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Quantum Test Found For Mathematical Undecidability
KentuckyFC writes "Philosophers have long wondered at the profound link between mathematics and physics, but how deep does this connection go? Pretty deep according to the results of a quantum experiment exploring the nature of mathematical undecidability. Here's how: any logical system must be based on axioms, which are propositions that are defined to be true. A proposition is logically independent from these axioms if it can neither be proved nor disproved from them; mathematicians say it is undecidable. In the experiment, researchers encoded a set of axioms as quantum states. A particular measurement on this system can then be thought of as a proposition which, if undecidable, yields a random result — which is what they found. 'This sheds new light on the (mathematical) origin of quantum randomness in these measurements,' say the researchers (abstract)."
No, really, they're serious.
The rules of math (which weren't so much invented as identified) seem oddly linked to the underlying physics. TFA mentions the unreasonable effectiveness of mathematics -- it's not so much that we can count the physics with the math, it's that the math predicts things which should be true, and are subsequently proven to be. The existence of things like a negative square root in an equation have predicted the existence of things like anti-particles, and those particles have been found experimentally.
It's precisely the fact that the math isn't independent of the physics that is at issue here That's a very startling proposition because it goes well beyond simply counting what is, it means the same rules which define the math in the first place underly the physical mechanisms.
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We use mathematics to quantify physics, but there is no "connection" between the two, except in the sense that we can count *anything* and say there's a connection. It's like saying, "How deep does the connection go between mathematics and bananas when I observe there are 10 bananas, and I add two more, and then I observe 12 bananas."
I'm glad you're so sure of yourself. However, the connection between *counting* (ring of integers) and, say, complex conjugation isn't so obvious. If you'd like to compete with Dirac (for example) and argue that he was dumb for taking so long to recognize antiparticles' existence, or that Green should have "obviously" recognized that there must be such things as evanescent waves because the Helmholtz equation has some complex roots for the wavenumbers, then be my guest.
I don't know what your background is, but such connections between mathematics and the "real world" are NOT always obvious, and it is a continued source of delight and puzzlement when one explores some neglected branch-cut in the maths, and it turns out to have real impact on the physics. Please, explain to all of we poor physicists how bananas can point us to truth.
Mathematics is an abstract game of counting, built up into great complexity.
Mathematics is a game of abstraction, played out in a wide variety of directions, counting being just one of them. The assumption that mathematics is just counting is rather frustrating. Yes, you can reduce mathematics to arithmetic, but then you can also reduce it to set theory, or to topos theory/category theory, and so on. The ability to express things in a particular way does not that that is what the the things are, especially given the profusion of different mutually interpretable "reductions" available.
1 + 1 = 2 will be true in any universe, under any god(s), in any circumstances. And all of mathematics is built up from that. It's universal truth.
Actually you can dream up universes where 1+1=2 doesn't hold. It can fail to hold for a variety of reasons. The various hypothetical universes vary with those reasons from completely uninteresting and trivial, through to, well, in this case, still relatively uninteresting. Of course there are other "fundamental truths" that you can drop (the law of excluded middle, for example, or DeMorgan's laws, which are both conceivably more fundamental than 1+1=2) and end up with remarkably rich and interesting universes. The absolute universality of mathematical truth is on rather shaky ground; certainly the mathematics we use seems pretty solid for our universe, but that doesn't make it universal over all possible universes.
We use mathematics to quantify physics, but there is no "connection" between the two
There is a connection to the extent that ideas developed in the abstract for purely mathematical reasons have often had surprising, unseen, and unlooked for applications to physics. It is the surprising aspect of that that makes philosphers question the apparently unreasonable effectiveness of mathematics.
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It's precisely the fact that the math isn't independent of the physics that is at issue here That's a very startling proposition
The word "math" refers to a huge collection of symbolic rule sets. These rule sets were not all invented at once by some magical mathematician in the past. They were produced over thousands of years of refinement.
One important point to note here is that many of these refinements were made specifically for the purpose of giving math a higher level of practical value. For example, the number zero, and subsequently the negative numbers, were added by most cultures only after they realized that they could derive a useful model of some aspect of reality by using these numbers.
I don't see why it would be surprising at all that a language which has been refined, over time, to describe reality would wind up describing reality.
I will further suggest that the truths of mathematics that seem intuitively obvious to us seem so only because our brains are structured such that these truths will seem intuitively obvious. What gave our brains this structure? Refinement-after-refinement due to the process of natural selection. So the reality which is being modeled by mathematics happens to be the same reality in which the inventors of mathematics (ie our brains) evolved. Who would have ever guessed that there would be some correspondence here?
I think the surprise only comes about when we forget the true origins of mathematics, and the true origins of the brains that understand mathematics and use it to represent reality.