Academic journal article Analysis and Metaphysics

In Defense of Mathematical Inferentialism

Academic journal article Analysis and Metaphysics

In Defense of Mathematical Inferentialism

Article excerpt


Mathematical realism asserts that mathematical objects, such as numbers, triangles, and functions, exist in the abstract world, and that a mathematical sentence is true if and only if the abstract world is as it says it is. This view is defended by such philosophers as Willard V. O. Quine (1948; 1980), Hilary Putnam (1971), Michael Resnik (1997), Mark Colyvan (2001), and Alan Baker (2005; 2009; 2012). In contrast, mathematical fictionalism asserts that mathematical sentences purport to be about the abstract world, but the abstract world does not exist, so they are all false. This view is defended by such philosophers as Mark Balaguer (1996; 1998; 2001; 2009), Gideon Rosen (2001), and Mary Leng (2005a; 2005b; 2010).

This paper defends a new position that I call mathematical inferentialism. The discussion proceeds as follows. In Section 2, I explicate mathematical inferentialism and display its virtues. I then reply to possible objections to this position. Next, I list the disadvantages of mathematical realism in Section 3 and the disadvantages of mathematical fictionalism in Section 4. It will become clear that mathematical inferentialism secures the objectivity of mathematics and explains the successful use of mathematics in empirical science without postulating the existence of the abstract world. (From now on, I drop the qualifier 'mathematical' in front of the name of a philosophical view.)



Instrumentalism in philosophy of science is the view that a scientific theory is not a description of unobservables but an instrument for organizing our thoughts about observables. The instrument is neither true nor false. It is only useful or useless, depending on whether or not it makes predictions that turn out to be true. Andreas Osiander (1498-1552) was an instrumentalist about the Copernican theory (Kuhn, 1957: 187). George Berkeley (1685-1753) was an instrumentalist about Newtonian mechanics (Downing, 2013). The insight of these instrumentalists is that a scientific theory can help us to make inferences about observables, i.e., it exhibits the inferential relationship between observational sentences.

As Pierre Duhem (1905) observes, an observational sentence cannot be derived from a scientific theory alone, and a scientific theory needs to be supplied with auxiliary assumptions to entail an observational consequence. What if all of its observational consequences are true? According to Bas C. van Fraassen (1980), such a theory is empirically adequate. On his account, a scientific theory is a description of both observables and unobservables. So it is true or false, depending on whether it correctly represents them. But the principle of economy dictates that we believe only that it is empirically adequate (van Fraassen, 1985: 294).

In my view, instrumentalism in philosophy of science is potentially applicable to the philosophy of mathematics. A mathematical sentence can perform the function of helping us to make deductive inferences from some concrete sentences to other concrete sentences, sentences that are rendered true or false by the concrete world. Consider the following three concrete sentences:

(i) John gave me an apple.

(ii) Jane gave me an apple.

(iii) Therefore, I had two apples.

Assume that I had no apples to begin with, and that John and Jane were my only sources for apples. The mathematical sentence, '1 + 1=2,' tells us that (iii) necessarily follows from (i) and (ii), thereby facilitating the deductive inference from (i) and (ii) to (iii). To generalize, a mathematical sentence can perform the function of helping us deductively organize our thoughts about the concrete world. This functional thesis about a mathematical sentence is the key constituent of what I call inferentialism.

Inferentialism proposes that a mathematical sentence is concretely adequate if and only if all of its concrete consequences are true, and that it is concretely inadequate if and only if a limited number of concrete consequences are true. …

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