I have noticed that among mathematicians there is a great diversity in style of expression. Some of these are exceptionally neat and elegant. What I mean by that is that proof, or argument takes few rows of text, is very concise and uses extraordinary well-defined vocabulary. I assume to acquire this those mathematicians have some systematic strategy that they are aware of when they acquire knowledge. What would be that strategy ? Furthermore living mathematicians develop even their own style which is not simply reproduction of the material found in books. These structures are so beautiful. Could somebody explain in what way could this be acquired ?
[Math] How to become eloquent mathematician
mathematicianssoft-question
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Wikipedia has a list of 21st-century mathematicians while the book Mathematicians : an outer view of the inner world has the following list in portrait form with their autobiography:
Adebisi Agboola -- Michael Artin -- Michael Francis Atiyah -- Manjul Bhargava -- Bryan John Birch -- Joan S. Birman -- David Harold Blackwell -- Enrico Bombieri -- Richard Ewen Borcherds -- Andrew Browder -- Felix E. Browder -- William Browder -- Lennart Axel Edvard Carleson -- Henri Cartan -- Sun-Yung Alice Chang -- Alain Connes -- John Horton Conway -- Kevin David Corlette -- Ingrid Chantal Daubechies -- Pierre Deligne -- Persi Warren Diaconis -- Simon Donaldson -- Noam D. Elkies -- Gerd Faltings -- Charles Louis Fefferman -- Robert Fefferman -- Michale Freedman -- Israel Moiseevich Gelfand -- William Timothy Gowers -- Phillip Griffiths -- Mikhael Leonidovich Gromov -- Benedict H. Gross -- Robert Clifford Gunning -- Eriko Hironaka -- Heisuke Hironaka -- Friedrich E. Hirzebruch -- Vaughan Frederick Randal Jones -- Nicholas Micahel Katz -- Robion Kirby -- Frances Kirwan -- Joseph John Kohn -- János Kollár -- Bertram Kostant -- Harold William Kuhn -- Robert Phelan Langlands -- Peter David Lax -- Robert D. MacPherson -- Paul Malliavin -- Benoit Mandelbrot -- William Alfred Massey -- John N. Mather -- Barry Mazur -- Margaret Dusa McDuff -- Curtis McMullen -- John Willard Milnor -- Maryam Mirzakhani -- Cathleen Synge Morawetz -- David Mumford -- John Forbes Nash, Jr. -- Edward Nelson -- Louis Nirenberg -- George Olatokunbo Okikiolu -- Kate Adebola Okikiolu -- Andrei Okounkov -- Roger Penrose -- Arlie Petters -- Marina Ratner -- Kenneth Ribet -- Peter Clive Sarnak -- Marcus du Sautoy -- Jean-Pierre Serre -- James Harris Simons -- Yakov Grigorevich Sinai -- Isadore Manual Singer -- Yum-Tong Siu -- Stephen Smale -- Elias Menachem Stein -- Dennis Parnell Sullivan -- Terence Chi-Shen Tao -- Robert Endre Tarjan -- John T. Tate -- William Paul Thurston -- Gang Tian -- Burt Totaro -- Karen Keskulla Uhlenbeck -- Sathamangalam Rangaiyengar Srinivasa Varandhan -- Michèle Vergne -- Marie-France Vigneras -- Avi Wigderson -- Andrew John Wiles -- Shing-Tung Yau -- Don Zagier.
In going from high-school to, say, graduate${}^\color{Blue}\dagger$ level math, the higher math being "nowhere close to the math that appealed to" you is probably a very real threat. However, if your pleasure in learning analysis and algebra is any indication - instead, it will be math that you love even more.
A few things that I've noticed changing as I've learned more math are:
- Generality: Everyone knows about integers, rationals, reals, maybe complex numbers, but the next step up, conceptually, is to look at rings and fields and then modules and so forth. In higher math, the generality of our constructs increases a lot. Often we then narrow our focus again and end up looking at 'cousins' of the things we were originally studying. Other times we run into problems and it becomes the question of the decade precisely how to successfully go about a particular campaign of generalization and overcome the relevant obstacles.
- Branching: What a lot of people don't understand is that mathematics isn't linear, and doesn't progress in a rigid sequence. It branches out into many different areas, and in exploring these branches they can "feel" radically different. You can have a crush on one branch while hating another branch. In some cases, you have a love-hate relationship, or "it's complicated," etc.
- Reinterpretation: With a little bit of dabbling in different areas of math, it's possible that a single problem/idea can be attacked/framed from many different angles, using very different concepts. Sometimes this can seem "natural" and easily "motivated," while other times alien and bizarre. Frequently this sort of thing is a bit of a pastime for some mathematicians.
- Richness: In summary, mathematics becomes richer. In scaling conceptual mountains, we build concept on top of concept on top of seventeen more concepts until we're left studying situations that are saturated with structure, and when we hike back down the other side we come across the exotic or pathological; in branching we discover a high degree of diversity we hadn't previously imagined, each with comparable feel and texture to them; and then when we study wide and far we find that even our familiar notions have multiple sides to them.
$~~ {}^\color{Blue}\dagger$Yeah, disclaimer: I'm not actually there yet. :-)
Best Answer
This is an expansion of my comment above.
The first step to literacy is reading. There are stylistic conventions for written mathematics, and they are not covered in any course, though there are some prescriptive books around. As you mention, there is more than one style, though in any given branch of mathematics, most of the papers probably form a homogeneous group.
The second step is writing. Writing by itself is immensely helpful, but it's even better if a "dialect expert" comments and corrects your output. This is especially important for non-native speakers of the target language (usually English). As for the comments, don't take them too seriously. There's a difference between conventions and style, and even regarding the former, sometimes it's good to be innovative.
Finally, one way to find enough material to write about (unless you're a phenomenal researcher even at this early stage) is to write expositions of proofs (or subjects) you like. There is often more than one way to explain a proof, and you can practice both your understanding of the material, your explanatory skills, and your literacy. You can complement it with lecturing on the material.