(no subject)

[prog]

Heh heh, pages upon pages of people going No... NO! Aaaaaagh you're all wrong shut up about the .999... thing. This is worse/more amusing than the time that the Monty Hall problem was AOTD.

I find it interesting that the text of the article actually predicts the belief-path the doubters take... when faced with simple and easily graspable proofs, they change their minds and state that obviously this means that the number system is broken.

Comments

[cnoocy]

Wow. That's too much fun. Must stop reading.

[dictator555.livejournal.com]

Hum. Admittedly I only read the original page and not the arguments. And frankly, I only skimmed once I got to the beginning of the calculus proof, because I don't remember calculus at all. (Happily.) But, ah, you know I don't believe in numbers, right? I mean, I think the system is broken because it's based on numbers, and all numbers are broken.

Can we still be friends?

[prog.livejournal.com]

wait what

[dictator555.livejournal.com]

Ah, have we not discussed this before? How I don't believe in numbers? I mean, obviously they're very useful sometimes, but that doesn't make them real.

[prog.livejournal.com]

We have not discussed this.

I should think we must disagree on the definition of "numbers", "real", or perhaps "sometimes".

[dictator555.livejournal.com]

Could be. We should have a nice long discussion about this in person some time. I think I'd enjoy it, and you'd end up pulling your hair out in frustration. :)

[dougo.livejournal.com]

Or "believe in".

[cnoocy]

I would agree that numbers aren't real in the same way oranges are. (We might disagree as to which are more real.)

[dictator555.livejournal.com]

You honestly think numbers are more real than oranges? Juicy, delicious oranges?

[chocorisu.livejournal.com]

"God created the integers; all else is the work of man" - Leopold Kronecker

Is that the sense you're thinking of? In a lot of ways I'd have to agree with you. The reals are a useful extrapolation of how things work in real life but they don't actually correspond to anything that *physically exists*. Reality is all about mind-bogglingly large whole numbers of extremely tiny things.

[dictator555.livejournal.com]

Yeah, I guess that's the general direction of my thoughts. Numbers and math are all invented by humans to simplify reality into a form that can be more easily manipulated for analysis. So even things like calculus can be useful, since scientists can use that type of math to find useful solutions to real life problems.

But just because they're useful doesn't make them real in any greater sense. There's no truth in math. Plenty of people treat the mathematical realm like it's some whole other dimension of reality. Like the whole Plato thing with the ideal world of mathematics being somehow a more pure version of the truth about the world.

But really, math is just a cool little trick people invented. As such, it has no objective truth other than how well it reflects reality. It's completely fallible. Theorums can easily be wrong if the assumptions input into them aren't based on reality.

So there's really no reason I should believe .999... = 1 just because some theorum says I should. If I don't believe decimals represent reality, then the whole concept is meaningless to me anyway.

I get cranky about it because I was raised by mathematically gifted women who found my lack of interest in math a bismirchment of the family name. :) But I probably shouldn't rant about it, especially since it's just my opinion. Sorry.

[chocorisu.livejournal.com]

Well you can play silly semantic games as much as you like, but the point of mathematics is that it's a self-consistent system. The rules of mathematics hold despite not directly representing reality, so whatever that MEANS, arguing that it's just that way because someone says so is kind of meaningless.

You don't have to believe 0.999... = 1, but if you choose to believe it's different, your understanding of mathematics is inconsistent. You can also say 1 = 2, in some other made-up system of numbers, but it doesn't actually solve any useful problems.

[wrog]

it's a self-consistent system

actually, we don't know that. Nor can we know that given Gödel's Theorem, which basically says that once you have a proof system that's rich enough to handle arithmetic, the only way you can prove its own consistency in terms of a bigger system, at which point it's turtles all the way down.

Best we can say is we have yet to find an inconsistency, or, rather, none of the inconsistencies found thus far (e.g., Russell's set of all sets that don't contain themselves) has proved to be irreparable. ZFC (everybody's favorite set theory) has held up pretty well over the last 80 years or so; on the other hand, the Axiom of Choice leads to a bunch of awfully strange theorems.

The real point of mathematics is that it's a way of thinking that (thus far ) has been remarkably successful at screening out bullshit. YMMV

[chocorisu.livejournal.com]

You're talking rubbish. Even Godel's theorem wouldn't work if the rules of the system were inconsistent. maybe it's incalculabale or there are things you can do to delibreately break it, but as far as actually gettnig useful results out of it, if we start making up silly rules just because we don't like the things that fall out from following proofs, we may as well just give up and go back to living in caves. SORRY ABOUT NONSENSE, VERY DRUNK.

[radtea.livejournal.com]

Even Godel's theorem wouldn't work if the rules of the system were inconsistent.

Which is why the response to Godel's theorem is ususally to say that mathematics is incomplete rather than inconsistent. What Godel showed is that any consistent axiomatic system that is rich enough to contain arithmetic contains true theorems that cannot be proven via deductive manipulations within the system. Such a system is said to be "incomplete".

To experimental scientists this was not exactly a big surprise, as we always viewed the kind of knowledge we created as more than just a hack to get at things that the theorists could get at by other means. The big surprise is that fifty years later pure mathematics has an even greater hold over theoretical physics than it did in Godel's day.

[chocorisu.livejournal.com]

Incomplete, certainly. But definitely not inconsistent.

[dictator555.livejournal.com]

I'm sorry you think I'm playing silly semantic games. It's actually an important philosophical distinction for me. I probably should have just talked to Jason offline about it, because I didn't mean to insult anybody with my "silliness".

Yes. My understanding of mathematics is inconsistent, or at least fairly uneducated. It's been a long time since I took calculus in high school, and as I said, I've managed to forget most of what I learned.

I was hardly attacking math. I admitted how useful it is. My point was that I don't believe in it because I don't think it represents reality. If I don't believe in infinity, the whole concept of .999... becomes meaningless. That's inconsistent with math, but it's not internally inconsistent with my own belief system. It's not my understanding that's lacking (although certainly there's some of that) but rather my agreement with some of the basic assumptions that mathematics relies on. So what if math is internally consistent? That doesn't make it right, just useful.

But it's just my opinion, dude, and seriously not that important. Try not to let my disbelief bother you too much.

[chocorisu.livejournal.com]

Oh, it really doesn't *bother* me as such. I'm certainly not insulted, and I hope you are not either. I simply find your position on mathematics surprising and it makes for an interesting discussion. I hope I didn't come across too combative!

[dictator555.livejournal.com]

You did kind of come off as combative, or maybe I was just being sensitive. Sorry if I over-reacted and was combative back. I'm glad we're bringing this back down to the level of interesting discussion. :)

[daerr.livejournal.com]

If you don't believe in 0.999... then the concept of 1=0.999... can't be expressed, no? So it would seem impossible to even make a statement about 1=0.999... when using a system that doesn't have 0.999....

I suspect that there are important aspects of how we model the world that rely on infinite sequences (and that they be infinite and not just very very long). That is, that we can predict things about how the world behaves given the existence of the infinite that we wouldn't be able to do so otherwise. If this is true then that would point toward the idea of infinity as being real at some basic practical level, no? (I would assume that π would be an example of this, but I couldn't really say.)

[dictator555.livejournal.com]

Yeah. Exactly. 1=.999... is meaningless if you don't believe in .999...

I don't really know much about infinity. Are there solid scientific reasons for believing in it? I'm not sure that saying it's useful mathetmatically is a good argument for the existence of infinity, merely that it's a useful hypothetical idea. But again, I don't know a ton about it really.

[tahnan.livejournal.com]

OK, now I take offense.

Not as a mathematician, though I am one; but as a semanticist. What do you mean "silly"? :-)

[radtea.livejournal.com]

Numbers and math are all invented by humans to simplify reality into a form that can be more easily manipulated for analysis.

Mathematics is language, and nothing but language. It has the same ontological status as English, which certainly exists.

[dictator555.livejournal.com]

Ooops. Meant to reply to this yesterday. I agree that mathematics is a language and, in so far as English exists math also exists.

But you can lie with language, or simply make stuff up. And in that sense, neither English nor mathematics are real. I mean, there's no larger Truth to language, English or Math. At best, they're a reflection of reality, not reality themselves.

You could make some arguments about the interactivity of language and life, like how learning a language changes the way people think. And in that sense they're real, too. I really only meant exist on a certain level. Loose language. My bad.

[dougo.livejournal.com]

I give up, what does AOTD mean?

[dougo.livejournal.com]

Oh, duh, Article of the Day. I was misparsing that sentence and expecting a different part of speech.

[metahacker.livejournal.com]

OMGSTFUN00B! The stupid! The self-righteous! It burns!

This must be the reason mathematicians are traditionally such loners...

[radtea.livejournal.com]

And like the Monte Hall problem, the failure of people to understand is really helped along by lousy pedagogy.

The Wikipedia article does a relatively good job of presenting the proof, but it would be far better if it started with a paragraph discussing multiple representations of numbers as a common occurence in all notational systems, so instead of seeming needlessly and pointlessly counter-intuitive when first introduced the equality of 0.999... and 1 would be presented to the reader as a previously unnoticed instance of a perfectly familiar and ordinary phenomenon.