Friday, May 2, 2008

Yes, Everything is Math

The student of physics cannot help but notice, and perhaps resent, the central place of mathematics in the curriculum. Doing physics is nearly synonymous with writing equations, and the centrality and sophistication of mathematical methods in physics have only grown over time.

Why is this? Why does mathematics seem to be the language of physics, and even of science as a whole? Is it an accident, something which might have been otherwise? Is it something even to be regretted - a bad joke played upon us by the creator, who could have created a more-fun universe but chose not to?

I believe it is not an accident. In the following I will argue that any possible universe must have a rigorous mathematical basis, just as we have discovered in our own.

Observe first that a universe must have rules. Donkeys don't fly, horses don't turn into supernovae, bugs don't become endless list of constraints operates at all times in our universe, and it seems clear that a similar list must apply in any universe. Any universe must contain things which are distinct from other things, and every distinction implies some kind of rule - thing A doesn't spontaneously change into thing B. No rules implies no things, i.e., no-thingness - nothing.

So where do these rules come from? Could we design a universe be by listing out each of them in "plain-english" form as above? You can try but - good luck.

For starters, you would need an infinite number of rules. More seriously, it isn't even possible to define any of the terms used in such rules. What, for example, is a bug? If a frog eats a bug, is it still a bug? When does it stop being a bug? Is a partially developed larva a bug? Is an animation of a bug a bug? Is a mutant or genetically altered bug a bug? Virtually all plain-english concepts are impossible to define, and therefore not adequate for specifying a universe.

For example, we said a bug can't become a buggy. If we define a bug as, say, something which has more than 4 legs and an exoskeleton, then we have not ruled out the possibility that a mutant bug with 3 legs and no exoskeleton could turn into a buggy. We also have not ruled out the possibility that a bug could turn into a Lincoln Continental; indeed, we can't even start this discussion without first defining the terms "leg" and "exoskeleton", which we will find impossible to do.

Philosophers have wrestled with these problems of definition for millenia, and there is no solution. The higher-level concepts embodied in plain-language terms cannot be fully defined, but are inherently fuzzy and subjective.

Now, why in reality does a bug not become a buggy? Obviously it is because the bug is made of atoms, and the atoms don't spontaneously rearrange themselves or change their characteristics. And why is this? Is it because we have some fundamental rules governing atoms, rules like "sodium can't spontaneously turn into chlorine"? No - because atoms are not fundamental building blocks either, and cannot be rigorously defined any more than bugs (is an ionized atom still an atom? Is an unstable atom still an atom?)

Atoms act the way they do because they are made from electrons, protons, and neutrons. Protons and neutrons, in turn, act the way they do because they are made from quarks.

And now we are getting someplace, because both electrons and quarks are fundamental, mathematical objects (at least in current theories). In other words, they can be defined completely. We can write down by means of equations exactly what they are and what rules they obey, under all circumstances, with no caveats or gaps. These are the kind of rules on which a universe can be based, and they are called the laws of physics.

So I would argue that the universe is built on mathematical objects because these are the only objects which can be comprehensively defined. No other kinds of objects can exist except as aggregated constructs of underlying mathematical building blocks (e.g., a bug is built from electrons and quarks). The underlying laws of physics are mathematical because no other kinds of laws exist. In creating a universe, the choice is not whether to base it on mathematics, but only which mathematics to use.

None of these arguments are original to me, of course, although I haven't heard them expressed in quite this form. The essential ideas, including my discussion of things not changing to other things, go all the way back to the original Materialists, as recorded by Lucretius. The original Materialists, interestingly enough, based their Materialism not on "scientific evidence", as is the custom today, but rather on exactly the philosophical arguments outlined above. Of course neither philosophy, nor science, nor any other technique can ever prove anything definitively, so we don't claim to prove that math must underly everything; however, we do claim that the case is pretty strong.

Recognition of the primacy of mathematics allows us to formulate a different conception of science and the scientific method, one which frames the debate with Creationists and other pseudoscientists in a different light. Science is the study of the underlying mathematical laws of the universe and the effort to connect all observed phenomena to them. The "scientific method" is nothing but common sense applied to this effort. There is no single method of science, just as there is no single type of argument in a legal case; however, there is a single goal to the endeavor of science, and it is by reference to this goal that we, in fact, distinguish science from pseudoscience. Science proposes explanations which are potentially connected to an underlying mathematical order; pseudoscience proposes explanations which are not. The concept of "refutability", which is very slippery to define in general, becomes crystal clear from this perspective: non-scientific theories are "irrefutable" because they can't be founded on mathematics and therefore do not follow any definable rules - and that which is not bound by rules can never be refuted.

We also find a new perspective on the concept of Materialism. Materialism is inseparable from Mathematics, and the "material" to which it refers can only be a mathematical construct - because no other construct is possible. This "Mathematical Materialism" is the necessary foundation of any universe. In doing science, we don't "discover" that the universe is mathematical, but merely what kind of mathematics it employs.


Anonymous said...

I would explain it in a different way. Physics is arguably the first field of academic study to create quantifiable categories such as mass, velocity, momentum, force, and energy, and to relate these categories to each other quantitatively. Newton's work depended heavily on astronomy - on the observations and models of Keppler and Copernicus. The quantitative nature of the models is mathematical in nature. We find the models believable and useful because they are so accurate in predicting things we care about.

Mathematics is the language of physics and engineering. We use it because it enables us to make very accurate predictions about physical situations.

One can, of course, describe physical things without mathematics. Some of the sense of things is conveyed; but much of the deeper meaning and most of the predictive qualities are lost.

I once argued that classical physics could be taught without calculus. And I think that one can get far with a command of algebra. I think physics is important enough that anyone who has taken high school algebra should also take a physics class.

Newtonian physics is important in its own right; but it is important also as an example of what science is - an interrelationship between mathematical model ( which in Hume's terms lives in the realm of ideas) and an empirical, observable world ( which in Hume's terms lives in the realm of fact.)

Ulla said...

I liked this. It suits my non-physist mind. :)

You forget the wave-particle duality. What you say is only of particles.

The way you think about the least stable matter is also very interesting. It is the quantum criticality. But the matter condensation? The dark matter? The fermione emergence? Where is the math in matter condensation or dark matter? Where is the 'nothingness?', the 'border'?

This is also big news: And now we are getting someplace, because both electrons and quarks are fundamental, mathematical objects (at least in current theories). In other words, they can be defined completely. We can write down by means of equations exactly what they are and what rules they obey, under all circumstances, with no caveats or gaps.

I do not know this is possible yet, only in TGD, and it is not accepted. Not at LHC can it be done. They act like a blind man.

A great blog. Congratulations.

Will Nelson said...

Thanks for your comments!
Regarding the wave/particle duality, and the other examples, I would argue that they are all mathematics. It is as mathematics that they make sense; they only seem to violate logic when stated in English.
Of course there are some things whose mathematics we don't fully know yet, like dark matter. We have many mathematical theories for them, and I presume that eventually one will be found to be correct.

freindlygeek said...
This comment has been removed by the author.
freindlygeek said...

I know it's 2 years old article, but I'd like to leave a comment.

I enjoyed reading this article! It's very easy to follow how you explained what you are trying to say.
I studied physics and biology in undergraduate. I think it is pretty apparent why the importance of role of mathematics in biology is increasing these days since to the bottom layer its all about physics which is described in math!

Also I think it is a clearness when people like math and physics, including me. Though I found it very interesting about uncertainty in quantum physics,which is against the ability of accurate prediction of mathematics. I could be wrong.

Anyhow thank you for a great article.

Will Nelson said...

Glad you liked the article! I, too, am in biology now, and it does seem like the reductionist approach is continuing to be the main avenue of progress there. Regarding uncertainty, I would argue that it's still basically mathematics, just the mathematics of probability. That is how it appears in Quantum Mechanics; there is no deeper, "non-mathematical" source of the uncertainties, at least not that we know of.

Will Nelson said...

About uncertainty, I would further point out that, although the outcomes are only defined probabilistically, the set of possible outcomes must be well-defined, and this also requires a mathematical foundation. An atom may decay to other particles, and there is some probability for doing that, but an atom will not simply vanish leaving a gaping hole in space, nor turn into Beethoven's 9th symphony. But one can't frame such constraints in "plain english", because it is not sufficiently precise to define anything. Only mathematics provides the well-defined objects one needs to define the possible outcomes, even in a probabilistic theory.

Anonymous said...

If the pseudoscience you've come across doesn't have mathematics, consider yourself lucky. There are entire fields of what looks like highly technical physics but is actually fake.

Ironically, the "TGD" Ulla mentioned is one of these.