Q&A: A Question on Faith
A Question on Faith
Question
Hello and blessings,
I’m at a crossroads, and there is a question that has been troubling me greatly for a long time.
In mathematical theory there is a system of axioms and rules of inference. True, even the various mathematical theories have not defined everything completely in a formal way from the ground up, but there have been successes in doing so for certain branches, and things have already been proven to be consistent.
I would expect that a system of halakhic ruling, supposedly grounded in the Torah of God, would be consistent. It is clear to me that in practice one cannot prove that the entire system is indeed consistent, but at the very least I would expect that a subsystem or some particular branch would be consistent and well-defined (for example, monetary law).
I would be very happy if it were possible to define even a small part in a mathematical way so that it could be consistent.
Unfortunately, I still haven’t come across any book that claims to do this. And unfortunately I think the entire system of halakhic ruling is not consistent, as evidenced by the many disputes and differing opinions. Accordingly, I doubt that our Torah was given by God, because I would expect a Torah given by God to be a consistent Torah.
Do you know of a book that deals with this topic? An attempt to define part of the halakhic system in a consistent way? Or alternatively, is such a thing even possible? And if not, why not?
Answer
You are assuming here that Jewish law has a structure parallel to an axiomatic system, and from that it would in principle be possible to prove its consistency. That is, of course, a baseless assumption. As far as I know, to this day no one has proved the consistency of any legal system. (Legend has it that when Kurt Gödel went to his exam for American citizenship, he proved the inconsistency of the U.S. Constitution. I assume that there too there was no mathematical proof of consistency.) One can of course try to find contradictions, but I do not see how, in principle, one could prove the consistency of such a system.
But this is not only in the realm of law. There are many areas of life—in fact all areas except mathematics—in which consistency cannot be proved, either because there is no consistency there in the mathematical sense, or because you cannot formalize them in a way that allows proof. That is not a deficiency in those fields but a result of their nature. Thus, for example, every halakhic rule requires interpretation, since we are not dealing with mathematical rules. Jewish law is full of disputes (whose existence says nothing against it, contrary to your strange assumption, and certainly does not mean that it is inconsistent or that it was not given by the Holy One, blessed be He, but only that it is difficult to prove consistency. At most, only each individual approach should be consistent on its own). So how can one prove consistency in a system of rules that do not say something unambiguous?
Think, for example, about the system of the 613 commandments. How would you want to prove consistency among them? On the face of it, they do not deal with the same subjects at all, and therefore do not “speak” to one another. Why does the commandment to be fruitful and multiply have anything to do with the prohibition against eating pork? Do they have any shared conclusions in the way the axioms of number theory or geometry do? It is trivially consistent.
Alternatively, here, I will prove to you the consistency of a subsystem of Jewish law (you said you would be satisfied with a subsystem). The commandment to be fruitful and multiply deals with obligations related to the pairing of two human beings and its outcomes (personal-status law in Even HaEzer). The prohibition against eating pork deals only with pork and is in no way related to Even HaEzer (but rather to Yoreh De’ah). Therefore, no situation will arise in which these two commandments have anything to say to one another. Therefore, no contradiction between them is possible. QED.
And now, for amusement, I will refute the proof. Suppose there are two people in the world. Adam marries Eve on condition that she eat pork. That is a halakhically valid condition. But Jewish law forbids her to eat pork. And at the same time Jewish law obligates him to marry her as an instrument for fulfilling the commandment of being fruitful and multiplying. We have proved inconsistency. QED. But in fact we have not proved inconsistency here. We have proved that there may be a situation in which not all the laws can be fulfilled together. That is a conflict, not a contradiction, and therefore it has nothing to do with consistency. Just as a positive commandment overriding a prohibition has nothing to do with consistency.
And another refutation: there is a commandment to eat matzah on the night of Passover. Suppose there is no more old grain. He must prepare it from new grain, but that is forbidden before the day of the waving offering. So he cannot fulfill the commandment of matzah.
Now I will comment on these two refutations: did we prove a contradiction? Absolutely not. We proved that there are situations in which a conflict is created and one must find a halakhic solution to it (regarding the second case: a positive commandment overrides a prohibition). Once the halakhic solution enters, it removes the contradiction and evaporates the proof that the system is contradictory. So the system is consistent, because it includes that solution—or are you discussing it without it?
Gödel’s theorem was based on several characteristics of the axiomatic system he was dealing with (equivalence to number theory). There is no reason to think that this applies to Jewish law. On the one hand, this may mean that Jewish law can be consistent and perhaps it may even be possible to prove this (because Gödel’s theorem does not apply to it), but on the other hand it is not built in a formal way that would allow this to be shown.
There is a feeling that if this is the nature of Jewish law, then it is not a serious system. In my view the opposite is true: if it could be formalized and its consistency proved, that would be what made it unserious. This also in no way means that it was not given by the Holy One, blessed be He. There are different interpretations of what was given by Him. Contrary to your assumption and naive expectation, Jewish law was created by human beings and not by the Holy One, blessed be He. The foundation on which it was built was what we received from Him. But that does not mean it is not binding. The authenticity of Jewish law is not a condition for its validity.
In short, with all due respect, this is really a childish argument. A kind of youthful excitement about a new field being revealed to you (mathematics and logic). I would take a short breath before drawing hasty conclusions from that excitement. If this is the main problem causing you doubts about faith and Jewish law, then our situation is excellent.
Discussion on Answer
“In short, with all due respect, this is really a childish argument. A kind of youthful excitement about a new field being revealed to you (mathematics and logic). I would take a short breath before drawing hasty conclusions from that excitement. If this is the main problem causing you doubts about faith and Jewish law, then our situation is excellent.”
This question really did come to me in my youth, but I still have not received a satisfactory answer. For what it’s worth, I am close to finishing a master’s degree in computer science, and this question still troubles me. I believe in the existence of God, and I can also accept the fact that empirical observations are not absolute truth—that is, I recognize that given a contradiction between science and Torah, it will not be possible to prove with absolute certainty who is right. But in my humble opinion, consistency is a necessary condition. Proving the correctness of a system from outside the system is impossible (as with any theory), but at the very least I would expect it to be possible to prove correctness from within, otherwise it would be possible to prove both a thing and its opposite. Likewise, it would not be possible to judge whether the halakhic rulings throughout the generations were correct at all—maybe religious figures throughout the generations simply made things up out of personal interests, etc.
“Once the halakhic solution enters, it removes the contradiction and evaporates the proof that the system is contradictory”
Why is there a need to create a solution each time? Is the Torah not absolute, but rather changing from time to time? Is the Torah that God brought down to the people of Israel at Mount Sinai not exactly the same Torah that we study today? And if the Torah really is absolute, and we are only “discovering” new things rather than “inventing” them, why do you think it is not possible to define some field from the ground up and judge it mathematically?
“There is a feeling that if this is the nature of Jewish law, then it is not a serious system. In my view the opposite is true: if it could be formalized and its consistency proved, that would be what made it unserious. This also in no way means that it was not given by the Holy One, blessed be He. There are different interpretations of what was given by Him. Contrary to your assumption and naive expectation, Jewish law was created by human beings and not by the Holy One, blessed be He. The foundation on which it was built was what we received from Him. But that does not mean it is not binding. The authenticity of Jewish law is not a condition for its validity.”
Sorry, but I believe that a system of halakhic ruling must be consistent, otherwise there is simply no guarantee of its correctness. According to my worldview, in this world there is only one thing that can be proven with complete certainty, and that is the consistency of a system and statements within it.
And if you think it is impossible to prove consistency at all, then explain this to me: why should I believe in something that cannot be proven to be correct, at least in and of itself?
Do you hold that this cannot be done at all?
“Here, I will prove to you the consistency of a subsystem of Jewish law (you said you would be satisfied with a subsystem)”
The system you proposed, in my humble opinion, is not rich enough. I am looking for something sufficiently rich.
Again, with all due respect, but you are insisting for no reason.
- I explained to you why this requirement is unfounded. So it does not help to repeat it unless you raise some argument against what I wrote. You speak about proofs for or against, and I claim that the format of mathematical or logical proofs is not relevant to this kind of material. I showed you the problem even in the simplest systems I demonstrated, which you called insufficiently rich. Even there you will not really be able to prove consistency or refute it. You will not be able to prove either a thing or its opposite in them, because proofs are not the format for dealing with such a system.
- It is not about producing a solution each time. When there is a problem, one looks for a solution. When you have a problem in science or in logic, do you not look for a solution? I do not understand this strange claim. Do you want a system that will not create problems, or a system that has no problems? The fact that it creates problems is our issue, not the system’s.
It is not that I think the halakhic domain cannot be described logically or mathematically. This cannot be done even in the natural sciences either (Tarski tried to do this partially—very weakly), and certainly not in “softer” fields. -
There is nothing to apologize for. A person is sometimes mistaken. That is legitimate. There are mathematical systems whose consistency cannot be proved. Are you willing to accept them or use them? If not, I do not know what you are doing in the field you mentioned. Not to mention systems whose consistency has not yet been proved (even if such a proof exists). If you believe in any values whatsoever, not necessarily religious ones or Jewish law, or in any legal field whatsoever, or in findings of psychology, physics, chemistry, and the like, I do not see how you do so if their consistency has not been proved. There is hardly any field of knowledge whatsoever whose logical consistency has been proved, except for a few simple mathematical systems. So if that is all you accept, good luck to you.
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You did not understand my argument. I showed you that even in this simple system your requirement is not well-defined.
And one more comment. Even if it were possible to build a computer program that would issue halakhic rulings, that would say nothing about its decidability or consistency. Certainly if we are talking about machine learning and not a classical program, because then the program would imitate human halakhic decision-making, and would probably also make mistakes from time to time. But human beings also make mistakes from time to time.
In principle, one can build a computer program that would provide psychological therapy to a person. That is already not very far from practical implementation. This is despite the fact that psychology not only does not pass tests of consistency, but is not even a science and is not systematic at all.