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Since a couple of days I have considerable problems with the following:
You should be cautious about pursuing "digit theory" within number theory too far, since it doesn't have a good reputation, the results of Lucas, Dickson, and Stickelberger notwithstanding. For instance, there is a review on MathSciNet about a paper involving digits that ends with the following remark: "There is also a list of serious number theory papers, by Lucas, Kummer, and others, that mention digits (usually to a prime base). But the reviewer is not convinced thereby that Smith numbers are not a rathole down which valuable mathematical effort is being poured."
(Emphasis as in original; except for potential error in copying.)
Part of KConrad's answer to this question 'Are there results in "Digit Theory"?' , which does not in any way ask for evaluation or guidanace in pursuing these questions (had it, I had never contributed an answer to this) but only existence of results. (A detail: this was added in the second revision, without the "within number theory" which was added in the third revision; I am however not sure whether this addition makes the thing better or worse. I downvoted, with initially brief comment [the comments are meanwhile deleted but a copy is in the meta-thread 'downvoting without comment is not constructive'], on the third revision, and there being is a fourth revision the author seems unresponsive.)
I already had decided to let this go. However, just now, I notice there is a (new) comment by OP of question on Mark Sapir's answer:
Thanks for bringing Bunjakovskiĭ to our attention. Do you refer to the first or the second formula? It seems that "digit theory" has existed for a long time in the mathematical underground, without ever becoming really respectable.
(My emphasis.)
Showing that this statement (KConrad's) seems to have an immediate effect on the perception of these types of questions.
This is one of these cases were I absolutely do not understand the standards of this site. How is it possible that apparently it is considered acceptable(see footnote) to introduce without any need (in addition in a somewhat flippant way) a negative evaluation of various fields of mathematical investigation. (Merely the quoting of this less than nice review seems problematic. For example, without having followed up in full detail on this, an author of the reviewed paper still published in 2011, so it seems possible they are reading the site; and this might not be such a nice experience then.)
So my question would be: why is the paragraph mentioned at the start widely considered acceptable? (On request I can recall, for comparison, several examples of things that were not considered acceptable that are in my opinion a lot less problematic.)
Footnote: The posting was frequently visible. Said answer has a current score of 13 (most of which arriving after the first revision, so this being part if it); as 16 upvotes and 3 downvotes, one from me as said, yet one might only be "general" as various parts of this question/answer got one downvote; so perhaps I am not alone, as there seems to be one other 'real' downvote, but still it seems the general opinion is this is acceptable. (Also I implictly referred to it on meta.) So it is not just 'nobody noticed'.
I don't know, quid. Two thoughts: the factual information given above the paragraph you find problematic seems, at a glance from my untutored eyes, to be good information. There is a wealth of helpful and scholarly detail there; I take it that's not in question.
The other thought is that, contrary to your belief "that this statement (KConrad's) seems to have an immediate effect on the perception of these types of questions" (citing OP's comment under Mark Sapir's answer as evidence), the unedited question seems to indicate that OP was already aware of the reputation of "digit theory" in the community as largely "recreational mathematics". I see no real cause to suppose that K Conrad's answer swayed the OP over to that belief.
Moreover, I think one could read K Conrad's answer as simply reporting facts (sociologically, it has been observed that "digit theory" doesn't have a high-standing in the community), not offering his own personal evaluation. One could view such a remark not in a negative light as being gratuitous or worse, but also positively as well-meaning or meaning to be helpful (or perhaps to give counterbalance against a possible misreading of his answer as endorsement of "digit theory"). Notice that K Conrad did not link to the Math SciNet article, nor did he name any names.
I think the first sentence of the question itself conveys a similar tone, with its reference to recreational mathematics. In particular, I think the question made an implicit invitation to make such evaluations.
These evaluations of a direction of study have appeared here once in a while, and they have occasionally led to some strained conversations between practitioners and skeptical people on the outside. I would not like it if MathOverflow became full of such evaluations, but I haven't noticed any particular recent trend in frequency.
Regarding your answer, I think it would get more attention if it were more cleanly structured, with each paragraph containing substantial mathematics, and with links rather than just keywords to search. In particular, the paragraph about the Annals paper could be improved with a brief description of the main result of the paper. If you added a link to, say, Mauduit's list of publications, it would help your assertion that there are people working with digits, who get substantial results (on digits) that are highly regarded in the mathematical community. In the current state, KConrad's answer is the only one that doesn't look like it was dashed off in a hurry.
There are several different issues being conflated here.
First, there is the question of whether merely stating that "digit theory doesn't have a good reputation" constitutes a negative evaluation of the subject. I do not think that it is. Whether something has a bad reputation is essentially a factual question, measurable for example by opinion polls. You can be a big proponent of a mathematical subject but still recognize, and bemoan, its poor reputation. Saying that the bad reputation is deserved, on the other hand, is a negative evaluation. If you accept this point, and you accept that digit theory does in fact have a bad reputation, then simply asserting this fact is neutral (but see my comments below).
Second, one can ask whether the reputation of digit theory is relevant to the question. I believe that it is. Part of the reason it is not so easy to find treatises on the subject is that it has a poor reputation. The comment is not gratuitous.
Having said that, I agree that KConrad's remark is not completely neutral, since the way it is phrased subtly conveys the impression that the subject's poor reputation is deserved. He could have instead said, "I happen to believe that digit theory is a wonderful area of mathematical study, which unfortunately has an undeserved reputation for being not a serious subject" (this is close to my own personal view of digit theory, by the way); this would have conveyed the same factual information about the reputation of the subject without implicitly agreeing with it. However, KConrad may not be personally enthusiastic about digit theory, in which case he can't make such a statement.
So maybe KConrad should have just said nothing at all about the reputation of digit theory? After all, for most branches of mathematics, people on MO generally do not make unsolicited comments about the reputation. The difference, I think, is that digit theory really does have an unusually poor reputation. Most professional mathematicians know this, but less experienced people, such as students and amateur mathematicians, often do not, perhaps because many of the professionals have the same scruples that you, quid, do, and feel obliged to say nothing about the topic lest they be accused of perpetuating the poor reputation. Because of this, there is a risk that a student may spend a lot of time working in this area in ignorance of the potentially risky career implications of such a choice. KConrad evidently felt that it was his duty to make sure that the original poster was pursuing this line of research with full knowledge of the potential career implications. At least, this is how I, and I think most others, perceive KConrad's remarks. I think that the reason people have not objected to KConrad's remarks is that he was not gratuitously denigrating a branch of mathematics but was just trying to provide some potentially helpful sociological context, in as neutral a manner as he could.
Finally, I do not think that KConrad's remarks have had or will have much effect on the perception of digit theory. The perception is already there; KConrad is just alerting the original poster to it. As for the effect on someone who is publishing in the area, such people are most likely already aware of the poor reputation of digit theory and will not be surprised or dismayed by KConrad's remark because they've already encountered it many times before.
Thanks to both markvs and Todd Trimble for the replies.
@markvs: The main emotion, which is perhaps still visible while I actually tried to write in a non-emotional way, is 'surprise'. Sometimes I feel completely detached of what I perceive as mainstream culture/opinion on MO. (This is not meant judgemental; it is just sometimes quite surprising if 'everybody else' seems to have a completely different opinion.)
Say, from memory (but I guess I could find the actual instances): for saying that the formulation 'pairwise distinct' was invented by a troll one gets two offensive flags; for saying that some claimed proof of the Goldbach conjecture is not a serious work [in a case where I think one would have a really hard time to find anybody minimally qualified considering it as even only moderately serious] one gets into some problem and for an approving (this assertion) comment also an offensive flag. And for this many upvotes. (Neither of the preceding things involved me directly.)
But, now, that I know your opinion, I already feel quite reassured. Thank you!
@Todd Trimble: The OP seemed uniformed, which is not a problem. [However, in principle I already had a problem with the question itself (for what it suggests); but then it seemed like an honest question. I believe there is an old precedent regarding something similar for 'functional equations', or so, but this just as an aside.]
Even if it were only reporting facts (to the extent something like this can be facts) it would still be inappropiate for MO (and off-topic as nobody asked for it). As you and KConrad in all likelihood know, there were not few discussion on how much or which (negative) 'community opinion' is to be reported or rather is not to be reported on MO. So, it seems quite odd to do so in a case where there does not even seem to be a direct benefit for anybody in having this opinion. In principle, I guess, one could view it like you sketch. But, if this were so, then KConrad could have replied to my comment, when editing the question, to clarify the confusion.
Actually, I, for one, would prefer the origin of the review were clearer, in particular if we should not read KConrad's opinion through this text, since then the 'authority' of the review's author on this matter becomes key.
Written before reading Scott Carnahan and Timothy Chow. I will follow up later.
@Scott Carnahan: Thank you for the explanation. Regarding the first sentence, please see the reply to Todd Trimble. Regarding your comments on my answer, thank you for your advice. To avoid a misconception, I have no problem with the reception of my answer; it is indeed a rather sloppy answer and had it been slightly shorter I would have left it as a comment; I even remember copying it in the comment box to see if it fits. The main point of it was to clarify the misconception that there are hardly results 'on digits' beyond recreational things and this result in Hensel. I likely could produce something better, but, in particular in this context, I most definitely will not start some sort of quasi-argument regarding 'digits' by expanding my answer; and also since I am against this in the first place. (Besides me not being overly qualified to act as the 'defender' here.) The most likely thing to happen is I will just delete my answer in due course, not to link some lines of investigation to this.
This is also a problem I have with this statement of KConrad what does the comment (in particular with the add-on) even apply to precisely; and what does it even mean to pursue something 'within number theory'. And, the question in the background what even is number theory and what not. For number theory specifically it is well-known that this is a touchy question.
@Timothy Chow: Thank you for the detailed analysis. One particular problem I have is the unspecific nature of the statement (see also the reply to Scott Carnahan) and that it is so open to interpretation. Yet, were this all like you said, I think one could have written this differently. And, if one did not (as an oversight) on first try one might at least have followed up in some way when somebody expresses having a problem with the answer. (The comment was there during an edit, so it is unlikely it was not noticed.)
In my mind, there is also a big difference between saying that some Subject S or type of Problem P is (at the moment) not part of the mainstream, and saying (or quoting someone saying) to investigate it is pourring water down a rathole, so, as far as I understand the metaphor, an in itself pointless activity.
And, regarding your final sentence, I am not sure I will suceed well to convey in English what I mean without running the risk of blowing the thing out of proportion, so I try it abstractly: I do not think it would be alright to mention some negative reputation some group has with some [we ignore here unjustifed or justified, as you explained], just since those affected must anyway be aware these opinions exist. (Of course this is not the same, but just to point out that the reasoning you give seems somewhat problematic to me.)
@Scott Carnahan, or also anybody else interested: While I do not wish (for abstract reasons) to expand my asnwer, if you should be genuinely curious regarding the result of Mauduit and Rivat there is for example an exposition by Ben Green for the Current Development in Mathematics 2007 conference ; the result is basically that half the primes have a odd sum of digits in base two.
For something else on digits by the same two authors and Drmota you could look at this MO question: Are there primes of every Hamming weight
A somewhat playful add-on, on the matter of 'recreational': As Lenstra said (according to Zeilberger's quotes page) "Recreational Number Theory is that part of Number Theory that is too difficult to study seriously." and a meanwhile unfortunatley inactive MO-colleague (it might be worth checking who) commented on above question that "This surely will be an open problem" But then it turned out not to be completely recreational as, well, it was essentially answered. ;D
I am somewhat disappointed by how we mathematicians are developing our own brand of political correctness here on MO (and not just in this particular situation -- the cases mentioned by quid for comparison are of a similar nature, unless someone is abusing the flagging system for the lulz). I fail to see anything remotely offensive in the last paragraph of KConrad's post; are we really worried that some HR zombie out there will be evaluating someone's publications on the sole basis of that post? In that case, the damage is long since done, with that review being out there (I guess MR reviews have a greater valuative weight than MO discussions) and (probably) countless similar claims made all across literature. Even the WP has a list of recreational number theory topics, even if (despite the WP's claims that "Listing here is not pejorative") this clearly debases a number of mathematical topics to the level of amateur research. I think that leaving out valuative opinions like this would deprive the reader of much of the useful context and background (in this case explaining, for example, the lack of literature on digits in NT -- even Lucas' theorem is often formulated without the word "digit").
@darijgrinberg: This is an interesting alternative approach.
First, regarding the detail of "abusing flags." While I know there are some "random" uses of flags here and there, I know for a fact in both cases that those were meant seriously (to be precise, in the case were two flags were raised, I know it for one, but one in each case this I know).
In the one case, the flag was acompanied by a comment right away, and in the other, when I saw it via tools I asked in a comment why this is flagged, as (me too) I was very surprised. This request resulted in somebody saying they flagged it and essentially 'maintaining' the flag. [The 'essentially' due to the fact that the situation was a bit complex, as the level of joke/serious of the post was unclear and got clarified in the interim more towards joking.] I should perhaps add that both flaggers were very active users whose opinion I value.
Second, regarding offensive. As I said abstractly in a different thread, I am not offended (personally) either. And, regarding your general point, I think if I demonstrated one thing on this site then it is that personally I have little problem to participate in opinionate debates. But do you think I would get away with actually starting a debate on main. Say, starting the way markvs did when quoting the review. (My formulation might have been unclear related to this, but I also had looked-up before the details of the review and arrived at some conclusion as to how relevant I consider the opinion of this reviewer.) And, I do not know if this particular review is on some sort of 'funny reviews list' and thus unusually visible or otherwise part of folklore in some circles, but merely that some MathSciNet review says something is really negligible in general. Also, this one is about a quater of a century old. Just the existence of this one review is basically irrelevant.
Third, I would not know where academic hiring decisions are made by some entity that I assume you would described by 'HR zombie.' What however happens is that some mathematicians yet not necessarily in the same field, will/can decide or participate in the decision on such things. And, there vague opinions on some field or other (can) come from some vague sources. I would go so far as to say 'KConrad wrote on MO' would by a lot not be the vaguest sources I can envision to potentially have some effect. So it is not so clear this is irrelevant. (Yet again, for me personally, I do not care.)
Fourth, I do not think that the amount of literature on something is so closely related to the reputation that this context really is inevitable or adds that much. (And, again, one could have expressed this differently.) There are also other reasons for such things. And, one thing should also not be overlooked while we talk about "Digit Theory" and I for example said it exists as a field of research this was picking up the formulation of the OP and answering in spirit. Even just all the things I mentioned do not all have that much to do with each other, except being somehow linked to digits. Just like there is not really so much of "the theory of prime numbers" encompassing everything that ultimately stems from this basic notion.
Ad First: Thanks for clearing this up! This is worrisome. But I think we should challenge rather than follow such precedents.
Ad Second: The way I understand KConrad's quoting of the review, it isn't supposed to be relevant per-se. Rather, it is used as a particularly pointed expression of a pervasive mindset among research mathematicians.
Ad Third: I was being hyperbolic with the "zombie", but for the sake of argument let's say I was talking about non-academic jobs, and replace "zombies" by "risk-averse personnel with little tolerance for deviations from a perceivedly mainstream biography". My point is that KConrad's post isn't likely to change much about the deleterious effects of publishing recreational mathematics to getting hired, and even if it would, we should not base our discussion policies on catering to hiring commitees. Heck, we'd have to censor the words "abstract nonsense" if we'd want to go that way; there are still lots of people who view category theory as not sufficiently profound to be a mathematical discipline, and their beliefs are reinforced by this notion.
Ad Fourth: I agree on this -- but did Keith (or the reviewer, even?) mention the amount of papers as a sign of lacking seriosity?
re 1: Somehow I feel I challenged enough lately. But, then this should not stop anybody else.
re 2: Yes. But, even admitting as Timothy Chow suggests KConrad felt it to be his duty to mention this fact (as he sees it), at least one might not have made it pointed in addition. Say, if in some other context, somebody would feel it to be their duty to say something against too much abstraction, I think it would not be optimal to do so mentioning that somebody doing otherwise is (only quoting of course!) "a pig broken in a beautiful garden" Or, even using the subsequent paragraph of the relevant source [which as many will know is Siegel's letter to Mordell].
re 3: I do not know how much in a non-academic context the subject really matters as long as it is not somewhat directly connected to the future task at hand. This could also be still more regional than what is or is not a pervasive mindset among research mathematicians. Naively, on could also think it could even help to have something to tell that is somewhat understandable and looks interesting. Again regarding 'recreational': in some sense various well-known things in number theory are 'recreational', but let me not elaborate else this might derail the discussion (but also I do not have any actually 'exotic' views here, it would be more playing with words).
Indeed, we should not let hiring comitees affect our mathematical discussions here, therefore we should not offer unsolicited advice on career perspective just when somebody asks for mathematical results.
re 4: no, they did not, however I understood you as claiming the the context is relevant for "in this case explaining, for example, the lack of literature on digits in NT" So, I meant to say since there is in my opinion not necessarily direct causation one way or the other of this (in general), I do not see why the context should be relevant to this end.
Added: Let me add that I really do not consider the quote I used as good (in some absolute sense), this is meant purely hypothetical.
quid: OP may have been uninformed about the sometimes fruitful applications of "digit theory" in mathematics, but surely he/she was not uninformed about the received opinion of digit theory in the community, since mention of this was made in the question. So again I doubt your assertion that KConrad's post had an immediate effect on OP's perception of the status of digit theory.
Michael: one could engage in a hair-splitting analysis of the words, but that "you should be cautious" could be read simply as "you should be aware that..." Maybe KConrad means to discourage research in this area, and also maybe not (but that he is sounding a warning). There are multiple possibilities for the purpose of that message, as I outlined earlier. Until he tells us what he really meant, I would give him the benefit of the doubt.
I expect this will be my last comment on the matter; I don't think I have the time or energy or passion to pursue the molehill any further.
Thank you both for the additional contributions!
@Todd Trimble: yes I understand this is already a bit long a discussion. So, just two short points:
If I ask (perhaps somewhat indirectly, but still) for clarification or say complain about it (which is perhaps the more accurate paraphrase of my comment), and the person ignores this, I am inclined to stop giving too much benefits of the doubt. (And, what would be the precise rational behind the later addition 'within number theory' [I have a personal theory, but let me not speculate] that as I replied to Scott Carnahan simply makes no sense to me or to put it slightly less confrontational makes the statement very unclear).
On mathematics. This is not mainly about 'sometimes fruitful applications' [Added: so again already this question/answer package seems misleading to me!]. I mentioned normal numbers, yet perhaps it is less universally then I thought known what this means: the question is on the distribution of digits of real numbers, x is called normal in base b if its base b digits are uniformly distributed, and normal if this is true for each base b. This notion was introduced by Borel, early contributions by Sierpinski and Besicovitch, then, e.g., Davenport & Erdős and various other see for example http://en.wikipedia.org/wiki/Normal_number,
For something specific: are the digits (base 10 or any base) of, for example, pi uniformly distribute? Is this an interesting question? Perhaps not everybody will find it interesting but then it seems not overly uninteresting or contrieved either.
Now, somebody might say this is not number theory, but then if one checks the people mentioned in "Notes" on the normal number page I linked to it will be already a bit harder to maintain this.
(Almost I am tempted to really expand my answer...)
I don't get why you are bringing normal numbers into the discussion, quid. Yes, I was aware of the notion (and btw my impression is that most questions about them on MO are notoriously open problems that we are very, very far away from having any idea about. E.g., normality of pi. Recall that notoriously open problems are not proper questions for MO.) Anyway, OP didn't ask about these. It sounds like you want to change the discussion entirely (question/answer package seems misleading to you?!) to something you find more interesting, or because of some mission to defend digit theory from perceived slights. Please clarify.
The first sentence of the question reads: "Results about numbers that are related to their decimal representation [...]." and it then asks for investigations related to this. Whether or not a number is normal (in base 10) is a result on a number related to its (decimal) digit representation. It is not clear to me where exactly our reasoning deviates.
Regarding the open-problem objection: Please note that the question does not only ask for results but also 'systematic treatement'. Coming up with some classifying properties and formulating conjectures related to them and establishing results as far as possible, is what I would call a systematic treatment (in this case one that is of course not yet completed, and very far from being so, indeed). But the page I linked, and would be easily findable just with the key-word, also mentions some actual result to base 10 normality. (By analogy, would somebody ask if a systematic treatement or results on the distribution of prime numbers exist, mentioning the Riemann Hypothesis and other open problems, and not only proved resullts, would seem quite appropriate to me.)
The reason why I though it was, perhaps in error, misleading is that I understood you as implying that OP was only uninformed about fruitful applications.While in my opinion they seem uninformed about quite a bit more of mathematics that exists here that is not what one would call typically recreational mathematics. And, so I thought the existence of this mathematics did not become clear so far. [removed a part]
As I said, I have no personal stake in the perception of "digit theory" and personally do not care so much about the perception of whatever field. While it is not only this (as there are various things that annoy my in this text) mainly for me this is a question of principle; roughly along the lines, how different are the standards that apply to different users on this site.
@quid: that was quite an ellipsis in your last comment! I read OP's inquiry as focusing on (decimal or other) representations of integers. Not irrational real numbers, which is (if I'm not mistaken) where the concept of normality applies. So the topic of normal numbers seems very off-topic to me. I don't really have an opinion on how-informed OP is; it seems safest just to stick to the topic.
@Todd Trimble: may I recall that you wrote (referring to OP) "surely he/she was not uninformed about the received opinion of digit theory in the community, since mention of this was made in the question", so I replied to this as the 'since' seemed like inferring too much to me; but I will be happy to leave these aspects away. [Added: I meanwhile removed this part.]
Well, In the entire question it says 'number'. Put let me not insist on formalities, and it is somehow true that one can get the idea the emphasis is more on integers. Yet two points, the second more relevant.
Then, consider the distribution of digits of the integers floor(10^n pi) as n grows instead.
The more serious point. The way I read the question is as first and foremost an idle-curiosity question around digits, where it is not at all clear what the actual goal is. Sombody knows some elementary/recreational/puzzle-like problems involving digits, never came across digits in a more 'serious' context, now saw the result in Hensel's book and decided they want to know if there is more existing work somehow involving digits. They might only have thought about integers when writing this, but it is not so clear whether they will not be also (and perhaps more than other things) be interested in these questions of normailty and more generally the complexity of digital representations of certain irrationals. (Sidenote: If OP were to follow up on this, they would arrive quickly at automatic sequence, and then since they appear, if this is not a similarity of names, to be interested in cellular automata, this might or might not be actually interesting to them. But the better advice is certainly to be cautious to pursue anything here.)
Thanks for sharing your opinions, quid. But here's me speaking to myself: basta! Enough speculation for me for this day. :-)
@Todd Trimble: I really appreciated you taking the time to discuss this. Thank you! And, you are quite right, I guess I also should do something else now.
@KConrad: Thank you for joining the discussion. However, your reply does in my opinion not at all adress the concern raised.
Added: I read meta first and did not realize you edited the question. Thank you for doing so. While I am for various reasons not quite satisfied with the situation, the edit in my opinion ameliorates sufficiently the situation that I will not pursue the matter further. Thank you again for your edit.
Final note: regarding the start of your meta comment here. Had you reacted days ago on my comment it would have been a lot simpler.
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