archimedes, viete and fermat

23 10 2009

Seems like the Greeks were limited in their appreciation of ∞. Without algebra, and dependent on pragmatics of geometry, Archimedes used Eudoxus method of exhaustion to work out the area circles and parabolas. Eg, inscribing and circumscribing circles with polygons, each being calculable as a set of triangles, getting a degree of accuracy as required.

Viete was the first european to create a series in 16th century, namely

infinite productSeems it is the dot dot dot which is significant, breaking some kind of psychological barrier, the idea of an infinite series.

Then came Descartes with his geometry, one of three appendixes to his philosophical work, and Fermat and Pascal. Then Newton and Leibnitz, of course. The book I am reading refers to ‘indivisibles’ whereas I am more familiar with the term ‘infinitesimals’. Zeno/s paradox, of an infinite number a formof steps, 1/2 + 1/4 + 1/8 + 1/16 etc will never reach one… the notion of an infinite series tending to a finite amount, in this case the limit is one.

The fact that it took so long for people in the west to approach this idea. I think it was accepted earlier, as it were, in the east. Not so much numerically, objectively with geometry, but rather the other applies side of maths, subjectively. As time tends to zero, subjectively experienced, strange things start to happen. Awareness of how one thought follows another. Uncovering the mechanism of preconscious thought, fiddling around with the projector such that the thoughts that arise are wholesome.

The strange thing is, by performing some clever tricks, transforming the shape of the expression, Fermat came round to defining a formula for the area beneath a parabola y=x^n by thinking of rectangles who bases form a geometric progression of a, ar, ar^2…


parabola1…which reduces to this when r  -> 1

parabola2…which happens to be the integration formula 0∫a x^n dx = a^n+1 / (n+1), thirty years before Newton and Leibniz.

That is, there is something about a technique used which the ancient greeks could not grasp that enabled the infinite to become comprehensible, or well behaved, manipulateable. What it makes me think of, in terms of the mind-drop solution, is that the dissolution of consciousness downwards into the preconscious processes, which is effectively infinite, somehow tends to a limit. The same for the other direction, upward to higher states of consciousness, perhaps. Allowing for thoughts that come to me while I write this, seems to be related to approaching the speed of light. That is, there is a barrier, which can not be broken if one continues with that line of thinking, but if one conceives of it, or becomes aware of it, at a higher/lower level, the barrier is simultaneously overcome, or perhaps conceived of, or perhaps there is a phase change of state of consciousness.

I am interested in the technique. The revolution of mind that enables the calculus to be derived.

Hmmm… upon editing, other things pop into my head. The conic sections wrt space-time…





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