Inspired by this thread on time travel, but this is a more general question about how much weight we should put on solutions that come out of the math of a particular problem in physics.

People may state things like

“There’s no mathematical reason why effect should follow cause, and not the other way around, so we should assume that both are possible in different regions or times of this or other universes”

“Action X can cause, according to the math, the universe to branch out into multiple universes”

“Physical laws have no time directionality, so we can’t rule out moving backwards in time”
But, we are the ones who came up with the equations that describe the laws of physics, and they are, at best, an approximation of what is “truly” going on in the universe, inside atoms, etc

So, just because our mathematical approximation is very good at describing what we observe, it does not mean that all the solutions that our mathematical approximation allows are physically realizable.

As an example, assume that someone tells you that an equation that describes the number of people in some room is the following:

You try to solve it, to see how many people are in the room, and find that N is either 2 or -4.

You, of course, will assume that N = 2

Just because the math allows N = -4, does not mean that you should start thinking “I wonder if it is possible to have -4 people in a room”

In the world of physics, though, people keep pondering the physical reality of seemingly ludicrous solutions to the mathematical approximations that we call the “laws of physics”.

If those “ludicrous solutions” can be tested in the lab and shown to actually occur (e.g. time dilation) then great!

But, if we can never observe or verify any of these solutions (like effect preceding cause), why are they even part of scientific debate?

There is a whole industry of people known in the physics community as theorists. Their job is basically to find all the possible situations that the math allows. Once a theory has been proposed, its the experimentalists job to evaluate it and determine if nature does indeed respect the math in that sense. Only then does the theorists’ proposal gain ANY credence. However, sometimes there seems to be only one theory that we can think of that would explain a given, established phenomenon, but for whatever reason the experiment can not or has not been done yet. In these cases, the theory might become extremely popular or a hot topic of research, but that does not mean that anyone should just believe it without further proof. String theory is a pretty good example of this.

And by the way, there is physical evidence that a certain physical phenomena do differentiate the direction of time.

Technically, the trivial solution in many differential equations and in linear algebra is “zero,” since things may be identically zero. This is considered the trivial solution, because it’s not informative- it is however one of the solutions, and since there’s other math that tells us how many solutions there may be, making note of it is appropriate.

But yes, in your example above, -4 would be a trivial solution. You are, however, mistaking the value of mathematical models.

Mathematical models tell us the limitations of results in, speaking roughly, purely logical space. In our universe, you can’t have negative people who aren’t pessimists or try to operate hair dryers while asleep in the bathtub. This doesn’t mean that in all possible universes there isn’t one where people are solely negative, this requires adding real-world limitations to the solutions.

We do it this way for completeness (to be sure we haven’t missed anything), and those inclusions of exceptions are often valuable long-term as solutions to problems later, when tech evolves.

As to multiple universes, it sounds like you don’t understand the science. Research Compton scattering of electrons, the double-slit experiment, and quantum interference, as well as the Born interpretation of the Schrodinger equation. Shortly, multiple universes was proposed- I want to say, but my memory is fuzzy, originally by Dirac- to explain effects such as Compton scattering.

It all depends on how good the mathematical model is. And you determine that by testing it, empirically, to see how good its predictions are. People were skeptical about Relativity until it’s predictions were born out by experiments.

Maybe I wasn’t clear. If mathematical theory X has proven to be correct about a number of its predictions, then I’m willing to give it more credence about things yet to be proven. If it’s a fresh new hypothesis with little or no testing done, then not so much.

I’m not sure there is really a whole lot to debate here, unless I’m misunderstanding your question.

Your thinking reminds me of medieval attitudes towards solutions of quadratic equations. If they encountered such an obviously “unphysical” root (because you can’t, after all, have -4 people), they ignored it.

But ignoring it deprive you of seeing the entire panoply of solutions, and of deriving general rules and trends that can be clearly sen by considering all such solutions.

It might not be possible to have -4 people, but there are real-life situations in which “-4” of something IS a real, physically realizeable solution. There are even situations in which “4i” of something (the imaginary number) is a physically rrealizeable solution, or even a complex number like “-3 + 4i”. Just look at the solutions for the motion of a driven damped harmonic oscillator, for instance.
The thing is, you don’t know what constitutes a physical solution or a useful result until you examine it. If you simply toss out what you don’t think is reasonable you may be depriving yourself of not only something useful, but something that yields a breakthrough insight.

The first de Broglie equation tells us the wavelength of a particle given its mass and velocity. It would be natural to regard it as an erroneous mathematical derivation if it hadn’t been confirmed over and over again.

I don’t think that’s a good example, because if you’re applying mathematics to any real-world situation, there are necessarily certain assumptions you’re making and restrictions you’re requiring. In the problem you list, the implicit assumption is that the answer is a natural number. You’d only allow negative numbers as a solution if you were talking about a vector quantity or some such.

Minor nitpick: -4 would not generally be considered a trivial solution. A trivial solution generally indicates an obvious solution. A solution such as the one indicated here would usually be called superfluous, or something synonymous

For one thing, that stimulates efforts to find some clever method that WILL let the phenomenon in question be verified. And because historically, claims that something can never be observed or verified and is merely a mathematical construct have often been proven wrong.

Also, I suspect that a lot of scientists don’t want to end up like the ones that said the same thing about atoms or subatomic particles, and quoted by future scientists in lines like “Dr Smith always insisted that ‘multiple universes could never be proved, and should be regarded as just a mathematical abstraction.’ Now I have a transdimensional scanner in my lab. Ha !”