I’m claiming that it’s not correct to say that there are people who think the continuation of the arithmetic sequence 1,2,3,4,5 is minus pi. .
Because what the Rabbi suggested, that there is someone who thinks in a determinantal way, doesn’t mean he thinks of a specific number; rather, he simply doesn’t understand at all what number is supposed to come next, since you can plug in any number and it will be “correct.” That’s not called a sequence, because the order of the numbers has no meaning at all in that way of thinking; any random substitution is correct.

If that’s also unclear then never mind, it’s not critical for me…

Besides that, in that same lesson the Rabbi spoke about rules of halakhic ruling that aren’t really rules, and that you can’t issue rulings based only on rules. Does the Rabbi have an article on that topic?
In the lessons I hear the Rabbi talk about it a lot, but I haven’t seen anything written out with sources in an orderly way.

Each person, according to his way of thinking, will be able to choose the most “natural” continuation for any number sequence. It seems that the questioner understood that there is one type who continues in the normal way, 1 2 3 4 5, and another type who has no particular continuation in mind at all, but any continuation seems acceptable to him. But that’s not so — I assume the intention was that each type has some continuation that seems most natural to him. One alien will continue the sequence with 19, another alien will continue it with 2 pi squared, and a third alien will say that several continuations seem equally natural to him and it’s hard for him to decide.
What determinants have to do with any of this, I have no idea. Maybe our cryptic poet meant that if you do interpolation with some set of functions (whose size, for simplicity, matches the number of points), then you build a matrix so that each row corresponds to a point, and the values in the matrix are the values of the function at that point (in row i, column j, you have the value of function j at the x-value of point i), and you multiply by the vector of unknowns (the coefficients of the functions) to arrive at the solution, which is the terms of the sequence; and then if the determinant of the matrix is 0, maybe there is no solution, etc’.

There is no such thing as thinking in a determinantal way, and little old me never said any such thing. Tolginus answered nicely.
Apparently you mean my claim that one can find a sequence that will give any continuation we desire. And I brought what Wittgenstein explained: that if a sequence has five numbers, one can find a sequence that is a polynomial in n (a combination of powers of n), and the coefficients are found by means of five equations with five unknowns. I said that if the determinant is not 0, one can find a unique solution. But that remark is not important to the discussion itself.

I’m currently working through column 247 (actually, it’s not easy for me to follow all the twists and hints there. It takes me more than one or two evenings, but it’s worth it. Really something.). And there in note 5 you wrote that the Talmud opposes positivism and wonders why a general law is needed (“your property” and “its guarding is upon you”). And I’ve also internalized that from what you said long ago in Two Wagons. But honestly it still isn’t settled for me, and I’ll use this corner to ask something on the margins. Seemingly, precisely because there are many possible generalizations, there is a lot of benefit in a rule and it is very useful. The Talmud isn’t surprised why the Mishnah brought the examples, because those are verses in the Torah and that’s where it all starts. And if the Mishnah added a common denominator, then obviously it came to add something (more than four separate derivations of a single paradigm, according to your classification), and the Talmud is simply asking what that thing is. So there you have it: the Mishnah gave the rule. Seemingly, “we do not derive from generalizations” applies when we know of an exception or it is likely that one exists, but ordinarily we do derive from generalizations and they are very useful, exactly like following the majority.

The polynomial doesn’t really add anything here (and I also don’t think Wittgenstein himself referred to it). Since the existing sequence can be sewn together with infinitely many polynomials (or other functions) and arrive at any continuation, you don’t gain anything from the natural-but-arbitrary choice of one particular polynomial over a direct choice of a natural-but-arbitrary “continuation.” An alien can simply feel that the natural continuation of 1 2 3 4 is 17 without any further justification, just as the alien who chose a certain polynomial chose דווקא that one without any further justification. It doesn’t seem that this step backward contributes anything intuitive. Seemingly the sting is that even if one explicitly defines a sequence, say f(n)=5n, and generates from it the numbers 0 5 10 15, in order to generate the “next number in line” one must first know who the next “n” in line is, and here the alien might think that n is 100 and then arrive at 500 as the “next” number in the sequence. Therefore if you already give him the sequence of n’s and tell him “apply fn to each of the numbers before you,” then he will indeed succeed in working by the rule (could you please answer this specifically, whether it is correct or not?). The moral is that if the Mishnah says “anything that is your property and whose guarding is upon you is liable for damages,” then clearly there is very great utility in this rule and we can follow it quite easily (unless there are special exemption laws according to their own rules and the Rabbi from Brisk, etc.).

(Correction: column 347)

Tolginus,
Thank you for the reference to Rabbi Michael Abraham’s article; that’s what I was looking for.
And if I understood correctly, your last question is my question, and once I know the formula of the sequence, the next result is already one and only one. Meaning, if someone comes and says that the natural continuation (in his eyes) of the sequence 1,2,3,4 is 17, then of course we will be able to prove that this is possible, but the next digit in the sequence will be one and only one according to the formula that forced the number 17.
And therefore the rule is essential in such a case.

No, even the sequence 1 2 3 4 17 has infinitely many continuations. There are infinitely many different ways to continue 1 2 3 4 with 17, and those possibilities will differ in how they continue afterward. If, in order to fit n points, you use a polynomial of degree n-1, then indeed there is one and only one polynomial (Lagrange interpolation), and the continuation will be fixed. But if you use a polynomial of degree n or n+1, then there will always be infinitely many possibilities. No matter how many numbers the alien “continues correctly,” it may always be that at the next number he pulls out some unclear thing (and it will turn out retroactively that the polynomial we thought he was using is different from the polynomial the alien is actually using). In any case, it seems to me that the whole polynomial business is just for illustration and not really important.

Who knows what the next number in the sequence is
6,21,107

I never said that rules are not useful. A person cannot think without rules. I only object to over-adherence to rules. Rules are an aid to thinking, but in the end our thought is built on inductions and analogies, not on deductions. The rules are meant to guide our analogies. The Talmud’s question is why the Mishnah saw fit specifically here to insert the rule, when its usual way is not to add it. In the Talmud there is a clear preference for analogies from particulars over deductions from rules.
And just go and see regarding this very rule itself, which requires that the damager be your property (according to the Rif’s version) in order to obligate you, whereas as is well known it is not true that the damager must necessarily be your property in order for you to be liable for it. There are quite a few exceptions: a bailee, a robber (who moved it, who drove it by shouting in its face), one who positions another person’s animal [to do damage] (in all the above ways), and the like. And here no exception at all is mentioned in the Mishnah. On the contrary, when no exceptions are mentioned, then it is obvious that we do not derive from generalizations. The novelty is that even when exceptions are mentioned, and then the wording would seemingly be precise, even then we do not derive from generalizations. And this is the Talmud’s own language: “even” in a place where “except” was stated.

By the way, as far as I recall (from many years ago already), Wittgenstein himself uses the polynomial in order to show that one can find a concrete function for any result one wants. Obviously there are many other possibilities, but it is hard to point to a general algorithm that will give you an explicit function that does the job. That is what the polynomial adds. Otherwise you need to draw the desired function, but it is hard to write an explicit expression for it. The polynomial route is completely general for any sequence of any number of terms and gives you an explicit answer.

Decisor: 264

A. I’ve desisted and do not opine regarding the rule itself. “One who positions” can be inferred using the rule itself, or else it is a case of a person causing damage, or reason treats it as though it were his property so as to obligate him for it. Why do you need the example for that liability? I didn’t understand the connection to a bailee and a robber; they are explicit in separate laws and not like the other damagers (even if a robber is treated as a damager in certain ways).

B. The point that isn’t clear to me is why one “needs” to produce a function. The alien function is simply: “for 1 I return 1, for 2 I return 2, for 3 I return 3, for 4 I return 4, for 5 I return 17, and good luck,” and that’s it. [I assumed Wittgenstein doesn’t refer to polynomials for the prosaic reason that when I read him (and didn’t understand much), I didn’t know about interpolations, and I don’t remember feeling that I lacked mathematical knowledge. Expertise, of course, I don’t have, and all the more so not ‘there’s no such Tosafot’.]

C. Decisor, in the spirit of that column, the required continuation is indeed 264, as the Rabbi answered, but even an alien can sometimes learn “how human beings think” if he’s not entirely alien from head to toe. So presumably you were expecting the answer “at least 47176780” according to https://oeis.org/search?q=6%2C21%2C107&language=english&go=Search. And I seem to recall that this isn’t the first time you’ve brought something connected to computability.

Ah, so your meaning is that even after there is a rule, one still uses inductions to extend it beyond what is included in the “positivist” interpretation of the text, and therefore in principle we could have applied that same extending process directly to the examples? In other words, the rule did not really exempt us from the need to compare cases to cases.

🙂
The next number is probably
47176870
and the number after it is unknown and probably will never be known, but it is bigger than
10 to the power of 36534

https://en.wikipedia.org/wiki/Busy_beaver

Actually I just wrote that for nothing. No need to answer again.

[Even though it is completely unimportant, I tried checking it just for the sake of the curious curiosity. I found a pdf (in English, unfortunately), and in a quick skim (paragraphs 140-240) I don’t see there the specific idea that you can fit any number sequence with a polynomial of sufficiently high degree. In the index there isn’t even the word polynomial or any part of it. He does use examples of “algebraic formulas” and also refers to them generally as a “rule” whose discovery provides a feeling of understanding. And it seems that in order to demonstrate possible divergences he uses simpler and more general tools there (“I saw that you add 2 each time, but I thought that was only until you reach 1000, and from there onward you start adding 4”). In any case, that’s what I managed to identify in a quick top-down search].

Tolginus,
A. There is no way to infer “one who positions” from this rule. One who positions another person’s animal is not its owner. Even if you adopt the yeshiva-style wording that we view him as though he were its owner, that’s a semantic trick. What it means is that it need not be your property in order to obligate you. Alternatively, when they told you the rule, it was not enough to infer from it how to use it, and we are back to the point that the rule does not replace analogies and inductions. Exactly as in your next comment.
B. I don’t remember what happens אצל Wittgenstein. Out of curiosity I went back and checked and indeed I didn’t find it. Maybe this is only my formulation of his argument.
When you produce a function, it helps you explain why there is “logic” behind the surprising continuation you are proposing. To your bare (Wittgensteinian) claim one can answer that you are an idiot and don’t understand, period. The fact that there is some idiot who is sure that 1+2=16.8 does not make that a legitimate and acceptable statement like the answer 3. But when you present a function, you show that your continuation is no less well-grounded than any other continuation. I don’t see what is unclear here.

As for your remark to the Decisor, you reversed things. Someone who would answer my answer would get 100 on the psychometric exam, and someone who would answer his answer would fail (and rightly so, unless this is a test for Turing machines that want to study computer science in John Searle’s Chinese room). My Wittgensteinian argument is needed in order to ground his answer. My answer needs no grounding. It is a simple and obvious intuition.

But everything is a “function,” not only smooth and pretty algebraic expressions. For my part I don’t see what a closed, formulated function adds, and I don’t think a bare continuation is any more bare than an arbitrary choice of function (and therefore if old man Wittgenstein really saw no need to get there, I understand why). So simply speaking, the argument applies very well to every rule in the world and not only to numerical rules (and there is no need to look for mappings from every collection of examples to a number sequence). But okay.

As for reversing things, indeed. I thought in passing that you had just made up a number to illustrate the claim that anything can be a continuation (and therefore I wrote what I wrote). Now I see that, just like one of Wittgenstein’s examples there, you were looking for the sequence of differences to be an arithmetic sequence. [Even so, according to your answer it is not clear what the Decisor wanted us to hear with this little riddle.]

(When I wrote that 264 is the required continuation, I didn’t know that it was “really” the required continuation. As stated, I thought you had just pulled a number out of nowhere, and I wrote that it was “required” only for the glory of alien-ness.)