The semantics of social constructivism

Abstract

This essay will examine some rather serious trouble confronting claims that mathematicalia might be social constructs. Because of the clarity with which he makes the case and the philosophical rigor he applies to his analysis, our exemplar of a social constructivist in this sense is Julian Cole, especially the work in his 2009 and 2013 papers on the topic. In a 2010 paper, Jill Dieterle criticized the view in Cole’s 2009 paper for being unable to account for the atemporality of mathematical existents. Cole’s 2013 paper addresses this objection, providing a modification of his 2009 paper allowing for atemporal mathematicalia. An unusual consequence of Cole’s account is that at least some existential claims about mathematicalia used to be false but now have always been true. By examining the semantics of such claims, we demonstrate that social constructivism is in fact, despite Cole’s attempts to rectify matters, incompatible with atemporal mathematicalia. In the course of examining these semantic details, however, an alternative hybrid view of fictionalism and social constructivism emerges. Those tempted by social constructivism, while perhaps disappointed by the negative results of the paper, may be encouraged by how much of their view can be recovered in this alternative account.

Appendix: What if we had more time (dimensions)?

Given Cole was willing to suppose we could collectively declare objects to have whatever temporal profile suited our purposes, I see no reason to suppose he would be unwilling to grant social-constructivism doubly-modal creativity. Thus, Cole could respond to all of the above by claiming that we can “Declare and collectively recognize that institutional facets of reality have whatever [doubly-]modal profile best serves our purposes.”

My response to this is roughly as follows: modifying all the above arguments, we could demonstrate that to making of the account that results from allowing socially-constructed doubly-temporal profiles demands moving to triply temporal logic. Cole could respond to this by proposing we adopt socially-constructed triply-temporal profiles, and the back and forth could go on ad infinitum.

Being more explicit, Cole could propose that social-constructivism’s modal creativity continues to the second temporal variable. Thus, the community responsible for introducing a social construct could declare and collectively recognize that it had whatever doubly-temporal profile suited their needs. Nonetheless, we would have to embrace the following:

(3) sounds at least confusing and possibly contradictory, so we need to introduce a third temporal variable to account for the semantics of (3). The arguments from before will carry over, mutatis mutandis, to establish that Cole must choose again between fictionalism and non-atemporal mathematicalia. We could then repeat this process to force a move to quadruply-temporal logic, etc.

Let’s skip to the end of this process and see what the results are. We introduce the language $\mathcal {L}^{\infty P}$ of infinitely temporal logic for this purpose.

1.1 Syntax

Alphabetically, $\mathcal {L}^{\infty P}$ differs from a base language $\mathcal {L}$ of propositional logic by the addition of a countable infinity of temporal modal operators $H_1,\, G_1$, $H_2$, $G_2$, etc. Grammatically, $\mathcal {L}^{\infty P}$ is the closure of $\mathcal {L}$ after the addition of all these operators. Natural language equivalents of most of these operators are too complex to be worth spelling out.

1.2 Semantics

An $\mathcal {L}^{\infty P}$-model is a triple $\langle \mathcal {T},<,V\rangle$, where

• $\langle \mathcal {T},<\rangle$ is a poset (partially ordered set—i.e. $<$ is an irreflexive and transitive relation on $\mathcal {T}$) representing the “times” or “moments” together with their ordering and

• $V$ is a function from $\mathcal {T}^\mathbb {N}=\{f:\mathbb {N}\longrightarrow T\}$ to the set of functions from ${{\mathrm{\mathbf {Prop}}}}$ (the set of propositions) to the two-element set $\{T,F\}$.

For $f\in \mathcal {T}^\mathbb {N},\, V(f)$ is written $V_f$. In symbols we have

$$\begin{aligned}&V:\mathcal {T}^\mathbb {N} \longrightarrow \{g:{{\mathrm{\mathbf {Prop}}}}\longrightarrow \{T,F\}\} \\&\qquad f \longmapsto ({{\mathrm{\mathbf {Prop}}}}\ni P\longmapsto V_f(P)\in \{T,F\}) \end{aligned}$$

To specify the notions of truth, validity, etc., we use the semantic details given in Fig. 6. Notice the final two clauses in this semantic theory are actually clause-schemes, one instance for each $n\in \mathbb {N}$.

Fig. 6

$\mathcal {L}^{\infty P}$ semantics

1.3 $\mathcal {L}^{\infty P}$ as a solution

The observation to make here is exactly the same as the one we made when analyzing $\mathcal {L}^P$ or $\mathcal {L}^{2P}$ as solutions. Suppose the modal creativity of social constructivism allows for us to declare and collectively recognize that for absolutely any function $f:\mathbb {N}\longrightarrow \mathcal {T},\, \mathcal {R},f\Vdash Q$. Then, it seems the sentence

demands we adopt yet another temporal perspective from which to analyze the semantics of Cole’s account. We would thus be forced to adopt the language $\mathcal {L}^{(\omega +1)P}$, whose semantic theory concerned valuations $V:\omega +1\longrightarrow \mathcal {T}$. Again, mutatis mutandis, the above arguments can be given, and again Cole will be forced to abandon one of the core elements of his theory or ascend to a yet more complex temporal language in which to state it.