How do we know that the physical universe operates on this sort of arithmetic, and not arithmetic modulo some obscenely large number?
What does it mean that the universe operates on a certain sort of arithmetic? A lot of descriptions of the universe uses conventional arithmetics, some theories use rather SU(3) group or Z2 group or whatever. (Arithmetic is so general that the other mathematical constructions we use are usually somehow reducible to arithmetics, but that we have a fairly large formal system which unites all branches of useful models isn't particularly surprising.)
How do you in principle decide whether the strange behaviour of large number of particles is a fact about under what kind of arithmetic the universe operates or whether it is an additional physical law governing putting large number of particles together?
Anyway, the referents of "2+2=4" are all sets of balls, fingers, planets etc. on which we perform counting. It is a contingent truth, only abstracted a lot. Universal truths are either a philosophical confusion, or theorems of arbitrary formal systems. Either way, there is no need for that category.
Perhaps I am not phrasing my question very well. I am not asking about the existnece of universal truths, but about the human intuition that such truths exist. When someone says "2+2=4", it feels as though they are asserting a necessary truth, something that cannot possibly be otherwise. See, for example, Sniffnoy's comment above, where he asserts that even if fingers and balls and whatnot counted by integers mod 3, the plain unmodded integers would still exist. This seems to me like an assertion that unmodded arithmetic is a universal truth that cannot be contradicted by any experiment. My question is, ought the thought experiment of a universe whose galaxies and stars are counted by arithmetic mod 3^^^^3 cause us to abandon this intuition?
When someone says "2+2=4", it feels as though they are asserting a necessary truth, something that cannot possibly be otherwise.
This is an illusion. If I say "37460225182244100253734521345623457115604427833 + 52328763514530238412154321543225430143254061105 = 8978898869677433866588884288884888725858488938" it should not immediately strike you as though I'm asserting a necessary truth that cannot possibly be otherwise.
My question is, ought the thought experiment of a universe whose galaxies and stars are counted by arithmetic mod 3^^^^3 cause us to abandon this intuition?
Counting is an algorithm, or really a sketch of an algorithm. In order to make this a coherent question, i.e. to imagine running an algorithm on that many galaxies and stars and coming up with a certain answer, and then thinking about the consequences, we would need at least
Each of these is preposterous!
If I say "37460225182244100253734521345623457115604427833 + 52328763514530238412154321543225430143254061105 = 8978898869677433866588884288884888725858488938" it should not immediately strike you as though I'm asserting a necessary truth that cannot possibly be otherwise.
It immediately strikes me that what you're asserting is either necessarily true or necessarily false, and whichever it is it could not be otherwise.
Why is the difference relevant? I honestly can't imagine how someone could be in the position of 'feeling as though 2+2=4 is either necessarily true or necessarily false' but not 'feeling as though it's necessarily true'.
(FWIW I didn't downvote you.)
I honestly can't imagine how someone could be in the position of 'feeling as though 2+2=4 is either necessarily true or necessarily false' but not 'feeling as though it's necessarily true'.
That seems to imply you think it would feel different than how you felt at first looking at my sum. Why, besides the fact that it's much simpler?
I sort of agree, in the sense that "2+2 = 4" is a huge cliche and I have a hard time imagining how someone could not have memorized it in grade school, but that's part of the reason why I regard the "self-evidence" of this kind of claim as an illusion. We take shortcuts on simple questions.
I believe that "2+2=4 is either necessarily true or necessarily false". I believe 2+2=4 is necessarily true (modulo definitions). I don't believe it's necessarily true that "2+2=4 is necessarily true".
There's some pretty strong evidence that the proof that 2+2=4 doesn't have a mistake in it (heckuva lot of eyeballs). I have good reasons (well, reasons anyway) to believe that mathematical truths are necessary. Thus most of my mass is on "2+2=4 is necessarily true". Yet, even if it's necessarily true that "2+2=4 is either necessarily true or necessarily false", and 2+2=4 is true, it still needn't be necessarily true that "2+2=4 is necessarily true", even though 2+2=4 is necessarily true.
If your eyes have glazed over at this point, I'll just say that Provable(X) doesn't imply Provable(Provable(X)), and if you think it does, it's because your ontology of mathematics is wrong and Gödel will eat you.
Not sure what work "necessarily" is doing, but mostly I'm with you. Still, I think this is mistaken:
I'll just say that Provable(X) doesn't imply Provable(Provable(X)), and if you think it does, it's because your ontology of mathematics is wrong and Gödel will eat you.
Though it is true and important that Unprovable(X) does not imply Provable(Unprovable(X)).
Observing that the universe functions by modular arithmetic would not contradict integer arithmetic; it would contradict the theory that counting objects in the universe can be expressed using integer arithmetic.
Well, perhaps my question is better phrased as, "What is the referent of mathematics which does not describe objects in the universe?"
Models are tested by reference to experiment. If the model only exists in my head, what is the experiment that tests it? If there is no external reference, then in what sense can anything at all be said to contradict the model?
Are models tested by reference to experiment? You can demonstrate that a model is inapplicable in some cases; if it's inapplicable in all cases, it is useless; I don't know if it means anything to say that the model itself is false.
This is doubly so when applied to mathematics; mathematics (specifically, logic) is the model that gives a context to the term "contradiction"; that tells what it means and how we know it applies.
prase has already pointed out the real problem with your question, but I'm going to go ahead and pick nits anyway, i.e., get the math straight. If "everything" works modulo 3, that doesn't mean 2+2 is not 4. It means that 4=1. 2+2=4=1; they are the same thing. You need to use a statement like 2+2!=1 rather than 2+2=4 for your example.
Then on top of that you have the fact that '2' the integer is not the same thing as '2' the integer mod 3, they reside in separate systems and the integers still exist regardless - but following this far enough will lead you to prase's objection, that we pick our varying mathematical systems to model the physical situations, the universe doesn't "run on" one system of numbers. (Well, at the base physical layer it can, in a sense, but not necessarily in a way where you could just substitute one for another and make things work.)
Things equal to the same thing are equal to each other; if 4=1, and 2+2=4, then 2+2=1.
Then on top of that you have the fact that '2' the integer is not the same thing as '2' the integer mod 3, they reside in separate systems and the integers still exist regardless
Ah. What does it mean to say that the integers (not mod 3) exist, if it is an observed fact that 2+2=1? That is, that what we think of as an integer number of objects is actually an integer mod 3? If I dispute your assertion, and say that the integers (not mod 3) do not actually exist, what experimental outcome will demonstrate that I am wrong?
Things equal to the same thing are equal to each other; if 4=1, and 2+2=4, then 2+2=1.
Yes. But it still equals 4. If you are trying to demonstrate a case where 2+2 isn't 4, this isn't it. You've demonstrated a difference, just not the difference you say you demonstrated. Hence, your original example was bad.
Ah. What does it mean to say that the integers (not mod 3) exist, if it is an observed fact that 2+2=1? That is, that what we think of as an integer number of objects is actually an integer mod 3? If I dispute your assertion, and say that the integers (not mod 3) do not actually exist, what experimental outcome will demonstrate that I am wrong?
There's are a few problems with your question. Firstly, when we ordinarily say "2+2!=1", this is implicitly a statement about integers, not integers mod 3. That is to say, "2+2!=1" isn't a complete statement at all; it only makes sense to the extent that we know what "2", "+", and "1" indicate. As a statement about integers, with the usual interpretations of thse symbols, it's true. As a statement about integers mod 3, with the usual interpretations of these symbols, it's false. It's not one statement being true in one universe and false in another, it's just two different statements.
Secondly, numbers are models. Remember, the universe does not run at the level of "objects", it runs at the level of particle fields (or something like that). We count objects using whatever system of numbers is appropriate for counting objects (to the extent that "object" is a sensible notion). But even if whole numbers were somehow not a good model for counting objects, there are still plenty of other things that they would be a good model for, and they come up pretty naturally mathematically even without that.
So aside from a proof of the inconsistency of mathematics, not much could convince me that the whole numbers don't exist. There are plenty of experimental outcomes, however, that could convince me that they aren't a good model for something we're currently using them for (e.g. counting objects, if 4 objects were indeed the same thing as 1 object, whatever that means).
The short stories "Luminous" and "Dark Integers" by Greg Egan would probably be of interest to you.
On a related note, I wonder if it would be possible or useful to rank posts depending on the total upvotes received by the comments to them -- essentially, finding posts that prompt interesting discussions.
Why can't 2+2=4 also be an observed fact? It's just not a fact that is localizeable in time or space.
I think instead of universal vs. contingent, it's better to think non-localizeable vs. localizeable. Or if you like, location-dependent vs. location-independent.
For integers modulo p, multiplication is modulo p-1. However, if you also include the roots of polynomials, multiplication is not modulo anything for all x there is an a=/=0 such that a^x=/=1.
So numbers mod p naturally produce the integers. There are more complicated processes going back the other way.
When someone says "2+2=4", it feels as though they are asserting a necessary truth, something that cannot possibly be otherwise.
This is an illusion. If I say "37460225182244100253734521345623457115604427833 + 52328763514530238412154321543225430143254061105 = 8978898869677433866588884288884888725858488938" it should not immediately strike you as though I'm asserting a necessary truth that cannot possibly be otherwise.
My question is, ought the thought experiment of a universe whose galaxies and stars are counted by arithmetic mod 3^^^^3 cause us to abandon this intuition?
Counting is an algorithm, or really a sketch of an algorithm. In order to make this a coherent question, i.e. to imagine running an algorithm on that many galaxies and stars and coming up with a certain answer, and then thinking about the consequences, we would need at least
Each of these is preposterous!
If I say "37460225182244100253734521345623457115604427833 + 52328763514530238412154321543225430143254061105 = 8978898869677433866588884288884888725858488938" it should not immediately strike you as though I'm asserting a necessary truth that cannot possibly be otherwise.
It immediately strikes me that what you're asserting is either necessarily true or necessarily false, and whichever it is it could not be otherwise.
This is intended to explore a a thought I had, rather than making any particular argument about truth.
The canonical example of a thing which is true without any obvious physical referent is the statement 2+2=4. It is true about fingers, sheep, particles, and galaxies; but intuitively it does not seem that any of those truths encapsulates the full meaning of the statement. Moreover, it certainly seems that there is nothing anyone could do to make the statement untrue; it seems that it would have to hold in any universe whatsoever.
Now my thought: How do we know that the physical universe operates on this sort of arithmetic, and not arithmetic modulo some obscenely large number? Suppose we repeat the experiment that convinces us 2+2=4 (and let's note that babies are presumably not born knowing this; they learn it by counting on their fingers, even if they do so at too young an age to express it in words), but with much larger integers. Perhaps we might find that, when we take 3^^^^3 particles, and add 1, we are left with 3^^^^3 particles without any awareness that any particles have disappeared. And what is more, if we take three sets of 3^^^^3 particles, and measure their mass separately and then together, we find that we get the same mass. After some long sequence of such experiments, perhaps we might convince ourselves that physics actually operates on integer arithmetic modulo 3^^^^3. (Which would be unexpected in that the physics we know operates on complex numbers, not integers, but perhaps that's an approximation to some fantastically-finegrained two-dimensional integer grid.)
What would this mean, if anything, for the truth of such statements as 2+2=4? It seems that it would then be a contingent truth, not a universal one; that there could in principle exist a universe whose physics operated on arithmetic modulo 3, so that 2+2=1. (Presumably such a universe would not have any sentient beings in it.) What if 2+2=4 is an observed fact about our universe on the same order as the electromagnetic constant or the speed of light?