Monday, February 26, 2007

More praise for error


Here's Richard 'two stage' Taylor:

Error is unavoidable in any process. There are various mathematical and physical proofs showing that error in measurement or in any quantifiable process is impossible to avoid.

A simple example is an attempt to measure the area of a circle. That is Area = pi x diameter. But pi is a transcendental number, so no exact measurement of a circle is possible. In the real world, engineers ultimately refer to 'probablity density functions' - this means that in all cases they are talking probablities, not certainties.

For most things of course there is (or can be) such a high degree of accuracy that error is greatly reduced, but it cannot be eliminated. This is a reality of the nature of the physical world. We would still be unicellular organisms without error.


And here's Trotsky:

The Aristotelian logic of the simple syllogism starts from the proposition that 'A' is equal to 'A'. This postulate is accepted as an axiom for a multitude of practical human actions and elementary generalisations. But in reality 'A' is not equal to 'A'. This is easy to prove if we observe these two letters under a lens—they are quite different from each other.

But, one can object, the question is not of the size or the form of the letters, since they are only symbols for equal quantities, for instance, a pound of sugar. The objection is beside the point; in reality a pound of sugar is never equal to a pound of sugar—a more delicate scale always discloses a difference. Again one can object: but a pound of sugar is equal to itself. Neither is this true—all bodies change uninterruptedly in size, weight, colour, etc. They are never equal to themselves.

A sophist will respond that a pound of sugar is equal to itself 'at any given moment'. Aside from the extremely dubious practical value of this 'axiom', it does not withstand theoretical criticism either. How should we really conceive the word 'moment'? If it is an infinitesimal interval of time, then a pound of sugar is subjected during the course of that 'moment' to inevitable changes. Or is the 'moment' a purely mathematical abstraction, that is, a zero of time? But everything exists in time; and existence itself is an uninterrupted process of transformation; time is consequently a fundamental element of existence. Thus the axiom 'A' is equal to 'A' signifies that a thing is equal to itself if it does not change, that is, if it does not exist.

At first glance it could seem that these 'subtleties' are useless. In reality they are of decisive significance. The axiom 'A' is equal to 'A' appears on one hand to be the point of departure for all our knowledge, on the other hand the point of departure for all the errors in our knowledge. To make use of the axiom of 'A' is equal to 'A' with impunity is possible only within certain limits. When quantitative changes in 'A' are negligible for the task at hand then we can presume that 'A' is equal to 'A'. This is, for example, the manner in which a buyer and a seller consider a pound of sugar. We consider the temperature of the sun likewise. Until recently we consider the buying power of the dollar in the same way.

But quantitative changes beyond certain limits become converted into qualitative. A pound of sugar subjected to the action of water or kerosene ceases to be a pound of sugar. A dollar in the embrace of a president ceases to be a dollar. To determine at the right moment the critical point where quantity changes into quality is one of the most important and difficult tasks in all the spheres of knowledge including sociology...

Vulgar thought operates with such concepts as capitalism, morals, freedom, workers' state, etc as fixed abstractions, presuming that capitalism is equal to capitalism. Morals are equal to morals, etc. Dialectical thinking analyses all things and phenomena in their continuous change, while determining in the material conditions of those changes that critical limit beyond which 'A' ceases to be 'A', a workers' state ceases to be a workers' state.

The fundamental flaw of vulgar thought lies in the fact that it wishes to content itself with motionless imprints of a reality which consists of eternal motion. Dialectical thinking gives to concepts, by means of closer approximations, corrections, concretisation, a richness of content and flexibility; I would even say 'a succulence' which to a certain extent brings them closer to living phenomena. Not capitalism in general, but a given capitalism at a given stage of development. Not a workers' state in general, but a given workers' state in a backward country in an imperialist encirclement, etc.

Dialectical thinking is related to vulgar in the same way that a motion picture is related to a still photograph. The motion picture does not outlaw the still photograph but combines a series of them according to the laws of motion. Dialectics does not deny the syllogism, but teaches us to combine syllogisms in such a way as to bring our understanding closer to the eternally changing reality.

1 Comments:

Blogger Richard Taylor said...

Trotsky writes well on this subject.

There is a fascinating "take" on this by J Brownoski in his great book THE ASCENT OF MAN - in the chapter "Knowledge or Certainty".* This deals with the Holocaust and the dangers of the arrogance of absolutism that,as weell as leaading to such view as thsoeof Hitler and his crew, affected some scientists - including some he knew personally such as J von Neumann - who proved (ironically) that pi is (Trotksy;s exmple is another way of looking at this - that pi is not an integer doesn't really matter) transcendental and a also predicted the DNA self reproduction process - he talks of many aspects of the paradox of knowledge and brings Guass's probability curve into the picture - he leads up to a discussion of the implictations of the dangers of absolutist arrogance - they equate in his documentary to Auschwitz - he stands at Auschwitz - scoops up the mud and ashes and comments that some of the people here almost literally that are now the ash and mud in his hands were his relatives (he is Jewish) - and this leads also to the dropping of atomic bombs on Japan by the US.

Also - science and maths and philosophy interact - and politics.

There is also Godel's paradoxical (but very real and true) theorem that shows that it is impossible to provce any logical propostion in maths or science. I dont think this can be affected by Russell's or Zeno's paradox etc

These things "can be shown".

* Also there is a lot on this kind of thing in popular books about the ideas of maths.

1:25 am  

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