Mathematician Definition

Moved from IsProgrammingMath

[What about a conference of applied mathematicians, statisticians, theoretical economists or theoretical physicists? By definition, someone with a first degree in mathematics (pure, applied, statistics, whatever) is qualified as a mathematician and entitled to use that label. Whether they work as a mathematician or pursue mathematics as a hobby is another matter.]

I disagree. Someone with a first degree in maths is no more a mathematician than someone with pre-med is a doctor. Someone with a first degree in physics is not a physicist. In both cases, the generally accepted labelling comes with a Ph.D. Also, the answer would be the same for a conference of the above mentioned types. Furthermore, even if you include *all* of these people, the numbers are still tiny compared to the field as a whole.

[Medical doctors, veterinary surgeons, etc. are special cases since their titles are not usually degree titles (e.g., in many countries, doctors or surgeons do not usually have a "doctor of" degree, so that "doctor" is a courtesy title). Most chemists do not have a Ph.D, nor most geologists, biologists, etc. Such a chemist would not be called a research chemist perhaps (unless actually doing research), but they're still a chemist - perhaps a teacher, for example. I would definitely call someone who teaches mathematics a mathematician, even if they have no degree at all (though then they wouldn't be called a qualified mathematician).]

MD was a perhaps a poor example, since the degrees do not map very well to each other. However, the point is still valid I believe; these degrees are the beginning of an education in the area, not an end. I would counter the above by saying that that very many people who work in geology, biology etc. are *not* chemists, biologists, or what have you. These differentiations are useful, as they give you an idea of what background somebody has, and what you can expect them to know. Thus it is not intellectual snobbery to insist on precision. I certainly would not call someone who teaches high school mathematics a mathematician, and similarly your other examples. The title 'research such-and-such' is useful for describing what somebody does, but it is not descriptive enough to break these types of jobs into only two classes. For example, it is entirely possible to be a mathematician by training, without being a research mathematician, but not vice-versa (regardless of whether or not you have a Ph.D). On the other hand, it is quite possible to spend the vast majority of your working life *using* mathematics, without being either of these. See below.

I have to disagree with most of that. The most specific title available is the one that's normally used, but the less specific title may also be true. In some circles, perhaps, possession of a Ph.D. may be so common that those without it are 'looked down on' by avoiding use of a justifiable term. Thus, theoretical physicists use a lot of mathematics, and may do little else. The more specific term is obviously more useful, but 'mathematician' would often also be true. Mathematics is a bit special in its own right as it's easy to do mathematical research with virtually no mathematical training whatsoever, and you might find out something 'useful' (where that word means 'of interest', since mathematical results don't need to have a direct practical use; they needn't even be provable). There may be a tendency to expect research experience in relation to certain sciences, but it's not necessarily justified. Certainly, I wouldn't refuse to call someone a statistician (if that's his work) merely because he's never done any research to advance known theory in the subject. Likewise for an economist. How many dental surgeons have done research? How would you classify Einstein? At what stage in his life did he first justify that classification? Is research necessary for a civil engineer, an architect, a pharmacist, an optometrist, or a meteorologist (to list just a few highly technical or scientific specialities which correspond directly to 'trades')? Do you know of any dictionary which suggests a chemistry teacher is not a chemist? (We could argue for ever over definitions if we're not prepared to use standardized ones.)

''Well, it is your prerogative to disagree :). I know that this is accepted usage in many circles of science, but of course there are many I don't know enough about to say. To address a couple of your points: Einstein was not a mathematician, and he did not claim to be. He himself made this distinction when he asked for help with the mathematics of GR. All of the specialities that you mention are fundamentally different from scientists, so it is a bit of an apples and oranges comparison. As for a dictionary, the OED has four entries for chemist, which I will not duplicate. The first is archaic, the second refers to a school of physicians, the fourth refers to the use in the UK of 'chemist' much as 'pharmacist' is used in the US. The closest to the meaning you want is the third: "One versed in the science of chemistry; one who makes chemical investigations". I suppose you could stretch this to include a high school chemistry teacher, but it is a bad fit. They do not make chemical investigations, and are not, with exceptions I am sure, what I would consider versed in the the science of chemistry. I expect that the average high school chemistry teacher has the sort of elementary background you would get from a minor or major in chemistry for a B.Sc, and probably hasn't used that much in years. What is wrong with the title 'high school teacher' for this person? It is much more descriptive of what they do, and what they know. Certainly the difference between them and a 'research chemist' or whatever you want to call them is night and day. They have more in common with a high school English teacher, in terms of the job they do, than a researcher.

Chemistry teacher is the more specific label, and so is more commonly used. I used to take your view on whether a chemistry teacher (with a chemistry degree, but not a Ph.D.) is a chemist, but I was corrected by such a chemistry teacher within my own family. The 'versed in' definition is the one I had in mind, and I would consider that the semicolon delimits it. I didn't say Einstein was a mathematician (though some might), but if he was a theoretical physicist, when did that label first apply?

I have a feeling that your chemistry teacher would similarly be corrected by most chemists. Comparing a high school teacher to a research scientist is very clearly an apples and oranges comparison. For similar reasons, I don't believe that this is sufficient to meet the 'versed' part of the above definition, semi-colon or no. In my opinion, they have started down that road, but have not finished the journey. But we need not quibble about that. As far as Einstein goes (and no, he did not consider himself to be a mathematician, so that should be enough for us), the answer is obvious. He became a theoretical physicist when he performed theoretical physics. In other words, when he first created new physical theory. Of course, this is a rather gray definition, but it is as good as you are likely to get. The accepted practice would probably be something like: when you start publishing in peer reviewed journals, but there are always exceptions (i.e. Ramanujan).

I wasn't comparing a chemistry teacher to a chemistry researcher, just saying they are both versed in chemistry to a sufficient degree for them both to be chemists.

Theoretical physics is mainly just a particular field of application of mathematics. Most of it wouldn't exist without the use and analysis of mathematical models. Einstein was famously modest in respect of his use of mathematics, which was sometimes beyond the day-to-day knowledge of the 'average' mathematician. But Einstein is often effectively referred to as a successor of Newton with regard to ideas about gravitation, etc., and Newton is commonly called (and was) a mathematician.

How would you define 'applied mathematician'? Does such a person instantly get relabelled according to the particular area he works in at the time?

[If someone is paid to do mathematics, and that is the main part of their job, they can correctly be called a professional mathematician (or statistician, etc.), whether they have formal qualifications or not. If, as a programmer, you prefer not to use GOTO statements, you are relying on a mathematical theorem which proves that the other statement types available to you will suffice. You don't need to be actually doing mathematics for the mathematical foundation to exist. Often, the mathematics isn't essential, just convenient. For example, you may write code to implement a particular sorting algorithm because you know (or assume) that the algorithm's worst-case efficiency has been mathematically proven to meet your needs. Thus the programmer does not need to be a mathematician, but certain aspects of what he does have a mathematical foundation. Sometimes, the mathematics goes unnoticed. Merely accessing an array correctly usually depends on the computer doing some simple arithmetic to calculate the array element address correctly. How often have you simply assumed that one of "a = b", "a < b" and " a > b" must hold, so that you need only test explicitly two of the three conditions? How often have you (unwittingly perhaps) employed a little Boolean algebra to simplify a logical expression? How often have you used a numeric variable of a particular type because you "know" that type of variable will be suitable for the arithmetic needed? Certainly, however, most programmers don't need mathematics training to enable them to cope with such things. Much of the mathematics the programmer relies on is very simple and/or may be hidden to some extent, and so no special mathematics training is required. That doesn't mean the underlying mathematics doesn't exist.]

We have been sidetracked into definitions here. Certainly the existence of a Ph.D is not necessary; you need only have the equivalent abilities and use them professionally. That is not what I meant. Many professions use mathematics daily (engineers, physicists, etc.) but they are not mathematicians. Similarly, merely using mathematics in your job does not make you a mathematician, even if that is all you do. If you want a more useful definition of what a *mathematician* is try this: A mathematician is a person who creates new mathematics. I also disagree with your other generalizations: there are many people working in chemical labs with or without undergraduate degrees who are not chemists. Chemical technicians or whatever you want to call them, but not chemists. I am sure similar differentiations exist for geologist and your other examples. Everyone I informally polled today agreed with this position: a research chemist is a person doing (perhaps commercial, of course) research in chemistry. A chemist is a person with similar training/capabilities who is not actually doing research (but has done some, at some point, since it is part of the training). There are a large number of other people who may work with chemicals, but they are not chemists. The mere fact of having an undergrad degree in chemistry and working in a chemical lab does not make you a chemist. It is certainly possible, although unusual, for such a person to self-train to the degree that they become a chemist.

When dictionaries support that view, I'll agree. At present, they don't. Until then, perhaps we've gone as far as we can with definitions, subject to your answering my latest questions! Perhaps we should stop calling programmers programmers, since most of them wouldn't be able to gain a Ph.D. in programming!

See above. The labelling of programmers as such is sufficient.

When developing mathematical algorithms, mathematicians are, in part, carrying out programming.

Many mathematician never construct algorithms, and for most it is certainly not central to what they do. Proofs are not algorithms.

My last use of Google to locate some recent mathematical work, found the following abstract: "We give a proof of the Poincare conjecture. The proof was inspired by the beautiful algorithm of Hyam Rubinstein for recognizing the 3-sphere and the proof of this by Abigail Thompson. The philosophy is that of the final chapter of Dicks and Dunwoody." I can't put a figure on just how common algorithmic work in mathematics is, but algorithms are certainly very common and very important in mathematics.

[There is plenty of empirical mathematics. Empirical mathematics has always been an important part of both practical mathematics and research mathematics. Let's not forget that long before computers were thought of, mathematicians were devising, testing and using algorithms. In other words, programming! Mathematicians were looking for 'bugs' (called errors or mistakes) in their work long before programmers started using that word. Mathematicians often break down hard problems into simpler ones, as do programmers. They also have a liking for elegance and simplicity.]

Mathematicians write proofs. From personal experience, I can state that mathematicians never create a design for a proof before starting, they immediately begin writing the proof.

Isn't a program just a proof/theory of how a problem is solved? Arguing about what to call the person who created it, or how they created it doesn't change the fact that what they created is a mathmatical construct. Just because "if that, then this, else that, or this and that" seems simple doesn't mean it's not still predicate logic, i.e. math. Would it help if code were written in chalk on a blackboard, would that make it any more math? I'm not a mathematician, but if math is the application of logic to abstract imaginary symbols to achieve a known outcome, then ProgrammingIsMath, there can be no argument, that's just what it is.

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