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The syllogistic method of presentation employed in the Euclidean geometry led to the belief that the more special theorems had been derived from the simpler ones, and these from the axioms, by a process of conceptual analysis; while the fact is that in mathematics all progress is by intuition alone, the syllogism serving merely to formulate and explain truths already attained, but not to supply new ones.
The following description will show that, directly we free ourselves from the onlooker-limitations of our consciousness in the way shown by Goethe and, in respect of the present problem, in particular also by Reid the ideal relationship between the two theorems is seen to be precisely the opposite to the one expressed in the above statement.
IV. Have I done anything charitably? then am I benefited by it. See that this upon all occasions may present itself unto thy mind, and never cease to think of it. What is thy profession? to be good. And how should this be well brought to pass, but by certain theorems and doctrines; some Concerning the nature of the universe, and some Concerning the proper and particular constitution of man?
Thus we are led to the striking conclusion that mythology has had a common root, both with science and with religious philosophy. The myth of Indra conquering Vritra was one of the theorems of primitive Aryan science; it was a provisional explanation of the thunder-storm, satisfactory enough until extended observation and reflection supplied a better one.
We have to alter one or two definitions. Definitions come first in geometry and in everything else. Get them right, and you will get your theorems and problems right.
It is such quite simple questions as these, and the quite plain facts which underlie them which will lead to sounder conceptions in this matter on the part of the peoples. It is not we who are the "theorists," if by "theorists" is meant the constructors of elaborate and deceptive theorems in this matter.
Laplace, indeed, attempted to prove, by means of two well-known theorems relative to the eccentricities and inclinations of the planetary orbits, that if those elements were once small, they would always remain so, provided the planets all revolved around the sun in one common direction and their masses were inconsiderable. It is to these theorems that M. Arago manifestly alludes in the text.
Hence the mist, the vague dimness that surrounds faith, baffling every effort to penetrate it; and within, a sense of rarefied perception that disquiets and torments unless humility born of common sense be there to soothe and set us at rest. Moral truths are not geometric theorems and multiplication tables, and it is not necessary that they should be.
Now this is also precisely what we mean, when we say that some tyrants are lawful sovereigns; which, therefore, is not a second proposition inferred from the first, any more than the English translation of Euclid’s Elements is a collection of theorems different from and consequences of, those contained in the Greek original.
These and similar propositions affirm resemblances, of the same nature with those asserted in the theorems of mathematics; but the distinction is, that the propositions of mathematics are true of all phenomena whatever, or at least without distinction of origin; while the truths in question are affirmed only of special phenomena, which originate in a certain way; and the equalities, proportionalities, or other resemblances, which exist between such phenomena, must necessarily be either derived from, or identical with, the law of their origin—the law of causation on which they depend.
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