View Full Version : Advanced math and probability

Kurald Galain

2008-06-25, 04:57 PM

Well, this came up in a recent thread and turned out to be more interesting than the thread's topic, so I'm forking it out. The way I figure, math and probability has a bearing on game mechanics, right?

Incorrect.

.999 repeating and 1.000 repeating are different numbers that are infinitely far apart. The fact that we can't represent the difference between 1/3 and .999 repeating/3 is irrelevant, the difference exists.

I'm afraid that http://en.wikipedia.org/wiki/.999 proves you wrong.

Tengu

2008-06-25, 05:25 PM

0.(9) is indeed the same number as 1. That's not really advanced math. Well, maybe for people who don't get xkcd jokes and other non-educated folk.

Dan_Hemmens

2008-06-25, 05:41 PM

Not done any advanced maths in years, but I take it that the proof is something like: 0.(9) is effectively the sum to infinity of 9/10 + 9/100 etc, which you can work out the same way you work out 1/2 + 1/4 ... ?

nagora

2008-06-25, 05:48 PM

Not done any advanced maths in years, but I take it that the proof is something like: 0.(9) is effectively the sum to infinity of 9/10 + 9/100 etc, which you can work out the same way you work out 1/2 + 1/4 ... ?

You could argue, I suppose, that the number .999... represents the difference between 1 and .9 divided infinitely, and the difference is therefore <anything>/infinity=0. But, yes summing the series would probably be the direct route for a mathematician.

Jack_Simth

2008-06-25, 05:50 PM

Not done any advanced maths in years, but I take it that the proof is something like: 0.(9) is effectively the sum to infinity of 9/10 + 9/100 etc, which you can work out the same way you work out 1/2 + 1/4 ... ?That's one way of doing it, yes.

Another is the conversion of infinite repeating decimals to fractions - results in 9/9 for .9999....

Another is subtraction; 1 - 0.9999... = 0.0000.... forever. There is no termination on the zeros.

Of course, the simplest is the practicalist view - if there is any difference at all, it is literally infinitely small of a difference, and doesn't make a difference. And a difference that makes no difference isn't really a difference, is it?

Emperor Tippy

2008-06-25, 05:53 PM

Well, this came up in a recent thread and turned out to be more interesting than the thread's topic, so I'm forking it out. The way I figure, math and probability has a bearing on game mechanics, right?

I'm afraid that http://en.wikipedia.org/wiki/.999 proves you wrong.

From your own link

Although the real numbers form an extremely useful number system, the decision to interpret the phrase "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in Mathematics: A Very Short Introduction that the resulting identity 0.999… = 1 is a convention as well:

However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.

0.(9) = 1 is necessary for some forms of math but in the end it is arbitrary.

Dan_Hemmens

2008-06-25, 05:53 PM

That's one way of doing it, yes.

Another is the conversion of infinite repeating decimals to fractions - results in 9/9 for .9999....

Problem is that the fraction method kind of feels like a trick (like the 1 + 1 = 3 thing)

Another is subtraction; 1 - 0.9999... = 0.0000.... forever. There is no termination on the zeros.

That's rather elegant, however.

Kurald Galain

2008-06-25, 05:58 PM

From your own link

0.(9) = 1 is necessary for some forms of math but in the end it is arbitrary.

And in your own sentence, "not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic".

Lots of conventions are arbitrary (e.g. that 0! = 1) but they are nevertheless well-accepted conventions. You might as well be arguing that you didn't believe in relativity because "it's only a theory" - that belies only a lack of understanding of the word "theory".

Dan_Hemmens

2008-06-25, 06:06 PM

Lots of conventions are arbitrary (e.g. that 0! = 1) but they are nevertheless well-accepted conventions. You might as well be arguing that you didn't believe in relativity because "it's only a theory" - that belies only a lack of understanding of the word "theory".

But it's just a *theory* that means it's *made up* like *fairies* and *Jesus*.

Also, I think your last sentence betrays a lack of understanding of the word "belie".

Pinnacle

2008-06-25, 06:08 PM

1/3 is 0.333... and 2/3 is 0.666...

1/3 + 2/3 is, of course, 1. 0.333... + 0.666... is 0.999...

If A+B=C and A+B=D is also true, C=D must be true.

Jack_Simth

2008-06-25, 06:08 PM

Problem is that the fraction method kind of feels like a trick (like the 1 + 1 = 3 thing)

You've seen the method before, right?

That's rather elegant, however.

Thanks.

Dan_Hemmens

2008-06-25, 06:10 PM

You've seen the method before, right?

1 + 1 = 3? Yes (a couple of ways). The 0.(9) = 1 thing I actually never saw until today, I was always rubbish at maths - I was a physicist.

Emperor Tippy

2008-06-25, 06:14 PM

And in your own sentence, "not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic".

Lots of conventions are arbitrary (e.g. that 0! = 1) but they are nevertheless well-accepted conventions. You might as well be arguing that you didn't believe in relativity because "it's only a theory" - that belies only a lack of understanding of the word "theory".

Just because an arbitrary convention is well accepted doesn't make it any less arbitrary. 0!=1 is useful, just like 0.(9)=1 is useful. So both are kept and used in theoretical math.

Physics theories are different from math theories. One is based on measurable, quantifiable, objective fact and the other is based on arbitrary conventions.

averagejoe

2008-06-25, 06:22 PM

I knew some neat proofs for this, but it's been so long that I can't remember them at this point. It's just one of those wierd mathematical things that completely blows people's minds. Like how if you take the graph f(x)=1/x from x=1 to infinity and make a solid by rotating the line around the x-axis, then you get an object with a finite volume but infinite surface area. That totally blew my mind, anyways. This isn't really advanced math, though.

From your own link

0.(9) = 1 is necessary for some forms of math but in the end it is arbitrary.

I would like to add my voice to Kurald in saying that, in the end, math is pretty arbitrary, and it's only how it is because of convenience. Anyways, achedemics like to be contrary, especially when it comes to well established things, because that's how you get ahead in achedamia. .(9)=1 is, however, a well-proven and useful result, and it would pretty much always be correct to say that it is the case that .(9)=1.

Edit:

Just because an arbitrary convention is well accepted doesn't make it any less arbitrary. 0!=1 is useful, just like 0.(9)=1 is useful. So both are kept and used in theoretical math.

Physics theories are different from math theories. One is based on measurable, quantifiable, objective fact and the other is based on arbitrary conventions.

Well, yeah. Mathematics is basically philisophical, it just happens to be a philosophy that models our world really well, so physics makes good use of it. .(9) does not in any way represent a physical object, it represents a mathematical one. However, it's silly to dismiss this solution as just an arbitrary convention, since that could be said for enough of math that it renders the statement basically meaningless. To use a better metaphor, you might as well say, "Einsteinien reletivity isn't proven, it assumes that the universe is observable and quantifiable."

Jack_Simth

2008-06-25, 06:28 PM

1 + 1 = 3? Yes (a couple of ways). The 0.(9) = 1 thing I actually never saw until today, I was always rubbish at maths - I was a physicist.

I'm referring to the general method of converting infinite repeating decimals to fractions, actually.

You take your infinite repeating decimal - let's go with .234234234234234... just for grins. And we'll say it is equal to b. I like b. We'll use b.

So:

b = 0.234234234234234...

I'm just assigning b here. Nothing magical - strictly mundane stuff.

Well, that also means:

1000*b= 234.234234234234...

Again - I'm just using the value I've assigned to b.

The magic starts to happen after this - see, I've now got a system of equations

b = 0.234234234234234...

1000*b= 234.234234234234...

If both sides are equal, we can subtract one from the other - lets subtract the first line from the second:

(1000*b)-b = 234.234234234234... - 0.234234234234234...

->

999*b = 234

Basically, I've just subtracted 1 from the multiplier on b, and canceled out everything after the first iteration (which I now have expressed as an integer).

Now we isolate b, by dividing both sides by 999:

b = 234/999

And now we have an integer over an integer. This works for basically any repeating decimal (you just choose 10^x, where x is the size of your iteration, rather than using 1000). All steps are nice and straightforward, no trickery beyond standard equation manipulations.

When you do that with .999999...:

b=0.99999...

->10*b=9.99999...

->(10*b)-b=9.99999...-0.99999...

->9*b=9

->b=9/9

->b=1

->0.9999....=1; QED.

Guildorn Tanaleth

2008-06-25, 06:30 PM

I would like to add my voice to Kurald in saying that, in the end, math is pretty arbitrary, and it's only how it is because of convenience.

Correction: Conventional Mathematics is the way it is in order to reflect our perceived reality. There are plenty of branches of Math dealing with theoretical number systems and other things based on different axioms, so you can choose almost any set of (internally consistent) conventions & still be doing Math.

Ka'ladun

2008-06-25, 06:32 PM

Here's the most thorough proof I've seen.

0.(9) can be represented of the sum of 9*.1^n, where n = 1 to infinity, which comes out to .9 + .09 + .009 + and so on. Now, this particular arrangement is a geometric series. The sum of a geometric series is a/(1-r), where a is the first term in the series, and r is the exponential factor of the series, in this case .1. Plugging in these variables yields .9/(1-.1) = .9/.9 = 1.

averagejoe

2008-06-25, 06:34 PM

Correction: Conventional Mathematics is the way it is in order to reflect our perceived reality. There are plenty of branches of Math dealing with theoretical number systems and other things based on different axioms, so you can choose almost any set of (internally consistent) conventions & still be doing Math.

Okay, fine, convenience or human interest. My point was that whatever rules are there for math are basically because someone chose to use those rules. I was just assuming that we were referring to conventional math, or there is no way of determining the truth of the statement .(9)=1 without different rules being presented.

Prophaniti

2008-06-25, 06:35 PM

Physics theories are different from math theories. One is based on measurable, quantifiable, objective fact and the other is based on arbitrary conventions.Sorry, but isn't math kinda the poster child for measurable, quantifiable, objective fact? Seriously, every advanced science keeps coming back to math. It's like the universal science that everything answers to. Damn math.:smallannoyed:

Dan_Hemmens

2008-06-25, 06:46 PM

I'm referring to the general method of converting infinite repeating decimals to fractions, actually.

I don't think I'd seen it explicitly actually.

Cool.

averagejoe

2008-06-25, 06:49 PM

Sorry, but isn't math kinda the poster child for measurable, quantifiable, objective fact? Seriously, every advanced science keeps coming back to math. It's like the universal science that everything answers to. Damn math.:smallannoyed:

No. Mathematics is not, in and of itself, measurable or quantifiable (though it is fairly objective). It's an interesting coincidence that science so often can be represented by math, and there is a lot that you can use math for in science, but in the end the two are entirely different animals. In science you can't really call something true until it has been observed. In math you can call something true when you've made a convincing proof. In other words, science says, "Let's try something and see what happens." Math says, "Let's assume these things, and see what logically pops out." Math is inherently philosophical, but with a definite structure and specific rules. If you cannot take something and say, "Look, this is true because we're seeing it," then it really isn't science.

Jack_Simth

2008-06-25, 06:54 PM

I don't think I'd seen it explicitly actually.

Cool.

Mind you, once you've seen it once, you can short-cut it as much as you like:

0.1234123412341234.... = 1234/9999

0.57893456789578934567895789345678957893456789... = 57893456789 / 99999999999

0.11111111....=1/9

... and so on, because you as a human you can look-ahead, so to speak, and recognize the pattern.

Emperor Tippy

2008-06-25, 06:59 PM

Sorry, but isn't math kinda the poster child for measurable, quantifiable, objective fact? Seriously, every advanced science keeps coming back to math. It's like the universal science that everything answers to. Damn math.:smallannoyed:

No, math isn't objective fact. It uses arbitrary postulates in a lot of the higher end stuff.

EDIT: It is objective in that if two people start off with the same base premise they will always get the same answer. It's not objective in that it doesn't exist independent of humans.

END EDIT

Are those arbitrary postulates useful and even necessary? Yes. But they are still arbitrary.

I understand full well all of the proofs of 0.(9)=1, I even understand the applications of 0.(9)=1 and why it is used.

I don't even disagree with its use in said math.

averagejoe

2008-06-25, 07:09 PM

*snip*

I don't get it then. In what way is it not true?

Jack_Simth

2008-06-25, 07:13 PM

I understand full well all of the proofs of 0.(9)=1, I even understand the applications of 0.(9)=1 and why it is used.

I don't even disagree with its use in said math.

...

Just to make sure I have this straight:

1) You agree with the proofs, that basically just rely on the number system as defined, and fairly basic mathematical manipulation.

2) You disagree with their conclusions despite (1), because everything involved is strictly arbitrary.

If both (1) and (2), there is no point in further debate, because you cannot be convinced. This is at the point of 2+2=5, because the meaning of the symbols 2, +, =, and 5 are arbitrary.

Prophaniti

2008-06-25, 07:15 PM

Isn't at least basic math pretty much universal? I mean, even if you called the numbers something else, when you have two *something* and you gain two more, you have four *something* now. I don't know much advanced math, but it's all based on the same basic principles, most of which are not only irrefutable but demonstratable, and definitely not dependant on humans to exist. Sure, nothing else would be around to appreciate or study this phenomenon of doubling, but it would still happen.

Does this basis in observable reality really decline that much in high level math? Math has always struck me as more of an explanation of science than anything else. Example: We see that objects fall to the earth unless acted upon. Math explains how, why, how fast and far, ect. All of which can be verified by observing falling things.

Somebody tell me if I'm way off base here, before I further demonstrate how utterly ignorant I am about mathamatical theory...

averagejoe

2008-06-25, 07:25 PM

Isn't at least basic math pretty much universal? I mean, even if you called the numbers something else, when you have two *something* and you gain two more, you have four *something* now. I don't know much advanced math, but it's all based on the same basic principles, most of which are not only irrefutable but demonstratable, and definitely not dependant on humans to exist. Sure, nothing else would be around to appreciate or study this phenomenon of doubling, but it would still happen.

Does this basis in observable reality really decline that much in high level math? Math has always struck me as more of an explanation of science than anything else. Example: We see that objects fall to the earth unless acted upon. Math explains how, why, how fast and far, ect. All of which can be verified by observing falling things.

Somebody tell me if I'm way off base here, before I further demonstrate how utterly ignorant I am about mathamatical theory...

Well, I can't answer for the underlying philosophy (that is, your first paragraph), but higher level math basically doesn't have any basis in the observable world, or, at least, is meant to generalize to things that aren't, for example n-dimensional space (or even infinite dimensional space.) Now it turns out that some of the weird things, like using functions as vector spaces, actually has use in quantum mechanics, which is part of what makes QM so weird. However, there are other things that don't have any analog in the real world than we imagine. Unfortunately these tend to be hard to explain, and some of these may even have application in unknown sciences. The thing is, we really don't know how closely math and science are connected, since we don't know enough about the universe to say. However, I will say that, for the most part, math and science evolved separately, and in fact it came as a surprise to many mathematicians that their field had any application beyond arithmetic and basic algebra.

kirbsys

2008-06-25, 07:37 PM

I had to do this for my freshman algebra two. Basically my arguement was that since .333R was the same as 1/3 and .333R times 3 was .999R and 1/3 times 3 was 1, then .999R was equal to 1.

How does the 1+1=3 thing work out?

Emperor Tippy

2008-06-25, 07:45 PM

...

Just to make sure I have this straight:

1) You agree with the proofs, that basically just rely on the number system as defined, and fairly basic mathematical manipulation.

2) You disagree with their conclusions despite (1), because everything involved is strictly arbitrary.

If both (1) and (2), there is no point in further debate, because you cannot be convinced. This is at the point of 2+2=5, because the meaning of the symbols 2, +, =, and 5 are arbitrary.

No. 2+2=4 is true in most math and in the real world. 0.(9)=1 is only true in math.

If you have '1' apple and then gain another identical apple you have '2' apples. Replace '1' and '2' with whatever random designation you feel like, it doesn't change the objective fact that 2 is twice the amount of 1.

0.(9)=1 isn't an objective fact or derived from objective fact.

Jack_Simth

2008-06-25, 07:48 PM

I had to do this for my freshman algebra two. Basically my arguement was that since .333R was the same as 1/3 and .333R times 3 was .999R and 1/3 times 3 was 1, then .999R was equal to 1.

How does the 1+1=3 thing work out?

Well, I suppose I could build one as an extension of the "proof" that 1=0 (which can thereafter be used to "prove" that any number equals any other number).

In general, though, proofs of that type tend to point out "holes" in the system used (1=0 proof generally makes use of canceling infinities, for instance - demonstrating that either there's some kind of order to infinities, or that infinity - infinity is undefined, or similar).

No. 2+2=4 is true in most math and in the real world. 0.(9)=1 is only true in math.

If you have '1' apple and then gain another identical apple you have '2' apples. Replace '1' and '2' with whatever random designation you feel like, it doesn't change the objective fact that 2 is twice the amount of 1.

0.(9)=1 isn't an objective fact or derived from objective fact.

Pour me 0.(9) cups of water. Then measure the difference between that and 1 cup of water. 0.(9) doesn't have a meaning outside of math anyway.

Or, alternately, 2 + 2 = 5, 2 + 2 != 4. I'm just not using the meaning of the symbol set [2, +, =, 5, !=, 4] that you're used to. In order to distinguish the "value" of 0.(9) and 1, you need to have some form of symbol set and symbol manipulation rules in use. In order to discuss the validity of 0.(9)=1, you also need to have some symbol set and symbol manipulation ruleset in use. What rules and symbol-set are you using such that 0.(9)=1 is not true?

You did not disagree with (1), and you did not disagree with (2). The disconnect is at the level of symbol-meaning, which fundamentally cannot be argued outside of empirical testing. Pour me 0.(9) cups of water, pour me 1 cup of water, then measure the difference between them. What is it?

averagejoe

2008-06-25, 08:04 PM

No. 2+2=4 is true in most math and in the real world. 0.(9)=1 is only true in math.

If you have '1' apple and then gain another identical apple you have '2' apples. Replace '1' and '2' with whatever random designation you feel like, it doesn't change the objective fact that 2 is twice the amount of 1.

0.(9)=1 isn't an objective fact or derived from objective fact.

See, now you're just being difficult. A ".(9)" doesn't even exist in the first place, nor does anything that requires some sort of infinity construct to work. Your statement doesn't make any sense. You inherently need a mathematical construct to have a ".(9)" and that construct will be arbitrary anyways.

Anyways, "2 is twice the amount of 1" isn't as objective as you think. For example, what constitutes an "apple?" It seems simple, but even on that fact you can get disagreement. Is it still an apple, for example, if some of it is missing? If so, how much of an apple is allowed to go missing before it stops being an apple? Does the nature or effect of the cut change things? etc. Then there's the fact that 2 is actually slightly less than or greater than twice 1, because the apples won't be the same size. And so on. That's not even getting into mathematical constructs where two isn't twice one, but no doubt you'll just dismiss those as non-objective and arbitrary and therefore wrong.

KillianHawkeye

2008-06-25, 08:05 PM

No. 2+2=4 is true in most math and in the real world. 0.(9)=1 is only true in math.

If you have '1' apple and then gain another identical apple you have '2' apples. Replace '1' and '2' with whatever random designation you feel like, it doesn't change the objective fact that 2 is twice the amount of 1.

0.(9)=1 isn't an objective fact or derived from objective fact.

That's only because 0.9999... is not really a measurable quantity, being that it goes on forever. Thus, at some point a scientist is going to have to cut it off (because they have to worry about things like significant digits and accuracy and such) and will either call it 0.99 or round it up to 1.00.

You keep saying that 0.9999.... = 1 is only true in math, but that's because it only exists in math. In the real world, it's just 1. 1 = 1, dude.

Yes, it is a convention.

The alternative to this convention is, quite honestly, mathematics that looks more like gibberish than mathematics.

In essence, the convention isn't arbitrary. It is a convention.

1+1=2 is also a convention that is not arbitrary. There are some reasonably deep reasons why 1+1=2 (as opposed to 1+1=3). There are also some reasonably deep reasons why 0.9999... = 1.

Claims that 0.9999... != 1 is a reasonable position to take produce the following simple reaction: show it. Give me the axioms you are using to prove that 0.999... != 1. I fully suspect you'll be unable to do this, or the axioms produced will be rather junkey. :-)

Chronos

2008-06-25, 08:06 PM

No. 2+2=4 is true in most math and in the real world. 0.(9)=1 is only true in math.

If you have '1' apple and then gain another identical apple you have '2' apples. Replace '1' and '2' with whatever random designation you feel like, it doesn't change the objective fact that 2 is twice the amount of 1.

0.(9)=1 isn't an objective fact or derived from objective fact.I don't see the distinction. We accept certain conventions in constructing representations of numbers. For instance, (given the conventional interpretation of the symbols "1" and "+"), we adopt the convention that the symbol "2" represents the same number as is represented by "1+1". We could adopt different conventions, of course, which would assign different meaning to those symbols, or no meaning at all, but those are the conventions we have chosen.

The thing is, though, under the same set of conventions under which we can say things like "1+1 = 2" and "2.73 + 1.44 = 4.17", we must inevitably also say that "0.9 repeating = 1.0". It's no different to change the set of conventions in such a way that point nine repeating doesn't equal one, than it is to change the set of conventions in such a way that one plus one doesn't equal two.

Patashu

2008-06-25, 08:15 PM

And in your own sentence, "not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic".

Lots of conventions are arbitrary (e.g. that 0! = 1) but they are nevertheless well-accepted conventions. You might as well be arguing that you didn't believe in relativity because "it's only a theory" - that belies only a lack of understanding of the word "theory".

An interjection: 0!=1 is not arbitary, it follows from n!=(n-1)!*n and 1! = 1, which is equivalent to the definition of the factorial function, and also follows from the observation that, in combinatronics, the number of ways you can arrange or choose from 0 objects is not 0 but 1.

Jorkens

2008-06-25, 08:35 PM

Isn't at least basic math pretty much universal? I mean, even if you called the numbers something else, when you have two *something* and you gain two more, you have four *something* now. I don't know much advanced math, but it's all based on the same basic principles, most of which are not only irrefutable but demonstratable, and definitely not dependant on humans to exist. Sure, nothing else would be around to appreciate or study this phenomenon of doubling, but it would still happen.

Does this basis in observable reality really decline that much in high level math? Math has always struck me as more of an explanation of science than anything else. Example: We see that objects fall to the earth unless acted upon. Math explains how, why, how fast and far, ect. All of which can be verified by observing falling things.

Somebody tell me if I'm way off base here, before I further demonstrate how utterly ignorant I am about mathamatical theory...

Interesting stuff here.

On the subject of 'basis in reality', no, I don't think maths can be said to have a basis in anything observable. Certainly if you take two rocks and put them next to two other rocks you've got four rocks, and every time anyone's ever done this, that's what happened. I think Mill put this argument forward, and then got slapped down by Frege who argued that it's a good thing for Mill that everything isn't nailed down, or we wouldn't know what two plus two is because we wouldn't be able to put stuff together. (He used the german word narglefest for this situation, which is great.)

I think the general approach that most mathematicians (or at least, most of the ones who think about such things) take is that 2+2=4 because starting from the very basic axioms of set theory and propositional logic, one can construct something that you consider to represent the number '2', something else that you consider to represent the number '4', and something that you consider to represent the operation '+', and then prove that given those axioms and those labels, 2+2=4. The things that we consider to be true mathematical statements are essentially statements in the theory developed from those axioms. They are 'true' in the limited sense that they can be got from your axioms by applying your given laws of inference.

But the axioms are still something we have to take on trust. For instance, we can't prove usefully that a given set of axioms are consistant (this is the content of Godel's theorem aiui - that to prove that a set of axioms is consistant requires the use of further axioms, which may or may not be consistant with the original axioms), and the axioms are abstract enough not to be considered 'self evident truths' so much as 'things we'd kind of like our logical system to obey to match up with what we expect a logical system to do.'

In fact, axioms other than ones that seem like self evident truths can often be useful. In Euclid's Elements, his 'self-evident' axioms include the observation that two parallel straight lines never meet and never get further apart. He constructed a theory based on his axioms that seems to accurately describe the behaviour of objects and so people believed for thousands of years that his axioms were indeed the only sensible ones. But then relativity came along, and found that you can actually get a geometric theory by taking a different axiom that describes the geometry of space time.

Another example is that while the collection of set theory axioms known as ZF (or ZFC) that give you the nice theory of arithmetic and the rest of maths as you'd expect it, other sets of axioms don't give you arithmetic so easily but can be very usefully applied to describe certain concepts in computer science. Even 1+1=2, while very useful for describing the behaviour of rocks or teacups may be less useful for describing the behaviour of rainclouds (1+1=1) or rabbits (1+1 = hundreds before too long).

Well, I can't answer for the underlying philosophy (that is, your first paragraph), but higher level math basically doesn't have any basis in the observable world, or, at least, is meant to generalize to things that aren't, for example n-dimensional space (or even infinite dimensional space.) Now it turns out that some of the weird things, like using functions as vector spaces, actually has use in quantum mechanics, which is part of what makes QM so weird. However, there are other things that don't have any analog in the real world than we imagine. Unfortunately these tend to be hard to explain, and some of these may even have application in unknown sciences.

This is a bit of an 'if all you've got is a hammer' situation, isn't it? It's not so much that there are Hilbert spaces out there in the real world so much as that Hilbert spaces are a handy mathematical tool that you can use to crank out predictions according to a certain theory. If we had a different mathematical toolbox, we'd probably come up with a different theory or a different representation of quantum. But a physical theory for which a mathematical formalism doesn't exist isn't really a physical theory since presumably you can't use it to make quantitative predictions.

I know what you mean about the weirdness, though - it's really taking the formalism a big step away from the stuff that you can observe, and I can't think of a previous science that did that. Maybe the earliest field theories seemed equally weird and abstract at the time?

The thing is, we really don't know how closely math and science are connected, since we don't know enough about the universe to say. However, I will say that, for the most part, math and science evolved separately, and in fact it came as a surprise to many mathematicians that their field had any application beyond arithmetic and basic algebra.

Yeah, even analytic number theory seems to be claiming to be 'quantum' these days, it's ridiculous. Meanwhile, everyone I know who claims to be working in quantum gravity spends the whole time doing unfeasibly abstract pure maths.

On the other hand, there has been a very long tradition of physics and maths feeding into each other - maths gives physicists tools to describe stuff, while physics gives mathematicians specific examples of interesting things (calculus, anyone?) which they can then wander off and generalize to their hearts' content. And a similar interplay seems to be starting with computer science - hence Donald Knuth's comment "computer science is now enriching mathematics - as physics did in previous generations - by asking new sorts of questions, whose answers shed new light on mathematical structures."

averagejoe

2008-06-25, 08:49 PM

This is a bit of an 'if all you've got is a hammer' situation, isn't it? It's not so much that there are Hilbert spaces out there in the real world so much as that Hilbert spaces are a handy mathematical tool that you can use to crank out predictions according to a certain theory. If we had a different mathematical toolbox, we'd probably come up with a different theory or a different representation of quantum. But a physical theory for which a mathematical formalism doesn't exist isn't really a physical theory since presumably you can't use it to make quantitative predictions.

I know what you mean about the weirdness, though - it's really taking the formalism a big step away from the stuff that you can observe, and I can't think of a previous science that did that. Maybe the earliest field theories seemed equally weird and abstract at the time?

I'm not really trying to say that there are Hilbert spaces out there, any more than I think there are literal ones and twos out there, ripe for the adding. I was just giving an example of less intuitive maths being used in a strong physical theory, for the sake of completeness.

Indon

2008-06-25, 08:49 PM

0.(9)=1 is only true in math.

0.(9) doesn't even exist in application. There's only 1.

To add to the proofs list: The reciprocal of 0.9 repeating is 1 (specifically 1.000...). The reciprocal of 1 is 1. Reciprocality is a 1:1 function (the proof of which is left as an exercise for the reader). Therefore, 0.9 repeating is 1.

Pelfaid

2008-06-25, 09:30 PM

EDIT: It is objective in that if two people start off with the same base premise they will always get the same answer. It's not objective in that it doesn't exist independent of humans.

END EDIT

Just an aside about the bolding. It can be argued that math does exist independently of humanity, depending on which philosophical model you support.

Vikazc

2008-06-25, 09:38 PM

I think the gist of the argument about the convention comes down to something fairly simple.

2+2=4 is based on real observable things, we established that.

1/3 + 1/3 + 1/3 = 1 Is also reality, comes from seeing reality.

.999R is a completely fabrication pulled out of someones ass to make math work. It has nothing to do with reality, it only exists to make it sound reasonable to have a decimal conversion of fractional numbers.

Repeating decimals are a deliberately broken system that exists only to connect two facts. Decimals and Fractions do not truly together, so they made some **** up.

ashmanonar

2008-06-25, 09:43 PM

0.(9) is indeed the same number as 1. That's not really advanced math. Well, maybe for people who don't get xkcd jokes and other non-educated folk.

:P Some of us just get educated about different things. Like Art History. And Post-Modernism.

monty

2008-06-25, 10:04 PM

It's simple. There is no 0.(9)

You can have an infinite geometric series, the infinite sum of .9*(1/10)^n. But that isn't a number, it's a value. A value that happens to be equivalent to 1, by the infinite geometric series formula S=a/(1-r). If you take .9, and add .09, and add .009, eventually you'll have to stop, in which case you won't have .(9), you'll have .999...999. Not the same. Even if you had infinite time, the limit of your additions would, in fact, be 1.

TheStagesmith

2008-06-25, 10:10 PM

This would be so much easier if we just used fractions instead of the terribly inaccurate things known as decimals...

Dervag

2008-06-25, 10:22 PM

0.(9) = 1 is necessary for some forms of math but in the end it is arbitrary.On the contrary.

"Arbitrary" means that the only reason for doing it that way is that someone decided to pull the result out of a hat.

The observation that 0.999... = 1 is exactly the opposite of "arbitrary." It follows logically from our definitions of things like 'infinity', 'decimal places', and '1'. It's not something we can replace with a new Pronouncement From On High and have everything be the same.

Driving on the right side of the road is arbitrary. The world would not be a better or worse place if we instead drove on the left. Or, if you are in some former member nations of the British Empire, the other way around.

Saying that 0.999... = 1 is not arbitrary, any more than saying that blue light has a shorter wavelength than green light.

If we wished, we could redefine "blue" so that blue is the color of strawberries, in which case "blue" light would have a longer wavelength than green light. Or we could redefine "shorter" to mean longer. Or we could redefine "wavelength." Because definitions are arbitrary, usually.

But it would be foolish and useless to change our definitions. Likewise, it would be foolish and useless to change our definitions in math. And as long as the definitions hold, it is not an arbitrary convention that 0.999... = 1.

Just because an arbitrary convention is well accepted doesn't make it any less arbitrary. 0!=1 is useful, just like 0.(9)=1 is useful. So both are kept and used in theoretical math.

Physics theories are different from math theories. One is based on measurable, quantifiable, objective fact and the other is based on arbitrary conventions.And yet the physicist use the "arbitrary conventions" of math and end up getting real facts out of them. They can, for example, use integral and differential calculus to predict the trajectories of spacecraft. Both those forms of calculus use "arbitrary" conventions such as "you can split up a finite object into an infinite number of infinitely small pieces that have effectively zero size except when we choose to zoom in on them." No real object can be divided that way. It's quite possible that not even the empty fabric of space can be divided that way.

And yet, using calculus works. So do a lot of other mathematical abstractions. Series expansions, delta functions, and imaginary numbers all work in physics.

So don't be so quick to dismiss math as a pile of arbitrary conventions. If they were truly arbitrary, Nature wouldn't conform to them so well, because Nature is very good at not doing what we want it to do.

No, math isn't objective fact. It uses arbitrary postulates in a lot of the higher end stuff.

EDIT: It is objective in that if two people start off with the same base premise they will always get the same answer. It's not objective in that it doesn't exist independent of humans.

END EDITIf it doesn't exist independent of humans, it is certainly implemented independent of humans.

For example, human beings have nothing to do with the reason stars shine. And yet, we have deduced that stars shine because of nuclear fusion. Nuclear fusion wouldn't work without quantum mechanics. In quantum mechanics you have to use imaginary numbers to describe things.

The practical upshot is that in the real universe, even the parts where we have no control whatsoever, there is plenty of evidence for things like complex numbers "existing." They have to, because without complex numbers quantum mechanics breaks down, and without quantum mechanics stars do not shine.

We may have named complex numbers. We may have thought of them without having any physical examples to look at. But we didn't create them, because physical things that obey laws that can only be expressed in terms of complex numbers existed long before human beings did.

This is a bit of an 'if all you've got is a hammer' situation, isn't it? It's not so much that there are Hilbert spaces out there in the real world so much as that Hilbert spaces are a handy mathematical tool that you can use to crank out predictions according to a certain theory. If we had a different mathematical toolbox, we'd probably come up with a different theory or a different representation of quantum.I doubt it. There are certain weird mathematical artifacts (like complex numbers) that simply do turn up in the real world. Their consequences are observable. It's not something you can get around easily.

This is a little like the idea that civilization would have 'invented' completely different laws of physics if history were different. It's not viable because the universe works the way it does work, and not the ways it might work. There are only so many viable ways to represent something in the math.

:P Some of us just get educated about different things. Like Art History. And Post-Modernism.If it doesn't help you to understand either the nature of people, or the nature of things, it doesn't count as education in my book. The webcomic xkcd alternates between jokes about human nature and jokes about the nature of things. Therefore, if you don't get any of the jokes, you probably aren't very educated in my book.

Of course, you may get the joke and not think it is funny, which is perfectly acceptable and reasonable.

This would be so much easier if we just used fractions instead of the terribly inaccurate things known as decimals...That would make things worse; among other things there are far too many numbers that can't be written in fraction form.

Eldritch_Ent

2008-06-25, 10:29 PM

Hah, nice to see my fun little math fact on the other thread spawned a new thread. :smallbiggrin:

I just talked to my brother on this, and he insisted that the difference between 1 and 0.9999... is 0.000...1, which as far as I've been taught, isn't a valid number, since the essence of having a repeating number is the decimal *never* ends, and by putting a 1 at the end of an infinite string of zeroes has, essentially, made infinite not infinite... Which simply shouldn't work, by nature of infinity. Right?

tyckspoon

2008-06-25, 10:46 PM

I just talked to my brother on this, and he insisted that the difference between 1 and 0.9999... is 0.000...1, which as far as I've been taught, isn't a valid number, since the essence of having a repeating number is the decimal *never* ends, and by putting a 1 at the end of an infinite string of zeroes has, essentially, made infinite not infinite... Which simply shouldn't work, by nature of infinity. Right?

You're correct. The difference between 1 and 0.999...9 is 0.000...1, but 0.999...9 and 0.999... are not the same number. One is a terminated decimal (of arbitrary, possibly mind-boggling length, but finite nonetheless) and the other is a neverending stretch of nines that will never see an end. They don't work the same. If you pick any point and say 'here, the difference is here, this is where the ...1 goes' you are necessarily not talking about 0.999... and your argument has no relevance to the question.

monty

2008-06-25, 10:46 PM

Hah, nice to see my fun little math fact on the other thread spawned a new thread. :smallbiggrin:

I just talked to my brother on this, and he insisted that the difference between 1 and 0.9999... is 0.000...1, which as far as I've been taught, isn't a valid number, since the essence of having a repeating number is the decimal *never* ends, and by putting a 1 at the end of an infinite string of zeroes has, essentially, made infinite not infinite... Which simply shouldn't work, by nature of infinity. Right?

Well, that number would be 0 + 0/10 + 0/100 + ... + 1/infinity, which is also 0. So the sum is zero.

Daracaex

2008-06-25, 10:46 PM

You think that's twisted? Check this out.

Let a=b.

Then a2=ab

a2+a2=a2+ab

2a2=a2+ab

2a2-2ab=a2+ab-2ab

2a2-2ab=a2-ab

2(a2-ab)=1(a2-ab)

2=1

(Note: While that does make sense at first, there is a fallacy involved. It's still fun though! See if you can spot the error!)

Frost

2008-06-25, 10:50 PM

You think that's twisted? Check this out.

Let a=b.

Then a2=ab

a2+a2=a2+ab

2a2=a2+ab

2a2-2ab=a2+ab-2ab

2a2-2ab=a2-ab

2(a2-ab)=1(a2-ab)

2=1

(Note: While that does make sense at first, there is a fallacy involved. It's still fun though! See if you can spot the error!)

Yes we all know that dividing by zero gives you whatever you want.

monty

2008-06-25, 10:52 PM

You think that's twisted? Check this out.

Let a=b.

Then a2=ab

a2+a2=a2+ab

2a2=a2+ab

2a2-2ab=a2+ab-2ab

2a2-2ab=a2-ab

2(a2-ab)=1(a2-ab)

2=1

(Note: While that does make sense at first, there is a fallacy involved. It's still fun though! See if you can spot the error!)

DIVIDE BY ZERO DOES NOT WORK THAT WAY!!!

mmmmm_PIE

2008-06-25, 10:58 PM

Adressing dervag, I would argue that the fact that the axiomic mathematical concept 'one' can be represented in two different methods by our (highly arbitrary) notation system is only made relevant by the fact that we must learn to work within that notation system while we lack something more appropriate (if such a thing really exists).

That '0.(9) = 1' lends nothing of importance to 'pure' mathematics, only our ability to deal with it.

sikyon

2008-06-25, 11:17 PM

The only difference between math and religion is that math is internally consistent.

Just an aside about the bolding. It can be argued that math does exist independently of humanity, depending on which philosophical model you support.

Exactly. Personally I really like the idea of an independent mathematic reality that humans are slowly discovering. I don't adhere to that philosophy 100% either.

A mildly related fact: your brain is much better at math than you are. For instance, the transformation that happens in the visual cortex (I hope I'm not misremembering where it happens) to make a 3D picture out of essentially 2D information is d*mn advanced mathematically.

mroozee

2008-06-26, 12:56 AM

There seems to be a lot of mathematical voodoo being discussed.

1) 0.999... = 1?

Depends upon the context, really. Base 11, this is clearly false. "Amount of ink used" again is false. But if we are identifying points on the real line and using the usual decimal representation, these two things are the same. For more detail, see: http://en.wikipedia.org/wiki/0.999...

2) 1 + 1 = 2 (or 2 + 2 = 4)

This is not true for large values of 1 (1.3 + 1.4 = 2.7 --> 3). Once we have defined what we mean by 1, 2, +, and =, this is pretty straight forward. One such construction uses the Peano Postulates nicely done here: http://mathforum.org/library/drmath/view/51551.html

3) Regarding science and mathematics...

Mathematics is loosely speaking the study of relationships (not in the Elan/Haley sense). The laws of physics manifest themselves through (often simplified) relationships between objects, time, space, and other physical quantities and thus lend themselves to mathematical analysis.

There are many types of relationships that do not occur in the "real world" but mathematics is still able to address. The notion of infinity (or degrees of infinity), for example, is not readily apparent in our physical experience yet mathematics is ready to address it.

One other point...

Advanced mathematics - and by this I mean the original mathematics researched by professional mathematicians - plays a vital, important role in present-day physics research. When physicists tell us that the universe must have 10, 11, or 26 dimensions (for flat spacetime) it is because these are the only ones that "fit" mathematically.

Rockphed

2008-06-26, 01:39 AM

2) 1 + 1 = 2 (or 2 + 2 = 4)

This is not true for large values of 1 (1.3 + 1.4 = 2.7 --> 3). Once we have defined what we mean by 1, 2, +, and =, this is pretty straight forward. One such construction uses the Peano Postulates nicely done here: http://mathforum.org/library/drmath/view/51551.html (http://mathforum.org/library/drmath/view/51551.html)

One significant digit means you are just making a guess without any means to measure your value. It would be like measuring height or measuring distance just by glancing. Hence adding up such things is frequently wrong as by the same logic you just used, 1 + 1 = 1, for small values of 1.

Chronos

2008-06-26, 01:58 AM

The practical upshot is that in the real universe, even the parts where we have no control whatsoever, there is plenty of evidence for things like complex numbers "existing." They have to, because without complex numbers quantum mechanics breaks down, and without quantum mechanics stars do not shine.Actually, it's quite possible to construct a formulation of quantum mechanics which doesn't use complex numbers. We could just as well say that the phase of a wavefunction is a separate quantity, represented by a separate real number. It's just that complex numbers already have all the properties we need to represent things more concisely, so we use them as a matter of convenience.

kamikasei

2008-06-26, 04:55 AM

There I thought that there would be... you know... some mention of probability in this thread, and perhaps an application to gaming... lacking that, I'll just link the most comprehensive page (http://qntm.org/?pointnine) I know of on the topic (which can't be edited by cranks to include fallacious contradictions) and leave it at that.

Before I came across that page I hadn't even known anyone debated this. Two proofs, 1.000... - 0.999... = 0.000... and 3(1/3) = (0.333...) = (0.999...) = 1 seem to me to handily satisfy any problems with intuitive understanding of the point.

Winterwind

2008-06-26, 05:10 AM

Why is this in Gaming? :smallconfused:

Anyway, another physicist chiming in here... I hope this hasn't already been said, but I didn't see it skimming through the thread:

Do you know how real numbers are defined in the first place in advanced mathematics? After all, when one builds mathematics from ground up, one has only natural numbers, which one expands to rational numbers, and so forth - the step to a number with infinitely many non-repeating digits is far from trivial. Hence, real numbers are defined as the numbers Cauchy-series converge to - series of numbers which approach some particular number more and more, so that after enough steps one gets arbitrarily close to said number.

Which means that 0.999... is not only identical to 1, it's actually the very definition of the real number 1.

nagora

2008-06-26, 05:19 AM

Which means that 0.999... is not only identical to 1, it's actually the very definition of the real number 1.

I think this is part of the problem: the Real number 1 is not the same, in terms of axioms, as the Natural number 1. So any discussion has to at least start of with making it clear that we are dealing with ONLY the Real number 1, which is obvious to someone who knows their number systems but is tricky for people for whom a number is just a number and the distiction between Real and Rational or Integer and Natural is a dim memory of a primary school lesson decades in the past.

In fact, 1 is itself an axiom in the Natural numbers, as far as I remember ("There is a number '1' which when multiplied by another number x gives x" or something like that).

Winterwind

2008-06-26, 05:48 AM

In fact, 1 is itself an axiom in the Natural numbers, as far as I remember ("There is a number '1' which when multiplied by another number x gives x" or something like that).Since I happen to still have my lecture notes from six years ago saved on my computer... let's see...

Ah, here we go, Peano's axioms for natural numbers:

1. The number 1 is a natural number.

2. To each natural number n there is a natural number n'=:n+1 as 'successor' of n.

3. The number 1 is not the successor of any natural number.

4. n'=m' => n=m

5. If set M that is a subset of N contains the number 1 and with each n element of N also the successor n', then M=N

Or in short, yeah, you're right.

I think this is part of the problem: the Real number 1 is not the same, in terms of axioms, as the Natural number 1. So any discussion has to at least start of with making it clear that we are dealing with ONLY the Real number 1, which is obvious to someone who knows their number systems but is tricky for people for whom a number is just a number and the distiction between Real and Rational or Integer and Natural is a dim memory of a primary school lesson decades in the past.True, but, since the real numbers are derived from natural numbers (with a few steps in-between), the real number 1 is still identical to the natural number 1.

Okay, so me saying 0.999... was the definition of 1 was maybe not entirely correct, since 1 is indeed introduced via axiom. The identity, however, still has to hold, or the real numbers would not be internally consistent.

nagora

2008-06-26, 05:57 AM

Okay, so me saying 0.999... was the definition of 1 was maybe not entirely correct, since 1 is indeed introduced via axiom. The identity, however, still has to hold, or the real numbers would not be internally consistent.

Ah, but are they internally consistent (duh duh duuuhhh!)? Do we have an expert on Godel here who can comment?

Kurald Galain

2008-06-26, 06:14 AM

No. 2+2=4 is true in most math and in the real world. 0.(9)=1 is only true in math.

That is false. Its real world equivalent is not just that "if it walks like a duck and quacks like a duck, it probably is a duck", but that "if it is so amazingly similar to a duck that nobody, even through sub-molecular analysis, will ever be able to spot any reason for which it might be considered not a duck, is a duck".

Have you heard of the Chinese Room theorem?

An interjection: 0!=1 is not arbitary, it follows from n!=(n-1)!*n and 1! = 1, which is equivalent to the definition of the factorial function, and also follows from the observation that, in combinatronics, the number of ways you can arrange or choose from 0 objects is not 0 but 1.

Yes, it's only arbitrary using Tippy's arbitrary definition of that word. Most other people would call it a "convention", just like "0.(9) = 1" is a convention.

Interestingly, the definition of factorials stops working at that point, i.e. 0! is not equal to 0 * (-1)! because that means that -1! = 1/0. Then again, factorials only make sense for natural numbers anyway, but this reminds me that one old math teacher of mine would go into lengthy monologues about whether zero is or isn't a natural number.

nagora

2008-06-26, 06:23 AM

That is false. Its real world equivalent is not just that "if it walks like a duck and quacks like a duck, it probably is a duck", but that "if it is so amazingly similar to a duck that nobody, even through sub-molecular analysis, will ever be able to spot any reason for which it might be considered not a duck, is a duck".

Have you heard of the Chinese Room theorem?

Surely that's a contrary example: the Chinese Room is not the same as an intelligent box, it mearly seems to be one and the intelligence is the result of its creator's care and attention to detail - so the true intelligence lies outside of the box. The Chinese Box is really an argument about practicality, not essence. Somewhat like the Turing Test.

Winterwind

2008-06-26, 06:26 AM

Ah, but are they internally consistent (duh duh duuuhhh!)? Do we have an expert on Godel here who can comment?Well, I sure hope they are! :smalleek::smallbiggrin:

On a more serious note, I doubt any possible inconsistencies would reveal themselves that easily and on such a crude level. The foundations of mathematics are not that shakey. :smallwink:

Kurald Galain

2008-06-26, 06:28 AM

On the subject of 'basis in reality', no, I don't think maths can be said to have a basis in anything observable. Certainly if you take two rocks and put them next to two other rocks you've got four rocks, and every time anyone's ever done this, that's what happened.

But, since you bring up quantum later in your post, it is not guaranteed to happen in real life :smallbiggrin:

the essence of having a repeating number is the decimal *never* ends, and by putting a 1 at the end of an infinite string of zeroes has, essentially, made infinite not infinite... Which simply shouldn't work, by nature of infinity. Right?

Precisely. A common beginner's mistake is missing the difference between "arbitrarily large" and "infinite", as the latter throws a number of curveballs at what might be considered intuitive common sense. And you get all sorts of fun things like ALN + 1 = some other ALN, but INF + 1 = the very same INF.

Why is this in Gaming? :smallconfused:

Because math relates to statistics, statistics relates to dice, and dice relate to gaming. Okay, that's a bit of a stretch perhaps, but I think a valid game-related point from this thread is that not every gamer understands the odds involved in the dice rolls he makes (not that doing so is prerequisite to enjoying the game). Some gaming systems like to mess with player's heads this way.

The canonical example is the Three Doors Puzzle; but perhaps a better one is the convoluted dice mechanics in some games. Pretty much everybody can tell you the odds of making an attack roll in D&D (16+ required? That's 5 out of 20, or 25%). Now try the same in White Wolf (what are the odds again of rolling three seven-or-highers on 6d10? And if I want to make this roll slightly more difficult, should I up the target number, remove one die, or require an additional success?)

The Babylon5 rpg has a mechanic of "roll a red die and a blue die; if the blue die is higher, add the red number to your skill value; if the red die is higher, instead subtract the blue number from your skill value". Where it boggles the mind that they didn't simply ask for a 2d6 roll and upped all the target numbers by 7.

Kurald Galain

2008-06-26, 06:31 AM

Surely that's a contrary example: the Chinese Room is not the same as an intelligent box, it mearly seems to be one and the intelligence is the result of its creator's care and attention to detail - so the true intelligence lies outside of the box. The Chinese Box is really an argument about practicality, not essence. Somewhat like the Turing Test.

Yes, that was my point. From a practical point of view, the chinese box is intelligent just as that from a practical point of view, 0.(9) = 1.

From a theoretical point of view, you'd have to follow the mathematical axioms and conventions, which once more prove that 0.(9) = 1.

Whichever way you put it, 0.(9) = 1, QED.

On the third hand, from a Zen point of view, 0.(9) = purple monkey dishwasher.

nagora

2008-06-26, 06:41 AM

But, since you bring up quantum later in your post, it is not guaranteed to happen in real life :smallbiggrin:

Ah, but now you've put your finger on something important. Because, that's a statement about physics and as such has to have the phrase "as far as we know so far" tacked onto the end. 1+1=2, given the appropriate axioms is always true, there is no need to qualify it beyond providing definitions of terms.

Real life is probably the same, but in real life we don't have the luxury of defining the terms, we're stuck with having to try to work them out, and that's really very hard indeed. Which is why saying that mathematics is the foundation of all science is true and false at the same time: it is true only where we have discovered all the rules that physics is based on - but that's meaningless since we can't know when or where we've done that, even assuming there is any aspect of physics where we HAVE managed to do that at all.

Maths is a toolkit for building an infinite range of realities, but which one of those realities corresponds exactly to ours is an open question and probably an insoluble one

kamikasei

2008-06-26, 06:49 AM

Surely that's a contrary example: the Chinese Room is not the same as an intelligent box, it mearly seems to be one and the intelligence is the result of its creator's care and attention to detail - so the true intelligence lies outside of the box. The Chinese Box is really an argument about practicality, not essence. Somewhat like the Turing Test.

Oh boy, don't get me started on the Chinese Room. If nothing else, it has little relevance to the foundations of mathmatics.

pendell

2008-06-26, 07:19 AM

Lemme see if I understand what is going on here.

Let me rewrite 0.9999 (infinite 9s) as

lim (1 - x) as x approaches infinity.

Since x is an infinitesimal -- is infinitely small -- the difference between

0.99999 (infinite) and 1 is literally a distinction without a difference. There is literally

no measurable or quantifiable way that you can treat the two numbers differently,

because whatever you do to 0.999999 -- no matter how far out you go --

the difference between it and 1.0 is still infinitely further out to the right. That's the

meaning of infinity.

So you can add, subtract , exponentiate, do whatever you like to 0.(999...) It will

always behave precisely the same as 1.0. The infinitesimal difference between

it and 1.0 cannot be measured, quantified, or used in any way. The idea that it is there at all is a leap of faith, since there's no way you can quantify or observe the difference.

Is that correct?

If so .. it's a neat parlor trick. How is it used?

Respectfully,

Brian P.

kamikasei

2008-06-26, 07:48 AM

Let me rewrite 0.9999 (infinite 9s) as

lim (1 - x) as x approaches infinity.

I think you mean "as x approaches zero" here.

Is that correct?

Not precisely. There is no difference between 0.999... and 1, any more than there's a difference between 4/4, 3/3, and 1. They're the same number written in different ways. No "leap of faith" is involved.

If so .. it's a neat parlor trick. How is it used?

It's not used, it just is. It's a fact, which it seems many people feel obliged to deny for some reason.

Indon

2008-06-26, 08:05 AM

Surely that's a contrary example: the Chinese Room is not the same as an intelligent box, it mearly seems to be one and the intelligence is the result of its creator's care and attention to detail - so the true intelligence lies outside of the box. The Chinese Box is really an argument about practicality, not essence. Somewhat like the Turing Test.

The Chinese Box is an argument about what intelligence is, and really has absolutely no bearing on a discussion about 0.999... Or math in general.

(P.S. - The Chinese Box is a distinct computing apparatus, it's just running completely inappropriate hardware to emulate its' very much Turing-Complete computing system)

It's not used, it just is. It's a fact, which it seems many people feel obliged to deny for some reason.

Like the Monty Hall problem! It took me so long to get how that worked.

Why isn't the Monty Hall problem in this thread, anyway? There's your probability right there.

Kurald Galain

2008-06-26, 08:19 AM

The canonical example is the Three Doors Puzzle; but perhaps a better one is the convoluted dice mechanics in some games.

Why isn't the Monty Hall problem in this thread, anyway? There's your probability right there.

There you go.

http://en.wikipedia.org/wiki/Monty_Hall_problem

I know a friend who's been getting lots of free drinks on people who didn't believe it (and thus figured they'd have an even change of getting a drink off him). Also, I have this ex-gf who was a math student and it took me hours to convince her...

pendell

2008-06-26, 08:28 AM

I think you mean "as x approaches zero" here.

Yes, I did. Thank you.

Respectfully,

Brian P.

pendell

2008-06-26, 08:35 AM

Follow up question.

Since the difference between 0.999..... and 1.0 is literally a distinction without

a difference, just as 4/4 and 1.0 are identical -- why would anyone write

0.999.... when they could write 1? Or 1.0? Could you arrive at 0.999.... through

computation? Is it used as a necessary component in an axiom, as i is needed

to discuss imaginary roots? Or is it simply a philosophical artifact, something to

argue about incessently while the DM is trying desperately to figure out how to get

her campaign back on the rails?

I suspect this question has already been answered, but I'm still trying to wrap my mind around the concept.

Respectfully,

Brian P.

kamikasei

2008-06-26, 08:42 AM

Like the Monty Hall problem! It took me so long to get how that worked.

Thing is, Monty Hall is actually deeply weird and counter-intuitive, like a lot of things about probability. The 0.999... thing is very clear-cut and I don't understand why anyone would try to disprove it.

Jorkens

2008-06-26, 08:44 AM

Follow up question.

Since the difference between 0.999..... and 1.0 is literally a distinction without

a difference, just as 4/4 and 1.0 are identical -- why would anyone write

0.999.... when they could write 1? Or 1.0? Could you arrive at 0.999.... through

computation? Is it used as a necessary component in an axiom, as i is needed

to discuss imaginary roots? Or is it simply a philosophical artifact, something to

argue about incessently while the DM is trying desperately to figure out how to get

her campaign back on the rails?

I suspect this question has already been answered, but I'm still trying to wrap my mind around the concept.

Respectfully,

Brian P.

It exists for the same reason that 4/4 and 8/8 exist - because the notation is defined in a way that's simple to explain and powerful enough to express any real number, and you can't get both of those positive properties without the slight drawback of having some numbers capable of being represented in two different ways.

I can't think of a situation where it's 'better' to write 0.(9) instead of 1.0, it's just that if you set up the notation so that you couldn't write 0.(9) instead of 1.0 you'd have to restrict it in a rather inelegant way.

Kurald Galain

2008-06-26, 08:47 AM

why would anyone write

0.999.... when they could write 1?

Generally, because they don't understand that they're the same thing.

Or 1.0?

That's physics; significant figures don't apply to math.

Is it used as a necessary component in an axiom, as i is needed

to discuss imaginary roots?

Nope.

Could you arrive at 0.999.... through

computation?

Yes, for instance through explaining limits, as Pendell suggested earlier.

Infinity is a funny thing, and rather counterintuitive. 1 + 1/2 + 1/4 + 1/8 repeating ad infinitum (sigma of 1 / 2^x where x ranges from zero to infinity) has a limit of two (that is, add up enough fractions in this fashion, where by "enough" I mean "infinite" and you end up at 2).

1 + 1/2 + 1/3 + 1/4 + 1/5 repeating ad infinitum (sigma of 1 / x where x ranges from one to infinity) is divergent (that is, add up enough fractions in this fashion, where again I mean infinite fractions, and you end up above any Arbitrarily Large Number you can mention).

Manga Shoggoth

2008-06-26, 09:05 AM

"God made the integers; everything else is the work of man."

- Leopold Kroenecker

Funkyodor

2008-06-26, 09:19 AM

I view this .(9) vs 1 issue as an exploitable aspect of the base 10 numbering system. You don't run into this situation if you change to base 12 where 1/3=.4, 2/3=.8, & 3/3=1

Nice non-infinitive fraction conversions. When you deal with infinitives then peoples minds break and Head 'Asplode!

kamikasei

2008-06-26, 09:30 AM

I view this .(9) vs 1 issue as an exploitable aspect of the base 10 numbering system. You don't run into this situation if you change to base 12 where 1/3=.4, 2/3=.8, & 3/3=1

And 1/11th? 1/5th?

The .999... = 1 identity applies exactly as much to other bases, the number just looks different. 0.BBB... = 1 in base 12.

Frosty

2008-06-26, 10:24 AM

Thing is, Monty Hall is actually deeply weird and counter-intuitive, like a lot of things about probability. The 0.999... thing is very clear-cut and I don't understand why anyone would try to disprove it.

I have to disagree. Initial confusion is due to the fact that the parameters of the game isn't stated clearly. After reading the exact statement of the Monty hall problem, it became clear rather quickly to me why switching is advantageous. You realize how simple it is when you realize these two facts:

1: The host always opens a door with a goat behind it the first time and it is not the door you picked.

2: The chances of you getting it right on your first pick is only 1/3

kamikasei

2008-06-26, 10:32 AM

2: The chances of you getting it right on your first pick is only 1/3

I understand that the argument is: when you first pick you have a 1/3 chance of choosing correctly. Then when you pick again you have a 1/2 chance of choosing correctly, so you should swap.

What I find confusing about it is twofold: once you've picked a door, there either is or isn't a prize behind it. Arguing about 1/3rd probabilities seems to imply that the prize acutally is there, or isn't there, or isn't there, three possibilities for a binary state. Secondly, once the choices are reduced to two - and you have already chosen one when it was a choice out of three - why does changing help you? Surely the probability that the prize is behind either remaining door is 1/2 and so it's 1/2 that it's behind the door you've already chosen and you might as well stick with that choice?

I'm not saying it's wrong. I'm saying it's counterintuitive. Maybe I'll find an explanation some day that gives me an intuitive understanding of how it works. On the other hand, to me there are several proofs for 0.999... = 1 which satisfy my intuition.

edit: And now that I read the Wikipedia article linked above I see an argument that does make sense to me and, while not "intuitive" (I needed to walk through the reasoning), doesn't seem wrong to me the way other arguments have. It seems I must go beat up my mathematician friends for confusing me with actually invalid arguments, and berate myself for writing off probability as intuitively incomprehensible as a result.

hamishspence

2008-06-26, 10:32 AM

we know that the difference between 0.999R and 1 is infinitely small But thats not quite the same thing.

still, if 1/3= 0.333R and 3 times 0.333R = 0.999R, and 3 times 1/3 = 1, it does make sense that 1 = 0.999R by simple substitution.

But we'd need proof that 1/3=0.333R for it to work, and we don't have it.

hamishspence

2008-06-26, 10:39 AM

the monty hall problem

You need to remember, first, that the host knows which it is, second, that the host will not open your door first, whether or not it has something behind it.

It works like this: you had a 1 in 3 chance of choosing correctly the first time, and the host opening either of the other doors does not affect this 1 in 3 chance.

So, with 1 door already open, your door still only has a 1 in 3 chance, but the other door has a 1 in 2 chance, since the 3rd door is already excluded.

So, you are choosing between a 1/3 chance (stick) and a 1/2 chance (swap) and thats why you should swap.

Kurald Galain

2008-06-26, 10:44 AM

I'm not saying it's wrong. I'm saying it's counterintuitive. Maybe I'll find an explanation some day that gives me an intuitive understanding of how it works. On the other hand, to me there are several proofs for 0.999... = 1 which satisfy my intuition.

In other words, "common sense isn't". What one person considers common sense is mystifying to another, and vice versa.

we know that the difference between 0.999R and 1 is infinitely small But thats not quite the same thing.

Mathematically speaking, yes it is.

But we'd need proof that 1/3=0.333R for it to work, and we don't have it.

Actually we do. The easy way of showing this is by making a long division.

kamikasei

2008-06-26, 10:48 AM

In other words, "common sense isn't". What one person considers common sense is mystifying to another, and vice versa.

Actually in this case I would say that common sense is common, and is also wrong. The vast majority of people apparently give the wrong answer to Monty Hall. The human brain seems hard-wired to screw up probability.

hamishspence

2008-06-26, 10:48 AM

so, infinitely small = 0, mathematically? odd, but I suppose it works.

Winterwind

2008-06-26, 10:51 AM

I understand that the argument is: when you first pick you have a 1/3 chance of choosing correctly. Then when you pick again you have a 1/2 chance of choosing correctly, so you should swap.Nope. The chance of you having originally chosen is 1/3; hence the probability of it being behind the remaining door after the host opened a door is 2/3, and this is why you should swap.

What I find confusing about it is twofold: once you've picked a door, there either is or isn't a prize behind it. Arguing about 1/3rd probabilities seems to imply that the prize acutally is there, or isn't there, or isn't there, three possibilities for a binary state. Secondly, once the choices are reduced to two - and you have already chosen one when it was a choice out of three - why does changing help you? Surely the probability that the prize is behind either remaining door is 1/2 and so it's 1/2 that it's behind the door you've already chosen and you might as well stick with that choice?

I'm not saying it's wrong. I'm saying it's counterintuitive. Maybe I'll find an explanation some day that gives me an intuitive understanding of how it works. On the other hand, to me there are several proofs for 0.999... = 1 which satisfy my intuition.Allow me to make that attempt...

There are three possibilities how this can go:

Case 1: Door A holds the prize, Door B is empty, Door C is empty

Case 2: Door A is empty, Door B holds the prize, Door C is empty

Case 3: Door A is empty, Door B is empty, Door C holds the prize

Let's say you chose Door A. This means that in Case 1, you win, and in Case 2 and 3, you lose.

Now the host eliminates a door that is empty, and you have:

Case 1: Door A holds the prize, either Door B or C remains and is empty

Case 2: Door A is empty, Door B holds the prize

Case 3: Door A is empty, Door C holds the prize

So if you switch, you win in Case 2 and 3, and lose in Case 1.

Hence, in two out of three cases, you win if you switch.

_____

Another way to illustrate this is to assume there are 1000 doors, and the host proceeds to open 998 doors that you have not chosen and that don't hold the prize.

Your initial chance of choosing the right one is negligible, the probability that the prize is behind one of the remaining doors is overwhelming. Then, the host reveals all of the empty ones from the remaining doors, and keeps one closed. If you switch, your chance of victory is 99.9%.

____

Even more simply put, by opening a door that is empty and that you haven't chosen, the host is basically saying "Look, the prize is either behind the door you have chosen, or one of the other doors. I'm going to tell you now behind which of the other doors it is, if it is behind the other doors.". Switching effectively amounts to you chosing all of the doors you did not choose initially at once.

____

I hope at least one of the above was somewhat intuitive. :smallredface:

So, you are choosing between a 1/3 chance (stick) and a 1/2 chance (swap) and thats why you should swap.Sorry, no. Swapping yields a 2/3 chance.

It has to, because the chances have to sum up to 1. If they did not, this would indicate there was a non-zero chance for the prize to not be behind any of the doors, which is contrary to the problem as posed.

hamishspence

2008-06-26, 10:58 AM

Oops, had right idea but wrong percentage. That does make sense.

kamikasei

2008-06-26, 10:59 AM

So, with 1 door already open, your door still only has a 1 in 3 chance, but the other door has a 1 in 2 chance, since the 3rd door is already excluded.

So, you are choosing between a 1/3 chance (stick) and a 1/2 chance (swap) and thats why you should swap.

Which leaves me with a 1/6 chance of... what?

Say rather (from Wikipedia) that when I first choose, I have a 1/3 chance of choosing the car and a 2/3 chance of choosing a goat. If I choose the car, then whichever door the host leaves behind will hide a goat, so switching will cause me to lose. If I choose a goat, the host will have to leave the door with the car, so switching will win in that case, which occurs with 2/3 probability. There; that paraphrase seems reasonably intuitive to me. I think part of the problem with this (beyond its initial counterintuitive nature) is that people explaining it often get it wrong (no offense; I'm pretty sure the flawed argument you gave was also given to me by a mathematician in the past).

edit: Thanks Winterwind, your first argument matches what I said here. Would you believe that my seeming incomprehension is due to the forum's instability? I read the Wikipedia article and got the idea before my reply to Kurald, but my posts and edits kept getting eaten. :smalltongue:

Frosty

2008-06-26, 11:00 AM

I understand that the argument is: when you first pick you have a 1/3 chance of choosing correctly. Then when you pick again you have a 1/2 chance of choosing correctly, so you should swap.

What I find confusing about it is twofold: once you've picked a door, there either is or isn't a prize behind it. Arguing about 1/3rd probabilities seems to imply that the prize acutally is there, or isn't there, or isn't there, three possibilities for a binary state. Secondly, once the choices are reduced to two - and you have already chosen one when it was a choice out of three - why does changing help you? Surely the probability that the prize is behind either remaining door is 1/2 and so it's 1/2 that it's behind the door you've already chosen and you might as well stick with that choice?

I'm not saying it's wrong. I'm saying it's counterintuitive. Maybe I'll find an explanation some day that gives me an intuitive understanding of how it works. On the other hand, to me there are several proofs for 0.999... = 1 which satisfy my intuition.

Here is the way I pictured it when I read the full problem.

So...first premise: The host always opens one of the 2 doors you didn't choose.

Ok, the first thing that came to mind was: Hey, the host's 2 doors has a bigger chance of having a car than the 1 door I've got.

Now, for the next premise: Out of those two doors, the host always chooses to open a door that does NOT contain the prize (the car).

Ok, so I thought: now his pool of doors is reduced to one. He's STILL GOT BETTER CHANCES THAN ME because he initially got 2 doors and I only got one.

A more logical way of thinking about it is that the fact he has now revealed that one of the doors does not contain a Car doesn't affect the probability that one of the two doors the host got has a car.

So now, the host offers me a choice. I can choose to stay with my original chance of winning, which is 1/3, or I can instead take the host's chances, which is 2/3.

----------------------------------------

Another way to think about it is like this:

Swtiching is better. Why? Because, the ONLY way I can win by NOT switching is if I was right the FIRST TIME. Since the probability that I was righ the FIRST TIME is only 1/3, it's better to switch.

kamikasei

2008-06-26, 12:22 PM

The canonical example is the Three Doors Puzzle; but perhaps a better one is the convoluted dice mechanics in some games. Pretty much everybody can tell you the odds of making an attack roll in D&D (16+ required? That's 5 out of 20, or 25%). Now try the same in White Wolf (what are the odds again of rolling three seven-or-highers on 6d10? And if I want to make this roll slightly more difficult, should I up the target number, remove one die, or require an additional success?)

I'm assuming you want three or more successes.

The answer is (sum from n to s of ((p^n)((1-p)^(s-n))(s-choose-n))), where n is the required number of successes (3), s is the size of the dice pool (6), and p is the probability of a success per die (.4).

http://i179.photobucket.com/albums/w310/kamikasei/whitewolf_probability_formula.png

I think. I'll see if I can render that as a formula. Reasoning below.

The chance of getting three or more successes is the chance of getting three successes, plus that of getting four, plus that of five, plus that of six.

For each number of success, the chance of getting that number n is p^n times (1-p)^(s-n), times the number of ways of choosing n successes from among the s dice. So say we're looking at two successes. The probability of getting successes on (only) the first two dice is .4*.4*.6*.6*.6*.6. Multiply this by the number of combinations of two successes among six dice - s-choose-n, 6-choose-2, or 15. So the probability of getting two successes in this case is about 0.311. Do this for all the numbers of successes that will satisfy, and add them up. I've just checked, and this approach does yield a total probability of 1, so I don't think I've screwed up. That gives a probability of rolling three success on six die of about 0.456. To see how you could make it slightly harder, you'd have to work out which parameter causes the smallest increase when altered.

monty

2008-06-26, 01:07 PM

Ok, here's another one for you:

First, I'm going to be using complex exponents, so you need to know (if you don't already) that e^([pi]i)=-1, which follows from e^(ki)=cos(k)+isin(k). Therefore, ln(-1)=[pi]i.

Anyway, on to the problem:

0=0 Identity property

ln(1)=0 A basic property of logarithms

ln((-1)^2)=0 1 is equivalent to (-1)^2

2ln(-1)=0 Power property

2[pi]i=0 As shown above

1=0 Divide both sides by 2[pi]i

x-y=0 Multiply both sides by x-y

x=y Add y to both sides

This holds for all complex values of x and y. Comments?

Kurald Galain

2008-06-26, 01:17 PM

I'm assuming you want three or more successes.

The answer is (sum from n to s of ((p^n)((1-p)^(s-n))(s-choose-n))), where n is the required number of successes (3), s is the size of the dice pool (6), and p is the probability of a success per die (.4).

Yes :smallbiggrin: my point is that it's kind of tricky to figure out on the fly. I did a large amount of math on it once, and as I recall, which of the three methods I suggested increases the difficulty most depends on the initial situation.

Of course, WW wised up to that and simplified their system in nWOD. And it doesn't really matter while playing anyway, most of the time. But it is not a very clear statistic.

Winterwind

2008-06-26, 01:26 PM

Therefore, ln(-1)=[pi]i.ln(-1)=[pi]i+2k[pi]i, for all integers k, actually; but alright, let's go with the main value.

Anyway, on to the problem:

0=0 Identity property

ln(1)=0 A basic property of logarithms

ln((-1)^2)=0 1 is equivalent to (-1)^2

2ln(-1)=0 Power property

2[pi]i=0 As shown above

1=0 Divide both sides by 2[pi]i

x-y=0 Multiply both sides by x-y

x=y Add y to both sides

This holds for all complex values of x and y. Comments?The power property does not apply to complex logarithms. Everything that follows after that line is false because, well, once you have introduced a mistake you can prove anything from there.

Thank you, by the way - I knew there was some riddle like this and had wanted to pose it to various people in the past, but couldn't remember its exact content.

kamikasei

2008-06-26, 01:27 PM

Yes :smallbiggrin: my point is that it's kind of tricky to figure out on the fly. I did a large amount of math on it once, and as I recall, which of the three methods I suggested increases the difficulty most depends on the initial situation.

True, thought it's not really advanced math, just tedious calculation. But you're right that a game designer should probably have enough mathematical acumen - or build off the work of someone who does - to ensure that, even if the exact numbers aren't visible at a glance, players and DMs can intuitively know whether tweaking a variable in one direction or another will increase or decrease the odds of success.

2[pi]i=0 As shown above

You could just stop here, or at 1=0 on the next line. The shenanigans are clear enough without bringing x and y in to it.

Tehnar

2008-06-26, 01:27 PM

Ok, here's another one for you:

First, I'm going to be using complex exponents, so you need to know (if you don't already) that e^([pi]i)=-1, which follows from e^(ki)=cos(k)+isin(k). Therefore, ln(-1)=[pi]i.

Anyway, on to the problem:

0=0 Identity property

ln(1)=0 A basic property of logarithms

ln((-1)^2)=0 1 is equivalent to (-1)^2

2ln(-1)=0 Power property

2[pi]i=0 As shown above

1=0 Divide both sides by 2[pi]i

x-y=0 Multiply both sides by x-y

x=y Add y to both sides

This holds for all complex values of x and y. Comments?

Would be great, if negative numbers can be the argument of a logaritam.

As they cant, you cant comput Log(-1)

mroozee

2008-06-26, 01:29 PM

Well, I sure hope they are! :smalleek::smallbiggrin:

On a more serious note, I doubt any possible inconsistencies would reveal themselves that easily and on such a crude level. The foundations of mathematics are not that shakey. :smallwink:

ZFC looks pretty good right now and has for quite some time. Of course, Gottlob Frege's formal system in the Grundgesetze looked solid for 10 or 12 years before Bertrand Russell poked a hole in it. But ZFC, like every other formal system that includes the natural numbers (and fits a couple of other standard requirements) is incomplete - meaning there are some statements that it can't voice a definitive opinion on (like the Continuum Hypothesis).

The results from Godel's Second Incompleteness Theorem are more to the point: If a (sufficiently strong) axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.

Simply put: "If mathematics is consistent, it cannot prove its own consistency."

monty

2008-06-26, 01:35 PM

Would be great, if negative numbers can be the argument of a logaritam.

As they cant, you cant comput Log(-1)

They can, actually, but as has been mentioned, the power property does not apply, which is why it fails. Take a graphing calculator, put it in complex mode, and see what ln(-1) gives you. On my trusty TI-84, I see 3.141592654i.

Jack_Simth

2008-06-26, 04:07 PM

Precisely. A common beginner's mistake is missing the difference between "arbitrarily large" and "infinite", as the latter throws a number of curveballs at what might be considered intuitive common sense. And you get all sorts of fun things like ALN + 1 = some other ALN, but INF + 1 = the very same INF.

Do note:

If Infinity + 1 = The same Infinity, then you've got one of two possibilities:

1) Infinity - Infinity = Undefined

2) 1 = 0 (and everything that follows - namely, that all numbers equal 0, and through it, that all numbers are equal)

If, instead, you treat Infinity as a number that is arbitrarily larger than any known number, but is itself an unknown number or a symbolic number, that issue goes away.

Jorkens

2008-06-26, 05:49 PM

ZFC looks pretty good right now and has for quite some time.

Certainly noone's found a problem with it yet, and it seems to keep producing useful results.

But ZFC, like every other formal system that includes the natural numbers (and fits a couple of other standard requirements) is incomplete - meaning there are some statements that it can't voice a definitive opinion on (like the Continuum Hypothesis).

My favourite fact about ZFC is that although ZFC has now been proved to be consistent relative to ZF (ie if ZF is consistent then adding the axiom of choice won't make it inconsistent), people have also proved that ZF + antichoice is consistent relative to ZF (ie you could also add the negation of the axiom of choice and get a consistent system.) I don't know how interesting antichoice setups are or what they look like, though.

A friend of mine once said that the axiom of choice is almost certainly true, the well-ordering principle is almost certainly false and noone knows about Zorn's lemma...

The results from Godel's Second Incompleteness Theorem are more to the point: If a (sufficiently strong) axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.

Simply put: "If mathematics is consistent, it cannot prove its own consistency."

Or alternatively, "to prove that a logical system is consistent, you have to use a bigger logical system. And then you don't know whether the bigger system is consistent, so you haven't really advanced much."

brian c

2008-06-26, 05:51 PM

Do note:

If Infinity + 1 = The same Infinity, then you've got one of two possibilities:

1) Infinity - Infinity = Undefined

2) 1 = 0 (and everything that follows - namely, that all numbers equal 0, and through it, that all numbers are equal)

If, instead, you treat Infinity as a number that is arbitrarily larger than any known number, but is itself an unknown number or a symbolic number, that issue goes away.

Let's say a few things about infinity.

The lowest-order infinity (for most purposes, the only "infinity"), denoted by c (equivalently, the sideways 8 we've all seen but there's no symbol for it on my keyboard) is defined as the number of natural numbers, ie the number of elements in the set {1,2,3,4,5...}

If any other set can be put into a one-to-one correspondence with the natural numbers, that set also has c elements; one would say the set has "cardinality c"

So let's go over this: take the set of all even positive integers, {2,4,6,8,10,...}. This set has cardinality c which can be seen by using the function f(n) = 2n from the naturals to this set.

Let's make a new set; if the even set from the previous paragraph is called E, this new set is E'. E' = {1,2,4,6,8,10...} so that it contains 1, and also every even positive integer. It should be clear that E' has one more element than E. However, take the function

f(n) = { 1 when n = 1

{ 2(n-1) else

This function sets a one-to-one correspondence between the natural numbers and E', so the cardinality of E' is c, the same as the cardinality of E. In this specific case, "infinity - infinity" equals 1.

With different choices of sets considered, "infinity - infinity" could equal anything you like, and thus formally speaking that value is left undefined.

For anyone who has done calculus, this is very similar in concept to taking a limit which tends to 0/0 or infinity/infinity. You can probably find a value for it (using L'Hopital's rule) but that value could be anything, and cannot be taken as "the" value of 0/0.

Jack_Simth

2008-06-26, 09:55 PM

Let's say a few things about infinity.

The lowest-order infinity (for most purposes, the only "infinity"), denoted by c (equivalently, the sideways 8 we've all seen but there's no symbol for it on my keyboard) is defined as the number of natural numbers, ie the number of elements in the set {1,2,3,4,5...}

If any other set can be put into a one-to-one correspondence with the natural numbers, that set also has c elements; one would say the set has "cardinality c"

Do note: This route has a lot of similarities to "big O" notation on the order of a function. You've end up with two completely different items sets that have the same "order".

So let's go over this: take the set of all even positive integers, {2,4,6,8,10,...}. This set has cardinality c which can be seen by using the function f(n) = 2n from the naturals to this set.

Let's make a new set; if the even set from the previous paragraph is called E, this new set is E'. E' = {1,2,4,6,8,10...} so that it contains 1, and also every even positive integer. It should be clear that E' has one more element than E. However, take the function

f(n) = { 1 when n = 1

{ 2(n-1) else

This function sets a one-to-one correspondence between the natural numbers and E', so the cardinality of E' is c, the same as the cardinality of E. In this specific case, "infinity - infinity" equals 1.

Yes, and there's also similar ways to collapse the set of all integers to the set of natural numbers, and ways to collapse the set of all ordered pairs of natural numbers to a single set of natural numbers (and, by extension, the set of all triplets of natural numbers, and so on), and several other things.

With different choices of sets considered, "infinity - infinity" could equal anything you like, and thus formally speaking that value is left undefined.

Yeah - which is why it works fairly well treating different infinities (not just infinites of different cardinalities, which can actually be hard to differentiate when people start getting clever on manipulations) as different symbols if you're going to be doing actual manipulation on them.

For anyone who has done calculus, this is very similar in concept to taking a limit which tends to 0/0 or infinity/infinity. You can probably find a value for it (using L'Hopital's rule) but that value could be anything, and cannot be taken as "the" value of 0/0.

Yep. When you start treating the infinities in equations as discreet values of exceedingly high, but unknown, value, such issues go away - but then, infinity +1 != the same infinity (it's a different infinity).

brian c

2008-06-26, 10:30 PM

discreet values

I feel bad for nitpicking, but they aren't discreet (unobstrusive) values, they're discrete (separate/non-continuous) values.

Jack_Simth

2008-06-26, 11:02 PM

I feel bad for nitpicking, but they aren't discreet (unobstrusive) values, they're discrete (separate/non-continuous) values.

Yes, well, Ay kant spel.

mroozee

2008-06-27, 01:15 AM

A friend of mine once said that the axiom of choice is almost certainly true, the well-ordering principle is almost certainly false and noone knows about Zorn's lemma...

Ok, that's pretty funny :smallbiggrin:

Let's say a few things about infinity.

The lowest-order infinity (for most purposes, the only "infinity"), denoted by c (equivalently, the sideways 8 we've all seen but there's no symbol for it on my keyboard) is defined as the number of natural numbers, ie the number of elements in the set {1,2,3,4,5...}

I don't agree with this. The cardinality of the Natural numbers is aleph-null while the cardinality of the real line is usually denoted c (or something close to that) and is equal to 2^(aleph-null) which, in turn, equals aleph-one if you buy into the Continuum Hypothesis. All three are very important and the undecideability of c = aleph-one is the crown-jewell example of Incompleteness.

Kurald Galain

2008-06-27, 03:59 AM

Do note:

If Infinity + 1 = The same Infinity, then you've got one of two possibilities:

1) Infinity - Infinity = Undefined

Precisely. Surely you didn't expect infinity minus infinity to equal something in particular? X - X = 0 holds true for real values of X (or, for that matter, complex values), which infinity is not.

If, instead, you treat Infinity as a number that is arbitrarily larger than any known number, but is itself an unknown number or a symbolic number, that issue goes away.

Incorrect. The point is that infinity is precisely not an arbitrarily large number. Whatever value you give to your ALN, it's easy to imagine a bigger ALN by simply adding one; for infinity, this is not possible (although yes, bigger infinities exist). Like I said earlier, one of the differences is that ALN + 1 = another ALN, and INF + 1 = INF.

Do note: This route has a lot of similarities to "big O" notation on the order of a function. You've end up with two completely different items sets that have the same "order".

The entire point of "big O" notation is that different items can have the same order. And second, no, two infinite values of the same cardinality are in fact the same value, just like .(9) = 1 which we discussed earlier.

Yes, and there's also similar ways to collapse the set of all integers to the set of natural numbers, and ways to collapse the set of all ordered pairs of natural numbers to a single set of natural numbers

Yes, but notably not to the set of real numbers. That is a different infinity (aleph-one according to some theories, but certainly not aleph-null).

Yep. When you start treating the infinities in equations as discreet values of exceedingly high, but unknown, value, such issues go away - but then, infinity +1 != the same infinity (it's a different infinity).

Nope. You're falling for the common beginner's mistake I mentioned earlier.

Kurald Galain

2008-06-27, 04:13 AM

Do note:

If Infinity + 1 = The same Infinity, then you've got one of two possibilities:

1) Infinity - Infinity = Undefined

Precisely. Surely you didn't expect infinity minus infinity to equal something in particular? X - X = 0 holds true for real values of X (or, for that matter, complex values), which infinity is not.

If, instead, you treat Infinity as a number that is arbitrarily larger than any known number, but is itself an unknown number or a symbolic number, that issue goes away.

Incorrect. The point is that infinity is precisely not an arbitrarily large number. Whatever value you give to your ALN, it's easy to imagine a bigger ALN by simply adding one; for infinity, this is not possible (although yes, bigger infinities exist). Like I said earlier, one of the differences is that ALN + 1 = another ALN, and INF + 1 = INF.

Do note: This route has a lot of similarities to "big O" notation on the order of a function. You've end up with two completely different items sets that have the same "order".

The entire point of "big O" notation is that different items can have the same order. And second, no, two infinite values of the same cardinality are in fact the same value, just like .(9) = 1 which we discussed earlier.

Yes, and there's also similar ways to collapse the set of all integers to the set of natural numbers, and ways to collapse the set of all ordered pairs of natural numbers to a single set of natural numbers

Yes, but notably not to the set of real numbers. That is a different infinity (aleph-one according to some theories, but certainly not aleph-null).

Yep. When you start treating the infinities in equations as discreet values of exceedingly high, but unknown, value, such issues go away - but then, infinity +1 != the same infinity (it's a different infinity).

Nope. You're falling for the common beginner's mistake I mentioned earlier.

Winterwind

2008-06-27, 05:30 AM

I would comment on this, but Kurald Galain has already posted everything I would have said, and more.

With regard to L'Hôpital's rule, there is no issue with it giving different results for "0/0", because it does not exist to calculate 0/0 in the first place - it exists to calculate the limit of specific terms, of which both the numerator and denominator converge to 0, which is not exactly the same as the whole term converging to 0/0, since they can converge at different speeds.

For example, x/sin(x). For x->0, the numerator and the denominator both converge to 0 seperately. Does the whole term converge to 0/0? No - LHôpital tells you that it actually converges to 1/1=1. And this is the true and only result of taking the limit, a most definite value, that cannot be "anything".

Of course, a different term where the numerator and denominator go to 0 would yield a different value, but that's because L'Hôpital is not concerned with 0/0, it's concerned with the properties of the particular terms in question.

Jack_Simth

2008-06-27, 06:33 AM

Precisely. Surely you didn't expect infinity minus infinity to equal something in particular? X - X = 0 holds true for real values of X (or, for that matter, complex values), which infinity is not.

Can you name any other type of number where subtracting it from itself

Incorrect. The point is that infinity is precisely not an arbitrarily large number. Whatever value you give to your ALN, it's easy to imagine a bigger ALN by simply adding one; for infinity, this is not possible (although yes, bigger infinities exist). Like I said earlier, one of the differences is that ALN + 1 = another ALN, and INF + 1 = INF.

I didn't say it was an arbitrarily large number; I said if you treat it as one, lots of the things that give people headaches go away. If you say all infinities of the same cardinality are equal, then you have to put in an extra rule that Infinity(cardinality k) - infinity (cardinality k) = undefined (plus a handful of others), otherwise you get results of the nature of 1=0 if you do basic manipulation on them. If you stop saying that infinity + 1 = same infinity, and instead say that infinity + 1 = a negligibly different infinity, you don't need the extra rules for comparing them ... and it's these extra rules that get people in trouble and cause headaches.

The entire point of "big O" notation is that different items can have the same order. And second, no, two infinite values of the same cardinality are in fact the same value, just like .(9) = 1 which we discussed earlier.

They have the same order, yes; but x^2 != x^2 + x except in a very small number of cases. When x is sufficently large, the difference is small enough that we don't really care about it... but they're very much not the same thing, even though the order is the same. I'd suggest that the cardinality of infinites ought to be treated similarly.

Yes, but notably not to the set of real numbers. That is a different infinity (aleph-one according to some theories, but certainly not aleph-null).

Nobody's figured out how to do the irrationals, at least; all the rational numbers are covered (as they're all ordered pairs of integers).

Nope. You're falling for the common beginner's mistake I mentioned earlier.

You're missing the "treating" and looking at it like it is an "is" - you're parsing wrong.

Kurald Galain

2008-06-27, 06:58 AM

Can you name any other type of number where subtracting it from itself

Perhaps if you could complete the sentence first.

I didn't say it was an arbitrarily large number; I said if you treat it as one, lots of the things that give people headaches go away.

Yes. And if you treat prime numbers as "numbers divisible by one and by themselves" (which is the elementary school definition) then some headaches will also go away, but the point is that you would be wrong. If you'd restrict fractals to real numbers, ignoring the complex parts or the concept of partial dimensions, then you'd also save yourself a lot of headaches, but again you'd be wrong. Yes, there exist figures that have two point five dimensions, rather than two or three.

If you say all infinities of the same cardinality are equal, then you have to put in an extra rule that Infinity(cardinality k) - infinity (cardinality k) = undefined

An extra rule isn't necessary as this follows from the existing axioms. Basic manipulation does not apply in boundary conditions. X / X = 1 except where X = 0, and so forth. Velocities can be added with basic manipulation, except where you're approaching light speed.

If you stop saying that infinity + 1 = same infinity, and instead say that infinity + 1 = a negligibly different infinity, you don't need the extra rules for comparing them ...

Yes, but then once again you'd be wrong. If you assume that bicycles are the same thing as automobiles, you won't need extra rules for traffic governing them. Doesn't mean it's a good idea though. Math isn't about making it easy for people and preventing their headaches. Math is about logic and fundamentals.

Nobody's figured out how to do the irrationals, at least; all the rational numbers are covered (as they're all ordered pairs of integers).

Yes, I know that. I was talking about reals, not rationals.

You're missing the "treating" and looking at it like it is an "is" - you're parsing wrong.

As my math teacher would respond, he would give you a mark that you may freely "treat" as an A+. Except that, you know, it actually "is" an F.

Jack_Simth

2008-06-27, 09:26 PM

Perhaps if you could complete the sentence first.

Ran out of time, didn't finish editing. What other number, subtracted from itself, doesn't equal 0?

Yes. And if you treat prime numbers as "numbers divisible by one and by themselves" (which is the elementary school definition) then some headaches will also go away, but the point is that you would be wrong. If you'd restrict fractals to real numbers, ignoring the complex parts or the concept of partial dimensions, then you'd also save yourself a lot of headaches, but again you'd be wrong. Yes, there exist figures that have two point five dimensions, rather than two or three.

But at the same time, in each of those examples, you gain something by keeping the thing your example loses. The imaginary portions of complex numbers give you shortcuts that greatly simplify wave calculations (and several other things), for instance. Here's a question:

What do you gain by "infinity + 1 = the same infinity", that you don't keep with "infinity + 1 = a negligibly different infinity"?

Or, to put it another way - I understand the convention, but have yet to see a reason why it's a good idea to have the convention as it is. Are you aware of one?

An extra rule isn't necessary as this follows from the existing axioms. Basic manipulation does not apply in boundary conditions. X / X = 1 except where X = 0, and so forth. Velocities can be added with basic manipulation, except where you're approaching light speed.

Actually, unless they're 0, the relativistic sums are still the ones that apply - it's just that in the normal cases the difference between the Newtonian velocity sum and the relativistic velocity sum is below most measurement threshold for the super-majority of instances a person will have reason to measure. A man on his horse weighs less than a man and his horse (due to the difference in height for the center of gravity). The difference is there, it's just negligible in the vast majority of cases where anyone cares.

This is actually a perfect example you brought up.

Yes, but then once again you'd be wrong. If you assume that bicycles are the same thing as automobiles, you won't need extra rules for traffic governing them. Doesn't mean it's a good idea though.

Legally speaking, on the road, bicycles are required to follow the exact same set of behavior rules as automobiles (at least in the United States of America). Bicycles are not taxed as much (but then, different models, types, and years of automobiles are often taxed differently), and they don't require licenses or insurance, but on a bicycle, legally speaking, you can get a ticket for things like running a red light, speeding (I've actually met people who have received speeding tickets for going too fast on their bicycles - in 25 mph zones, mind, not on the freeway), or failure to signal. There's actually very good reason for this kind of thing - while an out-of control bicycle is not going to directly cause anywhere near as much damage as an automobile, the automobiles that can't quite manage to safely avoid the cyclist that isn't behaving in an accepted manner will.

Don't you just hate metaphors?

Math isn't about making it easy for people and preventing their headaches. Math is about logic and fundamentals.

Actually, most bits of math were invented to make assorted things easier for various people and to get rid of their headaches.

But let's say it's about logic and fundamentals. What fundamental use or functionality does "infinity + 1 = the same infinity" get you that "infinity + 1 = a negligibly different infinity" does not? Logically, "infinity + 1 = a negligibly different infinity" lets you break a very large number of proofs for 1=0 (and similar). How does "infinity + 1 = the same infinity" actually help you?

Yes, I know that. I was talking about reals, not rationals.

Yes - and {the set of all rational numbers} union {the set of all irrational numbers} = {the set of all real numbers}. If some savant eventually figures out a way to properly convert arbitrary irrational numbers to fit some Integer n-space (where n is some fixed integer), the set of real numbers will collapse to natural numbers, too. The irrational numbers are the only members of the set of real numbers we can't store in the set of integers without loss of information.

As my math teacher would respond, he would give you a mark that you may freely "treat" as an A+. Except that, you know, it actually "is" an F.

Your math teacher is also stuck in that he must teach existing convention, regardless of whether or not it's a good convention.

What do you get from "infinity + 1 = the same infinity" that you do not get from "infinity + 1 = a negligibly different infinity"? What do you lose with "infinity + 1 = a negligibly different infinity" that you keep with "infinity + 1 = the same infinity"?

Or, why is the existing convention a good one?

brian c

2008-06-27, 10:51 PM

Ran out of time, didn't finish editing. What other number, subtracted from itself, doesn't equal 0?

A transfinite cardinal, such as aleph-null.

Yes - and {the set of all rational numbers} union {the set of all irrational numbers} = {the set of all real numbers}. If some savant eventually figures out a way to properly convert arbitrary irrational numbers to fit some Integer n-space (where n is some fixed integer), the set of real numbers will collapse to natural numbers, too. The irrational numbers are the only members of the set of real numbers we can't store in the set of integers without loss of information.

Okay, I don't understand exactly what you mean here, but there's a very major mistake. It is not possible to "collapse" the irrationals into the integers (or any other countably infinite set). It has been proven to be impossible (see Cantor's Diagonalization argument (http://en.wikipedia.org/wiki/Cantor%27s_diagonalization)).

There's a difference between something that is assumed true but might still be proven false, and something that has been rigorously shown to be false. The statement "There exists a one-to-one correspondence between the reals and the integers" has been rigorously proven to be false, so there's no reason to even discuss as an example that someone could find such a correspondence.

At any rate, yes, the treatment of infinity is a mathematical convention; so is the use of the = symbol or any other number of things. It is a very well accepted convention, having existed in more or less it's current form since Cantor about 130 years ago (although it took some time for his work to be accepted since back then a lot of people still didn't think infinity was even worth talking about). I don't mean to sound rude, but I assume you are an amateur mathematician; you're certainly well-informed and intelligent, but it does not seem as if you've had any real formal experience in logic, set theory and analysis. Unless you have had such experience at a University-level, then it's rather pointless for you to argue about this topic (though this is an internet forum, we argue about a lot of pointless things). It's a very interesting field of study, but if you talk about it without the proper background then you aren't doing yourself justice.

Arbitrarity

2008-06-27, 10:55 PM

Well, rigorously shown to be false under the standard principles of mathematics. Y'know, like 1 != 0, irrationals are fine, etc.

Jack_Simth

2008-06-27, 11:41 PM

A transfinite cardinal, such as aleph-null.

Gee... in a discussion of infinity, when I ask what other numbers fit a particular set of criteria, you come back with ... a type of infinity by another name. Are you serious?

Okay, I don't understand exactly what you mean here, but there's a very major mistake. It is not possible to "collapse" the irrationals into the integers (or any other countably infinite set). It has been proven to be impossible (see Cantor's Diagonalization argument (http://en.wikipedia.org/wiki/Cantor%27s_diagonalization)).

There's a difference between something that is assumed true but might still be proven false, and something that has been rigorously shown to be false. The statement "There exists a one-to-one correspondence between the reals and the integers" has been rigorously proven to be false, so there's no reason to even discuss as an example that someone could find such a correspondence.

A great many things have been proven impossible throughout history... until some random genius happens to find a counterexample or a way around it. The interior angles of a triangle always add up to 180 degrees; this is provable. Then someone invented non-euclidean geometry, and then there were some triangles that did not meet that proven rule. It may be true now, and there's a reasonable chance that it's actually true completely. It is not necessarily fully true, however.

At any rate, yes, the treatment of infinity is a mathematical convention; so is the use of the = symbol or any other number of things. It is a very well accepted convention, having existed in more or less it's current form since Cantor about 130 years ago (although it took some time for his work to be accepted since back then a lot of people still didn't think infinity was even worth talking about).

And I also note that you do not even attempt to answer a question I asked, directly, several times:

What do you get from "infinity + 1 = the same infinity" that you do not get from "infinity + 1 = a negligibly different infinity"? What do you lose with "infinity + 1 = a negligibly different infinity" that you keep with "infinity + 1 = the same infinity"?

Seriously. If infinity +1 = a slightly different infinity, then the things that offend people's sensibilities about Herbert's Hotel go away (the full hotel cannot shift everyone five rooms over to make room for five more people). You keep cardinality (in the same way you keep the order of polynomials while still noting that X^2 + 1 != X^2), the proofs using infinities that show 1=0 go away in a manner that's doesn't induce headaches, and you don't need to add a limit-case rule that infinity-infinity=undefined. Do you have a single example of something you gain by saying infinity + 1 = the same infinity, that one does not have with infinity + 1 = a negligibly different infinity? We already have broad size-differences between types of infinities; why not a finer granularity?

I don't mean to sound rude, but I assume you are an amateur mathematician; you're certainly well-informed and intelligent, but it does not seem as if you've had any real formal experience in logic, set theory and analysis. Unless you have had such experience at a University-level, then it's rather pointless for you to argue about this topic (though this is an internet forum, we argue about a lot of pointless things). It's a very interesting field of study, but if you talk about it without the proper background then you aren't doing yourself justice.

You're making absolutely no arguments about the subject itself in this section, just your opponent. What's the definition of an ad hominem fallacy, again?

Patashu

2008-06-28, 12:17 AM

That is false. Its real world equivalent is not just that "if it walks like a duck and quacks like a duck, it probably is a duck", but that "if it is so amazingly similar to a duck that nobody, even through sub-molecular analysis, will ever be able to spot any reason for which it might be considered not a duck, is a duck".

Have you heard of the Chinese Room theorem?

Yes, it's only arbitrary using Tippy's arbitrary definition of that word. Most other people would call it a "convention", just like "0.(9) = 1" is a convention.

Interestingly, the definition of factorials stops working at that point, i.e. 0! is not equal to 0 * (-1)! because that means that -1! = 1/0. Then again, factorials only make sense for natural numbers anyway, but this reminds me that one old math teacher of mine would go into lengthy monologues about whether zero is or isn't a natural number.

-1! IS 1/0. If you look at the gamma function and compare it to the factorial function that's what you get.

brian c

2008-06-28, 01:16 AM

A great many things have been proven impossible throughout history... until some random genius happens to find a counterexample or a way around it. The interior angles of a triangle always add up to 180 degrees; this is provable. Then someone invented non-euclidean geometry, and then there were some triangles that did not meet that proven rule. It may be true now, and there's a reasonable chance that it's actually true completely. It is not necessarily fully true, however.

I'm not sure you have a very good mathematical definition of "proof". Yes, it was proven millenia ago that the interior angles of a triangle add up to 180 degrees; however that proof is assuming Euclid's parallel postulate, which is itself not a sure thing. If the parallel postulate holds, then the interior angles must add up to 180 degrees. If the parallel postulate does not hold, then the angles do not add up to 180 degrees. In this sense, you're giving an analogy between the axioms of Euclidean geometry and the axioms underlying Cantor's theory of the transfinite numbers. I challenge you to find an unsound axiom in this case.

And I also note that you do not even attempt to answer a question I asked, directly, several times:

What do you get from "infinity + 1 = the same infinity" that you do not get from "infinity + 1 = a negligibly different infinity"? What do you lose with "infinity + 1 = a negligibly different infinity" that you keep with "infinity + 1 = the same infinity"?

Seriously. If infinity +1 = a slightly different infinity, then the things that offend people's sensibilities about Herbert's Hotel go away (the full hotel cannot shift everyone five rooms over to make room for five more people). You keep cardinality (in the same way you keep the order of polynomials while still noting that X^2 + 1 != X^2), the proofs using infinities that show 1=0 go away in a manner that's doesn't induce headaches, and you don't need to add a limit-case rule that infinity-infinity=undefined. Do you have a single example of something you gain by saying infinity + 1 = the same infinity, that one does not have with infinity + 1 = a negligibly different infinity? We already have broad size-differences between types of infinities; why not a finer granularity?

The aim of mathematics is not to be sensible, nor to be intuitive. The goal of mathematics is to be correct. While it certainly is nice when mathematics is sensible, many important results (whose correctness is beyond question) remain highly unintuitive.

What do we gain by saying that Infinity + 1 = Infinity?

Example: Suppose I ask you to count all of the natural numbers, starting with one. How long will it take you to finish?

You never will. It's possible in theory, since you can count 1, 2, 3, 4, ... and go on for as long as you can, each number one more than the last. However, there will always be more numbers afterwards; roughly speaking this means that there are infinitely many natural numbers.

Now let's suppose I ask you to count starting from 0. How long will it take?

You still never finish counting. There are infinitely many numbers in this set as well.

Question: Why did I tell you to start from 0? What was the rule to get this set from the previous one? It might be the same set, but with 0 added in. In that sense, the size of this set should be "infinity +1". On the other hand, maybe we just decreased every natural number by one. Then the number of elements should be the same, infinity.

Infinity + 1 = Infinity

You're making absolutely no arguments about the subject itself in this section, just your opponent. What's the definition of an ad hominem fallacy, again?

Perhaps I was not clear. What I meant to say was this:

You are claiming that a very well accepted idea in mathematics is fundamentally wrong, or that it could easily be improved. Let us suppose that this is true. Doesn't it seem odd that no established mathematician (of the thousands of philosophers, logicians, professors, doctoral students etc. who may be interested in this area) has come up with this solution that you have? I believe the most reasonable conclusion is, that while I may lack the technical understanding to tell you exactly what's wrong with your arguments, something most definitely is.

Awetugiw

2008-06-28, 01:37 AM

Actually, there is no reason why we can't have a bit of both Infinity + 1 = Infinity and Infinity + 1 != Infinity. We do, actually.

The difference is in whether we add in cardinal numbers or in ordinal numbers, Or, put it another way, Infinity + 1 != Infinity but #Infinity = #(Infinity+1).

"Infinity + 1 is not the same as Infinity, but they do have the same amount of elements."

(Note that unfortunately Infinity + 1 != 1+ Infinity = Infinity, so this does come with some extra trouble built into the system.)

The problem is of course that in most situations we are indeed talking about the cardinal numbers, so Infinity + 1 = Infinity. We can for example actually shift everyone five rooms to have room for five more people. It does actually work that way, whether we like it or not.

Kurald Galain

2008-06-28, 04:52 AM

Ran out of time, didn't finish editing. What other number, subtracted from itself, doesn't equal 0?

Infinity is not a number.

What do you gain by "infinity + 1 = the same infinity", that you don't keep with "infinity + 1 = a negligibly different infinity"?

What you gain is the concept of infinity. By insisting you treat it as a number you forget what infinity is about. I'd suggest you stick to Finitism instead.

Or, to put it another way - I understand the convention, but have yet to see a reason why it's a good idea to have the convention as it is.

It's not an arbitrary convention, it follows straight from the axioms.

If we treat infinity as an ALN, we get into a maze of weird definitions. Let's call this Jack Simth's Infinithing, or JSI. Supposing the amount of natural positive numbers equals JSI, then you posit that the amount of natural numbers including zero equals JSI + 1, and it would arguably follow that the amount of even numbers equals JSI / 2, the amount of whole numbers equals JSI * 2 + 1, and then the amount of rational numbers equals, what, JSI squared? Now what about the amount of prime numbers, how are you going to represent that?

Proof that all these variations of JSI are actually the same thing is easy. Given two countable sets, if a bijection can be written that matches each unique element of one set to precisely one unique element of the other, then those sets have the same size. It is trivial to demonstrate such a bijection for N and Z, or for N to even numbers, or for N to N-without-zero. Therefore all these sets have the same size, and thus JSI = JSI + 1 = JSI / 2. The bijection for Q is easiest if you draw it out. The bijection for primes is less trivial, but since it is proven that aleph-null is the smallest infinite cardinality, any infinite subset of a set with cardinality aleph-null must therefore also have the same cardinality. QED.

Actually, unless they're 0, the relativistic sums are still the ones that apply

You're missing the point (that by using the "simple and headache-saving" way you end up being wrong, even if in mundane speeds you will never notice that).

Legally speaking, on the road, bicycles are required to follow the exact same set of behavior rules as automobiles (at least in the United States of America).

I assume you mean "in whatever state you happen to live in", or did you look up all the variations of traffic laws throughout the US? At any rate, you already point out that "they don't require licenses or insurance", among others, so you're proving my point for me.

Don't you just hate metaphors?Not at all, you're just twisting logic in a failed attempt to discount them.

Actually, most bits of math were invented to make assorted things easier for various people and to get rid of their headaches.

Heh. Prove it. The closest thing I can think of where math was invented to make things easier on people, is the Alabama Pi.

If some savant eventually figures out a way to properly convert arbitrary irrational numbers to fit some Integer n-space (where n is some fixed integer), the set of real numbers will collapse to natural numbers, too.

This was proven to be impossible. The fact that you don't seem to understand why would explain why you don't understand the concept of infinity, and insist on treating it as an ALN instead.

The irrational numbers are the only members of the set of real numbers we can't store in the set of integers without loss of information.

The "only" ones, yes, but there are an infinite amount of them (an amount which is even more infinite than the amount of integers).

Your math teacher is also stuck in that he must teach existing convention, regardless of whether or not it's a good convention.

What is this, Teach The Controversy about math? I'd wager that my math teacher understands math better than you do. Once again, it's not a convention, it's a conclusion. If you don't understand what is to be gained by logical deduction, well, I hope you never end up in management.

Seriously, how do you think math works? That some Cantor-like figure sits at his inventing desk and thinks "hey, let's make a new law about calculating fractional numbers! And let's then vote on it in the Secret Math Teachers Cabal to make it an official convention so that we can sell new calculating devices to all stores"? Math Doesn't Work That Way. Good night.

Indon

2008-06-28, 01:20 PM

If, instead, you treat Infinity as a number that is arbitrarily larger than any known number, but is itself an unknown number or a symbolic number, that issue goes away.

Alternately, you just point out that most functions, such as addition, subtraction, and such, are undefined in regards to infinity (at least, in any useful math system).

Let's say a few things about infinity.

Let's not and say we did. Also, have you heard of the Banach-Tarski (http://en.wikipedia.org/wiki/Banach-Tarski) Paradox? Your attempt to treat infinity as an actual concept vaguely reminded me of it.

Ran out of time, didn't finish editing. What other number, subtracted from itself, doesn't equal 0?

0/0 is undefined, you know - because division by 0 is undefined.

Similarly, infinity-infinity is undefined, because subtracting from infinity is undefined (as is subtracting infinity, adding infinity, adding to infinity, multiplying infinity, dividing infinity, etc).

Our treatment of infinity is no more convention than our use of any function is - when a function returns useful results, we say that's part of the function. When a function does not, we say the answer is undefined because it's outside of the function.

It's not convention to say, for example, there's no point to trying to put a complex number on the real number line - it's just pointless.

Edit: Also, by bringing division by zero up in a discussion on the internet, I have math-Godwined this thread. It will die (or need to be locked) soon. You're welcome.

mroozee

2008-06-28, 03:58 PM

Sorry about this longish post, but sometimes I get carried away. This hopefully will be convincing and conclude the "Infinity + 1" and "Infinity - Infinity" debates, but if anyone has any questions or comments about anything in here, I'm always happy to talk maths.

VERY BASIC PRELIMINARIES

We won't go into detail about defining "sets" but for simplicity sake, think of a set as a box that contains "elements". For our examples, these "elements" will only be Natural numbers.

The analogy for "normal" addition in the cardinality of sets is the cardinality for the union of two (or more) distinct sets. Note: if you want to talk about some other "definition" of addition in the context of sets, that's fine, but you have to go through the work to define it so that everyone is speaking the same language.

Example (addition):

The Cardinality of {1,3,5,7} = 4 because there are 4 elements in this set.

The Cardinality of {2,4,6} = 3 because there are 3 elements in this set.

These two sets are distinct because there is no element present in both.

So 4 + 3 = |{1,3,5,7} U {2,4,6}| = |{1,2,3,4,5,6,7}| = 7

If the sets are not distinct, we can still define "addition" via the union, but it doesn't analogize as nicely to our "normal" arithmetic.

Similarly, "normal" subtraction in the cardinality of sets analogizes to the cardinality of one set after removing the elements of a subset.

Example (subtraction):

The Cardinality of {1,2,3} = 3. The Cardinality of {1,3} = 2. The second set is a subset of the first because every element in the second set appears in the first set.

So 3 - 2 = |{1,2,3} ~ {1,3}| = |{2}| = 1

If the second set is not a subset of the first, we can still define "subtraction" via the difference, but it doesn't analogize as nicely to our "normal" arithmetic.

We could go on and define all of the other stuff that we like: negative numbers, multiplication, exponentiation... but we won't right now.

INFINITY

Limiting ourselves to ONLY dealing with sets of Natural numbers and a single "addition" or "subtraction", the only infinity we care about is Aleph-Null. This can be loosely thought of as sets with: "countless", "endless", "unlimited" or "bigger than any 'normal' number" distinct elements.

There are many examples of infinite sets including:

N = The set of Natural numbers {1,2,3,...}

E = The set of even Naturals {2,4,6,...}

X = The set of even Naturals along with 1 {1,2,4,6,...}

O = The set of odd Naturals {1,3,5,...}

P = The set of prime number Naturals {2,3,5,...}

T = The set of Naturals beginning with a 2 {2,20,21,...}

etc.

NOW LET'S LOOK AT OUR QUESTIONS

a) What is Infinity + 1?

b) What is Infinity - Infinity?

a) So what is Infinity + 1?

Using our definitions...

We take a set whose cardinality is Infinity and a distinct set containing one element, take their union and then look at their cardinality. So which infinite set do we use?

If we choose N, we can't find a distinct element to get our addition right.

If we choose any of the others we can.

E + {1} = X, above and suggests that Infinity + 1 = Infinity

X + {3} = {1,2,3,4,6,...} which has cardinality Infinity

O + {2} = {1,2,3,5,7,...} --> Infinity

P + {4} = {2,3,4,5,7,11,...} --> Infinity

T + {7} = {2,7,20,21,22,...} --> Infinity

** In fact, for every case where it is meaningful to talk about Infinity + 1, the result is Infinity.

b) What about Infinity - Infinity?

Using our definitions, we take a set whose cardinality is infinity and a subset whose cardinality is also infinity, remove the latter from the former and check the cardinality.

So which sets do we use? N ~ E? Both sets have cardinality infinity and the even Naturals are a subset of the Naturals, so we can do our subtraction.

N ~ E = O, which has cardinality Infinity and we see that Infinity - Infinity = Infinity.

But what if we had picked a different pair of sets? Say, X ~ E?

{1,2,4,6,...} ~ {2,4,6,...} = {1} which has cardinality 1 and we conclude that Infinity - Infinity = 1.

If we had chosen P ~ P we would get the empty set {} with cardinality 0!

** Now we have a problem. If we just say, "Infinity - Infinity" we can get a wide range of answers... 0, 1, infinity... and actually anything in between if we work at it. So the question "What is Infinity - Infinity?" is not well defined and we have to answer "Undefined." If you specified WHICH infinite sets you were talking about, we could give a more definitive answer... but as is, we are stuck.

RebelRogue

2008-06-28, 04:15 PM

A great many things have been proven impossible throughout history... until some random genius happens to find a counterexample or a way around it. The interior angles of a triangle always add up to 180 degrees; this is provable. Then someone invented non-euclidean geometry, and then there were some triangles that did not meet that proven rule. It may be true now, and there's a reasonable chance that it's actually true completely. It is not necessarily fully true, however.

It is proven to be true within the given set of axioms! When triangles with non-Pi angle sums were found in spherical geometry it was because one of the axioms was no longer true. So, yes, you may find exception to a given proven rule if the set of axioms are changed! Sometimes this new set of axioms will be interesting - sometimes, not so much.

Muffin_Man

2008-06-28, 06:06 PM

Well, I can't answer for the underlying philosophy (that is, your first paragraph), but higher level math basically doesn't have any basis in the observable world, or, at least, is meant to generalize to things that aren't, for example n-dimensional space (or even infinite dimensional space.) Now it turns out that some of the weird things, like using functions as vector spaces, actually has use in quantum mechanics, which is part of what makes QM so weird. However, there are other things that don't have any analog in the real world than we imagine. Unfortunately these tend to be hard to explain, and some of these may even have application in unknown sciences. The thing is, we really don't know how closely math and science are connected, since we don't know enough about the universe to say. However, I will say that, for the most part, math and science evolved separately, and in fact it came as a surprise to many mathematicians that their field had any application beyond arithmetic and basic algebra.

If you're interested in Cience Philosophy, try reading Koyré, he has some great books about this.

Basically, is all Galileu's and Descartes fault.

Jack_Simth

2008-06-28, 06:42 PM

Actually, there is no reason why we can't have a bit of both Infinity + 1 = Infinity and Infinity + 1 != Infinity. We do, actually.

The difference is in whether we add in cardinal numbers or in ordinal numbers, Or, put it another way, Infinity + 1 != Infinity but #Infinity = #(Infinity+1).

"Infinity + 1 is not the same as Infinity, but they do have the same amount of elements."

Ah, so it's already out there, it's just almost nobody specifies what they're using. Good enough.

averagejoe

2008-06-28, 06:52 PM

If you're interested in Cience Philosophy, try reading Koyré, he has some great books about this.

Basically, is all Galileu's and Descartes fault.

I appreciate the offer, but I'm not. Philosophy seems to be, for the most part, a dead end.

brian c

2008-06-28, 07:09 PM

Let's not and say we did. Also, have you heard of the Banach-Tarski (http://en.wikipedia.org/wiki/Banach-Tarski) Paradox? Your attempt to treat infinity as an actual concept vaguely reminded me of it.

I'm familiar with the Banach-Tarski result (not in-depth) and I would count that as an example of how infinity - infinity = infinity, roughly speaking. Not sure if what you're saying about me should be taken positively or negatively, although I was intending to show that talking about infinity as a number is a practically worthless endeavor.

Indon

2008-06-28, 09:45 PM

I'm familiar with the Banach-Tarski result (not in-depth) and I would count that as an example of how infinity - infinity = infinity, roughly speaking. Not sure if what you're saying about me should be taken positively or negatively, although I was intending to show that talking about infinity as a number is a practically worthless endeavor.

I was drawing a parallel between your working with set theory, neither positive nor negative in that respect, just a little brain quirk on my part. As I'm sure you could tell from the rest of my post, I agree, I just think it's much more practical to just describe the phenomenon in terms of simple function domains (thus my, "how about we not and say we did?" comment :P).

Jack_Simth

2008-06-29, 01:21 AM

I'm not sure you have a very good mathematical definition of "proof". Yes, it was proven millenia ago that the interior angles of a triangle add up to 180 degrees; however that proof is assuming Euclid's parallel postulate, which is itself not a sure thing. If the parallel postulate holds, then the interior angles must add up to 180 degrees. If the parallel postulate does not hold, then the angles do not add up to 180 degrees. In this sense, you're giving an analogy between the axioms of Euclidean geometry and the axioms underlying Cantor's theory of the transfinite numbers. I challenge you to find an unsound axiom in this case.

Maybe I just haven't gotten enough sleep....

There might be an unstated one: all individual natural numbers are finite.

The upper bound to the value of a natural number is itself infinity - and it follows then that the upper bound of the number of digits in a natural number is also infinity.

Suppose we create a natural number with an infinite number of digits (I'll call it an Irrational Natural Number, or INN, for now). If we can do this, then we have a method by which to map an irrational number into Integer-2 space - it's an INN on the right of the decimal point (possibly reversing the digits for notational purposes, depending on how we construct or define our INN), and a regular integer to the left.

If numbers with an infinite number of digits and no decimal point are in the set of natural numbers, then we can collapse the set of Real numbers to an Integer-2 space, which can be collapsed to an integer-1 space, which can be collapsed to the set of natural numbers.

In which case, the set of real numbers are in aleph-null.

...

Definitely haven't gotten enough sleep.

Kurald Galain

2008-06-29, 06:52 AM

There might be an unstated one: all individual natural numbers are finite.

If numbers with an infinite number of digits and no decimal point are in the set of natural numbers,

That's a contradiction.

kjones

2008-06-29, 08:47 AM

The proof that there exists no bijection between the natural numbers and the reals (and thus that they cannot have the same cardinal number) is somewhat tricky - it helps to understand Cantor's Diagonalization Theorem, as linked previously. Basically, you can prove that you can map each natural number to its reciprocal, and you're still left over with a bunch of numbers.

I appreciate the value of challenging conventional wisdom, but at some point you have to realize that if you're running around claiming that 1 = 0, you need to take a step back and re-evaluate.

brian c

2008-06-29, 09:41 AM

In which case, the set of real numbers are in aleph-null.

...

Definitely haven't gotten enough sleep.

No, no, you probably haven't. Why don't you take the next few plays off champ. </Anchorman>

Jack_Simth

2008-06-29, 11:01 AM

That's a contradiction.

Do you have a definition of natural numbers that specifies it cannot have an infinite number of digits? We have equations that go to infinity (e.g., sumn=0 to infinity (Xn)*(10n) where Xn is a decimal digit); do you have any concrete reason why there cannot exist a natural number with an infinite number of digits?

Sure, you can't write them in normal decimal notation (but then, you can't write out any irrational in normal decimal notation, either); how's that a problem?

You can compare, add, subtract, multiply, and divide these numbers (all standard operations still technically apply - as you can define all the standard operations to operate recursively on digits and place-notation, which these have, just in infinite amounts).

The proof that there exists no bijection between the natural numbers and the reals (and thus that they cannot have the same cardinal number) is somewhat tricky - it helps to understand Cantor's Diagonalization Theorem, as linked previously. Basically, you can prove that you can map each natural number to its reciprocal, and you're still left over with a bunch of numbers.

Yes, it can also be proven that you can fit the set of integers into the set of integers with room to spare (multiply by any integer). It can also be proven that you can fit any Integer-n space into an integer-1 space.

I appreciate the value of challenging conventional wisdom, but at some point you have to realize that if you're running around claiming that 1 = 0, you need to take a step back and re-evaluate.

The 1=0 bit is just to point out that there's an issue with "infinity+1 = the same infinity". "Infinity+1 = the same infinity" requires that either a couple of operations not be defined (subtraction, specifically in the case of the 1=0 "proof", but there's a handful of others that result in similar issues) when used with the symbol, or it requires that 0=1. Fundamentally, those are the only two options for "infinity+1 = the same infinity." Now, if "infinity+1 = a negligibly different infinity" then you can potentially find a method by which define subtraction of infinities. I am not saying that 0=1, and I have not (at least, not in this thread); that's a method by which to point out a possible flaw with "infinity+1 = the same infinity."

Kurald Galain

2008-06-29, 11:37 AM

Do you have a definition of natural numbers that specifies it cannot have an infinite number of digits?

Actually, yes. It follows from the standard iterative definition of natural numbers (start with 0 or 1, then generate each number as the successor to the previous; each number has a finite amount of digits, each successive number has the same amount of digits or one more, both of which are also finite).

Sure, you can't write them in normal decimal notation (but then, you can't write out any irrational in normal decimal notation, either); how's that a problem?

Irrationals aren't natural.

kjones

2008-06-29, 01:09 PM

Yes, it can also be proven that you can fit the set of integers into the set of integers with room to spare (multiply by any integer). It can also be proven that you can fit any Integer-n space into an integer-1 space.

This may be, but your proof that "the set of all real numbers is in aleph-null" thus contradicts something that's very well understood and known to be true. Which do you think is more likely - that much of mathematics as we know it is incorrect, or that there's some flaw in your proof?

I'm not saying that you're literally trying to prove that 1 = 0, just that your argument is backing you into similarly logically inconsistent corners.

Your claim that "infinity + 1 = a slightly different infinity" is probably a response to the logical inconsistencies presented by the infinite hotel (http://en.wikipedia.org/wiki/Infinite_hotel) thought-experiment. But infinity really does work that way, and you can't treat it like an arbitrarily large number - it's not.

(One reason this discussion is difficult is that I don't know how much mathematical background you have, and I don't want to assume that you know things you don't. I've taken a freshman-year discrete math class, which covered a lot of the things discussed here, so take of that what you will.)

brian c

2008-06-29, 04:37 PM

(One reason this discussion is difficult is that I don't know how much mathematical background you have, and I don't want to assume that you know things you don't. I've taken a freshman-year discrete math class, which covered a lot of the things discussed here, so take of that what you will.)

I was thinking the same thing, and I didn't explain it very well. This shouldn't be a sort of competition to see who's more educated, and that very often doesn't correlate to being correct, but it can be hard to explain things when you don't know what your audience's background is. For the record, I'm going into my senior year as a math major; I've taken courses in discrete math and analysis (plus others that aren't germane to this particular topic)

Awetugiw

2008-06-30, 06:15 AM

Do you have a definition of natural numbers that specifies it cannot have an infinite number of digits? We have equations that go to infinity (e.g., sumn=0 to infinity (Xn)*(10n) where Xn is a decimal digit); do you have any concrete reason why there cannot exist a natural number with an infinite number of digits?

Well, in order for an infinite sum to be defined, it has to converge. If we represent numbers as sumn=-infinity to N Xn10n with each Xn between 0 and 9 (and N finite) we don't run into trouble. There is only a finite number of terms with positive n, and the terms with negative n decrease fast enough. Such a sum converges, and defines a real number.

sumn=-infinity to infinity Xn10n with each Xn between 0 and 9 will not converge, unless Xn=0 for all n>N for some finite N, and we might as well take the previous sum.

We could perhaps change the definition of distance in such a way that infinite sums can converge (for example, we could use 5-adic numbers. In this case a sum from -N to infinity would converge), but this really changes the way numbers work, making this give more trouble than it solves. (For example, in 5-adic numbers the sequence 5^n converges to 0. So if you take 5, 25, 125, ... it gets closer and closer to 0.)

If we only look at the natural numbers, we're in even more trouble. There is no way to define a distance on the natural numbers, so we cannot even define convergence. Unless one defines infinite sums in a completely new way, this means numbers with an infinite amount of digits don't work at all.

brian c

2008-06-30, 08:43 AM

Well, in order for an infinite sum to be defined, it has to converge. If we represent numbers as sumn=-infinity to N Xn10n with each Xn between 0 and 9 (and N finite) we don't run into trouble. There is only a finite number of terms with positive n, and the terms with negative n decrease fast enough. Such a sum converges, and defines a real number.

sumn=-infinity to infinity Xn10n with each Xn between 0 and 9 will not converge, unless Xn=0 for all n>N for some finite N, and we might as well take the previous sum.

We could perhaps change the definition of distance in such a way that infinite sums can converge (for example, we could use 5-adic numbers. In this case a sum from -N to infinity would converge), but this really changes the way numbers work, making this give more trouble than it solves. (For example, in 5-adic numbers the sequence 5^n converges to 0. So if you take 5, 25, 125, ... it gets closer and closer to 0.)

If we only look at the natural numbers, we're in even more trouble. There is no way to define a distance on the natural numbers, so we cannot even define convergence. Unless one defines infinite sums in a completely new way, this means numbers with an infinite amount of digits don't work at all.

Talking about p-adic numbers without defining them (or even with) is just going to confuse people who aren't familiar.

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