One of the fundamental assumptions underlying the human quest for knowledge is that we can somehow logically answer questions we have about anything. This is a very pragmatic assumption and has given rise to the amazing and often destructive spectacles that we as a species have developed, but please be aware that this assumption is just that – an assumption.To appreciate the true significance of how foundational this assumption is, please consider the following. In our modern world, mathematics lies at the foundation of everything. Physics relies on it, which in turn provides the foundations for chemistry, biology and all the other sciences that you can think about. So you can see an almost reverse pyramid here with mathematics at the bottom with everything else placed on top of it. Now for the twist, mathematics is both a formalism of and based upon logic. So what does it mean if logic is only an assumption?
First let me explain why I say that logic itself is an assumption. Towards the beginning of the 20th century a lot of work was done on the ideas of formalisms and building a set of mathematical axioms from which all mathematics could be derived. Around that time, this Austrian guy called Kurt Gödel showed up. He proved something called the incompleteness theorem. The implications of this theorem are what are important for our discussion. In short, it implied that in any in any formal system (of which mathematics is included) there are questions that cannot be solved using the axioms and logic of that system. Furthermore, it is not possible to know whether a problem is solvable beforehand. I’m not saying that you will get a wrong answer, but rather that an answer can never be obtained for some questions. So do you see the problem now? Logic, the foundation of mathematics, cannot always be used to solve problems.
So then again, where does this leave us? Am I suggesting that all the things that we have accomplished using logic, mathematics and science are irrelevant? Not at all, but to suggest that it can answer all our questions is flawed.
Also, here is an alternate perspective to consider: all of our ideas in logic, science and philosophy have all helped us understand our world and ourselves a lot better, but not necessarily because reality conforms to our underlying ideas in mathematics, but maybe because our ideas in logic and mathematics are based upon our own perceptions of reality. It is a subtle concept to grasp, but a fundamental one. To expand on this concept, consider that our logic developed to enable us to better perceive reality. We then started using logic to explain reality. Obviously reality will conform to the logic that we used. But to jump from that to using logic as a foundation for everything is a big leap in my opinion.
It is an interesting idea that our mathematics is dependent on our perceptions – difficult to comprehend, and I am sure opposition to it will be freely available, but I am not entirely convinced that it is false. If anything, the incompleteness theorem also implies that if such an idea is true, we would not necessarily be able to prove such questions within our own systems of logic, and a system outside ours might be necessary, which would also be subject to the incompleteness theorem.
But enough of speculation, one thing is certain, if someone argues that mathematics, science and philosophy can answer all of our questions, they do not truly understand the reality of logic. But it is a bit disturbing that mathematics itself can be inconsistent. If something as basic as mathematics and logic, floats upon such uncertainties, what absolute concrete foundation can we truly base our reality on – don’t be fooled, there are no absolutes!