In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita

In prase of laziness: Categorical non-foundationof mathematics
Minao Kukita
Graduate School of Informatics
Nagoya University

Symposium on the Categorical Unity of the Sciences,
March. 22-23, 2019

In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita‘I want to say, in all seriousness, that a great deal of harm isbeing done in the modern world by belief in the virtuousness ofwork, and that the road to happiness and prosperity lies in anorganized diminution of work.’
Bertrand Russell, “In Praise of Idleness”, 1935.

Varieties of normality
In prase oflaziness:
Categoricalnonfoundation ofmathematics

One way to think about the meaning of a λ term is by taking it
to be its normal form. Then, terms with no normal form haveno meaning, and therefore are all identified.

Minao Kukita

In a way, a term is thought to be normal if it cannot berewritten any further, i.e., cannot be reduced to any otherterm. A λ term can be rewritten if it contains beta redex(reducible expressions), a subterm of the form (λx.M)N, which
can be reduced to M[N/x] by the beta reduction rule.
However, there are other conceptions of normality. One canthink of a term as normal if it does not contain a head redex,redex that appears in the head position of a term. Such a term
is called in head normal form. This interpretation is calledstandard.

Varieties of normality
In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita

Yet another normality is weak head normal form. A term is in
weak head normal form if it is an abstraction, namely, of theform λx.M
This may sound too loose a conception of normality, but
Samson Abramsky showed that this could be quite reasonable(Abramsky, “The lazy lambda calculus”, 1987). For one thing,no functional programming language actually evaluates terms
to its normal forms or head normal forms. For another, thedomain semantics for the lazy lambda calculus has thenon-trivial initial solution, that is, not the one-point domain.

Meaning as normal form
In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita

If we take the meaning of a term to be its weak head normalform, then any term is meaningful if only it is reduced to anabstraction.
Example. Let Ω be (λx.xx)(λx.xx). Then, λx.Ω does not havehead normal form, but is in weak head normal form. So, in thelazy interpretation, Ω and λx.Ω are not identified, although, inthe standard interpretation, they are.

What is laziness good for?
In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita

Usual functional programming languages are lazy in the sensethat they do not evaluate inner redexes (redexes that are not inthe head position) or redexes within abstraction. This featureallow us to easily handle data structures such as stream(roughly, list of infinite length. More precisely, stream is thedual notion of list, in the sense that coalgebra is the dual ofalgebra).
Cf. Kees Doets and Jan van Eijck, The Haskell Road To Logic,
Maths And Programming, 2004.

Stream and corecursion
In prase oflaziness:
Categoricalnonfoundation ofmathematics

A stream is a data structure defined, for example, as follows:

Minao Kukita

One := (1, One)
This definition looks circular, and it is circular. If you are eager
to finish the evaluation, you will end up with running into an
endless loop. You have to be lazy and refrain from evaluating
it completely.
This kind of definition is called corecursion.

Top-down versus bottom-up
In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita

Greg Restall once said in a talk that, while semantics isbottom-up in nature, syntax is top-down. When engaged insyntax, we do not care about the details of subformulas butonly about the structures of formulas or of derivations.
Based on this intuition, Restall introduced the notion of‘coformula’, the dual of formula.

Top-down versus bottom-up
In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita

Abstract mathemaitcs is also carried out in the spirit of thetop-down, without regard to the particular properties ofindividual objects (cf. Michael Makkai, “Towards a CategoricalFoundation of Mathematics”, 1998).
In the argument against Geoﬀrey Hellman, Steve Awodey alsoemphasises the top-down nature of mathematics.

Awodey’s recommendation to philosophers ofmathematics (esp. to structuralists)
In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita

Awodey, “Structure in mathematics and logic: A categoricalperspective” (1996)
Category theory is particularly suitable for studying
mathematical structures.
Nevertheless, structuralists have not paid due attention tocategory theory.
Why not use category theory as a conceptual framework
for thinking about mathematical structure?

Hellman’s rejection
In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita

Hellman, “Does category theory provide a framework formathematical structuralism?” (2003)
Category theory is inadequate as a foundation ofstructuralism.
The axioms of category theory is not ‘assertory’ but‘definitional’.
Category theory does not answer the question about the‘home address’ of its objects (“where do categories comefrom and where do they live?”), which an adequate
structuralist foundation should.

Awodey’s rebuttal
In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita

Awodey, ‘An answer to Hellman’s question: “Does category
theory provide a framework for mathematical structuralism?” ’(2004)
Hellman is too foundationalistic.
Mathematics is in nature schematic and top-down.
Why is it hard for philosophers to follow Awodey’s
recommendation?

Benacerraf’s dilemma
In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita

Paul Benacerraf, “Mathematical truth”, 1983.
It is impossible to achieve the following two goals which havemotivated philosophers’ attempts to account for mathematical
truth.
To have a homogeneous semantics for the mathematicallanguage and the rest of our linguistic activity.
To have a homogeneous epistemology for themathematical knowledge and the rest of our knowledge.
For Benacerraf, such semantics should be ‘referential’, that is,model-theoretic or truth-conditional. He assumes that “truthconditions for the language (e. g., English) to whichmathematese appears to belong are to be elaborated muchalong the lines that Tarski articulated” (p. 410).

Austin’s complaint
In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita

Austin, “Performative utterances”, in Philosophical Papers,
1979.
Most philosophers tend to think that the sole interesting
business, of any utterance–that is, of anything we say–is to betrue or at least false’ (p. 233).

Foundation versus Non-Foundation
In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita

Foundationalist
Set Theory
Bottom-up
Eager
Assertion
Referential Utterrance
Implementation
Recursion
Individualism
Logicism
Iterative Conception of Sets
Concrete

Non-Foundationalist
Category Theory
Top-down
Lazy
Definition
Performative Utterance
Interface
Corecursion
Structuralism
Axiomatism
Typical Ambiguity
Abstract

These are two diﬀerent but supplementary aspects of ourlinguistic activities, including mathematical andcomputer-scientific activities.

We need both.
In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita“A formal system standing alone is an incomplete entity: itneeds its interpretation.”
Dana Scott,Rules and derived rules , 1974.

Feferman’s distinction of axioms
In prase oflaziness:
Categoricalnonfoundation ofmathematics

Solomon Feferman, “Does mathematics need new axioms?”,
1999.

Minao Kukita

Feferman distinguishes two kinds of axioms: ‘foundational’
axioms and ‘structural’ ones. The former assert obvious truth
about certain well-known mathematical entities such as sets ornatural numbers.
The latter have the function of defining a new class ofmathematical structures. According to Feferman, the value ofthese axioms lies in its ability to organise mathematician’s workand to ‘package and communicate our knowledge in digestibleway’.

What mathematics does
In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita

What is mathematics all about?
In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita

Mathematics is about literally anything that satisfies a set ofaxioms, not confined to so called mathematical objects ormathematical structures. Mathematical statements can beapplied (though approximately) to things in the real world suchas pebbles, planets, particles, water, human behaviours,
strategies for chess, moves of the Rubik’s Cube, and so on.
For me, it is one of the most irrational idea that mathematics
is all about sets or any other particular kinds of mathematicalentities whatsoever.

What is mathematics at all?
In prase oflaziness:
Categoricalnonfoundation ofmathematics

Andy Clark claims in his Natural-Born Cyborgs (2003) that
languages, among other artefacts, are part of our cognitive
faculties.

Minao Kukita

Similarly, we can say that mathematics is part of our cognitive,and communicative, faculties. Arguably, mathematics has been
the most eﬀective and reliable instrument for knowledge
acquisition and information communication, at least so far.

Tanimura’s tweet on physical theories
In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita‘I argued [...] that the success of physical theory in the sense of
explaining and accurately predicting real phenomena, inparticular the utility of mathematical descriptions, is not amiracle, but the result of more successful ones being selected
and thus leaving more copies.’
Exactly the same should be said of the mathematics as awhole, or even of science as a whole.

Theft?
In prase oflaziness:
Categoricalnonfoundation ofmathematics
Minao Kukita‘The method of “postulating” what we want has many
advantages; they are the same as the advantages of theft overhonest toil.’
Russell, Introduction to Mathematical Philosophy, 1919.

‘I want to say, in all seriousness, that a great deal of harm isbeing done in the modern world by belief in the virtuousness ofwork, and that the road to happiness and prosperity lies in an
organized diminution of work.’
Russell, “In Praise of Idleness”, 1935.