I see no reason why the semantic uniformity property should hold for mathematics. Having sentences of similar form certainly does not require that they have the same semantics even for sentences that have nothing to do with mathematics. English has polysemous words, and the meaning of “exists” in reference to mathematical concepts is different from the meaning of “exists” in reference to physical entities.
If this was for some reason a big problem, then an obvious solution would be to use different words or sentence structure for mathematics. To some degree this is already the usual convention: Mathematics has its own notations, which are translated loosely into English and other natural languages. There are many examples of students taking these vaguer translations overly literally to their own detriment.
This is definitely not a “big problem” in that we can use math regardless of what the outcome is.
It sounds like you’re arguing that semantic uniformity doesn’t matter, because we can change what “exists” means. But once you change what “exists” means, you will likely run into epistemological issues. If your mathematical objects aren’t physical entities capable of interacting with the world, how can you have knowledge that is causally related to those entities? That’s the dilemma of the argument above—it seems possible to get semantic uniformity at the expense of epistemological uniformity, or vice versa, but having both together is difficult.
Nobody has to believe that ordinary non mathematical langage contains a single well defined meaning of “exists”. And if “exists” is polysemous , then one of its meanings could be the meaning of mathematically-exists … it doesn’t have to have a unique meaning. Fictivism is an example: the theory that mathematically-exists means fictionally-exists, since ordinary language allows truth and existence to be used in reference to fictional worlds.
To provide a more concrete example, I would say that the claim “There are at least three Jedi older than Anakin Skywalker” is, in the interpretation of the sentence almost anyone who understands it would use, a true statement, even though Jedi certainly do not exist in the same sense that New York City exists. I would not say that the fact that Jedi and New York exist in very different ways really violates Semantic Uniformity in any substantial way.
I’m not super up-to-date on fictionalism, but I think I have a reasonable response to this.
When we are talking about fictional worlds, we understand that we have entered a new form of reasoning. In cases of fictional worlds, all parties usually understand that we are not talking about the standard predicate, “exists”, we are talking about some other predicate, “fictionally-exists”. You can detect this because if you ask people “do those three Jedi really exist?”, they will probably say no.
However, with math, it’s less clear that we are talking fictionally or talking only about propositions within an axiomatic system. We could swap out the “exists” predicate with something like “mathematically-exists” (within some specific axiom system), but it’s less clear what the motivation is compared to fictional cases. People talk as if 2+2 does really equal 4, not just that its useful to pretend that it’s true.
The main difference between mathematics and most other works of fiction is that mathematics is based on what you can derive when you follow certain sets of rules. The sets of rules are in principle just as arbitrary as any artistic creation, but some are very much more interesting in their own right or useful in the real world than others.
As I see it, the sense in which 2+2 “really” equals 4 is that we agree on a foundational set of definitions and rules taught at a very young age in today’s cultures, following those rules leads to that result, and that such rules have been incredibly useful for thousands of years in nearly every known culture.
There are “mathematical truths” that don’t share this history and aren’t talked about in the same way.
I see no reason why the semantic uniformity property should hold for mathematics. Having sentences of similar form certainly does not require that they have the same semantics even for sentences that have nothing to do with mathematics. English has polysemous words, and the meaning of “exists” in reference to mathematical concepts is different from the meaning of “exists” in reference to physical entities.
If this was for some reason a big problem, then an obvious solution would be to use different words or sentence structure for mathematics. To some degree this is already the usual convention: Mathematics has its own notations, which are translated loosely into English and other natural languages. There are many examples of students taking these vaguer translations overly literally to their own detriment.
This is definitely not a “big problem” in that we can use math regardless of what the outcome is.
It sounds like you’re arguing that semantic uniformity doesn’t matter, because we can change what “exists” means. But once you change what “exists” means, you will likely run into epistemological issues. If your mathematical objects aren’t physical entities capable of interacting with the world, how can you have knowledge that is causally related to those entities? That’s the dilemma of the argument above—it seems possible to get semantic uniformity at the expense of epistemological uniformity, or vice versa, but having both together is difficult.
Nobody has to believe that ordinary non mathematical langage contains a single well defined meaning of “exists”. And if “exists” is polysemous , then one of its meanings could be the meaning of mathematically-exists … it doesn’t have to have a unique meaning. Fictivism is an example: the theory that mathematically-exists means fictionally-exists, since ordinary language allows truth and existence to be used in reference to fictional worlds.
To provide a more concrete example, I would say that the claim “There are at least three Jedi older than Anakin Skywalker” is, in the interpretation of the sentence almost anyone who understands it would use, a true statement, even though Jedi certainly do not exist in the same sense that New York City exists. I would not say that the fact that Jedi and New York exist in very different ways really violates Semantic Uniformity in any substantial way.
I’m not super up-to-date on fictionalism, but I think I have a reasonable response to this.
When we are talking about fictional worlds, we understand that we have entered a new form of reasoning. In cases of fictional worlds, all parties usually understand that we are not talking about the standard predicate, “exists”, we are talking about some other predicate, “fictionally-exists”. You can detect this because if you ask people “do those three Jedi really exist?”, they will probably say no.
However, with math, it’s less clear that we are talking fictionally or talking only about propositions within an axiomatic system. We could swap out the “exists” predicate with something like “mathematically-exists” (within some specific axiom system), but it’s less clear what the motivation is compared to fictional cases. People talk as if 2+2 does really equal 4, not just that its useful to pretend that it’s true.
The main difference between mathematics and most other works of fiction is that mathematics is based on what you can derive when you follow certain sets of rules. The sets of rules are in principle just as arbitrary as any artistic creation, but some are very much more interesting in their own right or useful in the real world than others.
As I see it, the sense in which 2+2 “really” equals 4 is that we agree on a foundational set of definitions and rules taught at a very young age in today’s cultures, following those rules leads to that result, and that such rules have been incredibly useful for thousands of years in nearly every known culture.
There are “mathematical truths” that don’t share this history and aren’t talked about in the same way.
The motivation to me seems exactly the same as with fiction: we’re talking about things other than physical objects or whatever.
Talking about mathematics, qua fictions, cant possibly have less motivation than talking about fictions qua fictions.
People also tend to regard their own tribal myths as really true, as well.