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I am a junior of mathematics. My top post is: Weird distributive laws?

The answerer, "Theo Bendit", looks like an expert of algebra. This makes me wonder whether people who answer are actual mathematicians. So, how many actual mathematicians or, at least, people with a mathematics Ph.D. are here?

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    $\begingroup$ The educational attainment part of the the 2019 SO survey may have numbers that are roughly in the same ballpark as ours, though I'd expect us to have more graduate students, and fewer people with a bachelor's who are no longer students. We discussed conducting similar polls on the meta site at one point, but to my knowledge we never actually did it. $\endgroup$ – Alexander Gruber Sep 8 at 23:39
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    $\begingroup$ The r/math census might also be a good place to look for ballpark comparable numbers. $\endgroup$ – Alexander Gruber Sep 8 at 23:45
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    $\begingroup$ I am absolutely not an expert in algebra (though I am flattered)! I'm currently studying a PhD in non-linear geometric functional analysis. So, no PhD yet, but give it another year. $\endgroup$ – Theo Bendit Sep 9 at 2:52
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    $\begingroup$ @Theo, the definition of "expert in Algebra" is "anyone who has taken one more Algebra class than I have," which means you are an expert to many people who post on m.se. $\endgroup$ – Gerry Myerson Sep 9 at 3:03
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    $\begingroup$ There are only three mathematicians here, and we're using all the accounts. It's really tiresome, but we get compensated very very well. $\endgroup$ – Asaf Karagila Sep 9 at 6:28
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    $\begingroup$ I've never stepped foot in a university level math class ... $\endgroup$ – Roddy MacPhee Sep 9 at 11:00
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    $\begingroup$ What does it take to become a matematician? $\endgroup$ – Zacky Sep 9 at 12:30
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    $\begingroup$ @Arthur there is mentioned: "a matematician or atleast Ph.D of mathematics". By this I understand that a PhD of mathematics is not necessary a mathematician. $\endgroup$ – Zacky Sep 9 at 12:33
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    $\begingroup$ I'm a high school teacher with a Bachelor's Degree in Mathematics. According to that link, I meet the conditions of being a Mathematician. $\endgroup$ – Mark B Sep 9 at 15:18
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    $\begingroup$ In the 80s I heard the definition that to be a mathematician you need to have at least two papers listed in Math Reviews, Zentralblatt or the Russian equivalent. Ph.D. was not a requirement for graduate students (even exceptional undergrads) will often meet that requirement. Also that definition has a bit of lag. $\endgroup$ – Jyrki Lahtonen Sep 9 at 18:50
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    $\begingroup$ Math genealogy is an ambitious project to build a database of math Ph.Ds, with particular attention to the directed graph formed by advisor-student relations. I don't think it is as comprehensive as it would like: a bit of Western emphasis probably built into it, also not every math department (or individual) plays along and reports their data. $\endgroup$ – Jyrki Lahtonen Sep 9 at 18:53
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    $\begingroup$ A lot of professional mathematicians here use our real names and link to our professional websites. With a case-by-case approach, you can investigate the people who you come across. $\endgroup$ – Michael Burr Sep 10 at 17:07
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    $\begingroup$ I bet I could relate abstract nonsense to real life given enough time. $\endgroup$ – Roddy MacPhee Sep 11 at 13:48
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    $\begingroup$ Several comments concern the difficulties in defining "mathematician". I wish to point out also a possible ambiguity in the phrase, "are here". Users are more-or-less divided into those who ask one or more questions, but don't post answers, versus those who regularly reply to questions but rarely, if ever, ask any. I expect the great majority of users are in the first category, while the mathematicians, however defined, are largely in the second. One might also want to distinguish between those who appear briefly, then vanish, versus those in it for the long haul. $\endgroup$ – Gerry Myerson Sep 11 at 22:00
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    $\begingroup$ The fact that you get high quality answers is more important than the information that the answerer is/isn't a professional mathematician. $\endgroup$ – Paramanand Singh Sep 12 at 4:04
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As Paramanand says aptly in the comments :

The fact that you get high quality answers is more important than the information that the answerer is/is not a professional mathematician.

First, I will answer the question : By my estimation, I have been on the site for about three and a half years. I would like to mention that there are quite a few experienced and retired professors on this site. Without taking names, the one I know the best is a retired professor who loves probability from the bottom of his heart (I've had the pleasure of seeing him in person). He is a very frequent poster here, in the field of probability. There are some that are all rounders, and I must admit that having an interaction with any of them is an enlightening experience.

Some very famous people rear their heads often on this site. Did you know Noam Elkies is on this site? (If you don't know who Noam Elkies is, then find out : I had the pleasure of a chat with him on this site).

A lot of people on this site are actually probably just out of, or are in their PhD stage. What helps at that stage is that they are real all rounders in terms of their knowledge, but also have a shade more experience in some specific area of mathematics. In my case , that would be Markov decision processes and PDE.

Therefore, most answers you will receive to non-elementary questions will be from these people, or from people with more experience. My last seven or eight questions on this site have all been attended to or been answered by people who I know are in their PhD stage or are professors.

Ideally, posting a question of high quality, or with a specialized tag, gets the attention of the resident expert on that subject. For example, putting up Stochastic processes will ring somebody's doorbell(somebody who's a real expert by my admission).

Other than that, we also have people who are inexperienced by their own admission, but you know what : there's a difference between knowing something, and communicating that to someone else. That's why the best professors are not necessarily the best teachers (the best footallers are not necessarily the best football coaches, as an analogy?). The best teacher I know of a subject is not even on this site : she taught my neighbor origami, and she is now twelve years old!

So remember, there are a lot of experts on this site, and I am sure we are all friendly and can have a chat with you on whatever you desire : but the quality of the answer is all that matters in the end, and I can confirm to you that a non-professional is also very capable of giving excellent answers on elementary questions : occasionally better than experts , I opine.

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  • $\begingroup$ Ah, Noam Elkies. I know him as a contributor in Conway's Game of Life. $\endgroup$ – Dannyu NDos Sep 12 at 7:15
  • $\begingroup$ I often communicated between newbies and experts on mersenneforum after being taught set theory. $\endgroup$ – Roddy MacPhee Sep 12 at 11:49
  • $\begingroup$ @downvoters poor quality of post, or disagreement? $\endgroup$ – астон вілла олоф мэллбэрг Sep 15 at 9:59
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    $\begingroup$ Ideally, posting a question of high quality, or with a specialized tag, gets the attention of the resident expert on that subject. For example, putting up Stochastic processes will ring somebody's doorbell - As some evidence (not about Stochastic processes specifically), I am professional mathematician who is a casual user of this site, and don't browse much because there are so many questions, but have alerts set up to notify me of questions with a few tags that I am interested in. $\endgroup$ – Kimball Sep 16 at 7:05
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    $\begingroup$ Re your penultimate paragraph - I find it funny that the highest-voted among my answers is somewhat about origami (and to emphasize that it is about mathematics: The answer is totally useless in real life) $\endgroup$ – Hagen von Eitzen Sep 21 at 7:39
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I am not a mathematician and I am here. Therefore there is at least one non-mathematician here.

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    $\begingroup$ Written like a true mathematician. $\endgroup$ – Gerry Myerson Sep 11 at 21:55
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    $\begingroup$ This sounds like a quote out of "Gandhi." $\endgroup$ – Ron Gordon Sep 16 at 18:30
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    $\begingroup$ I'm pink therefore I'm spam... $\endgroup$ – copper.hat Sep 20 at 16:54
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    $\begingroup$ But if he proves that he's not a mathematician, he must be a mathematician. But then his proof is wrong and he is none. But then his proof shows he is one. But then .... $\endgroup$ – Hagen von Eitzen Sep 21 at 7:40
  • $\begingroup$ @HagenvonEitzen, "But if he proves that he's not a mathematician, he must be a mathematician." It only proves that he's a metamathematician. Not all metamathematicians are mathematicians. $\endgroup$ – Wlod AA Oct 3 at 3:17
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To cover an additional sector of people beyond those in астон вілла олоф мэллбэрг's answer. There are probably many people self-taught people with little to no formal credentials. (This is certainly true if PhD is the minimum bar.) Obviously, this is easily doable for something like high school algebra or undergrad calculus, but it's certainly not limited to that. While it's not realistically possible to become a surgeon, say, via self-study, it's very possible to learn "advanced" mathematics via self-study.

Mathematics is, along with computer science (CS), one of the most readily accessible fields as far as high-quality, freely available learning materials are concerned and, of course, little to no materials are required to actually do math or CS. For contrast, a lot more of the research literature of electrical engineering (EE) or chemistry is behind paywalls and learning resources are nowhere near as available. (Actually, EE isn't too bad about learning resources, though still much worse than math or CS.) There are also significant upfront and continuing costs (and risks...) to practicing EE or chemistry.

To be clear, if you want to contribute to the mathematical knowledge, it is extremely helpful to be in the academic mathematics community. It's clear a lot of progress comes from interactions between mathematicians. Also, a contribution will on really be a contribution if the ideas are disseminated which will often take more than simply publishing a paper that proves some theorem.

A non-professional can be very capable of giving excellent answers on even non-"elementary" questions. If your question isn't a research-level question (i.e. one that would be better on MathOverflow anyway), then it's very unlikely that it requires a practicing mathematical researcher to give a high-quality answer to.

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  • $\begingroup$ CS is a multi-disciplinary field. There are some parts of it which are fairly pure maths, and other parts which require significant materials (and, if you want to publish, an IRB) to do research. I suspect you're thinking primarily of theoretical CS. $\endgroup$ – Peter Taylor Sep 13 at 8:56
  • $\begingroup$ @PeterTaylor There is plenty of CS research that most wouldn't call "theoretical CS" but doesn't fall under human-computer interaction (HCI), e.g. the papers published on Google Spanner. I was also (poorly) trying to separate understanding, consuming, and applying ideas from math or CS (the "do") from contributing to the field. Learning enough group theory to understand homological algebra is a very different thing from understanding enough to make meaningful contributions to group theory. $\endgroup$ – Derek Elkins Sep 13 at 10:38
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    $\begingroup$ That said, a lot of HCI research I've seen is quite feasible for an individual to do. This 2015 paper which is highlighted on this ACM page uses Mechanical Turk and with $n=330$ has a cost of ~$100. $100 is less than a bottom-tier oscilloscope costs. Participant costs on the order of a couple hundred dollars is what I've usually seen in HCI research. Nothing about this is out of the reach of an individual. $\endgroup$ – Derek Elkins Sep 13 at 10:39
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    $\begingroup$ I really do not like the example surgeon vs learning advanced mathematics. The one is a about exercising a profession the other is about knowledge. I'd say it's not overly realistic either to be completely self-taught yet be authorized to say supervise PhD students. On the other hand I do not think it is overly difficult to acquire via self-study knowledge of anatomy at a par or even above that of a typical working physician. $\endgroup$ – quid Sep 14 at 0:09
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    $\begingroup$ @quid My point is that there are fields where some of the knowledge you need requires hands-on work but getting the means to do that hands-on work is limited. Surgery is one of those and so is supervising PhD students to a lesser extent. For EE or chemistry, there is a lot that you can learn from books but you'd be hopelessly incompetent if you'd never actually tried to build a circuit or go through a reaction. To get enough experience with that would require a non-trivial but doable investment in materials. $\endgroup$ – Derek Elkins Sep 14 at 1:04
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    $\begingroup$ I'm not saying you can become a professional mathematician via self-study; I'm saying you don't need to be a professional mathematician to have most of the knowledge be accessible (which is not to say it won't still take a lot of time and effort to acquire). You certainly don't need to be to competently answer a large swathe of "advanced" questions. Part of this is just the nature of the field, but a significant part of it is that mathematicians do go out of their way to make a lot of the research available and do produce a lot of good learning materials which I appreciate. $\endgroup$ – Derek Elkins Sep 14 at 1:05
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    $\begingroup$ I certainly agree that mathematics is more accessible than various other disciplines since it (at least in many sub-areas) does not need much resources. However, I'd argue that the knowledge needed to answer questions in the context of an SE, also those considered advanced in that context, is also accessible for many a subject. You might well be right that in mathematics huge parts of knowledge are freely available, even if not every document, while in other disciplines some things might be more restricted, but I don't really know. $\endgroup$ – quid Sep 14 at 16:13
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    $\begingroup$ I've certainly seen examples on EE.SE that would be difficult to answer without hands-on experience. As a hypothetical example, if someone presented a circuit diagram and oscilloscope output that was supposed to be a square wave but clearly wasn't, there are many possible reasons for it. Maybe it's stray capacitance, a loop in the circuit behaving like an antenna, a loop in the probes or power cables, a component going bad, poor probing technique, or a misconfigured oscilloscope. Most of these would be practically impossible to recognize given only book knowledge but doable given experience. $\endgroup$ – Derek Elkins Sep 14 at 20:02
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Would you say that Pierre de Fermat was a mathematician?

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    $\begingroup$ Yeah, he was definitely a mathematician $\endgroup$ – Donlans Donlans Sep 22 at 2:31
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    $\begingroup$ Woa... wait, is he on this site? $\endgroup$ – polfosol Sep 22 at 9:23
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    $\begingroup$ @polfosol Impossible for legal reasons ;-) $\endgroup$ – Paul Frost Sep 22 at 15:46
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    $\begingroup$ Pft! Real mathematicians use appropriately sized paper and don't try to fit all of their work into the margins! $\endgroup$ – Xander Henderson Sep 22 at 22:27
  • $\begingroup$ @XanderHenderson But some write postcards. books.google.de/… $\endgroup$ – Paul Frost Sep 22 at 22:42
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There is probably a somewhat large number of MSc engineering graduates having gone back and specialized in mathematics applied for their field on PhD level.

I am one of those, anyway.

For more "pure math" folks it seems to me that MathOverflow is a more of such a site.

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    $\begingroup$ To me, the principal difference is that MathOverflow is pitched at a higher level than math.SE (rather than it being a pure/applied split). So you will get many pure maths PhDs here helping to answer questions, but they will ask their questions on MathOverflow. $\endgroup$ – user1729 Sep 16 at 17:59
  • $\begingroup$ It's mainly a language split. In the sense that professors at engineering departments don't use the same language as professors at the math department. No stranger than if you insist on using correct algebraic names of everything you end up annoying and or confusing 75% of your students who expect useful names for an applied course. $\endgroup$ – mathreadler Sep 26 at 6:34

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