Descartes
His reasoning went as follows: I exist. My existence must have a cause. My existence cannot be its own cause because it just isn't up to snuff - my awareness seems to wax and wane with sleeping and waking, I have been unconscious altogether, my existence seems altogether ephemeral and insubstantial when viewed as a self-sufficient cause, so I must have a cause ``greater'' than myself. Hmm, cannot be anything in the world I see (historical evidence to the contrary, sorry Mom... you're just dubitable) as it isn't any better than I am, assuming that it is there at all. Even if my cause really IS something my mother and my father did one night long ago, they needed a cause, and that cause needed a cause, each cause greater than the one before. The whole world would need a cause. The whole Universe would need a cause. Must be something greater than me and the whole existing Universe too (if it exists, of course, which I find dubitable)!
Berkeley thought he had the answer. Following pretty much the same reasoning process as Descartes, he decided that Descartes might actually have been mistaken about having a body or a brain at all. The Universe might well be naught but an illusion. It was the act of perception itself that could not be doubted. The Universe consisted of (or existed in, was sustained by, was mathematically supported within) Mind, not matter. Matter is a figment of imagination made real by our perceptions of it. Thus we really are free - even freer than permitted by any degree of physical law.
By means of a process of intuitively guided, mathematically structured induction. We drop a penny a hundred times, and every time it falls according to an identical pattern. We therefore guess that if we drop it a hundred and first time, it will fall according to the same pattern. At some point, after some number of penny droppings, we conclude that there is a reason for the penny to fall, that this reason results in a reproducible, mathematically describable behavior, and that this reason persists and will act in the future as it has act in the past.
Sensible reasoning process? Sure, you bet! Try living without it! However, it is fundamentally flawed in the mathematical sense as it cannot be proven.
After all, why should the past and future be alike? Why should mathematical structure appear in reasons and causes? Most important of all, why should things have causes?
In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn't be proven either true or false using the rules and axioms ... of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you'll only create a larger system with its own unprovable statements. The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.
Gödel's Theorem has been used to argue that a computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths ... It plays a part in modern linguistic theories, which emphasize the power of language to come up with new ways to express ideas. And it has been taken to imply that you'll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself.
references
://www.phy.duke.edu/~rgb/Beowulf/axioms/axioms/node4.html
://www.miskatonic.org/godel.html
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