In addition to Hofstadter's wonderful writings, you might also be interested in work done on the relationships between mathematics and cognition (more generally than just consciousness). Take a look at these classics in that area: Rochel Gelman & C.R. Gallistel, The Child's Understanding of Number (Harvard University Press, 1986) George Lakoff & Rafael Nuñez, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (Basic Books, 2001) Stanislaus Dehaene, The Number Sense: How the Mind Creates Mathematics (Oxford University Press, 2011)

You asked what Goedel's incompleteness and inconsistency theorems state. Goedel proved two theorems known as his incompleteness theorems; I don't know of any called an "inconsistency" theorem (of course, he proved many other theorems, too!): Informally, the first one--perhaps it is also the most famous one--says that any formal system that is based on first-order logic plus Peano's axioms for arithmetic is such that: if it is consistent (that is, if no contradiction can be proved in it), then it is incomplete (that is, there is some proposition P in the language of the system such that neither P nor not-P can be proved in the system; presumably, only one of P and not-P is true; hence, there is some proposition in the language of the system that is true but unprovable in the system). Even more informally, an English-language version of the true-but-unprovable sentence can be expressed thus: This sentence is not provable. (If it is false, then it is provable, hence true. So...

I don't know anybody who claims that 0 is not a fraction. But I suppose it depends on what you mean by "fraction". If you mean a numeral (that is, a name or description of a number) that is normally written in the format: (integer numeral)/(integer numeral), e.g., 3/4, then, I suppose, strictly speaking, "0" is not a fraction. But you are correct to point out that "0/4" is a fraction. It is then common to identify integer numerals like 0,1,2, ...with fractions 0/1, 1/1, 2/1,... (and, of course, fractions like 2/1 are identified with fractions 4/2, 6/3, etc., just as 1/2 is identified with 2/4, 3/6, etc.). What's really going on here is that a single number , say 4, can be written as lots of different fractions: 4/1, 8/2, 16/4, etc., and these different numerals are then "identified" as naming or describing the same number . So, fractions are numerals . Perhaps what you have in mind are rational numbers , which are typically written as fractions. And, of course, 0 is a perfectly...

Using Gottlob Frege's theory of sense and reference, you might say that '3' is the name of the natural number that is the third successor of 0, and that 'the (positive) square root of 9' is a (definite) description of that very same number. The name and the description have different senses, but the same referent; the senses "get at" the same mathematical object in two different ways.

When an elementary-school student is learning how to multiply, the result of multiplying 2 by 2 is probably produced by a form of reasoning (perhaps repeated addition). When you or I do it, it's probably done by rote memory. But when any of us multiply two 6-digit numbers, it's almost certainly by "reasoning". (Maybe those with "savant syndrome" do it by some kind of memory-like process, or maybe it's just by very fast, unconscious reasoning.) But the "reasoning" we use for multiplying those larger numbers consists of applying the multiplication algorithm, among whose steps are instructions to multiply single-digit numbers together (like 2x2, or 9x8). And those multiplications are probably done by "memory" (what computer scientists call "table look-up"). That's because multiplication is a recursive procedure: We multiply large numbers by applying the multiplication algorithm, which requires us to multiply smaller numbers, eventually "bottoming out" in the base case of table-look up of the...

It doesn't imply that, because that's false. If pi had an infinite number of the same digits repeated, it would be a rational number. But it's an irrational number. See Wikipedia's article on pi .