Abstract
In this paper, I argue that there are cases of explanatory induction in mathematics. To do so, I first introduce the notion of explanatory definition in the context of mathematical explanation. A large part of the paper is dedicated to introducing and analyzing this notion of explanatory definition and the role it plays in mathematics. After doing so, I discuss a particular inductive definition in advanced mathematics—\({ CW}\)-complexes—and argue that it is explanatory. With this, we see that there are cases of explanatory induction.
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Notes
It should be noted that in Lange (2017), a revised account of the explanatory power of inductive proofs is presented. A reader particularly interested in explanatory proofs should refer to this work, but since I am only using criticisms of the Lange (2009) argument to motivate my own account I will not go into details about Lange’s revisions here.
Lange takes these two formulations to be equivalent in the sense that both characterize induction equally well. This is a point that Baker (2010) objects to by pointing out that a case can be made for the claim that PMI has significant theoretical advantages.
Others have made similar remarks. Baker (2010) points out that Lange seems to be relying on another assumption. Namely, the assumption that these two ways of formulating mathematical induction are theoretically equivalent. Baker goes on to argue that given some common assumptions about what we should expect from an account of explanation, these two are not equivalent. In particular, PMI has some explanatory virtues that Alt lacks. Hoeltje et al. (2013) also makes similar comments. A related criticism is pointed out in Baldwin (2016), and states that Lange’s argument does not only apply to proofs by induction, but also applies to universally axiomatized propositions. This is problematic since it seems that there are some such propositions that are, in fact, explanatory.
Giaquinto (2008) puts this point in terms of Russell’s distinction of kinds of knowledge when he points out that mathematics or mathematical objects are not the kinds of things that we can know by acquaintance.
Note that though I claim that this practice is less common in the sciences, I do not mean to suggest that it does not occur at all. In fact, there are plenty of instances in the history of science that a change in standard definition occurs. It is more commonly the case, however, that scientific changes occur as a result of theoretical changes. That is, a change in theory inspires or results in a change in definition. In mathematics, this is not what occurs, but instead the theory remains the same and the definition changes to illuminate some evasive feature of the theory.
I am not suggesting that the combination of explanatory definitions and explanatory proofs provides a comprehensive account of mathematical explanation. I would like to leave open the possibility that there are even still other forms of explanation in mathematics.
In particular, the literature on explanatory proofs, and Lange’s argument discussed above, takes explanations to be answers to why-questions.
The connection to the production of mathematical knowledge is something that will be discussed more later in this paper.
For a list of these different definitions see Thurston (1995, p. 30).
This is not something that I will discuss in depth during this paper, but I would like to point out that the notion of explanatory definition as I have presented it has not ruled out the possibility of a notion having more than one definition that counts as explanatory.
Note that this definition can be generalized to give the derivative as a function, rather than the value at a specific point, but we will use the definition of the derivative at a point for simplicity.
This exact limit definition does not appear on Thurston’s list though it is equivalent to what he calls the logical definition, and clearly formalizes what he calls the geometric definition.
Thurston calls this the symbolic definition (Thurston 1995, p. 30). It should be noted that this presentation of the derivative may be considered an algorithm suitable for certain cases (i.e., polynomials) and not a genuine definition. I do not want to discuss here whether or not we can legitimately classify it as a definition, but since it is often presented as a definition in introductory calculus classes at both the high school and college level, I will entertain it as such for this discussion. The purpose of this discussion is to highlight the values of the limit definition of derivative more than it is to highlight the failures of this algorithmic definition, and so taking this algorithm to be a definition does not harm my overall point.
To be precise these n-disks consist only of the interior and do not include the boundary (i.e., the surrounding sphere). For simplicity, I define disks to include boundaries, which will simplify the discussion of attaching maps. For a more detailed and precise account of these \({ CW}\)-complexes see Hatcher (2001, Chap. 0).
More specifically the topological space must be a Hausdorff space—i.e., you must be able to separate any two distinct points with disjoint open sets.
So, in general, \(X^n\) is the space resulting from attaching the n-disks to \(X^{n-1}\). Here by “attaching n-disks to \(X^{n-1}\)”, I mean that \(X^n\) is gotten by taking the disjoint union of \(X^{n-1}\) and the set of n-disks. The attaching maps are used to give the equivalence relation needed for defining these disjoint unions.
So a topological space that admits a \({ CW}\)-construction is called a \({ CW}\)-complex.
For more details on this particular example, and other examples see Hatcher (2001, p. 6).
Another term for what I have been calling “\({ CW}\)-construction” is “\({ CW}\)-decomposition”.
Here I am limiting the discussion to singular and simplicial homology.
A chain complex is given by
$$\begin{aligned} \cdots \rightarrow G_{n+1} \xrightarrow {\partial _{n+1}} G_n \xrightarrow {\partial _n} G_{n-1} \xrightarrow {\partial _{n-1}} \cdots \xrightarrow {\partial _2} G_1 \xrightarrow {\partial _1} G_0 \xrightarrow {\partial _0} 0 , \end{aligned}$$where each \(G_i\) is an abelian group, each \(\partial _i\) is a group homomorphism, and for every \(i \in {\mathbb {N}}\), \(\partial _i\partial _{i+1} = 0\).
For more details and some examples, see Hatcher (2001, pp. 137–146).
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Acknowledgements
I would like to thank both Curtis Franks and Tim Bays for their feedback on various drafts of this paper, and also the anonymous reviewers who provided numerous helpful comments on an earlier draft of this paper.
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Lehet, E. Induction and explanatory definitions in mathematics. Synthese 198, 1161–1175 (2021). https://doi.org/10.1007/s11229-019-02095-y
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DOI: https://doi.org/10.1007/s11229-019-02095-y