[whorleater]

Without proofs and rigor, aren't you just reducing mathematics to a set of rules to be memorized? Sure, you can memorize the basics of group theory, but when it comes down to constructing the Diffie-Hellman Key-exchange, you'll still the mathematical intuition derived from learning the proofs.

[Chris2048]

Maybe, but most people condense this by creating mental models of the problem, like visualisations. That's the aim.

For example, consider learning vectors, without the spacial/Cartesian visualisation as an aid. Or geometry without the visuals.

An "intuition" wrt skill can only come from experience - repeated exercises and practise. But before that another kind of "intuition" can come from a useful mental model. Maybe at some stage, working mathematicians stop using these models, but I recon:

- They helped to learn the subject, in the early stages.

- They help in simple cases.

- They are not simply abandoned, but replaced with more powerful mental models.

[whorleater]

> For example, consider learning vectors, without the spacial/Cartesian visualisation as an aid. Or geometry without the visuals.

This only works with visuals due to the relative simplicity of the topic, and simple visuals such as this are commonplace in modern textbooks and lectures. This [1], for example, is a visualization describing the one-way functions with hardcore predicates from a lecture.

However, these visualizations fall apart exponentially as you ascend the mathematical ladder of abstraction. Mathematical nomenclature becomes overburdened by many assumptions, and without proper rigor, becomes incredibly difficult and long-winded to explain. This is why newcomers find it impossible to pierce high level mathematics, each rung of the mathematical ladder builds upon the last. How would you suggest a visualization that is useful for the Kelvin-Helmholtz instability [2] for example? You can look at all the visuals and simulations you'd like on Wikipedia, but unless you're a mathematical savant you'll have to dig deep into mathematical rigor, borrowing work done by giants in the past [3]. There's really no easy shortcut to this.

> But before that another kind of "intuition" can come from a useful mental model

This mental model can be just as unhelpful as helpful. It is notoriously hard to fix false preconceived notions, and someone that develops an "intuition" that only applies as at basic level could easily lead them astray, a la the Dunning-Kruger effect. Beginning tabula rasa is often the path of least resistance, since once someone learns something /properly/ the first time, they're more likely to apply it correctly, rather than trying to apply a model that falls apart at higher abstractions. You can't really jump rungs in the math ladder, or even stave it off as a form of debt, telling yourself you'll learn it later.

[1]: https://i.imgur.com/q5KAelG.png

[2]: https://en.wikipedia.org/wiki/Kelvin%E2%80%93Helmholtz_insta...

[3]: http://www.rsmas.miami.edu/users/isavelyev/GFD-2/KH-I.pdf