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Updated: May 17, 2025
There are certain ultimate facts beyond which it is impossible to push our speculative inquiries; certain first or fundamental principles of Reason, which are in themselves indemonstrable, but which constitute the ground or condition of all demonstration; certain intuitive perceptions, which are widely different from rational deductions, but which determine and govern every process of reasoning and every form of belief.
For instance, the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions. But without a symbolism it would have been very hard to find this out. It is so obvious that two and two are four, that we can hardly make ourselves sufficiently sceptical to doubt whether it can be proved.
I am not going to demonstrate the indemonstrable to give you intellectual notions which, after all, will be but reflexes of my own peculiar brain, and so add the green of my spectacles to the orange of yours, and make night hideous by fresh monsters.
But to have also each of the premisses true is not merely to have syllogized, but also to have demonstrated," "so that if there is demonstration at all, there is an absolute necessity that there be something that is self-evident, which is called primary and indemonstrable."
Only the doctrine of the Ideas and of the knowledge of reason is transformed by the introduction and systematization of Jacobi's principle of the immediate evidence of faith. Reason, the faculty of Ideas, i.e., of the indemonstrable yet indubitable principles, is fully the peer of the sensibility and the understanding.
Others, on the contrary, which require no demonstration, but are of themselves manifest, are deservedly esteemed the first, the truest, and the best. Such indemonstrable truths were called by the ancients axioms from their majesty and authority, as the assumptions which constitute demonstrative syllogisms derive all their force and efficacy from these.
The criterion is this: if a man receives as fundamental facts, and therefore of course indemonstrable and incapable of further analysis, the general notions of matter, spirit, soul, body, action, passiveness, time, space, cause and effect, consciousness, perception, memory and habit; if he feels his mind completely at rest concerning all these, and is satisfied, if only he can analyse all other notions into some one or more of these supposed elements with plausible subordination and apt arrangement: to such a mind I would as courteously as possible convey the hint, that for him the chapter was not written.
Reason would be unable to satisfy her own requirements, if she passed from a causality which she does know, to obscure and indemonstrable principles of explanation which she does not know. According to the physico-theological argument, the connection and harmony existing in the world evidence the contingency of the form merely, but not of the matter, that is, of the substance of the world.
It is an indemonstrable assumption that oaks and beeches in the days of Socrates and Cicero were not slightly better trees than the oaks and beeches of to-day. Fontenelle saw the weakness of this reasoning. He saw that it was necessary to prove that the trees, no less than human brains, have not degenerated.
A hypothesis of this kind is as irrefutable as it is indemonstrable; but to speak of an infinite universe is to admit a perfect coincidence of matter with abstract space, and consequently an absolute externality of all the parts of matter in relation to one another.
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