Who do you consider the 3 greatest mathematicians ever?

Madhava of Sangamagrama discovered the infinite series of trig functions around the year 1400 without calculus
http://en.wikipedia.org/wiki/Madhava_series

 

Euler, Gauss, Hilbert

 

Newton (for Calculus, not to mention classical mechanics and optics)
Fourier (for his Fourier series and transform)
Galois (for group theory and abstract algebra)

In this case I interpreted "greatest" as "who came up with the most influential idea in my opinion", not necessarily most prolific or most brilliant (although they were probably all brilliant).

 

Very tough competition this one. I am just going to mention a few that died too early:
1. Evariste Galois (at 20, in a duel)
2. Niels Abel (at 26, tubercolosis)
3. Bernard Riemann (at 40, tubercolosis)
The first two would surely have contributed significantly more if they had been allowed to live to an older age. Riemann got 14 extra years compared to Abel but 40 was still too young. Conclusion: We do not like tuberculosis or duelling frenchmen.

Also I cannot resist including this one:
- Why did Newton not invent group theory?
- He was not Abel.

 

Newton, Leibniz, and myself.

I also independently invented calculus, but was 300 years late.
And being only 7 years old, no one could comprehend my notation.
They all looked like LEGO® building blocks.

Oh, good grief.......

 

Leibniz for the invention of calculus and his (arguably) superior notational system.
Descartes (and independently Fermat) for giving us Cartesian coordinates (making geometry possible through algebra)
Cauchy, for introducing rigor to calculus.

Cauchy could possibly be replaced, but I picked mine based not upon their skills as mathematicians, but how much they contributed to the mathematical world. So much in math relies on calculus, and our rectangular coordinate system has shown to be quite valuable.

 

Euler (most prolific by far), Gauss and Hilbert

 

Although people love appeals to authority, and I guess it's fair to say that it's always easier to remember a famous name than one less famous, I would still have to consider Euler, Laplace and Gauss as three of the greatest mathematicians.

According to historian Clifford Truesdell,

" … in a listing of all of the mathematics, physics, mechanics, astronomy, and navigation work produced in the 18th century, a full 25% would have been written by Leonard Euler. "

Also, I'm surprised no one has mentioned Laplace. In my opinion, he was not only a great mathematician, but above all a great discoverer. In every branch of astronomy, deep traces of his work are visible.

Finally, I believe Gauss deserves a spot, but many others do as well.

 

While questions like this are a bit silly and do not have definitive answers it is worth pointing out that mathematics as we know it today is so far beyond anything known before the 19'th century that comparing the two eras is a little like comparing apples and oranges.

Starting most likely with Gauss and then Riemann, mathematical thinking was transformed in a way that amounts to a Renaissance. Mathematical thinking suddenly became modern. Mathematicians generally consider Gauss and Riemann to be the founders of modern mathematical thought and therefore in some sense the "greatest" ,whatever that means.

After the discoveries of General Relativity theory and Quantum Mechanics, mathematics again took off and in the second half of the 20'th century went through what one might call a second Renaissance. Just as there is no comparison of the mathematics before and after the last half of the 18'th century, there is no comparison of the mathematics before and after say 1940. Many of the people who took part in this second Renaissance are still alive.

If you are really interested in where great ideas come from, you should be interested in cultural trends as well as specific individuals. Gauss and Riemann for instance lived in the classical period, a time of cultural fervor that includes Mozart and Beethoven, Benjamin Franklin, and Goethe. Why was this? What was the reason that so much creativity happened during this time? Would Gauss have been as great a mathematician if he had lived in 10'th century Iceland?

 

You could say that newton, euler, gauss, etc were born in times that were "ripe for discovery"- If it hadnt been them, it would have been someone else. For example, newton and Leibniz both discovered calculus independently.
This isnt to say they werent great geniuses, in particular you could say euler was ahead of his time, but most of these were great only because of the time period they were born in.
In my opinion, the truely remarkable discoveries are ahead of their time.

 

I'd say Euler and Erdos for the sheer amount of contributions they gave. And then Galois for his bad-donkeyery.

But none of these are my favorite. My favorite mathematician is Stanislaw Ulam.