Let me know if that was responsive to your questions.
I took your post to be an account of the meaning of mathematical claims, or of what it is that they assert about the world. In particular, you said that you would eliminate “the need for a non-physical, non-observable ‘Platonic’ math realm.” (Emphasis added.)
I take your comment here to be describing the sense of “need” that you were using in your OP: To need the concept of a platonic math realm means to need it to build an artificial being that can do math.
But I don’t think that many platonists would disagree. I’ve never heard anyone claim that calculator engineers need to learn mathematical platonism, or indeed any philosophy of mathematics at all, to do their job. Certainly none would say that we have to somehow program the calculator to be platonist for it to do its job. They wouldn’t even say that a human mathematician has to be a platonist to succeed at mathematics.
The problem with platonism isn’t that it keeps anyone from being able to build calculators. I’d say that the problem with platonism is that it convinces people that they can know about some things (ideal geometric objects, say) without interacting with them causally. This encourages some people to credit other mysterious “ways of knowing”, such as religious faith. And that, in turn, can get them so confused that they can’t succeed at certain tasks, such as building an AI. (Is that what you were getting at?)
If that sequence of confusions is the “failure mode” to avoid, then your success in your OP is to be judged by whether it actually keeps humans from feeling such a felt need for platonism.
But I don’t yet see that it does this, for the reasons that I gave in my previous comment. Someone could easily read your post, agree with the picture it paints, and yet say, “Yes, but just what kinds of things are these isomorphisms and operations and rules of math? I think that the most satisfying answer is still that they are inhabitants of some ideal platonic realm.”
I’d say that the problem with platonism is that it convinces people that they can know about some things (ideal geometric objects, say) without interacting with them causally. This encourages some people to credit other mysterious “ways of knowing”, such as religious faith. And that, in turn, can get them so confused that they can’t succeed at certain tasks, such as building an AI. (Is that what you were getting at?)
Agreed, but that was an implicit premise, not something I was trying to prove. That is, my article takes it for granted that you will not want to use an epistemology that implies that knowledge can arise without causal interaction, and that therefore you deem your epistemology flawed if and to the extent that it does so. So I assume the reader regards removal of the platonic realm dependency as desirable, for any of a number of reasons, including that one.
But I don’t yet see that it does this, for the reasons that I gave in my previous comment. Someone could easily read your post, agree with the picture it paints, and yet say, “Yes, but just what kinds of things are these isomorphisms and operations and rules of math? I think that the most satisfying answer is still that they are inhabitants of some ideal platonic realm.”
True: if you can’t implement a well-defined procedure (such as isomorphism or standard math) without positing its existence in an immaterial realm, then my article doesn’t have much that will change your mind on that matter (“you” in the general sense).
But I don’t see how someone would well-versed enough in rationality for this article to be relevant, yet still make such a leap. That kind of error occurs at a more basic level. Whatever reason suffices to make one feel the need to posit a platonic realm must have a broader grounding, right?
So I assume the reader regards removal of the platonic realm dependency as desirable
I think that this gets at the crux of my criticism. What kind of dependency on Platonism do you see your article as removing? That is, what kind of “need” for Platonism did you picture a reader feeling before reading your article, but being cured of after reading it?
Thanks, that does get to the heart of the matter. To borrow from one of the linked articles, I imagine someone in the role of Eliezer Yudkowsky here, being challenged by “the one” (bold added):
And the one says: “Well, I know what it means to observe two sheep and three sheep leave the fold, and five sheep come back. I know what it means to press ‘2’ and ‘+’ and ‘3’ on a calculator, and see the screen flash ‘5’. I even know what it means to ask someone ‘What is two plus three?’ and hear them say ‘Five.’ But you insist that there is some fact beyond this. You insist that 2 + 3 = 5.”
Well, it kinda is.
“Perhaps you just mean that when you mentally visualize adding two dots and three dots, you end up visualizing five dots. Perhaps this is the content of what you mean by saying, 2 + 3 = 5. I have no trouble with that, for brains are as real as sheep.”
No, for it seems to me that 2 + 3 equaled 5 before there were any humans around to do addition. When humans showed up on the scene, they did not make 2 + 3 equal 5 by virtue of thinking it. Rather, they thought that ‘2 + 3 = 5’ because 2 + 3 did in fact equal 5.
That is, a rationalist could avoid making obvious or large errors, but still believe “2+3=5”, above and beyond any physically-verifiable claim between two people, and above and beyond any specific model (map) of reality, physically instantiated in agents. My article says to that rationalist, no, you needn’t believe in this platonic “2+3=5″ apart from its implication in a commonly used model, and you can still elegantly and consistently handle all of the dilemmas associated with having to classify such abstract statements. In fact, you needn’t make a statement about anything non-physical.
Do you believe I’ve done so, and said something relevant to rationalists?
My article says to that rationalist, no, you needn’t believe in this platonic “2+3=5” apart from its implication in a commonly used model
Which implication is still a fact which seems to be non-physical, seems to have been true before there were any humans to do logic, etc. You’ve eliminated Platonic numerical entities and metaphysically privileged formal systems—which do seem to be improvements—but not non-physical a priori truths.
Any physical system that, as best you can infer, behaves with a known isomorphism to integer math.
**ETA: Oops, see zero_call’s correction; following the article, integer math actually corresponds to some widely-held conception—within human brains—of how numbers work. Since Tyrrell_McAllister’s point was that I was slipping in non-physicality, the rest of the exchange is still relevant, though.
Any physical system that, as best you can infer, behaves with a known isomorphism to integer math.
So, to make the pure physicality of all referents clear, should we label that node:
Physical system S outputs the string ‘4’ whenever it is fed the string ‘2+2=’
where S is the name of a specific concrete physical system such that the string ‘2+2=’ physically makes S output ‘4’ in a way that is isomorphic to the way that the rules of arithmetic logically imply that 2+2=4?
Yes, basically. I mean, I’d tweak it to read something more like
Physical system Sis regarded as outputting ‘4’ when interpreted per a specific known isomorphism M, whenever the query ‘2+2=’ is converted per M and applied to it.
but I don’t think that impacts whatever point you were trying to make.
I think that your tweak makes an important difference. And, if I may be so bold, I think that you want something closer to what I wrote :).
I’m trying to make good your claim that the causal diagram refers only to physical things. But your label refers to M, which is an isomorphism. What is the concrete physical referent of “M”?
I think that your tweak makes an important difference. And, if I may be so bold, I think that you want something closer to what I wrote :).
Why? The constraint that a system output a “string” is too strict; it suffices that they output something interpretable as a string.
I’m trying to make good your claim that the causal diagram refers only to physical things. But your label refers to M, which is an isomorphism. What is the concrete physical referent of “M”?
An isomorphism M is a one-to-one mapping between two phenomena X and Y. In this context, then, the physical referent of M is whatever physically encodes how to identify what in Y it is that the aspects of X map to.
Why? The constraint that a system output a “string” is too strict; it suffices that they output something interpretable as a string.
I agree now that “string” is too strict. I should have said “symbol”, where a symbol is anything with physical tokens. My proposed label is now
Physical system S outputs the symbol A whenever it is fed the symbol B
where
“S” is the name of a specific concrete physical system, and
“A” and “B” are the names of specific physically-manifested symbols,
such that a token of the symbol A physically makes S output a token of the symbol B in a way that is isomorphic to the way that the rules of arithmetic logically imply that 2+2=4.
I think that the work that you want to do by adding the word “interpretable” to the label is done by my conditions on what S, A, and B are.
An isomorphism M is a one-to-one mapping between two phenomena X and Y. In this context, then, the physical referent of M is whatever physically encodes how to identify what in Y it is that the aspects of X map to.
Then you should be able to make the label refer directly to that physical encoding of M. That is, instead of mentioning the isomorphism M, you ought to be able to refer just to some specific physical system T that “encodes” M in the same way that my physical system S above encodes the operation of adding 2 to 2.
However, if you’re still unhappy with my label, then you would probably be unhappy with this unpacking of your reference to M. But I can think of no other way to make good your claim to refer only to physical things.
(A strict platonist would say that even my label refers to nonphysical things, because it refers to symbols, only the tokens of which are physical. I’m happy to ignore this.)
Then you should be able to make the label refer directly to that physical encoding of M. … you ought to be able to refer just to some specific physical system T that “encodes” M … if you’re still unhappy with my label, then you would probably be unhappy with this unpacking of your reference to M. But I can think of no other way to make good your claim to refer only to physical things.
Well, I would need to permit more than just one physical encoding; I’d need to permit any physical encoding that is, er, isomorphic to an arbitrary one of them. But I don’t see this as being a problem—it’s like what they do with NP-completeness. You can select one (arbitrary) problem as being NP-complete, and then define NP-completeness as “that problem, plus any one convertible to it”.
So it appears I can avoid binding the meaning to one specific physical system, while still using only physical referents. And yes, your updated terminology is fine as long as you allow “symbols” and “fed” to have sufficiently broad meanings.
Incidentally, are you saying the same problem arises for defining “waves”? Do you think that referring to one particular wave requires you to reference something non-physical? Would you say waves are partly non-physical?
M as an isomorphism is just an interpretation between things (rocks, birds, etc.) and “math things” (numbers, etc.) Its physical referent is the human mental instantiation of that interpretation (e.g., in the form of neutro transmitters or what have you.) However, (see my comment a little above), I don’t think this is what you were getting at.
No, I thought the physical referent for the integer math was something like “Human mental instantiation of an idea that is reasonably agreed upon.” I believe you are referring to the physical referent of the preimage of the isomorphism (i.e., the physical system itself. A somewhat redundant thing to call a referent, since it is actually the explicit meaning of the statement.)
I took your post to be an account of the meaning of mathematical claims, or of what it is that they assert about the world. In particular, you said that you would eliminate “the need for a non-physical, non-observable ‘Platonic’ math realm.” (Emphasis added.)
I take your comment here to be describing the sense of “need” that you were using in your OP: To need the concept of a platonic math realm means to need it to build an artificial being that can do math.
But I don’t think that many platonists would disagree. I’ve never heard anyone claim that calculator engineers need to learn mathematical platonism, or indeed any philosophy of mathematics at all, to do their job. Certainly none would say that we have to somehow program the calculator to be platonist for it to do its job. They wouldn’t even say that a human mathematician has to be a platonist to succeed at mathematics.
The problem with platonism isn’t that it keeps anyone from being able to build calculators. I’d say that the problem with platonism is that it convinces people that they can know about some things (ideal geometric objects, say) without interacting with them causally. This encourages some people to credit other mysterious “ways of knowing”, such as religious faith. And that, in turn, can get them so confused that they can’t succeed at certain tasks, such as building an AI. (Is that what you were getting at?)
If that sequence of confusions is the “failure mode” to avoid, then your success in your OP is to be judged by whether it actually keeps humans from feeling such a felt need for platonism.
But I don’t yet see that it does this, for the reasons that I gave in my previous comment. Someone could easily read your post, agree with the picture it paints, and yet say, “Yes, but just what kinds of things are these isomorphisms and operations and rules of math? I think that the most satisfying answer is still that they are inhabitants of some ideal platonic realm.”
Agreed, but that was an implicit premise, not something I was trying to prove. That is, my article takes it for granted that you will not want to use an epistemology that implies that knowledge can arise without causal interaction, and that therefore you deem your epistemology flawed if and to the extent that it does so. So I assume the reader regards removal of the platonic realm dependency as desirable, for any of a number of reasons, including that one.
True: if you can’t implement a well-defined procedure (such as isomorphism or standard math) without positing its existence in an immaterial realm, then my article doesn’t have much that will change your mind on that matter (“you” in the general sense).
But I don’t see how someone would well-versed enough in rationality for this article to be relevant, yet still make such a leap. That kind of error occurs at a more basic level. Whatever reason suffices to make one feel the need to posit a platonic realm must have a broader grounding, right?
I think that this gets at the crux of my criticism. What kind of dependency on Platonism do you see your article as removing? That is, what kind of “need” for Platonism did you picture a reader feeling before reading your article, but being cured of after reading it?
Thanks, that does get to the heart of the matter. To borrow from one of the linked articles, I imagine someone in the role of Eliezer Yudkowsky here, being challenged by “the one” (bold added):
That is, a rationalist could avoid making obvious or large errors, but still believe “2+3=5”, above and beyond any physically-verifiable claim between two people, and above and beyond any specific model (map) of reality, physically instantiated in agents. My article says to that rationalist, no, you needn’t believe in this platonic “2+3=5″ apart from its implication in a commonly used model, and you can still elegantly and consistently handle all of the dilemmas associated with having to classify such abstract statements. In fact, you needn’t make a statement about anything non-physical.
Do you believe I’ve done so, and said something relevant to rationalists?
Which implication is still a fact which seems to be non-physical, seems to have been true before there were any humans to do logic, etc. You’ve eliminated Platonic numerical entities and metaphysically privileged formal systems—which do seem to be improvements—but not non-physical a priori truths.
It is a counterfactual claim about something physical. You can represent it in a causal diagram with only physical referents.
The causal diagram in your OP contains a node labeled “Integer math implies 2+2 = 4?”
What is the physical referent for “Integer math”?
Any physical system that, as best you can infer, behaves with a known isomorphism to integer math.
**ETA: Oops, see zero_call’s correction; following the article, integer math actually corresponds to some widely-held conception—within human brains—of how numbers work. Since Tyrrell_McAllister’s point was that I was slipping in non-physicality, the rest of the exchange is still relevant, though.
So, to make the pure physicality of all referents clear, should we label that node:
where S is the name of a specific concrete physical system such that the string ‘2+2=’ physically makes S output ‘4’ in a way that is isomorphic to the way that the rules of arithmetic logically imply that 2+2=4?
Yes, basically. I mean, I’d tweak it to read something more like
but I don’t think that impacts whatever point you were trying to make.
I think that your tweak makes an important difference. And, if I may be so bold, I think that you want something closer to what I wrote :).
I’m trying to make good your claim that the causal diagram refers only to physical things. But your label refers to M, which is an isomorphism. What is the concrete physical referent of “M”?
Why? The constraint that a system output a “string” is too strict; it suffices that they output something interpretable as a string.
An isomorphism M is a one-to-one mapping between two phenomena X and Y. In this context, then, the physical referent of M is whatever physically encodes how to identify what in Y it is that the aspects of X map to.
I agree now that “string” is too strict. I should have said “symbol”, where a symbol is anything with physical tokens. My proposed label is now
where
“S” is the name of a specific concrete physical system, and
“A” and “B” are the names of specific physically-manifested symbols,
such that a token of the symbol A physically makes S output a token of the symbol B in a way that is isomorphic to the way that the rules of arithmetic logically imply that 2+2=4.
I think that the work that you want to do by adding the word “interpretable” to the label is done by my conditions on what S, A, and B are.
Then you should be able to make the label refer directly to that physical encoding of M. That is, instead of mentioning the isomorphism M, you ought to be able to refer just to some specific physical system T that “encodes” M in the same way that my physical system S above encodes the operation of adding 2 to 2.
However, if you’re still unhappy with my label, then you would probably be unhappy with this unpacking of your reference to M. But I can think of no other way to make good your claim to refer only to physical things.
(A strict platonist would say that even my label refers to nonphysical things, because it refers to symbols, only the tokens of which are physical. I’m happy to ignore this.)
Well, I would need to permit more than just one physical encoding; I’d need to permit any physical encoding that is, er, isomorphic to an arbitrary one of them. But I don’t see this as being a problem—it’s like what they do with NP-completeness. You can select one (arbitrary) problem as being NP-complete, and then define NP-completeness as “that problem, plus any one convertible to it”.
So it appears I can avoid binding the meaning to one specific physical system, while still using only physical referents. And yes, your updated terminology is fine as long as you allow “symbols” and “fed” to have sufficiently broad meanings.
Incidentally, are you saying the same problem arises for defining “waves”? Do you think that referring to one particular wave requires you to reference something non-physical? Would you say waves are partly non-physical?
M as an isomorphism is just an interpretation between things (rocks, birds, etc.) and “math things” (numbers, etc.) Its physical referent is the human mental instantiation of that interpretation (e.g., in the form of neutro transmitters or what have you.) However, (see my comment a little above), I don’t think this is what you were getting at.
No, I thought the physical referent for the integer math was something like “Human mental instantiation of an idea that is reasonably agreed upon.” I believe you are referring to the physical referent of the preimage of the isomorphism (i.e., the physical system itself. A somewhat redundant thing to call a referent, since it is actually the explicit meaning of the statement.)
You’re right, I agree. I was being inconsistent with my article there.