Talk given to this meeting: https://filosofia.fflch.usp.br/eventos/7061

1. Unlikely bedfellows or
unholy alliance?
Dr Brendan Larvor
Unlikely bedfellows or unholy alliance? 1

2. Abstract
Lakatos demonstrates the possibility of philosophical hybridism. In his philosophy of
mathematics, he lists his influences as Hegel, Pólya and Popper.At least two of
these ought not go together, but in Lakatos these sources combine and point the
way to the eclectic condition of philosophy of mathematical practice now. In this talk,
I will discuss the tensions and synergies in Lakatos’s thought resulting from his
intellectual and literal migration. I will describe and illustrate some of the benefits of
philosophical hybridism, including a robustness criterion derived from finding the
same argument articulated in widely diverse idioms. Finally, I will discuss the
prospects for collaboration among the various projects gathered under the flag of
Philosophy of Mathematical Practices.
Unlikely bedfellows or unholy alliance? 2

3. Three wise men:
Unlikely bedfellows or unholy alliance? 3

4. Popper on Dialectic:
Unlikely bedfellows or unholy alliance? 4
Mind, Vol. 49, No. 196 (Oct. 1940), pp. 403-426

5. Dialectical philosophy of mathematics
• Focus on processes (not products)
• Seek rationality and integrity in the development of mathematics
• Concepts, not propositions
• Inside phenomenological stance:
o Changes in mathematical knowledge normally take place for mathematical reasons
o Takes the point of view of mathematics (rather than the individual mathematician)
• Nothing to say about ontology of mathematics
Unlikely bedfellows or unholy alliance?

6. Eclectic Robustness 1:
Naturalism in philosophy of mathematics
What is rational is real;
And what is real is rational.
Philosophy cannot teach the state
what it should be, but only how it, the
ethical universe, is to be known.
(Hegel, Philosophy of Right, preface)
Unlikely bedfellows or unholy alliance?
"If you're a 'naturalist', you
think that science shouldn't
be held to extra-scientific
standards, that it doesn't
require extra-scientific
ratification.“
Penelope Maddy

7. Eclectic Robustness 2: Fascism and Irrationalism
Russell: The Ancestry of Fascism
Unlikely bedfellows or unholy alliance?
Lukács: The Destruction of Reason

8. Strains
Hacking’s Hegelian
reading: progress as a
substitute for truth.
Post-Kantian Lakatos…
“Surely after Kant and
Bergson this is a
commonplace.”
Unlikely bedfellows or unholy alliance?
But!
Lakatos commits to
a correspondence
account of truth and
a ‘whiff of
inductivism’ to
reassure us that
science is heading
towards it.

9. Unlikely bedfellows or unholy alliance?
Not quite true:
Tamás Demeter (Budapest)

10. On the Sociology of Concept-
building in the Natural Sciences
Unlikely bedfellows or unholy alliance?
Lakatos, PhD dissertation,
1947, University of Debrecen

11. Aspray, W., Kitcher, P. (1988). History and Philosophy of Modern
Mathematics. University of Minnesota Press. p.17
Unlikely bedfellows or unholy alliance?

12. Mancosu, P. (2008). The philosophy of mathematical practice. OUP. p.3
Unlikely bedfellows or unholy alliance?

13. Van Bendegem, J. P. (2014). The impact of the philosophy of mathematical
practice on the philosophy of mathematics. In: Soler et al.: Science after the
practice turn in the philosophy, history, and social studies of science, p.215 &
221
Unlikely bedfellows or unholy alliance?

14. Hamami, Y., & Morris, R. L. (2020). Philosophy of mathematical practice:A
primer for mathematics educators. ZDM, 52(6), p.1113.
Unlikely bedfellows or unholy alliance?

15. Tensions
Can we work with the mainstream? (Corfield: no! Mancosu: yes. Larvor: not really,
though the mainstream is changing)
Cognitive science: normativity (rigour, proof, truth, etc.) drops out
History: relativises normative standards (but see Netz: cognitive archaeology of
successful thinking—in general history of science & maths cannot lose its normative
terms altogether without changing its subject)
Contemporary mathematics is too hard for philosophers to follow: (ABC conjecture,
Liquid Tensor Experiment). Looking at what mathematicians actually do… (the
mainstream doesn’t have this problem because foundations are all)
Unlikely bedfellows or unholy alliance?

16. Prospects
Lean! Available to all online and works at any level of mathematical sophistication
Corpus studies (Danish studies of proofs over time, analysis of crowd-sourced
theorems, tagged text for commands, etc.)
Studies of e.g. reviewing practices (is every line checked? Etc)
Value-questions: what is worth studying? What should be rewarded?Are prizes a
good idea? What about EDI? (Note: are we recording, rationally reconstructing or
participating in these discussions?)
Unlikely bedfellows or unholy alliance?

17. Thank you
Unlikely bedfellows or unholy alliance?