If in axiomatizing arithmetic we are ontologically committed to saying that 1 exists, 2 exists, 3 exists,etc., then we may say that there are numbers even if it is not axiomatic that 1, 2, 3, etc. are causally inert, nonphysical, etc.
Instead of being a platonist and treating numbers as abstract, you could treat them as occupying spacetime (like immanent universals or tropes), you could treat them as non-spatiotemporal but causally efficacious (like the actual Forms of Plato), or you could assert both. (You could also treat them as useful fictions, but I’ll assume that fictionalism is an error theory of mathematics.)
I think many of the views on which mathematical objects have some causal (or, if you prefer, ‘difference-making’) effect on our mathematical discourse are reasonable. The views on which it’s just a coincidence are not reasonable, and I don’t think abstract numbers can easily escape the ‘just a coincidence’ concern (unless, perhaps, accompanied by a larger Tegmark-style framework).
I don’t recognize a difference between universals and abstract objects but neither plays a causal role in the make up of the universe.
Let’s take the property ‘electrically charged’ as an example. If charge is a universal, then it’s something wholly and constitutively shared in common between every charged thing; universals occur exactly in the spatiotemporal locations where their instances are, and they are exhausted by these worldly things. So there’s no need to posit anything outside our universe to believe in universals. Redness is, as it were, ‘in’ every red rose. Generally, universals are assumed to play causal roles (it’s because roses instantiate redness that I respond to them as I do), though in principle you could posit a causally inert one. (Such a universal still wouldn’t be abstract, because it would still occur in our universe.)
If electric charge is instead an abstract object, then it exists outside space and time, and has no effect at all on the electrically charged things in our world. (So abstract electric charge serves absolutely no explanatory role in trying to understand how things in our world are charged. However, it might be a useful posit for the nominalist about universals, just to provide a (non-nominalistic) correlate for our talk in terms of abstract nouns like ‘charge’.
A third option would be to treat electric charge as a Platonic Form, i.e., something outside spacetime but causally responsible for the distribution of charge instances in our universe. (This is confusing, because Platonic Forms aren’t ‘platonic’ in the sense in which mathematical platonism are ‘platonic’. Plato himself was a nominalist about abstract objects, and also a nominalist about universals. His Forms are a totally different thing from the sorts of posits philosophers these days generally entertain.)
A natural way to think of bona-fide ancient Platonism (as opposed to the lowercase-p ‘platonism’ of modern mathematicians) is as cellular automata; for Plato, our universe is an illusion-like epiphenomenon arising from much simpler, lower-level relationships that are not temporal. (Space still plays a role, but as an empty geometry that comes to bear properties only in a derivative way, via its relationships to particular Forms.)
You’re taking metaphors way too literally. There is no “Realm”.
Hm? How do you know I’m taking it too literally? First, how do you know that ‘Realm’ isn’t just part of the metaphor for me? What signals to you when I stop talking about ‘objects’ and start talking about ‘Realms’ that I’ve crossed some line? (Knowing this might help tell me about which parts of your talk you take seriously, and which you don’t.)
Second, as long as we don’t interpret ‘Realm’ spatially, what’s wrong with speaking of a Realm of abstract objects, literally? Physical things occur in spacetime; abstract things exist just as physical ones do, but outside spacetime. Perhaps they occupy their own non-spatial structure, or perhaps they can’t be said to ‘occupy’ anything at all. Either way, we’ve complicated our ontology quite a bit.
If in axiomatizing arithmetic we are ontologically committed to saying that 1 exists, 2 exists, 3 exists,etc., then we may say that there are numbers even if it is not axiomatic that 1, 2, 3, etc. are causally inert, nonphysical, etc.
I’m still lost here.
Instead of being a platonist and treating numbers as abstract, you could treat them as occupying spacetime (like immanent universals or tropes), you could treat them as non-spatiotemporal but causally efficacious (like the actual Forms of Plato), or you could assert both. (You could also treat them as useful fictions, but I’ll assume that fictionalism is an error theory of mathematics.)
I’m not sure I would say Plato’s forms are causally efficacious in the way we understand that concept—but that isn’t really important. Any way, I have issues with the various alternatives to modern Platonism, immanent realism, trope theory etc. -- though not the time to go into each one. If I were to make a general criticism I would say all involve different varieties of torturous philosophizing and the invention of new concepts to solve different problems. Platonism is easier and doesn’t cost me anything.
I think many of the views on which mathematical objects have some causal (or, if you prefer, ‘difference-making’) effect on our mathematical discourse are reasonable. The views on which it’s just a coincidence are not reasonable, and I don’t think abstract numbers can easily escape the ‘just a coincidence’ concern (unless, perhaps, accompanied by a larger Tegmark-style framework).
Ah! This seems like a point of traction. I certainly don’t think there is anything coincidental about the fact that mathematical truths tell us things about physical truths. I just don’t think the relationship is causal. I believe causal facts are facts about possible interventions on variables. Since there is no sense in which we can imagine intervening on mathematical objects I don’t see how that relationship can be causal. But that doesn’t mean it is a coincidence or isn’t sense making. I Mathematics is effective because everything in the natural world is an instantiation of an abstract object. Instantiations have the properties of the abstract object they’re instantiating. This kind of information can be used in a straightforward, explanatory way.
universals occur exactly in the spatiotemporal locations where their instances are, and they are exhausted by these worldly things.
This is a particular way of understanding universals. You need to specify immanent realism. Plenty of philosophers believe in universals as abstract objects.
If in axiomatizing arithmetic we are ontologically committed to saying that 1 exists, 2 exists, 3 exists,etc., then we may say that there are numbers even if it is not axiomatic that 1, 2, 3, etc. are causally inert, nonphysical, etc.
Instead of being a platonist and treating numbers as abstract, you could treat them as occupying spacetime (like immanent universals or tropes), you could treat them as non-spatiotemporal but causally efficacious (like the actual Forms of Plato), or you could assert both. (You could also treat them as useful fictions, but I’ll assume that fictionalism is an error theory of mathematics.)
I think many of the views on which mathematical objects have some causal (or, if you prefer, ‘difference-making’) effect on our mathematical discourse are reasonable. The views on which it’s just a coincidence are not reasonable, and I don’t think abstract numbers can easily escape the ‘just a coincidence’ concern (unless, perhaps, accompanied by a larger Tegmark-style framework).
Let’s take the property ‘electrically charged’ as an example. If charge is a universal, then it’s something wholly and constitutively shared in common between every charged thing; universals occur exactly in the spatiotemporal locations where their instances are, and they are exhausted by these worldly things. So there’s no need to posit anything outside our universe to believe in universals. Redness is, as it were, ‘in’ every red rose. Generally, universals are assumed to play causal roles (it’s because roses instantiate redness that I respond to them as I do), though in principle you could posit a causally inert one. (Such a universal still wouldn’t be abstract, because it would still occur in our universe.)
If electric charge is instead an abstract object, then it exists outside space and time, and has no effect at all on the electrically charged things in our world. (So abstract electric charge serves absolutely no explanatory role in trying to understand how things in our world are charged. However, it might be a useful posit for the nominalist about universals, just to provide a (non-nominalistic) correlate for our talk in terms of abstract nouns like ‘charge’.
A third option would be to treat electric charge as a Platonic Form, i.e., something outside spacetime but causally responsible for the distribution of charge instances in our universe. (This is confusing, because Platonic Forms aren’t ‘platonic’ in the sense in which mathematical platonism are ‘platonic’. Plato himself was a nominalist about abstract objects, and also a nominalist about universals. His Forms are a totally different thing from the sorts of posits philosophers these days generally entertain.)
A natural way to think of bona-fide ancient Platonism (as opposed to the lowercase-p ‘platonism’ of modern mathematicians) is as cellular automata; for Plato, our universe is an illusion-like epiphenomenon arising from much simpler, lower-level relationships that are not temporal. (Space still plays a role, but as an empty geometry that comes to bear properties only in a derivative way, via its relationships to particular Forms.)
Hm? How do you know I’m taking it too literally? First, how do you know that ‘Realm’ isn’t just part of the metaphor for me? What signals to you when I stop talking about ‘objects’ and start talking about ‘Realms’ that I’ve crossed some line? (Knowing this might help tell me about which parts of your talk you take seriously, and which you don’t.)
Second, as long as we don’t interpret ‘Realm’ spatially, what’s wrong with speaking of a Realm of abstract objects, literally? Physical things occur in spacetime; abstract things exist just as physical ones do, but outside spacetime. Perhaps they occupy their own non-spatial structure, or perhaps they can’t be said to ‘occupy’ anything at all. Either way, we’ve complicated our ontology quite a bit.
I’m still lost here.
I’m not sure I would say Plato’s forms are causally efficacious in the way we understand that concept—but that isn’t really important. Any way, I have issues with the various alternatives to modern Platonism, immanent realism, trope theory etc. -- though not the time to go into each one. If I were to make a general criticism I would say all involve different varieties of torturous philosophizing and the invention of new concepts to solve different problems. Platonism is easier and doesn’t cost me anything.
Ah! This seems like a point of traction. I certainly don’t think there is anything coincidental about the fact that mathematical truths tell us things about physical truths. I just don’t think the relationship is causal. I believe causal facts are facts about possible interventions on variables. Since there is no sense in which we can imagine intervening on mathematical objects I don’t see how that relationship can be causal. But that doesn’t mean it is a coincidence or isn’t sense making. I Mathematics is effective because everything in the natural world is an instantiation of an abstract object. Instantiations have the properties of the abstract object they’re instantiating. This kind of information can be used in a straightforward, explanatory way.
This is a particular way of understanding universals. You need to specify immanent realism. Plenty of philosophers believe in universals as abstract objects.