1. pres_280611_lakatos.odt Heuristics in mathematical discovery 1/21
Imre Lakatos: Proofs and Refutations
"One of the things we're sure to discover is that the psychological processes don't
parallel the logical processes."
"[...] the steps of justification are not the steps of discovery."
(Larry E. Travis: The value of introspection to the designer of
mechanical problem solvers)
Topic:
• a historic account of how mathematical proofs are being found
• in contrast to 'formalist' accounts of metamathematics / philosophy of mathematics

2. pres_280611_lakatos.odt Heuristics in mathematical discovery 2/21
Imre Lakatos
Hungarian mathematician & philosopher
Born Lipschitz, later Molnár
translated Polya (how to solve it) into hungarian
Flew to GB after WW2
1960 at London School of Economics, where he met
Popper and other philosophers of science
Was friends with P. Feyerabend
Philosophy of science similar to Kuhn
Dissertation in Cambridge
Post-mortem published: “Proofs and Refutations”

3. pres_280611_lakatos.odt Heuristics in mathematical discovery 3/21
Metamathematics
formalized proofs (Eucledian methodology)
→ Mathematical notions are defined by a set of axioms, and theorems are derived by a
fixed set of inference rules .
Russel & Whitehead's Principia Mathematica (1910-1913) demonstrated the feasibility
of such an approach.
→ automated theorem provers possible
One consequence of the 'formalist' foundation of mathematics is that strong proponents
do only consider theorems derived by deductive proofs as mathematically meaningful.
→ therefore, “Gödelian theorems” would be mathematically not meaningful!
(Über formal unentscheidbare Sätze der Principia Mathematica ... 1931)

4. pres_280611_lakatos.odt Heuristics in mathematical discovery 4/21
Metamathematics
Does this form of proof reflect their discovery?
• Lakatos: No mathematician would embark into enumerating theorems.
• Automated proofs are syntactic, while human reasoning is semantic (but what is
“semantic”?)
• Examples and counter-examples play an important role in the development of
mathematical concepts
• Errors in faulty proofs may remain undetected (even over generations)
Lakatos e.g. complains that one is never being told how funding axioms arose. Math
books don't show how proofs are found, and they hinder to develop creative, explorative
thinking.
→ Lakatos' claim: Informal mathematics grows by a logic of proofs and
refutations.
→ mathematics in an experimental style

5. pres_280611_lakatos.odt Heuristics in mathematical discovery 5/21
Lakatos, I. (1976).
Proofs and Refutations: The Logic of Mathematical Discovery
1. A Problem and a Conjecture
2. A Proof
3. Criticism of the Proof by Counterexamples which are Local but not Global
4. Criticism of the Conjecture by Global Counterexamples

6. pres_280611_lakatos.odt Heuristics in mathematical discovery 6/21
Euler's formula for polyhedra
(1752)
V+E–F=2
Where does the initial conjecture come from?
Polya: “You have to guess a mathematical theorem before you prove it”
→ Lakatos starts where Polya ends.

7. pres_280611_lakatos.odt Heuristics in mathematical discovery 7/21
Cauchy's proof
(1813)
• Step 1: remove a face and stretch it to 2-D
• Step 2: triangulate the planar graph
• Step 3: remove triangles
▪ a) one edge & face
▪ b) one vertex, two edges, one face
→ decomposes the original conjecture into three lemmas

8. pres_280611_lakatos.odt Heuristics in mathematical discovery 8/21
socratic dialogue
Criticism of the proof by a local counterexample
• A proposal from the class: call it a theorem, not a conjecture.
• But the teacher prefers to call it a thought experiment / 'quasi'-experiment.
→ The students doubt proof steps 1, 2, 3
Gamma: A counterexample to step 3: Remove a triangle from the middle.
Only one face is subtracted !
Teacher: Right. Lemma 3 has to be defined more precisely.
But this doesn't mean that the initial conjecture does not hold. If possible,
proof step 3 could be modified.

9. pres_280611_lakatos.odt Heuristics in mathematical discovery 9/21
socratic dialogue
Criticism of the proof by a local counterexample
→ "local counterexample": one that refutes a lemma, and therefore
the proof, but not the main conjecture.
Teacher: Remove only external boundaries !
But Gamma finds a way to remove them in an order which changes V-E+F=2 .
Teacher: Ok, so I'll restrict removal order such that V-E+F=2 holds.
I claim that this is possible in general!
Gamma: But how to find that order? Can you proof that there is a
general way?

10. pres_280611_lakatos.odt Heuristics in mathematical discovery 10/21
socratic dialogue
Criticism of the proof by a local counterexample
→ what is the sense of such a proof? Aren't we worse of because we now have many
conjectures about which we are uncertain?
Teacher: No, we have embedded our conjecture into knowledge about graphs,
triangulation, and so forth and therefore have more “working points” for
our proof.
"decomposition opens new vistas for testing [...]
deploys the conjecture on a wider front so that our criticism has more
targets [...]
we now have at least three opportunities for counterexamples instead of
one"

11. pres_280611_lakatos.odt Heuristics in mathematical discovery 11/21
Criticism of the proof by a global counterexample

12. pres_280611_lakatos.odt Heuristics in mathematical discovery 12/21
Criticism of the proof by a global counterexample
Counterexample 1: A cube within a cube
(Lhuilier 1812-13)
V-E+F=2 does not hold ?
→ (a) Rejection of the conjecture. The method of surrender

13. pres_280611_lakatos.odt Heuristics in mathematical discovery 13/21
Criticism of the proof by a global counterexample
Idea: Define the initial concepts such that they exclude the counterexamples
Def. 1: A polyhedron is a solid bounded by polygonal surfaces
Def. 2: ... is a surface, on which a point can continuously move (topological).
The nested-cube is in fact two cubes!
→ (b) Rejection of the counterexample. The method of monster-barring

14. pres_280611_lakatos.odt Heuristics in mathematical discovery 14/21
Criticism of the proof by a global counterexample
More counterexamples: (connected tetrahedrons; picture frame; urchin)
→ V – E + F = 2 does not hold!

15. pres_280611_lakatos.odt Heuristics in mathematical discovery 15/21
Criticism of the proof by a global counterexample
More counterexamples: (connected tetrahedrons; picture frame; urchin)
Def 3: a) exactly two polygons adjunct at an edge;
b) every crossing an edge from on polygonn to the other does not cross a vertex
Def 4: for every point on the surface, there is a plane which dissects the polyhedron
such that the dissection is one single polygon
... - convex polyhedrons!
The method of monster-barring creates ad-hoc adjustments to every new counter-
example. But it cannot guarantee that there won't be any more counterexamples!

16. pres_280611_lakatos.odt Heuristics in mathematical discovery 16/21
Criticism of the proof by a global counterexample
Another idea: exceptions
“All polyhedra that have no cavities, tunnels, or multiple structure”
→ Exception-barring methods: piecemeal exclusions
“All convex polyhedra are Eulerian.”
→ Strategic withdrawal or playing for safety
→ (c) Improving the conjecture by exception-barring methods. Piecemeal exclusions.
Strategic withdrawal or playing for safety

17. pres_280611_lakatos.odt Heuristics in mathematical discovery 17/21
Criticism of the proof by a global counterexample
Idea: Monsters are misinterpreted. Re-interpret them correctly, and the conjecture and
proof will hold.
(Example: urchin)
→ (d) The method of monster-adjustment

18. pres_280611_lakatos.odt Heuristics in mathematical discovery 18/21
Criticism of the proof by a global counterexample
Idea: Establish a unity between proofs and counterexamples.
→ Lemma 1 does not hold
Solution: Adjust the lemma on the basis of the counterexample.
Teacher: The Euler conjecture holds good for 'simple' polyhedra, i.e. for those which,
after having had a face removed, can be stretched onto a plane.
→ (e) Improving the conjecture by the method of lemma-incorporation. Proof-generated
theorem versus naive conjecture
Pease et al.: “This method evolves into proofs and refutations, which is used to find
counterexamples by considering how areas of the proof may be violated.”

19. pres_280611_lakatos.odt Heuristics in mathematical discovery 19/21
Methods to cope with counterexamples
(a) Rejection of the conjecture. The method of surrender
(b) Rejection of the counterexample. The method of monster-barring
(c) Improving the conjecture by exception-barring methods. Piecemeal exclusions.
Strategic withdrawal or playing for safety
(d) The method of monster-adjustment
(e) Improving the conjecture by the method of lemma-incorporation. Proof-generated
theorem versus naive conjecture

20. pres_280611_lakatos.odt Heuristics in mathematical discovery 20/21
“simple pattern of mathematical discovery”: stages wrap-up
(1) Primitive conjecture.
(2) Proof (a rough thought experiment or argument, decomposing the primitive
conjecture into subconjectures of lemmas).
(3) 'Global' counterexamples (counterexamples to the primitive conjecture)
emerge.
(4) Proof re-examined: The 'guilty lemma' to which the global counterexample is
a 'local' counterexample is spotted. This guilty lemma may have been
previously 'hidden' or may have been misidentified. Now it is made explicit,
and built into the primitive conjecture as a condition. The theorem - the
improved conjecture - supersedes the primitive conjecture with the new
proof-generated concept as its paramount new feature.
Additional stages frequently occur:
(5) Proofs of other theorems are examined to see if the newly found lemma or
the new proof-generated concept occurs in them: this concept may be found
lying at cross-roads of different proofs, and thus emerge as of basic
importance.
(6) The hitherto accepted consequences of the original and now refuted
conjecture are checked.
(7) Counterexamples are turned into new examples - new fields of inquiry open up.

21. pres_280611_lakatos.odt Heuristics in mathematical discovery 21/21
“simple pattern of mathematical discovery”: holds for science, too
Lakatos generalises to science:
"Now while Popper showed that those who claim that induction is the logic of scientific
discovery are wrong, these essays intend to show that those who claim that deduction is
the logic of mathematical discovery are wrong. While Popper criticised inductivist style
these essays try to criticise deductive style.” (p. 143)
"Inductivist style reflects the pretence that the scientist starts his investigation with an
empty mind whereas in fact he starts with a mind full of ideas.” (p. 143)
• in discovery (e.g. of a conjecture and a proof) humans start with a problem, to which
they try to find a solution
• “problem” and “solution” are more psychological notions
To conclude, Lakatos gives several examples of mathematical proofs and their “heuristic
history”.