Well a related question, how many mathematicians are there?
Mathematics Genealogy Project comes up with 188357 records as of 5 May 2015
So, those are 188357 theses (each of those mathematicians has to have published a dissertation). Each one presumably original enough so you would need to read it and understand it, not duplicating something else.
MathSciNet has over 3 million citations now
3,152,772
Search Publications database
Let's suppose you can read a mathematical paper in ten minutes and understand it - now that would be a rather superficial understanding, and you would need to have an amazing memory to remember any of it - and read those papers for twelve hours every day (allowing some time for four hours a day eating, exercise, and breaks, and eight hours for sleep) - that would be 72 papers a day, 26,280 papers a year. So that would take you about 120 years.
If you suppose rather, that it takes you about as long to understand it as it takes for someone to present the paper in a mathematics talk - well a typical paper would correspond to maybe a 1 hour talk. So that's 12 papers a day, so then, that's 720 years to read all the papers.
Though in practice, I think few people even who are expert in a field could say they have thoroughly understood a paper or a thesis just from hearing a one hour presentation, in mathematics that is. You could in experimental science, but in maths the papers tend to be more specialized and harder to understand, so it would probably take more than just hearing someone give a one hour presentation on it + read their paper to say you have truly mastered it.
For the theses, let's suppose you can read two of those a day. I'm assuming someone who is a brilliant mathematician, top of their game, and fast reader. Then that's 258 years for those.
So anyway - that's a few numbers you can use, vary them depending on how long you think is needed to master the content of the average paper, or average thesis, and it may give you a first idea of the answer.
I think that a more practical answer, would be of the order of several millennia to master it all, except by the end you'd surely have forgotten most of it and need to start again.
On the other hand, if the aim is just to understand the highlights, the main results in each field, that could be a more manageable project. Not sure there are mathematicians with that capability for all of maths, but you can have that ability for sub topics.
My supervisor at Oxford, Robin Gandy, I think had a reasonably thorough understanding of pretty much the whole field of logic, in the sense that if anyone gave a talk on logic in our logic seminars - then he quite clearly understood what they were talking about, you could tell from his questions at the end. And also when he was supervising me for my thesis, he could remember talks he had been to even many years before and say - "you should read so and so's paper - that's relevant to this particular question" etc. Which also confirms that he did indeed understand the presentations. I don't think many people could say the same for the entire field of logic, though of course they master their particular topic area. Basically he had been in the field since the birth of modern logic almost. And had followed every development as it happened. There weren't that many people you could say that about (he was quite old, approaching retirement when he took me on originally as a student, I was one of his last three students).
But he wouldn't have mastered every intricate detail of every paper he heard a presentation about.
So, that is a thing one could aim to do, to understand some particular topic area in maths even an entire sub-field like logic, to the point where you can understand any paper that anyone presents on the topic, but not that you have read every paper - that might be a tall order now especially. Just that you could read them all, that you understand all the key concepts and ideas and the cutting edge challenges in the entire field to the extent that if you hear a presentation you immediately understand it, not just the equations but the context, relevance, what fundamental things it answers, and how it connects to other results in the field. And understand the kinds of challenges that were met in constructing the proofs too, and be able to appreciate the ingenuity and elegance of the solutions.
Here is a list of famous mathematicians, and there are several who made contributions to many different subject areas so quite a broad area of interest within mathematics. For instance Paul Erdos who had a broad understanding of combinatorics, graph theory, analytical number theory and many other areas of pure maths. Perhaps one of the closest we have to a "modern Euler" in recent times.
But we are probably not ever going to see in the future any polymaths able to understand all current mathematical knowledge of their time like e.g. Euler. That is - unless future humans have much longer lives than us. If mathematicians lived to a few million years for instance. But of course by then though they might have mastered all maths up to the C21, in those long lifetimes they'd be proving new long complex proofs, perhaps have proofs that go on for millions of pages :). Still, you asked about mastering all maths up to now, not future maths. Maybe some day someone in the future with a specialist interest in C21 maths, could do that, if also brilliant and with immensely long lives.
Mathematics Genealogy Project comes up with 188357 records as of 5 May 2015
So, those are 188357 theses (each of those mathematicians has to have published a dissertation). Each one presumably original enough so you would need to read it and understand it, not duplicating something else.
MathSciNet has over 3 million citations now
3,152,772
Search Publications database
Let's suppose you can read a mathematical paper in ten minutes and understand it - now that would be a rather superficial understanding, and you would need to have an amazing memory to remember any of it - and read those papers for twelve hours every day (allowing some time for four hours a day eating, exercise, and breaks, and eight hours for sleep) - that would be 72 papers a day, 26,280 papers a year. So that would take you about 120 years.
If you suppose rather, that it takes you about as long to understand it as it takes for someone to present the paper in a mathematics talk - well a typical paper would correspond to maybe a 1 hour talk. So that's 12 papers a day, so then, that's 720 years to read all the papers.
Though in practice, I think few people even who are expert in a field could say they have thoroughly understood a paper or a thesis just from hearing a one hour presentation, in mathematics that is. You could in experimental science, but in maths the papers tend to be more specialized and harder to understand, so it would probably take more than just hearing someone give a one hour presentation on it + read their paper to say you have truly mastered it.
For the theses, let's suppose you can read two of those a day. I'm assuming someone who is a brilliant mathematician, top of their game, and fast reader. Then that's 258 years for those.
So anyway - that's a few numbers you can use, vary them depending on how long you think is needed to master the content of the average paper, or average thesis, and it may give you a first idea of the answer.
I think that a more practical answer, would be of the order of several millennia to master it all, except by the end you'd surely have forgotten most of it and need to start again.
On the other hand, if the aim is just to understand the highlights, the main results in each field, that could be a more manageable project. Not sure there are mathematicians with that capability for all of maths, but you can have that ability for sub topics.
My supervisor at Oxford, Robin Gandy, I think had a reasonably thorough understanding of pretty much the whole field of logic, in the sense that if anyone gave a talk on logic in our logic seminars - then he quite clearly understood what they were talking about, you could tell from his questions at the end. And also when he was supervising me for my thesis, he could remember talks he had been to even many years before and say - "you should read so and so's paper - that's relevant to this particular question" etc. Which also confirms that he did indeed understand the presentations. I don't think many people could say the same for the entire field of logic, though of course they master their particular topic area. Basically he had been in the field since the birth of modern logic almost. And had followed every development as it happened. There weren't that many people you could say that about (he was quite old, approaching retirement when he took me on originally as a student, I was one of his last three students).
But he wouldn't have mastered every intricate detail of every paper he heard a presentation about.
So, that is a thing one could aim to do, to understand some particular topic area in maths even an entire sub-field like logic, to the point where you can understand any paper that anyone presents on the topic, but not that you have read every paper - that might be a tall order now especially. Just that you could read them all, that you understand all the key concepts and ideas and the cutting edge challenges in the entire field to the extent that if you hear a presentation you immediately understand it, not just the equations but the context, relevance, what fundamental things it answers, and how it connects to other results in the field. And understand the kinds of challenges that were met in constructing the proofs too, and be able to appreciate the ingenuity and elegance of the solutions.
Here is a list of famous mathematicians, and there are several who made contributions to many different subject areas so quite a broad area of interest within mathematics. For instance Paul Erdos who had a broad understanding of combinatorics, graph theory, analytical number theory and many other areas of pure maths. Perhaps one of the closest we have to a "modern Euler" in recent times.
"Erdös was a childhood prodigy who became a famous (and famously eccentric) mathematician. He is best known for work in combinatorics (especially Ramsey Theory) and partition calculus, but made contributions across a very broad range of mathematics, including graph theory, analytic number theory, probabilistic methods, and approximation theory. He is regarded as the second most prolific mathematician in history, behind only Euler."
The Thirty Greatest Mathematicians
But we are probably not ever going to see in the future any polymaths able to understand all current mathematical knowledge of their time like e.g. Euler. That is - unless future humans have much longer lives than us. If mathematicians lived to a few million years for instance. But of course by then though they might have mastered all maths up to the C21, in those long lifetimes they'd be proving new long complex proofs, perhaps have proofs that go on for millions of pages :). Still, you asked about mastering all maths up to now, not future maths. Maybe some day someone in the future with a specialist interest in C21 maths, could do that, if also brilliant and with immensely long lives.
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Robert Walker
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