Math that creates laws of physics ?

[oldtobor]

What is the mathematical item that has differential equations as a solution ?

I mean a normal function has a number as a solution, a differential equation has a function as a solution, then what is that item that has differential equations as a solution ?

Then the "operator equation " that has as a solution all the differential equations of physics would be the grand unified theory ? I am just guessing but is there something like this in mathematics ? Thanks for any ideas...

 

[droz]

I think you have an epistemological misunderstanding of mathematics. Mathematics is a wrapper/model/"toy" (i use toy here _very_ loosely). that describes reality. It is *not* 100% reality. So your subject is quite a loaded philosophical gun.

 

[oldtobor]

No problem. It is just an odd idea that I had. If you consider that a number is the solution to a normal equation, and in some cases you can have a series of numbers (like 3 different numbers for a third degree algebraic equation, etc.) and a function or more than one function can be the solution to a differential equation, then maybe someone created some kind of mathematical apparatus to describe equations that have as a solution a differential equation or even more than one differential equation.

Maybe some kind of symbology that corresponds to differential operators so that a solution can map to a differential equation. So since the set of all the laws of physics are ultimately just a set of a few tens or at most hundreds of differential equations, than the operator equation that leads to this exact set of differential equations becomes the ultimate grand unified theory. Just guessing, but is there something completely wrong in the concept ?

 

[Gib Z]

It may be on only very little consolation, but I remember that there are mathematical constructions that have differential equations as solutions, but can't recall it. However, we don't know, nature may be more complicated than even that "Operator equation" and need to be described by something that gives Operator Equations as solutions? Or perhaps even more complex? We don't know.

 

[Mephisto]

haha I like it! I guess first there would have to be some mathematical structure that would have DE's as solutions first, and then physicists would try to apply it to the real world somehow and see what they get :)
Hmm... is every physical law a differential equation though? You also need a couple of fundamental constants. Interesting idea though

 

[robert Ihnot]

Take S=1/2gt^2, S' = gt, S''=g, which represents gravity. I saw all this stuff when I took college Physics 1, but they did not mention differentiation.

Maybe all this simple stuff is missing the point?

 

[Hurkyl]

I mean a normal function has a number as a solution

Functions don't have solutions. Equations have solutions.

And unless your variables denote actual equations, you cannot have an equation as a solution to another equation.



Anyways, I imagine you're trying to think of functional analysis.

 

[HallsofIvy]

Differential equations can be thought of as vector fields on manifolds. The problem of determining the vector fields that have certain properties would then have differential equations as solutions.

In any case, the idea that "mathematics creates laws of physics", or than any mathematical theory could be "a grand unified theory" of physics is just non-sense. "Laws of physics" are "given" by the real world. Some mathematical theories may be better or worse approximations to specific laws of physics.

 

[Gib Z]

In any case, the idea that "mathematics creates laws of physics", or than any mathematical theory could be "a grand unified theory" of physics is just non-sense. "Laws of physics" are "given" by the real world. Some mathematical theories may be better or worse approximations to specific laws of physics.

That is true, however modern theories are becoming much more mathematical than physical, such as String Theory. What many physicists are looking for is a GUT, or a grand unified theory, or a Theory of Everything. Two interpretations of this is a theory that is the be all and end all of physical theories, it explains every part of physics there is the be explained, to the most precise form it is physically possible. The other is merely a theory that reconciles the differences between Quantum Mechanics and General Relativity, both which work extremely well on small and large scales respectively, but on a certain scale (called the Planck length) it is needed to take into accounts both theories. However Heisenberg's Uncertainty principle introduces conflicts with the Einsteins equations and currently it is not possible to explain things on this Planck Length to a satisfying amount.

 

[Crosson]

In any case, the idea that "mathematics creates laws of physics", or than any mathematical theory could be "a grand unified theory" of physics is just non-sense. "Laws of physics" are "given" by the real world. Some mathematical theories may be better or worse approximations to specific laws of physics.

It is certainly not non-sense to say "mathematics creates laws of physics", we just have not found any reason to believe that this is true (but it certainly has a sense).

I for one find it attractive to pursue the program of seeing fundamental physics as necessary in the same sense of "necessary" that is used in mathematics. This was the ideal from Aristotle through Kant, that mathematics consists of necessary truths (about our intuitions of) space and time, while theoretical physics should ultimately consist of similarly necessary truths about (our intuitions of) space and time with the additional presence of substance (matter).

But here in the 21st century we have supposedly discovered that Quantum Physics is unintuitive, while the rise of rigorous and general mathematics based on axiomatic set theory has obliterated the old notion of mathematics based only on our intuitions of space and time.

While the common opinion is that Kant was wrong as pertaining to necessary truths in mathematics and physics deriving from our intuitions of space,time and substance, I contend that in the end we would only need to make superficial changes to force a correspondence with what we will eventually see as the necessary laws in the universe. I admit that it is an unfalsifiable belief (although their are circumstances which would serve as its proof).

The main point I want to make is that it is not non-sense to hope or work towards
seeing the laws of physics as more and more necessary, and perhaps one day seeing them with similar necessity to that with which we see propositions of pure mathematics.

And finally, I would like to answer in advance the objection that physics is an experimental science, and that we would have no knowledge of non-intuitive 20th century physics if it were not for experimentation. I could then instigate a squabble over the premise of the experimentalist e.g. Einstein found SR from theoretical EM principles, not from Michelson-Morley experiments.

Instead I will grant the premise of the experimentalist, that his work is the cause of our knowledge. But the thing I would ask him to keep in mind is that the cause of our knowledge is not the same as our grounds for it, and developing this is the not-so-secret intention of the theoretical physicist. Michael Faraday said "I never believed a fact until I saw it with my own eyes", while a more theoretically minded person will contrarily hold out on accepting the results of an experiment (as long as possible) until it agrees with current theory (if the lag is too long then there is a grotesque limbo, this sometimes occurs in history and I don't think it makes for a pleasant generation of theoreticians).

 

[oldtobor]

Differential equations can be thought of as vector fields on manifolds. The problem of determining the vector fields that have certain properties would then have differential equations as solutions.

So then that vector field that has as a solution about 50 differential equations that correspond to the equations used in electromagnetics, general relativity and quantum physics would be a grand unified theory ? That field would be the most general solution describing our universe since it would have as a subset all the most important differential equations ?