I like to think of our exchanges as they relate to the future. Of course, the past is great too.
I feel bad that maybe I should let Mike pursue this line of discussion, and don't want to be ignorant by intruding in your discussion. However, since I've already committed the transgression, I'll take this opportunity to pursue our discussion further.
The abstractions came into being simply by the exercise of thought. Real things did not somehow get transmuted into abstractions. The abstractions of the real things became new "things" which are not real in the sense that they are not part of physical reality. They are simply ideas or objects of thought...The history traces how things became abstracted. The things themselves did not become abstract. Nor did they become abstractions. The abstractions came into being simply by the exercise of thought. Real things did not somehow get transmuted into abstractions. The abstractions of the real things became new "things" which are not real in the sense that they are not part of physical reality. They are simply ideas or objects of thought.... The chain does not reduce the abstract thing to a particular thing. The chain merely identifies the history of the thought process that led up to someone entertaining the notion of the abstract thing. Yes, such a history might exist and might even be discovered. But the thing you miss about abstraction, Harv, is that one may consider a real thing and then form an abstract idea that generalizes or expands on that real thing so that the abstraction "goes beyond" or you could say it breaks the connection to, the original real thing initially prompting or suggesting the thought. This has reached a point now where mathematicians routinely consider abstract objects that have nothing to do with reality. The infinitely tall hierarchy of orders of infinities is a good example.
Paul, I'm not talking about 'real things' being somehow transmuted into 'abstract things'. I'm talking about the concept of real things being transmuted into the concept of abstract things. For example, each of us has a concept of an apple we ate at some certain point in our lives. Perhaps our memory is a little hazy at this point since eating that particular apple, yet we might recall enough details to give us some sense of what eating that apple was like. Now, we can take this concept we have of eating that particular apple, and generalize (or abstract) from that particular experience by forming a concept of apples as a universal object. For example, we can form the notion that all apples have a similar shape, taste much like the apple we ate on that particular time, etc. What we have done is gone from a particular to a universal concept of an apple. Similarly, we can start to count the objects that match our newly adopted concept of apple, and we develop simple mathematics by playing with the concepts of apples. What we have done is move from a particular apple, to the concept of apples, to the concept of simple mathematics. We can move to a more universal approach and substitute the first apple we count with a number (1), and the second apple we count as a number (2), and so on. Once we have these new identities for the concept of apple (1, 2, 3, etc), we can just drop the term apple and concentrate on the new identities (numbers). As we do this, we find that we can count other universals (e.g., oranges, pears, boxes, people, etc). Finally, we take the additional step and not even associate numbers with any object whatsoever. We just think of number in an abstract way, and the "number has nothing to do with the real world".
Of course, this is a lie. The number is a concept that is tied (epistemologically) to universal objects (whether they be math objects or real objects), and even though we can fool ourselves that there is no tie (as if we made a divorce that eliminated the history of how we formed the concept of number), but such a divorce can never eliminate how the concept came about in the first place (either in our own childhood history or as in the human history of mathematics).
So, why do humans want to act as the divorce eliminated the history as if it never occurred? Well, I think, as Mike said, that we become so accustomed to thinking of abstract things, that we lose recognition of how we know of those abstract things. We fool ourselves in other words.
Let me prove it to you. Let's take the concept of set. When we say that element x belongs to set S, this is a valid way to talk about sets within math. However, as much as we want to say that element is a primitive term and set is a primitive term, the fact of the matter is that the word 'belongs' cannot be defined in mathematics, yet it must be understood that 'belongs' is the same as our English language concept of what it means if something belongs to something else. If it is undefined, then 'belongs' could mean 'revolves around', or any other term. We don't say that element x revolves around set S, do we? Of course not! The mathematical notions would be completely meaningless if we meant 'revolved around' instead of 'belongs'.
What this shows is that as a formal exercise mathematics can do without attachments to the world. However, be that as it may, it does not do away with the notion that mathematics is a human construction based on imputed meaning we obtain from the real world around us.
If you did, you would see that most of the new axioms postulated would seem ridiculous to anyone but a specialist in that particular narrow field. I don't mean just obscure, I mean RIDICULOUS.
The key phrase in your response is that it would 'seem ridiculous to anyone but a specialist'. The reason this is key is because the axioms that are being introduced are in context to what experience dictates as a legitimate axiom. This is not to say that anything can be an acceptable axiom, rather the range of acceptable axioms is a small percentage of conceivable axioms. For example, let me add a new axiom of mathematics that would be quickly rejected:
The Axiom of Infinite Theorems (AIT):
For every theorem, T0, there exists another theorem, T1, that is derived from T0 such that a theorem Tn+1 can always be derived from a theorem Tn.
From what I can tell, there is no reason that AIT should be rejected. I don't have to prove it because it is an axiom, and since there is no reason to reject this axiom, it is a viable axiom to consider. Yet, mathematicians wouldn't generally consider this axiom. Why? It has no significant meaning with regard to mathematics. There are little millions of such axioms that could be added to mathematics, but because they hold no human meaning (e.g., interesting mathematics to a human mathematician), they are rejected. An extraterrestrial mathematician might have a different perspective. They might have a collection of millions of axioms that we reject simply because the math is boring. The alien might feel differently and have no problem with doing so.