On Numbers: On Prime Numbers (1900-1949)

"I have sometimes thought that the profound mystery which envelops our conceptions relative to prime numbers depends upon the limitations of our faculties in regard to time, which like space may be in its essence poly-dimensional, and that this and such sort of truths would become self-evident to a being whose mode of perception is according to superficially as distinguished from our own limitation to linearly extended time." (James J Sylvester, Collected Mathematical Papers Vol. 4, [footnote] 1912)

"The theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics. The accusation is one against which there is no valid defence; and it is never more just than when directed against the parts of the theory which are more particularly concerned with primes. A science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life. The theory of prime numbers satisfies no such criteria. Those who pursue it will, if they are wise, make no attempt to justify their interest in a subject so trivial and so remote, and will console themselves with the thought that the greatest mathematicians of all ages have found it in it a mysterious attraction impossible to resist." (Georg H Hardy, 1915)

"To tell whether a given number of 15 or 20 digits is prime or not, all time would not suffice for the test, whatever use is made of what is already known." (Leonard E Dickson, "History of the Theory of Numbers", 1919)

"The mystery that clings to numbers, the magic of numbers, may spring from this very fact, that the intellect, in the form of the number series, creates an infinite manifold of well distinguishable individuals. Even we enlightened scientists can still feel it e.g. in the impenetrable law of the distribution of prime numbers." (Hermann Weyl, "Philosophy of Mathematics and Natural Science", 1927)

"No one has ever been able to prove, for example, that every even number greater than two can be expressed as the sum of two primes. Yet this is as well established by observation as any of the laws of physics. It is known that this and various other theorems are true if a certain hypothesis about the Zeta function, enunciated by Riemann nearly a century ago, is correct. No one has been able to prove this hypothesis. It has only been shown that all the consequences deducible if it is true are so far verified by experience. But any day a computer with little knowledge of pure mathematics may disprove it. Here then is a possible triumph for the mathematical amateur." (John B S Haldane, "Possible Worlds and Other Essays", 1928)

"Systems, scientific and philosophic, come and go. Each method of limited understanding is at length exhausted. In its prime each system is a triumphant success: in its decay it is an obstructive nuisance." (Alfred N Whitehead, "Adventures of Ideas", 1933)

"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgments which are not the result of conscious trains of reasoning. These judgments are often but by no means invariably correct" (leaving aside the question as to what is meant by 'correct'). Often it is possible to find some other way of verifying the correctness of an intuitive judgment. One may for instance judge that all positive integers are uniquely factorable into primes; a detailed mathematical argument leads to the same result. It will also involve intuitive judgments, but they will be ones less open to criticism than the original judgment about factorization." (Alan M Turing, "Systems of Logic Based on Ordinals", Proceedings of the London Mathematical Society Vol 45" (2), 1939) 

"Bernhard Riemann, whose extraordinary intuitive powers we have already mentioned, has especially renovated our knowledge of the distribution of prime numbers, also one of the most mysterious questions in mathematics. He has taught us to deduce results in that line from considerations borrowed from the integral calculus: more precisely, from the study of a certain quantity, a function of a variable s which may assume not only real, but also imaginary values. He proved some important properties of that function, but enunciated two or three as important ones without giving the proof. At the death of Riemann, a note was found among his papers, saying 'These properties of ζ(s) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it.' We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one. The question concerning that last one remains unsolved as yet, though, by an immense labor pursued throughout this last half century, some highly interesting discoveries in that direction have been achieved. It seems more and more probable, but still not at all certain, that the 'Riemann hypothesis' is true." (Jacques Hadamard, "The Psychology of Invention in the Mathematical Field", 1945) 

"The mystery that clings to numbers, the magic of numbers, may spring from this very fact, that the intellect, in the form of the number series, creates an infinite manifold of well-distinguished individuals. Even we enlightened scientists can still feel it, e.g., in the impenetrable law of the distribution of prime numbers." (Hermann Weyl, "Philosophy of Mathematics and Natural Science", 1949)