1 = .999...

[Crumb]

:wave:

So since the previous math controversy seems to have faded away, maybe I can drum up another one. This was posted over at IIDB once and became a big long argument that consisted of one side arguing that it was "a good approximation" and the other side saying "No, you don't get it: They are EXACTLY EQUAL!" So who here doesn't buy the simple fact that 1 is exactly equal to 0.999... an infinite sequence of nines?

1 = 0.999...

[beyelzu]

I think that they arent equal.

but I am not a math guy, so am open to proof.

[beyelzu]

however it just occurred to me that

2/3 is .6 recurring

and

1/3 is .3 recurring

and 1/3 + 2/3 is 1
and .3 recurring and .6 recurring is .9 recurring so maybe .9999 does equal 1

[Crumb]

Yes, that is one of the most convincing proofs.

Also:

.1rec = 1/9
.1rec * 9 = 1/9 * 9
.9rec = 1

[Gurdur]

You need to invite 99Percent from SecWeb for this subject.


Oooooooooo, what fun.

[Crumb]

Heh.

[JoeP]

What, no poll?

I agree that 1 = 0.999..

[Crumb]

What, no poll?
Damn! :kickscan:

[John Carter]

I see that two of the more basic proofs have been offered already. Some more arguments that .999... = 1:

x = .999...
10x = 9.999... (multiply both sides by 10)
10x - x = 9.999... - .999... (subtract the two equations)
9x = 9
x = 1

Given any two distinct real numbers a and b with a < b, there exists c such that a < c < b. In the case of a = .999... and b = 1, obviously there is no such c that is between them, and thus .999... and 1 are not distinct; that is, .999... = 1.

The most rigorous proof is this one (my version will not be as rigorous as possible, since I will not actually prove that the limit of a particular sequence has a specific value; if you've studied calculus the value of the limit is painfully obvious, and if you haven't, the argument is meaningless anyway):

When we write .999... what we really mean is the geometric series .9 + .09 + .009 + .0009 + ... or more precisely the sum as n goes from 1 to inifinity of 9(1/10)n. The sum of an infinite series is the limit of the sequence of partial sums. Thus, .999... is equal to the limit of the sequence .9, .99, .999,.... Obviously, the limit of this sequence is 1, so .999... = 1.

[ApostateAbe]

In my first calculus class, I was shown the evidence with the 2/3 + 1/3 proof, and it was amazing at the time. Then it just became like common sense later on as I learned more about limits.

[Shake]

Yeah, when I initially saw this, I was on the "no, .999... is just infinitely close to 1" camp, but then I saw some of the other proofs and was convinced. So long as you include the "..." then they are equal.

[Gurdur]

I oppose the arguments and proofs with :...... 99Percent !
I win. :P

Someone should really invite him over from SecWeb, just for this.

[John Carter]

Both .999... and 1 are symbols that represent some number; this is the distinction between a numeral, and a number. It just so happens that in this case the two symbols represent the same number. It is a mistake to get so wrapped up in the symbols that you forget that they are symbols, and not the actual number itself. Indeed, there are infinitely many ways to write any number.

[justaman]

I'd like to see this 99% guy argue with that lost post there John. :1thumbup:

[Gurdur]

I'd like to see this 99% guy argue with that lost post there John. :1thumbup:
He has no problems arguing with lost posts (or last posts) at all. Vociferously. On SecWeb, IIRC, he rather ignored or disregarded all the proofs.
Thus my small joke, which seems to have been misunderstood.

[Ari]

If .999... was infinitely close to 1 but not 1 what would,
1 - .999... = ?

[John Carter]

If .999... was infinitely close to 1 but not 1 what would,
1 - .999... = ?


Well, that's an interesting question, Ari. This relates to the second argument I gave in my first post on this thread. The answer is the result would have to be an infintesimal number*. But since the reals form an Archimedean field**, there are no non-zero infintesimal numbers. Thus for the reals, anyway, saying that .999... is infinitely close to 1 is the same as saying it is equal to 1.



* An infinitesimal is a number x such that for any natural number n, n*|x| < 1. Thus if y is a non-zero infintesimal, then 0 < |y| < |x| for all real numbers x.

** An Archimedean field is a field that obeys the Archimedean property: an ordered algebraic structure that has no non-zero infinitesimals. It easy to show that the reals are Archimedean using the least upper bound property of the reals (the fact that every set of reals that is bounded above has a least upper bound):

Let S be the set of all real infinitesimals. Since every positive real is an upper bound for S, S is bounded above, and thus has a least upper bound, c. Either c is infinitesimal or it is not. If it is, then so is 2c, which contradicts the fact that c is an upper bound for S, unless c is 0, in which case 2c = c. If c is not an infinitesimal, then c/2 is also an upper bound for S, which contradicts our assumption that c is the least upper bound for S. Thus S has only one possible element, namely 0.


ETA: You will often see the Archimedean property stated as : For every x in F, there exists a natural number N such that x < N. The two statements are equivalent, since if there was an x for which there is no N such that x < N, then the reciprocal of x would be an infinitesimal.

[Godwhacker]

I'd take a different angle and state that, since matter consists of waveforms, and thus everything has some variance, that, in actual real existence, nothing stays at 1 (whatever is being measured) for very long, and thus, 1=.999, and probably .998 and maybe even .997 at some times.

[Clutch Munny]

Gurdur's joke presupposes some background knowledge: The problem with 99Percent is not simply a mathematical one. 99Percent is possibly the dumbest "posting agent" (i.e., not sure whether s/h/it is a person or virtual Rand spambot) in the universe; readers fascinated with train wrecks and the like should take the opportunity to check out IIDB and see the action first-hand.

I must have laughed for ten minutes straight a few days back, when 99% interrupted her/his/its inane libertarian solecizing just long enough to pop into a thread on police tasering and say, "what more training do citizens need than to know that the police is the authority and should be obeyed?" It's a bad state of affairs when a posting agent can't even keep straight what brand of stupidity to parade -- libertarian or authoritarian.

Of course 99Percent was a long-time IIDB moderator...

[Abdul Alhazred]

If .999... was infinitely close to 1 but not 1 what would,
1 - .999... = ?

Flame war time. :airbox: :guns: :boxers: :fuming:

Cantorian versus Non-Cantorian set theory.

Is the concept of 'infinitesimal' mathematically valid?

I will take the Orthodox Cantorian position that the answer is no. There is only 'arbitrarily small' and 'limits'.

But if the foul heretics were right, the answer would be (font does not support), which is an infinitesimal.

[John Carter]

If .999... was infinitely close to 1 but not 1 what would,
1 - .999... = ?

Flame war time. :airbox: :guns: :boxers: :fuming:

Cantorian versus Non-Cantorian set theory.

Is the concept of 'infinitesimal' mathematically valid?

I will take the Orthodox Cantorian position that the answer is no. There is only 'arbitrarily small' and 'limits'.

But if the foul heretics were right, the answer would be (font does not support), which is an infinitesimal.


Ummm... what is "non-Cantorian" about non-standard analysis? Both the hyperreals and the surreals contain infinitesimals.

I think the divide you are looking for is not with Cantor's Set Theory. His set theory, btw, has several serious flaws, nobody accepts his formulation of set theory today, nor has it been for a century or so; see Russell's Paradox for just one example. It's also called Naive Set Theory for a reason!

The divide over the existence of infinitesimals is with Archimedes, and the Archimedean Principle. Note that there are plenty of examples of fields that are non-Archimedean, i.e., they contain infinitesimals. See, for example, the hyperreals (I'd also give the surreals as another example, but technically they do not form a field, sicne the are too big, in the Cantorian sense, to form a set. If we extend the notion of a field to proper classes, and call it a Field then the surreals form a non-Archimedean Field). Also, consider that is there were no such fields, it would be unnecessary (and therefore inelegant) to distinguish betwee Archimedea fields and non_Archimedean fields. But the reals are Archimedean, as I showed in my last post, so any subfield of the reals is also Archimedean.

[Abdul Alhazred]

Hmmm. It's been a while. I should have looked it up first insead of after.

[scigirl]

That is too weird. Now my brain is fuzzy from thinking about it.

[Crumb]

Welcome to :ff: scigirl! :wave: It's good to have you here. :D

[godfry n. glad]

however it just occurred to me that

2/3 is .6 recurring

and

1/3 is .3 recurring

and 1/3 + 2/3 is 1
and .3 recurring and .6 recurring is .9 recurring so maybe .9999 does equal 1

This "proof" fails because it includes an incorrect assumption.

"2/3 is .6 recurring" is incorrect. 2/3 is .666666.....6667, not ".6 recurring".

So, if .9999... is 1, is .99999....9998? Is that one as well? When does it stop being 1 and start being some other number? It seems to me if we carried this to a "logical conclusion", we should acquiesce to all numbers being the same as 1.

[Crumb]

Um... 2/3 is .666... No 7 unless you stop and approximate it.

[JoeP]

Yes, godfry, what are you talking about?

[godfry n. glad]

Okay. So how is .666666.... different than .7?

2/3 = .666666.....67, or .6667, or .7.

Same arguments apply as for .9999.... equalling 1.

Please answer the question I posed.

[John Carter]

however it just occurred to me that

2/3 is .6 recurring

and

1/3 is .3 recurring

and 1/3 + 2/3 is 1
and .3 recurring and .6 recurring is .9 recurring so maybe .9999 does equal 1

This "proof" fails because it includes an incorrect assumption.

"2/3 is .6 recurring" is incorrect. 2/3 is .666666.....6667, not ".6 recurring".


Not at all. .666...6667 is an approximation of .666...; it's called rounding. The correct and exact decimal representation of the rational number 2/3 is .666.... or as bey called it, .6 recurring.


So, if .9999... is 1, is .99999....9998? Is that one as well? When does it stop being 1 and start being some other number?


Of course it isn't 1. It stops being 1 as soon as you use some digit other than 9 in the expansion. Also, .99999...9998 < .99999...99989 < 1, so clearly they are all separate numbers.


It seems to me if we carried this to a "logical conclusion", we should acquiesce to all numbers being the same as 1.

I think you are confusing a finite approximation of a number with the number itself. It is the infinite number of recurring 9's that make .999... exactly equal to one; see my argument based on infinite series above. As soon as you stop, or round it off, you are no longer talking about exactly the same quantity.

[godfry n. glad]

however it just occurred to me that

2/3 is .6 recurring

and

1/3 is .3 recurring

and 1/3 + 2/3 is 1
and .3 recurring and .6 recurring is .9 recurring so maybe .9999 does equal 1

This "proof" fails because it includes an incorrect assumption.

"2/3 is .6 recurring" is incorrect. 2/3 is .666666.....6667, not ".6 recurring".


Not at all. .666...6667 is an approximation of .666...; it's called rounding. The correct and exact decimal representation of the rational number 2/3 is .666.... or as bey called it, .6 recurring.


So, if .9999... is 1, is .99999....9998? Is that one as well? When does it stop being 1 and start being some other number?


Of course it isn't 1. It stops being 1 as soon as you use some digit other than 9 in the expansion. Also, .99999...9998 < .99999...99989 < 1, so clearly they are all separate numbers.


It seems to me if we carried this to a "logical conclusion", we should acquiesce to all numbers being the same as 1.

I think you are confusing a finite approximation of a number with the number itself. It is the infinite number of recurring 9's that make .999... exactly equal to one; see my argument based on infinite series above. As soon as you stop, or round it off, you are no longer talking about exactly the same quantity.

Hey... You forgot something:

"It stops being 1 as soon as you use some digit other than 9 in the expansion. Also, .99999...9998 < .99999...99989 < .99999999.... < 1, so clearly they are all separate numbers."

Including .99999....

You are clearly confusing the finite approximation of .99999.... as 1.

[John Carter]

Okay. So how is .666666.... different than .7?

2/3 = .666666.....67, or .6667, or .7.

Same arguments apply as for .9999.... equalling 1.

Please answer the question I posed.

7/10 = .7, which, if you aew correct, would imply that 2/3 = 7/10. But two rational numbers a/b and c/d are equal if and only if a*d = b*c. This would mean that 20 = 21, so 2/3 is not equal to .7.

Another way to look at it is by long division; you should have learned how to do that in elementary school. Try it. You never get a 7 when doing the dvivsion, you keep getting nothing but 6's. Stopiing on some finite number of steps and calling the last digit 7 is nothing more than an approximation.

[John Carter]

however it just occurred to me that

2/3 is .6 recurring

and

1/3 is .3 recurring

and 1/3 + 2/3 is 1
and .3 recurring and .6 recurring is .9 recurring so maybe .9999 does equal 1

This "proof" fails because it includes an incorrect assumption.

"2/3 is .6 recurring" is incorrect. 2/3 is .666666.....6667, not ".6 recurring".


Not at all. .666...6667 is an approximation of .666...; it's called rounding. The correct and exact decimal representation of the rational number 2/3 is .666.... or as bey called it, .6 recurring.


So, if .9999... is 1, is .99999....9998? Is that one as well? When does it stop being 1 and start being some other number?


Of course it isn't 1. It stops being 1 as soon as you use some digit other than 9 in the expansion. Also, .99999...9998 < .99999...99989 < 1, so clearly they are all separate numbers.


It seems to me if we carried this to a "logical conclusion", we should acquiesce to all numbers being the same as 1.

I think you are confusing a finite approximation of a number with the number itself. It is the infinite number of recurring 9's that make .999... exactly equal to one; see my argument based on infinite series above. As soon as you stop, or round it off, you are no longer talking about exactly the same quantity.

Hey... You forgot something:

"It stops being 1 as soon as you use some digit other than 9 in the expansion. Also, .99999...9998 < .99999...99989 < .99999999.... < 1, so clearly they are all separate numbers."

Including .99999....

You are clearly confusing the finite approximation of .99999.... as 1.


What is finite about .999...? That's what the ellepsis means, godfry. That we never stop, that there are an infinite number of 9's.

[Skep]

I'd like to point out that a notation such as 0.999999...99998 is not a proper decimal representation of any real number.

[Ari]

A quick way to show that .667 is rounded and not exactly 2/3 is,
1/3 = .333...
.333... + .333... = .666...
.666... = 2/3
no 7 at the end.

[godfry n. glad]

however it just occurred to me that

2/3 is .6 recurring

and

1/3 is .3 recurring

and 1/3 + 2/3 is 1
and .3 recurring and .6 recurring is .9 recurring so maybe .9999 does equal 1

This "proof" fails because it includes an incorrect assumption.

"2/3 is .6 recurring" is incorrect. 2/3 is .666666.....6667, not ".6 recurring".


Not at all. .666...6667 is an approximation of .666...; it's called rounding. The correct and exact decimal representation of the rational number 2/3 is .666.... or as bey called it, .6 recurring.


So, if .9999... is 1, is .99999....9998? Is that one as well? When does it stop being 1 and start being some other number?


Of course it isn't 1. It stops being 1 as soon as you use some digit other than 9 in the expansion. Also, .99999...9998 < .99999...99989 < 1, so clearly they are all separate numbers.


It seems to me if we carried this to a "logical conclusion", we should acquiesce to all numbers being the same as 1.

I think you are confusing a finite approximation of a number with the number itself. It is the infinite number of recurring 9's that make .999... exactly equal to one; see my argument based on infinite series above. As soon as you stop, or round it off, you are no longer talking about exactly the same quantity.

Hey... You forgot something:

"It stops being 1 as soon as you use some digit other than 9 in the expansion. Also, .99999...9998 < .99999...99989 < .99999999.... < 1, so clearly they are all separate numbers."

Including .99999....

You are clearly confusing the finite approximation of .99999.... as 1.


What is finite about .999...? That's what the ellepsis means, godfry. That we never stop, that there are an infinite number of 9's.

Nothing, including equalling 1. If there are an infinite number of 9s out past the decimal point, it still never reaches one. It gets close, infinitely close, but it's still not 1. One could say that for all intents and purposes, it is 1, and I'd agree, but any such number, including the ones used by JC above are also a close enough approximations, for all intents and purposes.

[John Carter]

Here is a rigorous proof that does not depend on the decimal expansion of any rational number such as 2/3. I already posted this, but maybe you missed it:

When we write .999... this is short hand for the sum as n goes from 1 to infinity of 9*(1/10)n (or .9 + .09 + .009 + ...). This is an infinite geometric series, and the sum of the series is equal to the limit of the sequence of partial sums of the series. Thus .999... = the limit of the sequence {.9, .9 + .09, .9 + .09 + .009, ...} = {.9, .99, .999, .9999, ... }. Now, a sequence {S1, S2, S3, ... Sn, ... } converges to l iff for any e > 0, there exists an integer N such that for all n > N, |Sn - l| < e. Clearly, the sequence {.9, .99, .999, .9999, ... } converges and the value of this limit is 1. Therefore, the infinite sum .999... = 1.

[Crumb]

:bait1:

[John Carter]

Nothing, including equalling 1. If there are an infinite number of 9s out past the decimal point, it still never reaches one. It gets close, infinitely close, but it's still not 1. One could say that for all intents and purposes, it is 1, and I'd agree, but any such number, including the ones used by JC above are also a close enough approximations, for all intents and purposes.

Nope, if you allow something like thihs, letting one nunmber be "infinitely close" to another without being equal, then you have to allow non-zero infinitesimals, but as I proved in another post, the reals form an Archimedean field, and thus there are no non-zero real infinitesimals.

[godfry n. glad]

Here is a rigorous proof that does not depend on the decimal expansion of any rational number such as 2/3. I already posted this, but maybe you missed it:

When we write .999... this is short hand for the sum as n goes from 1 to infinity of 9*(1/10)n (or .9 + .09 + .009 + ...). This is an infinite geometric series, and the sum of the series is equal to the limit of the sequence of partial sums of the series. Thus .999... = the limit of the sequence {.9, .9 + .09, .9 + .09 + .009, ...} = {.9, .99, .999, .9999, ... }. Now, a sequence {S1, S2, S3, ... Sn, ... } converges to l iff for any e > 0, there exists an integer N such that for all n > N, |Sn - l| < e. Clearly, the sequence {.9, .99, .999, .9999, ... } converges and the value of this limit is 1. Therefore, the infinite sum .999... = 1.

Y'know... This all looks like a buncha fuckin' handwaving.

"Clearly" is bullshit. It ain't clear at all. It clearly closes on converging on 1 and the value of the limit is 1 because it's clear that it comes very, very close to 1. It's like saying that because it comes so infinitely close to 1, we might as well say it's as good as 1....ergo, it's one.

That's the same kind of bullshit that theists use to prove the existence of a god. "Clearly, it must be god."

Sounds like something that some egotistical spiritualist astrologer might come up with.

[Crumb]

So godfry, how do you explain this proof?

x = .999...
10x = 9.999... (multiply both sides by 10)
10x - x = 9.999... - .999... (subtract the two equations)
9x = 9
x = 1

[Skep]

:bait1:
Crumb, you are a naughty, naughty troll. :D

[godfry n. glad]

So godfry, how do you explain this proof?

x = .999...
10x = 9.999... (multiply both sides by 10)
10x - x = 9.999... - .999... (subtract the two equations)
9x = 9
x = 1

I don't. I'm not a mathematician.

However, I'd speculate that it could very well be a demonstration of the limitations of simple algebra at expressing anything to do with infinite or infinitesimal numbers or series of numbers, or zero.

Please explain the meaning of x/0, where x is any number, real or imaginary.

[JoeP]

Y'know... This all looks like a buncha fuckin' handwaving.

"Clearly" is bullshit. It ain't clear at all.
It ain't clear to you, godfry. Argument from personal incredulity.

John Carter's proof is rigorous, as he said - precisely the opposite of handwaving. I am flabbergasted that you are coming up with objections to this; I assumed you were quite intelligent.

[Crumb]

However, I'd speculate that it could very well be a demonstration of the limitations of simple algebra at expressing anything to do with infinite or infinitesimal numbers or series of numbers, or zero.
Why do you prefer this explanation over the obvious 1=.999... one?

Please explain the meaning of x/0, where x is any number, real or imaginary.
It is undefined. It has no meaning. What does that have to do with the issue at hand? :chin:

[John Carter]

I take it you never took basic calculus in school, or if you did, you did not understand what a limit is. Do you dispute that the value of an infinite series is equal to the limit of the sequence of partial sums of the series? If not, and you accept, as you said in your last post, that the value of the limit in this case is 1, then you have no logical choice but to accept that .999... = 1.

[John Carter]

However, I'd speculate that it could very well be a demonstration of the limitations of simple algebra at expressing anything to do with infinite or infinitesimal numbers or series of numbers, or zero.
Why do you prefer this explanation over the obvious 1=.999... one?

Please explain the meaning of x/0, where x is any number, real or imaginary.
It is undefined. It has no meaning. What does that have to do with the issue at hand? :chin:

If x is nonzero, then it is undefined and has no meaning. If x = 0, then it is indeterminate, which is subtly different from undefined.

After rereading godfry's posts, I think I agree with crumb.

[John Carter]

Oh, and the "egotistical spiritualist astrologer" (you left out alchemist, btw) never came up with limits. It was Cauchy, about 150 years later, who did that, and finally put calculus and analysis on a rigorous foundation.

[godfry n. glad]

Y'know...

In playing with that proof, I find it interesting that if one chooses to follow a slightly differing tactic, one gets an entirely variant answer.

Try multiplying both sides by 9, instead of 10. There seems to be some inconsistancy in what the value of x is to be, dependent upon what arbitrary integer is used to multiply both sides of the equation.

I get x =.99999.... and x = .88888.... Oops.

which is insensible.

My point with the x/o is that mathematics does not have and cannot explain many of the functions which are possible within mathematics.

[godfry n. glad]

[quote=godfry n. glad]However, I'd speculate that it could very well be a demonstration of the limitations of simple algebra at expressing anything to do with infinite or infinitesimal numbers or series of numbers, or zero.
Why do you prefer this explanation over the obvious 1=.999... one?

Because to declare that 1 = .9999... as "obvious" is analogous to claiming that it's "obvious" that a god exists. It ain't fuckin' obvious.

[Crumb]

Try multiplying both sides by 9, instead of 10. There seems to be some inconsistancy in what the value of x is to be, dependent upon what arbitrary integer is used to multiply both sides of the equation.

I get x =.99999.... and x = .88888.... Oops.

which is insensible.


I don't get that:

x = .999...
9x = 8.999...
8x = 8
x=1

How did you do it?

[Crumb]

Because to declare that 1 = .9999... as "obvious" is analogous to claiming that it's "obvious" that a god exists. It ain't fuckin' obvious.
The dichotomy as you laid it out seemed to be "1=0.999..." or "This algebra is not sufficient to deal with infinitely repeating decimals therefore this 'proof' is nonsense" I was wondering why you were partial to the latter.

[Skep]

Not a rigorous proof but food for thought:

1/9 = 0.111...
2/9 = 0.222...
3/9 = 0.333...
4/9 = 0.444...
5/9 = 0.555...
6/9 = 0.666...
7/9 = 0.777...
8/9 = 0.888...
9/9 = 0.999... = 1

:popcorn:

[JoeP]

How did you do it?
do not feed the troll
godfry is off his rocker here

[Crumb]

It's ok JoeP. It's the point of this thread. :) I don't think godfry's trolling, I just think he is wrong. :wink:

[JoeP]

Then what was your :bait1: smilie about? I am having serious cognitive dissonance here. It just doesn't seem like godfry. He's 0.0000... short of a full load.

[Crumb]

:bait1: = I hooked one.

[godfry n. glad]

Try multiplying both sides by 9, instead of 10. There seems to be some inconsistancy in what the value of x is to be, dependent upon what arbitrary integer is used to multiply both sides of the equation.

I get x =.99999.... and x = .88888.... Oops.

which is insensible.


I don't get that:

x = .999...
9x = 8.999...
8x = 8
x=1

How did you do it?

Well... I fucked up.

[godfry n. glad]

It's ok JoeP. It's the point of this thread. :) I don't think godfry's trolling, I just think he is wrong. :wink:

Nah... Just trying to stimulate some conversation other than all the smug patting each other on the back. However, I'm far from my comfort zone and obviously screwing up.....so I guess you can all go back to your mathematics circle jerk.

I'll just mosey on.

[Crumb]

Oh...don't go. It won't be any fun without you.

[John Carter]

[quote=godfry n. glad]However, I'd speculate that it could very well be a demonstration of the limitations of simple algebra at expressing anything to do with infinite or infinitesimal numbers or series of numbers, or zero.
Why do you prefer this explanation over the obvious 1=.999... one?

Because to declare that 1 = .9999... as "obvious" is analogous to claiming that it's "obvious" that a god exists. It ain't fuckin' obvious.

Ummm... I think you missed a quote tag somewhere. I never said that. Nor did I ever say or imply that 1 = .999... is obvuios. The only thing I said was obvious (and even then I said "clearly", though "obviously' works just as well in the context) was that the limit of the sequence {.9, .99, .999, ...} is 1. By that I mean that even though we can prove that this sequence converges and the limit has this value, it is a rather trivial result and anyone who has studied (and understood) basic 1st year calculus would agree to this without demanding that we show all our work.

[godfry n. glad]

[quote=godfry n. glad]However, I'd speculate that it could very well be a demonstration of the limitations of simple algebra at expressing anything to do with infinite or infinitesimal numbers or series of numbers, or zero.
Why do you prefer this explanation over the obvious 1=.999... one?

Because to declare that 1 = .9999... as "obvious" is analogous to claiming that it's "obvious" that a god exists. It ain't fuckin' obvious.

Ummm... I think you missed a quote tag somewhere. I never said that. Nor did I ever say or imply that 1 = .999... is obvuios. The only thing I said was obvious (and even then I said "clearly", though "obviously' works just as well in the context) was that the limit of the sequence {.9, .99, .999, ...} is 1. By that I mean that even though we can prove that this sequence converges and the limit has this value, it is a rather trivial result and anyone who has studied (and understood) basic 1st year calculus would agree to this without demanding that we show all our work.

Ignoring the misspelling, you contradict yourself.

I registered for calculus thrice, and dropped it thrice. It never made any sense to me. I managed a bachelor's and a master's degree without it and have lived a full and mostly rewarding life without it.

[Crumb]

I am the one who said this godfry, not John Carter.
Why do you prefer this explanation over the obvious 1=.999... one?

[godfry n. glad]

I am the one who said this godfry, not John Carter.
Why do you prefer this explanation over the obvious 1=.999... one?

Please note that the post which John Carter posted, stating that he had not used the phrase, it included him citing himself stating the phrase. I quoted his post in toto.

So be it.

I'm a math doofus.

So... D'ja all have fun kicking me around? Or y'all not done yet?

:fight:

[Crumb]

Please note that the post which John Carter posted, stating that he had not used the phrase, it included him citing himself stating the phrase. I quoted his post in toto.
Yes, and the first thing he wrote in that post was this:
Ummm... I think you missed a quote tag somewhere. I never said that.
So... D'ja all have fun kicking me around? Or y'all not done yet?
I had fun godfry, but I didn't think I was kicking you around. Was I?

[godfry n. glad]

I had fun godfry, but I didn't think I was kicking you around. Was I?

Why, sure... I expected that going into it. I knew I was waaaaaay outta my element (remember, calculus thrice, thrice). But you guys seemed to so miss 99percent, I thought I'd accomodate.

Tangentially, I'm in the midst of a book which is related. Zero: The Biography of a Dangerous Idea, by Charles Siefe (NY, Penguin, 2000). I'd just finished reading the whole Newton/Liebniz and the development of calculus (which, of course, went into the limits idea later) portion. The author describes Newton's abhorance of infinitessimals, and Liebniz' reveling in them and the consequent effect it had on thier variant notations. Is the term "fluxions" still used? It seems to me, from what I remember of three weeks of calculus, that Liebniz' notation must have prevailed.

Curiously, I believe he describes calculus as "not logical", but as a key to the cosmos. I've yet to finish the book. It's interesting to me, but must be pablum for the likes of this crowd.

He has, by the way, this little appendix:

Animal, Vegetable or Minister?

Let a and b each be equal to 1. Since a and b are equal,

b^2=ab (eq. 1)

Since a equals itself, it is obvious that

a^2=a^2 (eq. 2)

Subtract equation 1 from equation 2. This yields

a^2-B^2=a^2-ab (eq.3)

We can factor both sides of the equation; a^2-ab equals a(a-b). Likewise, a^2-b^2 equals (a+b)(a-b). (Nothing fishy is going on here. This statement is perfectly true. Plug in numbers and see for yourself!) Substituting into equation 3, we get,

(a+b)(a-b)=a(a-b) (eq.4)

So far, so good, now divide both sides of the equation by (a-b) and we get

a+b=a (eq.5)

Subtract a from both sides and we get

b=0

But we set b to 1 at the very beginning of this proof, so this means that

1=0 (eq.7)

This is an important result. Going further, we know that Winston Churchill has one head. But one equals zero by equation 7, so that means that Winston has no head. Likewise, Churchill has zero leafy tops, therefore he has one leafy top. Multiplying both sides of equation 7 by 2, we see that

2=0 (eq.8)

Churchill has two legs, therefore he has no legs. Churchill has two arms, therefore he has no arms. Now multiply equation 7 by Winston Churchill's waist size in inches. This means that

(Winston's waist size)=0 (eq.9)

This means that Winston Churchill tapers to a point. Now, what color is Winston Churchill? Take away any beam of light that comes from him and select a photon. Multiply equation 7 by the wavelength, and we see that

(Winston's photon wavelength)=0 (eq.10)

But multiplying equation 7 by 640 nanometers, we see that

640=0 (eq.11)

Combining equations 10 and 11, we see that

(Winston photon's wavelength)=640 nanometers

This means that this photon that comes from Mr. Churchill -- is orange. Therefore Winston Churchill is a bright shade of orange.

To sum up, we have proved, mathematically, that Winston Churchill has no arms and no legs; instead of a head, he has a leafy top; he tapers to a point; and he is bright orange. Clearly Winston Churchill is a carrot. (There is a simpler way to prove this. Adding 1 to both sides of equation 7 give the equation

2=1

Winston Churchill and a carrot are two different things, therefore they are one thing. But that's not nearly as satisfying.)

What's wrong with this proof?....

Charles Seife, from Zero: The Biography of a Dangerous Idea, pp. 217-219

I presume one can insert any celebrity one wishes.

[Crumb]

Why, sure... I expected that going into it. I knew I was waaaaaay outta my element (remember, calculus thrice, thrice). But you guys seemed to so miss 99percent, I thought I'd accomodate.
Does that mean you have changed your mind or just given up?
Tangentially, I'm in the midst of a book which is related. Zero: The Biography of a Dangerous Idea, by Charles Siefe (NY, Penguin, 2000).
This book is on my LONG list of ones I must read.

[John Carter]

I had fun godfry, but I didn't think I was kicking you around. Was I?

Why, sure... I expected that going into it. I knew I was waaaaaay outta my element (remember, calculus thrice, thrice). But you guys seemed to so miss 99percent, I thought I'd accomodate.

Tangentially, I'm in the midst of a book which is related. Zero: The Biography of a Dangerous Idea, by Charles Siefe (NY, Penguin, 2000). I'd just finished reading the whole Newton/Liebniz and the development of calculus (which, of course, went into the limits idea later) portion. The author describes Newton's abhorance of infinitessimals, and Liebniz' reveling in them and the consequent effect it had on thier variant notations. Is the term "fluxions" still used? It seems to me, from what I remember of three weeks of calculus, that Liebniz' notation must have prevailed.

Curiously, I believe he describes calculus as "not logical", but as a key to the cosmos. I've yet to finish the book. It's interesting to me, but must be pablum for the likes of this crowd.

He has, by the way, this little appendix:

Animal, Vegetable or Minister?

Let a and b each be equal to 1. Since a and b are equal,

b^2=ab (eq. 1)

Since a equals itself, it is obvious that

a^2=a^2 (eq. 2)

Subtract equation 1 from equation 2. This yields

a^2-B^2=a^2-ab (eq.3)

We can factor both sides of the equation; a^2-ab equals a(a-b). Likewise, a^2-b^2 equals (a+b)(a-b). (Nothing fishy is going on here. This statement is perfectly true. Plug in numbers and see for yourself!) Substituting into equation 3, we get,

(a+b)(a-b)=a(a-b) (eq.4)

So far, so good, now divide both sides of the equation by (a-b) and we get

a+b=a (eq.5)

Subtract a from both sides and we get

b=0

But we set b to 1 at the very beginning of this proof, so this means that

1=0 (eq.7)

This is an important result. Going further, we know that Winston Churchill has one head. But one equals zero by equation 7, so that means that Winston has no head. Likewise, Churchill has zero leafy tops, therefore he has one leafy top. Multiplying both sides of equation 7 by 2, we see that

2=0 (eq.8)

Churchill has two legs, therefore he has no legs. Churchill has two arms, therefore he has no arms. Now multiply equation 7 by Winston Churchill's waist size in inches. This means that

(Winston's waist size)=0 (eq.9)

This means that Winston Churchill tapers to a point. Now, what color is Winston Churchill? Take away any beam of light that comes from him and select a photon. Multiply equation 7 by the wavelength, and we see that

(Winston's photon wavelength)=0 (eq.10)

But multiplying equation 7 by 640 nanometers, we see that

640=0 (eq.11)

Combining equations 10 and 11, we see that

(Winston photon's wavelength)=640 nanometers

This means that this photon that comes from Mr. Churchill -- is orange. Therefore Winston Churchill is a bright shade of orange.

To sum up, we have proved, mathematically, that Winston Churchill has no arms and no legs; instead of a head, he has a leafy top; he tapers to a point; and he is bright orange. Clearly Winston Churchill is a carrot. (There is a simpler way to prove this. Adding 1 to both sides of equation 7 give the equation

2=1

Winston Churchill and a carrot are two different things, therefore they are one thing. But that's not nearly as satisfying.)

What's wrong with this proof?....

Charles Seife, from Zero: The Biography of a Dangerous Idea, pp. 217-219

I presume one can insert any celebrity one wishes.


What is wrong with that proof is that you have divided by a - b, which, since a = b, is 0.

[godfry n. glad]

Why, sure... I expected that going into it. I knew I was waaaaaay outta my element (remember, calculus thrice, thrice). But you guys seemed to so miss 99percent, I thought I'd accomodate.
Does that mean you have changed your mind or just given up?

It means that I remain technically agnostic and willing to offer conditional acquiescence to those who have more expertise than I. I'm also willing to listen and attempt to learn from them, if they see fit not to patronize.


Tangentially, I'm in the midst of a book which is related. Zero: The Biography of a Dangerous Idea, by Charles Siefe (NY, Penguin, 2000).
This book is on my LONG list of ones I must read.

I'm enjoying it as bus-riding material. Small book. Quick read for one who understands the concepts. It seems to be written for people like me, by somebody like Lone Ranger, but with a very historical bent. Mastery of the concepts is not necessary....obviously. :D

[godfry n. glad]

What is wrong with that proof is that you have divided by a - b, which, since a = b, is 0.

Exactamundo!

Note, though, that I haven't, the author has.

from Zero: The Biography of a Dangerous Idea, p. 219

What's wrong with this proof? There is only one step that is flawed, and that is the one where we go from equation 4 to equation 5. We divide by a-b. But look out. Since a and b are both equal to 1, a-b=1-1=0. We have divided by zero, and we get the ridiculous statement that 1=0. From there we can prove any statement in the universe, whether it is true or false. The whole framework of mathematics has exploded in our faces.

Used unwisely, zero has the power to destroy logic.

Is that what is referred to as a heuristic device?

[John Carter]

[quote=godfry n. glad]However, I'd speculate that it could very well be a demonstration of the limitations of simple algebra at expressing anything to do with infinite or infinitesimal numbers or series of numbers, or zero.
Why do you prefer this explanation over the obvious 1=.999... one?

Because to declare that 1 = .9999... as "obvious" is analogous to claiming that it's "obvious" that a god exists. It ain't fuckin' obvious.

Ummm... I think you missed a quote tag somewhere. I never said that. Nor did I ever say or imply that 1 = .999... is obvuios. The only thing I said was obvious (and even then I said "clearly", though "obviously' works just as well in the context) was that the limit of the sequence {.9, .99, .999, ...} is 1. By that I mean that even though we can prove that this sequence converges and the limit has this value, it is a rather trivial result and anyone who has studied (and understood) basic 1st year calculus would agree to this without demanding that we show all our work.

Ignoring the misspelling, you contradict yourself.

I registered for calculus thrice, and dropped it thrice. It never made any sense to me. I managed a bachelor's and a master's degree without it and have lived a full and mostly rewarding life without it.


How is this a contradiction? I never said what your quote tags say I did. Nor did I ever say that 1 = .999... is obvious. I did say that "clearly", the limit of the sequence {.9, .99, .999, ...} is 1.

This is very different from saying "Obviously, God exists", since I can, if necessary, produce a proof that this sequence converges to this value, but leave it out because it is trivial and I did not feel like typing it all out. It's no different from saying that 10*(3 + 2) = 50, without tediously saying 10*(3 + 2) = 10*3 + 10*2 = 30 + 20 = 50 (and even that, technically, leaves stuff out). Based on your comments about your own experiences with calculus, I very much doubt it would have made a bit of difference in wether or not you understood the entire argument.


I'd just finished reading the whole Newton/Liebniz and the development of calculus (which, of course, went into the limits idea later) portion. The author describes Newton's abhorance of infinitessimals, and Liebniz' reveling in them and the consequent effect it had on thier variant notations. Is the term "fluxions" still used? It seems to me, from what I remember of three weeks of calculus, that Liebniz' notation must have prevailed.

Cauchy's development of the limit made the use of infinitesimals unnecessary, and since there are no nonzero infinitesimals in the reals, this finally, after about 150 years, made calculus a rigorous discipline. No, fluxions are a part of history, and not used or referred to outside of History of Mathematics courses. Liebniz's notation for derivatives did win out for a very long time, but it is rarely used anymore, though sometimes you do still see it in engineering contexts. The modern notation is surprisingly similar to Newton's, as a matter of fact.

The limit is a deceptively subtle concept, and is notoriously difficult for most students to grasp fully at first. In fact, these days many schools do not even include a discussion of the formal definition of limits and the arguments necessary to prove them in their first year calculus courses. This is left to more advanced courses in pure mathematics. It's all a part of dumbing down our education, I think.

[John Carter]

What is wrong with that proof is that you have divided by a - b, which, since a = b, is 0.

Exactamundo!

Note, though, that I haven't, the author has.

from Zero: The Biography of a Dangerous Idea, p. 219

What's wrong with this proof? There is only one step that is flawed, and that is the one where we go from equation 4 to equation 5. We divide by a-b. But look out. Since a and b are both equal to 1, a-b=1-1=0. We have divided by zero, and we get the ridiculous statement that 1=0. From there we can prove any statement in the universe, whether it is true or false. The whole framework of mathematics has exploded in our faces.

Used unwisely, zero has the power to destroy logic.

Is that what is referred to as a heuristic device?

This is why division by 0 is undefined for all nonzero reals. 0/0 is a different animal, however, since any number satisfies x*0 = 0, we call 0/0 indeterminate rather than undefined.

0 may be dangerous in some sense, but it is crucial to modern mathematics. There are many ways to construct many different number systems, but in every single one, the first number generated, and which in turn is used to generate all the other numbers, is 0.

[godfry n. glad]

So how does "quantum mechanics" fit into all this? Or is that "quantum theory"?

Does topology still have active scholars working with it? Multiple dimesions? What sort of mathematics are used?

[John Carter]

So how does "quantum mechanics" fit into all this? Or is that "quantum theory"?

It doesn't. Mathematics is not concerned with physics, quantum or otherwise. Though it has been called "the Queen of Sciences", math is not really a science in any meaningful way. It is far more closely related to philosophy than it is to physics or any other science.


Does topology still have active scholars working with it? Multiple dimesions? What sort of mathematics are used?

Topology is still a very active area of research, and many areas use multiple dimensions. Advanced Calculus, for example, deals with calculus of functions in an arbitrary number of dimensions. I'm not sure what you mean by your last question.

[godfry n. glad]

So how does "quantum mechanics" fit into all this? Or is that "quantum theory"?

It doesn't. Mathematics is not concerned with physics, quantum or otherwise. Though it has been called "the Queen of Sciences", math is not really a science in any meaningful way. It is far more closely related to philosophy than it is to physics or any other science.

Thanks.

So, quantum theory is an application of mathematics (that is, mathematics is the tool used...right?) to specific physical questions. Or, all physical questions. Is mathematics used to do whatever calculations are necessary to test the hypothesis considered in physics? Is calculus the most advanced form of mathematics used, or are there other forms?


Does topology still have active scholars working with it? Multiple dimesions? What sort of mathematics are used?

Topology is still a very active area of research, and many areas use multiple dimensions. Advanced Calculus, for example, deals with calculus of functions in an arbitrary number of dimensions. I'm not sure what you mean by your last question.

Topology is evidently a form, or perhaps sub-discipline of mathematics. Is that correct? I'm obviously confusing mathematics and physics.

Do these advanced mathematics have applications with which I, a mathematics doofus, might recognize?

[John Carter]

I'd like to point out that a notation such as 0.999999...99998 is not a proper decimal representation of any real number.

You are, of course, correct. I took it to mean "some finite but arbitrarily large number of 9's, followed by an 8".

[godfry n. glad]

:deadhorse:

[Skep]

You've got that right, godfry. This topic has been beaten to death on just about every forum I've ever been a member of. On some forums the topic comes up with some regularity. I have never seen a convincing argument that 0.999... does not equal 1, including a paper written by a Florida university mathematics professor.

[John Carter]

So how does "quantum mechanics" fit into all this? Or is that "quantum theory"?

It doesn't. Mathematics is not concerned with physics, quantum or otherwise. Though it has been called "the Queen of Sciences", math is not really a science in any meaningful way. It is far more closely related to philosophy than it is to physics or any other science.

Thanks.

So, quantum theory is an application of mathematics (that is, mathematics is the tool used...right?) to specific physical questions. Or, all physical questions. Is mathematics used to do whatever calculations are necessary to test the hypothesis considered in physics? Is calculus the most advanced form of mathematics used, or are there other forms.

As I understand ot that is basically correct. Physicists, including quantum physicists, develop hypotheses using mathemtics as a tool, along with observation and previous theory, then test the hypothesis, and again use mathematics to analyze the reuslts, then repeat as needed.

As for calculus, it is, in fact, rather low on the totem pole as far as being "advanced" is concerned. It is included in the broad category of Analysis, however. Analysis is divided into three main categories: real analysis, complex analysis, and non-standard analysis. Real analysis is concerned with the study of the properties and characteristics of the real numbers, and can be thought of as a super rigorous calculus. Complex analysis is the study of the complex numbers, IOW, numbers of the form a + bi, where a and b are reals, and i is defined to be the number that satisfies the equation i2 = -1. Non-standard analysis is the sudy of alternative number systems like the hyperreals and surreals; these systems include the reals plus infinitesimals and infinite numbers. These are the broad divisions, there are many sub-disciplines within analysis. The same goes for algebra, topology, etc.

There are much more abstract mathematics involved in quantum mechanincs and other branches of modern physics, such as linear algebra, group theory, differential geometry and tensor calculus, to name a few.


Does topology still have active scholars working with it? Multiple dimesions? What sort of mathematics are used?

Topology is still a very active area of research, and many areas use multiple dimensions. Advanced Calculus, for example, deals with calculus of functions in an arbitrary number of dimensions. I'm not sure what you mean by your last question.

Topology is evidently a form, or perhaps sub-discipline of mathematics. Is that correct? I'm obviously confusing mathematics and physics.


Yes, it is. There are many many different branches of mathematics, and a very large number of sub-disciplines within these. Goliath, for example, is an algebraist. He worked with what is called Ring Theory, and in particular with what are called commutative rings, and factorization theory within that. Mathematics has become so diverse that a PhD in Analysis, for example, would have trouble understanding one of Goliath's papers in one reading.


Do these advanced mathematics have applications with which I, a mathematics doofus, might recognize?

Complex analysis is heavily used in electronics, for example. Game Theory is extensively used in strategy, in fact the discipline was originally developed during WW II and applied to anti-submarine warfare; a case can be made that game theory is ultimately responsible for the Allies winning the Second Battle of the Atlantic.

[John Carter]

You've got that right, godfry. This topic has been beaten to death on just about every forum I've ever been a member of. On some forums the topic comes up with some regularity. I have never seen a convincing argument that 0.999... does not equal 1, including a paper written by a Florida university mathematics professor.

A professor wrote such a paper?!?!?!? :eek: :eek: :eek:

[Skep]

You've got that right, godfry. This topic has been beaten to death on just about every forum I've ever been a member of. On some forums the topic comes up with some regularity. I have never seen a convincing argument that 0.999... does not equal 1, including a paper written by a Florida university mathematics professor.

A professor wrote such a paper?!?!?!? :eek: :eek: :eek:
Yes, it is on the web somewhere. I've been looking for it and cannot find it. It was linked to once when xouper claimed that no mathematician takes seriously arguments that 0.999... != 0. IIRC, that discussion took place on some gaming forum, the name of which which escapes me at the moment. I'll continue to look for it.

[John Carter]

Is this (http://www.math.fau.edu/Richman/HTML/999.htm) what you are talking about? Note that he is advocating skepticism, but presents no actual argument that they are not equal.

Looking at his home page, I see that this guy is also at heart a constructivist (mathematics that uses what is called intuitionist logic rather than classical logic), so I would expect that he would have unorthodox ideas. In fact his page on .999 = 1 is pretty much what I'd expect of a constructivist.

[Skep]

Yup, that's the guy. I just found it myself a few minutes ago. :)

[John Carter]

I notice that iin the section on Dedekind Cuts in that page, even Richman admits that the reals are specifically constructed in such a way that .999... = 1, so that paper does not in fact support the view that the equality does not hold, but rather critiques some of the more common proofs, including the one I gave earlier.


Dedekind then identified the cut {x in D : x < r} with the cut {x in D : x < r or x = r}, for each r in D, saying they were "only unessentially different." A similar move, made for example in [4, Definition 1.4], is to restrict to Dedekind cuts that do not have a greatest element, so {x in D : x < r or x = r} is not considered to be a cut. Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. Indeed, 0.9* corresponds to the cut {x in D : x < 1} while 1 corresponds to the cut {x in D : x < 1 or x = 1}. In general, we may identify an element d in D with the cut {x in D : x < d or x = d} (we call these principal cuts). So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning. That is why anyone who challenges that equation is, in fact, challenging the traditional formal view of the real numbers.


Emphasis mine

[ceptimus]

Exactly. Decimal representation is a human invention. It's not perfect. There happen to be two distinct ways of writing any terminating decimal number - reduce the terminating digit by one, and add an infinite string of nines after it.

I never understood why people get so worked up about this. Infinite strings of zeros after a decimal don't bother anyone.

By convention, we omit the infinite string of zeros, but if you put them in, it maybe helps to understand why the two numbers are equivalent.

1.000... = 0.999...

In general, any terminating decimal number can be changed to a different form: reduce the final digit by one, and add an infinite string of nines after it. For example the number three-and-a-quarter (here the dots refer to the digit zero or nine repeating, not the whole of the part after the decimal point):

3.25000... = 3.24999...

[John Carter]

Exactly. Decimal representation is a human invention. It's not perfect. There happen to be two distinct ways of writing any terminating decimal number - reduce the terminating digit by one, and add an infinite string of nines after it.

I never understood why people get so worked up about this. Infinite strings of zeros after a decimal don't bother anyone.

By convention, we omit the infinite string of zeros, but if you put them in, it maybe helps to understand why the two numbers are equivalent.

1.000... = 0.999...

In general, any terminating decimal number can be changed to a different form: reduce the final digit by one, and add an infinite string of nines after it. For example the number three-and-a-quarter (here the dots refer to the digit zero or nine repeating, not the whole of the part after the decimal point):

3.25000... = 3.24999...


Yup. Also, nobody seems to have a problem with multiple representations of the same rational number, such as 1/2 and 5/10, but present them with multiple decimal representations and suddenly the sky is falling.

[godfry n. glad]

Looking at his home page, I see that this guy is also at heart a constructivist (mathematics that uses what is called intuitionist logic rather than classical logic), so I would expect that he would have unorthodox ideas. In fact his page on .999 = 1 is pretty much what I'd expect of a constructivist.

Constructionist?

Intuitionist logic? (as in "I have a feeling that's logical?")

Classical logic? (Is that logic with Doric columns?)

Wow... Who'd a thunk that mathematics has sectarian factions?

Do you orthodox guys get together and hurl epithets at the unrepentant intuitionist constructivists? Can one be excommunicated from Analytical Mathematics? Or, any other mathematical field? Or, do you just have to live with the heretics, like the above mentioned intuitionist constructivist professor? How about censures or anaethemas? Inquisitions?

[John Carter]

There are 4 main schools of thought on mathematical philosophy: Platonism, Formalism, Logicism, and Constructivism. Platonism is the idea that mathematics is independent of the human mind, and almost all mathematicians prior to the mid-19th century were platonists. The advent of non-Euclidean geometry led to the realization that this was probably not true. If mathematical theorems are discovered and not invented, then how can there be not one but 3 systems of geometry, each of which is equally valid? Logicism was proposed by Frege, and was championed by Russell. Basically, it is the view that math is an extension of logic, and all of mathematics can be generated from pure logic alone. Formalism was founded by Hilbert at about the same time, and is the idea that math is basically just a mind game, played with formal symbols on paper. You can propose any system of axioms you like, and the only real criteria are that they be internally consistent and interesting to other mathematicians.

All mathematicians are anal as hell about rigor, proof and what constitutes a valid argument. Constructivists are even more anal than most. They argue that the only valid mathematical objects are those that can be directly constructed. Thus they are extremely suspicious of indirect proofs, especially what is called "proof by contradiction". Thus if we are trying to prove an particular type of thing exists, a constructivist will insist that you must provide a way to make (or construct) the object in order to prove it exists, iow you must produce it explicitly, or show how it can be produced. Most mathematicians, otoh, feel it is enough to show that if it doesn't exist, then we can find a contradiction.

Historically, constructivism in mathematics was founded as a reaction from some mathematicians to Cantor's work on set theory and Dedekind's work on irrationals.

There was a thread here not long ago where Clutch and I had a little side discussion on intuitionist logic. The major difference between it and more traditional logic systems is that it rejects the Law of the Excluded MIddle. This is an axiom of logic that states: P or ~P. This means that for any statement P, either P is true or P is false, there's no middle ground. In a post earlier in this thread, I proved that there are no nonzero infinitesimal real numbers. A constructivist, however, would say that my proof is not valid, since I invoked the Law of the Excluded Middle when I said the least upper bound of the set of all real infinitesimals is either an infinitesimal or it is not.

ETA: Most mathematicians today are probably logicists, formalists, or some combination of the two. There are very few platonists, in fact I have only ever met 1 person who had any mathematical sophistication that was a platonist (not incluiding physicists and other scientists, many of them would probably be platonists). The main reason there are few constructivists is the question of elegance. This is primarily an aesthetic thing, and in pure mathematics elegance is prized above all other considerations. Constructivist mathematics, however, is not at all elegant. It is, in fact, rather ugly to most.

[John Carter]

The page that skep was talking about does not actually argue against the fact that .999... = 1. It critiques some of the more common proofs from a constructivist point of view. In the end, he concedes that the traditional, formal construction of the reals requires that .999... = 1.

[viscousmemories]

There are 4 main schools of thought on mathematical philosophy: [...]
That was a fascinating and enlightening post, JC. I've always taken to mathematics like a duck to hot lava, but what little I've learned and retained makes a little more sense in that context.

[ceptimus]

Physicists and engineers are more likely to be Platonists because of the unfeasibly good agreement they observe between mathematical models and the real world.

If mathematics is just a game made up by mathematicians, why are we able to use it to predict things like solar eclipses years in advance, and the boiling points of compounds not yet actually made?

It does seem that while mathematicians are undoubtedly looking for the most elegant solution to their puzzles, the universe, at least sometimes, chooses to play the same game to the same rules.

[John Carter]

That was a fascinating and enlightening post, JC.


Thanks, vm.

Physicists and engineers are more likely to be Platonists because of the unfeasibly good agreement they observe between mathematical models and the real world.

If mathematics is just a game made up by mathematicians, why are we able to use it to predict things like solar eclipses years in advance, and the boiling points of compounds not yet actually made?

It does seem that while mathematicians are undoubtedly looking for the most elegant solution to their puzzles, the universe, at least sometimes, chooses to play the same game to the same rules.

This is the main argument in favor of platonism.

But your first sentence should read "...unfeasibly good agreement they observe between some mathematical models and the real world." Keep in mind that the math that physicists and engineers use in their work is only a tiny fraction of the entire scope of mathematics.

Math is, at root, about abstraction. It is no surprise, then, that if we apply this abstraction to the universe, we can find models that fit. Historically, the "games" that mathematicians played were directly related to this endeavor. Also, if there is some "absolute' mathematics that is independent of the human mind, why are there three mutually exclusive yet equally valid systems of formal geometry? Which is the "real" one, and where did the other two come from?

The other common argument in favor of platonism is Euler's Equation: eipi - 1 = 0. It's an incredibly elegant equation, and why, some argue, would 4 constants that were each developed at different times in seperate place by different people have such an elegant relationship?

[JoeP]

Euler's Equation: eipi - 1 = 0. It's an incredibly elegant equation, and why, some argue, would 4 constants
5 constants ... unless you believe either 1 or 0 is a variable... :P:D

[John Carter]

Euler's Equation: eipi - 1 = 0. It's an incredibly elegant equation, and why, some argue, would 4 constants
5 constants ... unless you believe either 1 or 0 is a variable... :P:D

D'oh! Either that, or I can't count very well! That's what I get for not proof reading that last post.

[Dragar]

But your first sentence should read "...unfeasibly good agreement they observe between some mathematical models and the real world." Keep in mind that the math that physicists and engineers use in their work is only a tiny fraction of the entire scope of mathematics.

A terribly tiny fraction. The last few hundred years have often seen a new area of physics is discovered - leaving a lot of very excited physicists - which is modelled sometimes very soon after because the mathematics to do so was already in place, as a result of sheer playfullness and curiosity on the part of the mathematicians.

Physicsts are usually (almost always) better at gathering the data, and coming up with ideas on ways to make sense of it. Sometimes theoretical physcists can even come up with equations out of those ideas they have, but it's usually the mathematicians who they hand those equations to, to be solved (unless they have been already - which often happens, too).

[Ensign Steve]

Euler's Equation: eipi - 1 = 0. It's an incredibly elegant equation

What does i mean in this equation? Is it the same i from i2=-1?

[Skep]

Yes, JD, one and the same i. :yup:

[Joshua Adams]

The other common argument in favor of platonism is Euler's Equation: eipi - 1 = 0. That should be eiπ + 1 = 0.

[Ensign Steve]

Hey, that's reall cool! I wikied it here. (http://en.wikipedia.org/wiki/Euler%27s_identity)

I suppose it's not all that surprising, though. I mean we sure do like that number 1, don't we? i^2=-1 by it's definition. Well pi has more to do with circles than it does with 1, but once you get sine and cosine in the mix (that has to do with a circle with a radius of 1, right?) you can cancel that sine and cosine out, and you're pretty likely to end up with 0, 1, or -1. And isn't e defined in such a way that some funcion (I forget which) of e is the inverse of itself? As in f(e)=1/f(e). I'm sure if you wiggle that one around a bit, you end up with 1.

Anyway, it all looks like a bunch more of that mathematical handwaving to me. But I still think it is really cool!

[Ensign Steve]

Wait. I mean f(e)=f(e)^-1. Looks like -1 is the magic number after all. ;)

[livius drusus]

That should be eiπ + 1 = 0.
Cool first post, Joshua. Or at least it seems cool from the my last math class was in 10th grade angle. :D

Welcome to FF. :welcome1:

[Skep]

And isn't e defined in such a way that some funcion (I forget which) of e is the inverse of itself? As in f(e)=1/f(e).
I think you're thinking of the fact that the derivative of the exponential function f(x) = e^x is e^x.

[Joshua Adams]

That should be eiπ + 1 = 0.
Cool first post, Joshua. Or at least it seems cool from the my last math class was in 10th grade angle. :D

Welcome to FF. :welcome1:Thanks for the welcome, though I must beg to differ with the notion that there is anything cool about correcting typographical errors in a math equation. :D

[Ensign Steve]

And isn't e defined in such a way that some funcion (I forget which) of e is the inverse of itself? As in f(e)=1/f(e).
I think you're thinking of the fact that the derivative of the exponential function f(x) = e^x is e^x.

:chin: Hmm... that is probably true (I'm taking your word for it) but that's not how e is defined, is it? I'm gonna wiki it right now... stand by.

ETA: Aha! I stand corrected. I was thinking e was somehow defined in terms of the natural logarithm* but a cursory glance at the wiki article makes me believe that the natural logarithm was defined in terms of e (I had it backward).

* anywho, the function I was looking for in my previous post, but was too lazy to look up) was: eln(x)=x

[Chatter]

ETA: Aha! I stand corrected. I was thinking e was somehow defined in terms of the natural logarithm* but a cursory glance at the wiki article makes me believe that the natural logarithm was defined in terms of e (I had it backward).There is no single definition for e and the natural logarithm function; it is down to personal taste. My course, for instance, defined the exponential function by its power series, defined e=exp(1), and then defined the natural logarithm function as the inverse of exp.

I have been told that the oft preferred definition is to define ln first, as you originally suggested. That is, you define ln(x) where x>0 as the integral of the reciprocal function from 1 to x. Identify ln as a logarithmic function and define e as its base, and define exp as its inverse or exp(x)=e^x.

[Ensign Steve]

I have been told that the oft preferred definition is to define ln first, as you originally suggested. That is, you define ln(x) where x>0 as the integral of the reciprocal function from 1 to x. Identify ln as a logarithmic function and define e as its base, and define exp as its inverse or exp(x)=e^x.

Neat! It's like the chicken and the egg. :giggle: I wonder, in the course of mathematical history, which was defined first. :chin:

ETA: :welcome: to :ff: Chatter!

[John Carter]

I have been told that the oft preferred definition is to define ln first, as you originally suggested. That is, you define ln(x) where x>0 as the integral of the reciprocal function from 1 to x. Identify ln as a logarithmic function and define e as its base, and define exp as its inverse or exp(x)=e^x.

Neat! It's like the chicken and the egg. :giggle: I wonder, in the course of mathematical history, which was defined first. :chin:

ETA: :welcome: to :ff: Chatter!

Historically, the natural log came first. e as a constant was first discovered by Jacob Bernoulli while trying to find the limit as n -> infinity of (1 + 1/n)n.

e is my favorite real number, since given f(x) = ex, f'(x) = f(x). Words simply cannot express how incredibly ubercool that is.

[John Carter]

To give a word of explanation for those who have not studied math beyond basic high school algebra and geometry about what I mean by "f'(x)":

That notation means the derivative of f(x). What's a derivative you ask? It's the "rate of change" of a function. To understand what that means, recall that the "slope" of a line is a measure of how steep it is. The smaller the magnitude of the slope of a line, the closer to horizontal it is, and the larger, the closer to vertical it is. With calculus, we can generalize this concept to any relatively smooth curve, and talk about how "steep" it is at any given point, and how its "steepness" changes over some interval, IOW its "rate of change", which is technically called the "derivative" of the function. Think of an almost vertical cliff face. It's basically a line, with a very large (steep) slope. With the derivative, we can describe a much more varied topography, say a whole mountainside, and how steep the incline is at any point on the mountain.

If you are one of those people who insist that these things be useful, in basic physics if you have an equation that describes the motion of an object as a function of time, you can use the derivative to analyze that equation, and get a new equation that describes the rate of change of its motion, otherwise known as it's velocity. Do it again to this new equation and you get the rate of change of it's velocity, and you now know its acceleration. You now have three equations with which you can tell the object's position, velocity and acceleration at any given time t.

The exponential function, ex, is extremely important in many applications, it describes compound interest, for one, and has many applications in biology and physics. How many times have you heard of exponential growth as something that is getting real big very quickly? Well, it turns out the rate at which the exponential function "grows", or changes, is itself described by the exponential function!

Oh, and before I forget, thanks for the correction, Joshua. That's a very embarrasing typo! And Dragar, I will reply to your last post soon; I have been too tired from work lately to give it the attention it deserves.

[Ensign Steve]

e is my favorite real number, since given f(x) = ex, f'(x) = f(x). Words simply cannot express how incredibly ubercool that is.

Agreed. :yup:

John, I confess I am not even going to try to read your explanation of f'(x). Usually I adore the way you school me in maths stuffs, but I already know derivatives, and I don't think reading about it could be near as fun as doing it. Even the way you write it. ;)

ETA: Okay, I lied. I did read it after all. Very well said. :clap:

I remember one time years ago, I used the rate of change of the mileage of my car (using Jiffy Lube receipts) to convince my insurance agent that I currently drive less than 10k miles per year, despite putting 60k on my car in the first 2 years I owned it.

[ex-xian]

Damn, I miss all the good stuff...

[Dragar]

Oh, and before I forget, thanks for the correction, Joshua. That's a very embarrasing typo! And Dragar, I will reply to your last post soon; I have been too tired from work lately to give it the attention it deserves.

Eep! I didn't expect a reply. :)

I'm sure you know more about how mathematics (and mathematicians) have helped physics along its bumpy road. I certainly have great respect for them. Many people, like Fourier, for instance, made sweeping advances in both areas.

[But]

Does anyone happen to have a link to the 20-ish page thread on 1.0 = 0.999.. at JREF?

[Skep]

Here ya go: http://www.randi.org/vbulletin/showthread.php?s=&threadid=29713

Here's one related to the above: http://www.randi.org/vbulletin/showthread.php?s=&threadid=30154

Here's another, unrelated to either of the above: http://www.randi.org/vbulletin/showthread.php?s=&threadid=23106


That oughta keep ya busy for awhile. :D

[John Carter]

The other common argument in favor of platonism is Euler's Equation: eipi - 1 = 0. It's an incredibly elegant equation, and why, some argue, would 4 constants that were each developed at different times in seperate place by different people have such an elegant relationship?


I'm going to jump on the bandwagon here and find yet another problem with this post. Actually, Euler's Equation* is this:

eix = cos(x) + isin(x)

When we let x = Pi in this equation, we get** Euler's Indentity:

eiPi + 1 = 0


*Some sources call this Euler's Formula, and use Euler's Equations to refer to a set of equations in Fluid Dynamics.


** cos(Pi) = -1 and sin(Pi) = 0, so we have:

eiPi = -1 + i(0) = -1
eiPi + 1 = 0

[John Carter]

I'm sure you know more about how mathematics (and mathematicians) have helped physics along its bumpy road. I certainly have great respect for them. Many people, like Fourier, for instance, made sweeping advances in both areas.

The current distinction between Mathematics and Physics is a fairly recent development. From the beginnings of physics as a science until the late 19th century I think all phycisists where also mathematicians in a very real sense, and most mathematicians were physicists as well. Newton comes to mind; in addition to his huge influence on physics, he is considered one of the three greatest mathematicians of all time. Gauss, as in Gauss' Law, was considered during his lifetime as the greatest mathematician ever (he was often referred to by his colleagues as "the prince of mathematicians") and holds that distinction to this day. The third member of the "triumvirate" is Archimedes, and he is credited with a few principles of physics as well.

[John Carter]

But your first sentence should read "...unfeasibly good agreement they observe between some mathematical models and the real world." Keep in mind that the math that physicists and engineers use in their work is only a tiny fraction of the entire scope of mathematics.

A terribly tiny fraction. The last few hundred years have often seen a new area of physics is discovered - leaving a lot of very excited physicists - which is modelled sometimes very soon after because the mathematics to do so was already in place, as a result of sheer playfullness and curiosity on the part of the mathematicians.


I once saw a list of the various branches and subdisciplines of mathematics that are used in physics. It was a fairly long list, but it was hardly a significant fraction of the entire scope of mathematics. For example, Group Theory was listed, but group theory is a very broad category, and while you certainly make much use of Lie Groups, there are huge areas of Group Theory that I seriously doubt are used much (I have trouble imagining that groups that rely on inaccessible cardinals have much application in physics). Foundations is one branch that is not applicable at all, except very indirectly (this includes proof theory, mathematical logic and set theory). Even the branch that is probably most universally applicable to all of physics, Analysis, has sub-disciplines that are not amenable to applications in physics. I'm specifically thinking of Non-standard Analysis here.

Even those branches that are very relevant have results that do not seem to be applicable to any possible real world situation. Consider this:

Definition: Two subsets A, B of Euclidean Space are equi-decomposable if there are finite partitions of A and B, Ui=1nAi, Ui=1nBi, such that for any i, Ai is congruent to Bi.

Theorem: Any two bounded subsets of Euclidean 3-space with non-empty interiors are equi-decomposable.

What this would mean in reality is that you could carve up a marble into a finite number of pieces and without any stretching or other deformation of any of those pieces reassemble them into a planet.


Physicsts are usually (almost always) better at gathering the data, and coming up with ideas on ways to make sense of it.

I'd say you could drop the "almost" in your parenthesis. Mathemticians are not generally interested in collecting data of any kind; mathmematics is not in any way an empirical endeavor.


Sometimes theoretical physcists can even come up with equations out of those ideas they have, but it's usually the mathematicians who they hand those equations to, to be solved (unless they have been already - which often happens, too).

That seems odd to me. Or are you lumping "Applied" Math in with "Pure" math? Even for mathematicians who work in branches that are concerned with equations of one type or another (and many do not), for most the concern is not to actually solve equations, but to determine under what conditions a given class of equations has a solution. Once these conditions are determined, the mathematician is satisfied and generally doesn't give a damn what the solution might actually be.

While proofing this post, I just realized that I probably misinterpreted your post, Dragar. Sorry, but I'm going to post it anyway; I put too much work into it not to.

[slimshady2357]

Gauss, as in Gauss' Law, was considered during his lifetime as the greatest mathematician ever (he was often referred to by his colleagues as "the prince of mathematicians") and holds that distinction to this day.

Gauss was awesome, he actually went out and tried to measure the curvature of space in... hmmmm I can't seem to find the relevant links. I'm starting to wonder if I'm confusing him with someone else.

None-the-less, he was awesome. The MacTutor History of Mathematics archive (http://www-groups.dcs.st-and.ac.uk/~history/index.html) has an excellent collection of short biographies on almost all even semi-famous mathematicians, e.g. Gauss (http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html). As well as much, much more interesting information on the history of many mathematical areas. It's a wealth of information :yup:

I was reading quotes (http://www-groups.dcs.st-and.ac.uk/~history/Quotations/Gauss.html) from Gauss and he once said "There have been only three epoch-making mathematicians, Archimedes, Newton, and Eisenstein." That's some praise for Eisenstein.

Adam

edit:

Herep (http://www.mathpages.com/rr/s8-06/8-06.htm) is article which discusses Gauss trying to measure the curvature of space. I haven't had time to read it all though.

[Dragar]

I can't really think of anything to disagree with in your post(s), John. The distinction between Mathematics and Physics certainly is a very recent one. Heck, the distinction between experimental physics and theoretical is a recent one - one of the problems of having human knowledge expand so far that even in one subject, it is impossible for one person to know even a tenth of it all any more.

That seems odd to me. Or are you lumping "Applied" Math in with "Pure" math? Even for mathematicians who work in branches that are concerned with equations of one type or another (and many do not), for most the concern is not to actually solve equations, but to determine under what conditions a given class of equations has a solution. Once these conditions are determined, the mathematician is satisfied and generally doesn't give a damn what the solution might actually be.

Perhaps I was mistaken here. I'll think on this, but I suspect you're right.

To echo other comments - Gauss was amazing. That man turns up everywhere in my physics books, and his deductions are simply remarkable.

[Chatter]

[QUOTE=John Carter]Gauss was awesome, he actually went out and tried to measure the curvature of space in... hmmmm I can't seem to find the relevant links. I'm starting to wonder if I'm confusing him with someone else.Miller (1972) claims this is a myth, and that Gauss was actually trying to determine how accurately maps can be constructed by taking the Earth as spherical rather than spheroidal.

While Gauss certainly made many contributions to physics and astronomy, he himself made a definite distinction between mathematics and the sciences.