Competing at the Limit

I participate from time to time at a site called Math StackExchange, where users ask and answer questions about mathematics.  Most often, the questions relate to a student's coursework, but there are some deeper questions as well.  It's one of a family of similar StackExchange sites devoted to a wide variety of topics, only some of which are academically inclined.

One question that comes up every now and then is the definition of a limit.  It looks like this:

limx→af(x)=L⇔∀ε>0,∃δ>0,∀x,0<|x−a|<δ⇒|f(x)−L|<ε

And it reads like this:

The limit of f(x) as x approaches a equals L, if and only if for every positive ε, there exists a positive δ such that whenever x is within δ of a (except possibly exactly at a), f(x) is within ε of L.

Understandably, to many math students starting introductory analysis, this looks like so much gobbledygook.  Textbooks typically try to aid understanding by drawing a picture of a function f(x) in the vicinity of some value x = a, showing that as x gets closer to a, f(x) in turn gets closer to its limiting value L (which might not in fact be f(a) itself, if that value even exists).

But what if the sticking point for students isn't always that notion of better and better approximations (central as that is to the definition of a limit)?  What if the sticking point is the interplay between the "for every" (symbolized by the upside-down A: ∀) and the "there exists" (symbolized by the upside-down E: ∃)?  The intent of this definition, first conceived of by the French mathematician Augustin-Louis Cauchy (1789–1857) and formalized by the Bohemian mathematician/philosopher Bernard Bolzano (1781–1848), is to ensure that we can always get as close as we want to the limiting value (without necessarily hitting it), simply by being as close as we need to be to the argument x = a.

We can represent this as a sort of (almost irredeemably nerdy) game between two players, the Verifier and the Falsifier.  The Verifier is trying to prove the limit is right by showing that everything near x = a maps to an f(x) that's close to L, while the Falsifier tries to disprove the limit by challenging the Verifier to get even closer to L.  For instance, if the function f(x) = 2x+3, the Verifier might be trying to demonstrate that the limit of f(x), as x approaches 5, is 13:

Falsifier.  I don't think it's true; I think the limit is not 13.

Verifier.  Well, if that's so, then you must think there's some neighborhood of 13 that I can't force f(x) to lie in.

Falsifier.  Right.  OK, I challenge you to get within 0.1 of 13.

Verifier.  Sure.  If x is within 0.05 of 5, then f(x) will be within 0.1 of 13: f(4.95) = 2×4.95+3 = 12.9, which is within 0.1 of 13, and f(5.05) = 2×5.05+3 = 13.1, which is also within 0.1 of 13.  [There is more to it than that, such as that f(x) is monotonically increasing, but we'll leave these details out for now.]

Falsifier.  All right, but can you get within 0.01 of 13?

Verifier.  Yes.  All I have to do is force x to be within 0.005 of 5: f(4.995) = 12.99 and f(5.005) = 13.01.  In fact, I can answer any neighborhood of 13 you challenge me with, simply by halving it to obtain my vicinity of x = 5.  If you want me to be within ε of 13, then all I have to do is be within δ = ε/2 of 5.  Then f(5–ε/2) = 2×(5–ε/2)+3 = 13–ε, and f(5+ε/2) = 2×(5+ε/2)+3 = 13+ε.  It's foolproof.

Falsifier.  Hmm, I guess you're right.  I'll have to concede that the limit is 13.

The exchange would have gone quite differently if Verifier had claimed that the limit was 12.  Then, for instance, when Falsifier challenged Verifier to get within, say, 0.1 of 12, Verifier would have been unable to choose a vicinity of x = 5 such that f(x) is between 11.9 and 12.1 over that entire vicinity, because any value of x very close to 5—close as we like—always has f(x) very close to 13, and that clearly doesn't fall between 11.9 and 12.1.  But if Verifier can always figure out the right vicinity to force the function to fall in Falsifier's neighborhood, then they can prove the limit to be correct.

This approach to proofs has much broader applicability; in game semantics, and in a kind of logic called independence-friendly logic, many demonstrations rely on this kind of interplay between a Falsifying universal quantifier (the "for every" ∀) and an existential quantifier (the "there exists" ∃).

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Now for a digression to something that will seem totally unrelated at first.

In the late 11th century, into the 12th, there lived a Breton named Pierre le Pallet who was a precocious philosopher.  He was initially trained by William of Champeaux, but quickly grew capable of duelling wits with his teacher, and ended by starting a school of his own, against the advice of William.  By all accounts, he was a self-proud man, convinced simultaneously that he was brighter than anyone else and that no one else was giving him proper credit for this.  In his defense, he was generally regarded as one of the leading philosophers of his time, his specialty being logic, a tool that he wielded in an almost competitive spirit in defense of positions that were then considered heretical.  It was during his late adolescence that he took on the name that we know him by today, Peter Abelard.

As Abelard, his fame grew considerably, and people from all around sought his counsel.  One of these was a canon in Notre Dame named Fulbert, who wanted Abelard as a tutor for his niece.  She was then in her early twenties (we think—there is significant uncertainty about her birthdate), and had demonstrated herself to be remarkably capable in classical letters.  She had mastered Latin, and Greek, and Hebrew, and had applied these to a study of Christianity, to which she was devoutly dedicated.

Her name was Heloise d'Argenteuil, and she and her relationship with Abelard were in time to become famous.  Both of them found the other attractive, and in or around 1115, they started an affair just out of the watchful eye of her uncle.  Ostensibly, Abelard was tutoring her, but this would be interrupted periodically by a bout of lovemaking.  When they were separated, they would exchange personal messages on wax slate (parchment being too expensive even for billet doux that would have to be discarded or hidden).  A message would be incised on a layer of wax mounted to a wooden back; this message could then be read and the wax melted and smoothed over to be used again and again.

The two lovers could not necessarily deliver the messages personally without incurring Fulbert's suspicion, and so would have to rely on the discretion of messengers.  But as the messages were typically written in Latin or Greek, which the messengers couldn't read, teacher and pupil could exchange their letters under the apparent guise of lessons.  Abelard and Heloise apparently exchanged over a hundred letters this way, letters we have access to only because Heloise seems to have transcribed them onto a scroll (now lost) which was found centuries later by a French monk named Johannes de Vepria.

The affair progressed as far as Heloise bearing a son by Abelard, whom she called Astrolabe, after the astronomical instrument, and about whom we know almost nothing at all.  Around this time, Fulbert caught wind of it, and managed to force them to marry, although Abelard extracted a promise from Fulbert not to publicize the marriage, so as to protect Abelard's reputation.

Fulbert, however, had had his own reputation damaged by Abelard over other matters, and so he began spreading rumors of the marriage.  Abelard had Heloise installed at an abbey for her own protection, a gesture that Fulbert misunderstood as Abelard trying to wash his hands of her.  So Fulbert hired some henchmen, and one night, they went to Abelard's sleeping quarters, and castrated him.

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Abelard went into seclusion, and it is unclear that he ever saw Heloise again after this time.  However, about a decade or two later, they exchanged a sequence of seven or so longer letters, instigated when Heloise somehow got her hands on a letter that Abelard had written to a monk about his life story.  That letter included a retelling of her own story, and the two lovers were reintroduced to one another in this way.

Except that by this time, Abelard had decided to impose a sort of pious asceticism on himself that extended to any romantic feelings he might have had for his one-time wife.  Heloise, in turn, wrote him back, entreating him to concede those feelings, feelings she was sure he still retained.  In the last pair of letters, Heloise appears to have relented, and buried herself in her religious life, and Abelard seems to have praised and encouraged this.  But these letters are permeated through and through with an almost overwrought subtext.

So who convinced whom?  As if in honor of these two, whose story has become synonymous with medieval romance, the roles of the Falsifier and the Verifier are often personified by the love-denying Abelard, whose initial is a convenient mnemonic for the universal quantifier ∀, and by the love-asserting Heloise, whose name is sometimes spelled Eloise, whereby her initial is a convenient mnemonic for the existential quantifier ∃—symbols ineluctably entwined in the cherished logic of Abelard's youth.