Felix Klein - Collected Quotes

"Even if a curve is not drawn nor it is assumed to be tracked by the eye, but we have a 'perception' of it, it has in any case a limited precision and does not therefore correspond to the exact concept of a function of precision mathematics but rather to the idea of a function stripe." (Felix Klein, 1873) 

"Analytic functions are those that can be represented by a power series, convergent within a certain region bounded by the so-called circle of convergence. Outside of this region the analytic function is not regarded as given a priori ; its continuation into wider regions remains a matter of special investigation and may give very different results, according to the particular case considered." (Felix Klein, "Sophus Lie", [lecture] 1893)

"[...]  the illustration of a space of constant positive measure of curvature by the familiar example of the sphere is somewhat misleading.  Owing to the fact that on the sphere the geodesic lines (great circles) issuing from any point all meet again in another definite point, antipodal, so to speak, to the original point, the existence of such an antipodal point has sometimes been regarded as a necessary consequence of the assumption of a constant positive curvature. The projective theory of non-Euclidean space shows immediately that the existence of an antipodal point, though compatible with the nature of an elliptic space, is not necessary, but that two geodesic lines in such a space may intersect in one point if at all." (Felix Klein, "The Most Recent Researches in Non-Euclidian Geometry", [lecture] 1893)

"Our science, in contrast with others, is not founded on a single period of human history, but has accompanied the development of culture through all its stages. Mathematics is as much interwoven with Greek culture as with the most modern problems in Engineering. She not only lends a hand to the progressive natural sciences but participates at the same time in the abstract investigations of logicians and philosophers." (Felix Klein, "Klein und Riecke: Ueber angewandte Mathematik und Physik" 1900)

"Everyone who understands the subject will agree that even the basis on which the scientific explanation of nature rests is intelligible only to those who have learned at least the elements of the differential and integral calculus, as well as analytical geometry." (Felix Klein, Jahresbericht der Deutsche Mathematiker Vereinigung Vol. 1, 1902)

"Mathematics in general is fundamentally the science of self-evident things." (Felix Klein, "Anwendung der Differential-und Integral rechnung auf Geometrie, 1902)

"Today it is no longer questioned that the principles of the analysts are the more far-reaching. Indeed, the synthesists lack two things in order to engage in a general theory of  algebraic configurations: these are on the one hand a definition of imaginary elements, on the other an interpretation of general algebraic concepts. Both of these have subsequently been developed in synthetic form, but to do this the essential principle of synthetic geometry had to be set aside. This principle which manifests itself so brilliantly in the theory of linear forms and the forms of the second degree, is the possibility of immediate proof by means of visualized constructions." (Felix Klein, "Riemannische Flachen", 1906)

“[…] 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”, 1908)

"Let us look for a moment at the general significance of the fact that calculating machines actually exist, which relieve mathematicians of the purely mechanical part of numerical computations, and which accomplish the work more quickly and with a greater degree of accuracy; for the machine is not subject to the slips of the human calculator. The existence of such a machine proves that computation is not concerned with the significance of numbers, but that it is concerned essentially only with the formal laws of operation; for it is only these that the machine can obey - having been thus constructed - an intuitive perception of the significance of numbers being out of the question." (Felix Klein, "Elementarmathematik vom hoheren Standpunkte aus", 1908)

“[…] the function of pure logic, when it comes to setting up new concepts, is only to regulate and never to act as the sole guiding principle; for there will always be, of course, many other conceptual systems which satisfy the single demand of logic, namely, freedom from contradiction." (Felix Klein, “Elementary Mathematics from an Advanced Standpoint”, 1908)

[…] theory of numbers lies remote from those who are indifferent; they show little interest in its development, indeed they positively avoid it. [..] the pure theory of numbers is an extremely abstract thing, and one does not often find the gift of ability to understand with pleasure anything so abstract. […] I believe that the theory of numbers would be made more accessible, and would awaken more general interest, if it mere presented in connection with graphical elements and appropriate figures.”  (Felix Klein, “Elementary Mathematics from an Advanced Standpoint”, 1908) 

"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions. […] A curve is the totality of points, whose coordinates are functions of a parameter which may be differentiated as often as may be required." (Felix Klein, "Elementar Mathematik vom hoheren Standpunkte aus" Vol. 2, 1909)

"I do in no wise share this view [that the axioms are arbitrary propositions which we assume wholly at will, and that in like manner the fundamental conceptions are in the end only arbitrary symbols with which we operate] but consider it the death of all science: in my judgment the axioms of geometry are not arbitrary, but reasonable propositions which generally have the origin in space intuition and whose separate content and sequence is controlled by reasons of expediency." (Felix Klein, "Elementarmathematik vom hoheren Standpunkte aus", 1909)

"It is my opinion that in teaching it is not only admissible, but absolutely necessary, to be less abstract at the start, to have constant regard to the applications, and to refer to the refinements only gradually as the student becomes able to understand them. This is, of course, nothing but a universal pedagogical principle to be observed in all mathematical instruction." (Felix Klein, "Lectures on Mathematics", 1911)

"It is well known that the central problem of the whole of modern mathematics is the study of the transcendental functions defined by differential equations." (Felix Klein, "Lectures on Mathematics", 1911)

"[…] the danger of a separation between abstract mathematical science and its scientific and technical applications. Such separation could only be deplored; for it would necessarily be followed by shallowness on the side of the applied sciences, and by isolation on the part of pure mathematics." (Felix Klein, "Lectures on Mathematics", 1911)

"The ordinary mathematical treatment of any applied science substitutes exact axioms for the approximate results of experience, and deduces from these axioms the rigid mathematical conclusions. In applying this method it must not be forgotten that the mathematical developments transcending the limits of exactness of the science are of no practical value. It follows that a large portion of abstract mathematics remains without finding any practical application, the amount of mathematics that can be usefully employed in any science being in proportion to the degree of accuracy attained in the science. Thus, while the astronomer can put to use a wide range of mathematical theory, the chemist is only just beginning to apply the first derivative, i. e. the rate of change at which certain processes are going on; for second derivatives he does not seem to have found any use as yet." (Felix Klein, "Lectures on Mathematics", 1911)

"The proof of the transcendency of π will hardly diminish the number of circle-squarers, however; for this class of people has always shown an absolute distrust of mathematicians and a contempt for mathematics that cannot be overcome by any amount of demonstration." (Felix Klein, "Lectures on Mathematics", 1911)

"The proof that π is a transcendental number will forever mark an epoch in mathematical science. It gives the final answer to the problem of squaring the circle and settles this vexed question once for all. This problem requires to derive the number π by a finite number of elementary geometrical processes, i.e. with the use of the ruler and compasses alone. As a straight line and a circle, or two circles, have only two intersections, these processes, or any finite combination of them, can be expressed algebraically in a comparatively simple form, so that a solution of the problem of squaring the circle would mean that π can be expressed as the root of an algebraic equation of a comparatively simple kind, viz. one that is solvable by square roots." (Felix Klein, "Lectures on Mathematics", 1911)

"If the activity of a science can be supplied by a machine, that science cannot amount to much, so it is said; and hence it deserves a subordinate place. The answer to such arguments, is that the mathematician, even when he is himself operating with numbers and formulas, is by no means an interior counterpart of the errorless machine. 'thoughtless thinker' of Thomac; but rather, he sets for himself his problems with definite, interesting, und valuable ends in view, and carries them to solution in appropriate and original manner. He turns over to the machine only certain operations which recur frequently in the same way, and it is precisely the mathematician - one must not forget this - who invented the machine for his own relief, and who, for his own intelligent ends, designates the tasks which it shall perform." (Felix Klein, "Elementary Mathematics from an Advanced Standpoint", 1919)

"As the objects of abstract geometry cannot be totally grasped by space intuition, a rigorous proof in abstract geometry can never be based only on intuition, but it must be founded on logical deduction from valid and precise axioms. Nevertheless intuition maintains, also in precision geometry, its irreplaceable value that cannot be substituted by logical considerations. Intuition helps us to construct a proof and to gain an overview, it is, moreover, a source of inventions and new mental connections." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"For establishing the laws of nature one resorts (not deliberately but involuntarily) to the simplest formulas that seem to describe the phenomena with reasonable accuracy. […] Even those laws of nature that are the most general and important for the world view have always been proved experimentally only in a confined ambit and with limited accuracy. […] The exact formulation of the laws of nature by simple formulas is based on the desire to master external phenomena with the simplest tools possible." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"In mathematics as in the other sciences, the same processes can be observed again and again. First, new questions arise, for internal or external reasons, and draw researchers away from the old questions. And the old questions, just because they have been worked on so much, need ever more comprehensive study for their mastery. This is unpleasant, and so one is glad to turn to problems that have been less developed and therefore require less foreknowledge - even if it is only a matter of axiomatics, or set theory, or some such thing." (Felix Klein,"Development of Mathematics in the 19th Century", 1928)

"In order to regain in a rigorously defined function those properties that are analogous to those ascribed to an empirical curve with respect to slope and curvature (first and higher difference quotients), we need not only to require that the function is continuous and has a finite number of maxima and minima in a finite interval, but also assume explicitly that it has the first and a series of higher derivatives (as many as one will want to use)." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"The course of the values of a continuous function is determined at all points of an interval, if only it is determined for all rational points of this interval." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"The different premises will possibly allow equally good explanations with respect to the imprecise nature of our sensory perception, because the sensory perception is, in fact, not concerned with issues of precision mathematics but of approximation mathematics." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"The difference between commensurable and incommensurable in its strict sense (and hence also the concept of irrational number) belongs solely to precision mathematics." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"The mistake from which todays’ science suffers is that the theoreticians are concerned too unilaterally with precision mathematics, while the practitioners use a sort of approximate mathematics, without being in touch with precision mathematics through which they could reach a real approximation mathematics." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"The most general definition of a function that we have reached in modern mathematics starts by fixing the values that the independent variable x can take on. We define that x must successively pass through the points of a certain 'point set'. The language used is therefore geometric […]." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"The weak point in all such reflections is that they depend on an arbitrary preference of certain ideas and concepts of precision mathematics, while observations in nature always have only limited precision and can be related in very different manners to topics of precision mathematics. It is more generally questionable whether we should be looking for the essence of a correct explanation of nature on the basis of precision mathematics, and whether we could ever go beyond a skillful use of approximation mathematics." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"When we look at a very large number of small objects that are close together the idea of continuum arises within us. […] Even if we believe to perceive a continuum in front of us, a more accurate observation will often convince us that we are only observing a dense succession of small particles." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

“But it should always be required that a mathematical subject not be considered exhausted until it has become intuitively evident […] ” (Felix Klein)

“[…] mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs.” (Felix Klein)

"Mathematics in our day reminds me of major small-arms production in peacetime. The shop window is filled with models that delight the expert by their cleverness and their artful and captivating execution. Properly speaking, the origin and significance of these things - that is, their ability to shoot and hit the enemy, recedes in one's consciousness and is even completely forgotten." (Felix Klein) 

"Projective geometry has opened up for us with the greatest facility new territories in our science, and has rightly been called the royal road to our particular field of knowledge." (Felix Klein)

"Regarding the fundamental investigations of mathematics, there is no final ending […] no first beginning." (Felix Klein)

"The axioms of geometry are - according to my way of thinking - not arbitrary, but sensible. statements, which are, in general, induced by space perception and are determined as to their precise content by expediency." (Felix Klein) 

"The developing science departs at the same time more and more from its original scope and purpose and threatens to sacrifice its earlier unity and split into diverse branches." (Felix Klein)

"To follow a geometrical argument purely logically without having the figure on which the argument bears constantly before me is for me impossible." (Felix Klein)

"Undoubtedly, the capstone of every mathematical theory is a convincing proof of all of its assertions. Undoubtedly mathematics inculpates itself when it foregoes convincing proofs. But the mystery of brilliant productivity will always be the posing of new questions, the anticipation of new theorems that make accessible valuable results and connections. Without the creation of new viewpoints, without the statement of new aims, mathematics would soon exhaust itself in the rigor of its logical proofs and begin to stagnate as its substance vanish. Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs." (Felix Klein)