PS. My own fault getting caught in gamification. But this site is also supposed to be useful, providing good math answers :)

@user21820 For further algebraic motivation see all Why is two to the power of zero equal to binary one?

[ SmokeDetector | MS ] No whitespace in body, no whitespace in title (161): DeletedDeletedDeleted ✏️ by ChaoticOwl on math.SE

@BillDubuque I had assumed that you know the details because you said in the past that you were familiar with logic, and my comment here is not directed at students, so you cannot compare my comments here to the answer there. Since you asked for explicit precise details, I will give them, but you should have asked earlier if you couldn't understand my point.

@BillDubuque I will start with this comment. You wrote:

> Let's consider your objection in additive form, e.g. adjoining a neutral element 0 (and/or additive inverses) to the additive semigroup (N,+) of naturals. Your claim translated there is that it is pedagically nonsense because "0 is not defined as n-n nor is it right to invoke laws of (N,+) to justify the value of 0". Do you really support that claim?

What you wrote is in fact wrong both technically (from a purely logical perspective) as well as from my pedagogical perspective. I shall start with the technical error. "−" is not in the language of the first-order structure (N,+), so it is simply meaningless to define 0 as "n−n". What we can hope for is a way to extend (N,+) in a way that makes "−" and "0" both meaningful symbols.

(Here you seem to be using N to exclude 0, which is not the convention in mathematical logic, but I will adopt your choice for these comments.)

Furthermore, the extension from N to Z can be done in two different ways, both of which are technically incompatible with your statement. Method 1 is via equivalence classes, in which case the integer 0 is certainly not n−n where n is a natural, because 0 is rather the equivalence class { (m,n) : m,n∈N ∧ m = n }. We can later embed N into Z, but that is a separate matter.

Method 2 is by defining an integer to be 0 (a new symbol), or a natural plus a sign bit. Here there are no equivalence classes, but 0 is still not n−n until we have defined "−" on Z, which we cannot do until after we have stipulated the elements of Z. So the integer 0 cannot be defined in terms of subtraction on Z.

Besides the logical issue that we cannot obtain the integer 0 in terms of "−", there is another logical issue: Even after we have defined (Z,+,−), we still cannot define 0 as n−n where n∈N (or n∈Z) unless we first prove that it is well-defined (i.e. n−n is the same for every n∈N).

@BillDubuque It's not like you don't know these things. You know that I know you do. But I have enough pedagogical experience with the general student population (including the weaker ones) to know that most students very much do not grasp all these.

When I said "pedagogically nonsense" it was because of all the above plus the fact that I have seen with my eyes many teachers (especially those in high-school or lower) who teach exactly these things about exponentiation wrongly, specifically in logically ill-defined manner. You should at least admit that x^0 = x^n/x^n is false for x = 0 and n = 1, and this is just one clue as to the issues with the post.

@BillDubuque By the way, I'm not sure why you link to that post of yours, because it (and the post linked from it) is exactly in line with what I am saying. In fact, you even gave evidence for why we need to care about logical rigour rather than waving our hands about, since you state:

> we know rigorous methods to construct such extended numbers systems in a way that proves they remain as consistent as the base number system). Further, as above, instead of appealing to ancient heuristics like the Hankel or Peacock Permanence Principle, we can use the axiomatic method to specify precisely what algebraic structure is preserved in extensions (e.g. the (semi)group structure underlying the power laws).

The pedagogical danger in hiding the logical rigour is clearly apparent from the quote from Peacock:

> "Whatever form is Algebraically equivalent to another, when expressed in general symbols, must be true, whatever those symbols denote." − Peacock (1830)

Which I hope is clearly false to everyone in this room.

@user21820 What you wrote has very little to do with the algebraic essence of the matter. As I mentioned a good place to learn that is Bergman's textbook on Universal Algebra. Since you seem to be more interested in misguided attempts to prove false remarks on topics that are clearly beyond your expertise, I won't be wasting any further time trying to explain them to you. Best of luck in your studies.

:52680435

@BillDubuque I don't know why you are always so antagonistic. I told you I wasn't talking about the algebra, but the pedagogy. And you still haven't admitted that x^0 = x^n/x^n is false for x = 0 and n = 1. Why not?

@user21820 I can only lead you to the fountain (of knowledge). I can't force you to drink. Again, best of luck.

Again, you do not admit the undeniable error in that post. Why don't you explain yourself regarding that? If you were to have written a post on that very thread, you would not have written it the same way as that poster, and I almost surely would find nothing wrong with that you write, just as I find nothing wrong with the post you linked to. So your refusal to admit the existence of the basic error in that post is mystifying.

@user21820 Seriously? You expect me to explain your claim that the post has "an undeniable error"? (among various other extremely vague claims you made above). I'm a mathematician - not a mind reader. I have no clue why you are so hung up on that answer, or why you can never admit your mistakes. Note that since you pointed it out here and downvoted it, a few others have upvoted it. So it seems like I'm not the only one who disagrees with your very harsh assessment.

@BillDubuque I said "explain yourself", not explain my claim. This is the third and last time I am asking you why you do not admit that the post is wrong in stating that x^0 = x^n/x^n without qualification since it is false (or more accurately meaningless) when x = 0 and n = 1.

And I have not made any mathematical statement here about that issue that you have shown to be false, so I have no clue why you say I can "never admit my mistakes".

I can explain my view in detail, but I can't force anyone to think, either.

@user21820 The natural setting to interpret the power laws is in (semi)groups. But you seems to be imposing some analytic context (e.g. real exponentials). Those are different beasts. I'm talking about the algebraic viewpoint not the analytic viewpoint.

@BillDubuque I understand and agree with you on the algebraic nature you have clearly explained in the post you linked. What I do not agree with you is the nature of the thread where I criticized that post.

And look, when x = 0 and n = 1, then x^n/x^n = 0^1/0^1 = 0/0, if it is meaningful at all. It is not. I really don't get why you seem not to get this.

I have no idea what makes you think I am talking about analysis by the way, since I never invoked anything to do with 0^0.

But even in the analytic case - the fact they did not mention an obvious exceptional case is no reason to make as big a fuss as you did here. Rather, it seems based on what you wrote above you have somehow convinced yourself that the author is a crank and you are intent on deleting his post for that reason alone, even though there is nothing seriously wrong with the answer. That's rather sad.

@BillDubuque I did not say the author is a crank! I don't think so either! Don't put words in my mouth...

I think, however, that the author does not know the logical foundations of how to define integers, rationals and reals rigorously from the naturals.

@user21820 And I think the same about you, i.e. you don't know the algebraic foundations. Logic is a servant of algebra, not vice versa. Attempting to understand algebra only from the underlying logical constructions is akin to trying to understand a computer algebra system by reading the assembly language spit out by a compiler - a hopeless endeavor.

Which brings to mind a quote of Morris Kline: On the banks of the Rhine, a beautiful castle had been standing for centuries. In the cellar of the castle, an intricate network of webbing had been constructed by mysterious spiders who lived there. One day a strong wind sprang up and destroyed the web. Frantically the spiders worked to repair the damage. They thought it was their webbing that was holding up the castle.

@BillDubuque Strawman. I never said anyone should only understand algebra via the logical constructions. Please stop arguing for the sake of arguing. You disagree with my viewpoint, so be it. Don't put others down as if your viewpoint is the only correct one.

@user21820 Have you ever studied universal algebra?

@BillDubuque Yes, and as I said, I agree with you that you can and should view such extensions as "adjoining an identity element to a commutative semigroup", and that "we can use the axiomatic method to specify precisely what algebraic structure is preserved in extensions", and that algebra is separate from the underlying logical constructions. I just disagree in when students should be taught this, in particular if they cannot even define rigorously natural exponentiation for arbitrary rings.

I don't say that you are wrong in your opinion; we just have to disagree on pedagogy that's all.

@user21820 But it is not at all unusual for students to learn subjects informally first, e.g. they learn rationals, reals, polynomials long before they encounter rigorous presentations of such. Would you likewise insist that they shouldn't learn these topics until they have the logical / foundational backgorund to do so rigorously? Of course that would be a pedagogical disaster.

@BillDubuque So you know that of course I don't do what you just wrote. In my approach, I teach about every structure via its axiomatization as its 'interface'. For real analysis, this is the full axiomatization of the reals. One does not need to know whether there exists a model of those axioms, but can start proving (in the ambient set theory) all the results of real analysis using those axioms.

@user21820 But students begin learning about these rings in grade school. The axiomatic method is beyond their knowledge (ditto for many grade school teachers).

@BillDubuque In my experience, the axiomatic method is totally within the capability of high-school students (grades 11-12), but unfortunately (as you noted) it is beyond the knowledge of high-school teachers. That can and should be rectified as quickly as possible in the intro courses at the undergraduate level. Indeed, using the axiomatization of the reals first before knowing the construction of the reals is what many teachers do at this level.

If you are referring to the lower levels, we should use this axiomatization of PA to emphasize the algebraic nature of the naturals, and not any version involving the successor symbol as primitive. This is my opinion.

The PA− axioms can be taught easily (without the symbols) at lower levels even before induction is ever introduced.

As is, the author is likely puzzled as to why there was a couple downvotes followed by more upvotes. It would be courteous for you to say something since it was your comment here that sparked that activity.

@BillDubuque That may be the case. I'm sorry if my 'much higher standard' was the sticking point that caused this disagreement. The reason I didn't post any answer there is because the question wasn't even about that, but by my guess due to the error of misusing the binomial expansion.

@user21820 Well we all have our pedagogical pet peeves. But I think we should strive to give convincing arguments here - not simply "wrong answer" (which may lead to robotic voting)

Sure. I hope you better understand my viewpoint now, whether or not you agree with my 'high standard'. (About voting, as you can see, there was no robotic voting; the only other downvote probably came from the one who starred my comment.)

(By the way, I never vote robotically, and it takes me quite some time to look at each post before I vote.)

@user21820 I think we both have high standards, but the difference has more to do with timing, i.e. what is the best level of rigor at various grade levels, etc. To be sure, I didn't mean to imply that you vote robotically (such thoughtless voting is one of my pet peeves).

Yes. I suppose that's why you didn't see a problem because you weren't looking for the same level of rigour that I was. It's also why I prefer that experts like yourself post answers, because even if you don't explicitly invoke rigour, you still generally phrase things in a way compatible with rigour. Anyway, I'm glad we seem to understand each other better now.

Yet another duplicate:

close This question is both a PSQ and a duplicate.

[ SmokeDetector | MS ] Offensive answer detected, potentially bad keyword in answer, toxic answer detected (159): Find the number of tangent lines to a curve by Ajinkya Kshirsagar on math.SE

@XanderHenderson Yes, I kind of did move on, almost, I see you put a critical comment on an answer (by user @user) neighboring the answer that I was complaining about yesterday, but you did not leave any comment on the answer that I was complaining about (by user @Henry). I am not suggesting that I know what you should or should not have done, I just wanted to make sure it was not some kind of a technical mistake or misunderstanding regarding which answer exactly I had suggested was inadequate.

One of the problems posed by the recent change from 5 to 10 in question upvotes is that it encourages users to pose uninterestesing mathematical questions, e.g. this recent numerological question. There does not seem to be an easy way to remedy this other than voting to close the question as lacking context, i.e. why should anyone be interested in such random representatations of primes?

Likely the change will also spark other analogous pseudo-mathematical questions too - so it would be a good idea to closely monitor such in the short term in order to manage it before it gets out of hand. I shudder to think of the battles if cranks manage to easily gain rep from such questions.

close 1, 2, 3

Speaking of numerology, I just noticed another question today containing a joke about such, where the OP didn't do the obvious search to learn the number theoretical significance of 561. Again another type of behavior (search laziness) that will likely be exacerbated by the recent score increases of questions. Is there anyone who disagrees that questions which can be immediately answered by an obvious web search should be downvoted and deleted?

Alas, it now takes 6 downvotes to penalize such laziness.

It's hard enough nowadays to nudge students to do their own legwork in order to learn good research skills. When major Q&A sites like SE revamp their gamification model in way that further encourages such laziness it is likely that will have global ramifications. Though there's little we can do about the global problem, we can address it locally by taking the necessary actions to discourage such extremely lazy questions. Please be vigilant (esp. for experienced askers - who should know better)

[ SmokeDetector | MS ] Mostly non-latin title (87): What is this B function in 𝑆𝑡𝑢𝑑𝑒𝑛𝑡′𝑠 𝑡−𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 function? by Es-Loop on math.SE

this looks like the answers are way too good to delete

also, the equivalence to the dupe isn't quite obvious; i wouldn't even dupe-close it (though i would point out the equivalence)

another case of significant work by answerer (i think; i don't know how the hell a sequence of distributions can converge pointwise)

Delete: DA, DB, DC, DD, DE, DF, DG, DH

Delete: DI, DJ, DK, DL, DM, DN, DO, DP

not sure whats wrong with most of these

is math.stackexchange.com/questions/973501/… = math.stackexchange.com/questions/1285958/… ? in that case, merge?

[ SmokeDetector | MS ] Offensive answer detected, potentially bad keyword in answer, toxic answer detected (159): Succesion of Probability Measures by john on math.SE

The question 3401393 Are there a finite number of trees with $k$ leaves and no vertices of degree $2$? is subject of the recent post on meta: A question bouncing in and out of the delete graveyard. This question has been mentioned both in CRUDE and in GENTLE.