Not meaning to nit-pick here (to use the word of the day!), but isn't the very fact of a claim being 'within pure mathematics itself' already something contingent, so that the truths ascertained are relative?


That's what I'm not so sure about (as before, I'm thinking aloud to an extent here), whether those two things can be kept distinct in terms of the type of point at stake here? Especially in terms of what you say about the truth or falsity of the proposition not being contingent upon anything but itself? Without the concepts, isn't the proposition meaningless? In a culture without a concept of multiplication (I don't know if there are some), how can the statement be said to be true (just as aesthetic truths concerning the 'beautiful' are meaningless in a culture without that concept)? I would suggest (this may be the Kuhnian in me coming out) that rather than there being a distinction between mathematics and 'philosophy of mathematics', that mathematics itself is a form of philosophy.

I remember having this sort of debate with an applied mathematician friend (funny breed them - they do soil the subject terribly by actually finding a use for it!  ) a year or two ago - if I remember rightly, her conclusion was that yes, strictly speaking all mathematical truths are contingent, but in the reality of day-to-day maths, one is so far removed from first principles that the question ceases to be important (typical applied mathematician, thinking about it so practically ). I think that  the first part of that conclusion is very much the view of one who essentially adheres to the type of radical scepticism that underlies an analytical philosophical tradition; I'm prepared to believe that there one could find a way of establishing certain mathematical truths as being uncontingent upon anything, but wouldn't that require a certain Kantian insistence on a priori categories as an essential precondition for all knowledge? I don't know how else it could be done (which is not of course to say that it couldn't).

Re group theory: Xenakis used it, in Eonta and other works. Not sure about other composers. Always wondered if anyone had found a musical application for Galois theory (the thing that was once suggested to me gained its notoriety mostly because of the story of Galois dying in a duel, rather more interesting than many other mathematicians' lives!).

Yes you are right Madame Pianiste - there are some well-known remarks about mathematical illumination or revelation made by the mathematician Henri Poincaré. He suggests that the vision reveals a kind of choice, and that the rules for such choice are "extremely fine and delicate. It is almost impossible to state them precisely; they are felt rather than formulated." He connects insight into mathematics with "emotional sensibility."

"Do not forget," he enjoins upon us, "the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true æsthetic feeling that all real mathematicians know, and surely it belongs to emotional sensibility."

This may turn into a competition to see whose nits are bigger. (Men, eh? You leave 'em alone for five minutes...)  But even if the first part of that is true (and I think it is only true in a trivial linguistic sense) I don't think the second part follows from it at all.

You could have different views as to what does or doesn't come within the purview of 'pure mathematics' (in particular the thorny question of whether mathematics and formal logic are co-extensive or not) but that wouldn't itself have any bearing on the issue of whether individual propositions within that scope are, or are not, necessarily true or false.   

FWIW I do think it is vital to keep a distinction between mathematics and metamathematics to avoid conceptual confusion ('confusion' also needing to be distinguished from 'radical scepticism' ). And, while I don't know whether there is some Polynesian island as yet unsullied by anthropologists where there is no concept of multiplication, it certainly seems unlikely, for example, that cows, dogs have such a concept. But it is perfectly possible, and essential, to distinguish that fact from questions of whether mathematical propositions are necessarily true or false. If you were pursuing the latter question, interviewing cows might be seen as, well, an eccentric approach   -   even by the standards of maths departments (Hmm, maybe not. Scrub that last little bit. )     


Thanks for that. To be added to the impossibly long list of things arising from these MBs to be followed up.

Well it's certainly old hat in one area we wouldn't necessarily think of as being musical composition, the creation of methods and touches in English-style bell ringing.  Ringers had grasped some of the concepts of Group Theory before mathematicians did (although obviously they didn't realise the potential wider implications).

I think the idea behind the Frege/Russell set-theoretical definition of number is that it's the most fundamental one they could think of - remember that Russell was trying to put the whole of mathematics on an unassailably logical basis. It needs only to define the concept of an empty set and of a successor function, and therefore treats a concept like "two" as not being a fundamental axiom.

However, if all maths (let alone life) was conducted from these kind of first principles nobody would get anywhere. Still I find this "purest of pure maths" very interesting.

There's a certain logic to the techniques used.  They are not, however *strictly* mathematical, as aesthetic considerations cannot, on the whole, be axiomatized in this sense.

As for my chunk on this talk of math revealing for us the beauty of music, I don't like that approach at all.  Physical, physiological, and psychological understanding are the only ways we can learn what different sounds cause us to experience; mathematical and other logical processes can be developed in order for composers to control how to get control of certain sounds/effects &c., or maybe helping people to discover how to produce new types of sounds.


*cough* I'm *really* skeptical about the application of these special ratios to large-scale musical forms in most cases where they're used.


This is more an application of mathematical reasoning to the visual arts; such things are not visually in and of themselves *that* mathematical. Purer examples exist, such as all these abstract isomorphism theorems and dualities that exist.


Very charming book!

Ian, so far as I've seen, the book Formalized music isn't exactly bullshit from what I've seen, just, from a mathematical perspective, extremely watered-down content-wise.

I think that some of the people here talking about metmathematics are talking more about the philosophical foundations of mathematics; metamathematics, to mathematicians, is a very definite field of mathematics.


Forte, I think, gives the most convincing applications of group theory to music; of course, this was all analytical work, and not strictly compositional.  One can conceptualize strict serial music quite well via group theory, of course.


I can notionally think of a few ways that people might get at applications of galois theory to music...one is the way one can use it to iteratively construct solutions to construct numbers (say intervals) with certain properties, or maybe the dual theory to Galois which is all about covering spaces. I ramble. But I don't think any has tried Just Yet. 

It was not, entirely; put that down to me having to catch up on so much stuff here when I caught up on the thread this afternoon.


I'm not entirely sure I understand what you are talking about you being comfortable with above; could you expand upon it a little?

Well there seems to be some sort of link, even if not an obvious one, jsut look at the number of people on this board popping up with maths degrees (me too incidentally)