[chill]

<cheap shot>And what is this substance of mathematics?</cheap shot>

Calling mathematics "incontrovertibly true" and stating that abstractions used in mathematics can be explained concretely in simpler terms are philosophical positions and the justifications for and against these positions are debated outside mathematics in philosophy. You praise the rigor of mathematics while ignoring the rigor in which philosophy searches for the justifications of assumptions for all fields of human knowledge including mathematics and science.

[gambling8nt]

The substance of mathematics is figuring out what is still true when one applies the highest standard for rigor that still permits the possibility of determining things not yet known to be true.

That any philosopher might continue to debate this issue is, while sadly true, irrelevant. A logical system without modus ponens can either define modus ponens, or is clearly not extensible enough to define anything. A logical system with modus ponens is either inconsistent, or at most as general as mathematics. Thus, there can be no extensible system of truth in which mathematics is false, validating my claim that it is incontrovertible.

As to whether or not mathematics can be explained in simple terms, I will suggest that this appears to be the case empirically, since some people become mathematicians, and they all seem to use the same terminology, more or less. To prove that every mathematical result in existence can be explained simply would require that I actually explain every mathematical result simply, and that is beyond the scope of an HN comment.

Finally, philosophers do not "search for the justifications of assumptions for all fields of human knowledge including mathematics and science" with the level of rigor I specified in my comment; indeed, the entire point of this discussion is that they tend not to use any level of rigor at all. Insofar as philosophy actually does anything to "justify the assumptions" for any field, it occasionally achieves useful results; however, much of philosophy is focused solely on a continuing failure to agree on semantics.

[chill]

Yet this failure to agree on semantics is in part because of the difficulty in providing solid, incontrovertible justifications for these meanings. Consider the new branches of mathematics that were [created|discovered] when Euclid's definition of a straight line were questioned.

That mathematics corresponds so well to the world we perceive is amazing. Why should this be the case? How can we be sure that mathematics and science holds for all cases which we do not observe or that they will continue to do so? Can rigorous justifications be given for these questions that do not rely on circular arguments and blind faith?

[gambling8nt]

We can't guarantee anything about the world. Empirically, science seems to work, and that is all we can ever get from it. Math is true regardless of its utility in science, and we use it in science because it is convenient to do so.

There are no solid, incontrovertible definitions. Only solid incontrovertible proofs (even there, we don't generally actually know whether or not a proof is incontrovertible, because proofs are rarely verified on that level...but, within mathematics, it is at least POSSIBLE to either verify a proof on that level, or provide a verifiable flaw). Definitions are a matter of convenience.

That the semantics of mathematics sometimes change does not imply that any of its terminology are ever inherently complex in the sense that those who don't understand it, "Don't get it." Indeed, since semantics in mathematics are only ever a shorthand, every mathematical construct could conceivably be expanded into the language of logic, and every mathematician (modulo human error) would agree that the expansion was valid. This underlying agreement on meaning is precisely what is missing from fields like much of literary criticism and philosophy.