In mathematics education we can find mention of *threshold concepts*. These are mathematical concepts that:
1) are notoriously difficult to grasp, both intuitively, conceptually and computationally; 2) are central and critical for furthering one's mathematical understanding; 3) eventually become quite easy ("How is it I could not understand it before?"), because we employ them so much that any mathematician will internalize them after a while.
Setting bar for these three criteria high, four threshold concepts come to my mind:
- basic algebra (it is well-known that many children struggle a lot with middle school maths when transitioning for arithmetics to algebra); - differentiation and integration (AFAIK, differentiation seems more difficult of these two for most students, because it makes them think about graphs in a novel way); - delta-epsilon arguments in real analysis (as 99% undergraduate students in maths can confirm :)); - forcing in advanced set theory (I know nothing about it myself, but I have read several places that it can be a backbreaker; but I am not sure whether it satisfies (3)).
Do you know other examples of such threshold concepts in mathematics?
hiAndrewQuinn3 days ago
I can basically confirm all of those. The delta-epsilon definition of a limit was what really kick-started my interest in mathematics, but it was also so brutal to wrap my head around.
I would say the concept of a group is another one for a lot of people. Abstract algebra is sometimes the first time you take a math class in undergrad where all references to numbers are very far away, and you have to accept that the definition of a group contains vast and untold multitudes beyond its straightforward definition. When it "clicked" for me, however, everything else in that class started flowing much better.
Gauge invariance in mathematical physics might be another good one, but that's a lot more niche in who actually learns it. Lyapunov stability from dynamical systems is something I'd point out as actually the opposite of this - it makes a lot of sense very quickly and helps make light of quite a few things you see in the dynamical systems course material leading up to it.
Someone3 days ago
“Infinities are weird” might satisfy your criteria.
It surfaces in such things as
- why do we consider 0.9̅ and 1 to represent the same number?
- why are there ‘as many’ even integers as there are integers, and ‘as many’ integers as there are natural numbers?
- why are not all infinites equal?
If you don’t know/accept these, you’ll keep using intuition that served you well for many, many problems on finite sets on problems involving infinite sets, with disastrous results.
usgroup3 days ago
Probability theory. It is closely aligned to my intuition now, but when I first learned it, it was difficult to accept beyond manipulating formulae.
Linearity aka most of linear algebra. Again, beyond manipulating formulae, many concepts eventually become intuitive with enough application, but its a hard won intuition to acquire.
blackbear_3 days ago
I've heard compactness described as one such concept.
https://math.stackexchange.com/questions/485822/why-is-compa...