Why is there something rather than nothing?

I recently discovered a very nice talk by Lawrence Krauss entitled "A Universe from Nothing". 

The actual talk starts at 12:48, and discusses a topic which used to be in the area of theologians and philosophers, but as the gaps of understanding shrinks science is now able to provide a plausible (but not yet definitive) explanation about how it all began. Even if you follow topics in astronomy, I think it is still awe inspiring to contemplate our place in the universe and realize that if you magically remove all visible matter this would hardly change anything in the evolution of our universe. Trillions of years in the future, the accelerated universe expansion would make all other galaxies disappear from our view and we would not have any experimental means to test the ideas of Big Bang and inflation. All of this would only be hearsay from distant past (our time now) if we somehow manage to survive that long, which is highly doubtful. 

So why there is something rather than nothing? Simply put, "nothing" is unstable because of  the combination between relativity and quantum mechanics. In quantum mechanics we have the Heisenberg uncertainty principle which shows that if we measure the position very precisely, the momentum must have a very large uncertainty. But this uncertainty would translate into very large speeds (even higher than the speed of light) and the only way out is to create particle pairs. Hence the vacuum is filled with virtual particles which pop in and out of existence short enough to obey the other Heisenberg uncertainty principle: the energy-time relations. But when you add gravity in the mix, this changes everything and the virtual particles become very real. And what would the total energy of such a system be? Precisely zero.  Then a natural question to ask is: what is the total energy of our universe? It turns out that this can actually be measured (we live in a special time when this is possible), and surprise, it is zero!

I won't spoil the video with additional information from it, please watch it, it is very nice, and instead I will focus on a part of it which I am disagreeing with: the role of the question "why?"

One common mistake people make is to think that correlation implies causation. This mistake is very easy to make because of our day to day experience. Krauss is not making this mistake, but another one which is also rooted in daily experience: why implies purpose. I will attempt to argue against this position and show that "why?" is a scientifically valid question which leads to genuine scientific answers.

Let me start with the concept of truth. In mathematics a statement is true if we can construct a proof starting from an axiomatic system. Now this concept of truth is not universal as it was shown by Godel. If your axiomatic system is rich enough to encompass arithmetic, Godel showed that any such system is incomplete and there are statements which can be neither proved nor disproved. If we augment the original axiomatic system with such a statement we create an enlarged axiomatic system. However, we can enlarge the axiomatic system with the negation of the statement and obtain another enlarged axiomatic system. Now the two enlarged axiomatic systems are incompatible with each other, and moreover, the process can be repeated. What this shows is that mathematics is infinitely rich and not axiomatizable. And so the concept of truth is parochial in mathematics.

But there is a second notion of truth which comes from nature: something is true if it is in agreement with reality. Physics uses this notion of truth because physics is an experimental science. Then a natural problem is to compare and contrast the two notions: the mathematical and the physical one.

When we answer "how?", we use the mathematical concept (and the "unreasonable effectiveness of mathematics"), but when we answer "why?" we use the physical approach. 

Let me illustrate with quantum mechanics. On the mathematical and how side, one starts with the usual axioms: a Hilbert space, observables are Hermitean operators, etc. However, on the physical and the why side, one can start from a physical principle: the invariance of the laws of nature under composition.

In the how, mathematical side, you build mathematical proofs, but on the why, physical principle side, you select distinguished mathematical structures from the infinite collection of the Platonic world of math which respect the physical principles. All mathematical structures are unique, but only a handful are "distinguished" and used by nature. "Why?" is a "distinguish-ability" question. Nature does not use hyperbolic composability for example because this violates another physical principle.

So why quantum mechanics, why special relativity, why general relativity? Because of the physical principles of invariance under composition, the physical principle of relativity, the physical principle of the equivalence between inertial and gravitational mass. 

And where are the physical principles coming from? Here I have a suggestion: they essentially encode the difference between the real world and the abstract world of math. Take hyperbolic quantum mechanics. It almost had a chance to describe something real, but it is a mathematical impossibility to construct a state space and hence to have an objective way to assign truth and make experimentally testable predictions. Positivity is a property of the physical world which is forbidden in the mathematical world by Godel's theorem.

So the question why is actually very meaningful and moreover it can create mathematical consequences which can be put to experimental tests. In physics why does not imply purpose but "distinguish-ability" of the mathematical structures which play a key role in the physical world. The reason Krauss disagrees with it has to do with the abuse of the question by theologians who answer why? with "because God". Unlike Europe, US is a very religious place where science is under constant attack by religious bigotry who enjoys significant political power (the video reference to Arkansas and Ohio was criticizing attempts to teach in public schools "intelligent design" as an alternative to evolution who is only "a theory": https://en.wikipedia.org/wiki/McLean_v._Arkansas https://en.wikipedia.org/wiki/Intelligent_design_in_politics).

I agree that theologians are "experts of nothing", but philosophy should not be lumped with the area which presumes the answer before you ask the question. However philosophy does not create new knowledge and only provides an interpretation of what science discovers. The physical principles behind special relativity and quantum mechanics were not uncovered by philosophical contemplation, but by solving concrete physical problems. It is also true that the change of the paradigms is not at all easy even if you have solved the concrete problem. Case in point, Lorentz discovered his transformations, but the proper paradigm was discovered by Einstein.