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Ask HN: How do I improve my command of mathematical language?

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Re: Ask HN: How do I improve my command of mathematical language?

#41

Post by agumonkey »

sinuhe69 wrote: post #25

One problem with mathematical difficulty is its “invisibility”. We can’t see how other people are thinking, how they solve the problem, how they struggle. And sometimes, we cannot even explain our thinking! That’s why making thinking visible is so important! Because only then we can learn from the others, they can learn from us and we can see where it’s right and where is wrong. One way to making thinking visible is…

I'm not a mathematician and not good at it either, but something that I see a lot is that mathematics in general are very keen on sharing the conclusions in concise, if not cryptic, forms.

A lot of metamathematics are lost in the process.. but I think that people that are really good in math like this way of life. But it kinda hides a part of what doing mathematics is. It should be written as books preludes.

Re: Ask HN: How do I improve my command of mathematical language?

#42

Post by nathanmcrae »

sinuhe69 wrote: post #25

One problem with mathematical difficulty is its “invisibility”. We can’t see how other people are thinking, how they solve the problem, how they struggle. And sometimes, we cannot even explain our thinking! That’s why making thinking visible is so important! Because only then we can learn from the others, they can learn from us and we can see where it’s right and where is wrong. One way to making thinking visible is…

My favorite math professor was great because any time he would forget the statement of a theorem or the outline of a proof he wouldn't just go to his notes to remember it, he would out-loud reason his way to the correct version. This would include the minor missteps he made on the way.

Best math lecturing experience I have had by far.

Re: Ask HN: How do I improve my command of mathematical language?

#43

Post by diffeomorphism »

> but if you asked me to explain what I'm doing I wouldn't be able to tell you, at least not using mathematical constructs/words

That can come later, the more important point is to explain "why" you are doing a specific step and not "what" and you are free to use just English words.

For example, say you want to find the maximum of f(x)= -x^2 + 4 x + 10. A bad explanation would be some mumbling about taking down the 2, ... , f'(x) =-2x +4, so x=2, f''(x)=-2, is negative, done.

Better: We know that at a maximum the slope of f has to be zero (otherwise we could go left or right to get larger), so let's find such points first. We thus compute the derivative function as f'(x)=-2x+4 and find that it is zero if x=2. But that does not give an answer yet, because it could still be a min/max or saddle point. .. draw a small picture of each. We thus compute the second derivative to see in which of these three cases we might be. f'(x)=-2 is negative so we locally look like a parabola opened downward, so this is indeed a maximum.

Re: Ask HN: How do I improve my command of mathematical language?

#44

Post by Ded7xSEoPKYNsDd »

Here in Germany early college math classes were mostly a rehash of everything from school, but with much more rigor and sometimes more detail.

So I'd recommend some college textbooks. If you live near a college, their library should have a ton of them to browse through and lend. You should also be able to find accompanying lectures on YouTube or college web sites.

Re: Ask HN: How do I improve my command of mathematical language?

#45

Post by zornthewise »

redvenom wrote: post #22

I am a mathematician. I would say the absolute best thing you can do is take the problems you DO know how to solve and write out the solutions, using complete sentences. If you can't explain some part, then you will quickly discover which parts you are following by rote. At that point you need to go back and understand the process. It will be a process of writing and writing, revising, and understanding. It's not an…

Mathematician too. I would say we still don't really know what a set is. ZFC "explains" sets in terms of... axioms and models of them which are also just sets? Should I worry about this recursion? Who knows, we just pretend things work.

Re: Ask HN: How do I improve my command of mathematical language?

#46

Post by nabajour »

tikej wrote: post #37

Maybe slightly radical idea for beginner, but since we’re on hackernews. Try to write down solutions to known, quite easy problems (that you are able to solve easily by hand) step-by-step (line by line) using SymPy ( https://www.sympy.org/ ) library (or any other computer algebra system; but SymPy is most well documented, free and most beginner friendly IMO). It may seem counterintuitive at first (you want to do lear…

If the issue is a lack of way to structure the explanation or missing the justification to go from one step to the next, having a structure that forces one to be explicit helps.

Other than sympy, a technique for formal proof might help.

One of my buddies showed me this talk from Leslie Lamport that I found interesting [1], albeit in french. I think this one [2] is an english version. And the paper behind is here [3], linked from Lamport's website [4].

Lamport makes a structured way to write proof so that all details can be written down. It seemed to me that a lot of knowledge and steps stay implicit in proofs. So if you're not familiar with the subject, you easily get lost.

[1] https://www.youtube.com/watch?v=k-i7y0R_-KE

[2] https://www.youtube.com/watch?v=uBiJpip9NVc

[3] http://lamport.azurewebsites.net/pubs/proof.pdf

[4] http://lamport.org/

edit: fixed links and text structure

Re: Ask HN: How do I improve my command of mathematical language?

#47

Post by Micoloth »

zmmmmm wrote: post #28
LudwigNagasena wrote: post #15

Earlier quoted context omitted.

> integrals and sums with no subscripts / unclear scoping, etc That’s just seems like bad writing. Out of math subjects I learned (discrete math, linear algebra, real analysis, statistics), only in the last one it may be unclear what’s going on, whether you are dealing with a random variable or its realization, and arguments of random variables are almost always omitted, so they don’t look like functions.

So here's an example on Wikipedia [1] https://wikimedia.org/api/rest_v1/media/math/render/svg/6c66... Does the sum include both terms or only the first? I happen to know what Covariance is, but if I didn't I wouldn't be clear. I'd probably guess right, but scroll down a bit to this: https://wikimedia.org/api/rest_v1/media/math/render/svg/e634... Is the +2 a multiplier on the second term or additive within the first s…

I absolutely do find infuriating when math is written down vaguely/confusingly/poorly, and ML papers are a great example of this happening way too often.

BUT, for your specific examples (in case that is an actual question):

The first one can be parsed using the fact that y_i in the second term wouldn't make sense if it was outside the summation, since the i index is only defined inside

The second (and actually, the first one, too) uses the very well established convention that multiplication always has precedence over summation, in every kind of expression. So the 2 multiplies the second sigma, and then the product is summed to the first sigma. This rule should make a lot of cases much clearer.

Re: Ask HN: How do I improve my command of mathematical language?

#48

Post by LudwigNagasena »

zmmmmm wrote: post #28
LudwigNagasena wrote: post #15

Earlier quoted context omitted.

> integrals and sums with no subscripts / unclear scoping, etc That’s just seems like bad writing. Out of math subjects I learned (discrete math, linear algebra, real analysis, statistics), only in the last one it may be unclear what’s going on, whether you are dealing with a random variable or its realization, and arguments of random variables are almost always omitted, so they don’t look like functions.

So here's an example on Wikipedia [1] https://wikimedia.org/api/rest_v1/media/math/render/svg/6c66... Does the sum include both terms or only the first? I happen to know what Covariance is, but if I didn't I wouldn't be clear. I'd probably guess right, but scroll down a bit to this: https://wikimedia.org/api/rest_v1/media/math/render/svg/e634... Is the +2 a multiplier on the second term or additive within the first s…

There is a reason Wikipedia is never in a syllabus in universities.
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