Definitions of Math

Here are excerpts from an article by Dr. Catalin Barboianu which talks about why it is difficult to give a good definition of math. I thank Dr Barboianu for allowing me to quote from his article.

Dr. Catalin Barboianu, is the author of the book “What is Mathematics: School Guide to Conceptual Understanding of Mathematics”.

The entire article is available at -

https://medium.com/@cb_67963/25-famous-definitions-of-mathematics-and-why-they-cannot-define-it-4eabc24aab42 - You can rest assured about the definition — mathematics, for which rigorous definitions stand as a firm principle, does not have a definition of its own!

We can “do” mathematics, and also we can learn mathematics, without knowing what mathematics is, just as we talk to one another without knowing everything about the nature of natural language, or as we use our human reason without contemporary science having revealed its nature and all its hidden secrets.

This latter comparison is not accidental, because the elementary mathematical concepts as well as the methods of mathematics are embedded in human reason in a sense as biological and evolutionary as possible.

Searching on the internet, you will find various definitions, some of them beautiful from a literary-stylistic point of view, or with great philosophical substance, but certainly you will not find a definition that reflects the nature of mathematics in its entire complexity.

My own definitions of mathematics, is given as the 25th.

1. “Mathematics is the music of reason.” (Paul Lockhart)

2. “Mathematics is the poetry of the universe.” (Jonathan David Farley)

3. “Mathematics is the queen of the sciences.” (Carl Friedrich Gauss)

4. “Mathematics is the discipline dealing with nontrivial and interesting things.” (Jody Azzouni)

5. “All that is correct thinking is mathematics.” (Grigore Moisil)

6. “Mathematics is a game played according to certain simple rules with meaningless marks on paper.” (David Hilbert)

7. “Mathematics is the door and key of the sciences.” (Roger Bacon)

8. “Mathematics is the science of what is clear by itself.” (Carl Jacobi)

9. “Mathematics is as much an aspect of culture as it is a collection of algorithms.” (Carl Boyer)

10. “Mathematics is the art of giving the same name to different things.” (Henri Poincaré)

11. “Mathematics is the supreme judge; from its decisions there is no appeal.” (Tobias Dantzig)

12. “Mathematics is not a deductive science — that’s a cliché. When you try to prove a theorem, you don’t just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.” (Paul Halmos)

13. “Mathematics is the tool especially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.” (Paul Dirac)

14. “Mathematics is not only real, but it is the only reality.” (Martin Gardner)

15. “Mathematics is an art of human understanding.” (William Thurston)

16. “Mathematics is pure language — the language of science. It is unique among languages in its ability to provide precise expression for every thought or concept that can be formulated in its terms.” (Alfred Adler)

17. “Mathematics, however, is, as it were, its own explanation; this, although it may seem hard to accept, is nevertheless true, for the recognition that a fact is so is the cause upon which we base the proof.” (Gerolamo Cardano)

18. “Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality.” (Richard Courant)

19. “Pure mathematics is, in its way, the poetry of logical ideas.” (Albert Einstein)

20. “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)

21. “The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.” (Bertrand Russell)

22. “Mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the actual world, but every possible world, must conform.” (Bertrand Russell)

23. “Mathematics is, I believe, the chief source of the belief in eternal and exact truth, as well as a sensible intelligible world.” (Bertrand Russell)

24. “Mathematics is the most beautiful and most powerful creation of the human spirit.” (Stefan Banach)

25. ‘Mathematics is simultaneously a discipline, a method, and a language, applicable to other disciplines or methods of knowledge, but also to mathematics itself. It is a mode of organizing, validating, and communicating knowledge, which uses the principles of logic and gives it a field of manifestation and application that is “pure,” free from contingency, and free from the confusion of natural language, in which the truths of propositions are universally valid and do not depend on any interpretation. It is the only discipline entitled to investigate concepts of any nature and any degree of generality, but also to be self-referential and self-applicable, as is in fact human reason.’ (Catalin Barboianu)

It is just another way to reflect the amazing complexity of mathematics.