Marx, late in life, made some note on Mathematical subjects which, however, weren't published till the 1930s. It would be unfair to make fun of a non-mathematician for jotting down random thoughts which occurred to him as he tried to make sense of an alien subject.
Nevertheless, such jottings may be of interest to mathematical economists working in the Marxist tradition. In particular, those who wanted to get rid of marginal analysis which Kantorovich had implicitly incorporated into his work on 'objectively determined valuations' or shadow prices, might have asked whether non-standard analysis of some type could offer a non-heretical way forward.
Consider the following note by Marx on 'Taylor Theorem, MacLaurin’s Theorem, and Lagrange’s Theory of Derived Functions’
Lagrange, towards the end of the eighteenth century, had made an attempt to found calculus purely on algebra. By about 1830, with the work of Cauchy, it became obvious to the mathematical community that this could not be done. However, this knowledge may not have percolated very far.
Newton’s discovery of the binomial (in his application, also of the polynomial) theorem revolutionised the whole of algebra, since it made possible for the first time a general theory of equations.
We would say this problem was solved completely by Galois in 1830. Prior to that there was no clear distinction between 'theory of equations' and 'algebra'. About 15 years previously, Argand had shown that the fundamental theorem of algebra (d'Alembert-Galois theorem) did not have a purely algebraic proof. By about the mid nineteenth century this was widely accepted. You can minimize the non-algebraic assumptions but you can't do away with them altogether. Marx may have thought this was still an open question.
The binomial theorem, however - and this the mathematicians have definitely recognised, particularly since Lagrange - is also the primary basis (Hauptbasis) for differential calculus.
Everyone could see that Newton needed it to get to calculus.
Even a superficial glance shows that outside the circular functions, whose development comes from trigonometry, all differentials of monomials such as xm, ax, log x, etc. can be developed from the binomial theorem alone
Logarithms predate the binomial theorem. Slide rules were being used 20 years before Newton was born.
It is indeed the fashion of textbooks (Lehrbuchsmode) nowadays to prove both that the binomial theorem can be derived from Taylor’s and MacLaurin’s theorems and the converse.
Apparently, Taylor's work didn't get much attention till Lagrange said it was the basis of differential calculus (which already existed before Taylor was born). I suppose he meant that his own treatment of classical mechanics drew much on Taylor.
Nonetheless nowhere - not even in Lagrange, whose theory of derived functions gave differential calculus a new foundation (Basis)
Lagrange sought to define derivatives algebraically from the Taylor series expansion, rather than through limits. Sadly this does not appear to be possible and this was well known by the mid nineteenth century.
- has the connection between the binomial theorem and these two theorems been established in all its original simplicity, and it is important here as everywhere, for science to strip away the veil of obscurity.
I suppose Marx knew that Newton was Master of the Mint. Maybe history had taken a wrong turn because a proper labour theory of value was suppressed by evil Capitalists.
Taylor’s theorem, historically prior to that of MacLaurin’s, provides - under certain assumptions - for any function of x which increases by a positive or negative increment h,76 therefore in general for f(x±h), a series symbolic expressions indicating by what series of differential operations f(x±h) is to be developed. The subject at hand is thus the development of an arbitrary function of x, as soon as it varies.
This is nonsense. Taylor's theorem gives a good approximation but only for differentiable functions. It is useful. It isn't magic.
MacLaurin on the other hand - also under certain assumptions - provides the general development of any function of x itself, also in a series of symbolic expressions which indicate how such functions, whose solution is often very difficult and complicated algebraically, can be found easily by means of differential calculus.
An approximation is useful enough. But Fourier Series, introduced in 1807, were often better. Pade approximants came into use by about 1890.
The development of an arbitrary function of x, however, means nothing other than the development of the constant functions combined with [power of] the independent variable x, for the development of the variable itself should be identical to its variation, and thus to the object of Taylor’s theorem.
Only differentiable functions are the object of the theorem whose utility is it makes it quick and easy to get a good enough approximation a lot of the time. But there were better methods.
Perhaps, Marx, in his old age was afraid that younger men would make innovations in his theory and get all the credit. Thus he wants the whole of calculus, indeed algebra, to be contained in the binomial theorem or the Taylor expansion. In this way, he remains the owner of every subsequent advance in his theory. Sadly, his theory was shit. No advance could be made.
It may now be asked:
Did not Newton merely give the result to the world, as he does, for example, in the most difficult cases in the Arithmetica Universalis, having already developed in complete silence Taylor’s and MacLaurin’s theorems for his private use from the binomial theorem, which he discovered? This may be answered with absolute certainty in the negative: he was not one to leave to his students the credit (Aneignung) for such a discovery. In fact he was still too absorbed in working out the differential operations themselves, operations which are already assumed to be given and well-known in Taylor and MacLaurin. Besides, Newton, as his first elementary formulae of calculus show, obviously arrived at them at first from mechanical points of departure, not those of pure analysis.
As for Taylor and MacLaurin on the other hand, they work and operate from the very beginning on the ground of differential calculus itself and thus had no reason (Anlass) to look for its simplest possible algebraic starting-point, all the less so since the quarrel between the Newtonians and Leibnitzians revolved about the defined, already completed forms of the calculus as a newly discovered, completely separate discipline of mathematics, as different from the usual algebra as Heaven is wide (von der gewöhnlichen Algebra himmelweit verschiednen).
The relationship of their respective starting equations to the binomial theorem was understood for itself, but no more than, for example, it is understood by itself in the differentiation of xy or (x/y) that these are expressions obtained by means of ordinary algebra.
The real and therefore the simplest relation of the new with the old is discovered as soon as the new gains its final form, and one may say the differential calculus gained this relation through the theorems of Taylor and MacLaurin. Therefore the thought first occurred to Lagrange to return the differential calculus to a firm algebraic foundation (auf strikt algebraische Basis). Perhaps his forerunner in this was John Landen, an English mathematician from the middle of the 18th century,
who contributed a column on mathematics to a ladies' journal!
in his Residual Analysis. Indeed, I must look for this book in the [British] Museum before I can make a judgement on it.
Why not just ask a Math professor? Can Calculus be given a purely algebraic foundation? No. Lagrange & Landen thought differently. They were wrong.
But, if Lagrange was wrong about math, might Marx not be wrong about political economy? This is the fear that motivates his 'mathematical' researches.
Lagrange’s great service is not only to have provided a foundation in pure algebraic analysis for the Taylor theorem and differential calculus in general, but also and in particular to have introduced the concept of the derived function, which all of his successors have in fact used, more or less, although without mentioning it. But he was not satisfied with that. He provides the purely algebraic development of all possible functions of (x + h) with increasing whole positive powers of h and then attributes to it the given name (Taufname) of the differential calculus. All the conveniences and condensations (Taylor’s theorem, etc.) which differentials calculus affords itself are thereby forfeited, and very often replaced by algebraic operations of much more far-reaching and complicated nature.
Marx is saying 'I am the Lagrange of Political Economy. I have given all History a foundation in purely economic analysis based on the labour theory of value. True, like Newton, I may not be aware of certain useful applications which flow from my own theory. This does not detract from my greatness.
2) As far as pure analysis is concerned Lagrange in fact becomes free from all of what to him appears to be metaphysical transcendence in Newton’s fluxions, Leibnitz’s infinitesimals of different order, the limit value theorem of vanishing quantities, the replacement of 0/0 ( = dy/dx ) as a symbol for the differential coefficient, etc. Still, this does not prevent him from constantly needing one or another of these ‘metaphysical’ representations himself in the application of his theories and curves etc.
Marx was aware that he wasn't 'Marxist' himself. He wanted to give an economic explanation for events but would get carried away by his own savage indignation at the philistinism of the bourgeoisie. The truth is, Math developed fast in places where the application of Math made the most money or most increased military security. The Indians and the Japanese and so forth had infinite series but they never bothered to develop calculus. Why? There was no pressing financial motive. Neither country was engaged in oceanic commerce. They weren't competing with neighbours who might invade if their fleet grow bigger thanks to their greater maritime commerce.
Competition isn't always a good thing. It can be wasteful or involve the production of nuisance goods. One way of reducing competition is to become a vertically integrated monopolist. You control every step in the production and marketing process and, hopefully, this keeps you safe from potential rivals. Marx faced competition from other radical political economists. If his system had an independent foundation within itself, then it was vertically integrated and relatively immune to attack. Equally important, Marx would have laid the foundations for the discovery of 'laws of motion' for Capital similar to the Euler-Lagrange equations. It was a pipe dream. Still, Marx became the name of an actual demon- unlike Laplace's merely hypothetical one. A little knowledge is a dangerous thing.
No comments:
Post a Comment