= 27d567bc2d48d9f40e9ccc20ea370b15)
Jack comments on What do professional philosophers believe, and why? - Less Wrong Discussion
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (249)
Quinean reasons. Tegmark's position, as far as I can tell, is that all abstract objects are also physically instantiated (or that the only difference between concrete and abstract objects is indexical). Which I think is plausible-- but I think abstract objects could be an entirely different sort of thing from concrete, physically existing objects, and still exist.
Do you think abstract objects have anything causally to do with the things (about our universe, or about mathematical practice) that convinced you they exist? My worry is that in the absence of a causal connection, if there weren't such abstract objects, mathematics would be just as 'unreasonably effective'. The numbers aren't doing anything to us to make mathematics work, so their absence wouldn't deprive us of anything (causally). If a hypothesis can't predict the data any more reliably than its negation can, then the data can't be used to support the hypothesis.
In general, I'd like to hear more talk about what sorts of relations these number things enter into with our own world.
No. But that is essentially true by definition. On the other hand, I think all causal claims are claims about abstract facts. E.g. when you say "The match caused the barn to burn to the ground" you're invoking a causal model of the world and models of the world are abstractions (though obviously they can be represented).
To me this is like hearing "If mass and velocity didn't exist Newtonian physics would be just as 'unreasonably effected'. Mathematical objects are part of mathematics. The fact that math is unreasonably effective is why we can say mathematical facts are true and mathematical entities exist. Just like the fact that quantum theory is unreasonably effective is the reason we can say that quarks exist. This is true of just everyday objects too. We say your chair exists because the chair is the best way of explaining some of your sensory impressions. It just happens that not all entities are particulars embedded in the causal world.
Causal claim may be expressed with abstract models, but that does not mean they are about abstract models. Causal models do not refer to themselves, in which case they would be about the abstract, they refer to whatever real-world thing they refer to.
Maths isn't unreasonably effective at understanding the world in the sense that any given mathematical truth is automatically also a physical truth. If one mathematical statement (eg an inverse square law of gravity) is physically true, and infinity of others (inverse cube law, inverse power of four...) is automatically false. So when we reify out best theories, we are reifying a small part of maths for reasons which aren't purely mathematical. There is not path from the effectiveness of some maths at describing the physical universe to the reification of all maths, because physical truth is a selection of the physically applicable parts of maths.
Sure, but it's not true by definition that numbers are abstract. Given your analogy to mass and velocity, and your view that mathematical objects help explain the unreasonable effectiveness of mathematics, it seems to me that it would make much more sense to treat these number things as playing a causal or constitutive role in the makeup of our universe itself, e.g., as universals. Then it would no longer just be a coincidence that our world conveniently accompanies a causally dislocated Realm of correlates for our mathematical discourse.
But it makes a difference to how our world is that objects have velocity and mass. By hypothesis, it doesn't make a difference to how our world is that there are numbers. (And from this it follows that it wouldn't make a difference if there weren't numbers.) If numbers do play a role as worldly 'difference-makers' of some special sort, then could you explain more clearly what that role is, since it's not causal?
I don't know what that means. If by 'mathematics' you have in mind a set of human behaviors or mental states, then mathematics isn't abstract, so its objects are neither causally nor constitutively in any relation to it. On the other hand, if by 'mathematics' you have in mind another abstract object, then your statement may be true, but I don't see the explanatory relevance to mathematical practice.
Sure, but it's also why we can assert doctrines like mathematical fictionalism and nominalism. A condition for saying anything at all is that our world exhibit the basic features (property repetition, spatiotemporal structure...) that suffice for there to be worldly quantities at all. I can make sense of the idea that we need to posit something number-like to account in some causality-like way for things like property repetition and spatiotemporal structure themselves. But I still haven't wrapped my head around why assuming numbers are not difference-makers for the physical world (unlike the presence of e.g. velocity), we should posit them to explain the efficacy of theories whose efficacy they have no impact upon.
The properties of quarks causally impact our quantum theorizing. In a world where there weren't quarks, we'd be less likely to have the evidence for them that we do. If that isn't true of mathematics (or, in some ways even worse, if we can't even coherently talk about 'mathless worlds'), then I don't see the parity.
Huh?
I don't recognize a difference between universals and abstract objects but neither plays a causal role in the make up of the universe.
You're taking metaphors way too literally. There is no "Realm".
It's not that complicated. We have successful theories that posit certain entities. I think believing in those theories requires believing in those entities. Some of those entities figure causally and spatio-temporally in our theories. Some don't. 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. It is very difficult to say this about mathematics since math appears likely to work the same way in all possible worlds. But that's a really strange reason to conclude mathematical objects don't exist. Numbers and quarks are both theoretically posited entities that we need to explain our world.
As far as I can tell everything you have said is just different forms of "but mathematical objects aren't causal!". I readily agree with this but since abstract objects aren't causal by definition and the entire question is about abstract objects it seems like you're begging the question.
(Edit: Not my downvote btw)
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.
We think the ones that don't figure causally or spatio-temporally aren't actually being posited at all. That's how you read physics. If you know how to read a map, you know that rivers and mountains on the map are suposed to be in the territory, but lines of lattitude and contour lines aren't.
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).
It's logically possible..like so many things.
Either these non physical things interact with matter (eg the brains of mathematicians) or they don't. If they do, that is supernaturalism. If they don't, they succumb to Occam's razor.
No. They don't. Stating scientific theories without abstract objects makes theories vastly more complicated when they can even be stated at all.
I didn't say delete numbers from theories. I mean't don't reify them. There is stuff in theories that you are supposed not to reify, like centres of gravity.
Centers of gravity are an even better example of a real abstract object. I'm definitely not reifying anything according to the dictionary definition of that word: neither numbers nor centers of gravity are at all concrete. They're abstract.
OK. So, in what sense do these "still exist", and in what sense are they "entirely different" from concrete objects? And are common-or-garden numbers included?
I think it might be best if you read the above-linked SEP article and some of the related pieces. But the short form.
C: We should believe in the existence of the abstract objects in our best scientific theories.
One and two seem uncontroversial. 3 can certainly be quibbled with and I spent a few years as a nominalist trying to think of ways to paraphrase out or find reasons to ignore the abstract objects among science's ontic commitments. Lots of people have done this and have occasionally demonstrated a bit of success. A guy named Hartry Field wrote a pretty cool book in which he axiomatizes Newtonian mechanics without reference to numbers or functions. But he was still incredibly far away from getting rid of abstract objects all together (lots of second order logic) and the resulting theory is totally unwieldy. At some point, personally, I just stopped seeing any reason to deny the existence of abstract objects. Letting them exists costs me nothing. It doesn't lead to false beliefs and requires far less philosophizing.
The concrete-abstract distinction still gets debated but a good first approximation is that concrete objects can be part of causal chains and are spatio-temporal while abstract objects are not. As for common-or-garden numbers: I see no reason to exclude them.
Quine has a logician's take on physics -- he assumes that the formal expression of a physical law is complete itself, and therefore seeks a purely formal criterion of ontological commitment, or objecthood. However, physics doesn't work like that. Physical formalisms have semantic implications that aren;t contained in the formalism itself: for instance, f=ma is mathematically identical to p=qr or a=bc, or whatever. But The f, the m and the a all have their own meaning, their own relation to measurement, as a far as a physicist is concerned.
The reasons are already part of the theory..in the sense that the theory is more than the written formalism Physics students are taught that centers of gravity should not be reified --that is part of the theory. No physcs student is taught that any pure number is a reifiable object, and few hit upon the idea themselves.
No philosophizing is required to get rid of abstract objects, one only needs to follow the instructions about what is refiable that are already part of the informal part of a theory.
I can't see how you can claim that Platonism doesn't lead to false beliefs without implicitly claiming omniscience. If abstract entities do not exist, then belief in them is false, by a straightforward correspondence theory. Moreover, is Platoism is true, then some common fomlations of physicalism, such as "everything that exists,, exists spatio-temporally" is false. Perhaps you meant Platonism doesn;t lead to false beliefs with any practical upshot, but violations of Occam's razor generally don't.
OK, but that means that centres-of-gravity aren;t abstract:: the center of gravity of the Earth has a location. That doesn't mean they are fully concete either. Jerrold Katz puts them into a third category, that of the mixed concrete-and-abstract. (His favoured example is the equator).
If you are going to include centers of gravity, and Katz's categorisation is correct, then there is still no reason to include fully abstract entities. And there is a reason to exclude centers of gravity, which is the informal semantics of physics.
There's that word again. I'm not reifying numbers. Abstract objects aren't "things". They aren't concrete. Platonists don't want to reify centers of gravity or numbers.
Platonism and nominalism don't differ in anticipations of future sensory experiences. The difference is entirely about theory and methodology. I've already replied to the Occam's razor thing: our theories that include abstract objects are radically simpler and easier to use than the attempts that do not exclude abstract objects.
I'm not sure they have a location in the same way that is generally meant by spatio-temporal: but the exact classification of centers of gravity isn't that important to me. I'm not claiming to have the details of that figured out.
There has to be some content to Platonism. You seem to be assuming that by "reifying" I must mean "treat as concretely existent". In context, what I mean is "treat as being existent in whatever sense Platonists think abstracta are existent". I am not sure what that is, but there has to be something to it, or there is no content to Platonism, and in any case it is not my job to explain it.
I am not sure what you mean by that. The difference is about ontology. If two theories make the same predictions, and one of them has more entities, one of them is multiplying entities unnecessarily.
And I have replied to the reply. The Quinean approach incorrectly takes a scientific theory to be a formalism. It is only methodologicaly simpler to reify whatever is quantified over, formally, but that approach is too simple because it leaves out the semantics of physics--it doensn't distinguish between f=ma and p=qr.
Such details are what could bring Platonism down.
Doesn't this imply that equivalent scientific theories may have quite different implications wrt. what abstract objects exist, depending on how exactly they are formulated (i.e. the extent to which they rely on quantifying over variables)?
Also, given the context, it's not clear that rejecting theories which rely on second-order and higher-order logics makes sense. The usual justification for dismissing higher-order logics is that you can always translate such theories to first-order logic, and doing so is a way of "staying honest" wrt. their expressiveness. But any such translation is going to affect how variables are quantified over in the theory, hence what 'commitments' are made.
I'm not sure what you mean by "equivalent" here. If you mean "makes the same predictions" then yes-- but that isn't really an interesting fact. There are empirically equivalent theories that quantify over different concrete objects too. Usually we can and do adjudicate between empirically equivalent theories using additional criteria: generality, parsimony, ease of calculation etc.