[Marxism] Fermat's last theorem - reply to Domnhall

Les Schaffer schaffer at optonline.net
Tue Apr 27 07:38:56 MDT 2004


Jurriaan:

i do not see how letting n 'equal' infinity gives anything worthwhile. 
after all, the equation would then be satisfied for __every__ x, y, and 
z  (with suitable respect for quadrants of the complex plane).

furthermore, x^n + y^n = z^n   (n = 1, 2, 3, ...) would be trivially 
true for x, y, and z = infinity as well, for any finite n

if the point is somehow to include "infinity" in some natural way with 
the rest of numbers, this has already been done ages ago; for example, 
the projective sphere, and stuff like that.

>So even at this
>very primitive level, things go wrong, because the authors neither (a)
>respect the simplest requirements of measurement theory
>

i dont see how your comments on fermat really lend any weight to this 
argument Jurriaan. perhaps you can tighten and re-express your basic 
argument?

>(B) This leads me to the second point to be made here, which is probably
>more familiar: namely, that quantification may provide the illusion of
>rigour and systematicity, while in fact the rigour and systematicity is
>spurious, because while the logic may be impeccable, it's not clear what
>objects the numbers and their manipulations actually refer to, or what
>ontology or semantic universe is being assumed. 
>

i would apply this argument to taking n "equal to" infinity in Fermat. 

>But, Marx says, that theoretical structure might itself be in
>part ideological, i.e. in merely reflects particular interests and needs
>which people had.
>  
>

it would be of more interest to me to look at the history of number 
theory and see what specific connections Fermat's type of work had with 
his material times. in this connection, the thread Fermat --> 
probability --> rise of insurance and the class of rich gambling 
gentlemen might be more fruitful to begin with.

also, Fermat's theorem traces back to Diophontane equations and ancient 
Greece. Anything interesting to look at there???
from Kline (Mathematical tought from ancient tto modern time, Volume I):

"... This theorem was stated by Fermat in a marginal note in his copy of 
Diophantus alongside Diophantus' problem: To divide a given square 
number into (a sum of) two squares. "

we know the Greeks were interested in geometry and the parceling of 
land. Kline continues:

"On the other hand it is impossible to seperate a cube into two cubes, 
..., or generally any power except a square into two powers with the 
same exponent. "

note the abstraction beyond simple geometric ideas. Klein a few pages later:

"... John Wallis, influenced by Vieta, Descartes, Fermat, and Harriot, 
went far beyond these men in freeing arithmetic and algebra from 
geometric representation."

this "freeing" surely makes an interesting double-edged sword. by 
abstraction, we gain utility (complex numbers, etc), insight, expressive 
power and new avenues for attack on problems. but with freeing from the 
geometric (read "traditional material applications")  allows the elite 
knowledged class to split further off  from its laboring roots. but i am 
thinking too of Bohr and Heisenberg and the Copenhagen interpretation of 
quantum mechanics, where the practical working formalism availed itself 
to some hazy philosphic meanderings from which we still suffer today.

and i know Charles Brown will be interested that Fermat was a lawyer 
(according to Struik, A concise history of mathematics).

les schaffer




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