Bill Wood wrote:

On Tue, 2007-08-14 at 16:02 -0700, Dan Piponi wrote: . . .

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On 8/14/07, Michael Vanier wrote:

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I'm reminded of a physics teacher who was having a similar problem explaining the concept of tensors, until he said that "a tensor is something that transforms like a tensor does!". Grrr...must...hold...my...tongue...

Dan, as a former student of a clone of that physics teacher, I am really interested in what you will say when you fail to hold your tongue.

-- Bill Wood

I have to admit I was wondering the same thing myself. Mike

Miguel Mitrofanov writes:

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Grrr...must...hold...my...tongue...

Dan, as a former student of a clone of that physics teacher, I am really interested in what you will say when you fail to hold your tongue.

-- Bill Wood

MV> I have to admit I was wondering the same thing myself.

So was I.

I'm guessing that Dan means that thinking of tensors as things that transform between co-ordinate systems in a certain way (e.g. via the Jacobian of the transition maps) isn't a terribly good way of thinking about them. Vector fields, co-vector fields and tensor fields are really co-ordinate independent notions and the transformation laws (if I may call them that) are a consequence of the way they transform under (smooth) maps. But perhaps this is better discussed on a differential geometry mailing list? Dominic.

On 8/15/07, Andy Gimblett wrote:

I assumed he was just trying not to sing the "Spider Pig" song.

I've been banned from singing that around the house. And the cat version. But I was mainly thinking about how the physicist's definition of tensor needn't be accepted as an irreducible given, but is a consequence of the definition of tensor product through its universal property: http://planetmath.org/encyclopedia/TensorProduct.html Having said that, I still completely agree with Michael that tensors are a great analogy for monads because I found the concept of a universal property tricky in the same way that I subsequently found monads tricky. BTW I think the concept of a universal property is probably the single most useful idea from category theory that can be used in Haskell programming. I recommend it to everyone :-) -- Dan

On 8/15/07, Dan Weston wrote:

"You too could have invented Universal Algebra and Category Theory". I nominate Dan Piponi to write it and eagerly await its release!

I've already started on it. Well, that's not the exact title and subject. And as an example I'll probably use the definition of tensor product that I linked to, not the even more compact and elegant one that you just gave. I'm a strong believer that lots of (but not all) tricky looking mathematics is just fancy language for intuitions that people already have. I suspect that most computer scientists already have much of the intuition behind the idea of a universal property, and that it is in fact easier to grasp for a computer scientist than a mathematician. :-) -- Dan

Dan Piponi writes:

On 8/15/07, Dan Weston wrote:

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"You too could have invented Universal Algebra and Category Theory". I nominate Dan Piponi to write it and eagerly await its release!

I've already started on it. Well, that's not the exact title and subject. And as an example I'll probably use the definition of tensor product that I linked to, not the even more compact and elegant one that you just gave.

I'm a strong believer that lots of (but not all) tricky looking mathematics is just fancy language for intuitions that people already have. I suspect that most computer scientists already have much of the intuition behind the idea of a universal property, and that it is in fact easier to grasp for a computer scientist than a mathematician.

-- Dan

Some universal properties (initial algebras and final co-algebras) are covered in Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire (1991) (http://citeseer.ist.psu.edu/meijer91functional.html). And of course there's Wadler's Theorems for Free. Dominic.