Before we conclude this chapter, let us turn to a possible objection one might make at this point.10 More specifically, one might make an objection against the exam- ple used to pump intuitions for semantic agnosticism. The example concerned the mathematician and the philosopher of mathematics and I claimed, in the example, that the mathematician would not let her work be influenced by the findings of the philosopher on the ontology of numbers. Therefore, I concluded, ontology and mathematics are two orthogonal issues. Here is where one might object and the objection might run along the following lines:
Objection. Brouwer held a certain view of mathematical objects and based his view of mathematics on a particular philosophy of mind (cf.
based on the observable features of language use. Quine goes so far as to reject notions such as ‘meaning’ and ‘intensions’ as “creatures of darkness” (Quine,1956, p. 180). I will not go so far, but it is interesting to see some similarities between Quine’s behaviouristic approach and these recent accounts of semantics.
10
Thanks to Franz Berto for raising this objection and helping me get clear on how to respond to it.
Van Atten 2015). He believed mathematical entities to be activities in the mind or mental constructions. His view of mathematical objects had great consequences for his mathematics and it led to a rejection of (large parts of) classical mathematics, while allowing for a form of constructive mathematics. Thus, the objector may conclude, it is not true that the ontology of mathematical entities does not affect the mathematicians work. Hence, the objector may conclude, the analogy at the beginning of the chapter with semantics, fails.
This is an argument against the analogy between mathematics and semantics in that in mathematics, it turns out, one’s ontologydoes affect one’s semantics.
To avoid this argument and restore the analogy, we have to discuss a subtlety that we have not discussed yet. Namely, the distinction between ‘standard’ mathe- matics and, what we will call, ‘non-standard’ mathematics. Standard mathematics is the textbook mathematics that you will find in the introductory textbooks on the topic and that is taught at high-school or early stages of university. Non-standard mathematics is mathematics that is non-classical (e.g., intuitionistic mathematics). What the objection shows, is that the arguments made in this chapter pertain only to standard mathematics. That is, standard mathematics is compatible with many ontologies and the mathematician can remain agnostic about the nature of mathematical objects. However, when doing non-standard mathematics, the ontol- ogy of mathematical objects might become of importance, as we see in the example of Brouwer.
As Van Atten (2015) notes, “Brouwer was prepared to follow his philosophy of mind to its ultimate conclusions [. . .]. In thus granting philosophy priority over
traditional mathematics, he showed himself a revisionist” (emphases added). It thus does not seem strange to classify his mathematics as non-traditional, or non- standard.11
With this distinction in place, the analogy remains. For standard semantics (cf.
Chierchia & McConnell-Ginet 1990; Heim & Kratzer 1998) the semanticist can re- main agnostic about the nature of (possible) worlds and keep using her world pos- tulates freely. However, also in semantics, there are non-standard semantics (cf.
Lewis 1968, 1980), where ontology and semantics do interact. For example, Lewis
(1968) has a semantics on which names are non-classically rigid,12deviating from the post-Kripkean (1980) picture that most standard semantics adhere to. Furthermore, Yagisawa’s (1988) extended genuine modal realism (which is an extension of Lewis’ account) does affect the semantics in that it allows contradictions at impossible worlds to spill over to contradictions at the actual world.
Chierchia & McConnell-Ginet(1990), who present a standard semantics, put the point very explicitly, when they say the following:13
11One might argue, that Brouwer’s non-standard mathematics arose because of his ontology of
mathematical objects. I do not want take sides on what caused what; what matters here is the distinction between standard and non-standard mathematics.
12
I call it ‘non-classically rigid’, for one might argue that his counterpart theory captures some aspects of rigidity. I mean to indicate that it differs from the rigidity thatKripke(1980) argues for.
Semantic Agnosticism in the Fictionalist Landscape|45
The formal apparatus of possible worlds, [. . .], was introduced in Kripke (1959) as a tool for investigating the semantic properties of certain formal systems. There has since been, and continues to be, much controversy in the philosophical literature over what assumptions that apparatus re- quires. In accepting Lewis’s point we do not deny that possible worlds might raise deep metaphysical issues, but we think that the formal ap- paratus can be adopted without resolving these issues, just as we can successfully use the notion of an individual in set theory and logic with- out resolving all the thorny problems it raises, for example, the mysteries of criteria for identifying individuals (p. 207, emphasis added) Distinguishing between standard and non-standard semantics or mathematics thus restores the analogy. In both cases, the mathematician/semanticist does not need to care about the nature of her objects of study when she works in the standard framework. However, when working in non-standard frameworks, the ontology might influence the semantics (e.g., Brouwer’s intuitionism and Yagisawa’s extended gen- uine modal realism).