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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.)
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