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.