Posted by: Dylan | January 23, 2009

This is Not Provable

We are perpetually driven to obtain more knowledge. People will toil their whole lives to discover the reason behind some occurrence, even if it has no pertinence. Humans want to know everything. It is because of this that we have come so far. People started as hunters and gatherers. They functioned as small and uncivilized tribes whose only goal was to survive and procreate. Now, in this present time, we have all sorts of fantastic machines and contraptions. These are a result of people’s insatiable desire to discover and create. Scientists and mathematicians want concrete and impenetrable proofs or descriptions as to why a system works, or why certain things happen. People rest comfortably in the knowledge that what we have discovered is true. But this safety is false. Our current understanding of the workings of our world could be far off from actuality. We might just not know it. The concept of perfectly true is an illusion. Nothing can be proven.

Systems:

For a time, it was thought that Newtonian physics was the correct scientific system. It worked under apparent circumstances, at least for a while. It was firmly grounded in mathematics and there were not excessive assumptions or axioms. As we all know, it did not hold true for everything. There were certain rare circumstances that Newtonian mechanics could not explain, such as how objects behaved as they approached the speed of light. From Einstein’s General Relativity came bizarre concepts, such as the fact that energy could be made into mass, and vice versa.

My point is, things that appear to be obviously true may not be so. There is actually no way to prove that a system is completely correct, or that an observation will always be true in the future. Every system and all research require a starting point. One may be able to prove the validity of this starting point using another system, but what lays at the foundation of this colossal tower? There is an assumption, grounded simply in observation and logical guesswork. These basic axioms appear completely correct, but there is actually no way to prove so.

Even a system whose axioms are taken to be true cannot prove everything in that system. Mathematically, no system can be complete and consistent. In 1931, Kurt Gödel proved this. A system is complete if every statement inside it may be proven using the given axioms. A system is consistent if there is no statement in it that is both true and false. This means that any formal system that is consistent has statements that are taken to be true but are not provable. Even something that is true cannot always be proven.

The Man and the Polar Bears:

Besides systems, there are observations. One may say that an object always falls towards the ground even without needing to use General Relativity or some other system to explain their reasoning.  Under normal circumstances, this has always been the case. Humans have been around and have been recording observations and taking records for a very long time. If you were to pick up a book and let go, you would say without doubt that it would fall to your feet, and it probably would. There is no way to prove that this always happens.

Consider a person who is stranded in the Arctic. He was born there and has lived there his whole life. He has also seen a lot of polar bears. This man would probably assume that all members of the bear family have white fur. To him, it would appear obvious. In his whole life he has never seen a bear of a different color. As we all know, this man is wrong. But we need to forgive him, for he has not seen any any other type of bear. We also have no reason to know that we have seen every type of bear. There may be a bear at some point in the future whose fur glows with a radiant orange color. To be certain that an observation is always true, one would have to live an infinite amount of time, and observe every possible situation. One would have to see every bear that has ever existed and every bear that will ever exist in order to prove that bears are always a certain color.

At best, something is probably true. This is an important idea to keep in mind. There is a danger in accepting something as true without questioning and investigating alternatives. We would not have developed this far if people clung fast to old beliefs. Everything was once thought to be composed of the four elements, fire, water, earth, and air. If people still believed this, all advances in microtechnology would be impossible. It is important to approach situations with an open mind and to be ready for something different. You may just prove something false. But you still can’t prove anything. Not even falsehood. Just improve the probability of it.

Of course, this is not an absolute truth. Nothing is provable.

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Responses

  1. Dylan:

    I was finally able to get on the net.

    You are way out there! I like the polar bear example. Still, for the time being, I am going to stay out from under pianos that are being lifted by fishing line.

    Keep the posts coming.

  2. I’d like to read more.


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