wssbckSemi-random notes on programming, math and other disciplines
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Mathematical Proof and Music<p><a href="http://www.reddit.com/r/math/comments/36nrfi/is_it_possible_that_there_exists_a_category_of/crfw5j1">Is it possible that there exists a category of proofs we have not yet discovered?</a></p>
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<p>That’s like asking if our list of musical genres is exhaustive, or if there could be music beyond jazz, funk, pop, soul, rock, …</p>
<p>What a proof boils down to is a process that takes you from known facts to a new conclusion based on reasoning, or the rules of logic. A valid proof could in theory be reduced to a step-wise list that starts with some of the axioms of your mathematical system, and deduces other formal statements by applying formal inference rules.</p>
<p>But that’s like saying that you could reduce a song to a time-series of frequencies. Sure, that would capture the song formally, but that’s not how a musician works with music. When a mathematician tries to establish the truth or falsehood of a statement, he can employ one or several heuristics or “strategies” that mathematicians have developed shorthand names for. Where a musician might tell another that he might be able to establish a desired musical effect through a plagal cadence, or through syncopation, or through any other shorthand a musician uses for a general musical strategy, a mathematician might advise someone to try establishing a fact by induction, or by contradiction, or by infinite descent, or by construction, or by establishing an invariant, or by…</p>
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Sun, 31 May 2015 00:00:00 +0000
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