talks in Belgium

1. The Logic(s) of Informal Proofs
Brendan Larvor
28 April / 12 May 2016

2. Presentation title here
Subtitle here

3. Alternative title:
Does Manders’ ‘The Euclidean Diagram’ offer a general model
of mathematical inference?

4. 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

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. One-part motivational argument
Against the Derivation Recipe model of proof
The view that a mathematical proof is a sketch of or recipe for a formal derivation
requires the proof to function as an argument that there is a suitable derivation.
This is a mathematical conclusion, and to avoid a regress we require some other
account of how the proof can establish it.

7. One-part motivational argument
Against the Derivation Recipe model of proof
Let P be a mathematician’s proof for a theorem C. Then, on the Derivation
Recipe model, P is not really a proof of C, but rather an argument to
convince the reader that:
Cʹ: there is a formal system S such that ⱵS γ where γ is the formula in S
corresponding to C
So, proponents and opponents of the Derivation Recipe view agree that P
is a compelling, rigorous argument for (a proof of) a mathematical
conclusion. They differ over whether P is a proof of C or Cʹ.

8. One-part motivational argument
Against the Derivation Recipe model of proof
Notice:
Cʹ is a Σ0 sentence, regardless of the complexity of C
Does this matter?
It offers the derivationist a hybrid model: every proof should be in principle
either a derivation or a construction of a Σ0 sentence

9. This may just be a matter of context:
– For doing maths, ask does it convince you of C?
– For reviewing, ask does it convince you of Cʹ ?
(for a journal which has some version of Mac Lane’s view as its
standard)
One-part motivational argument
Against the Derivation Recipe model of proof

10. But what is the target?
Maybe this:
“In the first-order predicate language”
• First-order?
“inferences from the axioms ZFC”
• i.e. in a mathematical domain with its own domain-specific moves (formation
of unions, etc.)
“a routine translation of this sketch into a formal proof”
• traduttore traditore!
“not by reference to the subject matter at issue”
• i.e. for Mac Lane, ‘formal’ entails ‘topic-neutral’
• For us, ‘informal’ means ‘inhaltliche’ (Lakatos P&R)

11. Outputs
Publications so far:
“How to think about informal proofs” 1-Jan-2011 Synthese p. 1-16.
“What Philosophy of Mathematical Practice Can Teach Argumentation Theory about
Diagrams and Pictures” 2012 The Argument of Mathematics Aberdein, A. & Dove,
I. (eds.). Springer p. 209-222.

12. In that Synthese paper:
Essentially informal arguments
formal arguments
(a) are expressed in a general logical language, the well-formed
formulae of which are explicitly defined (usually by recursion) and
(b) consist of successive applications of explicitly specified rules of
logical inference
Rather stupidly, this version doesn’t emphasise the content-
indifference of the inferential rules in (b)

13. In that Synthese paper:
Essentially informal arguments
Essentially Informal Arguments:
those informal arguments that would suffer some sort of violence or
essential loss if they were recast so as to satisfy (a) and (b).
PMP doctrine: almost all of the informal proofs that mathematicians
actually read, write and publish are essentially informal.

14. In that Synthese paper:
Essentially informal arguments
The via negativa definition of informal proofs leaves open the
possibility that all informal mathematical arguments could be fully
formalised without loss or violence.
(That’s presumably what Mac Lane means by ‘routine’.)
We need a positive account of the notion of an essentially informal
argument, that allows us to push this back.

15. In that Synthese paper:
Essentially informal arguments
Side-note on Mac Lane:
He 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’?

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

17. 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;…)

18. 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;…)

19. A ‘Cartesian’ Proof:

20. 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

21. 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

22. Arrow-chasing 2:

23. 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)

24. People who may possibly agree with me
II Luca San Mauro (SNS, Pisa) –presented in Helsinki (I think)
• “Church-Turing thesis, in practice: A case study in informal provability”

25. People who may possibly agree with me
II Luca San Mauro (SNS, Pisa) –presented in Helsinki (I think)

26. People who may possibly agree with me
II Luca San Mauro (SNS, Pisa) –presented in Helsinki (I think)

27. People who may possibly agree with me
II Luca San Mauro (SNS, Pisa) –presented in Helsinki (I think)

28. People who may possibly agree with me
II Luca San Mauro (SNS, Pisa) –presented in Helsinki (I think)

29. People who may possibly agree with me
III Dirk Schlimm & Andy Arana
As announced in Helsinki:
Geometric reasoning and geometric content
Dirk Schlimm
Philosophy, McGill University, Montreal, CANADA
“Some mathematical thinking is object-oriented“
They haven’t published yet…

30. 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

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

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

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

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

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

36. Great! Let’s just write all that up
Starting with Euclid (i.e. with Ken)
1. These are not proofs without words. Rather, the
diagram and the text work together. There is typically
a constructive phase, when the diagram is built up
using moves that are either specified in the
postulates or have been shown to be valid in earlier
proofs.
Are there examples of inferential actions on objects
other than propositions that are not constructions?

37. Great! Let’s just write all that up
Starting with Euclid (i.e. with Ken)
2. These constructive moves (adding lines or circles
to the diagram) are inferential actions. They are
strictly controlled by the first three postulates.
In looking for more of the same, we should look
for shared proof-practices governed by rules that
ensure reliable results—not every napkin-
diagram or thought-experiment will qualify.

38. 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

39. Great! Let’s just write all that up
Starting with Euclid (i.e. with Ken)
4. there is in this analysis no appeal to unanalysable intuition.
One might think that the rising successes of automated
proof finders and checkers would present a problem for an
approach to proofs rooted in the philosophy of (human)
mathematical practices.
But our claim is that some proofs involve actions on objects
other than propositions—there is no reason why these
actions should not be carried out by machines.

40. Great! Let’s just write all that up
Starting with Euclid (i.e. with Ken)
5. the argument that proofs in Euclidean geometry make
essential use of non-textual inferential actions depends on
the presence in the practice of diagrams.
So, will every case of non-propositional inferential action
depend on the use of a non-textual representation of
mathematical content?

41. Great! Let’s just write all that up
Starikova on Cayley graphs and Bernhard Krön’s re-proof of Stallings’ structure theorem
Stalling’ structure theorem: a finitely generated group
has more than one end if and only if it splits over some
finite subgroup.
Strictly speaking, groups do not have ends; graphs do. Moreover,
most finitely generated groups give rise to more than one Cayley
graph. But the number of ends is invariant over the graphs of a
given group, and therefore may be regarded as a property of the
group itself.

42. Great! Let’s just write all that up
Starikova on Cayley graphs and Bernhard Krön’s re-proof of Stallings’ structure theorem
This should be grist to the mill!
You can do things with Cayley graphs that you can’t do
with groups—these actions suggest themselves as
candidate inferential actions in our sense.
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.

43. Great! Let’s just write all that up
Starikova on Cayley graphs and Bernhard Krön’s re-proof of Stallings’ structure theorem
This should be grist to the mill! However…

44. 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).

45. Challenges
• Is it logic? 'logic' suggests systematisation, codification, which your view cannot supply. Also,
logical principles or rules of inference should be fundamental. Elements of logic should be in
some sense elementary. 'Divisibility' arguments are not elementary, because they can be
further analysed and explained.
• A general account? Didn’t we PMP people decide that it’s all particular, contextual, situated,
etc.?
• What should we say from this point of view about computers and proofs?
• What should we say from this point of view about foundations?
• The last claim about cultures and practices is a dodgy promissory note…