The Talmud [Menachot 99b] tells us that someone asked Rabbi Ishmael, “May one such as myself, who have studied the entire Torah, learn Greek philosophy [chochmat yevanit]?” The rabbi noted that the Tanach says that you must meditate on the Torah day and night [Joshua 1:8]. So he concluded: "Therefore, go and find a time that is neither day nor night, and at that time you may study Greek philosophy!" Obviously, he was telling him to take a spaceship and go far, far from the solar system, where there is no day and no night, and there, in the middle of nowhere, he could study Greek philosophy to his heart’s content. My halachic question is: How far must he be from the nearest star?
closed as off-topic by Double AA♦ Mar 4 '18 at 0:31
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To understand when Greek Philosophy may be studied we must figure out what was meant by day and night. There is a very early Scriptural source that tells us what day and night are.
וַיִּקְרָא אֱלֹהִים לָאוֹר יוֹם וְלַחֹשֶׁךְ קָרָא לָיְלָה וַיְהִי עֶרֶב וַיְהִי בֹקֶר יוֹם אֶחָד
And God called the light Day, and the darkness He called Night. And there was evening and there was morning, one day. (Mechon Mamre)
Thus, what R. Yishmael meant was that one can study philosophy at a time when there is neither light nor darkness. Now this brings us to the old philosophical debate about whether darkness is the opposite of light, or merely the absence of light. If the former, there could conceivably be a time at which there is no light and no darkness. According to the latter view, though, there can not exist such a time because a lack of light is automatically darkness.
The textual evidence in the Talmud does not indicate which school of thought R. Yishmael belonged to. If he belonged to the "opposite" camp then he was possibly allowing philosophy at certain times. If he belonged to the "absence" camp then he must have been saying that philosophy is never allowed.
Although the Talmud remains vague, we have a direct ruling from one of the earliest post-Talmudic sources. R. Saadia Gaon in Emunos V'deios Treatise I writes as follows:
Yet I cannot remain content at this point with all that I have mentioned until I have made it clear that this thing to which these proponents of dualism adhere so firmly – I mean [what they call the principle of] darkness – is not a principle opposed to that of light, but merely the absence of light. (Rosenblatt translation p. 64)
In Treatise II he states this again:
Ignorance, on the other hand, has no such source from which it is derived, being merely the absence of knowledge, as we explained in the matter of darkness that it constitutes the absence of light and not its opposite. (Rosenblatt translation p. 89)
Thus, we have a clear ruling from R. Saadia Gaon that there can be no such time where there is neither light nor darkness. As such, one is never permitted to study Greek Philosophy (which probably includes parts of the very Emunos V'deios from which this ruling is derived).
Let us take a closer look at the actual Talmudic text.
צא ובדוק שעה שאינה לא מן היום ולא מן הלילה ולמוד בה חכמת יונית
Go and search for a time that is not not of the day and not of the night, and learn in it Greek Philosophy.
The double negative clearly indicates that R. Yishmael is saying to find a time that is either day or night and learn philosophy. I.e. philosophy can be learned at any time.
I am taking issue with your quotation where you said, "Therefore, go and find a time that is neither day nor night, and at that time you may study Greek philosophy!" Rabbi Yishma'el is talking about a time, whereas your question is asking about a place. That contradicts what Rabbi Yishma'el was saying.
So, I will answer regarding when that time is. He is clearly referring to the Mashiach, of which Zacharia 14:7 says that this will be a day which is neither day nor night. At that time, he can study Greek philosophy.