That is because mathematics is based on a system of widely accepted assumptions.
I don't like the word assumptions here. It implies the possibility of error. Sort of like how Chuck Norris doesn't go hunting because it implies the possibility of failure. He goes killing.
We know that 5 is 5 because we all agree that is what it is.
I see what you're saying. I don't want to get into the complexities of mathematical logic, but the term we need to use here is "stipulation" rather than "assumption".
There is no way to verify mathematics empirically.
I know I'm going to lose track of what I'm thinking, and that this is going to seem an asinine point. But if I see a simplified proof for, say, Goldbach's conjecture, then that is empirical proof of the truth of Goldbach's conjecture.
Granted, this is not what we might call a "first proof" of the theorem. The empirical evidence (the writing on the board) must itself be justified by the semantic and syntactic rules of an accepted logical system - namely, that of arithmetic.
On the other hand, evidence which shows something to exist in the natural world would by necessity be empirically verifiable.
What about neutrinos? These are near-massless subatomic particles that are emitted from the fusion process within the sun. Because they are so small, it's hard to get them to interact with the physical world. In fact, the only proof we have of their existence are Chlorine pools deep underground - in gold mines. Supposedly, by the time the neutrinos have gone through enough matter, they can interact with the Chlorine to... I dunno... ionize it or something. We have an existence claim here that isn't directly verified. Instead, the effects of the presence of these posited particles are the only things that can be measured.
Thus, we have justification for the belief in the existence of these particles even though that justification is not foundational. It's not that first order kind that we want.
Deities are purported to exist, therefore they are subject to the same burden of proof which all else in existence is bound to. Namely, observation, verification, and duplication of results. Until such evidence is brought forth then there is no logical or rational reason to purport their existence as fact.
Seems like a decent response here would be to argue that if indirect existence in neutrinos is justified, then indirect evidence of God's existence should be accepted as justification as well. Now, people are going to go back and forth here for what counts as evidence for what and which events are merely coincidence. There may not be a consensus, but does that mean we should hold certain views as justified but not competing views?
The simple fact is that we stipulate all kinds of things. And these stipulations are analytic truths - they're true in virtue of meaning. Now I certainly wouldn't want to argue that we can stipulate existence claims, but why can't we stipulate conditionals?
Why is it that we can accept the following proposition N:
It is possible that neutrinos will interact with Chlorine, only if they exist.
But that's not the case for some proposition G:
It is possible that God will (insert some article of faith), only if God exists.
Anyone, I think, would accept these conditionals. Atheists like us, of course, would simply say that G is only arbitrarily true since the antecedent is false. But why are we allowed to reject G but not N? What if part of the system of beliefs is an inability to predict the outcome of certain events? I'd say most theists would be happy to accept that we cannot predict what God will do, and it's arrogant of us to think we can. Does this line of argument get the theist anywhere?
