Wednesday, March 06, 2024

Asymtotically we'll all be dead II

alternative title "avatars of the tortoise I"

Asymptotically we'll all be dead didn't get much of a response, so I am writing a simpler post about infinite series (which is the second in a series of posts which will not be infinite. First some literature "Avatars of the Tortoise" is a brilliant essay by Jorge Luis Borges on paradoxes and infinity. Looking at an idea, or metaphor (I dare not type meme) over centuries was one of his favorite activities. In this case, it was alleged paradoxess based on infinity. He wrote "There is a concept which corrupts and upsets all others. I refer not to Evil, whose limited realm is that of ethics; I refer to the infinite."

When I first read "Aavatars of the Tortoise" I was shocked that the brilliant Borges took Zeno's non paradox seriously. The alleged paradox is based on the incorrect assumpton that a sum of an infitite numbr of intervals of time adds up to forever. In fact, infite sums can be finite numbers, but Zeno didn't understand that.

Zeno's story is (roughly translated and with updated units of measurement)

consider the fleet footed Achilles on the start line and a slow tortoise 100 meters ahead of him. Achilles can run 100 meters in 10 seconds. The tortoise crawls forward one tenth as fast. The start gun goes off. In 10 seconds Achilles reaches the point where the tortoise started by the tortoise has crawled 10 meters (this would only happen if the tortoise were a male chasing a female or a female testing the male's fitness by running away - they can go pretty fast when they are horny).

So the race continues to step 2. Achilles reaches the point where the tortoise was after 10 seconds in one more second, but the tortoise has crawled a meter.

Step 3, Achilles runs another meter in 0.1 seconds, but the tortoise has crawled 10 cm.

The time until Achilles passes the tortoise is an infinite sum. Silly Zeno decided that this means that Achilles never passes the tortoise, that the time until he passes him is infinite. In fact a sum of infinitely many numbers can be finite -- in theis case 10/(1-0.1) = 100/9 < infinity.

Now infinite sums can play nasty tricks. Consider a series x_t t going from 1 to infinity. If the series converges to x, but does not converge absolutely (so sum |x_t| goes to infinity) then one can make the series converge to any number at all by changing the order in which the terms are added. How can this be given the axiom that addition is commutative. Now that's a bit of a paradox.

the proof is simple, let's make it converget to A. DIrst note that the positive terms must add to infinity and the negative terms add to - infinity (so that they cancel enough for the series to converge).

now add positive terms until Sumsofar >A (if A is negative this requires 0 terms). Now add negative terms until sumsofar The partial sums will cross A again and again. The distance from the partial sum to A is less than the last term as it just crossed A. The last term must go to zero as we get t going to infinity (so the original series can converge) so the new series of partial sums converged to A.

That's one of the weird things infinity does. I think that everything which is strongly counterintuitive in math has infinity hiding somewhere (no counterexamples have come to my mind and I have looked for one for decades).

Now I say that the limit of a series (original series of sum t = 1 1 to T) as T goes to infinity is not, in general, of any practical use, because in the long run we will all be dead. I quote from "asymptotically we'll all be dead"

Consider a simple "problem of a series of numbers X_t (not stochastic just determistic numbers). Let's say we are interested in X_1000. What does knowing that the limit of X_t as t goes to infinity is 0 tell us about X_1000 ? Obvioiusly nothing. I can take a series and replace X_1000 with any number at all without changing the limit as t goes to infinity.

Also not only does the advice "use an asymptotic approximation" often lead one astray, it also doesn't actually lead one. The approach is to imaging a series of numbers such that X_1000 is the desired number and then look at the limit as t goes to infinity. The problem is that the same number X_1000 is the 1000th element of a of an uh large infinity of different series. one can make up a series such that the limit is 0 or 10 or pi or anything. the advice "think of the limit as t goes to infinity of an imaginary series with a limit that you just made up" is as valid an argument that X_1000 is approximately zero as it is that X_1000 is pi, that is it is an obviously totally invalid argument.

An example is a series whose first google (10^100) elements aree one google so x_1000000 = 10^100, and the laters elements are zero. The series converges to zero. If one usees the limit as t goes to infinity as an approximation when thinking of X_999 then one concludes that 10^100 is approximately zero.

The point is that the claim that a series goes to x s the claim that (for that particular series) for any positive epsilon, there is an N so large that if t>N then |x_t-x} This tells us nothing about how large that N is and whether it is much larget than any t which interests us. Importantly it is just not true that there is a N so large that (for any series and the same N) if t > n then |x_t- the limit| < 10^10^(100)

Knowing only the limit as t goes to infinity, we have no idea how large an N is needed for any epsilon, so we have no idea if the limit is a useful approximation to anything we will see if we read the series for a billion years.

Now often the proof of the limit contains a useful assertion towards the end of the proof. For example one might prove that |x_t-X| < A/t for some A. The next step is to note that the limit as to goes to infinity of x_t is X. This last step is a step in a very bad direction going from something useful to a useless implication of the useful statement.

Knowing A we know that N = floor(A/epsilon). That's a result we can use. It isn't as elegant as saying something about limits (because it includes the messy A and often includes a formula much messier than A/t). However, unlike knowing the limit as t goes to infinity it might be useful some time in the next trillion years.

In practice limits are used when it seems clear from a (finite) lot of calculations that they are good approximations. But that means one can just do the many but finite number of calculations and not bother with limits or infinity at all.

In this non-infinite series of posts, I will argue that the concept of infinity causes all sorts of fun puzzles,but is not actually needed to describe the universe in which we find ourselves.


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