by Mordecai Plaut
(Page 3)
From the book "At the Center of the Universe".
There are two kinds of reasoning which are called "induction": informal induction and mathematical induction. The similarity between them does not go much beyond their name.
What we have called "informal induction" is the basis of all scientific knowledge of the physical world, such as there is. It is a principle of reasoning which starts with the specific, observed instances and leaps to general laws. Science likes to think of itself as accruing knowledge through applications of this type of reasoning operation. Its use was seriously questioned by the English philosopher David Hume over two hundred years ago. Notwithstanding that the areas of science which rest on it have developed prodigiously since then, no adequate rationalization for its use has yet been found. The scientists who use it daily must rely on a faith that it is good. So far, it would seem that this faith has not led them astray in their practice of science.
In contrast, mathematical induction is on as firm a foundation as is arithmetic. It is really a theorem of logic which says that if a collection is sequentially organized (well ordered), then we can show that the entire collection has a given property (P) by showing:
(IND) P (Xn) -> P (Xn+1) (ORG) P (X1)
where Xn is an arbitrary member of the collection and Xn+1 is its successor, and X1 is the first or initial member of the collection as it is organized.
In other words, this means that if we show that, granted an arbitrary member of the set has the property in question, then that implies that the next one in succession also has the property (IND) and also that we show that first member of the collection has the property (ORG), then these two things together imply that the entire collection has the property.
It may sound a bit complicated, but a little reflection will probably make it intuitive. First we establish that if a given member has the property then that implies that the next one has it too. If we can then determine that the first member has the property, we can show that the second has it too. But once the second is known to have it, we can easily show that the third has it too. Intuitively, we can continue this way ad infinitum, or until we have exhausted the set.
It might seem like an artificial theorem, but it is a result of the basic properties of what are among the most basic logical and mathematical structures. It has also proven extremely useful in formal investigations. Among the results that it gives are the commutative (a + b = b + a) and associative [(a + b) + c = a + (b + c)] properties of addition. If we keep all this in mind and return to our original topic, we can easily see that the transmission of knowledge via the Mesorah has the support of a structure which is like a mathematical induction!
Actually the people in the Mesorah form a treelike structure, since a teacher may have more than one student. However, if we pick a particular branch, we can show by mathematical induction that all have knowledge from the Mesorah.
First we consider the "induction step" (IND), above. If we assume that a particular (arbitrary) member has knowledge of the matter of the Mesorah then we must show that the next member also has knowledge. By hypothesis of hte Mesorah, (II) holds. This is so because (II) is the precondition for there to be a successor. That is, it is necessary that the successor (pupil) has satisfied himself that his (prospective) teacher is as an angel. Hence the successor does know that his predecessor (teacher) knows whereof he speaks. Therefore, by (I), the next member (student) also has knowledge of the Mesorah he receives.
It is also clear that the first element in the Mesorah, Moses, had knowledge. Since Moses received the material of the Mesorah from G-d, he would either have knowledge of it by (I), as we showed earlier, or perhaps experientially. Thus, we have shown that the first element has the property in question (knowledge) and that -- according to the stipulation of the Mesorah itself -- if a given member has knowledge then so does the next one. By the principle of mathematical induction, the entire Mesorah collection has knowledge, and this is what was to be demonstrated.
These considerations would seem to give the Mesorah an epistemological foundation which is readily understood within the Western tradition. In fact, an empiricist may find that the solidity of the justification of his own "true beliefs" suffers by the comparison. He may have plenty of ideas, but he certainly doesn't know what he is talking about.
It remains to observe that the mathematical induction which supports the Mesorah is not available for use by other systems such as the religions, for a variety of reasons. Although they may claim knowledge on the part of the initial element, they usually cannot establish the induction step. More fundamentally, however, those systems are typically based on belief, dogmatic authority, paradox, and/or mystery. None of these is at all adequate to support a structure similar to the one we have elaborated in this essay. For this task it is necessary to have knowledge such as that provided by the Mesorah. (6)
NOTES
1. Babylonian Talmud, Shabbos 31a
2. Chisholm, Roderick M., Theory of Knowledge (Englewood Cliffs, N.J.: Prentice-Hall, 1966), p. 57
3. Russel, Bertrand, Logic and Knowledge, Marsh, Robert C., ed. (New York: Capricorn Books, 1971), p. 367
4. Hintikka, Jaakko, Knowledge and Belief (Ithaca: Cornell University Press, 1962), p. 60ff
5. Babylonian Talmud, Chagigah 15b. The passage is an exegitcal comment on the verse from Malachi: For the lips of the cohen will keep knowledge, and wisdom and teaching, they should seek from his mouth, for he is an angel of the G-d of Hosts.
6. The argument here is perfectly general. Any system with a similar or equivalent origin and transmission structure would have available the support of mathematical induction. This essay is about the claim of the Mesorah and it is not interested in exhaustive comparisons. Suffice it to note that it is a contingent truth that there are no other such systems that are in a position to advance such a claim, and that the dynamics of induction make it unsuitable to use retroactively.
THE END
This essay is from the book "At the Center of the Universe". Order the book