How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things. In my opinion the answer to this question is, briefly, this: As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
"Geometry and Experience" An expanded form of an Address to the Prussian Academy of Sciences in Berlin on January 27th, 1921.
Does emerge of "Universal Sequence" violate above statement?
Update:
U-Sequence has no wiki page to cite so here is up to period 6 of this sequence:
1, 2, 4
6, 5, 3
6, 5, 6
4, 6
5, 6
Answer
No, the emergence of a "universal sequence" doesn't contradict the statement by Einstein. In fact, as Michael Brown rightfully said, this "universal sequence" doesn't qualitatively differ from any other piece of pure mathematics that was found to be relevant in reality so one could have mentioned thousands of other examples.
What Einstein wanted to say is that mathematics is allowed to assume axioms – so they're certain within mathematics – but they aren't guaranteed to be exactly, accurately, and certainly true in Nature. To prove the latter points, one needs to use the methods of natural sciences which are never quite 100% waterproof. Each experimental result only disproves some conjectures at some confidence level or may provide us with nontrivial evidence that some theory seems really good – but there's absolutely no way to be certain that a theory is exactly true for all conceivable tests.
So the certainty about propositions in mathematics – certainty boiling down to rigorous derivations from the axioms using the laws of logic – comes with an inevitable price: the axioms and their consequences (theorems) aren't guaranteed to be relevant for the real world.
In fact, pretty much every theory that has ever been proposed and studied in physics turned out to be just an approximation at the end. String theory is the first (and last) theory that may be a counter-example to the previous statement but we're not 100% certain about this statement, either. Because the "universal sequence" isn't string theory, it's guaranteed that it's most a good approximation for some phenomena in Nature, not an accurate fact about physics that certainly holds.
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