[Chris2048]

What I actually need is "learning math in a short time",

With a few, minimal yet illustrative examples (ala katas), plenty diagrams/illustrations and other mental aids, an no rigour - not a single bit of set-theory!

The proofs and rigor can come later...

Incidentally, I'm a dev in finance, looking to move into quant dev. I have a math degree (completed 2007) and I'm doing a "CQF" to catch up with the relevant quant knowledge. I think cyptography, and stats/data analytics might also be a good area for mathy-dev.

[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

[heimatau]

I agree with this sentiment. I'm currently pursuing a Bachelor's in Pure Mathematics (or called 'Theoretical Math', eventually I'd like to go far as a PhD in it). I think the ideas of Math could be taught in a condensed way. Maybe it's already done but...education needs to be disrupted in order to do this.

My current idea is that Math could be taught as a language and taught as a critical thinking class. A condensed class would like like 'this is an equation...here is what we can do with it (derivatives, areas/3D/4D, etc)...but...99.99% of you won't need to know it this way. You need to use math in a way that indirectly teaches you how to creatively look at problems in life.'

I'm not sure why everyone is forced to learn math without knowing WHY they are forced to know it. Creative problem solving is one of the best takeaways, imho, for the masses.

As for Adv. Math...I think it's not effective for most people's career paths and skillset they will require in the real world.

[mmarx]

That might suffice for solving real-world problems, but not for doing mathematics itself: Intuition will get you a long way, but for working out some of the finer details, you'll have to resort to rigour. Furthermore, without having gone through the rigorous training, you might not even know when your intuition doesn't reach far enough.

Terence Tao[0] put it this way:

»The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture.«

[0] https://terrytao.wordpress.com/career-advice/there’s-more-to...

[heimatau]

I think it's pretty clear, in my reply, I'm talking about the masses need for Mathematics. Which is for 'solving real-world problems'.

I agree with Terence Tao's sentiments.

Math, for the masses, is a great way to abstractly teach the masses how to critically think about things. Math, for the masses, shouldn't get bogged down in the rigour. But if one were to go on to Adv Math, then yes, rigour is needed and demanded of the mathematician.