Have you heard of Bertrand Russell, the famous mathematician and logician, and his quest to prove that 1 + 1 = 2? Well I’m simplifying a quite a bit, but Bertrand Russell was on a quest to explain the foundations of mathematics, In his book Principia Mathematica he had three aims: (1) to analyze to the greatest possible extent the ideas and methods of mathematical logic and to minimize the number of primitive notions, axioms, and inference rules; (2) to precisely express mathematical propositions in symbolic logic using the most convenient notation that precise expression allows; (3) to solve the paradoxes that plagued logic and set theory at the turn of the 20th century, like Russell's paradox.
While I will not discuss the points (2) or (3), I commend him on the attempt to tackle point (1). While I straight on confess my poor level of mathematical understanding, I share Russel’s agony on axioms, or postulates, as they are also called.
You see, us humans have a tendency to assume that certain things are self-evident, even though there are no proofs for them. While I couldn’t give an example of the postulates in mathematics, the postulate in physics that has caused me the most harm is the postulate that molecules move freely in gas.
This seems rather intuitive, as we have been taught this from a rather early age on. The problem is that we have had no way of knowing whether this is true or not.
One might respond, by saying that isn’t all modern physics built upon the foundation of freely moving molecules and doesn’t current physics offer an explanation to all known phenomena. The short answer is kind of. Quantum mechanics explains some phenomena and general relativity explains the rest. But to combine the two, you require the current string theory, and that requires ten or more dimensions.
One might even pose a question of whether there is a problem in one of the postulates in physics. And here we get to the free movement of molecules. In 1738 Daniel Bernoulli published the book Hydrodynamica, in which he stated that gases consist of great numbers of molecules moving in all directions (among other claims). Curiously, the theory wasn’t immediately accepted.
Gradually the theory was expanded to explain experimental data much better. As the theory matured, it’s power of explanation grew and gradually no one thought that the postulate of free movement of molecules should be revisited.
As I mentioned in my previous post, this apparently harmless postulate was carried on to Einstein’s explanation of the photoelectric effect and subsequently to quantum mechanics. While these theories require light to be a massless particle and the nature of reality to be probabilistic, no one thought this might be because of a faulty postulate, because it was easier to believe that the nature of reality doesn’t follow common sense than to assume that you can form mathematical models of the world, which could have perfect predictive power, but be nevertheless wrong.
Of course, the word wrong is rather subjective. If you consider rightness to be defined whether the theory leads to useful results, then the theory is right. However, if you believe the underlying philosophy of intangible energy levels and particles that have no exact location, then the theory is wrong.
This again leads to a conundrum: if something can be mathematically accurately described, but is still wrong, are the mathematics universal? While this appears to be a question of semantics, it really isn’t. If the true nature of reality is of Planck spheres traveling at the speed of light, can I be sure that the current mathematical notation, such as the Schrödinger equation or the Hamiltonian describe individual Planck spheres accurately? The answer is that I cannot. But neither can I say that they would be wrong.
The approach I took was quite unorthodox. Ignoring the time element for the simplest descriptions and only focus on the geometrical side of the interpretation of absorption peaks as representing true sizes of physical objects. What this temporary rejection of the Schrödinger equation does is ignoring whether the equation applies or not. If one is to assess the supramolecular orbital as collection of molecules, the molecules can be simplified into beads on a string. Only after it has been confirmed whether the supramolecular orbital model has better predictive power to describe one phenomenon, should it be tested in other phenomena.
This means that while the model appears correct in one case, there is still a possibility that it can be accidental or forced. However, if one cannot be sure of the predictive power of theories based on an incorrect postulate, a mathematical model based on this false postulate should not be the first tool to test the new postulate.
I compare this to a Norwegian correcting the essay written by a Swede. Yes, the two languages are almost the same, but there are enough differences that the essay might look wrong. But if you understand that you are not assessing the same language, then you are more forgiving and understand that the essay might be fine, but your own knowledge of the language might be insufficient.
Choosing far fewer postulates and rejecting at least some of the old ones makes me a lonely chemist. I don’t know the old mathematical language of physics, nor do I know the new language that is bound to be required. I’m more like the boy who realized that the emperor has no clothes.
So what are the current postulates?
1 The molecules in a gas are small and very far apart. Most of the volume which a gas occupies is empty space.
2 Gas molecules are in constant random motion. Just as many molecules are moving in one direction as in any other.
3 Molecules can collide with each other and with the walls of the container. Collisions with the walls account for the pressure of the gas.
4 When collisions occur, the molecules lose no kinetic energy; that is, the collisions are said to be perfectly elastic. The total kinetic energy of all the molecules remains constant unless there is some outside interference with the
5 The molecules exert no attractive or repulsive forces on one another except during the process of collision. Between collisions, they move in straight lines.
The thing is, when Daniel Bernoulli had the first idea of these (I don’t know whether they have changed at all in the past 300 years), his idea of atoms was very limited. If he had an idea that we could think of an atom, it would be a solid, indivisible object. However, when it was understood that atoms, let alone molecules aren’t solid objects, bells should have rung on the minds of people applying the kinetic theory of gases, asking whether we can retain these postulates. To be logically consistent, the postulates should have reduced to the smallest known particle as ever smaller elementary particles were found. Conversely, what was done instead was not to apply the kinetic theory of gases on these smaller particles, but rather a series of forces were introduced that acted between these particles without them touching. Because everything was mathematically consistent, perfect even, no-one thought to question this model.
But then came string theory. It was finally a theory that would be the theory of everything. It was just perfect, but with a teeny tiny caveat. It required a world with ten or more dimensions. Surely someone bright would have thought: “Hang on: do we have our postulates right? If our mathematics seems to be consistent, but only in a world with extra dimensions, there must be something wrong with the postulates.” To my understanding this never happened. Conversely, the physicists working on string theory became convinced that the world is even stranger than we ever thought, and most people not working on string theory thought that these people are probably out of their minds. But what I they’re right all along?
The thing is that in string theory everything is comprised of one-dimensional objects called strings. But what if this one-dimensional object would be a three-dimensional solid particle that doesn’t follow quantum mechanics, but whose properties generates quantum mechanics. This phrase might sound trivial but is actually very profound. You see, if the rules of the kinetic theory of gases are not for gases, but for these infinitesimally small spheres (most probably with a diameter of Planck length), then we don’t have to ditch the postulates, we just have to draw them much further back than when Bernoulli first thought of them. These revised postulates would be:
1. There are solid particles with a diameter of Planck length.
2. These are in constant motion at the speed of light.
3. Some of these particles are in direct contact and some collided with each other.
4. When collisions occur, they lose no kinetic energy; that is, the collisions are said to be perfectly elastic.
5. The particles exert no attractive or repulsive forces on one another. If not colliding, or being pushed, they move in straight lines.
That’s it. As far as I understand it, this is the theory of everything. Doesn’t sound very grand. There are no equations. There is a possibility that the first equation after these rules is the Schrödinger equation. But how I understand it, the equation is an unbelievably precise approximation of the truth, but whether it’s the whole truth, the jury is still out.
I’ll leave this post as it is for now, but I will probably revise it in the future and add some links.
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