**by Axel Arturo Barceló Aspeitia**

Arguments in favor of the existence of mathematical entities like numbers can be divided into two broad kinds: on the one hand, we have those that start from mathematical practice as a given and postulate the existence of mathematical objects as part of the best explanation for mathematical practice being as its, usually, either because of its success as a scientific enterprise or because of its importance for other successful scientific and technological practices. On the other hand, there are those that, instead, take everyday linguistic practice as starting point and postulate the existence of mathematical objects as part of the best explanation of our usual linguistic practices being as they are. The overall general strategy in both cases is to argue that without numbers, it would be very hard to explain why things that we accept to be true – or at least, to be successful as claims about the world, like simple arithmetical truths like seventeen being prime, complex physical laws like the superposition principle or just everyday assertions like there being twelve judges in the Supreme Court – are actually true. The basic idea is that something cannot be true unless the things it is *about* actually exist. Thus, if we know what something is true and that it is about some category of things, then we have good reasons to conclude that such things do exist. Bob Halle, for example, has argued:

If entities belonging to a certain ontological category just are what expressions of a certain logical category stand for, then we can argue for the existence of entities of that kind by arguing that there are true statements involving expressions of the relevant kind. If, for example, there are true statements incorporating expressions functioning as singular terms, then there are objects of some corresponding kind. If the singular terms are such that, if they have reference at all, they refer to numbers, there are numbers.

(Halle 2010, 406; quoted by Thomasson 2014, 133)

This line of reasoning brings forth the importance of linguistic analysis for the ontological enterprise of determining what sort of things conform our reality and, in particular, whether numbers do indeed exist. If Halle’s argument carries any force, the question of whether there are (literally) true sentences where numerals function as singular terms would have enormous ontological importance!

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