Download A taxonomy of mathematical mistakes

A taxonomy of mathematical mistakes
A taxonomy of mathematical mistakes
Marc Bezem
Department of Informatics
University of Bergen
July 2016
A taxonomy of mathematical mistakes
Proof assistants
I
I
Are claimed to help detect mistakes in mathematical proofs
Several questions arise:
1. What is a mathematical mistake?
2. Which mathematical mistakes can/not be detected by using
proof assistants?
3. How and when is a mistake found?
A taxonomy of mathematical mistakes
Resources
IL Imre Lakatos, Proofs and Refutations — The Logic of
Mathematical Discovery, ed. Worrall/Zahar, CUP, 1976
LL Leslie Lamport, How to Write a 21st Century Proof, 2011
JPS Jean-Pierre Serre, How to write mathematics badly,
YouTube, 2013
A taxonomy of mathematical mistakes
Not considered mistakes
I
Different philosophical positions, e.g., implying
I
I
I
I
Every real number is either = 0 or 6= 0
Every real function is continuous
Not considered here: imperfections in applied
mathematics, like numerical instability due to limited
precision arithmetic
We limit ourselves to pure mathematics:
1. Different philosophical positions should be fully axiomatized
including all the logical rules
2. Every calculation should be accompanied by a proof p of
an error estimation, like p : |⇡ 3.14| < 0.01
A taxonomy of mathematical mistakes
Mistakes can have value
I
Mistakes can be very productive: Frege’s inconsistent
Basic Law V has led to ZFC, QNF and to type theory
I
Differently, but equally productive: the failure of Hilbert’s
Programme (metamathematics, Gödel, de Bruijn)
I
Want impact? Make a (interesting) foundational mistake!
I
Normally, mathematics does not evolve through falsification
I
Cf. Lakatos: Proofs and Refutations — The Logic of
Mathematical Discovery
My position: ‘pragmatic formalism’
I
I
I
I
Creative process of invention not formalistic
Use formalism for verification and accountability
Best to leave ontological issues aside
A taxonomy of mathematical mistakes
Elementary mistakes
I
Typos, misformulations, miscalculations
I
I
I
Frequent in informal mathematics, not very interesting as
long as they can be repaired
Very annoying for the reader who is in the process of
understanding the math through close study of the text
Usually (and very usefully) captured by proof assistants:
I
I
In propositions only if the proposition becomes ill-typed
Otherwise usually captured in the proof of the proposition,
or later in the development of the theory (e.g., when the
mistake trivializes the proposition)
A taxonomy of mathematical mistakes
Definitions
I
Definitions can only be ‘wrong’ in that they define
something different than intended
I
I
I
I
I
sometimes inevitable by inherent weakness of the ambient
formalism, e.g., finiteness in FOL (explanation)
sometimes by evolving insight, e.g., the definition of a
polyhedron as to satisfy the Euler formula (Lakatos)
sometimes plainly wrong, e.g., the definition of substitution
under a binder (LISP, ALGOL, De Bruijn)
sometimes still subject to discussion, e.g., the circle S1 as a
HIT (discussion)
‘Wrong’ definitions are not captured by proof assistants
A taxonomy of mathematical mistakes
Mistakes in theorems
I
Probably oftest: hidden assumption
I
Only captured by proof assistants in the proof
I
Damage depends on the strength of the missing
assumption
I
Repairment can invalidate later applications of the
theorem: the hidden assumption may not hold there
I
A counterexample may exist, result is fatally flawed, even
with any reasonable extra assumption
A taxonomy of mathematical mistakes
Mistakes in proofs
I
I
A non-sequitur step, refuted by a local counterexample [IL]
Frequent: gaps
I
I
I
I
All captured by proof assistant (tedious but useful)
Degrees of seriousness:
I
I
I
conscious: ‘wlog’, ‘similarly’, ‘by induction’ (certainly not
always mistakes)
unconscious: e.g., a missing case (can be fatal)
light: an easy-to-find fix/addition proves the result
heavy: we don’t have a correct proof, result is open
Example (cf. [JPS]): We have 8x. 9y . P(x, y ). Denote this y
as f (x). The function f satisfies ...
A taxonomy of mathematical mistakes
Mistakes in proof systems
I
Axioms and rules define provability relation
I
Provability relation approximates truth
I
Special (very important) case of a definition
Possible shortcomings:
I
I
Too few provable formulas
I
I
I
inherently, by the desire to reduce truth to something simpler
incidentally, an axiom can be added (C to ZF, UA to MLTT)
Too many provable formulas, ultimately: inconsistency
I
Only the latter is considered a mistake
I
Proof assistants cannot do much here (cf. SAT solvers)
A taxonomy of mathematical mistakes
A taxonomy of mathematical mistakes (p.a. use)
I
Elementary mistakes (++)
I
Mistake in a definition, unintended, undesirable and
avoidable (-)
Mistake in a theorem:
I
I
I
I
Mistake in a proof:
I
I
I
missing assumption, fixed still useful (+, in proof)
global (cf. [IL]) counterexample (-)
un/conscious gap that can be closed (++)
un/conscious gap that cannot be closed (+/-)
Foundational mistake in the proof system (-)
A taxonomy of mathematical mistakes
Finiteness not first-order definable
I
I
I
Assume first-order theory T satisfies for all M:
M |= T () |M| is finite
V
Define for each n > 0: n := 9x1 , . . . , xn . 1i<jn xi 6= xj
expressing that there exist at least n elements
T is consistent with any finite number of
n
I
T is inconsistent with any infinite number of
I
The latter two conflict with compactness (and with
completeness)
I
NB: Infinity is defined by any infinite number of
cannot be defined by a finite first-order theory
(back)
n
n,
but
A taxonomy of mathematical mistakes
Discussion on the circle
I
In synthetic homotopy theory: S1 as a HIT
I
In ordinary homotopy theory: {z 2 C | |z| = 1}, R/Z, ...
I
en.wikipedia.org/wiki/Construction_of_the_
real_numbers (irony: Section Synthetic approach)
I
All approaches allow many models (ZFC has a countable
model if ZFC is consistent)
I
Is there an interesting property separating them?
(back)