As promised I found the quotes. The thread is here. The basic 'answer' is epistemological; if you analyse how human knowledge and reasoning works, and what such questions actually mean, you find that the answer is unknowable, and in fact that the question is meaningless.
I used to wonder about this a lot. I've since realised that, like most great metaphysical questions, it is a meaningless question, only asked because of a vast overestimation of the power of language and thought; the answer is outside of the domain of human knowledge or even meaningfulness. It's a great relief when you finally understand this statement.
The only thing that the human mind can do is observe local patterns in our little bubble of reality. Sometimes we can break those patterns down into more general patterns. That's what we do when we ask 'why' of something.
What we can't do is 'explain' something 'a priori', which means without assuming something else. To see this, just consider any explanation, and ask 'why' of it. For instance, why is there wind? Well, turns out it's down to the sun heating the globe unevenly. Why is the sun hot? Because there are hydrogen atoms fusing inside it. Why do hydrogen atoms fuse? Because the sun is so hot that the hydrogen nuclei overcome the barrier due to their positive charges and become bound by the strong nuclear force. Why is there a strong nuclear force? Probably unanswerable. There just is; some things are. And even if it wasn't unanswerable; what possible form could the answer take if it were to contain no assumptions? That's not what 'why' means; 'why' does not mean 'explain why something must be true', it means, 'explain why something happens as a result of something more general'.
So, taking the universe to mean that which exists, the question 'why does the universe exist' is not well formed. Bearing in mind the above, it does not mean anything.
Another way to look at it is to ask, 'why not'? Could the wind stop blowing one day? Why not? Maybe heat no longer causes air to expand, or maybe the sun no longer produces heat. Could this really happen? Why not? What is your a priori objection? All the way down to the strong nuclear force. Perhaps you push two neutrons together one day and they fail to attract each other. What objection could you possibly raise? Indeed, before you'd ever seen a pair of neutrons, how could you possibly have known by looking at them that they'd attract each other? Maybe they'd do nothing, or fly away, or spontaneously disappear, or maybe more would appear.
Why couldn't more appear? Why doesn't stuff appear out of nowhere all the time? (Tangential point but actually particles do appear out of nowhere all the time!). We can't provide any answer. We just know that in our domain of observation, stuff doesn't tend to appear out of nowhere very much. That's not a logical implication. Outside of our observed region of space, or our observed region of time (i.e. tomorrow perhaps), this could fail to be the case. And all of this of course also applies to the moment of creation.
Read this post again.
The first thing I'd say is that I think that the scientific discoveries of the 1900s should basically destroy our trust in any principle not being deduced from experience but rather being in some way universal (tangentially, this is why I also mentioned Darwin; he made clear to us our relationship with reality, as natural rather than spiritual beings. In the naturalistic paradigm, the concept of a principle not deduced from nature begs the question 'then where on Earth did the organism obtain the principle'? What form would the thought take in the physical neurons of the brain?). Logic is very basic and intuitive, yes, but, in my opinion, no more so than the concepts which were clearly struck down by the aforementioned events. For instance, our understanding of space. Take the statement, 'given a line and a point above it, there is one line through that point which never touches the first'. Or our understanding of time, 'if two events are simultaneous for me, they are simultaneous for you'. Or identity and causality, 'an object has a state, and that state determines its behaviour'. Are these statements not all just as intuitive as the logical example you gave? And yet they are all false. They are great models of reality for the domains that we normally experience, but outside of that domain they fall to pieces.
You are correct in your prediction of how I think we come to know the axioms. I actually think it's kind of obvious that we go through such steps, at least at first. I think a good analogy is with numbers. Many people put numbers on the same level that you put logic. But what actually is a number (let's consider whole numbers to keep it simple)? The number three is the thing in common between three trees and three clouds and three deer. But the idea that such disparate things have some entity in common is actually quite a high level of abstraction, and it must have taken many millennia of human history to develop it. If you asked one of these pioneers what two plus two was, how would they have gone about it? I think it's obvious you can't work with the abstraction itself. So you have to go down to the level of the physical substrate of the abstraction, either playing it out in reality or in your head, putting two objects together and finding that there are four. It only feels 'obvious' to us now because we are so used to working with the abstraction. And an important thing to note is the scope of meaning of the concept; it requires universe in which there are discrete objects, in order to think of this generalisation. But what's a discrete object..? There's not really such a thing. Stuff just clumps together in our universe and we notice the pattern.
I think the same goes through for logical axioms. You have to ask what the axioms are actually 'about'. They can't exist without reference to something; they must delineate some limited portion of reality. In the case of 'A is a B and B is a C therefore A is a C', we again need to have some model of discrete entities, and also we need to be able to put these entities into sets according to some adjective or another. And what's an adjective..? It's a pattern. And again, this statement may seem 'intuitively obvious', but perhaps there are objects which break it which we've just never been exposed to. And what is the neural process which allows us to conclude that it is true? In the naturalistic interpretation there must be one. Well, it seems to me that it must have come from observation. We observed every single object ever, from the time of babies (another tangent: read about Piaget's studies of infants, who turn out to be woefully poor at understanding basic logical concepts), behaving in this way. And so we formed an abstraction and now don't need to refer to the physical substrate, because we're so accustomed to it. I imagine clumps of neurons in our brain form from repeated exposure to the same pattern and become entities in themselves. In fact these concepts are so fundamentally basic that they probably constitute the fundamental essence of our being. But they're still not a priori.
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