Page:Carroll - Euclid and His Modern Rivals.djvu/227

. § 1.] I will now go through a few pages of 'this many-headed monster,' and make some general remarks on its style.

P. 4. 'A Theorem is the formal statement of a Proposition that may be demonstrated from known Propositions.These known Propositions may themselves be Theorems or Axioms.'

This is a truly delightful jumble. Clearly, 'a Proposition that may be demonstrated from known Propositions' is itself a Theorem. Hence a Theorem is 'the formal statement' of a Theorem. The question now arises—of itself, or of some other Theorem? That a Theorem should be 'the formal statement' of itself, has a comfortable domestic sound, something like 'every man his own washerwoman,' but at the same time it involves a fearful metaphysical subtlety. That one Theorem should be 'the formal statement' of another Theorem, is, I think, degrading to the former, unless the second will consent to act on the 'claw me, claw thee' principle, and to be 'the formal statement' of the first.

Nos. You bewilder me.

Min. Perhaps, however, it is intended that the teacher who uses this Manual should, on reaching the words 'a Proposition that may be demonstrated,' recognise the fact that this is itself 'a Theorem,' and at once go back to the beginning of the sentence. He will thus obtain a Definition closely resembling a Continued Fraction, and may go on repeating, as long as his breath holds out, or until his pupil declares himself satisfied, 'a Theorem is the formal statement of the formal statement of the formal statement of the'