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Tyrrell_McAllister comments on A wild theist platonist appears, to ask about the path - Less Wrong
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I included the qualification "the formal systems that you're thinking of". Were you thinking of that formal system when you wrote
?
This is a separate question from your original one. To answer your original question, the platonist need only point to a particular formal system (e.g., PA), and say that the nonexistence of the number 2 would mean [ETA: rather, would imply] that there would be no proof of 2's existence in that particular system.
But the platonist would also find your new question interesting. For example, when you wrote "There's a particular set of formal steps ...", how did you come to settle on that particular set of formal steps?
One position would maintain that some particular formal system is implicit in the statement of the twin prime conjecture (TPC). That is, when someone asks "Are there infinitely many twin primes?", they are speaking in an abbreviated fashion, and they really mean "Does PA prove that there infinitely many twin primes?", or "Does ZFC prove that ...", or something like that. This position would claim that number-talk has meaning only when the speaker has some specific formal system in mind.
The difficulty with this position is that people seemed to be making meaningful assertions about numbers for thousands of years before settling on any formal systems of arithmetic — indeed before they even had the concept of a formal system. (Euclid's is presumably too unrigorous to count.) Ancient number-theory texts such as the Introduction to Arithmetic by Nicomachus often didn't even include anything like what we would call a proof, not even in the sense in which Euclid's arguments are called proofs. Nicomachus pretty much just flatly asserts number-theoretic propositions, perhaps bolstering his claims with a few examples. Yet somehow readers would reflect on these propositions and agree with them, even though Nicomachus didn't communicate which formal system he was working within.
Another position would maintain that people absorb some formal system implicitly from their culture, and their number-talk is automatically in terms of that formal system. Even if that culture lacks the concept of a formal system, nonetheless all its number-talk is governed by some formal system.
But it is not even known whether the TPC is decidable in any of the standard formal systems. Suppose that the TPC were proved undecidable in, say, ZFC. Would the question really then lose all meaning? Consider a physical computer that brute-force factors one odd number after the next, and prints every consecutive pair of odd numbers that have no nontrivial factorizations. Would it really become meaningless to ask whether there is a bound on how many entries the computer would print, regardless of how many physical resources it were given? Granted, this is necessarily a counterfactual, but . . .
I am not a platonist, but I still can't call myself a formalist, because I can't bring myself to declare confidently that the TPC would become meaningless if it were proved undecidable in ZFC. Indeed, even if the TPC happens to be undecidable in every formal system whose axioms would seem to us to be "intuitively correct" of numbers, it still seems to me that our concept of number may suffice to determine a truth value for the above counterfactual. Either that computer would churn out pairs forever, or it wouldn't.
I wasn't, but I rather strongly opine that the word 'exists' should not be applied to a state that can change with the vagaries of what formal system I happen to be thinking of. At an absolute minimum, it should be qualified along the lines of "X exists within formal system Y" or "The existence of X is a theorem of formal system Y". At which point I can return to my original question: What is the formal system of which "God exists" is a theorem or axiom? I also note that "Given axioms X, God exists" is somehow a rather less impressive claims than a floating "God exists". Yet it's so much more specific and satisfactory.
This is at least an answer to the question, "What is meant by 'exist'?"; it gives us a definite procedure for deciding what does and doesn't exist. But I opine that it's not a very satisfactory one. Why that formal system and not some other one?
I don't think so, but I'm not sure I understand the relevance, but what I'm objecting to is the word 'exists'. Suppose you established that the computer was going to print these two numbers and then stop. That is an experimental prediction which we can test. (Updating our belief in the proposition upwards with every second that the computer prints nothing more.) I still don't see the value in asserting on these grounds that something exists. Why not stick with what is observable, namely that the computer halts, or doesn't halt?
If you want to say that 'exists' is a short form of "the computer halts", fine; but it does not seem to me that this is what platonists usually intend to say. And, to return to the original problem, it is still completely unclear what "God exists" is shorthand for.
The platonist certainly agrees. The test I described would only work for "accurate" maps of the territory. The platonist would consider PA to be an accurate (but incomplete) map of the actual natural numbers, while the formal system you described is not.
Actually, I was a little sloppy, there. When speaking on behalf of the platonist, I shouldn't have written "would mean", but rather "would imply". The point is that I wasn't defining what "the number 2 exists" means, but rather describing what the world would be like if the number 2 didn't exist.
At any rate, I don't mean to be giving a "definite procedure". For the platonist, PA is an accurate, but incomplete, map. Consider that there probably exist physical things of which we will never find any empirical trace. Similarly, the platonist expects that there exist mathematical objects that remain entirely unmapped by our formal systems or intuitions.
If you would be willing, on these grounds, to assert with some confidence that the computer will never print any more pairs, why would you demure from asserting, on these same grounds, and with this same confidence, "A largest pair of twin primes exists, and this most-recently printed pair is it."?
(Just to clarify: The program I described never halts, even if the TPC is true. The program continues to run; it continues to brute-force factor progressively larger odd integers. It just never prints anything further to its output tape beyond a certain point.)
Ok, but what is the test which distinguishes between accurate and inaccurate maps? It seems to me that the reasoning here has become circular: It is asserted that the number 2 exists because otherwise, formal system X would be unable to prove it; and also that formal system X is a good test because the number 2 exists. I feel that at this point, you ought to abandon the formal systems and just go for straightforwardly asserting that the number 2 exists in a mystical, intuitive sense which is not open to rational disproof, but which you can use to test formal systems for accuracy.
Because this does not seem to me to match the meaning of the word 'exists'. If I say that Planet X exists, I can point to it with a telescope and I expect in principle to be able to travel there. If I say that a species of animals exists, I am asserting the possibility of shooting one and eating it. What am I asserting when I say that a number exists? If it's that a particular computer will print so many numbers and then stop, then I think this is not the same class of assertion as in the two previous examples, and it ought to have a different word.
Both maps are accurate maps of different parts of the territory.
Yet they both make assertions about the number 0, its successor, and its successor's successor.
They use the word 'successor' to mean slightly different things.
I was a little sloppy on this point earlier, so I want to be more careful about it now:
Are you saying that "Species X exists" means exactly the same thing as "I could shoot and eat X in principle"? Or are you just saying that, if you show that you can shoot and eat X, you've shown that X exists? For example, is it meaningless to assert the existence of a species outside of your lightcone?
It becomes very tricky to talk without appearing to commit yourself to the existence of numbers.
For example, you say above that the computer is printing numbers. How is it doing that if numbers don't exist?
That might seem like nitpicking, and you probably could find a perfectly adequate rewording of that sentence that avoids even the appearance of implying the existence of numbers. But it has proved very difficult to give a fully satisfying nominalist rewording of all talk that is ostensibly about abstract objects.
ETA: The question you raise about how we could know, on a platonic account, that a given formal system is an accurate map of the natural numbers is a good one. It's a large part of why platonism is a nonstarter for me. But I don't think that it's incoherent in the same way that you seem to.
Well then, let it arrange groups of pebbles instead.
I gently suggest that you need to check your association maps, here. If you can give no account of this very fundamental thing, how we know that particular formal systems are the right ones to use, then isn't it time to go from "platonism is a nonstarter for me" to "platonism is wrong"?
How would I learn about such a species? It may not be meaningless, but I don't see how to connect it to any experience.
What do you mean by an "associate map" [ETA: oops, "association map"] in this context?
Unfortunately, there are many very fundamental things for which I am unable to give any account worth the time. Strangely, it seems that the more fundamental something is, the more difficult it is to account for it satisfactorily.
At any rate, you can take "platonism is a nonstarter for me" to imply "I think that platonism is wrong". (What else would "nonstarter" mean?)
As in Belief in the Implied Invisible, a physical theory with strong empirical support could commit you to believing in such a species with high probability.
I mean that you should check that you're not compartmentalising your beliefs. If we don't regularly test the implications and associations between our beliefs, we can end up asserting contradictory things, like the theist scientist. That said, this:
seems to indicate that the exercise isn't necessary for you here.
At any rate it apparently allows you to defend platonism to some extent. Perhaps 'non-finisher' would be a better term?
Yes, but then I could broaden my shooting definition slightly and say "At one time it was possible to shoot this species."; in other words I would give the past empirie that convinced me of the species' existence. Or alternatively, I could go for "in principle" and say that we just need some closed timelike curvies, or other means of extending the lightcone.
I didn't see myself as defending platonism, so much as defending a certain kind of existence-talk that I, along with the platonists, think is legitimate. I agree with them that one should be able to consider the possibility that a largest pair of twin primes exists. Furthermore, it should be possible to do this without having a formal procedure for deciding the question in mind, even implicitly.
Couldn't an empirically well-supported theory give high probability to the claim that there exists a species X whose future and past lightcones never intersect yours, and which is not accessible to you by any closed timelike curves or other means of extending the lightcone?
And besides, wouldn't it be kind of weird if your ability to use the word "exists" were constrained by how sophisticated your physics is? It would seem to follow that someone who believes in a classical Newtonian universe allowing arbitrarily fast travel can legitimately ponder the possibility that dragons exist 10^100 light years away, while you cannot because you (let's say) believe in a universe in which it is and was always impossible for you to interact causally with anything that far away.
Ok, but here is your original reply:
And all your replies since have taken the form "the platonist might say". Can I suggest that you should defend the theory you actually believe in? At this point you seem to agree with me that theorems within some particular formal theory is not a good reason to say that numbers exist. What then do you mean by asserting the existence of the number 2? No more platonism, if you please, since you don't believe in it.
No, actually, I don't see how it could. If, by construction, I never, either in past or future, interact with any of the species or with anything that it has interacted with, then I don't see how I can get an empirically supported theory of their existence.
Ok, just use the past interaction then.
This is a universal counterargument against saying that anything besides what we are currently observing exists.