Gödel and the limits of proof:
Consider the statement, “This statement cannot be proved.” If it could be proved, of course, it would be false. So it cannot be proved. So it is true. Since the statement is in English, one can conclude that for any system of thought that includes English, there will always be at least something in that system that is true but cannot be proved within that system.
There was a mathematician named Kurt Gödel (sounds a bit like “girdle”) who lived from 1906 to 1978, who managed to encode the statement into arithmetic. He published his findings at the age of 25. Since that time, it has been possible to say that any system of thought that permits arithmetic will include things that are true but cannot be proven within that system.
It is commonly said, that this does not mean it cannot be proven at all. It just means that it can only be proven by turning to a more powerful system. And of course we did exactly that, choosing a line of logic to prove what cannot be proven in simple declarative English. But having chosen a more powerful system does not get us off the hook, because the new system will introduce new and more baffling statements that in turn cannot be proven.
Since life includes arithmetic, it follows that there are things in life that cannot be proven. That is no surprise.
It is also true that science includes arithmetic. So at any one time, there will be things in science that are true but cannot be proven. One must turn to a more powerful system.
Before you let that bald statement make you too terribly happy or terribly unhappy, let me point out that the more powerful system might simply mean more science. There will be those who will believe that there is a system of thought that transcends science that science itself can never prove or disprove. I have no problem with that, but it is not where I am going.
Let us take it that at the very least there are things in science that cannot be proven except for another system, possibly more science. As science advances, it accounts for more observations, introduces new understandings and becomes ever more powerful. Well and good. But it also is incomplete and will forever remain incomplete, no matter how far it is taken.
At that point, I am aware I am still treading on some toes. And I do so willingly. There are advocates of science who take the position that science accounts for everything, that most things of importance have already been roughed out and that what remains is simply sweeping up a few odds and ends of clutter. Then we shall have “Understanding,” my caps and quotes.
Anyone who takes such a position simply has not done the homework. It ain’t so and it ain’t gonna be so. There will always be the true, the knowable but the unprovable. There will always be problems.
My own belief is that science not only will remain incomplete but that the problems will always get bigger. That is not because of some emotional eagerness for the unknown. It is just that ever bigger problems are the way it has always been.
Consider a couple of new “breakthroughs.” It is now said that the vast bulk of what is real in the universe is “cold dark energy.” This energy pushes galaxy clusters apart so that the universe is speeding up in its expansion. What nobody has ever mentioned is that cold dark energy should also have the opposite effect. It should have gravitational attraction of its own. After all, Einstein’s special theory of relativity holds that e = mc2, that energy equals mass times the speed of light (in a vacuum) squared. And energy is subject to gravitation. One early piece of evidence for relativity was the observation that the gravitational field of the sun bent light coming close to the surface. So packing the universe with energy should not simply push stars apart. That energy has mass, has its own gravity, and should also be pulling them back together.
Problem. The best observations indicate that the expansion of the universe is indeed accelerating. The way the physicists have handled it is to declare that energy is always negative. Lift a weight and you store energy. Lift it farther and you store more. Move it infinitely far away and you have stored all the system can hold. And that is defined as zero energy. Any closer and you are in negative energy. Very slick. Foul.
Watch a radioactive nucleus decay. Measure the mass of the nucleus before the decay and the mass of the fragments. (That’s including neutrinos, which had to be invented because the arithmetic didn’t work otherwise, but peace, they have been observed.) The mass of the pieces is less than the mass of the original. The shortfall is equal to the energy released in the decay. But this is perfectly normal energy, able to light lights and raise weights if it happens in a functioning nuclear reactor. If energy is negative, so is mass. Sorry. The problem remains.
At the other end of the size scale, there has been a new observation in quantum mechanics. For a hundred years we had been told that reality consists of ordinary matter that follows ordinary logic and of quantum objects, like photons, which only exist as a distribution of probabilities. The quantum object is not defined until it is observed. That led to the old riddle, “If a tree falls in the forest and nobody hears it, does it make a sound?” This was the best thinking for a very long time.
The position was powerfully supported by an idea called “Bells inequality.” Bell proposed an experiment what would prove whether a quantum object had properties that were simply unknown until measured or had properties that were undefined until measured, at which point it would cease to be a quantum object and join the ordinary universe. When the experiment was done, Bell was right and the quantum object was demonstrated truly to be undefined.
Yet within the past year, some people took a quantum object, a bit of electric charge I think it was, that could only exist on one of two states with a fifty fifty chance of being in either one. The charge was measured with the greatest delicacy and finesse. Then borrowing a page from magnetic resonance imaging, they reversed the whole thing and repeated exactly the same measurement. Then they reversed it again. In the end they had the original quantum object undisturbed but knew what its state was.
Did that prove Bell wrong? Not a bit of it. Bell’s work stands. It’s just that things have become far more puzzling at the most fundamental level. So I say as in the past, so in the future. There will always be new truth to be discovered.
So do not worry from an intellectual standpoint that what I have demonstrated changes everything we always thought we knew. That is standard. It will always happen.
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