Talk for symposium in Siberia

1. The rigour of real mathematical proofs
Brendan Larvor – 30 July 2019

2. What’s the problem?
A square is
divisible into a
finite number of
unequal squares

3. What’s the problem?
Task: satisfy yourself by
inward reflection that
the smallest square
cannot be at an edge.

4. What’s the problem?
How do proofs prove in practice?
“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 For Formal
Proof, 2012, p.x

5. But what is the target?
Maybe this:
A mathematical proof is rigorous when it is (or could be) written out
in the first-order predicate language as a sequence of inferences
from the axioms ZFC, each inference made according to one of the
stated rules… practically no one actually bothers to write out…
formal proofs. In practice, a proof is a sketch, 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.
(Mac Lane 1986: 377-8)

6. In that 2012 Synthese paper:
Essentially informal arguments
Mac Lane also says:
“proofs are not only a means to certainty, but also a means to
understanding. Behind each substantial formal proof there lies an
idea... it will not do to bury the idea under the formalism.”
(ibid.)
The idea is presumably topic-specific, yet its translation into FOL with
ZFC is ‘routine’?

7. John P. Burgess
A treatment of a given subject matter that is genuinely rigorous will ipso facto cease to be a treatment of that subject
matter (alone).
(p. 65)
If we make the study of the natural numbers or real numbers, say, completely rigorous, by reducing it to the deduction of
consequences from appropriate axioms, all results will automatically become applicable to a whole class of structures, the
class of models of those axioms.
(p. 112)
Rigor and Structure, OUP (2015)
(I don’t think it’s a real paradox and I don’t think Burgess does either.)
The ‘paradox’ of rigour

8. Progress in the philosophy of mathematical practice requires a
general positive account of informal proof (since almost all
mathematical proofs are informal in the strictest sense, even if they
are highly formalised);
The case against formal proofs as an account of how mathematical knowledge is
validated was a staple at PMP conferences:
• There are very few fully formalised proofs;
• Fully formalised proofs of significant theorems would be impossibly huge
(bigger than the solar system…)
• On the ‘formalist’ view, a regular proof is an informal argument that a formal
derivation is possible—which is a mathematical claim. So we agree that
regular proofs are informal mathematical arguments. The disagreement is
only about the conclusion.
• Explaining mathematical agreement by reference to non-existent derivations
seems like magical thinking

9. In that 2012 Synthese paper:
The Liberating Insight
In making arguments, we act on all sorts of items in addition to
propositions and well-formed formulae.

10. Articulating the dependency of informal inferences on their content
requires a reconception of logic as the general study of inferential
actions (in informal proofs, content, or representations thereof, plays
a role in inference as the object of such actions)
);This is not so radical:
• Formal logic offers a huge range of systems
• Formal logic has been extended to all manner of matters (tense logic,
deontic logic, modal logic, etc.)
• Consider arguments about moving furniture, or the possibility of a new
gymnastic feat
• Philosophy of experimental science—the experiment is no longer simply a
source of protocol sentences. It is a locus of rational action.
• Mathematical proofs (or rather, their texts) are full of imperatives. The
objects of these imperatives are often not propositions but rather
mathematical items or representations thereof
• These actions are often only available in certain domains (co-set counting;
divisibility arguments; ε-δ chasing; Euclidean diagram manipulation;…)

11. It is a decisive advantage of this conception of logic that it
accommodates the many mathematical proofs that include actions
on objects other than propositions;
);This is not so radical:
• Formal logic offers a huge range of systems
• Formal logic has been extended to all manner of matters (tense logic,
deontic logic, modal logic, etc.)
• Consider arguments about moving furniture, or the possibility of a new
gymnastic feat
• Philosophy of experimental science—the experiment is no longer simply a
source of protocol sentences. It is a locus of rational action.
• Mathematical proofs (or rather, their texts) are full of imperatives. The
objects of these imperatives are often not propositions but rather
mathematical items or representations thereof
• These actions are often only available in certain domains (co-set counting;
divisibility arguments; ε-δ chasing; Euclidean diagram manipulation;…)

12. Example from Polya:
Further, it explains the fact that mathematics is (aside from some
elementary mental arithmetic and simple spatial arguments)
essentially inscribed.
);
The real value of mathematical representations is not that they present
information clearly (though they do) but that they offer themselves for
manipulation.
Peirce: Icons

13. Arrow-chasing 1: Cayley graphs of finitely generated groups
(after Starikova)
You can do things to
Cayley graphs that you
can’t do to other
representations of groups
Starikova, Irina 2010 “Why do
Mathematicians Need Different
Ways of Presenting
Mathematical Objects? The
Case of Cayley Graphs” Topoi
Volume 29, Issue 1 , pp 41-51

14. Great! Let’s just write all that up
Starikova on Cayley graphs and Bernhard Krön’s re-proof of Stallings’ structure theorem
In general: the phenomenon of mappings of problems
into new domains may be (sometimes) explained as the
exploitation of inferential actions available only in the
target domain.
(e.g. turning a link into a braid)

15. People who may possibly agree with me
I Silvia De Toffoli & Valeria Giardino
“…it will be shown that knot diagrams are dynamic by pointing at the
moves which are commonly applied to them. For this reason, experts must
develop a specific form of enhanced manipulative imagination, in order to
draw inferences from knot diagrams by performing epistemic actions.”
“Forms and Roles of Diagrams in
Knot Theory”
Erkenntnis 79 (4):829-842 (2014)

16. People who may possibly agree with me
I Silvia De Toffoli & Valeria Giardino
Envisioning Transformations: The Practice of Topology. (2016) in Mathematical Cultures Larvor (ed.) Birkhäuser Science (Springer).
Transformation
rather than
construction

17. People who may possibly agree with me
I Silvia De Toffoli & Valeria Giardino
“Permissible actions help in defining what counts as
mathematical practice, because:
(i) they are accepted in a collective dimension;
(ii) they rely on the cognitive abilities of the practitioners and…
(iii) they refer to the use of stable systems of representations.”
Envisioning Transformations: The Practice of Topology. (2016) in Mathematical Cultures Larvor (ed.)
Birkhäuser Science (Springer).
What
about the
machines?

18. People who may possibly agree with me
IV Andrei Rodin
In the same session in Helsinki:
Constructive Axiomatic Method in Euclid, Hilbert and Voevodsky
Andrei Rodin
Institute of Philosophy, Russian Academy of Sciences, Moscow, RUSSIAN FEDERATION

19. People who may possibly agree with me
IV Andrei Rodin

20. People who may possibly agree with me
IV Andrei Rodin

21. People who may possibly agree with me
IV Andrei Rodin

22. Great! Let’s just write all that up
Is set theory part of logic?
Sir William Hamilton Lectures on
Metaphysics and Logic (1858–60)
The temptation to think so arises from
the historical centrality of categorical
(syllogistic) logic.
But this introduces sets in precisely the
way that gives rise to Russell’s paradox (if
we think of categories e.g. ‘human’
‘mortal’ as sets).

23. Great! Let’s just write all that up
Is set theory part of logic?
The actions possible in set theory (formation of unions
& intersections, etc. are only possible with sets. These
are as domain-specific as any other action in
mathematics.
They are actions on sets, not on propositions.
Actions known to lead to paradoxes are not permitted.
Translation into ZFC is translation from one specialised
domain of mathematics to another; NOT to a domain
of content-neutral actions. (i.e. is ZFC just the
language or also the logic?)

24. Great! Let’s just write all that up
Starting with Euclid (i.e. with Ken)
3. traduttore traditore!
“mathematicians are like Frenchmen; if one speaks to
them they translate it into their own language, and then
it will be very soon something entirely different.”
This is also true within mathematics
Including and especially, translation into a foundational
language.

25. Rapprochement from the formalist side
Alan Weir (2016) Informal proof, formal proof, formalism. Review of Symbolic Logic
“What is crucial is how we understand `formal rule'. The usual gloss is along
the lines of: `a rule which appeals to nothing outside the system'. This is
vague but greater precision is unlikely to be had in respect of a concept as
fundamental as this (the concept of `formal rule' not itself being part of
formalised mathematics)…
Note that I did not require that the formal rules of a derivation be what we
would recognise as logical rules.”

26. Rapprochement from the formalist side
Alan Weir (2016) Informal proof, formal proof, formalism. Review of Symbolic Logic
“e.g. basic rules for differentiation, power rules, product rules, quotient rules,
the chain rule and so on. These are completely formal rules. These sort of
examples show that derivations abound. There are concrete tokens of
mathematical derivations all over the place, in textbooks, exam scripts,
worked examples in engineers' rough notes, in the hardware of computers
computing values for various functions and so on.”
But (Weir and I agree, because it’s obvious) such derivations are not much
found in published proofs—they are precisely the bits professionals leave out.

27. Rapprochement from the formalist side
Alan Weir (2016) Informal proof, formal proof, formalism. Review of Symbolic Logic
“Hilbert's Thesis II:
In any cogent mathematical practice there is a systematic process of
transformation (not necessarily known to the practitioners) which turns
any correct proof into a (suitably related) finite derivation in a formal
system S. The system S in question is determined by the informal
practice and its transformation process; in particular, the formal rules of
S are rules which are implicit in the mathematical practice..”

28. Rapprochement from the formalist side
Alan Weir (2016) Informal proof, formal proof, formalism. Review of Symbolic Logic
“there is no restriction solely to linguistic systems—diagrammatic reasoning
can be formal as well. This is clear in systems such as Peirce's diagrammatic
proof architecture, where there are precise rules for what counts as a
legitimate proof step.
The status of other diagrammatic modes, ancient Euclidean demonstrations
or modern category-theoretic diagrams for example, is more problematic.
They perhaps stand to Peirce-style pictorial derivations as informal proof does
to linguistic derivations.”

29. The Russians have already calculated this
(From Andrei Rodin’s paper)
Vladimir Smirnov in his 1962 paper on the genetic method:
an adequate modern formalization of genetic theories can be achieved
not through their axiomatization (by which he understands a
presentation of such theories “in the form of axiomatic calculus where
rules of inference [..] control the transition from one proposition to
another”) but rather through a “direct formalization of recursive
techniques, i.e., of algorithmic processes”. To this end, one should
“broaden the scope of logic” by considering “stratagems of actions”
and “such forms of thought as prescriptions and systems of
prescriptions” as a properly logical subject-matter (op. cit. p. 275-276).

30. So why is the standard view standard?
• Because there was a crisis of rigour in the 19th century, just as we are told!
• Present-day dispute resolution
• Unification (how to connect our first proof about squares with the rest of mathematics?)
• Transfers of argument (see Burgess remark about numbers)
• The fully-formal proof is the extreme completion of a process that mathematicians engage in piecemeal and subject to
countervailing pressures.
Answers from mathematical practice