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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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    $\begingroup$ I don't think Badiou is making a statement about effective pedagogy, but rather foundational dependence. $\endgroup$
    – ziggurism
    Dec 18, 2017 at 3:24
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    $\begingroup$ 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 $\endgroup$
    – eepperly16
    Dec 18, 2017 at 3:25
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    $\begingroup$ 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. $\endgroup$
    – CyclotomicField
    Dec 18, 2017 at 4:44
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    $\begingroup$ @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. $\endgroup$
    – John Coleman
    Dec 21, 2017 at 14:24
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    $\begingroup$ "Why is set theory not taught at the outset of math education?" simple: because the easier topics are done before the harder topics. $\endgroup$
    – Gerald Edgar
    Dec 21, 2017 at 15:14

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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.

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  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)

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    $\begingroup$ 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. $\endgroup$
    – user507
    Dec 22, 2018 at 14:46
  • $\begingroup$ 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? $\endgroup$
    – Timothy
    Dec 24, 2018 at 17:37
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Note that set theory when used to present ideas in mathematics makes the work of reading and intuitively verifying these harder. Several "such thats" or "conditions thats" in any work may well enable and inform the work of subsequent metamathematics , but intuitions are needed to make maths intelligeable and applicable rather than formalisms , which still require further work to understand them!

This will perhaps be why in particular set theory is preferred as a TOOL In books where study of fundamental maths ideas is the Intention, rather than application or origination of math. A guess would be that Wikipedia's theoretically inclined approach to giving definitions in terms of sets has distorted perceptions of this?

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Set theory, as well as being a subject in its own right, is a convenient language in which most of mathematics can be expressed.

English, as well as being a subject in its own right, is also a convenient language in which most of mathematics can be expressed, but nobody would ask "Why isn't English taught at the outset of math education?"

When learning a new subject, it's useful to go somewhat in order of historical development. That's one reason why we learn whole numbers before trigonometry, for example.

Expressing the fundamentals of every part of mathematics in terms of sets is something that people have only really been doing for the last century or so. It's quite a sophisticated point of view which doesn't fit very well with understanding what mathematical objects actually are.

For example, the concept of ordered pair is much easier to understand if it's presented in the usual way ($(a,b)$ is a thing with the property that $(a,b) = (c,d)$ if and only if $a=c$ and $b=d$) rather than defining $(a,b)$ as the set $\{\{a\}, \{a, b\}\}$.

On a personal note, I can remember in primary school doing some set theory with strings and objects.

You put the objects on the floor and had to put loops of string around some of the objects to create sets. It all seemed rather boring and pointless. It wasn't until I started university, or maybe the final year of high school, that I realised what sets had to do with mathematics at all.

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