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Tyrrell_McAllister comments on The role of mathematical truths - Less Wrong
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I'm not yet seeing how this way of thinking about math contradicts platonism. It seems to leave unaddressed the questions that platonism purports to answer. That is, your account here is essentially independent of the ontological status of mathematical objects, operations, etc.
For example, you wrote:
This seems to leave unanswered the classical kinds of questions that gave rise to platonism, such as:
What kind of thing is this "isomorphism" of which you speak? Where does it live? It doesn't seem to be a physical thing itself, so what is it? And what about the mathematical operation that is isomorphic to the physical system? Is the mathematical operation another physical system? If so, which specific physical system is it? Is it, for example, some particular physical electronic calculator? If so, which one is it? It seems implausible that any particular physical calculator has the honor of being the mathematical operation of addition, say. But if the mathematical operation is not a particular physical system, what is it? Is it an isomorphism class of physical systems? But this gets back to the problem of what, physically, an isomorphism is, and adds the problem of what, physically, a class of physical things is. One might try to identify the class with the mereological sum of its elements, but there are well-known problems with this approach. And what about the "rules of math"? Which inhabitant of the physical universe is a "rule of math"? And so on.
All of the questions above are certainly confused to some degree. But I'm not yet seeing that you've made much progress on dissolving them.
[ETA: I don't mean to say that anything you said was wrong. It certainly seems to me to be the most promising way to approach the subject.]
If your "native hardware" can't understand and implement isomorphisms, you're no better off by positing an immaterial realm in which isomorphisms exist. At some point, you can no longer define your functionality in terms of a sub-specification. But this just means that an agent must have some level that acts automatically, without further reflection (cf Created Already in Motion), not that a being's ontology is insufficient on account of failing to posit some superset realm for the concepts it implicitly uses.
In case it wasn't clear from the previous paragraph, I look at this from the perspective of creating an artificial being that can do everything I can. If the deficiencies of the way of handling of math I've described (including failure to specify to ever greater precision the "rules of math") don't correspond to some kind of failure mode of the artificial being, then I have to ask if it really is a deficiency.
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."
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.