The Mathematician / Physicist in me finds this reduction to the simplest 'atoms'/particles attractive.  Basic learning may start with 1, 2, 3, 4, etc, but why define all numbers up to 10 (or 100) as axioms?  It's just inefficient, and illogical (where do you draw the line).

Frege/Russell got it down to an empty set and a successor function and thus derived 1.  Could you start with 0 and 1 as axioms and derive the successor function?  That would seem more natural - but I do grant you that 'natural' in this sense is probably dictated by how we learn numbers (i.e. 1, then 2, then 3....).

Sorry, this has strayed off the music side of things again.

Tommo

No we cannot find a good word for Russell. He was not a deep thinker. What little respect we may once have had for him evaporated immediately and absolutely upon inspection of his dreadful History of Western Philosophy wherein he dismisses Schelling in three sneering lines ("Schelling was more amiable [than Fichte] but not less subjective. He was closely associated with the German romantics; philosophically, though famous in his day, he is not important") while devoting a chapter each to the less worthy Kant and Hegel. If Russell could be so wrong and silly about Schelling, it is impossible to trust his judgement on any matter philosophical logical or mathematical.

Whitehead though was an astute and level-headed man; we were suckled on his Process and Reality.

You may be right about M'Lord Russell's character, Master Grew. His rather clumsy attempts to foist himself as a wet-nurse upon my grandmother were, according to whispered family legend, not the actions of a true gentleman. Sadly, we do not recall the presence of Process and Reality on our own nursery menu. In the austere post-war years Cow and Gate was the best that Nursie could find us and we suspect that was on the black market.

Also they had trouble with 2, from what I recall; they had issues with the fact that adding it to itself is the same as multiplying it by itself.  In addition, though the Indian dudes had a symbol for 0, it only really obtained it's full modern function as a number when the Arabs adopted it, I think.

There is also at least one (tribal) language that don't have numbers above three or four.

Ah yes.

http://www.mathematicalbrain.com/guard04b.html

This doesn't seem like an entirely inappropriate thread to post this thought on, given that I was going to give something of a compositional slant to it anyway; I was talking to a mathematician today, and he described to me the immense difficulty some problems that people are faced with posed, how they can cause intense stress/depression &c., but more than that, personally finding them intensely disturbing sometimes psychologically (possibly it might be reasonable to take the abstract nature of the problems as a source of this); do any of the composers on this board, or the musicians, ever find this when composing their own music, or in playing/learning the music of others?

For personal and emotional reasons, I cannot seem to bring myself to use the results of mathematical operations in my music, i.e., "generate" musical material from mathematical operations. For some reason, I have a block against the very word "generate". Instead, I try to set up situations that resemble improvisation, where I am confronted with musical possibilities whose structural provenance I cannot determine or cannot remember. Then I observe myself making decisions about how to shape that material, and I try to do so in an unsystematic, purely "intuitive" way, convinced that my intuition has its own structure and not caring to peek under the hood any further than that (i.e., figuring out the very mechanics of my own intuition -- great way to drive oneself crazy, that would be!*)

Though I am strong in mathematics, it took me a long time to realize that my understanding of that subject was actually getting in the way of my enjoyment of the composition process, even getting in the way of hearing (?). Math is a will-o-wisp, an attractive light that continually tempts me into the swamp of unreal order; I have to largely ignore it if I intend to make (finish!) something I can be happy with. After all, reality is messy, or the clash of conflicting realities is more 'real' than any order one imposes on it.

I'm not sure this helps the maths and music discussion, since it seems to disavow a fruitful relationship between the two. But I don't deny that they are linked, and predilections toward them do seem to correlate!

*since one is bound to fail.

(CD, I completely understand all that, and don't think I'm so different to you. I try to combine improv-related methods of creating with more systematic methods, in whatever proportion and in whatever way seems appropriate to the job in hand.)

Inky, by 'mince-machines' I simply mean parsing various pitch or rhythmic material, or some notions of large-scale proportion, through some kind of ordered process, however uninstinctive it initially seems, just to see what comes out the other end. Somtimes that can yield very illuminating results; ones which I/one can take away and 'play' with more intuitively, potentially, or indeed use as they are. Seems to me that often the process of creating 'art' can be one of tricking oneself into fecundity - often against the most unlikely odds.