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Now, referring to your post below, regarding the Birthday paradox and the quote:
But the main purpose of school education (and I would be wholly happy just with that) is to help students to realize that scientific logic is something different than everyday practical-empirical logic.
I've seen how pupils think, and I have helped some with wcience or mathematics. I do mean that the III circuit is best suitable for science, while many people probably try to comprehend science using only the first two circuits. For me, that is the source of the common "ununderstanding of science". The thing is, the "knowledge" of the first 2 circuits is imprinted (in structure) plus empirical (in content). The common intuition is to "feel" the facts, to associate with previous experiences.
The third circuit is "designed" for symbolic manipulations. You learn a language (especially the mother tongue) just by learning how things have to be said, and detecting formal generalizations, without worrying much why the words or grammar have to be like that. The III circuit is able to accept facts without confirmation from "intuition", een contrarily to intuition. Say, physisists use quantum mechanics without understanding it - philosophically they are still baffled with well-known paradoxes; but they can use the quantum machinery to design 90% of modern technology, still without objective contradictions. Similarly, pure mathematians don't spend much time "fully comprehending" every birthday paradox, since they won't get much further otherwise. It is nothing wrong with spending time digging into such a paradox - that does not mean that you don't use the 3rd circuit, it merely means that you let other circuits to play with that paradox. As far as the 3rd circuit is concerned, there is only so much to the birthday paradox... That is why I am optimistic about the 3rd circuit types: they can accept an assumption without fully "aggreeing" with it.
(If you want a bit of extra intuition for the birthday paradox, consider this: when you have 23 people, you have 23*22/2=253 pairs of people. You have 253 pairs, compared to 365 possible pairs of your birthday and any other birthday... 253/365 is already much bigger than 1/2, right? But the probability goes to 1/2 because the 253 are apparently not independent.)
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