Wilhelm Ahrens book of quotes
In 1904, Wilhelm Ahrens published Scherz und Ernst in der Mathematik, geflügelte und ungeflügelte Worte (Fun and seriousness in mathematics, well-known and less well-known words). The quotes in this books are in German, French and English. Clearly any reader was expected to know all three languages. On this page we give a selection of the German quotes translated into English by Tom Schnabel (Undergraduate, University of St Andrews) together with some of the quotes in English.
- Hermann von Helmholtz.
Anspr. U. Red. Geh. Bei der Helmholtz-Feier 2. XI. 1891 (Berlin 1892), p. 13.
The aim of science is the grasping of reality and, for the finite to be understood as a manifestation of the infinite, the law of nature.
- Carl Friedrich Gauss.
Erdmagnetismus und Magnetometer,
Werke, Bd. 5 (1877), p. 315-316.
For a scientist, explaining is analogous to tracing something back to a handful of desirably simple fundamental laws, which cannot be overcome, but must simply be taken for granted, in order to exhaustively explain a phenomenon.
- Hermann von Helmholtz.
Über die Erhaltung der Kraft,
Vortr. Phys. Ges. Berlin 28. VII. 1847. s. Ostwald's Klassiker d. exakt. Wiss. No. 1, p. 4.
The final goal of the theoretical sciences is to discover the last unchangeable causes of processes in nature. Whether all processes may really be traced back to such; that is to say whether nature must be completely comprehensible or whether there are changes in it that elude the law of causality and as such fall into the areas of spontaneity and freedom; is not the question at hand here. Regardless, it is clear that science, whose purpose it is to understand nature, must assume its comprehensibility and investigate and derive accordingly, at least until the day that irrefutable facts may demonstrate otherwise.
- Friedrich Wilhelm Bessel.
Über Wahrscheinlichkeitsrechnung,
Populäre Vorlesungen, herausg. v. Schumacher (Hamburgh 1848), p. 391.
A thunderstorm that obscures the sun is called a coincidence. A lunar eclipse however is not called a coincidence. Of one we know the cause, of the other we don't - but there was a time when an eclipse was also considered a coincidence - Many things that today are deemed a coincidence, will lose that name in the future. It is clear then that the term itself is relative. When Newton started spreading light in the world, many things were lifted from the dark realm of coincidence. Another Newton would uncover the causes of other things and a mind is conceivable for which little would be left to coincidence.
- Gustav Kirchhoff.
Über das Ziel der Naturwissenschaften,
Akad. Festrede Heidelberg 22. XI. 1865, p. 9.
If one were to know all forces of nature and the complete state of matter at any given moment in time, one would be able to mechanically determine the state of matter at any later time and derive how the various natural phenomena interacted. The highest goal of science must be to make the conditions outlined above a reality - that is to say to ascertain all laws of nature and the state of nature at one given moment and thus achieve the complete reduction of all natural processes to mechanics.
- Hermann von Helmholtz.
Das Denken in der Medizin,
Festrede Berlin militairärztl. Bildungs-Anst. 2. VIII. 1877. s. Vorträge u. Reden, Bd. 2 (1884), p. 185.
With strange ideas an author may quickly make a name for himself as an ingenious man. Given a large enough number of such ideas, some will surely be partially or fully correct. After all it would be a rather impressive feat to guess wrong every time. In such a lucky case, one can loudly boast to have first proposed the discovery. If not, any wrong proposals will soon fade into happy oblivion. Others who share a similar philosophy are often very inclined to help promote the value of the "first thought". The conscientious scientist meanwhile, who is reluctant to publish his ideas before all pages have been checked, all doubts have been quenched and the proof is completely robust, is thus at an unmistakable disadvantage. The current norm of resolving patent questions based purely on the date of publication, without taking into account the maturity of the work, has allowed this malpractice to blossom.
All the knowledge in the world, be it known or yet to be discovered, lies in the typecase of a book printer, if one only knew how to arrange the letters.
- Gustav Kirchhoff.
Vorlesungen über Mathem. Physik, Bd. 1 (Mechanik), (1876), p. 1.
Mechanics is the science of motion. We understand its aim to be to fully describe naturally occurring motion, in the simplest way possible.
- Otto Hölder.
Anschauung und Denken in der Geometrie (Leipzig 1900), p. 71.
Kirchhoff's claim that the science of mechanics purely describes naturally occurring motion has been repeated excessively. While one could certainly argue that every explanation is also a description, an explanation is nonetheless fundamentally different from a purely physical description. As such, we would do better to stick with this distinguishing word.
- Bernhard Riemann.
Fragment über "Neue mathem. Principien der Naturphilosophie,"
Werke, herausg. v. H. Weber, 2 Aufl. (1892), p. 528.
When is our perception of the world true?
When the connections in our imagination match the connections of reality.
The elements of our image of the world are completely different to the corresponding elements of reality. Our image of the world lies within, whereas reality is external. But the connections between the elements in our imagination and the elements of reality must match, in order for our perception to be true.
- Ludwig Boltzmann.
Über die Entwicklung der Methoden der theoretischen Physik in neuerer Zeit,
Vortrag Naturf.-Vers. München 1899. s. Deutsche Mathem.-Verein. Jahresber. 8, 1899, p. 85.
Countless questions that seemed unresolvable before are rendered meaningless by the following. Earlier one might have asked: how can a point of matter, that is purely imagined, exert a force, how can solid objects be formed merely from atoms, etc. Now we know that both the points of matter and the forces are purely abstract concepts. Points of matter can never be identical to solid objects, but they may be used to model them with arbitrary accuracy. The question whether matter consists of atoms or a continuum is hence reduced to the far clearer questions of whether the notion of infinitely many individual atoms or a continuum offers a better model of reality.
- Heinrich Hertz.
Die Prinzipien der Mechanik (1894), p. 9.
Why does no one ask after the nature of Gold or the nature of Speed? ... With the concepts of "Speed" and "Gold" we associate a large number of connections to other concepts, and in all these connections we find no contradictions to our understanding of the world. However, the concepts of "Force" and "Electricity" have been loaded with more connections than is healthy. As such, we seek clarity and proclaim our unclear desire to solve the mystery of the nature of force and electricity.
- Carl Friedrich Gauss.
Gottingen, 7. XL 1847. s. Briefw. Gauss-Schumacher, Bd. 5 (1863), p. 394.
In general, I would be tolerant of scientific fantasies and would merely argue against their inclusion in scientific astronomy, which must follow a completely different approach. Even Laplace's Cosmological Hypotheses fall in this class. Yes, I don't deny that I occasionally enjoy amusing myself in a similar manner, but I would never publish the like. Such musings include, for example, my thoughts about the inhabitants of other celestial bodies. I myself, contrary to popular belief, am convinced (at least what one would call convinced in these cases) that the bigger the celestial body, the smaller its inhabitants and other objects. For instance, on the sun, trees that are proportional to the sun's size in comparison to the earth's, could not exist, due to the much higher gravitational pull on the sun's surface. As such, all the branches would simply fall on their own accord, at least provided the composition of matter is different to that on earth.
- Paul du Bois-Reymond.
Über die Grundlagen der Erkenntnis in den exakten Wissenschaften, (Tübingen 1890), p. 17-18.
Science in its separate fields acts with utter inconsiderateness for the other fields of study. It is a common trend that with all fundamental laws derived, science tries to construct only a specific field of study, regardless of whether these may be used elsewhere. This is taken to the point where the same physical phenomenon is treated with completely different initial conditions when studied in different areas of science. ...
Even if direct contradictions (like for example continuous matter and the chemical molecule) should be viewed as monstrosities, generally this practice of science is certainly correct.
- Ludwig Boltzmann.
Über die Entwicklung der Methoden der theoretischen Physik in neuerer Zeit,
Vortr. Naturf.-Vers. München 1899. s. Deutsche Mathem.-Verein. Jahresber. 8, 1899, p. 82-83.
Kirchhhoff himself later reintroduced the term force, but not as a metaphysical term, but merely as an abbreviated term for certain algebraic expressions that are common in mechanics.
Kirchhoff did not fundamentally change classical mechanics; his reformation was purely a formal one. Hertz however went much further, but while most authors later adopted Kirchhoff's manner of representation, even if not completely his beliefs, I have never yet seen someone following the path Hertz started, despite the frequent praise it receives.
- Hermann von Helmholtz.
Vorrede zu Heinrich Hertz, Ges. Werke, Bd. 3 (1894), p. XIX u. XXII = Zeitschr. Phys. Chem. Unterr. 8, 1894/95, p. 28-29.
Hertz has tried to provide a consistently reasoned representation of a completely coherent system of mechanics and has striven to derive all laws of this field from a single fundamental law, which can logically only be viewed as a plausible assumption.
Naturally, many challenges still need to be overcome in the endeavour to explain the separate aspects of Physics using the foundations developed by Hertz. But on the whole, Hertz' book on the representation of the fundamental laws of Mechanics must greatly interest every reader who delights in a mathematically sound and brilliantly reasoned system of Dynamics. Possibly this book will be of great heuristic value in the future, as the starting point of the discovery of new general characteristics of mechanics.
- David Hilbert.
Mathematische Probleme,
Vortrag Mathem.-Congr. Paris 1900. s. Göttinger Nachr., Math.-phys. Kl. 1900, p. 261-262 = Arch. Math. Phys. (3) 1 (1901), p. 51-52.
In recent years, the potential insolvability of certain mathematical problems has gained increasing attention. As such, we have discovered that old conundrums, such as the proof of Euclid's fifth postulate, squaring the circle or solving quintic equations by factorization into radicals, have perfectly satisfying and rigorous answers, even if they may not be what we originally intended.
This strange fact, along with other philosophical reasons, is probably the cause for the belief that every mathematician shares, even though it has yet to be backed by proof. Namely that every mathematical problem can be solved, be it by finding the answer or be it by proving its insolvability and thus demonstrating the necessity of failure in finding a solution.
This conviction that every mathematical problem has an answer is a powerful motivator when we work. Within us we hear a constant cry: "Here is the problem, search for a solution. It can be found by thinking alone," because in mathematics there is no ignorabimus.
- Rudolf Lipschitz.
Bedeutung der theoretischen Mechanik,
Heft 244 der Virchow-Holtzendorffschen Sammlung gemeinverst. wiss. Vortr. (1876), p. 3.
The way of dealing with a mechanical problem by asking questions of nature has become a role model for the arts.
- Gustav Kirchhoff.
Über das Ziel der Naturwissenschaften,
Akad. Festrede Heidelberg 22. XI. 1865, p. 4-5.
There is a science, mechanics, whose role it is to determine the motion of bodies, given the causes of the motion are known.
Mechanics is closely related to Geometry. Both sciences are applications of pure mathematics and the certitude of the theorems of both lies at the same level. Just like geometric theorems, mechanical theorems must be credited with absolute certainty.
- David Hilbert.
Mathematische Probleme,
Vortrag Mathem.-Congr. Paris 1900. s. Göttinger Nachr., Math.-phys. Kl. 1900, p. 272 = Arch. Math. -Phys. (3) 1 (1901), p. 62.
Through our studies of the underlying fundamental principles of geometry it has become clear that we should study those disciplines of Physics in which mathematics already plays a key role, in a similarly axiomatic way. First and foremost among such disciplines are probability and mechanics.
- Otto Hölder.
Anschauung und Denken in der Geometrie,
Antrittsvorles. Leipzig Univ. 1899 (Leipzig 1900), p. 22.
Maybe, deduction in mechanics cannot be viewed as pure as in geometry. Maybe in mechanics, that has more concrete substance, along with the axioms, we also subconsciously use certain analogies based on experience.
- Felix Klein.
Über die Enzyklopädie der mathematischen Wissenschaften, mit besonderer Rücksicht auf Band 4 derselben (Mechanik),
Vortr. Naturf.-Vers. Aachen 1900. s. Physikalische Zeitschr. 2, 1900/1901, p. 93.
In England and Holland mechanics is simply classified as a part of physics.
On the continent, however, (with the exception of Holland) the view that mechanics, as a mécanique rationnelle, is a part of mathematics is far more prominent. Think only of the masters Laplace and Lagrange! What is more, there has never been a lack of effort to ascertain that the fundamental principles of Mechanics are given a priori, following the example of the axioms of geometry.
- Carl Friedrich Gauss.
Gauss an Olbers,
Göttingen, 28. IV. 1817. s. Gauss, Werke, Bd. 8 (1900), p. 177.
I am becoming increasingly convinced that the necessity of our geometry cannot be proven; at least not by a human mind or to the human mind. Maybe, in another life we will gain more of an insight into the nature of space, that has hitherto been unreachable. But until then, geometry should not be classed with Arithmetic, which holds a priori, but with the likes of Mechanics.
- Carl Friedrich Gauss.
Ideen (Nachlass), 27. IV. 1813,
Gauss, Werke, Bd. 8 (1900), p.166.
In the theory of parallel lines we have come no further than Euclid did. That is the partie honteuse [shameful part] of mathematics, that must sooner or later get an entirely new character.
- Carl Friedrich Gauss.
Göttinger Gelehrte Anz. 1816, April 20 = Werke, Bd. 4 (1880), p. 364-365.
Few topics in Mathematics will be as prominently covered in literature as the hole at the foundation of geometry in the justification of the theory of parallel lines. There is seldom a year in which some new attempt to fill this hole doesn't surface. And yet, if we are being completely honest, we cannot seriously claim that we have come any further than Euclid did 2000 years ago. Such a sincere and outright confession seems more befitting of science than the vain attempt to hide the unfillable hole behind an unsustainable web of deceiving, apparent proofs.
- Carl Friedrich Gauss.
Gauss an Gerling. Göttingen, 25. VIH. 1818,
Gauss, Werke, Bd. 8 (1900), p. 179.
I am pleased that you have the courage to express yourself in such a way as if to accredit the possibility that our theory of parallel lines, and thus our whole geometry, is wrong. But the wasps whose nest you rouse will buzz around your head.
- James Clerk Maxwell.
"On Faraday's Lines of Force",
cf Scientific Papers vol. 1 (1890), p. 156.
In order to obtain physical ideas without adopting a physical theory we must make ourselves familiar with the existence of physical analogies. By a physical analogy I mean that partial similarity between the laws of one science and those of another which makes each of them illustrate the other. Thus all the mathematical sciences are founded on relations between physical laws and laws of. numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers.
- Alexander Macfarlane.
"Peter Guthrie Tait, his life and works",
Bibl math. (3) 4 (1903), p. 189.
In future times Tait will be best known for his work in the quaternion analysis. Had it not been for his expositions, developments and applications, Hamilton's invention would be today, in all probability, a mathematical curiosity; and there are those who think that, now Tait is gone, such will ere long be its fate. But I venture to think that Hamilton himself will prove the better prophet: for he wrote to Tait: "Could anything be simpler or more satisfactory? Don't you feel, as well as think, that we are on the right track, and shall be thanked hereafter? Never mind when."
- W R Hamilton to A De Morgan.
Observatory (of Trinity College, Dublin), 6. I. 1852.
cf Graves, "Life of Sir William Rowan Hamilton",
Vol III (1889), p. 311/312 = Vol II (1885), p. 490.
In fact, with all my very high admiration ... for Gauss, I have some private reasons for believing, I might say knowing, that he did not anticipate the quaternions. In fact, if I don't forget the year, I met a particular friend, and (as I was told) pupil of Gauss, Baron von Waltershausen, .... at the Second Cambridge Meeting of the British Association in 1845, just after Herschel had spoken of my quaternions and your triple algebra, in his speech from the throne. The said Baron soon afterward called on me here, ... he informed me that his friend and (in one sense) master, Gauss, had long wished to frame a sort of triple algebra; but that his notion had been, that the third dimension of space was to be symbolically denoted by some new transcendental, as imaginary, with respect to "-1, as that was with respect to 1. Now you see, as I saw then, that this was in fundamental contradiction to my plan of treating all dimensions of space with absolute impartiality, no one more real than another.
- J J Sylvester.
("Trilogy"), Philosophical Transactions of the Royal Society of London,
Vol 154, Part III (1864), p. 579.
All roads are said to lead to Rome, so I find, in my own case at least, that all algebraical inquiries sooner or later end at that Capital of Modern Algebra over whose shining portal is inscribed "Theory of Invariants".
- J J Sylvester.
Address Meeting of the British Association Exeter 1869.
cf Report, Notices and Abstracts, p. 4.
In the "Fortnightly Review" [1869] we are told that "Mathematics is that study which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation" (Huxley, "Lay sermons, addresses and reviews" (London 1870), p. 185). I think no statement could have been made more opposite to the undoubted facts of the case, that mathematical analysis is constantly invoking the aid of new principles, new ideas, and new methods, not capable of being defined by any form of words, but springing direct from the inherent powers and activity of the human mind, and from continually renewed introspection of that inner world of thought of which the phenomena are as varied and require as close attention to discern as those of the outer physical world (to which the inner one in each individual man may, I think, be conceived to stand in somewhat the same general relation of correspondence as a shadow to the object from which it is projected, or as the hollow palm of one hand to the closed fist which it grasps of the other), that it is unceasingly calling forth the faculties of observation and comparison, that one of its principal weapons is induction, that it has frequent recourse to experimental trial and verification, and that it affords a boundless scope for the exercise of the highest efforts of imagination and invention.
- J J Sylvester.
("Trilogy"), Philosoph. Transactions of the Royal Society of London,
Vol 154, Part III (1864), p. 618; cf. also British Association Report 1869, Notices and Abstracts p. 7.
May not music be described as the mathematics of the sense, mathematics as music of the reason? the soul of each is the same! Thus the musician feels mathematics, the mathematician thinks music, - music the dream, mathematics the working life - each to receive its consummation from the other when the human intelligence, elevated to its perfect type, shall shine forth glorified in some future Mozart-Dirichlet or Beethoven-Gauss - a union already not indistinctly foreshadowed in the genius and labours of a Helmholtz!
- P A MacMahon.
Proc Roy. Soc. London 63 (1898), p. XVII.
During a conversation with the writer in the last weeks of his life, Sylvester remarked as curious that notwithstanding he had always considered the bent of his mind to be rather analytical than geometrical, he found in nearly every case that the solution of an analytical problem turned upon some quite simple geometrical notion, and that he was never satisfied until he could present the argument in geometrical language.