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What did a good academic job market for mathematicians look like and could it ever come back?
I'm about to enter a math PhD program at a pretty good school, but the job market for research university professors has been terrible as far as I can remember (since the late 2000s recession). I hear many stories of hundreds of excellent candidates applying for one or two openings.
Older professors tell me it wasn't always like this. What did the market look like when it was considered "good" and could it ever come back by the time I finish in 5-6 years? Or has there been a fundamental shift in academia preventing the good times as we knew it from ever returning?

Yes, the job market nowadays is not what it once was. On the other hand, if you like spending your days doing math, and don't care about getting rich, academic math is not the stupidest gamble, in the sense that, in the worst-case scenario, you have to get "a real job like everybody else". But, yes, "academic math" is not a "career choice", but more a way of converting a hobby/obsession into something that pays a bit, usually with health insurance.

And if you have a PhD in math, you should be able to find a fairly interesting "real job" more easily than looking for one with just a bachelor's degree.

@PeterShor - It obviously depends on your bachelor's degree and what you learned, and on what you mean by "interesting".  I can say my pre-grad school salary was, up to rounding error, the same as my current salary, and that's without accounting for 15+ years of (admittedly low) inflation.  I'm glad I made the decisions I did, but from a purely financial point of view...

I wonder how closely this relates to the computer industry job market.  Pure speculation, but could it be that people are transitioning into software engineering and the like from "pure math," which almost by definition is unmarketable?  (That is, if you define "pure" as "not applied to real-world or business problems.")  The computer industry job market isn't hurting that I can tell.

Maybe in the past, with scarcity of mathematicians, they could follow their passion and find a job. Now you kind of have to find what will be in demand in the futur (big data, ai, ...) and find your passion in that. You need to be a bit smarter in the type of research you'll be doing.

@Wildcard: I think many (U.S. domestic) computer engineers might disagree with you. I constantly see stories of people training H1-B visa holders to replace them, inability to get hired after age 45 or so, etc.

@DanielR.Collins - There are a lot of "software engineers" out there who, after many years of experience, sadly still don't have the ability to read (decently written) code and abstractly reason about what the code is doing without actually testing it and stepping through each variable.  These are the people who can't find jobs at age 45.  I would hope most people even contemplating math grad school won't have this problem.

@AlexanderWoo: On the contrary, ageism is a fairly well-documented point of bias in the software engineering trade, even for talented people: https://newrepublic.com/article/117088/silicons-valleys-brutal-ageism

@paulgarrett I'm probably missing the point of your last sentence. Do you mean that "something that pays a bit, usually with health insurance" is not a career?

@JiK, of course it depends what one means by "career". To my perception, the still-viable version of academic mathematics (which is what many of us always thought in terms of) is more like a hobby/obsession that quasi-miraculously pays modest bills, etc. Not a thing to pursue "as a job", since one would have to put more effort into it than is rewarded in any obvious way, etc. Not a "rational" choice, in many ways, even during the Cold War. E.g., if a person did not like math an awful lot, and needed lots of money, math was always a pretty dubious choice.

First, some historical perspective: Francis Sowerby Macaulay and Hermann Schubert, two mathematicians from the late 19th century whose names you still hear repeatedly these days if you come anywhere near algebraic geometry, both taught high school.
Second, a counting perspective: Research university professors probably average a doctoral student at least every five years, with some averaging more than one a year.  This means every professor produces somewhere between 6 to 40 replacements over the course of their career.  Assuming no change in the number of jobs, only one of those replacements will get a research university professorship.
Third, a practical perspective: It's hard to imagine research in mathematics, especially but not only pure mathematics, having direct practical benefit to many people.  That means the amount of funding purely for research tends to be rather small.  Even at research universities, teaching is an important rationale for paying a professor's salary, and the benefits to teaching are an important justification for research.
What happened between the 1950s and 1980s, particularly in the US, was an enormous expansion in higher education along with a massive increase in research funding.  The US went from having about 5% of its 20-year-olds going to college to about 50%.  Assuming a corresponding increase in the number of professors, this meant a professor could advise 10 PhD students and have every one of them get a job.  For obvious reasons, we will never see a 10-fold increase again.  Also, the US was enormously wealthy and had resources to spend on less practical purposes like mathematical research and providing high quality university education.  (The US still spends about 50% more per university student than Germany.  Most of that is going to smaller classes with more attention paid to helping students along.)

I would like to contradict you; lots of research in applied mathematics has direct practical benefit, even if the majority of it doesn't.

"It's hard to imagine research in mathematics ... having direct practical benefit to many people." How about RSA (which is based on number theory)? Or digital circuits (from boolean logic)? Or GPS (those working on non-euclidean geometry didn't think space worked that way)?

@PyRulez - I don't think any of those examples count as "direct".

@AlexanderWoo Whoops, missed a word. Sorry.

A lot of work done by people has little direct practical benefit to people. This is not specific to mathematics. Just don't sum positive numbers to $- {1 \over 12}$. [What??? Does Mathjax not work here?]

'That means the amount of funding purely for research tends to be rather small'. No. It is something specific to our current society that funding for research depends on 'direct practical benefits'. There is plenty of scope for funding things with only indirect practical benefits, or direct philosophical or spiritual benefits, or nebulous indirect impractical benefits, but we choose not to make that decision.

@PyRulez, the problem is perception, not reality.  Sure, if you pause to think about it, number theory gives you encryption, statistics gives you quantitative finance and the insurance industry, various branches of calculus give you high-frequency electronics and the semiconductor industry, and so on -- but the point is the applications aren't obvious, unlike something like fluid dynamics.

@PyRulez I think the distinction to be made is one in the present tense, not looking backwards with hindsight.  One cannot claim that one's work on number theory is in support of RSA until RSA comes into being (or is part of the development of the algorithm itself).  Before that point, it is hard to justify the expenditure of funds using RSA as your rationale.  After RSA came out, there was certainly some direct practical benefit coming from beating on the problem and making sure things are secure.  One of the great challenges universities and other higher education groups face...

... is trying to convince people to give them enough credit for building the underpinnings of the flashy inventions that people recognize and value.  It's a never ending struggle.  I've lost the speech, but there was a beautiful speech to congress by either Fenyman or Oppehnehimer which basically stated that they could not name a single practical use for the research they were championing, but that congress should pay for it anyway.  Their argument was beautiful.

@CortAmmon - and it's hard to imagine anyone giving the same speech nowadays, or not being laughed out of the room.

What are the "benefits to teaching" of research? I'm curious.

@RenéG 1. People who are also doing research have a more direct connection to the subject and so might be better able to teach it. 2. Paying people to do research as well as teach allows you to attract a better class of teacher.

Compare page 3 of the AMS annual survey from 1999 with page 4 of the 2008 edition and page 4 of the 2014 edition, in terms of employment for new hires we are actually much better currently than the early 90s.
Depending on the age of the individual you talked to: the good old days could be everything up until the late 80s, when rate of unemployment remain largely below 2%, and new graduates are often immediately hired on tenure-track positions. We are almost certainly not going to return to that level of employment: in the post-WWII era through the cold war there have been (for one reason or another, GI Bill for example) expansions in funding for academia. That growth was never sustainable. As a benchmark for the future you probably want to consider the data from 1990 on.
If you look in the 1999 data, of those obtaining employment in the United States, just under 30% (of the new PhD recipients) are hired in postdoc positions and a bit over 50% in permanent positions (30% in academia and the other 20% in industry/government). The 2008 data is similar. The postdoc percentage is higher in 2014 (38%) but not outrageously so.
You can spend some more time teasing out the data yourself; the annual survey results are available online for the past 15 years, and presumably the AMS has the old data dating back to the 50s available somewhere.
If you want to be more forward looking and consider hiring statistics overall instead of just for new PhDs, since the economic crisis the AMS has been running employment surveys every years. A quick glance at the 2014 survey suggests that things have improved slightly.

As a side remark: the large number of applications mathematics positions receive has in some part to do with the availability of MathJobs.org. The website makes applying to 50-100 positions in one hiring cycle relatively easier in mathematics compared to other academic fields without a similar centralized job database. So the inflation of number of applications received per opening compared to "old days" does not 100% correlate to the health of the job market.

Exactly. The GI-Bill-fueled expansion of college populations, and Cold-War-fueled expansion of federal research funding (and rationalizations about teaching loads), has not continued, and would not have made sense to continue. But the general structure and mythologies were/are grounded in that expansionist time... On another hand, in what line of work is everything always easy? The (falsely) "Golden Age" (which was unsustainable and substantially delusional) is over. I must say that I feel lucky to have caught that wave a bit, but even in the best of times academic math required... [cont'd]

[cont'd] ...  personal sacrifices, geographic location very significant among them. Especially prior to "internet", leaving family, friends, and maybe working spouses/partners behind... and more than once?

Don't forget the what a powerful force Sputnik was to in re-energizing governmental funding for the physical sciences and engineering in the US. I've talked to people hired in that spurt. Phrase like "a pulse and a Ph.D. or even ABD were enough to get hired" came up more than once. I like to claim that every other physics build on a US campus was built in the aftermath, though that estimate has very wide error bars.

@dmckee, ah, yes, indeed! "Sputnik" created a sort of "carte blanche" situation for physical sciences for a long time, maybe through the 1960s, at least.

So how many applications do mathematics positions in academia typically receive? I'm curious, because I've just applied for one myself.

@Significance It varies significantly between Universities (rank/popularity and if it's research or teaching focused). Some get well over a hundred applications per position, and some (especially smaller liberal arts and/or little state schools) fail their search because they didn't get enough applicants or no one took their offer.

@Significance: I work at a large public research university. Last year we advertised for one position and got over 400 applications. And I've heard of places with even worse ratios.

@Significance: I work at a top-5 math department, and last I heard I believe we got something like 600 postdoc applications. But as you can imagine, the number of serious candidates is much much smaller -- so these statistics, although striking, are ultimately rather uninformative. Most of these applicants are just checking off all the top universities on MathJobs, because after all, why not? (We still read every application, though!)

There are different levels of "good".  In 1969, as I was starting my last year in graduate school, a senior professor told me about job-hunting: "Well, you should decide where you want to go and assume that we can get you in there." I'd call that a good job market. But, in my case, it didn't work that way; apparently the greatest effects of the post-Sputnik boom in math had just faded away. So did I encounter a bad job market? By comparison with prior years, yes. But I sent out a total of three applications and got two offers. By comparison with more recent times, that was still an extremely good job market.  I see no realistic hope for things to get that good again.

In my opinion, there are two factors that define a "good" job market. One of them is NOT how many people apply for any given position. The number of people applying for a position depends on the number of people graduating, the number of people who want to switch universities, and the number of people who want to switch careers. I believe even in a good job market, you can still get 100s of people applying for positions. I do not believe a "good" job market means that everyone who wants a TT position with a 1-1 teaching load (i.e., 1 class in the fall and 1 in the spring) at an R1. I think a good job market means the "top people" (I don't want to define "top people") eventually get a job.
The first factor that I think defines a "good" job market is the number of positions "good" positions being advertised. I think of a "good" position in the US as a TT position with a teaching load less than 3-3 (i.e., 3 classes in the fall and 3 in the spring). My field is on the small side, so in a bad year, there would be less than 5 positions targeted to my specialty and in a good year 10+. In a good year there might also be another 10+ open calls and another 10 or so calls where if you squint enough you can argue you fit. So in a good year there might be 30 or so "good" positions to apply to and in a bad year less than 10.
The second factor is the number of positions that will help you progress. These could be soft money research positions or poorly paid one year visiting positions with a reasonable teaching load with colleagues that will help you become a better researcher. I do not consider 5-5 adjunct positions as helping you progress. Most of these progression positions, at least in my field, are a direct result of grant funding. They could come about from someone on the job market getting a grant themselves or from a current faculty member getting a grant to buy out of their teaching. In a good job market, there needs to be enough of these that the "top" people can avoid being unemployed and stagnating for a year.

I think this is a nice answer, but some points don't seem to apply to math. I've never heard of a soft-money research position in math, for example.

@TomChurch when I wrote the answer, the question was more general than it is now.


text:
Does "Science" encompass Mathematics?
Inspired by the question Does the term “science” encompass humanities and the social sciences?, I would like to ask a question that has been in my mind for decades.
Does "Science" include Mathematics? Or, put in another way, is Mathematics part of Science?
What I understand about Math are: Math is not humanities.  Math is not natural science. Math is not social science. If Math is part of science, what kind of science is it?
When people say STEM, do they mean Math is separate from Science? Or they just say it for the sake of convenience?

"Mathematics is the queen of sciences and arithmetic is the queen of mathematics" -- Gauss.

This question is a duplicate of a duplicate: http://math.stackexchange.com/q/288935.

@DanielR.Collins Sure, but, for instance, the queen of Australia isn't Australian.

@DanielR.Collins interestingly, Elizabeth II is queen of 16 countries but is apparently not legally a citizen of any of them (see here). So being queen of the sciences is perhaps not proof of being a science...

@FedericoPoloni beat me to it, but actually she is Australian (as the monarch she is "the embodiment of Australia" or some such nonsense), she is just not an Australian citizen. The legal and philosophical discussions on this question remind me of the question about how many angels can dance on the head of a pin. They are about equally logical and clear, and concern an equally useless question. Same for the current question about math and science, btw.

Some additional ideas: http://math.stackexchange.com/questions/287701/what-is-the-definition-of-mathematics

IMO, the response to Daniel's quotation is: yes, that's what someone who was one of the world's leading mathematicians and one of the world's leading scientists said over 150 years ago.  So it's certainly interesting, but the question pertains to today.  The fact that today there is no one who is viewed as a preeminent expert in science as a whole and no one who is at the top of a handful of academic disciplines (e.g. astronomy, geodesy, mathematics, physics, statistics) also seems relevant to the question.  The academic landscape is much different now than in Gauss's day.

Actually, if we follow history's lesson, the most common way to become "queen of sciences" is invading sciences with your troops, destroying them militarily and forcing them to surrender.

@DanielR.Collins: how can a queen have another (sub-)queen? That's as if there were a queen for the UK and another for Birmingham.

@DanielR.Collins: Given that Gauß was German and I can find German versions of that quote (implying that the English version may be a translation), you should take into account that the German term Wissenschaft commonly encompasses pretty much everything that can be researched, not just "natural sciences". In fact, that difference in meaning between Wissenschaft and science was what the original question this question was motivated by was about.

@DanRomik I think questions aren't considered duplicates just becase  they have been asked on other SE sites.

Relevant XKCD

Does science include mathematics for what purpose?  If you're asking about funding by the US National Science Foundation, then yes.  If you're asking about potential customers for your scientific lab-equipment catalog, then no.

Is the question very different from "Does science encompass English (or other language)?" But both questions might need to be differentiated from the study of math/English, which can be scientific.

I would say Mathematics is the language in which science can be expressed, I got this from an old lecturer that used to say 'Mathematics is the language of Physics'.

What definition of science do you want to use?

Depends who are the 'people' who 'say STEM' and the context within which they say it.

If one defines "science" as a process to understand the physical world through experimentation, which is a reasonable definition, then mathematics would not apply, since pure mathematics is not a study of the physical world and does not involve experimentation. The "scientific method" is not used in mathematics. Many things in mathematics are called "discoveries", the same way scientific concepts are called "discoveries", but they are not arrived at in the same way. There's a good case to be made for math being its own thing, since it's not merely a language either.

Or maybe "mathematics" encompasses "science".

@DanRomik: FWIW, The Queen would have held a passport prior to ascending the throne ("All other members of the Royal Family, including The Duke of Edinburgh and The Prince of Wales, have passports.")—and that document would have stated something under "citizenship"...

Related question on HSM: https://hsm.stackexchange.com/questions/54/when-did-mathematics-stop-being-one-of-the-sciences.

This is a good question, but there is no consensus as to a good answer.  Some people think mathematics is obviously a science, some people think it obviously isn't, and some just aren't sure.
It's common to include mathematics as a special case of science in general discussions.  For example, universities usually classify mathematics under the sciences, and "scientific publishing" would generally be understood to include mathematics.  However, there are exceptions, and abstract discussions of science often don't apply very well to mathematics.
If this distinction matters in a given case, then you'll have to discuss it explicitly, since you can never assume everyone will agree by default.

If Math is part of science, what kind of science is it?

The most compelling answer I've heard is that it is a formal science.  However, this terminology is somewhat obscure, and it is certainly not a consensus answer to your question.

When people say STEM, do they mean Math is separate from Science?

I don't think the intention is to assert that they are definitely separate, but rather just to have an inclusive acronym we can all agree on.

Super happy to be introduced to the term "formal science", thanks!

It's a "mathematical science" is a more tautological, but also common answer.  (Statistics is certainly also a mathematical science, and I'd also use that term to refer to much of CS and theoretical physics.)

Though if math is considered a formal science, I would disagree with the statement from the wiki link: the formal sciences do not involve empirical procedures.

Mathematics is also an exact science, so since you can completely go around empirical methods and arrive at definite conclusions with it that are not prone to error, it's safer to take a "fact" in a mathematical textbook and assume that it really is true.

The mistake is in thinking that mathematics is purely an accounting term when in fact it is the language of science. Without numbers, most science can achieve is a hypothesis. In mathematics, the question is asked, a model is made, and a solution is sought to explain the data. This must then be explained. How does it not fit the scientific method? Most science started in math. Newton melded mathematics and physics. Yes, mathematics is a science.

One response. Differential equations, statistics, and data science.

The first definition of science that Merriam-Webster gives is:

: knowledge about or study of the natural world based on facts learned through experiments and observation

Math doesn't investigate the natural world and thus isn't a science according to that definition.
In How to Think Straight About Psychology Keith E. Stanovich defines 3 main traits of science.

Three of the most important are that (1) science employs methods of systematic empiricism; (2) it aims for knowledge that is publicly verifiable; and (3) it seeks problems that are empirically solvable and that yield testable theories (the subject of the next chapter).

Math isn't employing empiricism and thus doesn't fulfill criteria (1) and (3).

On the other hand, "I have tried to encourage the reader to think of the computer as a physicist would his laboratory - it may be used to check existing ideas about the construction of the world, or as a tool for discovering new phenomena which then demand new ideas for their explanation." -- Tristan Needham, Preface to Visual Complex Analysis (1996).

Math doesn't investigate the natural world - I don't think it's so clear cut.  Discrete quantities occur (at least apparently) in the physical world, so counting, and thus combinatorics, investigates the natural world.  If you disagree with this logic, then by the same logic, you should also not consider much of theoretical physics a "science."

@Kimball : The math that describes discrete quantities can't be invalidated by empirical experiments because mathematical claims are analytic in nature. They are true in all possible worlds.

@Hurkyl Empiricism is a useful tool in a mathematician's toolkit, but empirical evidence doesn't establish a theory in mathematics.  Test millions of cases in an empirical science, and find that none of them violate the predictions of a hypothesis, and you start to establish a theory that people might accept and adopt.  In mathematics, people still recognize it as a conjecture (which might be useful to accept as a premise in some proofs), and might expect that it's true, but still aim to prove or disprove it.  Empirical methods are very helpful, but don't provide conclusions in mathematics.

This answer is so blatantly wrong it is laughable. What are you explaining then? Nothing. Mathematics is based almost entirely in the study of nature. Modern math came from Newton's study of physics. Saying this is like saying chemistry isn't science because its just a bunch of conversions. Math is empirical (if I experience something or have sensed a bunch of patterns I can then explain them) and it seeks solutions to patterns and nature that can be proven and tested.

@hurkyl yes, and I alluded to some of that type of use when I mentioned that we use unproven things as assumptions and work that way.  E.g., we don't know whether P = NP, but there's lots of work that looks at the large body of examples suggesting that they're distinct, and we can reason with that as an assumption.  In that sense there's a scientific theory based on P != NP with predictive power.  The actual fact is still something that people want to prove.  Empiricism can help guide,  but it doesn't settle the question.

@AndrewScottEvans : If you would have asked Newton, Newton wasn't just concerned with the natural world. Newton wrote about occult subjects and Bible interpretation as well as writing about natural science. One of his core ideas was that math is so universal that it can be both used to understand the heavens and earth.

@Christian So Newton's 'natural' world was a bit of a stretch but until the 20th century those two might as well have been similar. It is no different than a difficult to prove or incorrect scientific theory depending on whether you are atheist or not. If I write 1 it means little if I say 1 + 1 is 2, I may as well have said if I add 1 of something to another one of something then I will have two of something, because having one more of an object makes that object plus another which is 2. I then have to prove it. That is the entire idea behind most mathematical studies.

@AndrewScottEvans : The fact that you can prove something analytically without looking at the natural world, makes it an analytic inquiry. You can't prove anything about the natural world as Popper argued.

@Christian Physics is full of such 'discoveries' as are many sciences. Long ago we could only predict outcomes, even some astronomy is written on paper to be discovered later. In fact, why not think of mathematics as generating its own realm governed by laws and theory within its own sphere. You may be giving to much credence to the functional but even that is just a system where forces acting on variables x,y,z create the outcome f(x,y,z). The analytical part is a portion of the whole not the entire whole.

@Christian In fact, how can you prove that adding 1 to any number of objects is 2. You need observation or proof of other patterns. Induction, observation, and the scientific process are highly important. Once you move beyond the basic building blocks, mathematics is highly scientific.

@AndrewScottEvans : No, if I add two piles of apples together I get one pile of apples. A mathematician can assume that certain axioms hold and prove that certain axioms are true. In Science you can't assume that anything is certain. Nature might always prove you wrong.

The problem is you are trying to limit mathematics to its very basics and ignoring its many fields

Platonic mathematics
the kind of mathematics having no essential connection with the physical world. I say "essential connection" because links between nature and esoteric mathematical notions spring up in unexpected places, like the surprising connection between locusts and prime numbers.
Natural mathematics
the kind of mathematics arising from the uncanny efficiency with which mathematics describes nature.

@AndrewScottEvans : The fact that there a link between abstract concepts and the natural world doesn't mean that they are the same thing.

On my journey through Computer Science, Machine Learning / AI, and Cryptography I've often wondered the same thing. Here's the way I currently see the interactions between math and its related disciplines through the branches of mathematics that I've studied:

I imagine there's a fair amount of subjectivity here, and that other branches of mathematics will have their own unique place on the Venn diagram.

There's experimental moral philosophy. I think that's science, so the nonoverlap between Science and Philosophy doesn't make sense to me.

@Christian Having never heard of experimental moral philosophy, I would have a hard time putting it in the diagram :P. I'm not saying that Science and Philosophy don't overlap, just that they haven't overlaped in any of the disciplines I've studied.

Weird diagram, I would consider logic strictly mathematical.

@YoTengoUnLCD Interesting. I've heard it the other way; mathematics is a subset of logic - which includes several non-mathematical fields. Wikipedia describes Mathematical Logic (like prepositional logic) as "applications of formal logic to mathematics" ... implying that formal logic is outside mathematics.

Unfortunately the duplicate in mathematics hasn't got good answers, so I will give it a try.
The answer is: Mu.
It means that the question must be unasked or that neither "yes" nor "no" is right or wrong.
Mathematics is examining the properties of consistent mental models or structures. It started with numbers and geometric figures and it was used for applications (counting, area calculation) from the beginning. It would not be completely wrong to name it "number philosophy" although it has expanded greatly and examines now a innumer...a very great number of concepts.
An example to show the difference to science: Let's say a scientist would try to prove the Pythagorean theoreom without mathematics. He would find out
that if we draw squares on the side of a right-angled triangle, the smaller ones look like they have the same area as the big one. He would experiment with it and while they really look very similar, he will never achieve an equal result. The cuts are not completely straight, the material bends, the lines have always some extent. Even if he succeeds in determining that for all tested right triangles the values are mostly equal, he/she can never be sure that it will stay so.
A mathematician can prove that the sides are exactly equal. This is possible because s/he does not use real-world modelling and is therefore not limited by their applications. But while it is not the real-world, it still allows discoveries. It is not self-evident that right triangles have this interesting property.
Mathematics is a necessary part of science because it is a building block for any precise models we need to refine the work. Physical values are models by numbers (or matrices/tensors) and a concept of a dimension. In higher physics countle...a big number of mathematical models are used.
For the reason that mathematicians and other scientists are sharing many mental similarities (curiosity, challenging matters of course and relishing hard, but interesting problems) their faculties are often joined together which results in the STEM field.
So mathematics is "sciency".
But...it is not really a science because it is more fundamental: It does not require knowledge or experience from other scientific fields. If a mathematician travels back in time 20 000 years (the humans were on a comparable intelligence level as today) he would be able to teach a bright kid modern mathematics which is impossible for modern science because there is no infrastructure to replicate experiments. Mathematics also has no room for error or reevaluation. Once the axioms are set (yes, if the axioms change, the result can change), a result is valid for all times.
Mathematics is also used massively outside science including the humanities.
So mathematics is not a science.

Part of your description of math sounds more like logic than math to me, and I'm not sure about some of your penultimate sentence.  Some parts of math don't require understanding other fields, but the same is true in other fields of science.  And there has been lot of misunderstandings in math, and consequently reevaluation.

@Kimball "sounds more like logic than math to me". Logic is not only part of mathematics, modern mathematical logic is the base for Zermelo-Fraenkel set theory: the mathematical foundation. Modern logic is much more powerful and stringent than the old syllogistic Aristotelan logic (building sentences from propositions). So yes: Logic is one vital part of mathematics. The reason I use a general "consistent mental models" is the variety of mathematical structures with different propreties which exist: numbers, ,differential forms, groups, nodes, graphs, logical propositions.

@Kimball "Some parts of math don't require understanding other fields". If you mean other fields than math: All parts of what we call math do not require understanding other fields. None. Zero. It is really that brutal, sorry. If you do not believe that, give a counterexample. The misunderstandings and errors depend only on human defectiveness; in natural sciences all the imperfectness of senses, measurement errors and failure to reproduce something lead to failures even if you have made perfect deductions.

The answer to this question is perhaps not as clear-cut as it might appear, i.e. the issue is not merely definitional.
Some (for example the physicist Roger Penrose, in his book `The Emperor's New Mind') believe that mathematical structures exist in a non-physical, Platonic realm.
Conversely, the quantum physicist David Deutsch has claimed (notably in his book, `The Fabric of Reality') that, since the brains of mathematicians are physical objects, then the structures they can apprehend are constrained by the laws of physics (more specifically, to be computable by a quantum computer).
In `Where Mathematics Comes From', the cognitive linguists Lakoff and Nunez claim that, even if a transcendent Platonic mathematics existed, we would be unable to experience it, since our conceptions are analogised from our physical experience of space, force, motion etc.
Here is a quote from a recent article in the Notices of the American Mathematical Society by AMS President Robert Bryant:
People say that mathematics is logical, but the logical aspect is only part of it. Mathematicians usually don’t proceed logically. They make guesses, see patterns, do experiments, develop beliefs. Almost nothing in that process
is purely logical.
I'd personally say that this alone puts mathematics on a conceptually equivalent status to physics, but not everyone would agree with this.

I guess I don't really understand the relevance of most of what you say to the question at hand.  Is the existence of or contemplation of a "non-physical, Platonic realm" part of science?  I would think not.

Pete L. Clark - What I'm getting at is that the question of whether or not mathematics is Platonic is open (but possibly not falsifiable). If mathematics can be reduced to questions about the nature of consistent human pattern perception, then it is within the realm of physics. Certainly, if science can be considered to be "hypothesize, experiment, potentially revise hypothesis, repeat", then Bryant seems to be saying that he thinks mathematics is a science, which I would personally agree with.

"What I'm getting at is that the question of whether or not mathematics is Platonic is open (but possibly not falsifiable)." Sorry, I didn't get it.  I said that the Platonicity of mathematics is not a scientific question, and you hint that it might not be.  But if it isn't, why is it relevant to the answer?  "If mathematics can be reduced to questions about the nature of consistent human pattern perception, then it is within the realm of physics." This doesn't make a lot of sense to me...

If, as Deutch claims, all possible mathematics can be simulated by a quantum computer then mathematics is a sub-discipline of physics as a matter of fact rather than being an issue of definition.

...That which lives in the physical universe is governed by the laws of physics.  That applies to all human activity and in particular to all academic disciplines.  In this way I can also reduce all academic disciplines to chemistry, biology, psychology and so forth.  But these are trivial remarks, unhelpful both intellectually and in terms of a practical understanding of contemporary academia.  I wonder whether I have misunderstood what you're saying.

It's not trivial in the context of academic practice because, culturally and historically speaking, pure mathematicians have tended to resist thinking that the laws of the physical universe apply to them. The notion that physical/statistical notions have a bearing on expressible pure mathematics (Deutch,Chaitin) is not common currency.

"It's not trivial in the context of academic practice because, culturally and historically speaking, pure mathematicians have tended to resist thinking that the laws of the physical universe apply to them." Well, now I am sure that I don't understand what you're saying.  Which mathematician has expressed skepticism that the laws of the physical universe apply to him or her?

Let us continue this discussion in chat.

More specifically, I mean "Don't apply to their mathematical conceptions", e.g. a perfect circle.

Sorry, your comments are too enigmatic for me to be drawn into a discussion. Just FYI, your practice of writing "more specifically" and then writing something brief that changes the meeting entirely caused me to give up.

Pete L. Clark - That's a pity. Just to clarify for the record: in response to your question about `laws of the physical universe applying to mathematicians', I mean: most mathematicians are happy to consider a perfect circle.

Max Tegmark has also put forward the hypothesis of a mathematical multiverse. To me this makes sense, you get rid of many philosophical problems if you dump the notion of a physical universe. If all that exists of our universe is nothing more than its formal mathematical description, then that description is a timeless entity, there is then nothing to explain about something having come into existence out of nothing.

I'll go for a semi-yes (maybe more).
When one says mathematics is not real, as compared to physics, I generally question the person about the reality of an electron. How does he know an electron is real? Ever seen one? Or guessed through models and measurements?
I long believed mathematics was not a science. However, parts of mathematics have become quite computational. Think about proofs for the 4-color theorem. Mathematics can teach about real-world problems, see the Pentium problem. And I am seduced by Alain Connes views on the existence of an

archaic mathematical reality outside space-time yet as inexhaustible as normal physical reality

discussed in details on SE Archaic mathematical reality:

Take prime numbers, for example, which as far as I'm concerned,
  constitute a more stable reality than the material reality that
  surrounds us

So, along with the increasing use of some mathematics in nowadays data engineering (data science/big data buzz words), I am more and more convinced that large parts of mathematics can be considered science... at least no less that string theory (Why String Theory Is Not A Scientific Theory).

Are you a Popperian falsificationist? Then you believe that only those statements that can be disproved by experiment can be classified as science, and as such, mathematics is not a science.
Of course, there are those that believe that mathematics is an experimental science, and the lab is the computer. If you are of this opinion, then math is a science in the Popperian sense, but most people think that software and hardware bugs are so common that this stance is untenable.
If you find Paul Feyerabend's epistemogical anarchism convincing, then you will claim that mathematics is a science, as it produces tangible value.
I personally wouldn't classify mathematics as a science. The best definition I can offer is mathematics is a set of techniques for thinking clearly about quantity and shape. But even that definition can be challenged with little effort.

Experiments are performed in the brains of mathematicians, not just `in the lab'.

While there are certainly some mathematical statements that are not falsifiable (Godel, Turing), surely you are not claiming that this is in general the case?

"quantity and shape" — math is the study of formal objects, not high school geometry and algebra.

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text:
Good book on English for academic writing and speaking for non-native speakers
I'm not native speaker so I always have some problems with academic writing and speaking. Any good book for English in the academic context generally or for Computer Science in particular (writing or speaking)?

For English for non-native speakers I strongly recommend:

Day, R.A. & Sakaduski, N.., 2011. Scientific English. A guide for scientists and other professionals. Third edition. Greenwood, Santa Barbara CA.
Glasman-Deal, H., 2012. Science research writing for non-native speakers of English. Imperial College Press, London.

Both cover general writing in English. In addition, it is difficult to avoid

Strunk, W. Jr & White E.B., The elements of style. Fourth Edition. Longman, New York.

The latter is a classic usually referred to as just Strunk & White.

+1 for the classic Strunk and White. Superb book for concise, easy to understand writing.

It would be useful if you wrote your research area. For a mathematician, Nicholas Higham's "Handbook of Writing for the Mathematical Sciences" is a good choice.

Computer Science

Add it to your question, as not everyone will be interested in comments under one of the answers.

I can recommend

English for Writing Research Papers
English for Presentations at International Conferences
English for Academic Correspondence and Socializing

all written by Adrian Wallwork and published by Springer.
The author shows examples from his work as a reviewer and a language teacher/professor and corrects them, so he writes an original version and the revised version. In some cases he mentiones specific problems of people from several countries with the English language.
The author wrote also a grammar book, but I don't know it.

Loved Alley's book because it focuses on scientific/technical writing as a craft and highlights quality criteria of scientific writing:

Alley, M. (1996). The Craft of Scientific Writing (3rd ed.). New York: Springer.

You may consider the following books:

On Writing Well : William Zinsser;
The Elements of Style : Strunk and White;
Sin and Syntax : Constance Hale;

Source: SciWrite: Writing in the Sciences.
You may also take the online course i.e. Writing in the Sciences at the following website:
https://class.stanford.edu/courses/Medicine/SciWrite/Fall2013/about

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text:
Is it plagiarism to copy formulations from others?
As a non-native speaker of english, I often struggle with always finding new, well-sounding, non-repetitive descriptions for the same thing and I also, to be honest, find it a waste of time of always having to do something different. For example:
Section 1 describes the X while section 2 is about the Y.
The Z is explained in section 3 and section 4 refers to A. The next section is about B...
Is it ok to just copy the formulation of someone else (of course my X,Y,Z,... are completely different) and always use the same thing?

You might also be interested in this question of mine about the expression that you cite: http://academia.stackexchange.com/questions/3501/whats-the-point-in-the-paper-is-structured-as-follows

Sentence patterns are not intellectual property; otherwise, every author who wrote "To X or not to X" would be plagiarizing Shakespeare. (They are "riffing" off of him, but not plagiarizing!)
The example you are citing is perfectly innocuous, particularly since you are not doing anything more than summarizing the paper contents. The only thing that would make it wrong would be to copy those sentences directly from someone else's work.

@kirdie In case you are not familiar with "To be or not to be", please refer to Shakespeare's famous phrase

Well I'm not a native speaker but I know that phrase :-)

There is a fine line between copying and plagiarism. In general the answer would be no, copying would be plagiarism. However, with certain fomulations there may be limited ways to vary. If the text you copy has some intellectual value, the result of someones inventiveness more than just lining up words to form a sentence, then the formulation has intellectual value and should be referenced, not for th eEnglish but for its content. A trivial sentence is just language and it is not unlikely that one would formulate a sentence identically to someone else. We also learn language from looking at how others (who we believe are better than ourselves) may express themselves. This is not plagiarism.
So for me the critical issue is if there is content other than linguistics that is being copied, if so then plagiarism is round the corner. So as a final statement, I would say: better safe than sorry; don't copy stuff, try to use the linguistic formulation but write the sentence with your word. Learn grammar and speling by checking what others do and emulate, not copy.

If in doubt, just repeat yourself.
One of the things that stands out about reading papers written by poor English speakers is the ugly contortions they go to in order to avoid repeating themselves. "X is about Y. Z is about W. A is about B" may be repetitive but it's perfectly understandable. I would suggest that you do this rather than attempting to copy the form of another writer, I'd also note that your assumption that you can simply lift the same form each time and have it look right may well be wrong anyway.
I realise this doesn't answer the question you actually posed but I think it deals with the question you need answering.

^this. With some effort you will have a number of these descriptive phrases that you can re-use and at least you know they are yours. The example you give seems to be a very obvious case to "copy" someone else, but it is also an example where there is no actual need to do so. You would pretty much end up with the same phrase if you start writing by yourself. However once you start copying others, you may end up crossing the line without noticing. Better to build up the discipline of processing someone else's thought and then writing it in your own words.

No, your example is definitely not plagiarism.
More generally, you can refer to the book "Writing for Computer Science" by Justin Zobel (Second edition, p.65, Section Quotations) to decide when to quote and when to merely copy:

"note that it is not essential to quote such a dull statement as
  (...); paraphrase, or even simply omitting the quote symbols, would be
  more appropriate. Omisission of quotation marks in this case is
  acceptable--that is, not plagiarism--because (this) statement is a
  natural way to express the concept."

The case Zobel is talking about is completely different.  His example deals with whether it's necessary to put quotation marks around "open sets are of insufficient power" in the sentence "Crosley [2000] argues that open sets are of insufficient power".  In Zobel's example sentence, I agree that it would not be plagiarism to omit the quotation marks even if "open sets are of insufficient power" was taken verbatim from Crosley, but I don't see how this is relevant to the question asked here.  Zobel isn't saying that you can just copy a "natural way to express the concept" with no attribution.

I did understand exactly that, that copying "a natural way to express the concept" was not plagiarism. (Which does not mean that the concept itself should not be correctly be attributed and referenced.)


text:
Why does the copy editor want Nature to be “Nature (London)”?
I have noticed that the copy editors of at least two publishers (the American Institute for Physics and American Physical Society) do something weird when they copy-edit my submitted papers. For all references to papers in Nature, which I include in my references as such:

J. W. Doe, Nature 197, 412 (1974)

they replace “Nature” by “Nature (London)”:

J. W. Doe, Nature (London) 197, 412 (1974)

I do not understand why they do that! I know it is customary in they style for books to have a city next to the publisher name (though I don't think it is very relevant in this day and age), but why do it for journals? Are there multiple different editions of Nature?

The first thing for me to say is I do not know, but I assume it is because they need to differentiate with another journal. According to the Web of Science list of journal abbreviations Nature should be listed as"Nature" and nothing else. There is another journal called Nature but with the abbreviation "Cah Rev The". I think yu can strongly argue for nature as "Nature" based on the fact that it is the Web of Science abbreviation and that "(London)" is not part of it. It would at least be interesting to know why they chose a non-s

For these journals, the copy editor is correct.  "Nature (London)" is listed in the American Physical Society's house style for abbreviations, which makes it the standard abbreviation for APS journals, and it is also in the American Institute of Physics's list.  It's not part of a general pattern of including locations, but rather a special case.
Presumably the inclusion of "London" was originally intended to avoid some long-ago ambiguity, perhaps with La Nature.  It sounds like La Nature was more of a popular magazine about science than a modern scientific journal, but then again so was Nature in its early history.
There is no serious ambiguity about the name Nature today, but publishers are reluctant to change abbreviations, partly out of fear that if you've been using a specific abbreviation for many decades, a careful reader may wonder whether a different abbreviation is a mistake or even refers to another journal.

Indeed, according to both AIP style guide, as well as APS style guide, references to journal articles should be referred to without a place of publication. However, in the ACS style guide on page 9, you can read the following:

For some periodicals whose CASSI abbreviation includes a place of publication, you need not add the place of publication unless its omission would
  create ambiguity. If CASSI lists only one journal with a given main title, there is no ambiguity in omitting the place of publication.

And indeed, the CASSI tool entry for Nature reads as follows:

Displaying Record for Publication: Nature (London, United Kingdom)

Thanks… though it makes me wonder why CASSI has Nature (London, UK) while it clearly is nowhere to be found in the official title in the Nature journal or its web pages…


text:
Using "Smith et al" when another Smith is also referenced
My name is John Smith and I'm writing a paper. In the discussion of the existing literature, I cite quite a few papers from another team, including many whose first author was a David Smith. I usually write this using heavily the “et al” style:

Einstein et al. first established in 1976 a possible plan for eradicating world hunger by massive beet culture in Antartica,1 but it took 20 years before Wiles et al. clearly delineated the challenges of such a prospect.2 The earlier analyses, by Smith et al.,3–7 held the narrow view that climate3–5 and transportation issues6–7 would be the limiting factors, forgetting to address the marketing aspects and negative implications on consumer image of the brand. In this paper, we present …

(the journal style calls for superscript numbers for citations)
Now, it seems somewhat likely that the reader may think the Smith from “Smith et al.” may actually be me. How should I help avoid this?

Not worrying about it.
Use first name or initial, “David Smith et al.” or “D. Smith et al.”
Choosing another author, like the last author, as in “Professor et al.”?
Some other formulation?

#2. This is the way I've seen it on other papers.

#1. The list of references should contain at least initials, which hopefully make clear this is another Smith.

I like your example, too!

Working through your list:

Using just the standard "Smith et al." is the usual standard that I've seen in my fields (physics, materials science, chemical engineering).
If you feel the need to indicate explicitly that this is not your work, then you can choose to use a variant that includes the full initials of the author:

"D. A. Smith et al. found. . . ."

Using a formulation "Professor et al." is incorrect usage of et al., which is normally used to designate in "actual" order the authors listed. The better formulation would be "Jones and colleagues" or "Jones and co-workers"; however, if the same first author is responsible for all of the papers, then using one of the other authors as the "focal point" is very misleading.
Other formulations, I believe, would be much less common than any of the other variants you've listed.

However, you could always try to just avoid mentioning Smith's name by referring to the contents of the work directly without saying "Smith et al. did X," by writing "X has been observed under conditions Y" or something similar.

If your references clearly can be traced to a unique paper then the name is not (or should not be) confusing. Yes, someone may mistake you for someone else or vice versa but then their checking of sources is out of sub-par. You can safely continue referencing the (standard) way you do it.
If two publications exist as Smith yyyy (Smith et al. yyyy) then it is common to use letters so that the references become Smith yyyya and Smith yyyyb (Smith et al. yyyya; Smith et al. yyyyb). So not even in this case is it necessary to add initials. You may find that initials are used in older puiblications but by introducing the letters to distinguish several similar references initials have become obsolete.
It is also possible to add indicators in the text that allows the reader to understand where your work is referenced by using "we" or "I" (as the case may be) when discussing a particular reference/result, that is use an active voice instead of passive.

This is how I have always seen it done.

Problem is that citation style in my field is most often superscript numbers (and references ordered at the end)… that doesn't help with disambiguation!


text:
How to cope with gap after PhD?
I submitted my PhD recently and am waiting for the oral examination. Though I have started looking for postdoc positions, the possibility of being without a job, even if for a small period after the viva, is quite frightening. How can I cope with it?

I'm facing the same situation. You could go on vacation. You could go job hunting. Depending on your skills, you could do a freelancer job.

Write papers, apply for jobs, relax.

It is not easy to relax when you have to pay the bills at the end of the month...

One possibility to check about—which I took advantage of at the end of my own graduate career—is that of being hired short-term as a teaching assistant. The good news is that you would have extra time to find a new position. The challenge is that you would be making a commitment to teaching for an additional semester. It would have significant ramifications if you were to  skip out on this duty if you obtained a job offer to start immediately at a company, or a postdoc offer at another university.

Very good, realistic, useful suggestion.


text:
Should I decline unrelated work assigned by PhD advisor?
During the past years of my PhD program in biology, my advisor has developed a habit of lending my service to other labs/groups for potential collaborating opportunities. In addition to helping fellow students in the lab with their projects, I've helped a handful of other labs/groups with various data analysis and statistical testing tasks that were totally unrelated to my PhD project; I rarely got anything in return other than practicing my relevant skills.
There is a lab in our department that I helped with data analysis in June. I've submitted my PhD thesis in August. This lab now came back to my advisor asking for a further work involving testing the correlation between the expressions of 200ish genes and different phenotype in 20ish RNA-seq datasets. My advisor has handed the job to me again.
It is not a small task to me, and since the submission of my PhD thesis my scholarship has stopped so I have to work to pay the bills. Also, I feel unappreciated because the other lab doesn't even know I am the person doing the job, and they will probably list my advisor as a co-author should there be a paper.
I'd like to know if I should turn down the request, even if it will potentially result in my advisor rejecting a recommendation letter? My advisor once jokingly mentioned that they are holding their recommendation letters hostage so I have to continue working for them (for free). I've done tons of unrelated and unpaid works for them and this project feels like the last straw.

"My advisor once jokingly mentioned that they are holding their recommendation letters hostage" -- that's unethical behavior.  Even joking about it is highly unethical regardless of whether you mean it, since (as in your case) those listening may take it as a real threat.

Get yourself listed as a co-author on any paper produced from your work. That might be the best you can hope for and might also be reasonable to them.

In some circumstances you can dodge out of these obligations by saying you just got hired part-time for a very lucrative position and will be getting promoted to full-time as soon as your degree clears. You would need to play off questions about your new job by saying you signed an NDA (in particular, do NOT claim to be working at a specific company). Obviously this only works if you are a good liar, and are reclusive enough that the advisor couldn't easily fact-check your vague claims. And you don't want the lie to snowball into a sitcom episode...

Just to clarify, when you say "holding their recommendation letters hostage", do you mean that that other group would refuse to provide you a recommendation letter, or that your advisor would refuse to write you one, if you didn't do the required work?
My first advice would be to talk to your advisor, especially since your scholarship has run its course. No reasonable person would insist under the circumstances, but you should also consider that there might a misunderstanding, e.g. that your advisor feels that you benefit from such an arrangement, so an honest conversation might clear everything up. However, if the latter part of my question is the case, that goes beyond unethical and I wouldn't be surprised if your problems wouldn't matter to a person like this.
There are a few options you have. First, is to unconditionally refuse to do the assigned work and take the consequences, whatever they might be. Second, to conditionally refuse to work, i.e. you won't do it unless they compensate you for it (pay, authorship, etc.). Third, to suck it up, as you are about to graduate soon and hopefully leave that institution, it might be worth considering that this is the last time you have to do something like that.
Now, what worries me the most is the possibility that your graduation could be sidelined or delayed in order to force you to work more for them (you didn't comment whether or not that was a possibility). If that is the case, all the other options are rather useless and in comes the fourth option, that I deliberately avoided above as being the most destructive: escalate the whole thing to the higher-ups (the department level, the university level, maybe even an attorney) and hope that in some future your case will be resolved.
Based on the question, I can't gauge the scope of the corruption, i.e. is only the advisor corrupted, does the other group know that you are essentially forced to work for them, for free, does the department condone such behavior? That and the location your based at, should help you define the impact of the fourth option.
Bottom line, my advice is: get out of there as soon as you can. Make a cost-benefit analysis of your next actions. Ask yourself questions like: What would it take just to graduate? What would it take to graduate and ensure reasonable recommendations? Can I afford it (and here I don't mean just the financial part, but also the price in time, mental health, opportunities, etc.)?

Thanks so much for your answer. To answer your questions: 1)  It is my advisor who will probably refuse to provide me with a recommendation letter if I decline the job; 2) There is no misunderstanding: my advisor doesn't give me the job believing I will benefit from it. The job falls into the area of my expertise; no other members in the lab including my advisor is able to get the job done; 3) Other labs will probably list my advisor as a co-author, without caring who actually did the job in our lab; 4) My institution is in Australia where a viva or defense is not required for PhD students.

@Well_uneducated Thanks for the clarification. So, as I see it, you already graduated by submitting your thesis. If that is the case, what is keeping you there? Do you want to stick around for the LoR? If so, your advisor might hold that forever over your head.

My thesis needs to be examined by examiners and I’ve been waiting for the outcome. I don’t go to school now, my advisor is expecting me work remotely for him. Anyway I’ve decided to tell my advisor that they can either pay me or list me as one of the co-authors. Thank you for your suggestions!

There is absolutely no way your adviser should be asking you to do this without a pre-existing case-iron guarantee from the other group that YOU will be listed as a co-author on any paper. You should simply refuse to do anything without this. As a bioinformatician, who sees this sort of thing happening all the time, it makes me SO mad.

Very politely notify them your scholarship and RA position (or whatever it is called that requires you to do work in return for money) has ended, and offer a consulting rate to continue. Make sure there's an agreed-upon number of hours for each task, so they aren't thinking they can give you one hour of work at a time, when the emails alone take longer than that. Research in most fields doesn't happen unless there is a fountain of money behind it, so this is not as beyond the pale as it may seem from your perspective. Just be careful to make clear it is simply the economic situation of the real world (which apparently they forgot they live in) that is the issue, not that you are making these demands to be unhelpful and make them go away.
If they can't pay you, then you can offer to help someone else take over the work, for example by giving them a list of steps and answering their questions via email.
Or you can offer to help in exchange for co-authorship, but then make sure you get into the loop so it is clear to all that they are violating journal terms by not including you as a co-author. If they are dependent on your analysis for their paper at this point, they may put in a position where you are compelled by professionalism to help them finish, but of course this needs to be out in the open with a promise of at least co-authorship in return.
Everyone with valuable skills has to deal with this kind of thing. People will push until you push back. The overwhelming majority aren't necessarily as cutthroat as the occasional bad joke might imply; they just convinced themselves your time is near-worthless while theirs is precious. Clarify to them this is not the case anymore.

Thank you for your advice. I totally agree with your comment especially the last paragraph. Instead of simply rejecting the task, I will offer my help in exchange of a fee or co-authorship.

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text:
What should you do when your professor does not pay you for completed research work?
I had an oral agreement with a supervisor that I was to complete research work for him. We established how many hours I was to work, and my hourly rate, when the work was to be done, etc. I completed the work, sent it off to him, he gave me some feedback and asked me to correct some citations and so, I incorporated the feedback, made fixes to my citations and then never heard back from him.
I recently emailed him about not being paid; he told me that he was not able to use the work and instead offered to find me alternative funding opportunities in the future. When I insisted that I did the work and deserved to get paid for my time, he threatened to report me for academic integrity issues. I'm only a Master's student, and this is my first research assistant position. Is this normal? Who should I speak to about this?
I am not part of a union, and my supervisor approached me to do this work. I do have email correspondence that confirms we have an agreement, but not confirming the specific terms (e.g. he wrote emails where he told me, submit your work and your time/hour log and I will pay you). He later revealed he had used the grant money to pay a different student to do different work.
The oral contract was made in front of others. It was in a lab setting, so others definitely overheard. Another clarification was that another student was to take over the work I began because I was going on an exchange to a different school. So there is more proof of a contract existing between my supervisor and me. My #1 concern is the threat of reporting me for academic integrity. My #2 concern is getting paid. I've contacted my faculty and was advised to report this to the dean of my faculty, which I'll be doing shortly.

Which country is this in?

@Scared-scarecrow And you still have the time log, and record of submitting it? How much are we talking about here? Not to say you aren't owed it no matter how much, but approaching getting $400 is very different than $5000

I have edited the post to transfer some useful clarifications from the comments into the post itself. I also removed a ton of answers-in-comments; please use the answer box for answers.

On a related note, what did we learn about oral contracts?

I recommend getting hard copies (i.e. print-outs) of all of the e-mail correspondence (in case your e-mails disappear from their servers - not likely, but possible).

"This is my first research assistant position": that is not a "research assistant position", it is just an intellectual property theft!

Do you have any idea what sort of "academic integrity issues" he's threatening to report you for?

I am also interested in what the "academic integrity issues" are supposed to be. (Also: let us know, how this turned out.)

@Valorum As George R R Martin put it: "Words are wind".

Is this normal?

No.  In most places, wage theft is a serious crime.

Who should I speak to about this?

The HR department.  Possibly also the department chair.  Provide a detailed written statement of what happened and when.

+1 for going to the institution's HR department. It's a chronic problem that academics and researchers will make ad-hoc hires without going through proper process, and HR will have people whose whole jobs are to sort out the messes that they create. You may end up being owed benefits because the nature of your work falls into a category which they've already categorized. Best case for you (worst ever for the supervisor) is if your tasks fell into a category within the scope of a contract of an existing union.

Good point, if there's a union you might speak to them too.  Though you might need to pay a membership fee.

Also contact student support, the person in charge of your masters, and any personal tutor that you may have. The PI, in addition to acting appallingly unethically by not paying you and by threatening you, may well have broken the University's own rules by employing you without contract and proper process.

Should contact both parties mentioned in the order mentioned. Talking to the student union would probably also be beneficial. Once HR is involved the chair won't have option but to investigate. However, keep in mind that HR is NOT your friend.

I was in a similar situation (1 month's coding work for a professor of a different university with a view for a grant together down the line), but HR, not the professor, was the issue. HR was definitely not my friend. The verbal contract was in conflict with all sorts rules and regulations so they could not pay and the professor was too busy to fight it —so I politely declined to submit the grant and cut my losses.

An Oral Agreement, with presumably no third parties to witness the agreement. Good luck collecting anything. This will just become a mud-slinging fest, with the student's word vs. the professor's.

As others have suggest - the prof almost certainly did not have the power to make a contract between you and the university (you would have been employed by the university, not by the professor). The likely outcome is that the professor will be in deep trouble with the university, but in the end the professor is not in a position to make a contract on the university's behalf, and so it is probably not valid.

@IanSudbery: I'm not so sure about your last statement - at the very least it will depend very much on legislation. Here in Germany, I'd think it quite likely that the student has a valid employment contract with the university.  But even if not, they are still entitled to either fulfilment or damages (which would be the same since they already fulfilled their part of the contract).

Almost certainly depends on the location, and IANAL, but in the UK its not possible for just any old employee to enter into a contract on behalf of an organisation. Would you think the same if it were a postdoc saying this rather than a professor? If so, what would be the legal difference?

@IanSudbery: At least at the universities where I have been, it is quite possible for the professor to do most of the correct administrative work without the student seeing any of this, in partiular for the small HiWi (student assistant) contracts. Legally, a professor (or a postdoc) can have implicit or explicit permission to hire a student on behalf of the university. A professor saying "I'll hire you as student assistant." would not be unusual at all - even if the actual employer is the university. "Expect  administration to take some months until this all is sorted out, and several more

more months until you actually see any money" would be a usual follow up sentence, but you really cannot conclude anything about the professor being in bad faith from this not being said. A professor over here is a public official whose job is to teach the student how academia works. That is a very strong position for the student assuming good faith. Postdoc offereing a job: Two relevant differences are that a professor always has at least some budget, and that the postdoc is subordinate to a professor - while the professor is not subordinate to anyone (though of course still bound by rules).

So the postdoc should say or show that they have a budget and permission to hire someone, but then: why not. E.g. if there's an ad on the whiteboard "student help wanted for ..., Dr Postdoc" Also, the professor is usually the "contact" employee at the university that manages the student help. HR would certainly expenct the professor to be the one to tell a student to not work if they erraneously show up for work in order to prevent a valid employment contract forming by conduct implying intent. The example we were warned aganist by administration is: someones fixed term contract expires. New

contract has not yet arrived. The "[non]employee" comes to work, and the PI/professor does not tell them to go home, or even worse, actively hands out work or collects results. This starts a valid permanent working contract and is the nightmare of HR.

I guess its location dependent. Yes a prof might say what your prof said, but they would then initiate the process with the admin - but its the admin that actually does the "hire" even if the student never sees this. The university has not agreed to employ someone until HR says so. A prof here is absolutely subordinate to someone - their HoD, their head of faculty and ultimately the senate and executive board. They are not a public official, they are an employee of a corporation like any other employee - same legal rights and responsibilities as the cleaner.

The university might have budgeted for the Prof to spend money, but the money remains the property of the university. Even if the Prof can request they spend it on something (probably through an automated mechanism for a simple purchase), they still retain the final say and can say no. Only the university can raise a purchase order with a supplier for example, the prof can't on their own.

No, this is not normal. If an agreement about number of working hours and an hourly salary is made, the salary is normally supposed to be paid according to that agreement.
It is not clear to me if a contract was written up, or if everything was oral agreements. If there is no contract, and no written correspondence to confirm the agreement, you will unfortunately have a hard time lifting this case. But in any case, the correct person to contact (which is your question), is your union, or a local union representative at your institution - that is, if unions are common in your country, which you don't mention.
If you are not a member of a union, you should become so if that is a possibility.
You should also start looking for another supervisor, as this person clearly don't have your best interest in mind.

You should report this to a some authority.
The chances are good (but we do not know this) that the person has done this before and will do it again.  Ethically if you have evidence enough to make a formal complaint you should to prevent them exploiting people.  Ethically and practically are, however, two different things - you will have to make the decision yourself.

he wrote emails where he told me, submit your work and your time/hour log and I will pay you

That is sufficient grounds to take a complaint case.  It is borderline whether it is enough to successfully take a legal case for failing to pay against them.  It is not quite a contract (IANAL) I think, as the terms would be explicit (e.g. how much per hour or in total) for a contract.  If you have corresponding emails now saying he will not pay that would be much stronger.
Strictly speaking you need legal advice specific to you country from your own lawyer.
Do not expect to ever get paid.  You can possibly put an end to this person's behavior (by maybe getting them fired), but getting paid is not so easy.

I recently emailed him about not being paid;

Email is your friend here.  Hopefully the replies were in email.
Make records of any emails, e.g. print them, copy them to disk.  Get copies of texts by phone or whatsapp fs whatever if possible and try and get recordings of any phone messages left for you.
Courts and formal review processes will need evidence like this.

he told me that he was not able to use the work

This is rarely a legitimate complaint in a labor case (which is what you would probably have here - again IANAL).  If your work was OK up until you asked for payment a court would typically take that as a demonstration your work was OK and without fundamental flaw.  There would have to be a history of consistent complaint from the other person to justify not paying.  They should also have stopped using you to assist them to be able to defend their position.  Stopping after you ask for payment they refuse to make is too late.

and instead offered to find me alternative funding opportunities in the future.

That is known as "bait and switch", it's not considered reasonable.  You expected to be paid, but unfortunately did not have an explicit agreement on the form of that payment, so it might be a slight weak point.

When I insisted that I did the work and deserved to get paid for my time, he threatened to report me for academic integrity issues.

If he was foolish enough to do that in email, you have an extremely serious case against him for abuse of power.  He was acting an an employee of an institute and your supervisor so that is a case you can directly take against them.  Typically you would make a formal complaint first via the institute's normal procedure.  I would expect that to be enough to put this individual in danger of loosing their job - I'd personally fire someone for doing this (when good evidence existed), but your institute and country's normal practices are unknown to me.
Make sure you have good legal advice for this process.  Join a union and request their help and advice - they typically have a legal representative they can at least put you in touch with, sometimes for reduced fees or free as an initial step.  If that is not an opinion get your own lawyer.

I have to remind you that even VERBAL agreements are contracts - or contract-like .. unfortunately harder to prove than written contracts - except made in front of third parties

@eagle275 My understanding is that the OP has emails, so verbal contracts or agreements would have practically no significance here.  Any written agreement will override a verbal one and you cannot amend a written agreement (or an agreement in a similar form) with a verbal one.  So if we agreed by email to do X for Y, then later neither of us would be able to enforce a verbal change to that agreement - it would have to be by email or in writing.  That's the law as I understand it, but IANAL.

No my comment is based on the post of OP - he says that the exact amount and conditions were agreed upon only verbally - which still constitutes a contract - then he has emails about "correct X and hand in your hours and I'll pay you" which supports the argument of the verbal agreement/ contract ..

@eagle275 As I said in my answer, the emails are not quite an explicit contract, but my point was that the supervisor's emails effectively acknowledged the contract in writing and trying to undo the contract (and make threats) means that the supervisor's claims will carry practically zero weight compared to the written evidence of emails.

@eagle275 IANAL, but professors are not employers, universities are. Given that a professor has no authority to make contracts on the university's behalf, I doubt the contract is valid. For example someone who worked front of house at McDonald's couldn't make a contract with a customer for MacDonald's to employ them at head office. I doubt the situation is any different with a prof and a student,  legally (but not ethically) speaking.

@IanSudbery Really hard to know what interpretation a court would take on these details as it's very jurisdiction dependent and also depends on the standard practices and reasonable expectations people can have.  This is why so many of us have recommended the OP speak to a local lawyer, preferably via a union as they'll have more specific knowledge of local case law on this.

@StephenG Well,  I'd definately always recommend joining and using a union if one is available.

@IanSudbery If the contract is not valid for the reason you state, might the student have a case of fraud against the professor? He presented himself as a representative of the university with the power to hire and pay people. If the student does not in fact have a contract with the university, then it would seem that the professor's indication that he had the power to make that contract was fraudulent in the first place. I'd have no recourse against McDonald's for someone who pretends to enter into contracts on their behalf, but I'd expect to have recourse against that person.

@NuclearWang - That would be hard to call fraud unless the professor explicitly said that the university would pay the student. In the absence of such an explicit statement it could also be interpreted as a private contract between the professor and student (outsourcing the work, essentially), forcing the professor to pay out of his own pocket.

@IanSudbery A professor can absolutely be an employer. The professor can have problems getting the university to pay his employee, which means he or she would have to pay the employee out of their own pocket. Most professors are wise enough to avoid that situation. And the way this particular contract is described, I see no indication that the university would be involved in any way.

@NuclearWang: (for jurisdiction Germany) the student has the right to get fulfillment of the contract or damages. From the professor, if there was no valid contract with the university. However, there may actually be a valid contract with the university here - this would depend on the details. E.g. if the university has let the professor hire students in a similar manner before, that would count as the university granting the professor the power to hire on their behalf.

tl;dr: Try your local small claims court or arbitration service to claim your salary, and make a formal complaint to your university regarding the professor's threat.
Disclaimer: I'm not a lawyer, I'm an academic too. Try to look up guidelines specific to your jurisdiction. Don't worry about what may be common in academia. Pay disputes are very common in every field of life, and your situation isn't that different to if a car wash or a restaurant tried not to pay you for work already completed.
The full answer
There are two issues here - firstly getting the money you are owed, and secondly dealing with your professor's threat of retaliation.
Although the exact processed will vary from country to country, I would suggest this as a somewhat universal blueprint to get the money you are owed.

Compile all evidence in writing. Make an annotated PDF of all the emails where your agreement is mentioned. Email people who were present at the meeting when you agreed the hourly rate, asking "Hey, just sorting out some project stuff. You were there when I set an hourly rate if X Euros with professor Y. Just wanted to check you remember that this was the amount we agreed on." From their replies, you now have evidence of the oral agreement in writing too.
Email a formal final request for payment to your professor. Keep it brief (don't go into your whole dispute), and include payment details and a date by when you expect payment.
If the professor still refuses to pay, file a claim with your local small claims court or arbitration service. Small pay disputes are very common, and these kinds of services are designed to be cheap and efficient to use. For example, in the UK the fee to submit a claim is ~20GBP. Evidence (i.e. your annotated set of emails) is submitted online to an arbiter who will try to reach an agreement between the parties. No need for lawyers, and most cases can be resolved online or by phone, with no need to go into a court. In general you won't have the opportunity to call witnesses in this kind of arbitration, so make sure you have all your evidence (including the emails from witnesses) in a PDF or on paper.

The threat of academic retaliation by your professor is another issue, and a very serious one. Was this threat made in writing? If so, I would immediately make a formal complaint to the university. This will pre-empt any possible retaliation on the professor's part. If the threat was not made in writing, you may not want to make a complaint at this stage - however by formally pursuing the money you are owed, you will have demonstrated that he has a conflict of interest if he ever does decide to make a complaint against you.

In the UK specifically, to initiate a case like this you must go through the ACAS service (Advisory, Conciliation and Arbitration Service). They have a very helpful phoneline you can phone to be guided through the process. They'll also be able to tell you exactly what the law is as it relates to the case.

all_licenses: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
Why say source code is available with an ArXiv paper when it is not?
I sometimes come across ArXiv papers where the authors claim the source code for the project discussed can be obtained, but on checking the link provided the source code comprises no more than a place holder. So the code is not available, and on subsequent checking months later (in case the authors were in a rush to publish the paper before the code was quite ready to release) the code is still unavailable.
Now I understand there is no obligation to provide source code with commercial potential, unless maybe the project was subsidized with public funds. But what do authors have to gain by claiming it is available when it is not actually available?

The question has been edited according to the comments and this conversation has been moved to chat.

There is no advantage to claiming the source code is available and then not doing that. All it will achieve is irritating the really interested readers.
The most likely reason is it’s a simple mistake. Something like they wanted to do that but forgot. Or other reasons, maybe department policies came up that prevented the author from doing that. Or maybe the author found another more suitable place to publish the code and forgot to update the link in the paper. Or...
The simplest solution is to contact the author directly.

or, LaTeX mangled the link, since it contained a tilde character, a percent character or something similar... lots of stupid issues can be blamed on technology.

There is an obvious and self-evident advantage, which is that the paper gets into a journal mandating open data, without actually making your data open. Contrary to what might be projected many, many scientists see advantage in hiding their data

Or the link just had a typo in it or the author forget to hit "publish" on the source code.

@benxyzzy: Is that based on concrete personal experience, and if so, in what field?  At least in my field (pure math/theoretical CS — so not much experimental data, but lots of code) I’ve really never seen anyone who seemed to be trying to hide their code.  And I don’t think I’m saying that out of idealistic naïvety: I’m aware of plenty of other academic sins happening in my field (plagiarism, ghost authorships, clear and deliberate salami-slicing, and so on), but I’ve never seen anything that seemed like dishonestly/inappropriately hiding code.

I've come across machine learning papers with links to Github repos that I cannot access on a not infrequent basis. It's not clear to me why - it could be the owner inadvertently set it private, or have made it private at some stage since the paper was published, or created the repo as a placeholder and never got around to populating it. It's quite frustrating to be honest.

I don't know about arXiv, but on bioarXiv, people will often upload their manuscripts at the same time as they submit them for peer review. People will put in links for data and code that they intend to make available when the peer-reviewed form of the manuscript is published.
Some times this is because they want to protect their data/code until the last possible minute. Other times they tell themselves they'll get round to sorting it out before the peer reviewed form is published, but then forget.
Note: I am saying what happens. I'm definitely not saying this is good or acceptable practice.

I think what they should have written is that "it will be available upon publication". Many people will provide the source code only after the paper is published. It is possible they intended to release the code upon publication, and they may need to get approval from their funding source etc. They should use proper language though.