Yes, I agree that “pure” math is 100% a priori and I agree that there is different issue of applicability.

But there are people in philosophy that think there is no such thing as certain knowledge, that everything is revisable in the light of new empirical data. Even math and things like laws of non-contradiction. So, I was wondering what would it mean for the math to be revisable like that.

It’s an analytic truth that 2+2=4, but it’s an empirical question whether after putting two apples in a basket, and then two more, there will be four apples in the basket.

Take a look at this post, which I wrote some time ago, for a more complete explanation.

I think you are right when you say that if ‘2+2=4′ is empirical is really related to what we mean by ‘2+2=4′. And your example is that such a case would be where we mean by it that “if we add two things to two things, we yield a collection of four things”. If THAT is what we mean, I agree that it is empirical issue, and it will depend on the nature of the things in question, and what we mean by “add”. One example (maybe not best related to your example meaning, but possibly to other meaning of 2+2=4) would be addition of speeds in special relativity. In that case it is not true that ‘2+2=4′ (again, in this now very specific meaning), but we have another formula for addition of speeds.
As far as I’m concerned this answers my question what would it mean for ‘1+1=2′ to be empirical. It simply depends on what we MEAN by ‘1+1=2′.

That aside, I think there is a meaning of ‘1+1=2′ which can not turn empirical. And that is my proposed meaning that ‘when we have two things, we have one and one more thing and vice versa’. Given THIS meaning, I can’t for the best of me figure out what would it mean for the truth of this statement to be empirical. I think however based on your next example (with QM), you would disagree with my claim in this paragraph (or maybe not?), so let me comment on that one now…

I’m returning here after writing the following part. It kind of turned too long. Sorry for that. Please ignore it if you find it unrelated to the issue at hand…

I don’t think there can be contradictions in the reality. Contradictions are simply impossible. There are just apparent contradictions which come from our bad conceptualization of the phenomenon. So to say, if we come to an apparent contradiction, we are to blame our thinking, and not reality or thinking in general. To return to position-momentum, I think the problem is not the impossibility to comprehend this, but merely the underlying assumed metaphysics – Atomism. If we see properties of the particles as aspects of the particle (or of a particle system really, as it is hard to talk about position or momentum of an alone particle), it would be even weird if the aspects are unconnected….

The analogy would be a 3d object that casts two shadows. If we take shadows to exist as self-subsistent properties of the object, we would find it weird when changing the one shadow (actually the object, as the shadows don’t exist as things that can be changed), we also change the other shadow. But if we see the properties as aspects, there is nothing strange that they are related – what is measured is the particle itself, and different aspects of the particle have to be related – they don’t exist by themselves.

Anyway, on the point at hand, I think we can stipulate that mathematical theorems are “empirical” all and only to the extent that they are construed as statements that apply to (among other things) empirically describable states of affairs. So the proposition ‘2+2=4′ is an empirical fact just to the extent that it is taken to mean that a collection of any two things added to another collection of any two things will in every case yield a collection of four things. (Here I’m really just sort of restating part of the thought by Einstein I quoted under Greg’s post.)

Now, ‘2+2=4′ admittedly seems like a necessary truth. But it may only be a case of what I’ll call necessary truthiness. (Apologies to Stephen Colbert.) How would we know the difference?

A possibly instructive example come from (where else?) quantum mechanics — viz., the case of quantum uncertainty.* Try to conceive of a particle that does not have a jointly determinate position and momentum. Can you do it? Probably not. In fact, I think doing so may be psychologically impossible. And yet, if the predominant interpretation of quantum uncertainty is to be credited, particles do not have a determinate position-momentum.

This confounding empirical discovery, combined with the necessary-truthiness of determinate position-momentum, is probably what led Feynman to declare that “no one understands quantum mechanics.” I think we’d be in a similar situation with respect to mathematics, were such a compelling series of countervailing discoveries to undercut it: No one would understand it.

It’s no surprise, then, that we can’t imagine how it might be so.
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* I’m setting aside several possible objections here, among them: that the quasi-necessary status I attribute ex ante to the thesis of determinate position-momentum is contestable; that notwithstanding the consensus view, some scientists and philosophers insist that uncertainty is epistemic and not ontological (though this may be further evidence that the intuition underlying the determinacy thesis is irresistible); and that unlike mathematical statements, the determinacy thesis is not a formal theorem.

You say it is true by definition independently of how the world is.
But if that is independent of how the world is, how can it be empirical?

To be empirical it means that it can be true or false depending on how to world is. But really there are no ONEs or TWOs in the world, and certainly the relation “whenever there are two things, there is one and one more thing” is not encountered as a thing in the world on which we can make experiments.

I can imagine two men in white coats, but I can’t imagine them being two men in white coats, and still not be one and one more man in a white coat.

The demon may affect what we see (like in matrix), but he can’t present us with abstract ONE and TWO, so the problem is here that he can’t lie to us in this way. Whenever he presents us with TWO things, those will be ONE and ONE more thing. There is nothing the demon could do! He can make us make mistakes I guess in our reasoning, but you seem to agree that there is not much he can affect in the simple thought that whenever there is two, there is one and one more.

And also, you seem only to have intuition that it is possible that we are irrational about 1+1=2. But this is an intuition about complex issue (because it includes issue of rationality, the relation between thought and world, and so on). Why do you trust your intuition that this is possible, more than the direct intuition that when there is two things, there is one and one more thing (and vice versa) ?

Such stuff is true by definition, independently of how the world is. But that depends upon our natural presumption of our own rationality. And one might discover empirically that one was not rational. E.g. one finds oneself sitting in a room with men in white coats, who are explaining this to one. Of course, one would probably not, in such a scenario, see how reasonable their explanation was. But one might…

Irrationality is a mixed bunch. One might be irrational in some ways whilst being quite rational in other ways. So one might eventually come to know that one’s arithmetical intuitions were not to be trusted. One would have discovered, empirically, that arithemtic (as we know it, which is by definition what it is) is false. (The demon scenario just works by the demons fucking one up sufficiently, but of course I’ve no idea how precisely they would or could do that.)

I still don’t understand what would it mean for arithmetic to be empirical.

I agree that we might be mistaken about something in our a priori thought. There are cases where something was considered as proven in math, only to be shown that it is wrong by more closer analysis. However in those cases we didn’t figure out that we are wrong by observation of the world, it was proven it was wrong again with a priori thought. One can really see if a theorem holds by trying out some specific numbers, but that is not an empirical evidence, it is again a priori thought, just less abstract.

Not sure how the demon would work. In the case of perception it is intelligible that a demon could lie to me about how the world is. The scenarios like Matrix are easily imaginable.

But how would demon lie to me about arithmetics? What would that mean? Would it be like a teacher lying to me that whenever there is *two things*, there is *one* and *one* more, and still *one* more thing? Of course. I can believe in what someone told me, and that to be a lie. But this is not a case here, I don’t believe that whenever there are two things, there is one and one more thing because someone told me. The truth of it is discovered from my own thought, in a direct identity contained in *one* simple unity (that is why it is *identity*, I’m not thinking of two different things, and “joining” them in thought, I’m seeing that one thing can be seen this and that way). So, where would the demon intervene?

Now, as I said, we may make mistakes, the more complex are theorems the more space there is for a mistake (we might fail to see some interrelation, or mistake some relation for some else and so on). And demon or hypnotist may affect my thought in such a way. But this is not a case of arithmetic being empirical still. It is just example of how I may be wrong in my thinking.

So what would be a case of empirically refuting arithmetic? Also how would demon scenario work?