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Sunday, August 12, 2018

Fundamentals of mathematics


Photo source: Wikimedia Commons, Louvre Museum

Photo license: CC BY-SA 2.0 FR


This post is a collection of quotes about mathematics. There are 17 quotes divided into 6 sections.


A. Mathematics is the study of numbers, lines and shapes (3)

B. Mathematics is not the material that reality is made out of (2)

C. Mathematics is a catalog of representations (3)

D. Geometrical descriptions cannot be exact to physical reality (3)

E. Mathematical truths are discovered within experience (3)

F. Mathematical equations are logic (3)


A. Mathematics is the study of numbers, lines and shapes


W. E. B. Du Bois (1868-1963, sociologist):

1. "When you have mastered numbers, you will in fact no longer be reading numbers, any more than you read words when reading books. You will be reading meanings." (


John Dee (1527-1608, mathematician):

2. "It is by the straight line and the circle that the first and most simple examples and representation of all things may be demonstrated..." (Monas Hieroglyphica, 1564)


Graham Flegg (mathematician):

3. "Geometry is the most visual of the mathematical disciplines." (Numbers: Their History and Meaning, 1983)


B. Mathematics is not the material that reality is made out of


Ernest Rutherford (1871-1937, physicist):

4. "When we have found how the nucleus of atoms is built up we shall have found the greatest secret of all - except life. We shall have found the basis of everything - of the earth we walk on, of the air we breathe, of the sunshine, of our physical body itself, of everything in the world, however great or however small - except life." (Quoted in The Wit and Wisdom of the 20th Century by Frank Pepper)


Freeman Dyson (1923-now, physicist):

5. "The real world is muddy and messy and full of things we do not yet understand." (Missed Opportunities, 1972)


C. Mathematics is a catalog of representations


Tobias Dantzig (1884-1956, mathematician):

6. "The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit." (Number: The Language of Science, 1930)


Michio Kaku (1947-now, physicist):

7. "Mathematics... is the set of all possible self-consistent structures..." (Hyperspace, 1995)


John von Neumann (1903-1957, mathematician):

8. "A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so." (The Role of Mathematics in the Sciences and in Society, 1954)


D. Geometrical representations cannot be exact to physical reality


William Whewell (1794-1866, philosopher):

9. "And so no force however great can stretch a cord however fine into a horizontal line which is actually straight." (Elementary Treatise on Mechanics, 1819)


John von Neumann (1903-1957, mathematician):

10. "If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." (Remarks at the Association for Computing Machinery Annual Meeting, 1947)


Richard von Mises (1883-1953, mathematician):

11. "I am prepared to concede without further argument that all the theoretical constructions, including geometry, which are used in the various branches of physics are only imperfect instruments to enable the world of empirical fact to be reconstructed in our minds." (Probability, Statistics and Truth, 1957)


E. Mathematical truths are discovered within experience


John Locke (1632-1704, philosopher)

12. "No man's knowledge here can go beyond his experience." (An Essay Concerning Human Understanding, 1689)


G. H. Hardy (1877-1946, mathematician):

13. "I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations', are simply our notes of our observations." (A Mathematician's Apology, 1941)


Leonhard Euler (1707-1783, mathematician):

14. "Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations." (Quoted in Induction and Analogy in Mathematics by George Polya)


F. Mathematical equations are logic


Bertrand Russell (1872-1970, philosopher):

15. "Pure mathematics consists entirely of assertions to the effect that, if such and such proposition is true of anything, then such and such another proposition is true of that thing." (Mathematics and Metaphysics, 1917)


Gottlob Frege (1848-1925, philosopher):

16. "Arithmetic thus becomes simply a development of logic and every preposition of arithmetic a law of logic albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction." (The Foundations of Arithmetic, 1893)


Robert Chambers (1802-1871, geologist):

17. "The study of mathematics, generally, is also of great importance in cultivating habits of exact reasoning; and in this respect it forms a useful auxiliary to logic." (Chamber's Information for the People, 1875)