Goedel's Theorem and Aquinas
James Chastek over at Just Thomism had an interesting post today about the futility of demanding proof of everything. thomism.wordpress.com/
[They say everything must be demonstrated] because of apaedeusiam, i.e. lack of proper education or lack of sound learning. For it is a lack of proper education tht makes a man not know what things he must seek a demonstration of, and what things not, for all things cannot be demonstrated. For if all things might be demonstrated, then, since the same thing is not demonstrated by itself, but by another, there would be a circle in the demonstration. This cannot be, because then the same thing would be more and less known, as is clear in the first book of the Posterior Analytics. Or else, if the demonstration proceeded infinitely, there would not be a demonstration, because the conclusion of any demonstration traces its certitude by reduction to the first principles of demonstration, which would not exist if the demonstration proceeded upwards to infinity.
Sententia libri Metaphysicae, lib. 4 l. 6 n. 12
Et hoc propter apaedeusiam, idest ineruditionem sive indisciplinationem. Est enim ineruditio, quod homo nesciat quorum oportet quaerere demonstrationem, et quorum non: non enim possunt omnia demonstrari. Si enim omnia demonstrarentur, cum idem per seipsum non demonstretur, sed per aliud, oporteret esse circulum in demonstrationibus. Quod esse non potest: quia sic idem esset notius et minus notum, ut patet in primo posteriorum. Vel oporteret procedere in infinitum. Sed, si in infinitum procederetur, non esset demonstratio; quia quaelibet demonstrationis conclusio redditur certa per reductionem eius in primum demonstrationis principium: quod non esset si in infinitum demonstratio sursum procederet.
It is tempting for a one-time mathematician to see in the boldface an early intuition of Goedel: that not all statements withing a discourse can be proven within the discourse. But Aquinas is really being more general than that. He is mocking the old recreation of dictionary chasing, in which the sophomore discovers that each word in the dictionary is defined in terms of other words, and that these definitions can be chased indefinitely. Aquinas might be better translated as "not all things can be demonstrated" since to say "all things cannot be demonstrated" sounds too much as if nothing can be. What he meant was that somewhere there had to be first principles, axioms, definitions, etc. that were accepted as being more certainly known.
(BTW, it is good to see i.e. in its original incarnation as id est [idest].)
Chastek makes the same point. ( Collapse )
While on the ancient and medieval account of science (which still survives in science to this day), the goal of science is to move forward from one's starting points (principles, theorems, theories etc), for the ill educated man, the goal of education is exactly opposite: he must go backwards from presuppositions and show how they continually betray more presuppositions that must also in turn be destroyed
Which explains much of the post-modern dilemma. Notice how people, in order to question something simple and obvious, often bring up things that are flimsy and insubstantial -- like "potential persons" or "multiple universes" or [somehow] "emergent property" hand-waving. Hume, because he did not like the way arguments from causation ended up, decided to deny causation entirely.