"P and Q: A Geometric Noir" by Ricardo L. Nirenberg

P.  –Q!  My dear Q!  What brings you to this neck of the woods?

Q.  –Hush my dear P!  I’m travelling incognito.  If it’s found out that I left my post, I’m as good as barbe-Q.

P.  –I’ll keep it low.  In any case, I’m so happy to see you.  We’ve been through so much, you and I.

Q.  –Yes, from the very beginning there has been a most direct and intimate line of communication between us, so much so that it is still in everyone’s mouth; like, “There’s a unique straight line through P and Q.”

P.  –More than that: nothing, absolutely nothing distinguishes us from each other, only our posts.  Leibniz used to say that there cannot be two leaves exactly identical, or two drops of water exactly alike.  Why?  Because God would not admit redundancy or waste in His creation.  He should have met us.  The saddest thing is, Leibniz had met us, and he knew us well, as the great geometer he was.  But he was a politician before all.

Q.  –I don’t know much about Leibniz.  I’ve often thought one could say that we are identical twins.  Except, what would be the point?  We weren’t born.  We points are always-already, as Althusser has taught the world.

P.  –So good that you came by, and just in time, because I would like to ask you a most delicate and confidential question, my dear Q.

Q.  –Go ahead, shoot.

P.  –Tell me: are you entirely at ease with your condition?

Q.  –How do you mean “your condition”?

P.  –You must be aware that you and I are not the only points who are distinguished only by our posts.  The same goes for R and S, for R and you, or S and me.

Q.  –Yes, of course, we’re all points in the same Euclidean space.  So, what’s new?

P.  –Well, doesn’t it ever bother you, this absence of individuality, of particularity…  Let me come to the point a little bluntly: aren’t you sickened by this renouncing of personality and sticking to our posts for the sake of—how should I put it—an abstraction?

Q.  –Funny you should ask that question. Only this morning I had a little talk with T...  But hush! let it stay strictly inside the straight-line segment between you and me.

P.  –I’ll keep it safe, don’t worry.  I’ll keep it right in the middle, as in Euclid’s Proposition number one.

Q.  –Well, I trust you.  T thinks most highly of himself.  Everyone knows him for a snob, a hoity-toity, and a fool.  I greeted him and he ignored me; if points had parts one would have thought he showed me his backside.  I interpellated him (if you don’t mind my use of Althusserian terms): “T, will you be kind enough to tell me who the hell you think you are?”  He sniffed, looked down on me, and said, “Why, in point of fact and since you ask, I am the noble point whose coordinates are γ + 2,038,967.38…, exp(-9,570.15…), and π - 3.07…  On all sides a most distinguished parentage.”

P. –Did he?  I can’t believe it.  Where has he been these last three centuries?  Has it never occurred to him, has no one taught him, that systems of coordinates are not up to us points, that they are arbitrarily imposed on us, and that anyway they didn’t exist before Descartes invented them?

Q.  –Not only that.  “Listen, you dummy,” I said to him, “any number x can be written as x = e + x – e, where e is the most noble base of natural logs, so what’s the big deal about those coordinates you have assumed?”

P.  –And what did he say to that?

Q.  –Nothing.  You think it made the least impression on him?  Ideology will trump all arguments, even the most conclusive.  Althusser wrote that.

P. –But what kind of ideology are we talking about in the case of T?

Q.  –Feudal, my friend.  T still sticks to his ancestry and to his points of honor.  Laughable, but he is perfectly harmless.

P.  –Hm… Perfectly harmless?

Q.  –Ideology is the main obstacle to progress and justice in Euclidean space.  That’s true.  But a feudal ideology like T’s has lost all its virulent power: it is old and discredited.

P.  –I’m not certain I understand.  If you don’t mind, my dear Q, could you tell me what do you mean by “ideology”?

Q.  –Ideology, my friend, means delusion; not any delusion, however, but a delusion about our own identity.  The ideologue has a false belief about himself.

P.  –So, for example, Oedipus the famous King of Thebes?

Q.  –Well, no, not necessarily…  Being mistaken about his filiation doesn’t make Oedipus into an ideologue.  On the other hand, when King Oedipus proudly declares that he is the child of his own works, then, yes, he’s being an ideologue, one who has absorbed the bourgeois false consciousness of the self-made man.  And so it is just and proper that he should be punished as horribly as he is, or even worse.

P.  –I think I’m beginning to understand.  Ideology is the holding of a false belief about oneself: not just any false belief, but a false belief about one’s independence or autonomy.

Q.  –That sums it up pretty well, my dear P.

P.  –So, if someone is convinced that he has no independence, that he is completely subjected to the will of another because he’s been bewitched, should we say that he’s an ideologue?

Q.  –Listen, P, it seems to me you’re trying to pull my curlicue.  You know perfectly well that, as Marx and Engels have written in their German Ideology,  geometry is continually evolving out of the existence of points, but of points not as they may appear to their own fancies – who cares about our fancies! – but as we really are, objectively; i.e. as we operate, produce geometrically, that is, as we produce lines, straight or curved, planes, surfaces of every shape and genus – in short, as we points work in practice—mark my words, P, as we work—under definite geometric rules, limitations, presuppositions and conditions independent of our personal will.

P.  –You speak of points as they really are, objectively.  But what are they, what are we, really and objectively?  I have never met a serious definition of a point.  The old Euclidean one, “A point is that which has no parts,” is a bad joke.  “That which”!  the effrontery!  It’s like saying, “A diesel engine is that which has no carburetor” to people who have no idea of what on earth may be an engine or a wheel.  And now, at present, and for a long time, the authorities have given up on Euclid’s idea of definition, and they leave us points totally undefined.  I have no idea what I am.  Do you?  Where should we turn for help in finding out?  To Socrates, perhaps?  No, Socrates worried enough about who he was, but never, as far as I have heard either from Plato or Xenophon, gave a hoot about us points.  And don’t even think of David Hilbert, that Euclid for our time, who deliberately and cruelly eluded all allusion to our nature or individuality.  So I turned in desperation to the poets: perhaps they, as mouthpieces of the muses, would have something to say about us points, just as the Greek tragic poets had blarneyed humans by telling them what wondrous, awesome beings they are.  But all I found was humiliating and degrading comparisons, like this one by Jonathan Swift, he of Gulliver and of the Tub: “The Mathematicians steer’d a middle Course between the Naturalists and the MetaPhyz-icians; they own’d a fart to be a Quantity yet Indivisible, and gave it the name of Mathematical Point, as having neither Length, Breadth, nor Thickness.”  Some, apparently, found that funny.  Finally, my dear Q, after much toil and heartache, I came upon a sage who had diligently inquired about us points and had risked his prestige, his career, and indeed his life, waging on our existence, our importance and our individuality, demonstrating how it’s us points and only us who make gasses move—all gasses, Q, mark my words, and not only Swift’s farts.  For that, Ludwig Boltzmann was ridiculed, tormented, and demonized by his colleagues the physicists, so much so that he put an end to his life by hanging himself at Duino, a sea resort. 
            No, my dear Q, search for no help outside: there is none, and nothing to expect from others, except jeers.  We can only turn our inner vision downward; and when I do, I see no ground, I cannot find a bottom, and it appears to me that my soul has unfathomable depths.  Something similar, to be sure, happens when I turn my eyes outwards and up, to infinite space and all my fellow points.  If I didn’t misunderstand you, you were telling me I should pay attention to the infinite out there, but not to this infinite I feel I carry within: you might as well tell a point to pee out of only one kidney.

 

***

Secret Report to the Geometric Authorities.

            The above is a faithful transcript of my conversation with point P, during one of my regular secret visits of inspection, according to the established protocols of the Office of Geometric Certainty and Security (OGCS).  In my opinion, P is the carrier of dangerous germs of individualism and subjectivism, which so far seem not to have spread around but which may do so if not checked.  P is a singular point and I recommend that this unexpected and unfortunate singularity should be resolved with dispatch.
Signed,
Q.