# Why is set theory not taught at the outset of math education?

A beginner in math, reading Badiou, I found the following quote on set theory in Being and Event:

The axiomatization consists in fixing the usage of the relation of belonging, $\in$, to which the entire lexicon of mathematics can finally be reduced, if one considers that equality is rather a logical symbol.

If Badiou and so many others have such high esteem for the fundamental importance of set theory, why is it not being taught at the very outset of math education? I understand Badiou is, after all, a philosopher. I am just an idiot without a teacher, wanting to comprehend.

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• I don't think Badiou is making a statement about effective pedagogy, but rather foundational dependence. – ziggurism Dec 18 '17 at 3:24
• Most applications of math and indeed most "pure" math is done at a much higher level of abstraction than fundamental set theory. In principle, for example, functions $f : X \to Y$ are sets, $f \subseteq \mathcal{P}(X \times Y)$, but in practice we rarely think of them as such – eepperly16 Dec 18 '17 at 3:25
• Decisions of curriculum for public education are political and subject to the whim of politicians and constituents alike. Here you can graduate from high school without even knowing what mathematics is, let alone doing any of it, because that's how people vote. – CyclotomicField Dec 18 '17 at 4:44
• When I was in U.S. grammar school in the 70's, basic set concepts/notation were taught around the 4th grade or so. This got slagged later as "new math" because people are dumb. – Daniel R. Collins Dec 20 '17 at 19:42
• @DanielR.Collins The phrase "new math" was invented by the proponents of teaching set theory to young children, not a later insult coined by their opponents. I learned such things as early as the first grade in the late 60s. I rather liked it but can understand the arguments against it. Since leading critics of the new math included Richard Feyman and Morris Kline, I hesitate to dismiss then as simply "dumb". I do think that there was an overreaction against it. Introducing it in elementary school and waiting until college are not the only choices. – John Coleman Dec 21 '17 at 14:24

Not a complete answer (could there be one?), but too long for a comment.

why is it [Set Theory] not being taught at the very outset of math education?

It has been tried, most likely still is in certain places at different degrees. For some history and background, lookup the New Math of the '60s, possible keywords Belgium, Willy Servais, Georges Papy. Unfortunately, many of the relevant papers are behind paywalls now, but quoting from one which is not (Geert Vanpaemel - Belgian contributions to the New Math movement in Europe ...):

Servais emphasized the concept of a pédagogie ouverte, an open approach to the learning process. [...] He wrote with enthusiasm about the choice of set theory as the foundation of mathematical education, and in particular lauded Papy’s use of Venn diagrams and arrow-graphs, which were, in his opinion, ideally suited to the mind of young children. He also endorsed their attempts, criticized by some, to bring the New Math to primary schools. [...] Servais was convinced that the basic unity, acquired by the use of set theory, provided the solution to the pedagogy that he had in mind: as the goal of mathematics teaching was to activate the mind of the child towards grasping the mathematical structures in the world around him, it was necessary to define these structures and to make them the backbone of the whole syllabus.

1. Because many worthwhile topics in math don't require set theory to learn them and apply them. You can get a lot of good stuff done with arithmetic, algebra, trig, calc, etc. that is uninformed by set theory.

2. Because set theory is hard for 5 year olds just because of the development of their intellect.

3. Because it is easier to learn set theory already having some experience in other areas of math. (E.g. real, rational, complex numbers)

• Because set theory is hard for 5 year olds just because of the development of their intellect. I don't think this is quite accurate. There's nothing inherently that hard about the baby set theory that they used to teach in grade school when I was a kid in the US in the 70's. – Ben Crowell Dec 22 '18 at 14:46
• Do you think it's because they don't want to teach Naive set theory because it's contradictory and Zermelo-Fraenkel set theory and New Foundations are so much harder to teach? – Timothy Dec 24 '18 at 17:37