Sentence examples for mathematical utterances from inspiring English sources

These three views make controversial claims about how the language of mathematics should be interpreted, and Platonists rebut their claims by carefully examining what people actually mean when they make mathematical utterances.

This standpoint, however, threatens to collapse into structuralism, into viewing mathematical utterances as schemata implicitly generalising over a range of (in general) abstract structures which satisfy the schemata.

And Yablo has made some relatively strong claims about an analogy that he thinks holds between mathematical utterances and metaphorical utterances, or figurative utterances.

One might well think that the game formalist should treat mathematical utterances, on that view just strings of meaningless marks, as unsinnig, not just sinnlos.

Thus, Yablo's particular version of fictionalism is open to objections to the effect that mathematical utterances are in fact not similar or analogous to metaphorical utterances.

Given this, and given (what seems obvious) that ordinary people don't have positive intentions for their mathematical utterances to be interpreted nonliterally e.g., as expressing conditional propositions it seems to follow that we ought to interpret our mathematical utterances at face value.

Similarly, given use fictionalism all that is needed properly to understand what is communicated in ordinary utterances of mathematical sentences is an ordinary compositional semantics together with principles linking the literal contents of mathematical sentences to what the fictionalist says the utterances ordinarily convey.

If paraphrase nominalists admit that platonists and fictionalists are right about the meanings of real mathematical utterances i.e., the utterances of actual mathematicians then (since they also want to maintain that there are no such things as abstract objects) they will be committed to the claim that the utterances of actual mathematicians are untrue.

For since paraphrase nominalists don't believe in the existence of mathematical objects, if they admit that ordinary utterances of '3 is prime' are best interpreted as being about mathematical objects, or purporting to be about such objects, then they will have to admit that such sentences are literally untrue, as fictionalists maintain.

Yablo thinks that something like this is true in connection with typical utterances of (pure and mixed) mathematical sentences, e.g., sentences like '3 is prime' and 'The number of Martian moons is 2.' So Yablo is certainly proposing a hermeneutic nominalist view, but it's not clear that his view is best thought of as a kind of hermeneutic fictionalism.

But what makes it awkward to take Yablo's view as a version of fictionalism is that he seems to think that what (pure and mixed) mathematical sentences really say or, more precisely, what typical utterances of these sentences really say is true and nominalistic in content.