Bernhard Riemann

Bernhard Riemann

The answer to these questions can only be got by starting from the conception of phenomena which has hitherto been justified by experience, and which Newton assumed as a foundation, and by making in this conception the successive changes required by facts which it cannot explain. Researches starting from general notions, like the investigation we have just made, can only be useful in preventing this work from being hampered by too narrow views, and progress in knowledge of the interdependence of things from being checked by traditional prejudices.
This leads us into the domain of another science, of physic, into which the object of this work does not allow us to go to-day.

One now finds indeed approximately this number of real roots within these limits, and it is very probable that all roots are real. Certainly one would wish for a stricter proof here; I have meanwhile temporarily put aside the search for this after some fleeting futile attempts, as it appears unnecessary for the next objective of my investigation.

Natural science is the attempt to comprehend nature by precise concepts.

The word hypothesis has now a somewhat different significance from that given it by Newton. We are now accustomed to understand by hypothesis all thoughts connected with the phenomena.

Newton was far from the crude thought that explanation of phenomena could be attained by abstraction.

No mathematician is more associated with the midnineteenth-century transition from algorithmic to conceptual thought than Riemann. (The Princeton Companion to Mathematics)

It is clearly a preliminary note and might not have been written if L. Kronecker had not urged him to write up something about this work (letter toWeierstrass, Oct. 26 1859). It is clear that there are holes that need to be filled in, but also clear that he had a lot more material than what is in the note. What also seems clear: Riemann is not interested in an asymptotic formula, not in the prime number theorem, what he is after is an exact formula! (Atle Selberg comments about Riemann’s paper On the Number of Primes Less Than a Given Magnitude)

黎曼的这篇1854年的论文，是非常重要的，也是几何里的一个基本文献，相当一个国家的宪法似的。爱因斯坦不知道这篇论文，花了七年的时间想方设法也要发展同样的观念，所以爱因斯坦浪费了许多时间。… 不过很有意思的是我想 Riemann-Christofell 曲率张量 是一个很伟大的发现，黎曼就到法兰西科学院申请奖金。科学院的人看不懂，就没有给他。… 得不得到奖不是一个很重要的因素，黎曼就没有得到奖。他的 Riemann-Christofell 张量在法兰西的科学院申请奖没有得到。 (什么是几何学, 陈省身)

Before Riemann introduced the notion of what is now called a ‘Riemann surface’, mathematicians had been at odds about how to treat these socalled ‘many-valued functions’, of which the logarithm is one of the simplest examples. … Riemann taught us we must think of things differently. (The Road to Reality, chapter 8 Riemann surfaces and complex mappings, Roger Penrose)

Only the genius of Riemann, solitary and uncomprehended, had already won its way by the middle of the last century to a new conception of space, in which space was deprived of its rigidity, and in which its power to take part in physical events was recognized as possible. (Albert Einstein)

His contemporaries were to see nothing of him while he waded through Cauchy’s output. Several weeks later Riemann resurfaced, declaring that ‘this is a new mathematics’. What had captured Cauchy and Riemann’s imagination was the emerging power of imaginary numbers. (The Music of the Primes, Riemann’s Imaginary Mathematical Looking-Glass, Marcus du Sautoy)

Euclid in Alexandria. Euler in St. Petersburg. The Göttingen trio — Gauss, Dirichlet, Riemann. The problem of prime numbers had been passed on like a baton from one generation to another. The new perspectives of each generation provided impetus for a fresh surge along the track. Each wave of mathematicians left its characteristic mark on the primes, a reflection of their era’s particular cultural outlook on the mathematical world. (The Music of the Primes, chapter 5 The Mathematical RelayRace: Realising Riemann’s Revolution, Marcus du Sautoy)

Selberg spoke about his view on the Riemann Hypothesis. Although he had made a major contribution on the way to a proof. he stressed that there was still very little to support its truth. ‘I think the reason that we were tempted to believe the Riemann Hypothesis then was essentially that it is the most beautiful and simple distribution that we can have. You have this symmetry down the line. It would lead also to the most natural distribution of primes. You think that at least something should be right in this universe.’
Some misinterpreted his comments, thinking that Selberg was casting doubt on the validity of the Riemann Hypothesis. Yet he was not as pessimistic as Littlewood who believed the lack of evidence meant the Hypothesis was false. ‘I have always been a strong believer in the Riemann Hypothesis. I would never bet against it. But at that stage I maintained that we didn’t really have any results either numerical or theoretical that pointed very strongly to its truth. What the results pointed to was that it was mostly true.’ (The Music of the Primes, chapter 7 Mathematical Exodus: From Göttingen to Princeton, Marcus du Sautoy)

Many of the mathematicians who have contributed to our understanding of the primes have been rewarded with long lives. Having proved the prime number theorem in 1896, Jacques Hadamard and Charles de la Vallée-Poussin both lived into their nineties. People had begun to believe that their having proved the Prime Number Theorem had made them immortal. The belief in a connection between longevity and the primes has been further fulled by Atle Selberg and Paul Erdös, whose alternative elementary proof the Prime Number Theorem in the 1940s saw both of them live into their eighties. Mathematicians joke about a new conjecture: anyone who prove the Riemann Hypothesis will indeed become immortal. (The Music of the Primes, Chapter 12 - The Missing Piece of the Jigsaw, Marcus du Sautoy)

While for many mathematicians “intuitive work” can be hit-or-miss, Riemann’s mathematical intuitions were incredibly acute, and his results generally turned out to be correct. (The Millennium Problems: the Seven Greatest Unsolved Mathematical Puzzles of Our Time, Keith Devlin)