Mathematical space is a collection of objects with a structure defined over them. This definition as a whole, as well as the words used in it, and their relations may seem too vague, broad, and ambiguous. What is “object”? What is “collection” and “structure”? Why the structure is “defined”, and not intrinsically belongs to these objects or their collection? And, if it’s defined, who and why defines it?
Instead of intuitively guessing answers to these questions, we might have wanted to build our understanding of the vague and complex concepts from a limited basis of the simple, narrow, well defined, clear and distinct concepts, as Rene Descartes, one of the framers of the modern philosophy and algebra, have wanted. However, the same time he noted that human mind is a very poor inventor. It can’t create anything genuinely new. It’s very good, though, in mixing and combining blocks already known to it. If we go to the path of excessive reduction of the number, richness, and generality of these blocks, we may end up unable to span complex (if at all) concepts from the simple, distinct, and clear blocks.
Therefore, we may be better off defining simple, well defined, and clear rules of combination of the potentially opaque, complex, and only intuitively comprehensible building blocks. To add insult to the injury, postmodernism pointed out that intuition depends on the personal experience, therefore each person will have different from others innate concepts. The question ‘is your “red” the same as my “red”?’ does not necessarily expects an affirmative answer. Still, we may define labels or tokens (word “red”) to the real-world phenomena, hoping that everyone would sort out relations of their inner mental images with those labels by themselves, likely on sub/unconscious level.
In this context, Chomsky’s LAD (Language Acquisition Device) could be also decoded as the “Label” Acquisition Device, because those innate concepts of the real-world understanding are imbedded into natural human languages. For example, the very concept of number 1 (and other numbers than 1) presents in all(?) grammars of human languages in some form, or even does the dynamic of going from 1 to many via 1+1=2 (through dropped in the explicit way by many modern languages).
Therefore, we may not need to go into a rabbit hole of a purely analytical ontologies, risking to end up with nothing: this is/consists of that and that, which, in their turn, are/consist of even smaller and simpler things, until they disappear. Instead, on some level we may start thinking in terms of a more synthetic ontologies of the type: this could be thought of as that (and that, and that) in their combinations and relations (as well as equally true other combinations of other “that”s).
Based on those deep, even language embedded concepts of natural numbers, formal mathematical axioms, like Peano’s axiom, come quite naturally: for each natural number x there is only one successor y = s(x) = x + 1; if successors are equal then do equal their predecessors; there is only one number – 1 – that is not a successor of any number; 1 with its successors spans the whole collection of natural numbers N.
Mathematics has been always an object of amusement, that such a seemingly artificial construct out of the fantasy world could be so useful being applied to problems of the real world. Meanwhile, mathematical concepts are always based, on the one hand – one the real world phenomena, and on the other hand – on humans’ patterns of the world comprehension. One can reasonably expect that wherever those concepts lead us, their range will be the same – humans’ understanding of the world, i.e. humans may always find areas of their application.
The above implies that extraterrestrial, or artificial intelligence, with potentially different ways of observing and comprehending real world, may develop quite different mathematics. Even humans, using different then the “usual” set axioms have developed quite different mathematical theories, but we may expect ET’s and AI’s difference on a much larger scale.
However, let’s come back in the next chapter to the ways of thinking about “collections of objects” and “mathematical structure defined over them” from the original definition…