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What is a manifold?

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Re: What is a manifold?

#131

Post by ChadNauseam »

lisper wrote: post #111
seanhunter wrote: post #84

Earlier quoted context omitted.

> But I would go with something more like: tensors are a way to represent vectors so that the representation of a given vector is the same no matter what basis (or coordinate system) you choose for your vector space. That's just incorrect though for a couple of reasons. Firstly, a vector in the sense in which it is used in physics is a rank 1 tensor so it has this transformation behaviour just like other higher order…

> That's just incorrect though Quite possible. But that's in no small measure because I have yet to find an actual cogent definition of "tensor" that distinguishes a tensor from an array. (I have a similar problem with monads.) > So what I mean when I talk about the reality of the tensor I mean whatever it is the tensor is expressing in the physical universe OK, but then "the reality of a tensor" not depending on the…

Could you elaborate on your problem with monads?

Re: What is a manifold?

#132

Post by lisper »

ChadNauseam wrote: post #131
lisper wrote: post #111

Earlier quoted context omitted.

> That's just incorrect though Quite possible. But that's in no small measure because I have yet to find an actual cogent definition of "tensor" that distinguishes a tensor from an array. (I have a similar problem with monads.) > So what I mean when I talk about the reality of the tensor I mean whatever it is the tensor is expressing in the physical universe OK, but then "the reality of a tensor" not depending on the…

Could you elaborate on your problem with monads?

I don't understand them. None of the introductory materials I've read on them ever made any sense to me. I kinda sorta understand "state" and "maybe" but the general idea that links these two things together and (I am given to understand) leads to other cool stuff just eludes me.

Re: What is a manifold?

#133

Post by lisper »

ChadNauseam wrote: post #130
lisper wrote: post #39

Earlier quoted context omitted.

> You might find it circular reasoning but it is not Um, yes it is. "A foo is an object that transforms as a foo" is a circular definition because it refers to the thing being defined in the definition. That is what "circular definition" means .

Of course, there are some definitions that reference themselves that are not circular, or alternatively, some circular definitions that are useful. For example, the definition of "ancestors" might be "your parents and your parents' ancestors" (with the implication that if you don't exist, you don't have any ancestors). But I agree that this is not a useful definition for "tensor".

> "your parents and your parents' ancestors"

That's recursive, not circular. These are not the same thing despite being closely related. Recursive definitions are useful. Circular ones are not.

Re: What is a manifold?

#134

Post by ChadNauseam »

lisper wrote: post #133
ChadNauseam wrote: post #130

Earlier quoted context omitted.

Of course, there are some definitions that reference themselves that are not circular, or alternatively, some circular definitions that are useful. For example, the definition of "ancestors" might be "your parents and your parents' ancestors" (with the implication that if you don't exist, you don't have any ancestors). But I agree that this is not a useful definition for "tensor".

> "your parents and your parents' ancestors" That's recursive, not circular. These are not the same thing despite being closely related. Recursive definitions are useful. Circular ones are not.

I was using the definition you gave for “circular”

Re: What is a manifold?

#135

Post by ChadNauseam »

lisper wrote: post #132
ChadNauseam wrote: post #131

Earlier quoted context omitted.

Could you elaborate on your problem with monads?

I don't understand them. None of the introductory materials I've read on them ever made any sense to me. I kinda sorta understand "state" and "maybe" but the general idea that links these two things together and (I am given to understand) leads to other cool stuff just eludes me.

Easiest way to understand is to write code using them IMO. (Writing a parser for some language with a parser-combinator library is a good place to start.) Otherwise it's like trying to understand how to play a board game by reading the rulebook. It's easier to just try playing the game

Re: What is a manifold?

#136

Post by lisper »

ChadNauseam wrote: post #134
lisper wrote: post #133

Earlier quoted context omitted.

> "your parents and your parents' ancestors" That's recursive, not circular. These are not the same thing despite being closely related. Recursive definitions are useful. Circular ones are not.

I was using the definition you gave for “circular”

Do you really need me to explain the distinction between "circular" and "recursive", and the nature of HN comments as an informal medium with an audience that generally does understand these things and does not need to have them explicitly called out?

Re: What is a manifold?

#137

Post by ChrisGreenHeur »

BigTTYGothGF wrote: post #30
BigTTYGothGF wrote: post #20

Earlier quoted context omitted.

> you can put a cd shaped object on You're thinking of open sets.

In particular, consider two intersecting planes. You can put all the discs you like on that surface, but it's not a manifold because on the line of intersection it's not locally R2.

The position of the points on the surface can be changed without worrying about the topology and its the topology we are worried about

Re: What is a manifold?

#138

Post by ChrisGreenHeur »

dist-epoch wrote: post #57
ChrisGreenHeur wrote: post #10

A manifold is a surface that you can put a cd shaped object on in any place on the surface, you can change the radius of the cd but it has to have some radius above 0.

Including the hole at the center of the CD?

there can be no hole on the cd shaped object, technically disc is the word to use but the point of the sentence is to provide a vivid picture
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