Tomas Petricek, fsharpWorks & Alan Turing Institute
tomasp.net  tomas@tomasp.net  @tomaspetricek
Crash course in philosophy of mathematics
The existence of mathematical objects is independent of us, our language, thoughts and practices.
The Romance of Mathematics makes a wonderful story, but it intimidates, it helps to maintain an elite,
it rewards incomprehensibility.Lakoff, Núñez (2000)
Mathematics does not grow through increase of the number of established theorems, but through improvement by specu lation and criticism, by the method of proofs and refutations.
Lakatos (1976)
"I turn aside with a shudder of horror from this lamentable plague of functions which have no derivatives."
Culturally specific ideas often find their way into
the very fabric of mathematics itself.Lakoff, Núñez (2000)
The only mathematics we know or can know is
a brainandmindbased mathematics.Lakoff, Núñez (2000)
It is up to cognitive science to apply the science of mind to human mathematical ideas.
Cognitive science of mathematics
Cognitive science [showed that], abstract concepts [are]
understood, via metaphor, in terms of more concrete concepts.Many mathematical ideas are ways of mathematicizing ordinary ideas,
as when derivatives mathematicize the idea of instantaneous change.Lakoff, Núñez (2000)
Innate arithmetic
Babies have some mathematical capacities
Conceptual metaphors
Links concepts via neural conflations
Layering metaphors
Explain more abstract mathematical concepts
Linguistic examples
Add onions and carrots to the soup
Which is bigger, 5 or 7?
Equational properties
Adding A to B gives the same result as
adding B to A for object collections
Limitations of the metaphor
Zero in terms of collections?
Lambda calculus, category theory and functional programs
Programs

Proofs

Categories

The idea of program verification is what philosophers call "category mistake". Program verification is, literally, a form of nonsense.
Fetzer (1988)
Carefully constructed to fit well via the
method of proofs and refutations
All three are product of the same network of mathematicians, solving the same problem.
Searching for foundations of mathematics, formalising reasoning based on inference that could be done mechanically.
All three are derived from the same embodied experience using a number of conceptual and layering metaphors.
What is the embodied experience?
Cognitive science and lambda calculus
Modus Ponens
Given two Container schemas A and B and an
object X, if A is in B and X is in A, then X is in B.
Function Application
Given two types \(A\) and \(B\) and a value \(x\),
if \(f : A\rightarrow B\) and \(x:A\) then \(f(x):B\)
reduce, verb (used with object), reduced, reducing.
Cognitive science of extraterrestrial beings
No notion for direction
Function application is directional!
Perhaps only reversible computations?
There is only one being in the world
Would it have more numbers than one?
There are no boundaries in chaos!
There is no inside and outside
No container schema metaphors
Would aliens understand lambda calculus?
Is lambda calculus discovered or invented?
Platonism is just one (religious) belief
Philosophy of mathematics and computer science
Social, cultural enterprise, product of embodied mind
So, would aliens understand lambda calculus?
Stretch your imagination! Boring aliens might...
Aliens with circular language and time
Not your grandma's sentient being
How mathematics actually works
Cognitive account of mathematics via metaphors
Category mistakes and dissenting voices in the community