The above translation is mine. You may notice a few peculiarities in my translation. Namely, that I translate gematriyot as ‘mathematics’ (though ‘geometry’ would be more accurate) and not as numerology, and that I translate parpera’ot as ‘appetizers’ (as evidenced by the Mishna in Pesachim 10:2). These peculiarities give a totally different picture of what the Mishna is talking about.
משנה מסכת אבות פרק ג משנה יח
רבי אליעזר בן חסמא אומר קינין ופתחי נדה הן הן גופי הלכות תקופות וגמטריאות פרפראות לחכמה:
R’ Eliezer b. Chisma says: nests and menstrual onsets are the very corpus of halakhot. Astronomy and mathematics are appetizers for Torah.
[While I was writing this, I came across this piece by one Jonathan Baker which pretty much says what I’m about to say. I wonder if this Jonathan Baker is the same as ThanBook.]
Kinim is the area of halakha which discusses what happens when sacrifices get mixed up with each other. The final Mishna in Kinim is the single most difficult Mishna in all of Shas (the first Mishna in Yevamot doesn’t even touch its bootstraps). It inspired a mathematical commentary by Prf. Moshe Koppel. It’s hard core stuff. Pitchei Niddah, while not quite as complicated, requires a pretty decent working knowledge of arithmetic and the calendrical system, which form the bulk of the content of these halakhot.
These two areas of Halakha require a working knowledge of ‘outside’ = non-Halakhic disciplines. Nevertheless, studying them is studying halakha – they are part of the Halakhic corpus. Mathematics and astronomy are, however, prerequisites – what the Mishna calls ‘appetizers’ – for these laws. If the gematriyot in question are numerologies, then we’re talking about cutesy things which can pepper Torah discussions. An appetizer is a much more serious part of the meal.
This also explains the two groupings – 2 areas of Halakha which rely heavily on non-Halakhic knowledge are nevertheless gufei halakhot, whereas the disciplines that must be mastered before the halakhot are ‘appetizers’ – part and parcel of the meal, necessary to get to the main course in the appropriate state of hunger, but still not the main show. This Mishna is thus an elegant paradigm for the relationship between Torah and Madda.