What is foundations of mathematics ? And why isn't it taught more often?

I have just come across "Foundations of Mathematics". Wikipedia describes it as

Foundations of mathematics can be conceived as the study of the basic mathematical concepts (number, geometrical figure, set, function, etc.) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their models giving a meaning to formulas, definitions, proofs, algorithms, etc.) also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics.

Now this seems a bit strange to me. If foundations of mathematics really do deal with the study of basic mathematical concepts shouldn't they be taught to us as the first thing ? Also in theory wouldn't this mean that someone who has a good grasp of foundations of mathematics will have a very good understanding of mathematics as a whole ?

If that is true then why isn't foundation of mathematics taught at a earlier stage in life and more often ?

I apologize for the over simplification, but I am not an ace mathematician. Hence I want some insight into what foundations of mathematics are.

• "why isn't foundation of mathematics taught at a earlier stage in life ?" Because "basic" math concepts are quite abstract and not intuitive, and thus quite difficult to grasp before having a deep knowledge of math. – Mauro ALLEGRANZA Apr 18 '17 at 15:48

In ancient Greece, Euclid proposed an idea mathematicians have liked (in principle) ever since: have assumptions called "axioms" we take for granted, then prove "theorems" from them. In theory, you could write a list of all the axioms of modern mathematics, then take it from there. Euclid had a few axioms of geometry, as if to show us how it's done.

However, until c. 1900 mathematicians rarely took that approach. They were very careful about the way results were proven, but pretty much intuited the starting points. Nowadays, they can say, "Number theory has the Peano axioms" or "All mainstream mathematics has a model in ZFC" (a version of set theory), or something like that; but such axioms were designed to capture what we were already doing, and today when you teach people these axioms it's the familiarity of the implications that makes them palatable.

There was actually a teach-foundations-to-children experiment called the New Math, but it went badly. The basic problem was that, even though theoretically you'd think the easiest way to understand the subject is to start from axioms then see where they lead, in practice that's not a particularly intuitive way to do it, if only because you need some quite heavy-handed machinery even to state the starting points when they're done properly. For example, how do you found the idea of real numbers as a continuum? One option is as Dedekind cuts on rational numbers; another is as equivalence classes of Cauchy sequences of rational numbers. Yeah: good luck explaining that in primary school, even if rational numbers have been explained or taken for granted.

We've had much more success teaching mathematics a very different way that more closely (though not exactly) parallels the history of ideas' development. First you teach natural numbers, then you introduce fractions, then you get to surds and complex numbers and so on. (I'm skipping topics other than "types of numbers" to simplify the timetable.) The reason this works, presumably, is because each level of the resulting education provides the right context to understand the problems that motivated the next development in mathematical theory. But to motivate the foundation of mathematics the same way, it turns out you have to natter about some very abstract and complicated ideas like consistency and decidability.

• I don't think we know that axiomatics originates with Euclid. Indeed, we know very little about the context of Euclid. We have other examples of axiomatics that are approximately contemporaneous without attribution to Euclid: Archimedes states the axiom we normally name after him as an axiom in On the Sphere and the Cylinder, and he attributes it to Eudoxus, on which Euclid's theory of proportion is definitely based. There are several references to pre-Euclidean "Elements", but we have absolutely nothing about what their structure is. – Chappers Apr 17 '17 at 18:36
• Just because one extremely-poorly-thought-out approach failed does not imply that there is no good approach. Just saying... – user21820 Nov 7 '17 at 10:27

Let me put it this way: if you had to learn precisely how an engine works before you could drive a car, there will be very few drivers on the road. Mathematicians mainly drive cars rather than analyzing the functioning of the engine. For some details on the metaphor of set theory as the automobile, see this article.

• I used to think mathematicians were the ones analyzing the engine, while the "non-math" people were just driving the car. – ng.newbie Apr 21 '17 at 13:34
• Well, it depends what you mean by the car. If the car is, say, calculus, then sure, you have plenty of physicists, engineers, and other scientists exploiting it rather than tinkering with the "engine". On the other hand, if the car is set theory, then most practicing mathematicians today are mostly ignorant of it. For some details on the metaphor of set theory as an automobile see the article linked in my answer. – Mikhail Katz Apr 23 '17 at 8:05