There is no royal road to science and only those who do not dread of its fatiguing climb have a chance of gaining its luminuos summits.
-Karl Marx

Dec 5, 2012

Negation of Negation

     With most insects, this process follows the same lines as in the case of the grain of barley. Butterflies, for example, spring from the egg by a negation of the egg, pass through certain transformations until they reach sexual maturity, pair and are in turn negated, dying as soon as the pairing process has been completed and the female has laid its numerous eggs. We are not concerned at the moment with the fact that the process does not take such a simple form with other plants and animals, that before they die they produce seeds, eggs or offspring not once but many times; our purpose here is only to show that the negation of the negation really does take place in both kingdoms of the organic world.
        Furthermore, the whole of geology is a series of negated negations, a series in which old rock formations are successively shattered and new ones deposited. First the original earth crust formed by the cooling of the liquid mass was broken up by oceanic, meteorological and atmospherico-chemical action, and these fragmented masses were stratified on the ocean bed. Local elevations of the ocean bed above the surface of the sea subjected portions of these first strata once more to the action of rain, the changing temperature of the seasons and the oxygen and carbon dioxide of the atmosphere. These same influences acted on the molten masses of rock which issued from the interior of the earth, broke through the strata and subsequently cooled off. In this way, in the course of millions of centuries, ever new strata were formed and in turn were for the most part destroyed, ever anew serving as material for the formation of new strata. But the result of this process has been a very positive one: the production of a soil out of a mixture of the most varied chemical elements and in a state of mechanical pulverization, which makes possible the most abundant and diversified vegetation.
        It is the same in mathematics. Let us take any algebraic quantity we like: for example, a. If it is negated, we get -a (minus a). If we negate that negation by multiplying -a by -a, we get + 2, i.e., the original positive quantity, but at a higher degree, raised to its second power. It makes no difference in this case that we can obtain the same 2 by multiplying the positive a by itself, thus likewise getting 2. For the negated negation is so securely entrenched in 2 that the latter always has two square roots, namely a and -a. The fact that it is impossible to get rid of the negated negation, the negative root of the square, acquires very obvious significance as soon as we come to quadratic equations.

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