Math used in Physics versus other sciences

[Ken Ucarp]

I've read about complex mathematical things like Tensors, Hilbert Space, and so on. To my uninformed mind it appears they are used as if they are special and specific to Physics. But then I read another thread in this forum where an example was given applying to biology. And I think I've seen discussions of economics using similar terms. I have to admit I've always thought these things were special and only related to the "big physics" I was never smart enough to grasp. But now it seems like they're really just generic man-made as it were, tools that could apply to all kinds of mundane things.

Somebody restore my feeling that physicists (whom I regard as special and gifted people for the bigness of their subject) considering the universe and space and time and fundamental particles aren't using the same tools as economists studying the buying habits of populations of shoppers at malls.

 

[jedishrfu]

They are mathematical concepts that extend the notion of matrices and are often applied to differential geometry, general relativity and other areas. They became well known when Einstein used them in GR. Now Differential Geometry has found many other uses in a lot of fields.

As an example, one such tensor describes the pythagorean distance in a space of any dimensions:

https://en.wikipedia.org/wiki/Metric_tensor

Other uses are listed at the end of this article on Tensors:

https://en.wikipedia.org/wiki/Tensor

 

[Ken Ucarp]

They are mathematical concepts that extend the notion of matrices and are often applied to differential geometry, general relativity and other areas. They became well known when Einstein used them in GR. Now Differential Geometry has found many other uses in a lot of fields.

As an example, one such tensor describes the pythagorean distance in a space of any dimensions:

https://en.wikipedia.org/wiki/Metric_tensor

Other uses are listed at the end of this article on Tensors:

https://en.wikipedia.org/wiki/Tensor

So I think you are confirming my loss. Although your examples contradict your words.

 

[fresh_42]

I've read about complex mathematical things like Tensors, Hilbert Space, and so on. To my uninformed mind it appears they are used as if they are special and specific to Physics. But then I read another thread in this forum where an example was given applying to biology. And I think I've seen discussions of economics using similar terms. I have to admit I've always thought these things were special and only related to the "big physics" I was never smart enough to grasp. But now it seems like they're really just generic man-made as it were, tools that could apply to all kinds of mundane things.

Of course do these tools apply to a lot of occasions. At its kernel it is often a quantity which varies in time under some external constraints. If this is the flux of a fluid in a pipe, a virus spreading the world, the behavior of shoppers on a black Friday or simply the amount of gas to drive a certain distance doesn't really matter. They are just different set-ups.

Somebody restore my feeling that physicists (whom I regard as special and gifted people for the bigness of their subject) considering the universe and space and time and fundamental particles aren't using the same tools as economists studying the buying habits of populations of shoppers at malls.

The relationship between physics and mathematics is for many reasons an especially close one. This means that they influence each other and many mathematical concepts have been especially developed to solve physical problems. On the other hand, there are purely mathematical concepts which were adopted by physics afterwards: graded algebras have been around long before someone used them for string theory. For both directions of influence exist many examples. The result is that mathematicians will look for physical examples, if they try to imagine applications, and physicists will look up mathematical insights first, if they find or assume new concepts. But mathematical structures play as well an important role in genetics or macro economics. It's just that biologists and economists are far less used to look at what mathematics has to say, than physicists are. This has a couple of reasons. However, these facts are not suited to conclude, that there are mathematical tools which are exclusively available for physics.

 

[mfb]

Somebody restore my feeling that physicists (whom I regard as special and gifted people for the bigness of their subject) considering the universe and space and time and fundamental particles aren't using the same tools as economists studying the buying habits of populations of shoppers at malls.

The tools are often related, the mindset is even more similar, and it is common that particle physicists study things like buying habits of people, traffic flow or similar things after a MSc/PhD.

 

[jedishrfu]

Reminds me of Waxahachie!

 

[Ken Ucarp]

Thanks all for the great responses. My mellow is fully marshed. I for one think you should change the names of these things so as not to confuse unsuspecting young physicist wannabees. Tensor sounds too much like Tension Object. In general the ending -or sounds like a real thing. How about Multifunc? Hilbert Space. Sounds too physical. How about Hilbert Megamatrix. Think about it! :) (I'm kidding of course, please don't flame me).

 

[fresh_42]

Tensor sounds too much like Tension Object.

How appropriate!

But my guess is, that it had more to do with stretching linearity to multilinearity. Btw. David Hilbert was a mathematician and no physicist and Hilbert spaces and tensor spaces have a priori nothing to do with each other. I still try to find out the physicists' usage of the term tensor. To me it is as if they always use it, as soon as any product and often even direct sums come into play. It makes me wonder, that they don't use it for the multiplication of ordinary numbers.