[hinkley]

> author describes tensors as things behaving according to tensor transformation formula

In grade school it drove me nuts when the homework required us to describe a word without using the word (or it’s Latinate siblings). And yet as an adult there are few enough weeks that go by where some grownup doesn’t try to pull that same trick.

If you think developers are guilty of circular logic, check out some of the math pages on Wikipedia. You can get lost in moments.

[jhokanson]

Speaking of math pages on Wikipedia ... and math text more generally

Is it just me or are we horrible at teaching advanced math? Where are the examples (with actual numbers)? Where is the motivation? Where are the pictures?

[hinkley]

Randall Monroe has a comic about how most people need enough math to be able to handle a birthday dinner where the guests split the bill for the birthday boy/girl evenly and pay for their meals and tip separately.

That’s a pretty good bar and I wonder if we could just cut to that chase earlier. But I also believe that people need enough math to see when they’re being cheated, and I feel like you could just tell middle schoolers that and they would pay attention. Maybe even primary school.

You told Billy he could have three apples, and now there are two left. Did Billy take more apples than he should have?

It’s always how do you share your cookies fairly with your friends and if they’re my cookies why do I have to share them at all? Screw “fairly” I’m keeping the extras at least. That sort of sharing is a socially advanced concept they don’t entirely get just yet.

[groby_b]

Except that humanity desperately needs a better understanding of probabilities and non-linear relationships. We don't use more than division because we haven't succeeded teaching more, not because nobody needs it.

[eternityforest]

Do we actually need to know how it works, or do we just need to really deeply understand that common sense is not scientific evidence and everything we actually care about can't be predicted just by "Being smart and thinking about it"?

Actually being able to do stuff with Bayes law by hand is going to be not only hard to teach, but probably impossible to remember for those of us who don't actually do math in real life. People forget stuff after a few months or years.

I highly doubt the average person is interested in checking the math on a science paper, so if you want the general public to understand statistics you... have to show us all a reason to, and also teach us all of the related skills needed to make it useful. Or else.... we will all just forget, even with the best teacher in the world.

Most of us aren't doing random game engines as a hobby project or testing things on bacteria cultures.

Maybe they should teach it in context of how to understand a scientific paper, since that's one of the more relevant things for non-pros. If you just teach statistics alone people will say

"Ok, now I know that it's easy to lie to yourself if you don't use any numbers but I don't have collections of large numbers of data points in my life to actually analyze"

[hinkley]

"Thinking Fast and Slow" makes a quite extended argument that our brains have a faulty intuition about probabilities, and I nodded along thinking of all the bad decisions I've watched my teams make over the years (either noticed retrospectively, or presaged by myself or some other old hat).

If you have a way to fix this, you would be set for life, going around playing a sort of corduroy jacketed Robin Hood, keeping the rich from stealing from the poor.

[groby_b]

I have a corduroy jacket... so, halfway there? ;)

Realistically, it is (somewhat) fixable in small contexts. I've worked (and continue to work) on teams that are somewhat decent at risks and probabilities, but it's definitely an exceptional experience.

I don't know how to widely teach that. But I'm not yet ready to give up and say "it can't be taught", because those folks on my team are the counterexamples.

[GoatOfAplomb]

> Where are the examples (with actual numbers)?

In upper-level undergraduate math, I made a game of seeing how many pages I would go before seeing 7 printed anywhere. It was usually 10 pages, if I included the page numbers.

[ravi-delia]

What are we calling advanced math? There comes a point where I personally find it much easier to avoid examples until I'm problem-solving, since otherwise I'll get stuck in a loop of wondering if the thing I noticed generalizes. Could just be that my working memory is poor, but when I see a real honest number I know I'm in for a grueling day.

[joppy]

Wikipedia is a terrible place to learn advanced mathematics, for the reasons you raise (and more). There are lots of terrific short books, and many terrific lectures online.

[chobytes]

This is definitely a problem! Having a large set of interests and problems to draw examples and intuition from are how I deal with it. I suspect this is why so many mathematicians are also into physics.

[exdsq]

100%! For those of us who need to learn from practical examples through to generalized intuition maths can be really really hard to learn depending on the source. Wish I was one of those people who finds it easier to learn from abstract first through to implementations second.

[xyzzyz]

It's not just you, we are horrible at teaching advanced math. However, the reason for it is that advanced math is, as far as we can tell, just really, really, really hard. It's not that mathematicians don't care about teaching others (they very much do, and they try their best to get their understanding across to others), or that Wikipedia authors are particularly bad at clear exposition (they are, if anything, above average). Quite simply, we know of no royal road to understanding mathematics, you have to put in many hours to bite it in very small pieces.

Here's an example:

https://en.wikipedia.org/wiki/Homology_(mathematics)

It has motivation, examples, and even actual numbers (though they're really just 0 and 1. most of the time). In my opinion, it's very good and clear exposition, for an encyclopedic article. However, I strongly suspect that people without enough mathematical knowledge (and "enough" in this case is something in the neighborhood of "enough to obtain an undergraduate degree in Mathematics") will simply not get anything about it beyond "it's about number of holes" (and that's not even remotely close to the whole picture: homology theories are important and useful in context of things with no "holes" to speak of). If you think otherwise, but not know what a quotient group is, you're just fooling yourself.

This is something I observe on HN a lot: people don't understand advanced mathematics, and are dumbfounded by the fact, trying to blame weird notation mathematicians insist on, or lack of motivation/examples/pictures etc. I never see people here do the same with advanced physics ("if the Standard Model is so standard, why can't they briefly and clearly describe what it is" is not something I ever see), molecular biology, or material science. People seem to know their limits and understand that really grokking these fields requires many years of deep study.

I think it's because many people on HN have good experience learning mathematics at school: it was something they always grasped really easily, and were easily able to figure out how to calculate derivatives, integrals, get matrices into normal forms etc. I don't want to rain on anyone's parade, because these things are still relatively difficult, and it does require more intellectual ability and effort that probably 3/4ths of the population aren't capable of. However, relative to advanced mathematics, undergraduate calculus is really rather trivial stuff.

Point is, if you don't understand modern advanced mathematics, you shouldn't get any more disappointed than you are about not being able to play violin. These things just don't come easy.

[chobytes]

I think a lot of circularity occurs in mathematics because we don't typically qualify our utterances when it can be implicitly understood.

Eg "Numbers (formal) are those objects which behave like numbers (informal)."

[dboreham]

Bertrand Russell turns over in grave.

[CatsAreCool]

I'm working on a language MathLingua (www.mathlingua.org) whose goal is to precisely describe mathematics using a format that is easy to read and understand to help address ambiguity in mathematical texts written using natural language.

It is still a work in progress, but does it help address some of the problems you see in learning mathematics? Any feedback is greatly appreciated. Thanks.