a talk about mathematical proof from 2015

1. How the philosophy of mathematical practice can be logic
by other means
8 April Deductive and Mathematical Cognition
University of Bristol

2. In previous papers, I have argued that:
1. 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);
2. informal proofs are arguments that depend on their matter as well as their logical form (in
other words, ‘informal’ is a poor English translation for inhaltliche);
3. 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);
4. 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;
5. further, it explains the fact that mathematics is (aside from some elementary mental
arithmetic and simple spatial arguments ) essentially inscribed.

3. I now claim that:
5. this conception of logic facilitates an intimate connection between logical questions about
rigour and the study of mathematical cultures and practices (since the logical constraints on
inferential actions are enacted as cultural norms).
• This is what we need, because at the moment, in the philosophy of mathematical practice,
we have on the one hand a stock complaint about formal logic as an explanatory model of
mathematical proof, and on the other an increasingly rich literature of studies of specific
mathematical practices. The model I present can draw on the latter to supply the deficiency
identified by the former.

4. Conference Questions
Philosophy of Mathematics
• How do we acquire arithmetical knowledge?
• How do we acquire geometrical knowledge?
• What role does the historical development of mathematical notation play in determining the
nature of mathematical knowledge?

5. Conference Questions
Philosophy of Logic
• What is the subject of logic?
• Are we deductively rational?
• If not what are the implications for the prescriptive role of logic as a guide to correct
reasoning?
• Should we construct and assess our logics using data from the study of deductive
reasoning processes?

6. Picture and slide - Slide title in Arial
Slide subtitle in Arial
• Text ranged left, one paragraph space
between lines of text in black
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b.p.larvor@herts.ac.uk

7. Picture and slide - Slide title in Arial
Slide subtitle in Arial
• Text ranged left, one paragraph space
between lines of text in black
• Text ranged left, one paragraph space
between lines of text in black
• Text ranged left, one paragraph space
between lines of text in black
• Text ranged left, one paragraph space
between lines of text in black
b.p.larvor@herts.ac.uk