There are two main reasons why the achievements of our modern mathematical superstars are not as conspicuous as those of the great mathematicians of the past.
• In the past two centuries, mathematical discoveries have grown at an exponential rate.
• Much mathematical research today is conducted through collaboration, so credit is shared among several people.
Exponential Growth of Mathematics
In 1900 there were a few dozen journals dedicated to mathematical research. These included Mathematica, Acta Mathematica, Journal de Mathématiques Pures et Appliquées, Mathematische Annalen and the American Journal of Mathematics.
By the year 1950, there were hundreds of journals publishing mathematical discoveries. These included Mathematics of Computation, Pacific Journal of Mathematics, and the Bulletin of the American Mathematical Society.
At the end of the 20th century, there were thousands of mathematical journals involving new fields of mathematics that had never been imagined in previous centuries. Advances in computer technology had further expanded the range of mathematical investigation.
Historian of mathematics, E. T. Bell, conjectured in 1950 that Henri Poincaré, who died in 1912, was probably the last mathematician who had some knowledge of all branches of mathematics.
So with this explosive growth in mathematics, the greatest minds are isolated in deep arcane research that only those specialists in their field understand. Every so often, a mathematician or physicist working in one field discovers a result in another mathematical field that connects with their work. This brings a consolidation of results and often an expansion of their knowledge base and a cross-pollination of ideas. This happened when Einstein shared his work with Grossmann, a geometer who helped him formulate the General Theory of Relativity in a curved space.
Collaboration
Before 1950, most research papers in mathematics reported discoveries or advancements in knowledge that had been made by an individual, because a university department usually had only enough money to support a small number of mathematicians whose specialties were in different fields. As university staffs increased in size there was more collaboration, and joint publications. resulting in shared achievements and distributed credit. (It is estimated that over 90% of Paul Erdös’s papers had co-authors.) Such collaboration is particularly evident in computer science where teams of mathematicians advance artificial intelligence and individual recognition is sacrificed to team recognition.
With the arrival of the Internet at the end of the 20th century mathematicians in the same field share their research with colleagues in distant realms. This enables mathematicians to build on the work of others in their search for a solution to a problem. For example, among Terrence Tao’s most cited papers are those dealing with the sophisticated field of compressed sensing. For example, his paper titled “Exact signal reconstruction from highly incomplete frequency information” published in 2006, has over 14,000 citations, and addresses the problem of reconstructing a signal from incomplete information. This paper and several papers of his papers on compressed sensing have been co-authored with Emmanuel Candès.
The achievements of the today’s most capable mathematicians have equalled those of the past. The proof of Fermat’s Theorem (that Gauss he resolved not to undertake) by Sir Andrew Wiles, and the proof of the Poincaré Conjecture by Grigory Perelman were achievements of a magnitude comparable to those of the past.
