10 Theses on the Nature of Mathematics
10. Mathematics is a language.
The universe...stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics ... Galileo: Il Saggiatore (1623).
... pure mathematics is not a separate field of knowledge, rather a completion of the general language. Niels Bohr, cited in Blechman, Myskis, Panovko: Angewandte Mathematik.
Mathematics is an important language, as many findings - especially in physics - are virtually impossible to state in a non-mathematical way. It has a very difficult grammar, as some conjectures need centuries to be proved. Because of these difficulties there is enough work for "pure" mathematicians studying "only" the internal relations without any contact to applications. Still, studying only "pure" mathematics is a bit like learning a language but never using it to talk to real people. There are many who believe that mathematics is more than just a language, that mathematical truths have an independent existence and that truths can be found by pure reasoning, without any contact to the real world:
In the pure mathematics we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist there when the last of their radiant host shall have fallen from heaven. Edward Everett, (1794-1865) Quoted by E.T. Bell in The Queen of the Sciences, Baltimore, 1931.
There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation. Hermite, Charles (1822 - 1901), cited in The Mathematical Intelligencer, v. 5, no. 4.
Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. "Immortality" may be a silly word, but probably a mathematician has the best chance of whatever it may mean. Hardy, Godfrey H. (1877 - 1947). A Mathematician's Apology, London, Cambridge University Press, 1941.
One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them. Heinrich Hertz, Quoted by ET Bell in Men of Mathematics, New York, 1937.
It is easy to come to the conclusion that circles, prime numbers, Riemann's zeta function ... have an independent, eternal existence and any real world objects are only imperfect resemblances of these eternal "pure ideas". The philosophical consequence is idealism, which was perfected by the German philosophers Kant and Hegel. In fact, Kant uses mathematics as an example to make his point:
The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience. Emmanuel Kant, (1724-1804).
There is a nice simile by Dantzig on the surprising applicability of purely theoretic reasoning:
The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and delight. Dantzig.
9. Mathematical theories are not true, but advantageous in a certain epoch.
...by natural selection our mind has adapted itself to the conditions of the external world. It has adopted the geometry most advantageous to the species or, in other words, the most convenient. Geometry is not true, it is advantageous. Jules Henri Poincaré, (1854-1912) Science and Method.
A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it. Max Planck (1949)
The mathematic, then, is an art. As such it has its styles and style periods. It is not, as the layman and the philosopher (who is in this matter a layman too) imagine, substantially unalterable, but subject like every art to unnoticed changes from epoch to epoch. The development of the great arts ought never to be treated without an (assuredly not unprofitable) side-glance at contemporary mathematics. Oswald Spengler, (1880 -1936) The Decline of the West.
8. Mathematics is a library.
Mathematics is like a library, which is used by scientists and engineers of diverse fields in search for solutions to their problems. The key contribution of mathematics is to reformulate the problems (and solutions) found in distant fields in a neutral language and thus act as a clearinghouse of structural knowledge about problems and their solutions. Mathematicians guide "readers" through the library as well as write books about problem areas that are often asked for.
Mathematics is neither the "Queen of Sciences" (Gauss) nor the "Handmaiden of the Sciences" (Eric Temple Bell (1883-1960)), but its librarian role is an important one.
7. The solution of a problem usually precedes its rigorous mathematical treatment.
The deeper understanding and more rigorous treatment of a certain problem structure by mathematicians usually comes after its understanding and solution by mathematicians, physicists, engineers, ... This view is best expressed by a quote from "How to Write Mathematics" by Halmos:
The heart of mathematics consists of concrete examples and concrete problems. Big general theories are usually afterthoughts based on small but profound insights; the insights themselves come from concrete special cases.
(I learned the German version of this quote from a book by Blechmann, Myskis, and Panovko: ``Angewandte Mathematik''. The origin of the quote was communicated to me by Zenon Kulpa.)
By citing examples of mathematical developments that needed decades for their implementation and application, some mathematical text books create a feeling of superiority and supremacy among students of mathematics. This is deceptive and dangerous. Mathematics is often very much involved in technological breakthroughs. But these breakthroughs are not brought about by mathematicians alone, especially not by those who believe in the lagging behind of applications.
Should I refuse a good dinner simply because I do not understand the process of digestion? Heaviside, Oliver (1850-1925). [Criticized for using the calculus of Laplace transforms formally, without fully developing and understanding the theory of Laplace transforms.]
6. Absolute rigor is a myth.
Rigor is perceived as the most distinguishing feature of mathematics by many mathematicians as well as non-mathematicians. As a matter of fact, rigor is relative. Levels of rigor range from the rather informal presentation of an idea at conferences, its more rigorous form in a journal article to its expression in a formal language that can be checked by an automatic theorem prover. Second, there is a trade-off between more rigorous and more intuitive reasoning. Most mathematical theorems are guessed before they are proved. Most statistical relations are found by exploratory techniques before their significance is considered:
Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigor should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere. W.S. Anglin, "Mathematics and History", Mathematical Intelligencer, v. 4, no. 4.
The essential point in science is not a complicated mathematical formalism or a ritualized experimentation. Rather the heart of science is a kind of shrewd honesty that springs from really wanting to know what the hell is going on! Saul-Paul Sirag
The use of mathematics allows to increase the level of rigor in a certain field. Nothing more and nothing less. The "absolute rigor" of mathematics as a whole is a myth.
5. There is an uncertainty principle involving rigor and real world relevance.
On top of the fact that there is a trade-off between more rigorous and more exploratory reasoning, there is a trade-off between rigor and real world relevance:
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Albert Einstein, (1879-1955) In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
While mathematics is a convenience in relating certain types of order to our comprehensions, it does not give us any account of their actuality. Euclid's geometry was once supposed to be an exact description of the external world. The only world of which it is an accurate description is the world of Euclidean geometry. Whitehead.
If scientific reasoning were limited to the logical processes of arithmetic, we should not get very far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability. Vannevar Bush
The propositions of mathematics have, therefore, the same unquestionable certainty which is typical of such propositions as "All bachelors are unmarried," but they also share the complete lack of empirical content which is associated with that certainty: The propositions of mathematics are devoid of all factual content; they convey no information whatever on any empirical subject matter. Hempel, Carl G.: "On the Nature of Mathematical Truth" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
4. The mystified beauty and simplicity of mathematical truths are simply consequences of mathematics' main function: to solve problems.
Beauty and simplicity of mathematical results are often admired and mystified:
The mathematician is fascinated with the marvelous beauty of the forms he constructs, and in their beauty he finds everlasting truth. J. B. Shaw, In N. Rose Mathematical Maxims and Minims, 1988.
The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connexion of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concerned, these, and such like, are the surest grounds of the title of mathematics to human regard, and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance. J.J. Sylvester Presidential Address to British Association, 1869.
Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness. Eric Temple Bell (1883-1960)
Mathematics has beauties of its own -- a symmetry and proportion in its results, a lack of superfluity, an exact adaptation of means to ends, which is exceedingly remarkable and to be found only in the works of the greatest beauty. When this subject is properly ... presented, the mental emotion should be that of enjoyment of beauty, not that of repulsion from the ugly and the unpleasant. J. W. A. Young, In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
There are two points here: analytical thinking and ease of computation. A key to the solution of complex problems is to analyze the several aspects of the problem at hand by abstracting from real world features. Analytical thinking leads to simple (and hence beautiful) truths about the different aspects of a problem. Its main effect, however, is to reduce complexity. Whenever mathematics is applied, it boils down to some computations. Since humans were the only "computers" before the 20th century, the "difficulty", "cost", or "complexity" of a problem was directly related to the "simplicity" and "beauty" of its available solutions. Hence the importance of "closed-form solutions", "analytic solutions", and "special functions".
What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details. Jules Henri Poincaré, (1854-1912) In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
The aim of science is to seek the simplest explanations of complex facts. We are apt to fall into the error of thinking that the facts are simple because simplicity is the goal of our quest. The guiding motto in the life of every natural philosopher should be, "Seek simplicity and distrust it." Alfred North Whitehead, The Concept of Nature, 1926.
3. The advent of electronic computers changed mathematics forever. Algorithms replaced formulas as the smallest unit representing structural knowledge.
As argued above, "closed-form solutions" are algorithms for "human computers" to solve problems. When the way to do basic computations changed, the "implicit objective function" of mathematics changed. Instead of looking for simple (=beautiful) formulae one should look for compactly formulated, fast algorithms.
It is a pity that the theory of algorithms and the theory of complexity - which are now as fundamental to mathematics as are basic algebra and analysis - is neither taught in math classes in high school nor in math departments at universities. [Footnote 1]
The abundance of computing power leads to a potentially broader applicability of mathematical results and a larger impact on society as a whole.
2. The distinction between pure and applied mathematics is artificial.
As said above, rigor is relative. The same holds for applicability. How to solve an integral equation may be "theory" for an engineer and "application" for a mathematician studying operator theory. Virtually any mathematical result will be regarded by some as "rather applied" and by others as "rather theoretical".
Mathematicians have always had problems with applicability. Some acknowledge the need for applications but maintain that one should not look for it. Some deny it outright. Some look down on "dirty" applications and "inferior" applied mathematicians:
Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle. Jacques Hadamard, In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world. Nikolai Lobachevsky, In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Pure mathematics, may it never be of any use to anyone. Henry John Stephen Smith, (1826 - 1883) [His toast:] In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. Jules Henri Poincare.
It is true that Fourier had the opinion that the principal aim of mathematics was public utility and explanation of natural phenomena; but a philosopher like him should have known that the sole end of science is the honor of the human mind, and that under this title a question about numbers is worth as much as a question about the system of the world. Carl Jacobi, In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
I am interested in mathematics only as a creative art. Godfrey H. Hardy, (1877 - 1947) A Mathematician's Apology, London, Cambridge University Press, 1941.
Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics. Godfrey H. Hardy, (1877 - 1947)
Applied mathematics is bad mathematics. Paul Halmos. Mathematics tomorrow (L.A. Steen, ed.), 9-20, Springer, New York.
Well, a science without any connection to the real world is dead. Or you have to call it art and find people who enjoy it. In Chebyshev's words:
To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls. Chebyshev In G. Simmons, Calculus Gems, New York: Mcgraw Hill, Inc., 1992, page 198.
[...] es gibt eine Neigung zu vergessen, daß die gesamte Wissenschaft an die menschliche Kultur überhaupt gebunden ist und daß wissenschaftliche Entdeckungen, mögen sie im Augenblick auch überaus fortschrittlich und geistreich und unfaßlich erscheinen, außerhalb ihres kulturellen Rahmens sinnlos sind. Eine theoretische Wissenschaft, die sich nicht dessen bewußt ist, daß die Begriffe, die sie für relevant und wichtig hält, letztlich dazu bestimmt sind, in Begriffe und Worte gefaßt zu werden, die für die Gebildeten verständlich sind, und zu einem Bestandteil des allgemeinen Weltbildes zu werden - eine theoretische Wissenschaft, sage ich, in der dies vergessen wird und in der die Eingeweihten fortfahren, einander Ausdrücke zuzuraunen, die bestenfalls von einer kleinen Gruppe von Partnern verstanden werden, wird zwangsläufig von der übrigen Kulturgemeinschaft abgeschnitten sein; auf lange Sicht wird sie verkümmern und erstarren, so lebhaft das esoterische Geschwätz innerhalb ihrer fröhlich isolierten Expertenzirkel auch weitergehen mag. Erwin Schroedinger.
It seems as if mathematicians are simply better in proving theorems if they do not think of applications but rather believe that mathematical ideas have an independent existence, a divine beauty, pure math is superior to anything else (the queen of sciences), math is the science of absolute rigor, and so on. This may explain some of the above opinions by "pure" mathematicians. Sometimes, however, the arrogance of some mathematicians evokes replies:
Bridges would not be safer if only people who knew the proper definition of a real number were allowed to design them. Norman David Mermin, (1935 -) "Topological Theory of Defects" in Review of Modern Physics, v. 51 no. 3, July 1979.
1. The essence of mathematics is the multitude of the influences it has on society.
Mathematics is not defined by beauty, rigor, analytic thinking, and such. These are aspects. Its essence is the multitude of influences it has on society, which each generation of mathematicians defines anew. Its essence is that
- it is a language indispensable to many scientists,
- it is a clearinghouse of structural problems and their solutions used by scientists and engineers,
- it is a hidden key technology, (key technology because many recent technological breakthroughs went hand in hand with mathematical or computational breakthroughs; hidden, because users rarely see the math involved in high tech devices),
- it has an increasing influence on technology because of the abundance of computing power and the advances in software engineering technology, and last but not least,
- it is to some people an art.
1. Mathematics departments have already lost some of the most exciting areas to other scientific communities/departments: the theory of algorithms and complexity, systems theory, biostatistics, time series analysis, neural networks... It seems to me that mathematics is at a crossroads. Either mathematicians embrace all these mathematical ideas generated outside math departments and try to unify mathematics again. Or mathematics takes the same path as philosopy, which was once an important field but lost more and more content to other fields.
- The MacTutor History of Mathematics archive
- Bob Jacobs (originally at www.chemistrycoach.com/quotations.htm)
- Rasmus Pagh (originally at www.daimi.au.dk/~pagh/mathquote.html)