Conjecture and proof: a primer

Warning: This article contains a lot of mathematical references. While I am trying to be considerate of those with number allergies, there are times when I cannot help myself. I apologize in advance.

As everyone knows, in the court of law and even most socially accepted justice systems, it is insufficient for one to make accusations in order to establish another’s guilt; he must have proof that the other indeed did commit this wrongdoing. It is thus important to understand what exactly constitutes proof.

A proof is, in the strictest mathematical sense, grounded on statements that have been established to be true. Anyone who remembers high school geometry would remember the two-column proof, which logically linked statements together. A conjecture, on the other hand, is a guess based on observations.

For a more down-to-earth example of a conjecture proven wrong, consider the following problem: Place n points on the circumference of a circle such that, if line segments are drawn between every pair of points, no three line segments are concurrent (i.e. intersect at one point). Into how many regions is the circle divided? Let’s take a look at the first few values of n:

Looks like the number of regions doubles as each new point is added, doesn't it?

You may think that the number of regions doubles as each new point is added, and thus guess that for six points, there would be 32 regions. However, as the figure below shows, there are actually 31:

Yep, count 'em as many times as you like. There's nothing missing.

(On a side note, the correct formula is actually \frac{n^4 - 6n^3 + 23n^2 - 18n + 24}{24}. For the mathematically intrepid, I leave it to you to prove it as an exercise; a hint is that this sum is expressible as the sum of numbers of the form \binom{n}{k}. For the less intrepid, Google is your friend.)

What went wrong? We have not established that the next number has to be 32. We might as well have asked a friend to write a series of whatever integers he wants, and we would try to guess what he would write next; he may very well write 1, 2, 4, 8, 16, then write 31 just to troll us. There would be no reason for him to be obliged to write 32. The same is true for the problem of the regions in the circle.

At the moment, I cannot remember to whom this quote was originally attributed, but it goes something like “It takes only one counterexample to disprove a statement.” Indeed, that is true; however, as has been shown, it is impossible to prove a statement using examples alone. The Pólya conjecture is an oft-cited case of a conjecture that appeared to be true for the first 900,000,000 numbers or so, but was found to be false after a search yielded a very large counterexample.

Theoretical mathematics places emphasis on rigor. However, reality (if I may) does not exactly subscribe to this paradigm. As a matter of fact, the “scientific method” that is taught in schools is but glorified conjecture. For example, it was conjectured, based on observation alone, that the sun literally rose and set. However, advances in astronomy, which allowed us to look beyond our own planet, eventually proved this to be false. Note that the scientific method’s emphasis on observed phenomena as evidence, as proof if I may, is what makes our prided scientific findings shaky at best; it is the reason our scientific paradigms change every so often.

This is, of course, not to say that conjecture is totally worthless. Many conjectures actually go on to become theorems after people investigate them and actually end up proving their truth. One can argue that the infamous Fermat’s Last Theorem, proved by Andrew Wiles, was not exactly a theorem but a conjecture: Pierre de Fermat, being an amateur mathematician, probably found the proofs for a few cases and decided to claim that he had the elementary proof for all; however, the problem resisted elementary attacks for so long, and all the tools that had been used to crack it had not been invented at the time, that Fermat’s “proof” may well have been a mirage. Nonetheless, Fermat’s conjecture gave mathematicians a place to start investigating, until, eventually, Andrew Wiles, building on previous efforts, finally proved it. And, of course, the scientific method has, despite its shortcomings, served us well for the most part.

At this point, you may be wondering what all of this has to do with the hunt for Crystal Eagle. After all, you came here to find leads, not for a Ma195g.1/Sci10 class.

Alas, if you were looking here for proof, you’ll be disappointed. Proof for this kind of case, especially, when most evidence is electronic and secured, is difficult to come by. Most of my findings are conjecture at best. However, while they do not on their own definitively establish guilt, they do give the investigation a good place to start. I have seen numerous accusations, but most of them are baseless and, at best, are highly unlikely; at worst, they are virtually impossible and not worth investigating. While it is quite difficult to outright reject the possibility that any given person is our culprit, it is easy to identify who are likelier to be the culprits, and thus direct the search for hard evidence accordingly.

You may now continue to my next article (if it has already been written).


2 responses to “Conjecture and proof: a primer

  1. Pingback: Sonya | Mathematiciowned

  2. Pingback: A Humbling Math Problem | Conejos Creations

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