Model-Based Reasoning in Mathematical Practice

Abstract

The nature of mathematical reasoning has been the scope of many discussions in philosophy of mathematics. This chapter addresses how mathematicians engage in specific modeling practices. We show, by making only minor alterations to accounts of scientific modeling, that these are also suitable for analyzing mathematical reasoning. In order to defend such a claim, we take a closer look at three specific cases from diverse mathematical subdisciplines, namely Euclidean geometry, approximation theory, and category theory. These examples also display various levels of abstraction, which makes it possible to show that the use of models occurs at different points in mathematical reasoning. Next, we reflect on how certain steps in our model-based approach could be achieved, connecting it with other philosophical reflections on the nature of mathematical reasoning. In the final part, we discuss a number of specific purposes for which mathematical models can be used in this context. The goal of this chapter is, accordingly, to show that embracing modeling processes as an important part of mathematical practice enables us to gain new insights in the nature of mathematical reasoning.

Appendix

In this appendix, we briefly recall some notions from (monoidal) category theory. Classical references for category theoretical notions and constructions are Borceux [24.33] and Mac Lane [24.34]. We start with some basic notions from set theory. Let us consider two nonempty sets A and B and a set-theoretical map (or function) f between them. This situation can be depicted as follows:

that is, this process f can be visualized by an arrow; the only requirement being that one must be able to tell for every element of the departure set A where it is going to. Now, let A,B,C be three sets and consider two functions f and g as follows:

We can now consider the composition, denoted by $g\circ f$:

The composition of functions has the associative property: whenever f,g,h are functions (that can be composed), one has $(h\circ g)\circ f=h\circ(g\circ f)$. Remark also that for any set A, we can consider the function that maps any element of A onto itself:

This function 1 _A is called the identity function on A. It has the property that for any function $f:A\to B$, the following holds: $f\circ 1_{A}=1_{B}\ \circ f=f$.

Let us now consider a more general scenario, not necessarily set-theoretic. Let A and B be objects:

and replace the set-theoretical notion of map by just an arrow between these objects:

We can now give an idea of the notion of category:

Roughly speaking, a category $\mathcal{C}$ consists of objects and arrows (between objects) such that there is a composition $\circ$ for the arrows and an identity arrow 1 _A for any object A of $\mathcal{C}$. These ingredients have to satisfy some conditions that mimic the associative behavior of composition of functions between sets and the above-mentioned property of the identity function on any set.

In this sense, following Awodey [24.35], category theory might be called abstract function theory.

We give some basic examples of categories:

• $\mathcal{C}=\textsf{Sets}$

  objects: sets

  arrows: functions between sets

• $\mathcal{C}=\textsf{Vect}_{k}$ (k being a field)

  objects: k-vector spaces

  arrows: k-linear maps.

Now we would like to illustrate the adjective monoidal in the term monoidal category. Therefore, let us introduce monoids.

A monoid $(M,*,1_{M})$ consists of a set M, a function $*:M\times M\rightarrow M$ and an element $1_{M}\in M$ such that:

• We have $1_{M}*m=m*1_{M}=m$, for any $m\in M$

• For any three $m,n,p\in M$ the following holds

  $$(m*n)*p=m*(n*p)\;.$$

We are ready to sketch what a monoidal category looks like. Very roughly speaking, a monoidal category is a category $\mathcal{C}$, in which we can multiply objects and arrows (this multiplication is denoted by $\otimes$) and in which a unit object exists (denoted by I) such that this $\otimes$ (resp. I) imitate the behavior of the operation $*$ (resp. the element 1 _M ) from the monoid structure $(M,*,1_{M})$.

We will denote such a monoidal category $(\mathcal{C},\otimes,I)$ briefly by $\underline{\mathcal{C}}$ in the sequel.

Actually, in technical terms, the definition of monoidal category is precisely the categorification of the definition of monoid (here categorification aims at the name for the process as it was coined by Crane and Yetter in [24.36]).

Here are some examples of monoidal categorical structures:

• $(\mathcal{C},\otimes,I)=(\textsf{Sets},\times,\{*\})$, where:

  • × is the Cartesian product of sets.

  • $\{*\}$ is any singleton.

  We will briefly denote this monoidal category by $\underline{\textsf{Sets}}$.

• $(\mathcal{C},\otimes,I)=(\textsf{Vect}_{k},\otimes_{k},k)$, where:

  • $\otimes_{k}$ is the tensor product over k

  • k is any field.

This example will be briefly denoted by $\underline{{\textsf{Vect}_{\textsf{k}}}}$.

Now we have a vague idea of what it means to be a monoidal category, in order to illustrate an example, we wish to glance at certain objects in such categories. More precisely, we start with considering monoids (sometimes called algebras in literature) in a monoidal category $\underline{\mathcal{C}}$. The idea is that these objects mimic the behavior of classical monoids (i. e., sets with an associative, unital binary operation), the language of monoidal categories offering a natural setting to do so; this is an instance of the so-called microcosm principle of Baez and Dolan [24.37], affirming that “certain algebraic structures can be defined in any category equipped with a categorified version of the same structure.”

A monoid in $\underline{\mathcal{C}}$ is a triple $A=(A,m,\eta)$, where $A\in\mathcal{C}$ and $m:A\otimes A\to A$ and $\eta:I\to A$ are arrows in $\mathcal{C}$ (such that two diagrams – respectively mimicing the associativity and unitality condition – commute; we refer the reader to [24.6, Sect. 5.3.1] for instance).

Many algebraic structures can be seen as monoids in an appropriate monoidal category; we present some examples here, for details and more examples we refer the reader again to [24.6, Sect. 5.3.1] e. g.:

• Taking $\mathcal{C}$ to be the monoidal category $\underline{\textsf{Sets}}$, one can easily verify that, taking a monoid in $\mathcal{C}$, one recovers exactly the definition of a classical monoid, as one expects.

• Similarly, a monoid in $\underline{\textsf{Vect}_{\textsf{k}}}$ gives precisely the classical notion of (an associative, unital) k-algebra.

• A more surprising example is given by the octonions. The octonions $\mathbb{O}$ are a normed division algebra over the real numbers. There are only four such algebras, the other three being the real numbers, the complex numbers, and the quaternions. Although not as well known as the quaternions or the complex numbers, the octonions are related to a number of exceptional structures in mathematics, among them the exceptional Lie groups. For more details, we refer to the excellent paper by Baez on this subject [24.38]. One of the properties of the octonions is that they are nonassociative (that is, considered as monoid in $\underline{\textsf{Vect}_{\textsf{k}}}$). They can be seen, however, as an (associative) monoid in the monoidal category constructed by Albuquerque and Majid in [24.16].

In case a monoidal category $\underline{\mathcal{C}}$ exhibits moreover a braided structure (whatever this means), we denote $\underline{\mathcal{C}}$ equiped with this braided structure as $\widetilde{\underline{\mathcal{C}}}$. In this case, one can not only consider monoids in $\widetilde{\underline{\mathcal{C}}}$, one can impose more structure on the definition of monoid, obtaining such notion as Hopf monoid in $\widetilde{\underline{\mathcal{C}}}$. To be a bit more precise, a Hopf monoid in $\widetilde{\underline{\mathcal{C}}}$ is a bimonoid (which is a monoid also having a so-called comonoid structure, both structures being compatible), having an antipode. The reader is referred to [24.6, Sect. 5.3.2] for more details.

The categories $\underline{\textsf{Sets}}$ and $\underline{\textsf{Vect}_{\textsf{k}}}$ can be given a braided structure, which we denote by $\widetilde{\underline{\textsf{Sets}}}$ and $\widetilde{\underline{\textsf{Vect}_{\textsf{k}}}}$ respectively, such that – without going into the details – the notions of group and Hopf algebra can be recovered as being Hopf monoids in $\widetilde{\underline{\textsf{Sets}}}$ and $\widetilde{\underline{\textsf{Vect}_{\textsf{k}}}}$, respectively.