[nkrisc]

> The next paragraph about how mathematics was closely coupled to reality for most of history and only recently with our understanding of infinite sets became too abstract is not really at all accurate of the history of mathematics. Euclid's Elements is 2300 years old and is presented in a completely abstract way.

I may be off-base as an outsider to mathematics, but Euclid’s Elements, per my understanding, is very much grounded in the physical reality of the shapes and relationships he describes, if you were to physically construct them.

[empath75]

Quite the opposite, Plato, several hundred years before Euclid was already talking about geometry as abstract, and indeed the world of ideas and mathematics as being _more real_ than the physical world, and Euclid is very much in that tradition.

I am going to quote from the _very beginning_ of the elements:

Definition 1. A point is that which has no part. Definition 2. A line is breadthless length.

Both of these two definitions are impossible to construct physically right off the bat.

All of the physically realized constructions of shapes were considered to basically be shadows of an idealized form of them.

[kannanvijayan]

Another point to keep in mind is that a lot of mathematics that's not considered abstract _now_ was definitely considered "hopelessly" abstract at the time of its conception.

The complex number system started being explored by the greeks long before any notion of the value of complex spaces existed, and could be mapped to something in reality.

[griffzhowl]

I don't think we can say the Greeks were exploring complex numbers. There's something about Diophantus finding a way to combine two right-angled triangles to produce a third triangle whose hypotenuse is the product of the hypotenuses of the first two triangles. He finds an identity that's equivalent to complex multiplication, but this is because complex multiplication has a straighforward geometric interpretation in the plane that corresponds to this way of combining triangles.

There's a nice (brief) discussion in section 20.2 of Stillwell's Mathematics and its History

[mrguyorama]

Hell, 0 used to be considered too abstract!

[griffzhowl]

Plato was only about a generation before Euclid. Their lives might have even overlapped, or nearly so: Plato died in 347BC and Euclid's dates aren't known but the Elements is generally dated ~300BC