Is the following argument sound?Â
Mathematical sentences like â4 is evenâ should be read at face value; that is, they should be read as being of the form âFaâ and, hence, as making straightforward claims about the nature of certain objects; e.g., â4 is evenâ should be read as making a straightforward claim about the nature of the number 4. But
If sentences like â4 is evenâ should be read at face value, and if moreover they are true, then there must actually exist objects of the kinds that they're about; for instance, if â4 is evenâ makes a straightforward claim about the nature of the number 4, and if this sentence is literally true, then there must actually exist such a thing as the number 4. Therefore, from (1) and (2), it follows that
If sentences like â4 is evenâ are true, then there are such things as mathematical objects. But
If there are such things as mathematical objects, then they are abstract objects, i.e., nonspatiotemporal objects; for instance, if there is such a thing as the number 4, then it is an abstract object, not a physical or mental object. But
There are no such things as abstract objects. Therefore, from (4) and (5) by modus tollens, it follows that
There are no such things as mathematical objects. And so, from (3) and (6) by modus tollens, it follows that
Sentences like â4 is evenâ are not true (indeed, they're not true for the reason that fictionalists give, and so it follows that fictionalism is true).
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