Open thread, August 4 - 10, 2014

polymathwannabe

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If it's worth saying, but not worth its own post (even in Discussion), then it goes here.

Notes for future OT posters:

1. Please add the 'open_thread' tag.

2. Check if there is an active Open Thread before posting a new one.

3. Open Threads should be posted in Discussion, and not Main.

4. Open Threads should start on Monday, and end on Sunday.

It's "Here's a sequence of symbols. Should we assign this sequence of symbols the value of 1, or not?" Which is just a silly argument to have.

It's not. The "0.999... doesn't equal 1" meme is largely crackpottery, and promotes amateur overconfidence and (arguably) mathematical illiteracy.

Terms are precious real estate, and their interpretations really are valuable. Our thought processes and belief networks are sticky; if someone has a crap interpretation of a term, then it will at best cause unnecessary friction in using it (e.g. if you define the natural numbers to include -1,...,-10 and have to retranslate theorems because of this), and at worst one will lose track of the translation between interpretations and end up propagating false statements ("2^n can sometimes be less than 2 for n natural")

the correct response (unless they have sufficient real analysis background) is not "Well, here's a proof that of that claim", it's "Well, there are various axioms and definitions that lead to that being treated as being equal to 1".

It would be an accurate response (even if not the most pragmatic or tactful) to say, "Sorry, when you pin down what's meant precisely, it turns out to be a much more useful convention to define the proposition 0.999...=1 such that it is true, and you basically have to perform mental gymnastics to try to justify any usage where it's not true. There are technically alternative schemas where this could fail or be incoherent or whatever, but unless you go several years into studying math (and even then maybe only if you become a logician or model theorist or something), those are not what you'll be encountering."

One could define 'marble' to mean 'nucleotide'. But I think that somebody who looked down on a geneticist for complaining about people using 'marble' as if it means 'nucleotide', and who said it was a silly argument as if the geneticist and the person who invented the new definition were Just As Bad As Each Other, would be mistaken, and I would suspect they were more interested in signalling their Cleverness via relativist metacontrarianism than getting their hands dirty figuring out the empirical question of which definitions are useful in which contexts.

It's not. The "0.999... doesn't equal 1" meme is largely crackpottery

A lot (in fact, all of them that don't involve a rigorous treatment of infinite series) of the "proofs" that it does equal 1 are fallacious, and so the refusal to accept them is actually a reasonable response.

You seem to making an assertion about me in your last paragraph, but doing so very obliquely. Your analogy is not very good, as people do not try to argue that one can logically prove that "marble" does not mean "nucleotide", they just say t... (read more)

3KnaveOfAllTrades12y

Actually, I could imagine you reading that comment and feeling it still misses your point that 0.999... is undefined or has different definitions or senses in amateur discussions. In that case, I would point to the idea that one can makes propositions about a primitive concept that turn out to be false about the mature form of it. One could make claims about evidence, causality, free will, knowledge, numbers, gravity, light, etc. that would be true under one primitive sense and false under another. Then minutes or days or month or years or centuries or millennia later it turns out that the claims were false about the correct definition. It would be a sin of rationality to assume that, since there was a controversy over definitions, and some definitions proved the claim and some disproved it, that no side was more right than another. One should study examples of where people made correct claims about fuzzy concepts, to see what we might learn in our own lives about how these things resolve. Were there hints that the people who turned out to be incorrect ignored? Did they fail to notice their confusion? Telltale features of the problem that favoured a different interpretation? etc.

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