"Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it is the truth." (Benjamin Peirce [in William E Byerly, "Benjamin Peirce: II. Reminiscences", The American Mathematical Monthly 32 (1), 1925]
"Mathematics, indeed, is the very example of brevity, whether it be in the shorthand rule of the circle, c = πd, or in that fruitful formula of analysis, e^iπ = -1, - a formula which fuses together four of the most important concepts of the science - the logarithmic base, the transcendental ratio π, and the imaginary and negative units." (David E Smith, "The Poetry of Mathematics", The Mathematics Teacher, 1926)
"Meantime, there is no doubt a certain crudeness in the use of a complex wave function. If it were unavoidable in principle, and not merely a facilitation of the calculation, this would mean that there are in principle two wave functions, which must be used together in order to obtain information on the state of the system. [...] Our inability to give more accurate information about this is intimately connected with the fact that, in the pair of equations [considered], we have before us only the substitute - extraordinarily convenient for the calculation, to be sure - for a real wave equation of probably the fourth order, which, however, I have not succeeded in forming for the non-conservative case." (Edwin Schrödinger, "Quantisation as a Problem of Proper Values" , Annalen der Physik Vol. 81 (4), 1926)
"Our bra and ket vectors are complex quantities, since they can be multiplied by complex numbers and are then of the same nature as before, but they are complex quantities of a special kind which cannot be split up into real and pure imaginary parts. The usual method of getting the real part of a complex quantity, by taking half the sum of the quantity itself and its conjugate, cannot be applied since a bra and a ket vector are of different natures and cannot be added." (Paul Dirac, "The Principles of Quantum Mechanics", 1930)
"I recall my own emotions: I had just been initiated into the mysteries of the complex number. I remember my bewilderment: here were magnitudes patently impossible and yet susceptible of manipulations which lead to concrete results. It was a feeling of dissatisfaction, of restlessness, a desire to fill these illusory creatures, these empty symbols, with substance. Then I was taught to interpret these beings in a concrete geometrical way. There came then an immediate feeling of relief, as though I had solved an enigma, as though a ghost which had been causing me apprehension turned out to be no ghost at all, but a familiar part of my environment." (Tobias Dantzig, “The Two Realities”, 1930)
"In his desire to consider at any cost the propagation phenomenon of the waves ψ as something real in the classical sense of the word, the author had refused to acknowledge that the whole development of the theory increasingly tended to highlight the essential complex nature of the wave function." (Edwin Schrödinger. "Mémoires sur la mécanique ondulatoire", 1933) [author‘s comment in the French translation]
"There is thus a possibility that the ancient dream of philosophers to connect all Nature with the properties of whole numbers will some day be realized. To do so physics will have to develop a long way to establish the details of how the correspondence is to be made. One hint for this development seems pretty obvious, namely, the study of whole numbers in modern mathematics is inextricably bound up with the theory of functions of a complex variable, which theory we have already seen has a good chance of forming the basis of the physics of the future. The working out of this idea would lead to a connection between atomic theory and cosmology." (Paul A M Dirac, [Lecture delivered on presentation of the James Scott prize] 1939)
"There is a famous formula, perhaps the most compact and famous of all formulas developed by Euler from a discovery of de Moivre: It appeals equally to the mystic, the scientist, the philosopher, the mathematician." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
“[…] imaginary numbers made their own way into arithmetical calculation without the approval, and even against the desires of individual mathematicians, and obtained wider circulation only gradually and the extent to which they showed themselves useful.” (Felix Klein, “Elementary Mathematics from an Advanced Standpoint”, 1945)
"It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one." (Jacque Hadamard, "An Essay on the Psychology of Invention in the Mathematical Field", 1945)
[French: "On a pu écrire depuis que la voie la plus courte et la meilleure entre deux vérités du domaine réel passe souvent par le domaine imaginaire." (Jacques Hadamard, Essai sur la psychologie de l'invention dans le domaine mathématique, 1945)]
"It is a curious fact that the first introduction of the imaginaries occurred in the theory of cubic equations, in the case where it was clear that real solutions existed though in an unrecognizable form, and not in the theory of quadratic equations, where our present textbooks introduce them." (Dirk J Struik, “A Concise History of Mathematics” Vol. I, 1948)
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