Rabbinic Attitudes towards Mathematics
A D'var Torah on Avot 3:23
by Jonathan Baker
23 Ap 94
Mishnah: Rabbi Eliezer ben Chisma omer kinin u-pitchei niddah
hen hen gufey halakhot, tekufot u-gematriot parparaot l'chochmah.
Mishnah: Rabbi Eliezer ben Chisma says: Bird-offerings and the
commencement of menstruation; these are the body of the halakhot.
Astronomy and geometry are condiments to wisdom.
At first glance, this seems fairly straightforward: sacrifices
and the laws of ritual purity are examples of the sort of
halakhot one should study, even if they are not necessarily
interesting. Mathematics and natural philosophy should only be
studied as peripheral elements of one's learning.
This is essentially how the Artscroll siddur reads it. One
should only progress to gematria and calendar calculations after
one has sated themselves on standard halakhic materials.
The peshat-oriented commentators support this reading. Rashi,
for example, reads "gematriot" as we would, that is, as
numerological linkages between different Torah passages.
"Tekufot" are planetary orbital mechanics, which are
mathematically complex. While these areas of study appear
interesting and tempting, they yield no reward. The regular
halakhot, however, while initally seeming dull, conceal great
depths and important ideas, which will be revealed the serious
student.
Rabbi Ovadiah of Bertinoro similarly reads "parpara'ot" as
"dessert". Only after the main course of the central halakhot,
may one delve into natural philosophy and mathematics.
This attitude raises a serious question, however. Much Talmudic
material, the core of halakhic study, requires a fair amount of
mathematics. The laws governing round sukkot, surveying Sabbath
limits, the calculations for the fixed calendar: all of these
require mathematical and astronomical knowledge. How can the
Mishnah relegate mathematics to the periphery of halakha?
Let us look at the author of this mishnaic statement, R' Eliezer
Chisma.
The major story concerning R' Eliezer Chisma emphasizes his
prowess as a mathematician and the high regard in which his
teachers held him, partly for his mathematical skill.
The Talmud in Horayot 10b tells us that R' Gamaliel and R' Joshua
were traveling together on a ship. R' Gamliel had brought bread,
while R' Joshua had brought flour as well as bread. This turned
out to be necessary, as the trip took longer than expected. R'
Joshua explained why he had brought the extra flour: There is a
star which appears once in 70 years, which draws navigators off
course. [Some have seen this as Halley's Comet] I brought extra
flour in case this happened. R' Gamliel exclaimed: You are so
smart, yet here you are on this ship going abroad to earn a
living! R' Joshua replied: You think I'm in a bad situation?
Consider my students R' Eliezer ben Chisma and R' Yochanan of
Gudgoda, who can calculate the number of drops in the ocean, yet
they have no food nor place to work!
This story indicates that R' Joshua knew astronomy (tekufot), and
that his students were even greater mathematicians than he. This
mathematical knowledge was important enough for R' Gamliel to
offer them jobs at his academy or yeshiva.
Would such a person have expressed a disdain for mathematics, as
the above commentators have read his statement? The answer must
be, "No."
I had originally thought, based on the beginning of the Tosafot
Yom-Tov's commentary on this passage, and on a comment by the
Lubavitcher Rebbe on this mishnah, that this was a statement
about levels of mathematical competence. The laws concerning
bird offerings and the onset of niddah both involve simple of
arithmetic, to calculate the number of birds to offer if an
initial offering became invalid, and to calculate the length of a
woman's period. The laws of tekufot (solstices and equinoxes)
and geometry both require more mathematical sophistication than
most people had. Perhaps R' Eliezer said that everyone should be
able to deal with arithmetic, so that one can study those laws
which are the "body" of halakhot, the more substantial laws.
Only the student of astronomy, eruvin and sukkah design would
need more mathematical, especially geometric, ability. But of
course, such knowledge would season, would deepen one's
understanding of halakhic issues.
But no! Tosafot Yom Tov and Rambam go even further, each
according to the demands of their own philosopies.
The Tosafot Yom Tov was R' Yom Tov Lippman Heller. Basing
himself on a statement of an unidentified Maggid, R' Yom Tov
interprets "parparaot" as "appetizers", not as "desserts". This
changes everything.
R' Yom Tov indicates that one should learn geometry (his
translation) and astronomy before one learns the depths of Torah
and Halakha. As he says, "Through knowledge of the spheres of
heaven, one will know and recognize the Creator, may his name be
blessed, for he drives the heavens. For from that which is
driven, you will come to know the driver." Math and astronomy
are prerequisites to Torah and to mystical knowledge of God.
In a magnanimous turn, R' Yom Tov acknowledges the translation of
"gematriot" given by Rashi and Bertinoro: numerological
manipulations also yield knowledge about the Creator. This is
the premise of early Kabbalistic works, such as the Sefer
Yetzirah (Book of Formation). Both geometry and gematria lead to
knowledge of God.
This reading of "parparaot" as "appetizer" provides the key to
understanding the Rambam. On this particular mishnah, the Rambam
merely refers the reader to his introduction. The following
passage is indicative of Rambam's attitude towards mathematics
and halakha (from "Eight Chapters", the introduction to his
commentary on Pirke Avot, chapter 5).
The real duty of man is, that in adopting whatever measures he
may for his well being and the preservation of his existence
in good health, he should do so with the object of maintaining
a perfect condition of the instruments of the soul, which are
the limbs of the body, so that his soul may be unhampered, and
he may busy himself in acquiring the moral and mental virtues.
So it is with all the sciences and knowledge man may learn.
Concerning those which lead directly to this goal, there is
naturally no question; but such subjects as mathematics, the
study of conic sections, mechanics, the various problems of
geometry, hydraulics, and many others which do not tend
directly towards that goal, shoudl be studied for the purpose
of sharpening the mind, and training the mental faculties by
scientific investigations, so that man may acquire
intellectual ability to distinguish demonstrative proofs from
others, whereby he will be enabled to comprehend the essence
of God.
The Rambam advocates the study of mathematics as a tool for
properly appreciating the Creator. Not only should one study His
works, but also one should sharpen the mind in order to devote
one's whole being to the service of God.
In fact, the structure of Maimonides' epochal Code argues
strongly for this point. Two or three chapters in the first
section of the first book, Laws of Fundamentals of Torah in the
Book of Knowledge, discuss the physical and chemical structure of
the world as it was known in his time: the spheres of the
heavens, the four elements air, earth, water and fire, etc. One
cannot properly approach God and the study and practice of His
Torah without a proper scientific knowledge of His Universe.
To summarize: Rashi and Rabbi Ovadia of Bertinoro were both
masters of peshat, finding the simplest, most straightforward
meaning for each phrase in the Mishnah. But here, it seems, they
might have missed the mark. Rambam and the Tosafot Yom Tov read
the mishnah conscious of the author of the statement. For these
two, the peshat of the mishnah was not so obvious. To them, as
to R' Eliezer, Torah and physical science are closely
intertwined.