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Having the pleasure of reading some original text, I was wondering if someone can translate two small statements on the second half of page 11 from http://archive.org/stream/mathematicalanal00booluoft#page/11/mode/1up

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It regards Aristotle's "architectonic" of knowledge. We can say roughly that Philosophy give us the causes (i.e. the explanation : "the why") of the facts (the "real existence") while Mathematics give us only a "description" (the "how that is") of them. –  Mauro ALLEGRANZA 2 days ago
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I minored in classics in college and majored in philosophy taking several history of philosophy courses and looked at some translations of texts like Spinoza's Ethica. I don't recall anyone saying it, but I felt it rather obvious that the convention goes that anytime you see something like "pleasure (laetitia)" the translator has put the "vernacular" language (here that's English) next to the original language which it appeared in. I don't see how it could make any sense to do so otherwise. So, "why" translates "to dioti"... which transliterates the Greek, and "that" translates "to hoti". I find those terms rather vague. I suggest to get a better understanding of what W. Hamilton, who I think it fair to call an Aristotelian, means as cited by Boole, you'd do well to at least find some translation of "Aristotle" (Wikipedia indicates that we probably don't have his actual writings) or the Aristotelians where some translator has those Greek terms. I would guess that you'd want to read some translation of The Organon.

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The correct transliteration of τò δίοτι is tò díoti. –  Brian M. Scott Nov 23 '12 at 4:02
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@panny: This page gives a good idea of what Aristotle meant. –  Brian M. Scott Nov 23 '12 at 4:09
    
Funny coincidence, printed (out of curiosity) "Organon - Katêgoriai" some years ago, did not read. Maybe I will have a peak now. To note on the side, above essay is surprising just ~100 pages easy digestable content, done reading. If only more (of the significant) literature was a cake like this. –  panny Nov 23 '12 at 4:33
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Here's a logical digression founded on these Greek phrases.

In Greek you can form abstract nouns very easily, by taking an idea such as "good" or even "why", and sticking "the" on the front to get "The Good" or, here, The Why and The That.

It's slightly risky translating these into English. The Good is standard, and I can explain the Whys and Wherefores of a situation, but "The That" doesn't work.

The Good means, roughly "goodness"; but it also means "that thing which is good above all other good things" or "the thing which is good if anything is". Plato shows the possible confusion arising here in the Meno. Is The Good itself good? (yes) Is Goodness? (well, no: it's just an idea, not a human who can be virtuous or evil).

Hilbert proposed a logical quantifier "epsilon". Given a formula $\phi(x)$, you can form the term $\epsilon_x\, \phi(x)$, and the rule is that this refers to an element of your domain which satisfies $\phi$ if one exists---or a random element otherwise.

Then $\forall x \phi(x) \equiv \phi ( \epsilon_x\, \not\phi(x))$, so we don't need the usual quantifiers. But epsilon is stronger than the usual first order language, because to interpret it requires the Axiom of Choice---if $\phi(x,y)$ has another free variable we have to choose elements for every value of $y$.

This is an example of where the boundary between logic and mathematics isn't necessarily obvious. I'd say it was a case where maths (having Choice as a separate axiom) is less dangerous than logic (rolling Choice into the formalism). So I side against Hamilton here.

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Lawvere points out that in some categories Choice is true, in others it's false, so it's not axiomatic. See, eg Lawevere and Rosebrugh "Sets for Mathematics" –  alancalvitti Nov 23 '12 at 15:12
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