness of the theorem
proved on the basis of the background assumptions
expressed by the axioms and rules of the system in
which the proof is formalized.
(Solomon Feferman, 2012)
Resistance:
The passage from informal to
formalized theory must entail
loss of meaning or change of
meaning.
(Reuben Hersh1997, p. 160).

10. Statements of the ‘Standard View’
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On the standard view, [informal proofs] are only
high-level sketches that are intended to indicate
the existence of formal derivations. But providing
less information only exacerbates the problem…
(Jeremy Avigad 2020)
…an informal proof is a form of data
compression. Coding schemes for data
represent the most common patterns concisely,
reserving extra bits for those that are unusual
and unexpected.
(Op. cit.)
Resistance:
Comparing proof to a narrative allows
us to draw on intuitions regarding the
distinction between the plot and its
syntactic presentation. After reading
Pride and Prejudice, we can reliably
make the claims about the characters
and their motives, but we cannot
reliably make claims about the
number of occurrences of the letter
‘p.’
(Op. cit.)

11. Statements of the ‘Standard View’
FPMW15 : 15ème French PhilMath Workshop 11
…the notion of translation in the standard view is
quite different from the one of linguistic
translation. …a better analogy would be with the
process of compilation, that is, with the
‘translation’ of a computer program written in a
high-level programming language into machine
language
(Yacin Hamami 2019)

12. Avigad’s Heuristics for reliable informal proof
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1. Reason by analogy
2. Modularize
3. Generalize
4. Use algebraic abstraction
5. Collect examples
6. Classify
7. Develop complementary approaches
8. Visualize

13. What to do with these various senses of ‘proof’?
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14. What to do with these various senses of ‘proof’?
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Weber (2014) Cluster model
1. A proof is a convincing argument that
convinces a knowledgeable mathematician
that a claim is true.
2. A proof is a deductive argument that does
not admit possible rebuttals.
3. A proof is a transparent argument where a
mathematician can fill in every gap (given
sufficient time and motivation), perhaps to
the level of being a formal derivation.
…
…
4. A proof is a perspicuous argument that
provides the reader with an understanding of
why a theorem is true.
5. A proof is an argument within a
representation system satisfying communal
norms.
6. A proof is an argument that has been
sanctioned by the mathematical community.
Proofs are not mathematical concepts; they are discursive concepts.
There is no property that distinguishes proofs from non-proofs. (Keith Weber)

15. What to do with these various senses of ‘proof’?
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Birth
Nurturance
Female
Legal
Guardian
Father’s
wife
Lakoff’s
cluster
model of
motherhood

16. What to do with these various senses of ‘proof’?
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…all ideas, in which a whole
process is promiscuously
comprehended, elude definition;
it is only that which has no
history, which can be defined.
(GM II 13)

17. Thank you
FPMW15 : 15ème French PhilMath Workshop 17