Proofs, narratives, rhetorics, style and other
mathematical elements
Jean Paul Van Bendegem
Vrije Universiteit Brussel
Centrum voor Logica en Wetenschapsfilosofie
Universiteit Gent
Starting hypothesis
Mathematics is a heterogeneous activity
A first analysis produces (at least) five different types:
 Type I: The search process to find a solution to a mathematical
problem typically falls apart in different stages that require
different strategies, including reformulations of the problem,
 Type II: The proximity of mathematical problems need not be
related to a proximity of the corresponding proofs or similar
problems can require different proofs and distinct problems
can be solved by similar proofs,
 Type III: A mathematical problem can change drastically when
transposed from one mathematical background to another, up
to the point of its disappearance or becoming uninteresting,
 Type IV: Mathematical theories that form the background for a
(set of) problems are to be considered different because of the
different proof strategies and related concepts that are being
used,
 Type V: Mathematical explanation, whatever it might be,
depends (though perhaps not solely) on proof and it thus
‘inherits’ its heterogeneity.
x³ + y³ + z³ = n,
for 29  n  33
 x³ + y³ + z³ = 29 has at least two simple solutions:
x = 3, y = z = 1 and x = 4, y = –3, z = –2,
 x³ + y³ + z³ = 30 has a smallest solution: x = –283059965,
y = –2218888517, z = 2220422932,
 x³ + y³ + z³ = 31: no solutions,
 x³ + y³ + z³ = 32: no solutions,
 x³ + y³ + z³ = 33: as it happens this problem is still open
and, apparently, no one seems to have
an idea how to handle it.
How were these results found?
 The cases n = 29 and 30 are the outcome of computer searches
 The cases n = 31 and 32: elementary mathematics
m
0
1
2
3
4
5
6
7
8
m²
0
1
4
9
16
25
36
49
64
m³
0
1
8
27
64
125
216
343
512
m³
0
1
-1
0
1
-1
0
1
-1
(mod9)
 Case n = 33 still unsolved (though a solution for n = 52 is known)
Additional curious features:
 For the case n = 1 (Mahler 1936):
(9t4)³ + (3t − 9t4)3 + (1 − 9t3)3 = 1
 For the case n = 2 (Werebrusov 1908):
(1 + 6t³)³ + (1 − 6t³)³ + (−6t²)³ = 2
‘Oddest’ feature: turn the equation around and a new problem
emerges:
n = x³ + y³ + z³
A problem about decomposition of natural numbers in powers
Thesis put forward
Aspects of mathematical practice such as
 Beauty
 Style
 Rhetoric
inherit this heterogeneity
Two consequences
 Each of these aspects supports the others
 Mathematical foundations serve to reduce the heterogeneity
 Mathematicians care about beauty
 Poincaré, Hardy, Hadamard, Atiyah, Papert, Rota and
many others
 However: not that many systematic approaches
 Most often referred to: Gian-Carlo Rota, “The
Phenomenology of Mathematical Beauty” (1997)
Importance of making distinctions
 Beauty between mathematics and art
 Beauty between mathematicians and nonmathematicians
 Beauty within mathematics:
 Theorems
 Proofs
 Steps in a proof
 Theories
 Definitions
Importance of making distinctions
 Beauty between mathematics and art
 Beauty between mathematicians and nonmathematicians
 Beauty within mathematics:
 Theorems
 Proofs
 Steps in a proof
 Theories
 Definitions
Importance of making distinctions
 Beauty between mathematics and art
 Beauty between mathematicians and nonmathematicians
 Beauty within mathematics:
 Theorems
 Proofs
 Steps in a proof
 Theories
 Definitions
David Wells in The Mathematical Intelligencer, 1988-1990,
produces a list of 24 theorems ranked by beauty
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
ei = -1 (7,7)
V + F – E = 2 (7,5)
The number of primes is infinite (7,5)
There are 5 regular solids (7,0)
1 + 1/2² + 1/3² + ... 1/n² + ... = ²/6 (7,0)
The fixed-point theorem (6,8)
2 is irrational (6,7)
 is transcendental (6,5)
The four-colour theorem (6,2)
Any prime of the form 4n+1 is uniquely the sum of 2
squares (6,0)
David Wells in The Mathematical Intelligencer, 1988-1990,
produces a list of 24 theorems ranked by beauty
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
ei = -1 (7,7)
V + F – E = 2 (7,5)
The number of primes is infinite (7,5)
There are 5 regular solids (7,0)
1 + 1/2² + 1/3² + ... 1/n² + ... = ²/6 (7,0)
The fixed-point theorem (6,8)
2 is irrational (6,7)
 is transcendental (6,5)
The four-colour theorem (6,2)
Any prime of the form 4n+1 is uniquely the sum of 2
squares (6,0)
What if theorem and proof are related?
George David Birkhoff (1884-1944)
“Aesthetic Measure” (1933)
 Not taken very seriously
 Started formal-mathematical aesthetics
 Linked to information theory (Shannon)
 Also applied to ethics
M=O/C
M = measure of beauty
O = order
C = complexity
Applied to geometrical figures, tilings and ornaments
and to music
Note: explicit argument to “derive” the formula
starting from M = f(O, C) to M = f(O/C) and, by
simplicity, to O/C
Example: the beauty of a square
M = (V + E + R + HV – F)/C
V = vertical symmetry = 1
E = 1 if V is 1 (corresponds to equilibrium)
R = rotational symmetry/2 = 4/2 = 2 (turns)
HV = 2 iff the polygon fits in a horizontal-vertical grid
F = 0 (if the figure is sufficiently regular)
C = 4 = number of distinct straight lines = number of sides of
the polygon
Hence M = (1 + 1 + 2 + 2 – 0)/4 = 6/4 = 1.5
How to apply to theorems and proofs?
Most often heard: “Most beautiful is a simple proof
for a deep result”
Suggest an interpretation:
O = measure of what a “deep” result is
C = simplicity of the proof
M = the beauty of a proof = O / C
Some direct consequences
 A shorter proof for the same theorem increases its
beauty
 A similar proof for a deeper result is more beautiful
Problem:
Since only the ratio is considered, an unimportant
result with a really short proof could be just as
beautiful
Some more detail:
What is a deep result?
Core concept: connectedness
Example: If many theorems have been proved of the
form “If A, then B”, then A is deeply connected
But how does it relate to proof ?
Theorem: √2 is irrational
Theorem: √2 is irrational
Proof: suppose √2 = a/b
such that gcd(a,b) = 1
Proof: suppose √2 = a/b
a2 = 2b2
hence a = 2c
2c2 = b2
hence b = 2d
thus gcd(a,b) ≠ 1
Hence a2 = 2b2
Number of prime factors
in a2 is even, in 2b2 odd,
which is impossible
Another proof:
 Rather an oddity
 Rather “surprising”
a
b
This measure of beauty corresponds to the beauty of
the efficient, economic problem-solver
Hence it expresses a basic feature of the activity of a
mathematician
Thus serves to identify exemplars
Hence it is essential and not a mere “side-effect”
Some further consequences
1. Demonstrates the importance of proof methods
 The reductio proof
 Proof by infinite descent
 Proof by cases
 Mathematical induction
 Career induction
 (Partial) computer proof
 “Experimental” proof
2. Shows the importance of research programmes in
mathematics to identify the well-connected problems
and the favoured proof methods
3. Explains why outside mathematics other aesthetic
standards occur
4. Explains why beauty can change over time
5. In principle extendable to theories and definitions
As mentioned this cannot be the whole story
Another approach to understand the diverse forms of
beauty is to make the connection with style
However style turns out to be equally heterogeneous:
 Applied to a person
 Applied to a connected group of persons
 Applied to a cultural setting (time, place)
Style related to different phases in the mathematical process
(1) Style of discovery
(2) Style of translation in written text
(3) Style of the finalized text, a proof
François LE LIONNAIS (1901-1984): “Les grands courants de la
pensée mathématique “ (1948)
Distinction between romanticist and classicist attitude to
mathematics, where
romanticist corresponds to (1)
classicist corresponds to (2)
Ad (1): Problem-solving involves



“discovering” new problem domains
covering new grounds
“uncharted” territory
Better term: complexity-seeking style
Classic examples:

Introduction of imaginary numbers

Cantor’s theory of infinities

Riemann’s hypothesis
Ad (2): Notes, roughly noted ideas, letters
generative style (?)
Ad (3):
complexity-reducing style
simplicity-seeking style
unification-seeking style
Best known and best studied at the present
Bourbaki style is a perfect example
Set a format for producing mathematical texts
(related to foundational studies and their role in mathematical
practice)
Enter rhetorics
Again a complex story:
 Inside or outside of mathematics?
Within mathematics:
 Belongs to the formal or the informal?
 Belongs to particular elements or to the structure?
 Applicable to proofs or to other mathematical
elements?
“The myth of totally rigorous, totally formalized mathematics is
indeed a myth. Mathematics in real life is a form of social
interaction where “proof ” is a complex of the formal and the
informal, of calculations and casual comments, of convincing
argument and appeals to the imagination and the intuition.”
(p. 68)
Philip J. Davis & Reuben Hersh: “Rhetoric and Mathematics”. In: John S.
Nelson, Allan Megill & Donald N. McCloskey (eds.): The Rhetoric of the Human
Sciences: Language and Argument in Scholarship and Public Affairs. Madison: The
University of Winconsin Press, 1987, pp. 53-68.
Structure = narrative?
“The ludic is a treatise of the ideal type […]: a work based on
obtaining results in surprising, intricate ways, where the author
brings out his own voice in rich, modulated ways, and where the
textual surface is often made deliberately opage by, say, long
passages of calculation.” (p. 108)
Reviel Netz: Ludic Proof. Greek Mathematics and the Alexandrian Aesthetic.
Cambridge: CUP, 2009.
(a) basis: suppose that n = 0, then (a1) 1 + 2 + 3 + … + n = 0 and (a2)
n.(n+1)/2 = 0 and (a3) so they are equal.
(b) induction step: suppose the statement holds for n, so:
1 + 2 + 3 + … + n = n.(n+1)/2
(c) add to both sides n+1:
1 + 2 + 3 + … + n + (n+1) = n.(n+1)/2 + (n+1)
(d) the right-hand side can be transformed into:
(d1) n.(n+1)/2 + (n+1) = (n+1).(n/2 + 1)
(d2) n.(n+1)/2 + (n+1) = (n+1).(n + 2)/2
(d3) n.(n+1)/2 + (n+1) = (n+1).((n+1)+1)/2
(e) put this together, and one finds:
1 + 2 + 3 + … + n + (n+1) = (n+1).((n+1)+1)/2
which is precisely the statement to be proven for n+1.
(f) by mathematical induction, for all n, 1 + 2 + 3 + … + n = n.(n+1)/2.
Introduces the broader field of semiotics
 Semiotic analysis by Brian Rotman
 Semiotic analysis by Paul Ernest
 Not restricted to the educational setting
 Important forerunner: the Significs
The Signific Movement
 Members: Gerrit Mannoury, L.E.J Brouwer, Frederik
van Eeden, van Ginniken, …
 First quarter of 20th century
 Linked to the Wiener Kreis
 Core thoughts:
 Language is a “living” thing
 Primarily a social process
 Should be studied as such
 Distinction between speaker and hearer
 Also applicable to mathematics
Where does this leave us?
 In a semiotic framework all mentioned aspects can be brought
together: beauty, style, rhetoric, …
 In such a framework these aspects depend on one another
 Each of these elements inherits the heterogeneity of
mathematical practices
Suggestion:
Better to talk about ‘varieties of X’ rather than simply X
And explanation?
Thesis: the same holds for explanation
Support:
 There are at least two basic proposals (Steiner & Kitcher) that
do not seem to exclude one another
 The on-going discussion shows that not everyone is satisfied
with these two accounts
 Inside or outside mathematics?
 If inside, related to particular practices or not?
Foundational studies
Homogeneity as compensation
Along different dimensions:
 A foundational theory (ZFC, category theory, …)
 An idealized mathematician (Brouwer’s creative subject)
 An idealized community of mathematicians (logical modelling
of groups of epistemic agents)
 A standardized form of proof (related to computer proofs)
 …
Final thought:
Does not the heterogeneity imply that every possible
theory of mathematical practices will be necessarily
incomplete?
Mathematicians are familiar with “surprises”

1
4
2
1
1
  k (



)
8k  1 8k  4 8k  5 8k  6
k  0 16
PSLQ algorithm (Helaman Ferguson)