Saturday, April 19, 2014

Does believing in god accomplish anything?

This is a bit off-topic for my blog but I just find it very interesting. What do people get out of believing in god?

Belief in god doesn't, by itself, provide any moral guidance. For this one needs a whole set of more detailed beliefs about god's nature, and these beliefs are just free-standing moral beliefs that don't really have any connection to whether god exists or not. This is the famous Euthyphro dilemma of Socrates.

Belief in god also doesn't provide any meaningful explanation of ultimate origins. One may say the universe exists because god created it, and then the obvious followup is what created god? The typical answer to this is that god always existed and didn't need to be created; to which the equally obvious response is, then why can't the universe have always existed without ever being created?

Belief in god provides no hints to any of the remarkable things that have discovered through science. To learn about these things the believer has to study the exact same books as the atheist, and with the exact same amount of dedication.

Lastly, belief in god doesn't provide any understanding of the purpose of existence. Most religions explicitly state that god's purposes are beyond our understanding, hence not explicable to either believers or non-believers. If a religion does provide some concept of purpose, then this again is a free-standing belief relating to values, similar to a moral belief, without any clear connection to the existence of god.

I'm certainly not the first person to ever point these things out. Many believers are more or less aware of these problems as well, yet belief persists. Why?

My best guess is that, even though one doesn't gain any direct information about anything, belief in god still provides a feeling that things might be meaningful or purposeful in some way that isn't possible without god. It is difficult for people - including me - to accept that the whole universe has no more meaning than a bunch of math equations, and that we, ourselves, are just bunches of atoms of no significance to existence as a whole.

Nevertheless, having a feeling that things might be meaningful doesn't tell you what the meaning is, and having a feeling that morals really matter doesn't tell you how to act morally. Belief in god doesn't seem to add anything but confusion to human-scale discussions of any of these topics. So next time someone asks whether you believe in god, answer them with a question: "what difference does it make?"

Why the speed of light is constant in relativity

This a perennially confusing question which I attempted to answer on Quora ( I also wrote a whole book, "Relativity Made Real", to provide a more detailed answer. Contrary to popular expositions, there is no really short and meaningful answer to this question, but here was my effort on Quora to condense the issue (and my book!) to its essence:

Special relativity is best divided into two conceptual pieces. The first, and most important, is a qualitative observation that all objects must be affected by motion. This is really not very surprising when one considers the structure of matter, i.e., electrons and nuclei held together by electric forces. The electric forces take time to transmit between the particles, so when the object is set into motion then obviously there are some very complex changes to the forces inside the matter. It would be remarkable indeed if its shape and size did not change, along with the rate of any processes it happens to be experiencing (e.g., ticking, if it is a clock). 

A good analogy is to imagine the individual electrons and nuclei as tiny people holding megaphones. They try to arrange themselves in a nice crystalline structure by shouting back and forth at their neighbors through the megaphones, and estimating their distances based on the response times and the volumes of the voices. The sound waves they exchange are analogous to the electric field, which also is transmitted by a type of wave, namely electromagnetic waves. 

If we now set this "object" made from tiny people into motion, it will be thrown into disarray, because now the sound waves will take more or less time getting from one person to the next, the voice volumes will be changed, and all the calculations will be off. The collection of people will not be able to maintain the same shape, and likewise a collection of moving atoms cannot maintain its same shape.  

But if all objects are physically changed by motion, then so are measuring devices like clocks and rulers, which immediately implies that moving observers will also measure different values for almost every quantity. There is nothing sacred about measurements; they are carried out by ordinary physical objects, and the results they produce are dictated not by prior principles or philosophy, but by the physical system they are embedded in. 

The situation we just described could clearly become extremely complicated, with arbitrarily complicated motion effects. It is indeed very easy to write down arbitrarily complicated laws of this kind, and we wouldn't (and probably couldn't) live in such a universe. Here is where the second piece of the relativity puzzle comes into play: the "principle of relativity" postulates that our actual laws come from a subset of this larger, possible set, a subset which has a very special property.

When one studies the effect of motion on objects which are "held together by waves", as described above, one finds that there is a very surprising mathematical possibility for their behavior. The various waves involved can be structured in such a way that the effects on moving objects are exactly calibrated so that all observers will always measure the same speed for the waves ("speed of light"). It is extremely non-obvious that this is possible, and it certainly is not necessary in any way; our universe could have been built otherwise. (And it could also have been built without any waves at all, in which case Einstein's relativity would be impossible, and we would be discussing only the relativity of Newton/Galileo).  

This "principle of relativity" essentially postulates that our universe has the simplest kind of laws it can have, given that it is built on a foundation of waves. It places great restrictions on the allowed wave laws, to the point that many effects can be computed without even knowing anything else about those laws. That is why the "principle of relativity" appears to function as a free-standing law on its own, even though it is really a property of the underlying quantum wave laws of physics (for example, one can write down the Standard Model of particle physics without ever saying the word "relativity"). 

So the principle of relativity does, indeed, imply that everyone measures the same speed for light; in fact, that is the entire content of the principle. But under the hood what it is doing is picking out a certain very special subset of the enormous collection of possible wave theories, and postulating, rather hopefully, that our universe is described by only these kinds of wave. It certainly didn't have to be that way, but if it wasn't then it would be so complex that living creatures would probably never have evolved to discuss it.

Monday, April 14, 2014

Why Quantum Mechanics Requires Complex Numbers

Why do complex numbers feature so prominently in quantum mechanics, when classical mechanics got by just fine without them? 
Scott Aaronson gives this explanation, which revolves around the idea of wanting to assign a meaning to "negative probabilities":

I don't find this convincing, because I don't see a reason why nature should care about using one sort of probability over another. There's no physical gain from doing this, in the sense of enabling a universe that we can live in. 

Rather, I would argue that the wavefunction has to be complex in order to have enough information to encode both position and velocity of particles into one function. A real-valued function works for position or velocity separately, but to have both in one function one needs the complex phase.

But why does one want to stick both x and p into one wavefunction in the first place? Here is where the "physical" benefit comes in. Having both x and p encoded into one single wavefunction partially removes their independence, by making them connected through the uncertainly principle. This has very profound effects at the microscopic level, and most importantly it allows things in the universe to be stable.

Let's back up for a second and try to imagine a world built entirely using classical physics. Atoms would then be like little solar systems - and this would be terrible because classical orbiting systems are all different, so no two atoms would be alike, and they are also generically unstable. For example, classical particles can orbit as close as they like to the center, so over time they will give up bits and pieces of energy (e.g., through weak interactions with other atoms) and gradually fall to the center. It would be completely impossible to evolve living creatures using this kind of inconsistent and unstable building block.  

Similar problems afflict the classical theories of fields, in particular electromagnetism. There is an infinite range of frequencies, and classical physics allows each frequency to hold any amount of energy, however small. All the energy in the universe would then leak gradually into higher and higher frequency electromagnetic waves, and would become essentially useless. This is the so-called "ultraviolet catastrophe" which Max Planck was trying to solve when he discovered the quantum. 

So, classical physics is just not suitable as the underlying theory for a universe that can support life, because its components simply have too much freedom. The particle motions are not constrained enough to form consistent and stable building blocks, such as atoms, and the fields are infinite energy sinks that drain away all available energy. 

These problems are solved by quantum mechanics, and in particular what solves them is to encode position and velocity both into one wavefunction. Being entertwined in one function means they are not fully independent, and in fact the relationship is exactly the famous uncertainty principle (see e.g. 

The uncertainty principle for particles means that squeezing a particle into a smaller space causes it to have higher velocity. Now remember the problem (one of them) with atoms in classical physics, namely that the electrons can orbit as close as they want to the center. This can't happen anymore because squeezing the electron close to the center makes it move faster, which carries it away from the center again. In other words there is a minimum size for the electron orbit - the "ground state" - and moreover its size and shape is completely determined by the uncertainty principle, hence is exactly the same for all atoms. This creates the stable and consistent building blocks needed to evolve life. (Of course things get more complicated for additional electrons and the higher energy orbits, but the principle is the same). 

Now consider the electromagnetic field. Here the uncertainly principle implies that a mode with small wavelength ("squeezed into a small space") must oscillate faster, i.e., have higher energy. Again there is a tradeoff and the result is that for each wavelength there is a minimum unit of energy that it can transfer - the quantum. The smaller the wavelength, the larger the unit, and this prevents energy from dribbling bit by bit into that infinite pool of wave modes, because for short wavelengths the mode can only accept large chunks of energy at a time. Lesser amounts of energy are therefore stabilized and don't get drained way. 

 In short, it’s just very hard to build stable systems on a classical foundation because classical particles (and fields) have too much freedom. Hence the subject of classical chaos theory, which has no really QM analog. quantum mechanics solves this stability problem, and the complex-valued wavefunction lies at the heart of the solution. I haven’t seen any proof that QM is the *only* way to solve the stability problem, but I haven’t seen any other way either. 

Wednesday, April 9, 2014

Demarcating the difference between science and non-science

Here is an article lays out pretty clearly the current conventional wisdom about what distinguishes science from non-science, the so-called "demarcation problem":

In my opinion this is, unfortunately, not an adequate view. I use the term "unfortunately" here with full intent, because it really is unfortunate that the true distinction between science and non-science is something more abstract and not easily comprehensible to a layperson.
In fact the only possible demarcation between science and non-science is mathematizability. Scientific theories are those which could, in principle, arise from an underlying fully mathematical structure. This obviously includes evolution which arises inevitably from the molecular basis of life, which in turn arises from the purely mathematical theory of elementary particles.
By contrast, any theory which involves a god is inherently not reducible to mathematics. Indeed, this could be taken as the definition of a god; I doubt any believer's conception of god corresponds to an entity whose every aspect and action is governed by mathematical formulas.  
If, on the other hand, the universe is not described by mathematics at the deepest levels, then there is no underlying structure and no point even talking about science. If there is no underlying structure then anything is possible at any time. Any regularity that we happen to observe and study with "science" is not a reflection of underlying order, which by hypothesis does not exist, but rather is just the whim of gods or something like that. 
Personally, I think this scenario is not only incompatible with the existence of science, but actually  impossible, because regularity is necessary for existence, and regularity comes only from mathematics. Hence, in my opinion, all universes which can possibly exist will have science, and all for the same reason, namely that they are founded on mathematics. 
What about falsifiability? In the view I propose here, this is closely entwined with science, but not absolutely essential in all cases. 
Note first that, absent a mathematical foundation, falsifiability is clearly impossible since it is impossible to formulate a falsifiable statement that is not compatible with reduction to mathematics.  A falsifiable statement is something like "95% of chicken eggs have one end more pointy than the other". This statement is compatible with reduction to mathematics because, in fact, it arises in our world through the mathematical theory of the atoms from which chicken DNA and whole chickens are built. In general, any falsifiable statement  must reflect a regularity of the universe which we can observe, and any such regularity is compatible with reduction to mathematics, since mathematics consists of the study of all regular structures. 
Non-falsifiable statements, unfortunately, are not quite so simple. They fall into two types (that I am aware of). The first involve entities such as gods which, by definition, are not reducible to mathematics. They can never be falsified because the entities involved are not, by definition, governed by any kind of mathematical laws, and hence they don't obey any rules that could conceivably be tested. They just "do what they want" (except that, as I mentioned above, I don't believe that can exist at all).   
There are, however, other sorts of non-falsifiable statement which are compatible with mathematics. As we push the possible bounds of human knowledge, we are running into such statements in the form of multiple universe theories and the anthropic principle. Multiple universes are easily and naturally predicted by many mathematical theories, yet it is difficult to see how we could ever have clear evidence for their existence. Nevertheless they clearly make sense, and could exist, and hence our  notions of science must expand to encompass these ideas. 
The reason such ideas remain science, rather than pseudoscience, is precisely  that they are compatible with a fully mathematical theory of the universe. Indeed, the best evidence we are likely ever to have for the multiverse is that is seems to be an inevitable outcome of some extremely compelling mathematical theory that explains many other things that are fully falsifiable in our own universe.  
Falsifiability, therefore, retains its central role, in the sense that we can never believe any theory, no matter how mathematically amazing it is, if it doesn't make some predictions that we can actually test. However, it need not have only falsifiable consequences, and we must expand our thinking to include this possibility. Part of this expansion requires that philosophers of science and laymen both must finally come to accept the absolutely central, and completely non-accidental, role that mathematics plays at the deepest levels of physical existence.