Math and Reality

For the past few days, I've been bothering my TA about thermodynamics, specifically about the following three equations:

U = T*S - P*V + µ*N
dU = T*dS + S*dT - P*dV - V*dP + µ*dN + N*dµ
S*dT - V*dP + N*dµ = 0

Basically, my TA stated that all the variables depend on one another, so the integration cannot be done directly. Also, she stated that the "complete" forms of all the variables (e.g. S(U,V,N)) are unknown.

I replied saying that, from a mathematical perspective, if T, P, and µ are constant, all three equations are automatically satisfied. Thus, if S, V, and N are only taken for specific T, P, and µ, the S, V, and N must adjust themselves accordingly to continue to satisfy the equation. Otherwise, the original equation for U cannot possibly be valid. Also, if points in S, V, and N are found in a specific T, P, and µ, then:

U2-U1 = T*(S2-S1) - P*(V2-V1) + µ*(N2-N1)

is valid for all S, V, and N, given that they actually have equations. My TA subsequently stated that she didn't trust my math. This started a debate between us, in which I think she became quite annoyed.

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What I said is below:

Let's consider a very simple function:

f(x) = sin(x) + x

At small x, f(x) is very approximately 2x, but for large x, x overtakes sin(x) and makes it seemingly "disappear". Therefore, the function we use for minuscule x is f(x) = 2x, which is reasonable. The function we use for large x is f(x) = x, which is also reasonable. However, that doesn't mean that f(x) = sin(x) + x is not the complete solution.

To translate this to a more realistic example, at low speeds, we generally measure lengths of L, because the sqrt(1-v^2/c^2) adjustment in L*sqrt(1-v^2/c^2) is insignificant at low speeds. However, that doesn't mean that sqrt(1-v^2/c^2) adjustment should cause the equation to fail for length measurements at low speeds.

My belief is that if a mathematical theory is complete, no adjustments are necessary to cause all the equations to work. (Note that I'm using "complete" differently from "correct".) All parts of the equations should work, and the only portions of the equations that should ever be thrown out are those that are significantly smaller than the other parts of the equation.

Thus, if a theory is constantly relying on approximations, there is a problem with its definition.

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Her reply:

People are not mighty enough to figure out that the real form your f(x) is actually sin(x)+x. But we still want to push forward our knowledge and live better, so we make approximations. There is nothing wrong with approximations. Of course there are problems, but can you solve it? Scientists are trying to solve those problems all the time, but it doesn't prevent us from using approximations to benefit us now. Look at the engineering products. Don't you think the science they used are all approximations? We all benefit from the "approximation products". In making an airbag in cars, should the engineers use the ideal gas law or complicated statistical mechanics gas laws? The ideal gas law works just fine and it saves lives. Isn't it great?

And yes, people do make simple modifications of the ideal gas law (like Van Del Waals equation), because life is a tradeoff. To achieve certain goals we use certain tools. If you only want a rough estimation why do we want to go to those more accurate but complicated laws?

I think in your learning, you should be able to acknowledge the importance, the usefulness, and also the limitations of those laws. You should appreciate the efforts that humanity has made so far. If you really want a perfect and unbeatable law, you can try to develop one yourself. I hope that your way of thinking does not lead you to reject the knowledge humanity has accumulated in the past thousands of years. Humanity is always trying to do its best, so don't blame it on limited intelligence.

And science is never just science itself. It has human factors. It has historical and cultural backgrounds. I think you isolated science from other things, and you think that science should be perfect and it's a sole refection of the truth in nature. That was what I thought at first, but now I figured that this idea is not true.

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What do you think? Is nature governed by mathematics? If nature is governed by mathematics, are people intelligent enough to be able to discover the true forms of nature's functions? If all of the "true" physical laws are discovered, will the approximation solutions be more useful? Why is science so oriented towards making research data work than trying to find actualities?

mathmatics are just a way of relating natural processes. one doesn't govern the other. in fact they don't relate directly at all as one is entirely abstract. And the reason why science is so oriented towards making research data work is because that's what the people with the money shell out for. Scientists more often set out to find something specific as opposed to just finding something true.

I apologize if i just killed your god.

Your argument sounds strikingly for the existence of the Theory of Everything. I suggest you read about Godel's conjectures.

Why is science so oriented towards making research data work than trying to find actualities?

Are you suggesting that researchers create conclusions out of thin air? Traditional science is based upon observation. I'm not understanding what you mean here.

Quality of data in many cases means only a first order approximation (linear fit, possibly log linear) can even be made.

It's not a complete model of everything, but the more you take into account, the more complicated your model gets.

And the problem there is: Your results only have a certain level of accuracy. The more variables you split that up into, the larger the uncertainty on each variable.


I'll give a simple example, that's not likely to come up, but demonstrates the principle.

You have a fit f(x) = kx.
As is fairly normal when you fit a line, you can calculate a range of values for k that fit the model well.
So you have [k-t, k+t] where you're 95% confident the true value of k lies.
If you do t/k, you can normalize the amount of error. If your value of k=100, and t=1, it's a fairly good fit (1% spread on either side). Conversely, if k=1 and t=1, it's a very loose fit (100% error possible easily).

You could, if you liked, include several constants:
f(x) = ix + jx
But looking at i and j independently, each still has a range of [i-t, i+t].
If i~=j, then you have a worse fit in every case compared to the original model - from i=j=50, a 2% spread on each side, from i=j=0.5, a 200% spread on each side.

Adding extra parameters to a model only works if you have proportionally more data available. You lose certainty every time you add variables.


When you look closely at most of nature's laws, you find yourself in a field of stochastic responses. Fluids are, ultimately, a huge number of individual particles bouncing off each other stochastically, and the best "correct" gas model would include all these collisions, and use them to predict what happens to the gas next. This approach has two problems:
1) hard to measure the position and trajectory of 10^20+ particles at once
2) hard to simulate 10^20 particles
The ideal gas law reduces 10^20 variables to approximately 10, and still gives "good" results. When you have a 19 orders of magnitude reduction in computational complexity, with a corresponding loss of accuracy that averages out to very little, you just have to accept that the simplified model is "good enough" in most cases.


If you want to find out what happens in a physical situation exactly, the perfect mathematical model is the world itself. Nothing else can ever be as completely accurate as the original.

All models are wrong. Some models are useful. ~George Edward Pelham Box

Noah

And apart from the main question, can anyone tell me if I'm correct in my assumption that for any given constant T, P, and µ, the other three variables must adjust themselves accordingly for the equations to hold?

My TA keeps telling me that this is wrong, but she won't tell me why.

Why is science so oriented towards making research data work than trying to find actualities?

Actualities don't make money, stuff that works good enough, does.

Actualities is something for people with too much time, or future generations.

Oh btw, that scientist is called "van der Waals", not "Van Del Waals".