It appears to me that the natural sciences' essential difference emerges because it is relatively easy to combine individual cases of observation and experience into unconditionally valid general laws, whereas for the humanities this is insurmountably difficult. Indeed, in mathematics, the first general principles which are given pride of place as axioms are so few in number and of such infinite scope and such immediate evidence that they do not need a single proof. Consider that the entire field of pure mathematics (arithmetic) is developed from the following three axioms:
"If two quantities are equal to a third, then they are equal to each other."
"Equals added to equals, gives equals."
"Unequals added to equals, gives unequals."
The axioms of geometry or theoretical mechanics are no more numerous. The above-mentioned disciplines are developed from these few basic premises, in that conclusions are drawn from them in ever more intricate ways. Arithmetic does not limit itself to adding the manifold sets of a finite number of quantities; in higher analysis, it even theorizes an infinite number of sums to be added, whose quantities grow or diminish according to the most varied laws. Thus, arithmetic seeks to solve problems which could never directly be realized on earth. Here we see the conscious logical activity of our mind [Geist] in its purest and most complete form; we can acquaint ourselves with its great efforts, the tremendous care with which it must advance, the precision which is necessary to determine the scope of the general propositions, the difficulty in forming abstract concepts and to understand, but also how to have trust in the security, consequences, and fruitfulness of this kind of intellectual work.
This is even more apparent in applied mathematics disciplines, namely in mathematical physics, which includes astrophysics. After Newton had recognized, through a mechanistic analysis of planetary movements, that all matter attracts with a power that is in inverse proportion to the square of the distance, this simple law was sufficient to calculate planetary movements in the furthest reaches of the past and future with the utmost accuracy, if the location, speed, and mass of all the individual heavenly bodies in our system are given for any particular instance. Indeed, we can recognize the effects of this power in the movements of twin stars whose distance apart is so great that it takes years for their light to reach each other, so far apart that our efforts to measure it have failed up to this point.
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