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Comment by mtizim | Hacker News Reader

With no mathematical rigor there is no mathematical understanding. You are robbing yourself, as the concepts are meaningless without the context.

Truly appreciate the power of linear approximations by going through algebra, appreciate the tricks of calculus, marvel at the inherent tradeoffs of knowledge with estimator theory, and see the joy of the central limit theorem being true. All of this knowledge is free, and much more interesting than a formal restatement of "it was not supposed to rain, but I see clouds outside, I guess I'll expect light rain instead of a big thunderstorm".

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> With no mathematical rigor there is no mathematical understanding. You are robbing yourself, as the concepts are meaningless without the context.

I will think more about this, but I'm not sure I agree. I have enjoyed reading Feynman talk about twins and one going on a supersonic vacation without understanding the math. Verisimilitude allows a modeling of understanding with a scalar representation of scientific knowledge, so why not?

Of course I would like to understand the math in its purest forms–just the same as I wanted to read 1Q84 in Japanese to be able to fully experience it in its purest form, but my life isn't structured in a way were that is realistic even if the knowledge of the Japanese language is free.

> Truly appreciate the power of linear approximations by going through algebra, appreciate the tricks of calculus, marvel at the inherent tradeoffs of knowledge with estimator theory, and see the joy of the central limit theorem being true.

I can't even foil so the journey toward understanding can feel unattainable in the time resources I have. This absolutely may be a limiting belief, but the concept of knowledge being free ignores the time cost for some exploring these outside of academia or professional setting.

Indeed everything has an opportunity cost, and every life has its own priorities.

Since you mention Feynman, I would like to observe that many expositors who target the lay audience have the skill of making the audience believe that they have comprehended(1) something of an intellectual world that they have no technical grounding to truly comprehend(2). In my view these are two distinct types of comprehension/understanding. So long as the audience is clear on which type of understanding they are getting, and is not wasting time unwittingly pursuing one type at the expense of the other then I see no harm.

There is a risk however, that the pop expositors will put you in a headspace where even if you are faced with accessible, but type 2, material you will not be familiar with what really constitutes understanding. As a mature age student it took me quite a few years of maths exams to switch from 1 to 2. Nowadays I am more comfortable with admitting that I don't understand some piece of math (for that is the first step on the path to learning) than being satisfied with a pop-expository gist.

I've thought a lot about this exact topic. You need both to do well.

You need handwavy and vague versions of things to understand the shape of them and to build intuition.

Then you need to test the intuition and build up levels of rigor.

Especially in the context of the Kalman Filter. I just helped a bunch of middle school students build a system for field localization and position tracking. They don't have all kinds of knowledge. They don't have linear algebra or a real understanding of something being gaussian and have to have a bazillion variables. They understand that their estimates and the quality of stuff coming off their sensors have different qualities based on circumstances, and that gain needs to vary. They'll never hit the optimum parameters.

But: their system works. They understand how it works (even if they don't know how to quantify how well it works). They understand how changing parameters changes its behavior. When they learn tracking filters and control by root locus and all kinds of things later, they'll have an edge in understanding what things mean and how it actually works. I expect their intuition will give them an easier time in tackling harder problems.

Conversely, I've encountered a bunch of students who know what "multimodal" means but couldn't name a single example in the real world of such a thing. I would argue that they don't even know what they're talking about, even if they can calculate a mixture coefficient under ideal conditions.

There's a lot of fluffly language here that isn't saying much.

Linear algebra is not something that takes years of patient study to gain basic competency. It had almost no prerequisites and can be understood enough to understand least squares in a focused weekend or two.

Thank you for the encouragement. I'll will take a week or two and spend some time with some focused learning. Do you have any recommendations where to start?

> With no mathematical rigor there is no mathematical understanding

While I appreciate rigor to really know deep details, is not only not a requirement for understanding, but a hurdle. A terrible insurmountable hurdle.

To first have understanding, I need some kind intuition. Some explanation that makes sense easily. That explanation is btw, what typically the inventor or discoverer had to begin with, before nailing it down with rigor.

> With no mathematical rigor there is no mathematical understanding. You are robbing yourself, as the concepts are meaningless without the context.

You don't need to know what Gravity is to calculate the time it takes for an apple to fall from a tree. You just need to accept that g=9.8m/s2.

You also don't need to understand the chemistry of flour, salt, sugar, sodium, milk and eggs to bake a cake.

None of these are needed, or even useful, for understanding the Kalman filter.

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