# Troll or skeptic?

One of us has been getting involved in lengthy and, some would say, pointless discussions on various questions. See, e.g., 59709, 58174, and several others. Is this person raising serious mathematical issues, or is this person just a troll - and, if the latter, should something be done about it?

• I've seen at least two people here give up on talking to him... my take-away on the conversations was that the person in question insists on his own interpretation and wonders why people don't see it his way. Aug 24 '11 at 4:24
• The person you mention is entitled to his opinions and interpretations and it's fine with me when these are expressed in separate threads, as in the first question or in threads where such considerations are on-topic. What I'm having trouble with is when I think of poor user Sara asking an innocent question in the second post you mention. I hope she was wise enough to ignore the comment thread to her question that will have been confusing at best.
– t.b.
Aug 24 '11 at 5:28
• @Theo: Dear Theo, I agree completely; your comment summarizes the points I tried to convey in my answer below. Best wishes, Aug 24 '11 at 5:35
• @Matt: Dear Matt, thanks. I only saw your answer after I finished editing my comment and, indeed, we seem to agree fully. Best wishes,
– t.b.
Aug 24 '11 at 5:41
• I want to believe this person is acting obtuse rather than irreconcilably oblivious in the known threads, but he even blogged about one of my comments (linked by the Gerry in the OP here) and another question of his on magmas on his public and very old xanga site (began 2002), so I don't think he's playing the contrarian simply to jerk our strings - I think he's engaged in what he deems as serious business. Since he has his own blog, he should definitely be made aware of when thread derailments don't belong on math.se if it comes up again in the future....
– anon
Aug 24 '11 at 5:42
• ....but as far as creating arguments and discussions that cause consternation, there isn't really a real problem: people will either ignore him or respond to him of their own free will, and beyond apparently ignoring many points repeatedly made to him, he has been conversing completely civilly. (And I also don't mean to imply all of his opinions expressed aren't serious business - I just believe a few issues he's taken up have been totally pedantic and semantics-based.)
– anon
Aug 24 '11 at 5:47
• Did somebody notify the person in question of this discussion? I haven't looked very hard but I didn't see a comment to that effect.
– t.b.
Aug 24 '11 at 5:55
• I just thought it might be worthwhile to link to the edit suggested by Doug for the second post (visible to 10k+ users). I'm not suggesting that the edit should have been approved, I would have rejected it as well.
– t.b.
Aug 24 '11 at 7:26
• @Theo: The edit suggestion is visible for everybody. And it is very enlightening, because Doug wanted to make it "rational numbers a, b, c, d", whereas Sara meant the rational numbers $\frac ab$ and $\frac cd$. Aug 24 '11 at 7:34
• @Gerry Perhaps the title should be "Troll, skeptic or philosopher". As DS writes on his blog " I graduated from Bowling Green in 2002 with a degree in Philosophy". You've probably encountered some of the many analogous posts by philosophers over the years on sci.math (cont'd) Aug 24 '11 at 14:03
• I don't believe we have established any consensus on how to deal with meta-level topics such as philosophy of mathematics. But I think we should be very careful not to make any decision while under the influence of not the most interesting philosophical discussions. There are many mathematical philosophers whose expositions would be of interest to a large segment of the mainstream mathematics community, including our main site. Aug 24 '11 at 17:24
• @Bill: sure. I think it is absurd to call the inane disquisition about the fact that $2\cdot(3\cdot 5)$ and $(2\cdot 3)\cdot 5$ are not the same string of pencil marks on a page and its putative relationship to the fundamental theorem of arithmetic, philosophy of mathematics! Aug 24 '11 at 19:31
• @Mar That's but one of many matters DS has discussed here. Whether or not you consider those interesting philosophical points is something you should take up with DS, not I. There do exist mathematical philosophers who have written interesting papers on semiotics of mathematics and related matters. Aug 24 '11 at 19:53
• @Doug: In mathematics one proves, or assumes, conditions which allow him to consider different strings as if they are the same. In the above case, associativity. That is, if the theory proves $\times(\times(x,y),z)=\times (x,\times(y,z))$ then we can assume that $\times(\times(x,y),z)=\times(x,\times(y,z))$ since the interpretation is the same.
– Asaf Karagila Mod
Aug 30 '11 at 22:08
• @Doug: Dear Doug, I think most mathematicians regard $xyz$ (multiplication of three numbers) as a ternary operation; at least I do. It is defined as either $(xy)z$ or $x(yz)$ --- as you know, these give the same answer. Regards, Aug 31 '11 at 2:11

I am more concerned by the behaviour in the second-linked post than in the first. In the second linked post (the one about multiplying rational numbers), this person's comments led to a comment thread filled with more-or-less off-topic nonsense, which is surely unpleasant and distracting for the OP, and for others browsing the site. (Indeed, it damages the credibility of the site when someone asking about multiplying rational numbers gets a slew of nonsense comments, rather than simple-and-to-the-point comments and answers.)

The first linked post is a different case, since it was asked by the person in question, and other users of the site are free to ignore it if they want to, or to vote it down (as they in fact did).

Would it be too heavy-handed for the moderators to request that this person not use the comment threads in other people's questions as a soap-box?

• "moderators to request that this person not use the comment threads in other people's questions as a soap-box?" - yes, please, count me in with this request. Aug 24 '11 at 5:33
• Would it appropriate for one of us to clear the comment thread? We can copy it over if people want to see the history, but in the name of keeping it clean for actual site usage, we do have the option. I'm aware of some issues that have been had in the past with deletions, so even though your description, to me, sounds like it warrants deletion, I figure it'd be wise to check in, first.
– Grace Note StaffMod
Aug 24 '11 at 20:59
• @Grace Please let OUR moderators decide. Evaluating such matters often requires specific knowledge of the mathematical matters at issue. Moroever, we have our own conventions here that may be quite different from SO or other SE sites. There's been a lot of confusion here recently on related issues, e.g. above and here, so the discussion there may prove enlightening to some readers, even if it goes beyond the OP's concerns. It's only a dozen comments, some with useful pedagogical value. There are many comment threads here that are much worse. Aug 25 '11 at 1:23
• @bill I agree that 12 comments isn't necessarily a huge problem, but I would be deeply concerned if a particular user kept engaging people on this site in the same sort of pointless-ish debates that are at best tangential. Get enough of that and you will start driving away your best users. Aug 26 '11 at 2:45

Doug is very difficult to talk to, but I can't make up my mind whether he's actually done anything wrong (as opposed to simply misguided). I am convinced he is not a troll. I would recommend that anyone who doesn't want to get involved just ignore everything he says.

• +1 I agree he is not a troll. I haven't yet figured out why he places so much emphasis on matters that most of us deem trivial. These matters are important for presenting formal proofs to automated theorem provers, but not for informal proofs presented to mathematical peers. So perhaps he's working on automated theorem proving and his questions represent his struggles attempting to formalize some of the math he's learned. Speaking from experience, this can be extremely difficult if one does not have the appropriate mathematical background - which requires very diverse skills. Aug 25 '11 at 1:43
• @Bill: based on what I've read I think he just equates mathematics with formal mathematics. Aug 25 '11 at 1:46
• Indeed. Above I was speculating on possible motives why that might be so. Browsing his numerous blogs may yield further insight. Aug 25 '11 at 1:59

IMHO, the problem here is that some people do not try hard enough to understand the question and sometimes use their answers/comments to express their opinions in place of trying to answer the question that has been asked, independent of the intention of the person asking the question.

Unless explicitly mentioned otherwise, it is common practice in mathematics to assume that we are talking in the language of classical mathematicians (who are usually not interested in philosophical discussions), we write informal arguments with enough details that the other person can complete by themselves, but not too much details (which is a skill/art by itself).

Say I ask a question about how to prove AC from Zorn lemma. If someone comments on my question by saying that AC can be problematic in constructive mathematics he/she might be saying something correct, but that is irrelevant to what I am asking! Understanding questions (what is being asked? what is the context? what is the right level of detail? ...) is an important part of answering questions.

• Is this meant to apply to DS's comments/answers or, rather, to comments by others to DS's questions, or is it meant to apply universally? Without knowing which it is difficult to interpret what the votes mean (currently +4,-0). Sep 1 '11 at 18:31
• @Bill, I am trying to explain the underlining problem, I think it does apply to some of DS's comments/answers but my point is more general. Sep 1 '11 at 18:39
• And it also applies to many of the comments by others to DS's questions. Sep 1 '11 at 18:42
• @Bill, it can, if his intention is to ask a question about philosophy of mathematics and someone comments by saying that most mathematicians accept some classical mathematics assumption, then that comment is not relevant to the question (although it is probably correct). ps: I should add that I don't think this applies to the first question linked above in my opinion, because the questions is asking about the meaning of a classical mathematical result. Sep 1 '11 at 18:46
• Right. My main point was to explicitly emphasize that because your answer refers to a general issue, the votes that it receives cannot be interpreted as community feedback on DS's specific behavior. Rather they can only be interpreted as applying to the general issue that you raise. Sep 1 '11 at 18:54
• Kaveh, that the issue arises does also apply to comments by others to my questions. As a good case-in-point see this thread: math.stackexchange.com/questions/60259/… Carl's answer got upvoted, but it was irrelevant. I asked about independence of logical theorems (axioms), and Carl responded by talking about rules of inference! I decided to ignore that since he said something I found interesting, but the question was clearly misunderstood since rules of inference are metalogical, and theorems (axioms) are logical. Sep 13 '11 at 1:35
• @Doug, I think the presumption here is that one is asking a question in mathematics not about it, so if you want philosophical answers you should better be more specific about the kind of answers you are interested in. In that question you are asking what independence means (by people who use the word i.e. mathematicians/logicians), and therefore Carl's answer is completely relevant IMO and I am one of the users who has up-voted it. Sep 13 '11 at 2:44
• @Kaveh I asked about independence of axioms. In other words, the independence of a particular axiom from a set of axioms. All axioms fall under the same category. Carl answered "We could say that A is independent of a set of axioms B if A is not an admissible rule over B, or if A is not a derivable rule over B." Now, A refers to a rule. Thus, his response talks about the independence of the rule A from the set of axioms B. They don't happen in the same category. If they do, the distinction between the object language and metalanguage is obliterated. In logic this does not happen. Sep 13 '11 at 3:20
• @Doug, FYI, an axiom is a special case of a rule. In any case, this is not relevant to my post here. If you have a problem with his answer comment on his answer not here. Sep 13 '11 at 4:15
• @Kaveh An axiom basically comes as a formula of a certain kind. A rule of inference is not a formula. For any axiom x you can write "|- x", and no axiom can have anything to the left of the turnstile. On the other hand rules of inference can have things to the left of the turnstile. Axioms are not special cases of rules. Sep 13 '11 at 12:20
• @Doug, you have the bad habit of discussing issues in the wrong places. As I said this is not relevant to my post above. If you have a question ask it on the main site, as I said this is not the place for it. (and no! you are wrong. If you want to know why you are wrong ask a question on the main site and I will answer it.) Sep 13 '11 at 12:38

So far, the volume seems small, so I don't see it as a problem. Those who want to ignore these posts will do so, others will answer. I suspect the odds of OP satisfaction are low, and the population that ignore these posts will increase. Unless the volume gets high enough to be a problem I would ignore it.

• Any reason why you chose to reopen this discussion 2 months after the rest of us lost interest? Nov 17 '11 at 12:08
• @GerryMyerson: No, I forget why it came up. I thought I posted this longer ago than last night, don't know why it is here now. Nov 17 '11 at 16:37
• "About four years ago I---no, it was yesterday..." -Steven Wright Nov 19 '11 at 6:42
• "I suspect the odds of OP satisfaction are low, and the population that ignore these posts will increase." If you had posted this long ago, it definitely has turned out false in the case of post 59709. Nov 21 '11 at 14:23

I wasn't aware of this thread until now (did I miss someone informing me of it?). If the following isn't too self-indulgent, I'll try and explain why I place emphasis on matters others deem trivial.

I remember once reading Richard Feynman talking about some physics paper he wrote up and how his colleague responded to it. It contained a bunch of difficult mathematics to make some point about the physical world. His colleague looked at the paper and basically said "How does anyone know that isn't a bunch of "junk"?"... except the exact word wasn't "junk", of course. Feynman decided he would do better to stick to the simpler mathematics in his papers after that.

Somewhat similarly, if I look at some mathematical piece of writing which purports to have a proof or other mathematical reasoning, how do I know that the argument given isn't a bunch of junk? I know some people might refer to the authority of "the mathematical community" for this, but it still comes as possible that the entire community has gotten mislead in some way (this is NOT a claim that this is happening). As I understand it, in the course of mathematical history "false proofs" have gotten accepted by "the mathematical community", as at least I feel sure some of Euclid's "false proofs" got accepted by "the mathematical community" of a time long ago. I realize some other might respond to my query by saying "well a proof is just meant to be intuitively convincing." Well, my experience and observation of others informs that it at least seems possible that, in principle, one can feel convinced of absolutely anything. If that holds, and proofs just need to come as convincing, then it actually comes as fair to say that mathematical proofs and mathematical reasoning in general qualifies as a bunch of junk.

The only way I've ever seen to resolve that problem comes as to make it so that proofs in mathematics don't fall into such difficulties comes as to make sure that a proof, or assertion, can get formalized. And the only definition of proof that I think actually works comes as that a proof is either a formalized statement, or can get precisiated into a formalized statement. I don't mean to assert that it always comes as necessary or desireable to formalize mathematical proofs and assertions, and no I don't equate mathematics with formal mathematics exactly. But, if there ever exists any doubt that a proof can get formalized, then it should get formalized. Of course that depends on which logic one takes for granted also, but any formalization makes it clear, I think, that if the logical system gets taken for granted, the rest will follow. At present, that's the only way I can see to resolve this problem.

I also will add that many people take mathematics as having classical logic in the background, and I generally make comments and answers trying to assume that background. This has significance, as to paraphrase Lukasiewicz, classical logic is not a heap of stones. If the very, very slightest error exists in some claim made in the context of classical logic, the claim ends up completely false, and the structure trying to get built completely collapses. The same holds for mathematics where we have classical logic in the background. Consequently, even very, very trivial matters need to get looked at, or in other words, nothing is trivial. Of course, though, if you don't have classical logic in the background and say have a logical system that allows for minor contradictions, this sort of issue doesn't arise.

And that's my take. I feel sure many will disagree. If this answer isn't appropriate, I will delete it. And I have no clue if that all makes me into a "skeptic", a "troll", or "philosopher", because I don't know what sort of meaning you want to imbue to those terms, and I do think there exist different, valid, not necessarily consistent, ways to think about those terms.

• Formalization solves nothing - how is your thought process when reading a "formal" proof any more trustworthy than your thought process when reading a natural language proof? Perhaps you have a brain tumor that causes every 10000th statement you read to appear true to you, regardless of it's correct truth value? Even computerized formalization solves nothing; perhaps your automatic theorem checker has a bug in it. As you say, one can feel convinced of absolutely anything, including that one has correctly checked a formal proof. Aug 30 '11 at 15:30
• You may find the discussion on this MO question helpful. In particular, I like Tom Goodwillie's comment that "If you're looking for utter certainty, then even mathematics is not entirely the right field", and of course Terry Tao's answer that "ultimately, mathematicians are not really after proofs, despite appearances; they are after understanding." Aug 30 '11 at 15:32
• Terry Tao's blog post on the pre-rigorous, rigorous, and post-rigorous stages of one's mathematical education is also relevant here: "It is of course vitally important that you know how to think rigorously, as this gives you the discipline to avoid many common errors and purge many misconceptions. Unfortunately, this has the unintended consequence that “fuzzier” or “intuitive” thinking Aug 30 '11 at 15:39
• @Zev The thought process isn't necessarily more trustworthy. Yes, one can incorrectly feel that one has correctly checked a formal proof. So, if there exists any doubt about such a check as correctly done, one can re-check the proof again to see that it yields the conclusion. Since the formal proof has stable symbols independent of the meanings which we assign to them (or at least considerations of meaning get pushed near to 0), the proof can get checked until the symbols degrade. Though formalization might not solve everything as you point it, this in no way implies it solves nothing. Aug 30 '11 at 15:42
• What if you never think to doubt a particular faulty check? What if every double-check you run suffers from the same error, indefinitely? The point that formalization lets us discard the meanings of our symbols, while correct, is irrelevant. You're under the mistaken impression that mathematics somehow enables us to gain absolute knowledge. You experience everything, whether it be page after page of symbols, or a computer readout of an automatic check, through your untrustworthy senses. Aug 30 '11 at 15:46
• I'm sorry if I misunderstood your position, I didn't intend to make a straw man argument. At any rate, I'm glad that you agree that there is no certainty, even with "formal" proofs. But, if you were to ask a mathematician which would convince them more of a statement's validity, a natural language proof or a computer printout claiming to verify some formalized version of the statement, I think that a significant proportion, even most, would choose the natural language proof. Aug 30 '11 at 16:22
• Two things: 1. I feel bad that you weren't notified and had to find out about this thread by accident. I apologize for not notifying you. 2. I don't want to be dragged into this debate, but I wanted to point you to Bettina Heintz's Die Innenwelt der Mathematik (hoping you read German) which I found very interesting. I'm sure you're aware of Lakatos's Ph.D. thesis. Both these books are not exactly addressing the issues here, but are closely related, in my opinion.
– t.b.
Aug 30 '11 at 16:23
• You may also want to consider Paul Siegel's comment from the MO question: "I don't worry about such things for the same reason that I still walk to work every day even though I could get hit by a car at any minute. If I spend the rest of my life trying to convince myself that what I think is a proof really is a proof and one day I actually succeed, then I will just wish I had spent all that time thinking about geometry instead." It would take far too much effort to formalize their work for most mathematicians (e.g., all mathematicians who don't work on formalization) to bother - Aug 30 '11 at 16:28
• @Doug: I think I should leave it at that, for now. With Theo, I would also like to express my apologies for your not being notified of this thread, it may have been a case of everyone assuming that someone else had done so. While most people (including me) would argue about the relevance of your concerns to the actual practice of mathematics, I consider it important that your concerns are addressed, as this is key to getting "unstalled". I hope that what I've said helped in some small amount. Hopefully others will chime in with their thoughts and opinions about the practice of mathematics. Aug 30 '11 at 16:37
• @Zev Re: your first comment that "formalization solves nothing". Do you really think that the rigorization of calculus and, more recently, the formalization of intuitive Italian geometry by modern algebraic geometry "solved nothing"? Aug 30 '11 at 17:12
• @Bill: You misunderstand - I mean that when we already have rigorous foundations, writing proofs using little or no natural language and convoluted strings of logical symbols (or using the silly two-column format) solves nothing. Obviously, the process of developing rigorous foundations where there were none (e.g. calculus, Euclidean geometry, Italian school algebraic geometry) is an essential part of the development of mathematics. Aug 30 '11 at 17:14
• @Bill: I certainly don't claim to understand modern algebraic geometry, but I have enough of a sense of it to say that the geometric intuition simply takes a bit more time to see; if the Italian geometers had time to get training in the newer theory, I am sure they would come around. It would be their failure if they ignored demonstrable errors simply because they didn't see the intuition right away. Aug 30 '11 at 17:38
• To quote Lockhart (who is a bit hyperbolic, but makes excellent points), "Rigorous formal proof only becomes important when there is a crisis - when you discover that your imaginary objects behave in a counterintuitive way; when there is a paradox of some kind." There was such a crisis, and modern algebraic geometry was developed as a result. Perhaps there will be future crises. But in the meantime, "A proof should be an epiphany from the Gods, not a coded message from the Pentagon. This is what comes from a misplaced sense of logical rigor: ugliness." Aug 30 '11 at 17:40
• (in reference to a two-column geometry proof): "The spirit of the argument has been buried under a heap of confusing formalism. No mathematician works this way. No mathematician has ever worked this way. This is a complete and utter misunderstanding of the mathematical enterprise. Mathematics is not about erecting barriers between ourselves and our intuition, and making simple things complicated. Mathematics is about removing obstacles to our intuition, and keeping simple things simple." This is the essence of my, and I think most others', criticism of Doug's approach to mathematics. Aug 30 '11 at 17:41
• @Zev I don't necessarily disagree with you. Rather, I just wanted to point out that one has to be careful about making such very general statements since they could easily be misinterpreted by those who are not experienced mathematicians. Aug 30 '11 at 17:43

Edit: I do agree that the lengthy comment thread at Sara's question is not really relevant to that question.

However, let me point out that Doug really has a very good point in his first comment to Sara's question (maybe without realizing the point himself ...). The thing is that of course one has to justify $b^{-1}\cdot d^{-1}=(b\cdot d)^{-1}$. For this one first of all wants that $b\cdot d \ne 0$ if $b\ne0$ and $d\ne0$, and this does not follow from commutativity and associativity alone; you need distributivity.

Now I do agree that he could have made this clear himself, and I'm not even sure that this really was the point he wanted to make. But I somehow can understand his line of communication. In one of the later comments he wrote

Look, write out the proof for b^(-1)d^(-1)=(bd)^(-1) with a proof analysis citing every step along the way.

Well, I did, and came to the conclusion that you need distributivity unless you're willing to allow that $b\cdot d=0$ in $(b\cdot d)^{-1}$.

• So, first you declare that Doug has a very good point, then you explain what the very good point is, and finally you state that you are not sure that the point you explained really is Doug's point. That's interesting... More to the point (if I may say), distributivity as I know it involves two binary operations en.wikipedia.org/wiki/Distributive_property but here I see only one. You might wish to explain.
– Did
Aug 24 '11 at 9:18
• (Re: And this does not follow from commutativity and associativity alone!) Well, the statement in question is true in any monoid (where it doesn't even make sence to talk about distributivity). The point is, you don't need to use that $b\cdot d$ has inverse, you prove it (showing that $b^{-1}\cdot d^{-1}$ is such inverse). Aug 24 '11 at 9:41
• @Grigory: Yes, it follows that $b\cdot d$ has an inverse, but the point is still that you want $b\cdot d\ne0$, and this does not follow from commutativity and associativity alone. I've edited my answer accordingly; it was indeed misleading. Aug 24 '11 at 9:44
• @Didier: You have a point there :-) (in your first sentence). I see the second operation when I write $0$, which is the neutral element of addition. Aug 24 '11 at 9:48
• @Grigory: Then please prove that $0$ has no inverse. Aug 24 '11 at 9:53
• @Hendrik Sorry, you're right, one needs distributivity to show that $b\cdot d\ne0$. But the point is, it's not relevant to Sara's question, is it? Aug 24 '11 at 9:55
• @Hendrik, right. So what is bothering you is that the statement "if neither b nor d is zero then bd is not zero either" is unproved? But since one exhibits an inverse of bd (under the assumption that neither b nor d is zero), this step is as rigorous as can be, no? (Grigori's point, in fact.) And really, the only assumption used here is that one is in a group where b and d both have an inverse and the conclusion is that bd has an inverse as well (which is how I thought about the matter and explains that I did not see why the neutral element of the addition was needed at all).
– Did
Aug 24 '11 at 9:59
• $\mathbb{Z}$ is the prototypical integral domain. This is for most purposes a given in elementary arithmetic. Observe Doug's issue in the thread was basically that the existence of an identity and inverses were never explicitly posited, even though as Gerry pointed these two assumptions are as evident as they need be for Sarah and non-pedants simply by the form of the setup. In the end, insistence on excessive levels of formality can be superfluous or even counterproductive to learning.
– anon
Aug 24 '11 at 10:11
• @Didier: Well, you have a point again :-), but I'm being completely serious with my trouble about this. I taught the axioms of the real numbers in an Analysis 1 course recently, and in the field axioms it says that every number $x\ne0$ has an inverse, denoted $x^{-1}$. (It does not say that 0 does not have an inverse.) Then I said that $x^{-1}y^{-1}$ is an inverse of $xy$, but I didn't write it as $(xy)^{-1}$ since I didn't have $xy\ne0$ yet, and the notation $0^{-1}$ is undefined. This is clearly somewhat pedantic, but I think it is an important piece of pedantry. Aug 24 '11 at 10:12
• @Grigory: I do agree that the comment thread at Sara's question is not really relevant to the question. Aug 24 '11 at 10:17
• Hendrik, how can it possibly be the case that $xy=0$ if the inverse of $xy$ demonstrably exists? You're making no sense. You don't even need to explicitly write $(xy)^{-1}$ before you show $xy$ has an inverse anyway, so there's no issue here.
– anon
Aug 24 '11 at 10:24
• Hendrik: The original question this all sprang from was decidedly not at the level of analysis, it was just basic algebra, where (as I already said above) no zero divisors is taken as a given, and rightly so for pedagogical reasons. It's true I likely stretched the discussion into formalism with my first comment over at the thread, but I certainly did not take the torch this far. As for the present discussion, while it's true I was not aware of my own assumption that $0$ has no inverse, I'm still curious as to how such an inverse would invalidate the proof of $b^{-1}d^{-1}=(bd)^{-1}$?
– anon
Aug 24 '11 at 19:23
• @Hen 1 There's no reason to jump through hoops to avoid mention of $0^{-1}$. For if $0$ is invertible in R then $1 = 0\:r = 0$, a contradiction if R $\ne$ {$0$} (recall that $1\ne0$ by definition in a field or domain). But if R = {$0$} then $r = 0^{-1} = 0$. No problem. When R $\ne$ {$0$} that is a valid way to conclude a proof by contradiction. If you restrict your definition of inverse to nonzero elements then you prevent such proofs. That was my key point in this recent question on a proof in Rudin - see my comments to Asaf in my answer. Cont'd in Hen 2 below. Aug 24 '11 at 21:44
• @Hen To say that a proof about rings depends on the distributive law is not saying much. For any proof that does not use the distributive law is either a proof about a ring's additive group or multiplicative monoid. The two have no connection whatsoever without the distributive law. Thus any proof not using the distributive law is really a proof about commutative groups or monoids. Aug 24 '11 at 21:59
• @Hen 6 Re: 1. If you delete the question then no one $< 10k$ can see the comments. They are very valuable. Re: Rudin. As I explained, it is rigorous to write $\ 1 = 0/0 = 0\:0^{-1} = 0\$ in the $0$ ring (the context of Rudin's proof). But presenting this proof (or Rudin's) is ill-advised for an intro Analysis course because it invites confusion - as we've seen. I hope to explain these matters at length in my answer when some time frees up. This will probably result in some upvotes here. I just upvoted since the downvotes are undeserved. Alas, there is much confusion on such matters. Aug 29 '11 at 21:31