Differences between Mathematics and Physics/Engineering

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I am collecting examples of where maths and the physical sciences differ over something.

Examples offered would be gratefully received.

For instance

cos (z) = 3 has no solution in the real world and this fact is of vital importance in creating transistors.

However in the mathematical world the equation has complex solutions.

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Interesting exposition about whether it is meaningful in physics to talk about particles possessing negative mass : http://en.wikipedia.org/wiki/Negative_mass

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I am collecting examples of where maths and the physical sciences differ over something.

Examples offered would be gratefully received.

For instance

cos (z) = 3 has no solution in the real world and this fact is of vital importance in creating transistors.

However in the mathematical world the equation has complex solutions.

How about any irrational number? For instance, π has no mathematical termination, but circles are closed. Same for $\sqrt{2}$; no termination but squares have diagonals.

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Bill Angel

Thank you, Bill for the thought. Negative mass is indeed an interesting subject.

Certainly sticking a negative sign in front of something that Physics takes as non negative but is allowed in Maths is worth thinging about.

Unfortunately I'm not sure it shows a difference for negative mass since negative (effective) mass is already in use in Physics and important in semiconductor Physics amongs other places.

But there may well be some negatives that are OK in Maths but not Physics.

Acme

Thanks for the contribution Acme,

Again unfortunately I don't see where Physics forbids irrational or trancendental numbers.

Both the radius and circumference of a circle are real world objects.

Incidentally this thread was stimulated by preparing a response to a question about the origin of energy bands in semiconductors.

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Circle, sphere, line etc. are only at high enough distance/scale.

At quantum level, circle would be made of single atoms, and wouldn't be circle anymore, but approximation of circle.

Similar like circle on computer monitor screen is made of pixels. It just appears round from distance.

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...

Acme

Thanks for the contribution Acme,

Again unfortunately I don't see where Physics forbids irrational or trancendental numbers.

Both the radius and circumference of a circle are real world objects.

...

You're welcome. I was thinking that the ongoing decimal expansions imply that the physical thing those decimals describe go on & on too, which clearly the physical 'things' do not.

Circle, sphere, line etc. are only at high enough distance/scale.

At quantum level, circle would be made of single atoms, and wouldn't be circle anymore, but approximation of circle.

Similar like circle on computer monitor screen is made of pixels. It just appears round from distance.

In a similar bass-ackwards sort of way, a mathematical polygon with sufficiently many sides becomes indistinguishable from the physical circle. See Chiliagon

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Look for Banach Tarski paradox. In summary, you can cut a ball of volume 1 into some pieces in such way that after some rearrangement of those pieces you obtain two identical balls of volume 1. That kind of paradox made mathematicians question, whether or not the axiom of choice is legitimate since using it creates paradoxes completely contradictory to nature and physics. (conservation of mass)

There are also functions which are continuous and yet not differentiable in any point. You are quite unlikely to encounter them in nature, because at least from my limited knowledge in physics I am pretty convinced that most of the functions you encounter are infinitely differentiable. (by parts at least)

In the same category you also have Devil's staircase. It's a function which you can differentiate almost anywhere and it's derivative will by always 0 and it's increasing. In some sense you move up with 0 speed.

That's all that comes to my mind right now.

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Surely the Banach Tarski would only violate common perceptions of a conservation of volume - there is no viloation of the conservation of mass. The paradox "works" because the porosity ends up screwing with the everyday notion of volume a/o density; there is no contention that I have seen that extra mass would be created. This is not saying it is not strange and counterintuitive.

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Look for Banach Tarski paradox. In summary, you can cut a ball of volume 1 into some pieces in such way that after some rearrangement of those pieces you obtain two identical balls of volume 1. That kind of paradox made mathematicians question, whether or not the axiom of choice is legitimate since using it creates paradoxes completely contradictory to nature and physics. (conservation of mass)

There are also functions which are continuous and yet not differentiable in any point. You are quite unlikely to encounter them in nature, because at least from my limited knowledge in physics I am pretty convinced that most of the functions you encounter are infinitely differentiable. (by parts at least)

In the same category you also have Devil's staircase. It's a function which you can differentiate almost anywhere and it's derivative will by always 0 and it's increasing. In some sense you move up with 0 speed.

That's all that comes to my mind right now.

Thank you , Keen, for your thoughts.

I think you will find that the Tarski fractal decomposition you mention is the principle that lies behind the fact that any piece of a hologram contains the entire image.

Playing around with infinities like Cantor's staircase or Peano's space filling curve (do you know that one?) is fun, but they all fit into a solid block just as well as a perfect circle, parabola etc.

But keep the thoughts coming, they are interesting.

It is also interesting that all the offerings, so far, are for something in mathematics that is not in the physical world.

I actually had in mind the other way round.

Edited by studiot

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cos (z) = 3 has no solution in the real world and this fact is of vital importance in creating transistors.

Can you explain the relevance of that to transistor design?

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Can you explain the relevance of that to transistor design?

The solution of the Schrodinger equation applied to a periodic structure such as a semiconductor lattice leads to a complicated trigonometric equation which has real solutions that work in transistors and imaginary solutions that have no physical impact.

For example the simplest approach is the Kronig-Penny model.

You are probably aware of the technique in elementary physics of 'equating real and imaginary parts and discarding the imaginary results'.

There is a particularly clear working of this in Prof Atkins' book, Molecular Quantum Mechanics, p269ff.

Edited by studiot

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Surely the Banach Tarski would only violate common perceptions of a conservation of volume - there is no viloation of the conservation of mass. The paradox "works" because the porosity ends up screwing with the everyday notion of volume a/o density; there is no contention that I have seen that extra mass would be created. This is not saying it is not strange and counterintuitive.

That's actually a good point: and a reason I do not really consider it a real paradox. If we want to use notions such as mass or volume, we can only work with sets that are measurable, because anything else doesn't really make sense and the sets involved in this paradox aren't measurable.

With that being said, you can't really physically obtain such sets, but that's probably irrelevant, because as far as I know, on microscopic levels you have so many strange things that are going on, that measurability it's the least of your concerns.

Thank you , Keen, for your thoughts.

I think you will find that the Tarski fractal decomposition you mention is the principle that lies behind the fact that any piece of a hologram contains the entire image.

Playing around with infinities like Cantor's staircase or Peano's space filling curve (do you know that one?) is fun, but they all fit into a solid block just as well as a perfect circle, parabola etc.

But keep the thoughts coming, they are interesting.

It is also interesting that all the offerings, so far, are for something in mathematics that is not in the physical world.

I actually had in mind the other way round.

The other way around for me is quite difficult as I have little to none experience with physics and engineering, so I can just think about some strange stuff I've seen in mathematics and tell myself "That kind of thing can't exist in real world".

With that being said mathematics is a model and just a formal way to describe the laws you can observe, so as soon as in physics you require some formal approach that doesn't have a mathematical description yet, you can be pretty sure that a mathematician creates a new field in mathematics. Lots of mathematical notions actually originated in problems in physics.

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I am collecting examples of where maths and the physical sciences differ over something.

Examples offered would be gratefully received.

For instance

cos (z) = 3 has no solution in the real world and this fact is of vital importance in creating transistors.

However in the mathematical world the equation has complex solutions.

Math and physics differ over almost everything. Math is imaginary; it was/is invented. Physics (all real science) is discovery; we discover only what is real. Math accepts no limits. Physics searches for/defines limits.

Math provides the virtual state of an object. Physics cannot provide the real state of an object because only the current state can be observed. Place (and identity) are all that are observable currently. The first and second derivatives of place, speed and acceleration, as well as all other relative constructs, such as time and position, require multiple observations of the object and its surroundings which cannot be simultaneous. Thus physics provides an approximate history while math gives an exact virtual current state. The key here is the definition of virtual.

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Math and physics differ over almost everything.

To my mind there are lots of close similarities and sometimes separating the two is not so clear.

Math is imaginary; it was/is invented.

This people debate. It is rather a philosophical question, but I tend to favour mathematics being discovered.

Math accepts no limits.

While at the same time mathematics is not a 'free for all'. It seems that one cannot do whatever one wants and still have a consistent and interesting structure.

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Thank you , Fred, for responding to my question.

I was really seeking specific examples in preference to generalities and would welcome any that you can offer.

Edited by studiot

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