Wisdom From Below
Part 2 – Axiom Stories * In connection with 20 Menachem Av, Yartzeit of Rabbi Levi Yitzchok Schneerson

By Prof. Shimon Silman, RYAL Institute and Touro College


“In mathematics as well as in every intellectual matter, there must be two elements: 1) elementary concepts and axioms; for example, that the whole is greater than any of its parts, and 2) all the [mathematical] concepts and theories that are built on the above. These two categories are called chochma and bina. Chochma is basic concepts and bina is those concepts built upon the basic concepts. This can also be referred to as the ‘ban’ of ‘Tohu’ that preceded the ‘mah’ of ‘Tikkun.’ ‘Ban’ is similar to the basic concepts and ‘mah’ is similar to the concepts that are built on them.” (From a letter written by Rabbi Levi Yitzchok to his sons, the Rebbe Melech HaMoshiach and Rabbi Yisroel Aryeh Leib, Likkutei Levi Yitzchok, Igros Kodesh, p. 250)


In Part 1 we explained that every logical system is of necessity based on fundamental assumptions called axioms, which themselves are not logical but are adopted by convention (i.e., people who like these axioms agree to use them).


Let’s take a break from the abstraction for a moment and tell some stories - axiom stories:


 A man I know was raised in a non-observant environment. His exposure to Yiddishkeit was minimal and the many questions that he had went unanswered. Around the time of his bar mitzva, believing that there was nothing more to Yiddishkeit than what he had already seen, he made an important decision: after his bar mitzva he was going to leave Yiddishkeit and become an atheist. And so he did.


After studying the doctrine of the atheists for some time, a certain uneasiness began to overtake him. He realized that everything they said was based on certain assumptions they could not prove. This made him feel very uneasy because the whole reason he left Yiddishkeit was because it had certain basic ideas and beliefs he felt could not be proven. But now he was finding out that the doctrine of the atheists was not logically self-contained either. They had their own set of assumptions upon which they based their philosophy and those assumptions could not be proved. Rather, their belief was a matter of faith. If you accepted the assumptions then the rest followed. But if you didn’t, then there was no system. “I am back where I started from!” he told himself. “Why are the non-provable assumptions of the atheists any more valid than the fundamental beliefs of Yiddishkeit?” He then began to reexamine Yiddishkeit, and after a period of time became a complete baal teshuva - and a Lubavitcher Chassid.


 The realization that any logical system is ultimately based on some assumptions, or axioms, that are not “logical” but must simply be assumed, takes the wind of absoluteness out of any system of thought, scientific or philosophical.


 A young Jewish engineer once wrote a letter to the Rebbe Melech HaMoshiach about his life and background. In response, the Rebbe wrote:


“There is a well-known saying of the Baal Shem Tov that we heard from [the Rebbe Rayatz] - that everything that a Jew sees or hears certainly has some instruction for him in his service of Hashem... I see from your letter that you are a mehandes [from the Hebrew word handasa, which was used classically to mean geometry. Thus a mehandes would be a geometer. In more modern usage, handasa is used to mean any kind of engineering, and a mehandes would be an engineer]. But it is not exactly clear what professional work you do. Is it building construction or various types of measurement - surveying, etc.? In any case, at the foundation of all of these is the discipline of handasa - geometry. What lesson can be learned from this discipline?


“Geometry has characteristics of an exact mathematical science and also of an applied science. It follows that it is not so precise. L’havdil, on an infinitely higher level, the same thing applies to our Torah. On the one hand, it is the wisdom of Hashem and thus the ultimate of truth and precision - “No man can fathom its worth and it is hidden from the eyes of the living.” On the other hand, its ultimate purpose, as its name Torah indicates, is instruction in the daily life in this coarse material world. Therefore, in [analyzing] the contrast between these two characteristic extremes, we can find the fundamental and infinite distinction between the Torah, which is called “our wisdom and understanding in the eyes of the nations,” and the wisdom and understanding of the nations themselves, or the intellect of the ‘animal soul’ of the Jew.


“The distinction is as follows: Human intellect, even that of the sciences that are considered exact sciences, is based on foundations which science itself has nothing to say about. This is so because science, especially exact science, accepts as a conclusion only those things that have proof; and the foundations of all sciences and mathematics, including geometry, have no proof. Thus a person is free to accept them or to reject them. This is especially emphasized in the case of geometry, in which, as is well known, there are three primary theories, each based on a number of assumptions (axioms) - and the axioms of each theory contradict those of the others. In other words, science does not have the ability to make an absolute statement, only conditional statements, such as: ‘If you accept these axioms as true and you also accept the methods of reasoning and proof, then you will get the following results...’


“Two major points follow [from the above discussion]: 1) It is up to the person whether or not to accept the axioms. 2) Even if he does accept them, he cannot be forced to do anything in accordance with the results that follow because the whole process merely says, ‘If you take the following course of action, the results will be as follows.’ If one does not care what harm may possibly come to him by following a certain course of action, there is nothing that compels him not to act in that manner. In other words, science does not give instruction in life, but rather ‘tells a story’ or predicts the future, and says, ‘Based on our experience until now and based on the axioms which we now want to accept as true, the sequence of events will be as follows...’


“In total contrast to this is our holy Torah. Being the wisdom of the True Existence - Hashem - it is necessarily absolute. It is absolute truth - both in its basic statements and in the ‘principles of the Torah’ that direct the manner in which the basic statements are discussed [the rules of reasoning]. Since this is the wisdom of the Creator of the entire universe, which includes man, it follows that all its conclusions compel man to act in accordance with those conclusions and in no other manner at all.


“This is one of the points that, as an engineer, should be embedded in your mind - that it is impossible to raise any question whatever based on science against the Torah, l’havdil, since the Torah is absolute truth. According to the way science defines itself, [science] is not absolute, but rather dependent on the assumptions that a person wants to make. Furthermore one has the freedom to establish contradictory theories which may all be maintained in accordance with the will of various people, such as the three theories in geometry - that of Euclid, that of Lobachevsky, and that of Riemann.”


(Igros Kodesh, Vol. 6, pp. 145-147)



(To be continued. In Part 3, b’ezras Hashem, we will discuss these three theories of geometry.)




Questions & comments should be directed to


Let’s take a break from the abstraction for a moment and tell some stories – axiom stories...




“In total contrast to this is our holy Torah. Being the wisdom of the True Existence – Hashem – it is necessarily absolute. It is absolute truth...”




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