# Exactly what is considered to be a mathematician?

• Math100
In summary, there are three individuals who are being discussed. The first person has a Ph.D. in mathematics and is currently a professor at MIT. The second person has a Bachelor's degree in mathematics and has published papers on simpler and more efficient methods for solving certain math problems. The third person has proven theorems and conjectures and has received the prestigious Field's Medal. Based on the discussion, it can be inferred that all three individuals can be considered mathematicians, with the third person being a particularly distinguished research mathematician."f

#### Math100

Suppose there are three people.
1) Obtained a Ph.D. in mathematics and is a mathematics professor at MIT.
2) Obtained a Bachelor's degree in mathematics, found/discovered/invented simpler/easier/shortcut ways of doing certain math problems and publishes those math papers.
3) Proved theorems/conjectures and won Field's Medal.
Which one of them is considered to be a mathematician?

wrobel and PeroK
The one who doesn't question AC.

dextercioby, berkeman and Klystron
Suppose there are three people.
1) Obtained a Ph.D. in mathematics and is a mathematics professor at MIT.
2) Obtained a Bachelor's degree in mathematics, found/discovered/invented simpler/easier/shortcut ways of doing certain math problems and publishes those math papers.
3) Proved theorems/conjectures and won Field's Medal.
Which one of them is considered to be a mathematician?

The one who doesn't question AC.

Given mathematics as the subject of the discussion, I reckon acronym 'AC' refers to the 'axiom of choice' devised by Ernst Zermelo in proofs of set theory. If this is an accurate reading of AC, then the first sentence of the original post may be rephrased
Suppose there are three mathematicians.
obviating the subsequent query. Well done.

russ_watters and berkeman
Suppose there are three people.
1) Obtained a Ph.D. in mathematics and is a mathematics professor at MIT.
2) Obtained a Bachelor's degree in mathematics, found/discovered/invented simpler/easier/shortcut ways of doing certain math problems and publishes those math papers.
3) Proved theorems/conjectures and won Field's Medal.
Which one of them is considered to be a mathematician?
Ooo, ooo, I know!

pinball1970, russ_watters and Klystron
Suppose there are three people.
1) Obtained a Ph.D. in mathematics and is a mathematics professor at MIT.
2) Obtained a Bachelor's degree in mathematics, found/discovered/invented simpler/easier/shortcut ways of doing certain math problems and publishes those math papers.
3) Proved theorems/conjectures and won Field's Medal.
Which one of them is considered to be a mathematician?

How much coffee do these three drink?

Alfred Renyi said "a mathematician is a machine for turning coffee into theorems".

pinball1970, dextercioby, malawi_glenn and 5 others
How much coffee do these three drink?

Alfred Renyi said "a mathematician is a machine for turning coffee into theorems".
Thank you for the correct quote! Esp. for not giving it to Erdös, as it is done so often.

dextercioby
Suppose there are three people.
...
Which one of them is considered to be a mathematician?
That three does not seems to be a good fit for this type of question.

Maybe you should try using the 'duck test' instead? That's also based on 'three'

russ_watters, Klystron and berkeman
That three does not seems to be a good fit for this type of question.

Maybe you should try using the 'duck test' instead? That's also based on 'three'
Now name three ducks.
Wait, I know that one
Hughie, Dewey and Louie
or
Donald and Daisy Duck. Uncle Scrooge makes three.

Astronuc and berkeman
Suppose there are three people.
1) Obtained a Ph.D. in mathematics and is a mathematics professor at MIT.
2) Obtained a Bachelor's degree in mathematics, found/discovered/invented simpler/easier/shortcut ways of doing certain math problems and publishes those math papers.
3) Proved theorems/conjectures and won Field's Medal.
Which one of them is considered to be a mathematician?
What is their respective Erdős number?

Klystron, fresh_42 and berkeman
That three does not seems to be a good fit for this type of question.

Maybe you should try using the 'duck test' instead? That's also based on 'three'
Three mathematicians walk into a bar. You'd think the third one would have ducked.

russ_watters, marcusl and Klystron
What is their respective Erdős number?
Did you just sneak into the community with your 3?

DrClaude and berkeman
Did you just sneak into the community with your 3?
Don't despair, Ramanujan also had a 3.

fresh_42
Assuming the OP actually was hoping for an answer, I suggest that all three candidates are mathematicians, since they all "do mathematics", in the sense of thinking about math problems and questions, and making original contributions to their solution.

dextercioby, russ_watters, Kaguro and 1 other person
Assuming the OP actually was hoping for an answer, I suggest that all three candidates are mathematicians, since they all "do mathematics", in the sense of thinking about math problems and questions, and making original contributions to their solution.

I might say further that perhaps 1) is a professional, or perhaps academic mathematician; 2) is an amateur mathematician (as was Fermat); 3) is an especially distinguished research mathematician.

dextercioby and Math100
and to connect to other responses, I suppose someone knows who said a comathematician is a device for changing cotheorems into ffee?

Kaguro
I might say further that perhaps 1) is a professional, or perhaps academic mathematician; 2) is an amateur mathematician (as was Fermat); 3) is an especially distinguished research mathematician.

Thank you for your genuine response.

Exactly what is considered to be a mathematician?
A specialty logician or practicioner of their efforts whose domain is quantity.

Given these two general kinds of people can combine their study as it applies to other subject domains, this might be a simlified GENERAL description of the class we call, "mathematicians".

Math100
whose domain is quantity
Quantity can be used to give a description of structure, but other descriptions can and do occur.

Math100 and fresh_42
Exactly what is considered to be a mathematician?
obtaining results in math what else

dextercioby and Math100
Exactly what is considered to be a mathematician?
A specialty logician or practicioner of their efforts whose domain is quantity.

Given these two general kinds of people can combine their study as it applies to other subject domains, this might be a simlified GENERAL description of the class we call, "mathematicians".

wrobel's answer raises another question, namely when one ceases to obtain results does one cease to be a mathematician? or only when one ceases trying? or never, after having once succeeded, or tried? i.e. did he mean "having obtained"? Scott Mayers' answer also implies that one is [currently] practicing ones art. So maybe I should say I was once a research mathematician, and am now an amateur or even a student of mathematics. Indeed a closer look at the original categories reveals that only 2) explicitly states current research activity, although past activity at least is implied in 1) and 3).

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My best answer, sorry 2ct, would be the same slogan I once heard about Rock:

Mathematics isn't science, mathematics is a state of mind!

This state of mind was best demonstrated by Spock in the original version. It means that mathematicians first want to know all rules before jumping to conclusions. They always point to hidden assumptions in debates, and it is a special torture for them to listen to e.g. politicians because hidden assumptions are their daily business. There is not really this one truth for mathematicians, only valid and invalid deductions.

Quantity can be used to give a description of structure, but other descriptions can and do occur.
This would be a more specific goal that all mathematicians, as a class, cannot be asserted without imposing it on all others. I agree to this sub-goal of particular mathematicians. I personally think that the subject math, as a subset of logic, is best differentiated from other forms of logical inquiry by the nature of focusing on quantity. Originally, the ancient Greeks opted to use geometry to prove mathematical truths . In this way, this is about 'structure' but they had no choice at the time given the lack of acceptance of the use of "0" to represent the structural conception of "nothing". People, including mathematicians, will differ on whether they think that numbers represent something 'real' or not. So while you and I might like to favor the concept of math as including "structure", many inversely think that math is only a relatively artificial construct where "structure" to them is about the measured realities, the literal physical representation of structure.

I've heard another that defines math as the subject of patterns. But to me, this is more generically applicable to logic itself, not to a subset of it. Also, "science" acts as a subset as "the logic of observation", when just considering the general topic without considering the institution of science or "what scientists do". As this might demonstrate, there are more than one way to use the terms, like mathematicians.

What I'm wondering is if you are looking for "what should a 'mathematician' do?" and not what one means by it colloquially?

How much coffee do these three drink?

Alfred Renyi said "a mathematician is a machine for turning coffee into theorems".

No wonder I never became a mathematician. I don't like coffee.

wrobel and Scott Mayers
ok this discussion has me functioning more like a mathematician. i have been struggling to read a section in Dummit and Foote on discrete valuation rings, in particular the theorem that a local domain is a DVR iff every non zero fractional ideal is invertible. These purely algebraic results make my head spin, but i read it. In one direction they assumed every fractional ideal was invertible, applied it to the maximal ideal and deduced the maximal ideal was principal, hence the ring was a DVR. Then I noticed that the proof actually proved more, namely that in any local domain, every invertible ideal is principal. I.e. the proof did not use that they were arguing about the maximal ideal, just that it was invertible. The idea is that invertible ideals in a ring are apparently sort of an analog of codimension one subvarieties of an algebraic variety, and those are the ones that locally are defined by one equation, i.e. their ideals are (locally) principal. DVR's are one dimensional local rings, but this fact is true in any dimension local ring. The point is that once you stop just mindlessly reading a proof, and start thinking about it and its implications, and analyzing just which facts are used and which ones are not, you get a new statement, and you are behaving like a mathematician. Then tedious, uninspiring reading becomes more enjoyable and more illuminating. thank you!

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I might say further that perhaps 1) is a professional, or perhaps academic mathematician; 2) is an amateur mathematician (as was Fermat); 3) is an especially distinguished research mathematician.
Fermat was not an amateur mathematician. This stupid claim originating from one ignorant modern author and propagated then by everyone has to cease.
Please, read the two books of Tannery "Oeuvres de Fermat", the numerous letters he has written to his contemporaries, and especially the overwhelming number of number theoretic problems studied by Fermat at the end of the second tome (especially the variety of problems is amazing).
Fermat was an extremely intensive researcher, who produced many great discoveries that were published, and corresponded intensively with its contemporaries in mathematics and physics. Lagrange, Euler and Gauss (who understood the Latin very well) all described Fermat as a "grand savant", "great man", "the sagacious Fermat" etc. It is now known that Newton was inspired by the work of Fermat for his discovery of infinitesimal calculus.

The right point: Fermat was the only real professional mathematician (in the modern sense of the term) of his time in France.

dextercioby and malawi_glenn
Depends on what you mean by amateur. Was Fermat ever paid wage for his work on math?

Depends on what you mean by amateur. Was Fermat ever paid wage for his work on math?
I don't think this is what is intended when one says "amateur". Do you think Archimedes was an amateur mathematician? and Euclides?

@coquelicot: This confusion illustrates the universal need for definitions if one wishes to be understood. I certainly meant by "amateur" the meaning its word origin implies: namely one who does the activity out of love, rather than to earn a living by it. Thus to me there is no contradiction in calling a great mathematician also an amateur.

Since you ask, it seems to me that Archimedes earned his keep more by his applied works, such as his construction of defensive military engines in the defense of Syracuse, but his pure mathematical results were apparently done for the love of the research. He was certainly a great mathematician. Of Euclid's life we know very little, but it is generally agreed that he was not a great mathematician, although in my opinion he did write a great text book, whose results seem mostly to be the fruit of other mathematicians' research.

There is no challenge offered here to your opinion that Fermat was a great mathematician, but
it is easy to find fairly knowledgable references to Fermat as an amateur:

https://artofproblemsolving.com/wiki/index.php/Pierre_de_Fermat
"Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French magistrate and government official. He, however, is most famous for being an amateur mathematician."

Perhaps you might grant that Stephen Wolfram is not entirely stupid:
https://scienceworld.wolfram.com/biography/Fermat.html
"French lawyer who pursued mathematics in his spare time. Although he pursued mathematics as an amateur, his work in number theory was of such exceptional quality and erudition that he is generally regarded as one of the greatest mathematicians of all times."

Here also is one source to back up your opinion that greatness should not be called amateurism:
https://simonsingh.net/books/fermats-last-theorem/who-was-fermat/
"He was a truly amateur academic and E.T. Bell called him the Prince of Amateurs. However, when Julian Coolidge wrote Mathematics of Great Amateurs, he excluded Fermat on the grounds that he was ‘so really great that he should count as a professional.’ "

Even this reference however makes clear that Coolidge did consider Fermat as technically an amateur.

But I think you can relax. I.e. even people who call Fermat an amateur do regard him as truly great.
Cheers.

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dextercioby, Klystron and malawi_glenn
I don't think this is what is intended when one says "amateur".
Yes it is, look up the definition of the word.
You need to read the context.
Same as professional, which means that opposite - and can mean two things as well.

There are many amateur mathematicians on this site too :) Just doing math for the sheer joy of it, but are not payed to do so ;)

dextercioby
@coquelicot: This confusion illustrates the universal need for definitions if one wishes to be understood. I certainly meant by "amateur" the meaning its word origin implies: namely one who does the activity out of love, rather than to earn a living by it. Thus to me there is no contradiction in calling a great mathematician also an amateur.

Since you ask, it seems to me that Archimedes earned his keep more by his applied works, such as his construction of defensive military engines in the defense of Syracuse, but his pure mathematical results were apparently done for the love of the research. He was certainly a great mathematician. Of Euclid's life we know very little, but it is generally agreed that he was not a great mathematician, although in my opinion he did write a great text book, whose results seem mostly to be the fruit of other mathematicians' research.

There is no challenge offered here to your opinion that Fermat was a great mathematician, but
it is easy to find fairly knowledgable references to Fermat as an amateur:

https://artofproblemsolving.com/wiki/index.php/Pierre_de_Fermat
"Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French magistrate and government official. He, however, is most famous for being an amateur mathematician."

Perhaps you might grant that Stephen Wolfram is not entirely stupid:
https://scienceworld.wolfram.com/biography/Fermat.html
"French lawyer who pursued mathematics in his spare time. Although he pursued mathematics as an amateur, his work in number theory was of such exceptional quality and erudition that he is generally regarded as one of the greatest mathematicians of all times."

Here also is one source to back up your opinion that greatness should not be called amateurism:
https://simonsingh.net/books/fermats-last-theorem/who-was-fermat/
"He was a truly amateur academic and E.T. Bell called him the Prince of Amateurs. However, when Julian Coolidge wrote Mathematics of Great Amateurs, he excluded Fermat on the grounds that he was ‘so really great that he should count as a professional.’ "

Even this reference however makes clear that Coolidge did consider Fermat as technically an amateur.

But I think you can relax. I.e. even people who call Fermat an amateur do regard him as truly great.
Cheers.
So, why I've never seen that Archimedes, Euclides or Appolonius were called "amateur". In french, my mother tongue, amateur don't mean "not paid" but "a person generally of low level in the branch, doing that for his pleasure, but of relatively low level". Admittedly, this word of french origin may have got a different meaning in english. I maintain that calling Fermat an amateur is still absurd, because there were no paid mathematician in France in this time: Descartes, Pascal and Robertval were all amateur? that is non sense in my opinion.

Edit: As can be understood in my answer above, the problem is not really calling Fermat "an amateur", the problem is that it is the only great man of his category who is called "an amateur", even among far less great men like Pascal. Again, this negative term (with respect to the greatness of the man) comes from one person (I don't remember his name) who thought Fermat became famous because of one or two enigmas in number theory he found by luck. Nothing is more wrong. The sole work of Fermat in geometry would suffice to categorize him among great men of the past. Add to to that his work in combinatorics and probabilities, in the theory of elimination (the first method of elimination ever published), in analysis and infinitesimal calculus, and you can categorize him among the greatest men. Add to that his overwhelming number of results and problem he studied in number theory, of which he sent many proofs to Carcavi, Robertval, Mersenne and Pascal (most of them were lost), and you get a Giant. This humiliating term of "amateur" which is addressed only to Fermat (among the greatest men) is really ridiculous.

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dextercioby and Klystron
In french, my mother tongue, amateur don't mean "not paid" but "a person generally of low level in the branch, doing that for his pleasure, but of relatively low level".

"Amateur" is just a french word.

"Amateur" is just a french word.
Why is it in an English dictionary then?