When you say “in a world where there weren’t quarks” I have no idea what you’re talking about. It appears to be some kind of possible world where the laws of physics are different. But now we’re making statements of fact about abstract objects.
No, when I say ‘in a world where there weren’t quarks’ I mean in an imagined scenario in which quarks are imagined not to occur. I’m not committed to real non-actual worlds. (If possible worlds were abstract, then they’d have no causal relation to my thoughts about them, so I’d have no reason to think my thoughts about modality were at all on the right track. It’s because modality is epistemic and cognitive and ‘in the head’ that I can reason about hypothetical and counterfactual situations productively.) I’m a modal fictionalist, and a mathematical fictionalist.
In imagined scenarios where we sever the causal links between agents and quarks, e.g., by replacing quarks with some other mechanism that can produce reasoning agents, it seems less likely that the agents would have hypothesized quarks. When we remove abstract numbers from a hypothetical scenario, on the other hand, nothing about the physical world seems to be affected (since, inasmuch as they are causally inert, abstract numbers are in no way responsible for the way our world is).
That suggests that positing numbers is wholly unexplanatory. It might happen to be the case that there are such things, but it can’t do anything to account for the unreasonable effectiveness of mathematics, because of the lack of any causal link.
Abstract objects play a similar role in current physical theories to that which luminiferous aether used to play. The problem with aether isn’t just that it was theoretically dispensable; it was that, even if we weren’t smart enough to figure out how to reformulate our theories without assuming aether, it would still be obvious that the theoretical successes that actually motivated us to form such theories would have arisen in exactly the same way even if there were no aether. Aether doesn’t predict aether-theories like ours, because our aether theory is not based on empirical evidence of aether.
(Aether might still be reasonable to believe in, but only if it deserves a very high prior, such that the lack of direct empirical confirmation is OK. But you haven’t argued for platonism based on high priors, e.g., via a Tegmark hypothesis; you’ve argued for it empirically, based on the real-world successes of mathematicians. That doesn’t work, unless you add some kind of link between the successes and the things you’re positing to explain those successes.)
Modern-day platonists try to make their posits appear ‘metaphysically innocent’ by depriving them of causal roles, but in the process they do away with the only features that could have given us positive reasons to believe such things. It would be like if someone objected to string theory because it’s speculative and lacks evidence, and string theorists responded by replacing strings with non-spatiotemporal, causally inert structures that happen to resemble the physical world’s structures. The whole point of positing strings is that they be causally or constitutively linked to our beliefs about strings, so that the success of our string theory won’t just be a coincidence; likewise, the whole point of reifying mathematical objects should be to treat them as causally or constitutively responsible for the success of mathematics. Without that responsibility, the posit is unmotivated.
math appears likely to work the same way in all possible worlds.
What do you mean by “work the same way”? I can pretty easily imagines world where mathematicians consistently fail to get reliable results. There may even be actual planets like that in the physical universe, if genetic drift eroded the mathematical reasoning capabilities of some species, or if there are aliens who rely heavily on math but don’t relate it to empirical reality in sensible ways. If such occurrences don’t falsify platonism, then our own mathematicians’ remarkable successes don’t verify platonism. So what phenomenon is it that you’re really claiming we need platonism to explain? What kind of ‘unreasonable effectiveness’ is relevant?
When we remove abstract numbers from a hypothetical scenario, on the other hand, nothing about the physical world seems to be affected (since, inasmuch as they are causally inert, abstract numbers are in no way responsible for the way our world is).
I can come up with possible worlds without quarks (in a vague, non-specific way). I have no idea what it means to “remove abstract numbers from a hypothetical scenario”. I don’t think abstract objects have modal variation which is closely related to their (not) being causal. But in so far as mathematics posits abstract entities and mathematics is explanatory than I don’t think there is anything mysterious about the sense in which abstract objects are explanatory.
Abstract objects play a similar role in current physical theories to that which luminiferous aether used to play. The problem with aether isn’t just that it was theoretically dispensable; it was that, even if we weren’t smart enough to figure out how to reformulate our theories without assuming aether, it would still be obvious that the theoretical successes that actually motivated us to form such theories would have arisen in exactly the same way even if there were no aether. Aether doesn’t predict aether-theories like ours, because our aether theory is not based on empirical evidence of aether.
I disagree. I think the problem with aether is entirely just that it was theoretically dispensable. And I think the sentences that follow that are just a way of saying “aether was theoretically dispensable”.
Modern-day platonists try to make their posits appear ‘metaphysically innocent’ by depriving them of causal roles, but in the process they do away with the only features that could have given us positive reasons to believe such things.
Their utility in our explanations is sufficient reason to believe they exist even if their role in those explanations is not causal. Your string theory comparison doesn’t sound like a successful scientific theory.
What do you mean by “work the same way”?
As in we can’t develop models of possible worlds in which mathematics works differently. This has nothing to do with the abilities of hypothetical mathematicians.
As in we can’t develop models of possible worlds in which mathematics works differently.
Or we can’t develop models of mathematically possible worlds where maths works differently. Or maybe we can, since we can image the AoC being either true or false Actually, it is easier for realists to imagine maths being different in different possible worlds, since, for realists, the existence of numbers makes an epistemic difference. For them, some maths that is formally valid (deducable from axioms) might be transcendentally incorrect (eg, the AoC was assumed but is actually false in Plato’s Heaven).
No, when I say ‘in a world where there weren’t quarks’ I mean in an imagined scenario in which quarks are imagined not to occur. I’m not committed to real non-actual worlds. (If possible worlds were abstract, then they’d have no causal relation to my thoughts about them, so I’d have no reason to think my thoughts about modality were at all on the right track. It’s because modality is epistemic and cognitive and ‘in the head’ that I can reason about hypothetical and counterfactual situations productively.) I’m a modal fictionalist, and a mathematical fictionalist.
In imagined scenarios where we sever the causal links between agents and quarks, e.g., by replacing quarks with some other mechanism that can produce reasoning agents, it seems less likely that the agents would have hypothesized quarks. When we remove abstract numbers from a hypothetical scenario, on the other hand, nothing about the physical world seems to be affected (since, inasmuch as they are causally inert, abstract numbers are in no way responsible for the way our world is).
That suggests that positing numbers is wholly unexplanatory. It might happen to be the case that there are such things, but it can’t do anything to account for the unreasonable effectiveness of mathematics, because of the lack of any causal link.
Abstract objects play a similar role in current physical theories to that which luminiferous aether used to play. The problem with aether isn’t just that it was theoretically dispensable; it was that, even if we weren’t smart enough to figure out how to reformulate our theories without assuming aether, it would still be obvious that the theoretical successes that actually motivated us to form such theories would have arisen in exactly the same way even if there were no aether. Aether doesn’t predict aether-theories like ours, because our aether theory is not based on empirical evidence of aether.
(Aether might still be reasonable to believe in, but only if it deserves a very high prior, such that the lack of direct empirical confirmation is OK. But you haven’t argued for platonism based on high priors, e.g., via a Tegmark hypothesis; you’ve argued for it empirically, based on the real-world successes of mathematicians. That doesn’t work, unless you add some kind of link between the successes and the things you’re positing to explain those successes.)
Modern-day platonists try to make their posits appear ‘metaphysically innocent’ by depriving them of causal roles, but in the process they do away with the only features that could have given us positive reasons to believe such things. It would be like if someone objected to string theory because it’s speculative and lacks evidence, and string theorists responded by replacing strings with non-spatiotemporal, causally inert structures that happen to resemble the physical world’s structures. The whole point of positing strings is that they be causally or constitutively linked to our beliefs about strings, so that the success of our string theory won’t just be a coincidence; likewise, the whole point of reifying mathematical objects should be to treat them as causally or constitutively responsible for the success of mathematics. Without that responsibility, the posit is unmotivated.
What do you mean by “work the same way”? I can pretty easily imagines world where mathematicians consistently fail to get reliable results. There may even be actual planets like that in the physical universe, if genetic drift eroded the mathematical reasoning capabilities of some species, or if there are aliens who rely heavily on math but don’t relate it to empirical reality in sensible ways. If such occurrences don’t falsify platonism, then our own mathematicians’ remarkable successes don’t verify platonism. So what phenomenon is it that you’re really claiming we need platonism to explain? What kind of ‘unreasonable effectiveness’ is relevant?
I can come up with possible worlds without quarks (in a vague, non-specific way). I have no idea what it means to “remove abstract numbers from a hypothetical scenario”. I don’t think abstract objects have modal variation which is closely related to their (not) being causal. But in so far as mathematics posits abstract entities and mathematics is explanatory than I don’t think there is anything mysterious about the sense in which abstract objects are explanatory.
I disagree. I think the problem with aether is entirely just that it was theoretically dispensable. And I think the sentences that follow that are just a way of saying “aether was theoretically dispensable”.
Their utility in our explanations is sufficient reason to believe they exist even if their role in those explanations is not causal. Your string theory comparison doesn’t sound like a successful scientific theory.
As in we can’t develop models of possible worlds in which mathematics works differently. This has nothing to do with the abilities of hypothetical mathematicians.
Or we can’t develop models of mathematically possible worlds where maths works differently. Or maybe we can, since we can image the AoC being either true or false Actually, it is easier for realists to imagine maths being different in different possible worlds, since, for realists, the existence of numbers makes an epistemic difference. For them, some maths that is formally valid (deducable from axioms) might be transcendentally incorrect (eg, the AoC was assumed but is actually false in Plato’s Heaven).