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Updated: May 19, 2025
Some examples of the substitution of construction for inference in the realm of mathematical philosophy may serve to elucidate the uses of this maxim. Take first the case of irrationals.
Theodorus's proof was evidently not general; and it was reserved for Theaetetus to comprehend all these irrationals in one definition, and to prove the property generally as it is proved in Eucl. The solution, attributed to Plato, of the problem of the two mean proportionals by means of a frame resembling that which a shoemaker uses to measure a foot, can hardly be his.
Superstition is a domestic enemy which he always carries within himself: those who will seriously occupy themselves with this formidable phantom, must be content to endure continual agonies, to live in perpetual inquietude: if they will neglect the objects most worthy of interesting them, to run after chimeras, they will commonly pass a melancholy existence, in groaning, in praying, in sacrificing, in expiating faults, either real or imaginary, which they believe calculated to offend their priests; frequently in their irrational fury they will torment themselves, they will make it a duty to inflict on their own persons the most barbarous punishments: but society will reap no benefit from these mournful opinions from the tortures of these pious irrationals; because their mind, completely absorbed by their gloomy reveries, their time dissipated in the most absurd ceremonies, will leave them no opportunity of being really advantageous to the community of which they are members.
In old days, irrationals were inferred as the supposed limits of series of rationals which had no rational limit; but the objection to this procedure was that it left the existence of irrationals merely optative, and for this reason the stricter methods of the present day no longer tolerate such a definition.
We gather from one of these titles, 'On irrational lines and solids', that he wrote on irrationals. Democritus realized as fully as Zeno, and expressed with no less piquancy, the difficulty connected with the continuous and the infinitesimal. Is it, said Democritus, equal or not equal to the base?
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