in your opinion?
Based on what criteria? Without specifying that it is like asking who is the greatest athlete of all time. There are simply too many different sports which are all very different to be able to answer without further specifications. (Does chess count as a sport? Do you base it on dominance in the sport? (Including a factor for how much of a random element is involved.) Do you weight it based on how popular the sport is? On how many people are practicing the sport?)
Yeah, criteria here is important. Do you mean who understood math the best, who moved the field forwards the most, who used it best, who's the most important? I'd give different answers for each. These are the ones I would consider most important, as well as my personal favorites, not necessarily greatest.
So this is a typical example of criteria being important! I already disagree ... (Not with the people in the list being important, that much is obvious, but there are others I would put before.)
Edit: It also becomes difficult to judge contributions from people who lived very different amounts of time as tended to be relatively common historically. How do you compare the lifetime achievements of giants like Gauss with those of Galois or Abel who died in their 20s? What would those two have achieved if they had lived to be 77 (i.e., like Gauss)?
I'd be interested to see the professions of the different people who answer. I chose what I thought of as most important with the bias of an computer engineer.
Edit based on @Orodruin's edit: Also people who lived in extremely different cultures and had access to different tools. For example, what would Imhotep have accomplished if he had lived in England at the time of Newton and what could Newton have accomplished if he had lived in Egypt of the time of Imhotep?
the criteria for this question is who moved the fields of math the most?
"And [Mathematics] builds from generation to generation. You have to really know what you're doing to do [mathematics] - and when you really understand something, you can explain it to someone else. The greatest [mathematicians] of one century ago, the brightest names that are still spoken with reverence, their powers are as nothing to the greatest [mathematicians] of today." - hpmor [almost]
In terms of influence, I think the top three are:
Joseph Fourier (Fourier Series, laws of heat transfer, Fourier transforms)
Muhammad ibn Musa al-Khwarizmi (invented algebra)
Rene Descartes (more known for philosophy, but also significant for devising the coordinate plane)
Aristotle (formal logic)
Neils Abel (group theory)
Bernhard Riemann (most of the mathematical framework on which general relativity was built)
But my personal favorite mathematician is Tarn Adams, the man behind Dwarf Fortress, who apparently has a math PhD.
In terms of those who have advanced the field of mathematics the most the top 5, in no particular order, are:
Runners-up include Rene Descartes, al-Khwarizmi, Niels Abel, Bernhard Riemann, David Hilbert, Alan Turing, Kurt Godel, Pierre-Simon Laplace, Joseph Fourier, Joseph Louis Lagrange, Gottfried Leibniz, Nikolai Lobachevsky, and Henri Poincare.
The last one was mainly horsing around. He did brew a mean cup of coffee.
So you could say he advanced maths a lot by helping mathematicians focus for hours on end
Joseph Fourier, i dont know much about him but a few people mentioned him
what were his accomplishments?
Fourier Analysis used in the mathematical analysis of periodic functions. simply a periodic varying function can be expressed as a sum of sines and cosines.
He discovered the mechanism behind the greenhouse effect. And also that you can write functions as infinite series of trigonometric functions.
He was also the PhD advisor of the PhD advisor of the PhD advisor of the PhD advisor of the PhD advisor of the PhD advisor of the PhD advisor of the PhD advisor of the PhD advisor of my PhD advisor.
I think that Felix Klein and Emmy Noether were great mathematicians (And Felix Klein was the PhD advisor of....).
i think Grothendieck would probably be the most influential of the 20th century
Since I have only read a few of them, including Euclid, parts of Riemann, and very little of Gauss, and almost none of the other famous people, I do not feel qualified to answer the question as asked. I would have to have mastered the works of these people to rate them, plus studied some history of how their works influenced others. If you will take as an answer which ones impress me the most, with my limited awareness, I would mention Archimedes, Gauss, and Riemann, (and Euler, HIlbert, Poincare', and Grothendieck).
In his book "Mathematics Queen and Servant of Science" E.T. Bell states
Today a mathematician has the advantage of centuries of formal development of the field. Thus the quote "If I have seen further it is only by standing on the shoulders of giants" belies the question of who is the greatest mathematician or even the three best.
While we may grossly rank mathematician according to contributions in the subfields of mathematics it is difficult to if not impossible to say if A is greater than B. In his book Bell lists over a hundred mathematician (applied and pure) who contributed both directly and indirectly to Physics. Without studying all their contributions how can one rank them. and by what criteria e.g. most cited in the literature? That would be questionable because of the advantage that present day mathematician have since there are probably more mathematician alive today than have existed who can cite a paper. This is not meant to minimize any contemporary mathematicians contributions for even Newton had the advantage of Euclid's work.
Euclid, Gauss, Newton, Euler, Galois, Riemann, Cantor, Hilbert, Noether, Grothendieck, Laurent Schwartz, von Neumann, ...
No way I can pick three
Separate names with a comma.