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Why tensors? A beginner's perspective

mfaizan.github.io

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Re: Why tensors? A beginner's perspective

#81
post #78
post #74

Earlier quoted context omitted.

Which mathematician disagrees with what exactly? Tensors are introduced by physicists to ensure various physical quantities (which involve coordinates and their derivatives) do not depend on the arbitrarily chosen coordinate system. This is ensured through the transformation properties of tensors. The name tensor itself comes from the theory of elasticity, Cauchy stress tensor, which BTW is uniform in many practical…

Interesting, you use the word matrix differently than me. The way I use it, a 2d array isn't necessarily a matrix. It's only a matrix if it represents a linear map between vector spaces with respect to chosen bases. Then again, I'm not much of a programmer, but I've taught linear algebra a few times. My head is just in a different place I guess.

Any 2D array of data is a valid matrix.

But not any multidimensional data is a valid tensor.

Re: Why tensors? A beginner's perspective

#82
post #81
post #78

Earlier quoted context omitted.

Interesting, you use the word matrix differently than me. The way I use it, a 2d array isn't necessarily a matrix. It's only a matrix if it represents a linear map between vector spaces with respect to chosen bases. Then again, I'm not much of a programmer, but I've taught linear algebra a few times. My head is just in a different place I guess.

Any 2D array of data is a valid matrix. But not any multidimensional data is a valid tensor.

I'm not sure I buy this after you just used an example of a 2d array of you're favorite colors just a few minutes ago. Maybe I'm missing something. What kind of linear transformation does that represent, and between what vector spaces?

Re: Why tensors? A beginner's perspective

#83
post #29

Earlier quoted context omitted.

> 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 pa…

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?

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.

Re: Why tensors? A beginner's perspective

#84
post #82
post #81

Earlier quoted context omitted.

Any 2D array of data is a valid matrix. But not any multidimensional data is a valid tensor.

I'm not sure I buy this after you just used an example of a 2d array of you're favorite colors just a few minutes ago. Maybe I'm missing something. What kind of linear transformation does that represent, and between what vector spaces?

(the exchange between you two is a perfect example of the disconnect between how mathematicians see linear algebra, and how physicists do)

Re: Why tensors? A beginner's perspective

#85
post #82
post #81

Earlier quoted context omitted.

Any 2D array of data is a valid matrix. But not any multidimensional data is a valid tensor.

I'm not sure I buy this after you just used an example of a 2d array of you're favorite colors just a few minutes ago. Maybe I'm missing something. What kind of linear transformation does that represent, and between what vector spaces?

My favorite colors are 1, 2, 3 and 4 in some 8-bit grayscale color space. Here's a matrix

  [ 1 2 ]
  [ 3 4 ]
This represents some linear transformation, and is a matrix. Are you trying to argue otherwise?

FYI, matrices transform between vectors within a vector space.

Re: Why tensors? A beginner's perspective

#86
post #29
post #14

That was explanation from a perspective of someone acquainted with modern physics. As such, it will make sense to physicist, but no sense to most everyone else, including mathematicians who don’t know modern physics. For example, in the beginning, author describes tensors as things behaving according to tensor transformation formula. This is already very much a physicist kind of thinking: it assumes that there is som…

> 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 pa…

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.

Re: Why tensors? A beginner's perspective

#87
post #14

That was explanation from a perspective of someone acquainted with modern physics. As such, it will make sense to physicist, but no sense to most everyone else, including mathematicians who don’t know modern physics. For example, in the beginning, author describes tensors as things behaving according to tensor transformation formula. This is already very much a physicist kind of thinking: it assumes that there is som…

I think it’s a little funny you said that

>[the explanation in the OP] will make sense to physicist, but no sense to most everyone else, including mathematicians

And then went on to describe tensors in a way that is unfriendly to non mathematicians by saying

> tensors are a sort of universal objects representing multi linear maps: bilinear maps V x W -> R correspond canonically one-to-one to regular linear maps V (x) W -> R, where V (x) W is a vector space called tensor product of V and W, and tensors are simply vectors in this tensor product space.

Re: Why tensors? A beginner's perspective

#88
post #85
post #82

Earlier quoted context omitted.

I'm not sure I buy this after you just used an example of a 2d array of you're favorite colors just a few minutes ago. Maybe I'm missing something. What kind of linear transformation does that represent, and between what vector spaces?

My favorite colors are 1, 2, 3 and 4 in some 8-bit grayscale color space. Here's a matrix [ 1 2 ] [ 3 4 ] This represents some linear transformation, and is a matrix. Are you trying to argue otherwise? FYI, matrices transform between vectors within a vector space.

You have a list of labels you've conveniently decided to conflate with numbers. It's fine to use an enum for your data, but no, this data structure is not meant to convey a transformation.

You bring up a good point though, if this were meant to be a transformation, then we're talking about modules (Z/2^8Z being the underlying ring) and not vector spaces, which is fine. I was needlessly narrow when I said "vector spaces" earlier.

Re: Why tensors? A beginner's perspective

#89
post #84
post #82

Earlier quoted context omitted.

I'm not sure I buy this after you just used an example of a 2d array of you're favorite colors just a few minutes ago. Maybe I'm missing something. What kind of linear transformation does that represent, and between what vector spaces?

(the exchange between you two is a perfect example of the disconnect between how mathematicians see linear algebra, and how physicists do)

Haha, maybe so. I'm a mathematician, but based on some other comments I seem to be playing the role of physicist I think? Not sure.

Re: Why tensors? A beginner's perspective

#90
post #37

Earlier quoted context omitted.

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?

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 middl…

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.
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