1. What is the philosophy of
mathematical practice?
Brendan Larvor
University of Hertfordshire
MP40 – July 2009

2. Motivation
Foundational and metaphysical attention:
• Is too often motivated and/or conditioned
by extraneous concerns
• Washes out differences of practices
• Makes a mystery of human mathematical
success
• Obscures (or leaves subjective) questions
of value and development
• May be parasitic on experience of
mathematical practice that it cannot
theorise

3. Context
• Mature, professional history of
mathematics and science
• Emerging ethno-mathematics
• Cognitive science/psychology/evolutionary
accounts of natural mathematical abilities
• Broad shift to practice (e.g. in philosophy
of experimentation, ethics, philosophy of
language).

4. What and what? PMP2007
• Jens Høyrup (Roskilde): What did the abbacus teachers really do
when they (sometimes) ended up doing mathematics?
• Roman Murawski & Izabela Bondecka-Krzykowska (Poznan):
Categorical structuralism in the contemporary philosophy of
mathematics
• Erich Reck (Riverside, CA): Dedekind and the emergence of
structuralist understanding in mathematics
• Lieven Decock (Amsterdam): Locke and Hume among the
Mundurukú. Equinumerosity versus number concepts as basic
mathematical concepts
• Helen De Cruz (Brussels): Towards an informed argument from
geometry: The role of intuitive notions of space in the history of
Euclidean and non-Euclidean geometries
• Danielle Macbeth (Haverford, PA): Diagrammatic Reasoning in
Euclid’s Elements
• Ad Meskens (Antwerp): Reading Diophantos

5. …PMP2007
• Bernd Buldt (Fort Wayne, IN): Mathematical practice and
Platonism
• Jessica Carter (Odense): Mathematical objects as abstract objects
• José Ferreirós (Seville): Mathematical knowledge and the interplay
of practices: The case of sets and natural numbers
• Yehuda Rav (Paris): The axiomatic method in theory and in practice
• Ahti-Veikko Pietarinen (Helsinki): Peirce’s pragmaticism as an
antifoundationalist philosophy of mathematics
• Filippo Barra (Siena): The intuitionistic standpoint in the philosophy
of mathematics as a legacy for a philosophy of mathematical
practice
• Ronny Desmet (Brussels): The serpent in Russell’s paradise
• Catherine Womack (Bridgewater, MA): Tacit inference in visual
proofs – in defense of informal mathematics
• Anthony Peressini (Milwaukee, WI): Invisible mathematics:
numerical analysis and its role in mathematical application
• Koen Vervloesem (Leuven): Computer proofs with high-level
concepts

6. …PMP2007
• Ian Dove (Las Vegas, NV): Quasi-empiricism and luck: Extrinsic
justifications in the history of mathematics
• Andrew Aberdein (Melbourne, FL): Learning from our mistakes –
but especially from our fallacies and howlers
• Madeline Muntersbjorn (Toledo, OH): Construction, articulation
and explanation: Phases in the growth of mathematics
• Benedikt Löwe, Thomas Müller & Eva Wilhelmus
(Amsterdam/Bonn): Knowledge ascriptions in mathematical practice
• Dirk Schlimm (Montréal): From domains of being to systems of
axioms, and vice versa: On the role of axiomatics in the discovery of
lattices
• Roy Wagner (Tel-Aviv): What can post-structural philosophy teach
us about the history of mathematical logic?
• Eduard Glas (Delft): A historical note on the quasi-empirical view of
mathematics
• Jeremy Gray (Milton Keynes): 19th century analysis as philosophy
of mathematics
• David Corfield (Tübingen): Mathematics as stories or histories

7. What and what? Books

8. Drawing on Serfati:
Extending functions
to domains where
they seem
nonsensical

9. Seven Extensions
1. Factorial of a positive real number
2. Exponential of a complex number or
square matrix
3. Trigonometric functions of complex
numbers
4. Matrix pseudo-inverses
5. Derivative of a non-differentiable function
6. Derivative of a function on normed
vector-spaces
7. Union and intersection of r-partitions

10. Case 1: Factorial of a positive
real number
• How can n! mean anything unless n is a
natural number?
• Ask Euler!
• n!= n.(n-1)!
)1(.)( −Γ=Γ xxx

11. ( ) ∏
∞
=
−














+





+=Γ
1
1
1
1
1
2
1
n
x
n
x
n
x
( ) ( )
( ) ( ) 





−++
−
∞=Γ x
n
nxxx
n
nx
1...1.
1...2.1
lim
…which satisfies the functional equation and
equates to x! when x is a natural number

12. Case 4: Matrix pseudo-inverses
• Can we give a sense to A-1
when A is not
invertible?
• Yes, and there are several options (unlike
in physics…)
• All coincide with the usual inverse in the
case of invertible matrices

13. Moore-Penrose pseudo inverse
Four trivial identities for invertible matrices:
AA-1
A = A
A-1
AA-1
= A-1
(AA-1
)* = AA-1
(A-1
A)* = A-1
A
Become the definition of the pseudo-inverse
(* is the adjoint)

14. Moore-Penrose pseudo inverse
Penrose proved that for every matrix A there
is a unique matrix X such that:
AXA = A
XAX= X
(AX)* = AX
(XA)* = XA

15. Another pseudo-inverse
Provide the vector space of matrices with
a norm. Note that if a square matrix is
invertible, its inverse is the unique matrix
B such that:
0=− IAB
Again, we take a trivial identity in the
original case, and make it the definition of
the extension.

16. Another pseudo-inverse
Now let A be any complex matrix. The
right-hand pseudo-inverse is the unique
matrix B such that:
Note how this reduces to the
original case (=0), and proving
uniqueness is again the hard part
IAB − is minimal