but I think abstract objects could be an entirely different sort of thing from concrete, physically existing objects, and still exist.
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.
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.
We should believe our best scientific theories
Our best scientific theories make reference to/quantifier over abstract objects—mathematical objects like numbers, sets and functions and non-mathematical abstract objects like types, forces and relations. Entities that theories refer to/quantifier over are called their ontic commitments.
Belief in our best scientific theories means belief in their ontic commitments.
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.
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.
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.
Letting them exists costs me nothing. It doesn’t lead to false beliefs and requires far less philosophizing.
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.
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.
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).
As for common-or-garden numbers: I see no reason to exclude them.
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.
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.
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.
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.
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.
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).
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’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
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.
Platonism and nominalism don’t differ in anticipations of future sensory experiences. The difference is entirely about theory and methodology.
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.
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.
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.
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.
You seem to be assuming that by “reifying” I must mean “treat as concretely existent”.
Oh come on now. That’s literally what the word means. It’s the dictionary definition. Don’t complain about me assuming things if you’re using words contrary to their dictionary definition and not explaining what you mean.
In context, what I mean is “treat as being existent in whatever sense Platonists think abstracta are existent”.
As I’ve said a thousand times I think all there is to “being existent” is to be an entity quantified over in our best scientific theories. So in this case treating abstract objects as being existent requires scientists to literally do nothing different.
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.
Neither nominalism nor platonism make predictions. Scientific theories make predictions and there are no nominalist scientific theories.
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.
Honestly, I don’t see how this is relevant. I don’t agree that the Quinean approach leaves out the semantics of physics and I don’t see how including the semantics would let you have a simple scientific theory that didn’t reference abstract objects.
Such details are what could bring Platonism down.
Obviously it is possible that there are arguments that could convince me I’m wrong. I’m not obligated to have a preemptive reply to all of them.
As I’ve said a thousand times I think all there is to “being existent” is to be an entity quantified over in our best scientific theories.
The point of Quinean Platonism is to inflate the formal criterion of quantification into an ontological claim of existence, not to deflate existence into a mere formalism.
So in this case treating abstract objects as being existent requires scientists to literally do nothing different.
It requries them to ignore part of the informal interpretation of a theory.
Neither nominalism nor platonism make predictions.
Then one of them is unnecessarily complicated as an ontology. You see to think Platonism isn’t ontology. I have no idea what your would then think it is.
there are no nominalist scientific theories.
Whether theories are nominalist, or whatever, depends on how you read them. They don’t have their own
interpretation built-in, as I have pointed out a 1000 times.
I don’t agree that the Quinean approach leaves out the semantics of physics a
nd I don’t see how including the semantics would let you have a simple scientific theory that didn’t reference abstract objects.
Theories can include numbers and centers of gravity, and reference them in that sense, and that is not
the slightest argument for Platonism. Platonism requires that certain symbols have real referents—whichis another sense of “reference”.
Looking
at a symbol on a piece of paper doesn’t tell you that the symbol has a real referent. Non-Platonism isnt the claim that such symbols need to be deleted, it is an interpretation whereby some symbols get reified—have real world referents—and others don’t. Platonism is not the claim that there are abstract symbols in formalisms, it is an ontological claim about what exists.
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.
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)?
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.
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.
We should believe our best scientific theories
Our best scientific theories make reference to/quantifier over abstract objects—mathematical objects like numbers, sets and functions and non-mathematical abstract objects like types, forces and relations. Entities that theories refer to/quantifier over are called their ontic commitments.
Belief in our best scientific theories means belief in their ontic commitments.
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.
Oh come on now. That’s literally what the word means. It’s the dictionary definition. Don’t complain about me assuming things if you’re using words contrary to their dictionary definition and not explaining what you mean.
As I’ve said a thousand times I think all there is to “being existent” is to be an entity quantified over in our best scientific theories. So in this case treating abstract objects as being existent requires scientists to literally do nothing different.
Neither nominalism nor platonism make predictions. Scientific theories make predictions and there are no nominalist scientific theories.
Honestly, I don’t see how this is relevant. I don’t agree that the Quinean approach leaves out the semantics of physics and I don’t see how including the semantics would let you have a simple scientific theory that didn’t reference abstract objects.
Obviously it is possible that there are arguments that could convince me I’m wrong. I’m not obligated to have a preemptive reply to all of them.
The point of Quinean Platonism is to inflate the formal criterion of quantification into an ontological claim of existence, not to deflate existence into a mere formalism.
It requries them to ignore part of the informal interpretation of a theory.
Then one of them is unnecessarily complicated as an ontology. You see to think Platonism isn’t ontology. I have no idea what your would then think it is.
Whether theories are nominalist, or whatever, depends on how you read them. They don’t have their own interpretation built-in, as I have pointed out a 1000 times.
Theories can include numbers and centers of gravity, and reference them in that sense, and that is not the slightest argument for Platonism. Platonism requires that certain symbols have real referents—whichis another sense of “reference”.
Looking at a symbol on a piece of paper doesn’t tell you that the symbol has a real referent. Non-Platonism isnt the claim that such symbols need to be deleted, it is an interpretation whereby some symbols get reified—have real world referents—and others don’t. Platonism is not the claim that there are abstract symbols in formalisms, it is an ontological claim about what exists.
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.