Seiryo Versus Skysaber: Intelligence answers the stupid

[Dracos]

The first in a vast production by Bjorn Christianson...

I am proud to be able to host...

Bjorn Versus Skysaber
The Intelligent Answer to the Stupid Idiot.

Preserved here for all to see will be Bjorn's brilliant and satirical rebuttals to Skysaber's works.

First, to start us off, the initial shot:
Dividing by Zero

Edit: fixing broken link

[Anonymous]

I will sacrifice my firstborn son to Bjorn Christianson!

[Anastasia]

Not bad.  A bit of it was rather mathy for my tastes, but it's good work.

And the counter tone used was priceless.

[Bjorn]

Well, no.  Your assumption was that t(n) (the time to compute the n'th computation) was t(n)=t(n-1)/2.  Thus, expanding the recursion, we have t(n)=t(0)/(2^n).  Taking the limit as n goes to infinity, we have that t(infinity) is 0.  And since, by assumption, we're trying to actually compute an infinite number of computations, that means that we attain the limit.

No, it does not, because there is no 'infinity'th computation, and the statement that some computation takes 0 time is only valid if there was. In calculus terms, you're treating the supremum as it was a maximum. There are only computations at steps n=0,1,...; the natural numbers.

On the other hand, let s(n) be the time elapsed after the nth computation (which is simply t(0) + t(1) + ... + t(n), as you defined t(n) to be above). It becomes rather obvious that the limit of s(n) is 2*t(0), twice the time it takes for the first computation (any halfway decent calculus class will tell you so, and any good calculus class will prove it so). Thus, the time that has elapsed once all the computations are completed is simply 2*t(0).

Or, more correctly, the supremum of the set of "the time to compute n computations" is 2*t(0).  As you point out, the supremum is not the maximum.  And guess what?  The set is open.  I'm not the only one treating the supremum as a maximum.

More colloquially, "you can't have your cake and eat it too."  Look at what you're saying carefully:  you claim you can compute an infinite number of additions without computing an infinite number of additions.  As I said, *if* you get to the "infinity'th" computation, then you reach a computation time (for a single operation) of 0.  On the other hand, if you don't reach a computation time of 0, then you haven't finished computing an infinite number of additions.

As long as it's only taking a finite amount of time to do the computations, you *haven't* done an infinite number of computations.  You're still approaching the limit, and hence still in the finite domain.

You seem unfamiliar with the concept of infinite series. Geometric series in particular.

No, in fact, I'm quite familiar with them.   If you'd like to swap academic credentials, feel free to privately message me.

An infinite series is the limit of an ordered set of parametrized additions and/or multiplications.  You *cannot* add or mulitiply together an infinite number of elements.  This is nonsensical.  You *can*, however, take the limit of this set as n tends towards infinity.  This gives you the supremum of the set, which is conventionally said to be the "sum" or "product" of the infinite series.   It is *not*, however, the "sum" or "product" in the conventional sense of the terms.  Rather, it is what the sum/product *would be* if it was possible to do an infinite number of computations -- which is not, in fact, possible.

Not so! The fact that a set of real numbers bounded from above has a least upper bound (supremum) is the basis of all calculus (this is frequently taken as an axiom ("Axiom of Completeness" or "Axiom of Supremum") in analysis texts, but it taken be proven as a theorem from a suitable definition of real numbers (Dedekind cuts or Cauchy series over the rationals are two of the most popular ways)).

From this, the result that infinitely (but countably) many positive, non-infinitesimal addends can yield a finite sum is almost trivial.

The result that the set of (individually finite) additions can have a finite supremum is "almost trivial."  It does not guarantee that the supremum is, in fact, a maximum.

A series is a particular example of a sequence.  It does not receive any special handling.  Particularly, for the limit of any sequence to be a maximum rather than a supremum (assuming the case of a monotonically increasing sequence, though the outlines of the argument are independent of that assumption), the set must be closed -- to wit, there must be some element in the set which actually attains the maximum.  In the example you gave above, there is no element in the set of "time to compute n additions" which requires 2*t(0) to compute.  Hence, the set is open, and 2*t(0) is a supremum.

Edit:  Just to be entirely clear, I refer you to Rudin, "Principles of Mathematical Analysis," Chapter 3 (particularly definition 3.21).

I sincerely hope that the statement "a*inf = inf, a!=0 finite" was only meant as an analogy for products of limits, since taken outside of that context it would commit exactly the same fallacy as Skysaber.

lim(n->inf) [a*n] = inf, for a finite and nonzero.  Does this make you happier?

We're dancing around in circles here that I, at least, find particularly uninteresting.  None of this past your first argument (which we've chalked up to a misunderstanding, and hence my fault) has any relevancy whatsoever to the essay that prompted this.  Right now, we seem to be arguing whether it is mathematically possible to compute an infinite sum in finite time.  I think we can both safely say that, as long as we're talking about computing it without use of theoretical arguments, this is impractical if not impossible.

Unless I've grandly mistaken the situation, I think it would be best if we took this to email or private messages.  

Bjorn

[Kuroneko]

We're dancing around in circles here that I, at least, find particularly uninteresting.  None of this past your first argument (which we've chalked up to a misunderstanding, and hence my fault) has any relevancy whatsoever to the essay that prompted this.  Right now, we seem to be arguing whether it is mathematically possible to compute an infinite sum in finite time.  I think we can both safely say that, as long as we're talking about computing it without use of theoretical arguments, this is impractical if not impossible...
Unless I've grandly mistaken the situation, I think it would be best if we took this to email or private messages.  

Quite right -- none of this has any relevance to the original essay. Personally, I found it somewhat interesting regardless. If you're interested in continuing this discussion, please tell me over private message or preferably email (at my profile), and I shall reply to the rest of your message.

Then again, we can both just drop it if you're uninterested.
---
Regards,
  FukySin

[Rezantis]

Thank you muchly, I was about to play the mod card after the next post.  ^_^

[metroid composite]

Hmm, should I nitpick?  Ah why not...I mean this is a forum with "evil" in the title....

Suppose I asked you to subtract 1x10^(-100,000)
from eight.  The answer is, of course, not eight, but it is
so close to eight that there really is no point in making
any distinction.  That would probably be true for the first
thousand or so times you did that subtraction, as well, and
if you were the child in your example, you might very well
conclude that the answer would never be different from
eight.  The truth, however, is that eventually the answer
wouldn't be all that close to eight at all; it just takes
doing a hundred thousand subtractions to actually see that.
So who are we to say that, in fact, if you actually sat
around and somehow managed to subtract zero from eight an
infinite number of times that you wouldn't be left with
seven?

Only a hundred thousand?  I think you're mixing up 10^100000 and 10^5....

which, in
turn, means that zero times infinity will be equal to zero.
(I realize that deducing that requires algebra a bit beyond
the grade-school level, and I hope I haven't lost you.)
That makes sense, of course, since anything times zero is
zero.  But on the other hand, we also know (since we know
about infinity) that anything times infinity should be
infinity, which implies that zero times infinity should be
infinity, which means that zero times infinity plus eight is
infinity, so where does that leave us?

Ironically enough, in Lebesgue Integration zero times infinity does get defined to be zero...in a sense (all we need to be able to integrate a function is for it to be finite "almost everywhere").  Lebesgue Integration is the only case I can think of offhand where the operation of zero times infinity is defined in any sense at all, so the definition isn't as unreasonable as it initially seems from a theoretical perspective.

Edit:  Just to be entirely clear, I refer you to Rudin, "Principles of Mathematical Analysis," Chapter 3 (particularly definition 3.21).

Yay Rudin!  He's so badass....


Erm anyway, good work in general.  It's not like I don't make plenty of mistakes myself when I'm acutally writing out math.

[Dracos]

Um.

It wasn't that he was making mistakes there.  It was that he was intentionally dumbing it down.  Using the correct English there would've been way over the head of Skysaber and looked silly in context besides.  The writing used in a proper mathematical paper doesn't really go well in a sarcastic retort. =)

Dracos

[Bjorn]

Hmm, should I nitpick?  Ah why not...I mean this is a forum with "evil" in the title....

Only a hundred thousand?  I think you're mixing up 10^100000 and 10^5....

Er, so I did.  (Well, actually, I think I put -100,000 where I meant to put -6, but whatever).  Still, I stand by my point.  You might be able to see the difference after doing a hundred thousand subtractions.

Probably not, though.

Ironically enough, in Lebesgue Integration zero times infinity does get defined to be zero...in a sense (all we need to be able to integrate a function is for it to be finite "almost everywhere").  Lebesgue Integration is the only case I can think of offhand where the operation of zero times infinity is defined in any sense at all, so the definition isn't as unreasonable as it initially seems from a theoretical perspective.

Er.  Unless I'm mistaken, you're confounding "set of measure zero" with "zero."  Lesbegue integration isn't multiplication, and Lesbegue integration over sets of measure zero isn't multiplying by zero.

If I want to stretch definitions, I can find more cases where "zero times infinity" is defined -- the intersection of the empty set with a (un)countable set is the empty set, the union of a countable set with the empty set is a countable set, etc, etc.  But we're talking about arithmetic, and number theory doesn't care about these other cases. ;)

Yay Rudin!  He's so badass....

Yes, he is.  He'd be even more badass if he'd give the names for theorems which had names, though.

[metroid composite]

Er, so I did.  (Well, actually, I think I put -100,000 where I meant to put -6, but whatever).  Still, I stand by my point.  You might be able to see the difference after doing a hundred thousand subtractions.

Probably not, though.

Depends what you mean by "see".  The most accurately computed physical constant to date is 14 digits IIRC.  On the other hand, if you're willing to write out 100,001 digits, then you'd notice the difference as soon as you did the first subtraction.  (and it's not like the difference between writing 100,001 digits and writing 99,996 digits is all that noticeable).

Er.  Unless I'm mistaken, you're confounding "set of measure zero" with "zero."  Lesbegue integration isn't multiplication, and Lesbegue integration over sets of measure zero isn't multiplying by zero.

*looks up definition*
Nope; it does get used in the definition for integration of simple functions (which in turn are used with limits to define integration of all fuctions) since integration of simple functions is defined as
SUM ( f(set) * measure(set) )

If you want a reference, Gerrald B. Folland 2nd Ed, p. 49 (section 2.2 Integration of Nonegative Functions) after writing the equation of an integral for simple functions has the line "(with the convention, as always, that 0*infinity = 0)."

Though yes, there's plenty of definition-stretching cases (like an infinite sum of zeros; it's not actually multiplying zero by infinity as the sum is computed via a limit, not multiplication).

[Bjorn]

Depends what you mean by "see".

See, in this case, I meant it ironically.  I made a mistake; the orders of magnitude between subtrahend and number of repititions is far too great.  Hopefully my general point is not invalidated, however.

Nope; it does get used in the definition for integration of simple functions (which in turn are used with limits to define integration of all fuctions) since integration of simple functions is defined as
SUM ( f(set) * measure(set) )

If you want a reference, Gerrald B. Folland 2nd Ed, p. 49 (section 2.2 Integration of Nonegative Functions) after writing the equation of an integral for simple functions has the line "(with the convention, as always, that 0*infinity = 0)."

I don't have this particular reference, but it seems like an awfully sloppy way of doing things.  Rudin (in both the little blue and the big green) explicitly defines simple functions as having finite value.  That, and the fact that Lesbegue integration is defined via a supremum, lets you deal with integration of infinite-valued functions without having to do some unmotivated 0*infinity hand-waving.

More importantly, that approach does not "define" 0*infinity to be 0; it allows you to show that sets of measure zero are negligible *regardless* of the value of the integrand.

Anyways.  Fun as it is to flash back to my undergraduate, I once again suspect that this conversation is a bit beyond the interests of most of the board, and the mods are eyeing me with blood in their mind and knives in their hand.  So if you want to keep discussing this, perhaps we should switch to pmsg.

[metroid composite]

Anyways.  Fun as it is to flash back to my undergraduate, I once again suspect that this conversation is a bit beyond the interests of most of the board, and the mods are eyeing me with blood in their mind and knives in their hand.

Coooooooooool! >_>

And sadly I think my sister is currently holding my copy of Rudin in New Jersy (I'm Rudin-less *sniffle*) so I can't check.  But yes, I can see how it could be approached differently...that case just stuck out in my mind as unique treatment (it was backed up by the prof too...but then the guy was like 75 and remembers the good ol' days when some mathematicians wanted to treat infinetesimals as numbers, so I wouldn't be surprised if his ideas were outdated).  As for the text Folland itself, I like it on the whole (being one of only two-three graduate texts I found readable...and one of the others was Hatcher who just sets the record for all kinds of awesomeness; text free to download off the Internet?  Duuuuuuude).  So yeah, Folland works fairly well as a resource for me (doesn't mean it won't have any mix-ups, but *shrug*).