When One Door Closes, Others Open

In his book, Ignorance: How it Drives Science, Stuart Firestein wrote:
“In science there are so far two well-known instances where knowledge is shown to have limits.”
He was referring to the famous Uncertainty Principle from quantum mechanics and Gödel’s Incompleteness Theorem in maths. The former says it is impossible to know both items in certain pairs of properties of objects.

What does Gödel’s Incompleteness Theorem say? Simply put, it about any system of axioms (statements taken to be true, as being “obvious” without a formal proof) and proofs built using those axioms. No matter how you much progress you make with this system, the theorem says that there will always be true statements that cannot be proven.

Aha, you think, but does this just mean that one needs to add another fairly obvious axiom to the list? Would that then make all true statements provable? Go ahead, said Gödel, add another axiom to the list. I’ll then find a different true statement that can’t be proven with your new expanded list. So you decide to add yet another axiom to “fix” the new problem; and Gödel throws the next unprovable but true statement at you. And on and on you go. So who would “win” this battle? As Steven Landsburg wrote in his book, The Big Questions:
“Gödel did in fact prove that he can go on and on forever.”

But if you thought these limitations imposed by the Uncertainty Principle from quantum mechanics and Gödel’s Incompleteness Theorem would mean we’ve hit a brick wall in our quest of knowledge, think again. Sure, they’ve told us of limits in certain directions of knowledge. But, at the same time, writes Firestein, both these instances opened up other areas of knowledge. They have led to concepts like entanglement in physics, and to “unconsidered ideas about recursiveness, paradox, algorithms and even conciousness”!

To use an analogy: we are floating on the river called Knowledge, it looks like the limitations on it imposed only change the course of the river, and prevent us from going in certain directions, but they don’t stop the river altogether.

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