I'm just starting up the academic year (yes, it starts in February here in the Southern Hemisphere) teaching a 2nd-year Introduction to Pure Mathematics class.

For general background, I would like to explain the general concept of axioms and the role of axiomatisation in Pure Mathematics, and refer them to a suitable reasonably-accessible essay for more information.

Googling has led me to an oldish paper by De Villiers (mzone.mweb.co.za/residents/profmd/axiom.pdf) which is quite interesting. In particular he distinguishes between "constructive axiomatisation" in which (essentially) the axioms come first and then logical deduction creates new mathematics and "descriptive axiomatisation", where new mathematics that has arisen in an untidy, organic, uncontrolled way is reorganised and systematised with the benefit of hindsight (in other words, which ideas led somewhere, which didn't).

In particular, he mentions criticisms of the "axiomatic-deductive" method in TEACHING (i.e. Lecture 1 gives the axioms for a vector space and proves that 0 is unique, etc etc) because it conveys - inaccurately - to students that mathematics is just the dry mechanical manipulation of statements starting from some "god-given" or "magic" axioms that are simply presented as a fait-accompli, rather than a creative, iterative, untidy process.

But I can't just give the students this link, because it assumes that the reader already KNOWS examples of descriptive and/or constructive axiomatisation, and it also uses language that I predict my students would not enjoy. (Example: Freudenthal (1973, 451) refers to this dramatic change in style of thinking, as the "cutting of the ontological bonds.")

To a large degree, time constraints will force me to continue teaching essentially used the axiomatic-deductive method, but I would really like to alert the students to the fact that this is just one side of the coin.

Anyway, it seems to me that this topic is something that surely has faced many many teachers of Pure Mathematics before, and that there should be an essay somewhere that is aimed at beginning students, rather than their professors.

So my question is: Is there an essay, aimed at beginning students, that explains in simple terms the role of axioms and axiomatisation in mathematics?

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    $\begingroup$ Great question. I like it that you're trying to show students that mathematics is not "god-given". $\endgroup$ Feb 28, 2015 at 9:57
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    $\begingroup$ Here are two references that do not answer your query, as they are aimed more toward professors. Still, interesting: Richard Wells, "Mathematics and Mathematical Axioms," 2006 (PDF download). Kenny Easwaran, "The role of axioms in mathematics," 2008 (JSTOR link). $\endgroup$ Feb 28, 2015 at 18:59
  • $\begingroup$ How about RL Wilder's classic Introduction to the Foundation of Mathematics? (If this does the trick, then maybe I will post it as an answer.) $\endgroup$ Jun 24, 2015 at 4:24

1 Answer 1


Only one thing came to my mind reading your post: Concepts of Modern Mathematics written by Ian Stewart. It was an eyeopener for undergraduate me, and it is now for many of undergraduate students I teach. It is about mathematics as a whole, including a section on axioms and axiomatisation (or, as it is "Axiomatics"). To make you curious, here is how that section starts:

Only an elephant or a whale gives birth to a creature whose weight is 70 kilograms or more. The President's weight is 75 kilogram. Therefore, the President's mother was either an elephant or a whale.

  • $\begingroup$ I'll certainly look that one up.. I quite like Ian Stewart's writing and the table of contents reveals lots of interesting and relevant chapter titles. $\endgroup$ Mar 1, 2015 at 9:14

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