Differences between Mathematics and Engineering/Physics

Thread Starter

studiot

Joined Nov 9, 2007
5,003
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.
 

Papabravo

Joined Feb 24, 2006
12,301
"Real world" and "mathematical world" are meaningless nonsense terms in a mathematical context. What you meant to say was:

cos (z) = 3 has no solutions if z ∈ ℝ, the set of real numbers.
cos (z) = 3 has solutions if z ∈ ℂ, the set of complex numbers.

BTW - How would you prove the second statement?

Do not fall into the trap of thinking that the use of complex numbers implies we are talking about things which are not physically realizable. That may be so in some cases, but certainly not in others. For example, the impedance of an antenna over a frequency range is a complex valued function of a real argument. Does this fact imply that the antenna does not exist? Of course not, and we have an example of something real in the physical world describe by a mathematical construct that many people have a hard time wrapping their heads around.

In control theory we have LTI systems described by a transfer function. It is a complex valued function of a complex variable. Does this mean such a system cannot exist? Of course not.

And so proceed ad infinitum.

IMHO the physical world and mathematics are in complete harmony.
 

Papabravo

Joined Feb 24, 2006
12,301
And the problem I have with that example is that we are not finished with the search. Lack of evidence does not imply lack of existence. It does however raise the bar for evidence that is collected. Extraordinary claims require extraordinary proof. I think what Studiot is asking for is places where the physics and mathematics posit contradictory results. That is actually a much tougher proposition.
 

nsaspook

Joined Aug 27, 2009
6,221
And the problem I have with that example is that we are not finished with the search. Lack of evidence does not imply lack of existence. It does however raise the bar for evidence that is collected. Extraordinary claims require extraordinary proof. I think what Studiot is asking for is places where the physics and mathematics posit contradictory results. That is actually a much tougher proposition.
That is true as mathematics has evolved to explain and probe what exists physically inconsistent mathematical theorems are abandoned.
Maybe infinity is a better example. https://web.math.princeton.edu/~nelson/papers/warn.pdf
 

Thread Starter

studiot

Joined Nov 9, 2007
5,003
Not necessarily contradictory.

We solve wave equations and throw out the imaginary part of the solution.

No one has asked me about my comment on transistors I note.
 

MrAl

Joined Jun 17, 2014
6,417
I think the closes thing we have are magnetic monopoles. Symmetric Maxwell's equations hint they exist but no physical existence of 'real' monopoles has every been detected.

http://home.web.cern.ch/cern-people/opinion/2014/02/observation-authentic-make-believe-monopoles
Hi there,

I am not sure how you got monopoles from Maxwells equations, as one of them implies just the opposite because there is never a break in a flux line. So for me, Gauss's law for magnetism tells me that there are no true monopoles.

Interesting that this came up in this particular thread though, because i think there has been what we might call 'mathematical' monopoles observed in nature but they are not actually true monopoles. So we have yet another example of a math entity vs reality.
 

Papabravo

Joined Feb 24, 2006
12,301
Not necessarily contradictory.

We solve wave equations and throw out the imaginary part of the solution.

No one has asked me about my comment on transistors I note.
I gave you the benefit of the doubt that what you said was factually correct. If you would care to explain the relationship between cos(z) = 3 and the transistor, I'm all ears.
 

nsaspook

Joined Aug 27, 2009
6,221
Hi there,

I am not sure how you got monopoles from Maxwells equations, as one of them implies just the opposite because there is never a break in a flux line. So for me, Gauss's law for magnetism tells me that there are no true monopoles.

Interesting that this came up in this particular thread though, because i think there has been what we might call 'mathematical' monopoles observed in nature but they are not actually true monopoles. So we have yet another example of a math entity vs reality.
I know it does but as usual some very smart guy (much more than me) finds a way to do it (maybe).
http://www.hcs.harvard.edu/~jus/0302/song.pdf
 
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Thread Starter

studiot

Joined Nov 9, 2007
5,003
If you put sin^-1(3) or cos^-1(3) into Wolfram Alpha you will generate the complex number that solves this.

Alternatively use the complex identity

\[{e^z} = {e^a}\left( {\cos b + i\sin b} \right)\]

Where z= a+ib, a nd b real

As for the transistor, the question arose because I was explaining to someone who asked why the enrgy levels form bands in semiconductors.

This involves solving the Schrodinger equation for the semiconductor and leas to the equation

\[\left( {\frac{{{\beta ^2} - {\alpha ^2}}}{{2\alpha \beta }}} \right)\sinh \beta b\sin \alpha a + \cosh \beta b\cos \alpha a = \cos k\left( {a + b} \right)\]



Now the details are unimportant but the right hand side in interesting because the physicists say

-1 <= cos x <= +1

So these are the bounds on the left hand side.

The end result gives the energy bands for a given level (period k) but the electron is dis allowed outside these bands.

This explains the bands and the gaps in the real world.

However the mathematicians say, gaps, what gaps?
Just fill them with complex numbers.
 

WBahn

Joined Mar 31, 2012
24,581
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 is that of "vital important" in creating transistors?
 

WBahn

Joined Mar 31, 2012
24,581
Not necessarily contradictory.

We solve wave equations and throw out the imaginary part of the solution.

No one has asked me about my comment on transistors I note.
I just did -- I didn't see this thread until just now.
 

WBahn

Joined Mar 31, 2012
24,581
Now the details are unimportant but the right hand side in interesting because the physicists say

-1 <= cos x <= +1

So these are the bounds on the left hand side.

The end result gives the energy bands for a given level (period k) but the electron is dis allowed outside these bands.

This explains the bands and the gaps in the real world.

However the mathematicians say, gaps, what gaps?
Just fill them with complex numbers.
If the bounds are that -1 <= cos x <= +1, then how does that enable mathematicians to ignore those bounds?
 

amilton542

Joined Nov 13, 2010
496
@studiot

I think very highly of you and the work you contribute to AAC, but is the purpose of your thread to generate a dividing line amid applied and pure math that already exists?
 

WBahn

Joined Mar 31, 2012
24,581
One of the math profs here has an XKCD (I think) cartoon on their door where two guys are explaining how if we assume that the speed of light were just 64 orders of magnitude faster in the early universe that this problem and that problem would go away and when someone in the audiences says that you can't just arbitrarily make light travel faster than c, the response is, "No, YOU can't!". The caption is "Why mathematicians aren't invited to physics conferences."
 

cmartinez

Joined Jan 17, 2007
6,380
One of the math profs here has an XKCD (I think) cartoon on their door where two guys are explaining how if we assume that the speed of light were just 64 orders of magnitude faster in the early universe that this problem and that problem would go away and when someone in the audiences says that you can't just arbitrarily make light travel faster than c, the response is, "No, YOU can't!". The caption is "Why mathematicians aren't invited to physics conferences."
I was going to mention klein's bottle, the möbius strip and some topology problems as things that have no counterpart in the "real" world... but then I remembered that some of those things have found applications in quantum mechanics (how "real" are they anyway?) ...
Years ago I found an article that has now either been moved or deleted... it was titled "mathematical reality", and it was, at least for me, fascinating reading.... if I ever find it again I'll post it here.

This discussion is most probably beyond my abilities... but I'm normally drawn to this sort of thing for that same reason. ;)
 

nsaspook

Joined Aug 27, 2009
6,221
There are many 'mathematical reality' solutions that have practical applications. Hyper-dimensional objects are used to navigate spacecraft/aircraft/missiles as they travel at high speed and calculate the motions of physical objects with computer graphics. The position of a point/vector can be specified by XYZ coordinates in space using 3D Rotation matrix transformations but they are missing a critical element directly involving timing. So what done is to add an extra dimension (w rotation) that gives a possible spiral to movements in space that maps to a hypersphere with unit quaternions with Quaternion rotation operations.

Don't ask me how it works at the detailed mathematical level but in my limited exposure in training robotic movements it allows fluid motions with wrist joints and arms that are very difficult to do with normal angular inputs.
It takes some effort to get a good mental picture but it's worth the time.
 
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cmartinez

Joined Jan 17, 2007
6,380
There are many 'mathematical reality' solutions that have practical applications. Hyper-dimensional objects are used to navigate spacecraft/aircraft/missiles as they travel at high speed and calculate the motions of physical objects with computer graphics. The position of a point/vector can be specified by XYZ coordinates in space using 3D Rotation matrix transformations but they are missing a critical element directly involving timing. So what done is to add an extra dimension (w rotation) that gives a possible spiral to movements in space that maps to a hypersphere with unit quaternions with Quaternion rotation operations.

Don't ask me how it works at the detailed mathematical level but in my limited exposure in training robotic movements it allows fluid motions with wrist joints and arms that are very difficult to do with normal angular inputs.
It takes some effort to get a good mental picture but it's worth the time.
It's funny you should mention quaternions. I worked with them a few years ago when I developed a rather complex piece of software for a customer that translated AutoCAD drawings into motion sequences for ABB robotic arms... Those sequences worked with every available degree of freedom that the robot had. It was fascinating work. But perhaps just as hard was designing a friendly (as friendly as this sort of thing can be) and clear user interface that worked within AutoCAD itself.
 

wayneh

Joined Sep 9, 2010
16,102
I am collecting examples of where maths and the physical sciences differ over something.
I'm having trouble understanding the request. Math is a set of tools invented in the mind of man that have useful application in the real world but can also manipulate ideas that have no physical counterparts. Just as I can use metalworking tools to make a wrench that fits no nut, math tools can be built that fit no physical reality.

I see this like a Venn diagram; math and the real world overlap quite a bit, but there is a lot of math outside the physics circle, and a sliver of physics still outside the math circle. (I'm assuming we don't yet have all the math we need to describe the physical universe.) I wouldn't say that math and physics "differ" just because they don't completely overlap.

But, here's the first thing that popped into my head: Pi. It seems to me to be a failure of math that something so physically simple - the ratio of a circle's circumference to its diameter - is a never-ending number.
 

WBahn

Joined Mar 31, 2012
24,581
But, here's the first thing that popped into my head: Pi. It seems to me to be a failure of math that something so physically simple - the ratio of a circle's circumference to its diameter - is a never-ending number.
I don't see how that is a "failure" of math. It is what it is.

Having said that, we could presumably fix that and just use a base-pi number system. But then someone would point out the "failure of math" that something so physically simple as the ratio of the diagonal of a square to the length of its size is a never ending number.
 
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