philosophy of mathematics

1. After Lakatos, what?
Brendan Larvor
b.p.larvor@herts.ac.uk

2. Plato’s Divided Line
Republic Book VI (§§509-513)
Intelligible is to Visible
As
Philosophy is to Mathematics
As
Material objects are to their shadows and
reflections

3. Intelligible Visible
Philosophy Mathematics
(Geometry)
Material
Objects
Shadows,
Reflections,
etc..
1. Takes hypotheses and
examines them
dialectically to reach
unconditional first
principles
2. Makes no appeal to
images or diagrams
1. Treats hypotheses as
established truths and
derives their
consequences
2. Appeals to diagrams and
images

5. Hegel on Mathematics:
[Axioms] are commonly but incorrectly
taken as absolute firsts, as though in and
for themselves they required no proof... If,
however, axioms are more than
tautologies, they are propositions from
some other science... Hence they are,
strictly speaking, theorems,…
The Science of Logic vol. two, section 3, chapter 2 A (b) ‘The Theorem’ (p. 808).

6. Hegel on Mathematics:
[Mathematical] proof,... follows a path that
begins somewhere or other without indicating
as yet what relation such a beginning will
have to the result that will emerge. In its
progress it takes up these particular
determinations and relations, and lets others
alone, without its being immediately clear
what the controlling necessity is; an external
purpose governs this procedure.
(Phenomenology of Spirit §44, p. 25)

7. Proofs and Refutations
Overturns the view that mathematics is
undialectical, lifeless, incapable of self-
movement, dependent on extraneous
energies, by setting the finished proof in its
heuristic context. i.e. by showing us the
total practice of mathematics.

8. Remarks on P&R
• All the heuristic patterns degenerate
eventually (methodological anarchism).
• The heuristic styles in P&R are not
dialectically unified.
• P&R offers no mathematical equivalent of
absolute knowledge: there is no complete
system of heuristic rules to be had.
• Therefore judgment (‘taste’) must be
exercised.

9. Criticisms of P&R
• The history is distorted by the focus on just
one conjecture.
• Lakatos over-plays the search for proofs
and refutations at the expense of
mathematical description.
• His logical shapes are not specific to
mathematics.
• He failed to spot the dialectical life in
relatively formalised mathematics.