It's not the sauce, it's the poultry and other metamathematical considerations

So yesterday I linked to an article by a Czech fellow named Luboš Motl, as part of a larger discussion of nonsense to which he is no part whatsoever.ⁱ Today I actually got around to reading what the man said.

At issue is the following problem : while in some approximations the sum of all positive integers will necessarily be an integer and positive, in some other approximations the sum of all positive integers will be quite exactly -1/12. The reasons why this should be so are not quite unrelated to how we ended up with the spin of particles being -1, +1 or +1/2, and are in fact discussed in some detail in linked material at the linked article. We won't be going into it here, at any rate.

Obviously some people will, on the basis of their very strongly held intuitions as to what reality "really" is, pick the more pedestrian approximation from the set and declare everything else quite heretical. This is neither news nor particularly noteworthy, and I'll daresay there's nothing in principle wrong with it.ⁱⁱ It does meet the valid objection that as far as the comings and goings of heretics are concerned, the opinions of the righteous are about as relevant as my minority view that the real Bitcoin block #281620 really should have hashed to
0000000000000001de6958c3b16d4ba804eec8bd45849a3c9d72946274a65748 instead of 0000000000000001de3958c3b16d4ba804eec8bd45849a3c9d72946274a65748, the more prosaic value it actually ended up hashing to. I don't happen to hold such a view, of course, but in the end that makes very little difference. Which in turn meets the equally valid objection that as far as the righteous are concerned, the insane preoccupations of the heretics couldn't matter less. Which in turn and so on and so forth and again... failure to state a case.

The one point worth the mention in all this discussion is that Motl proposes the theory whereby the definition of sums used by the pedestrianly righteous is at fault, in that their sumation is arbitrarily defined in an inconvenient manner. There's no particular reason a finite-set-cum-Taylor worldview and approach to summation should be employed instead of say Borel, or Ramanujan, or at least two dozen other candidates that the complexly heretic know about. In point of fact, there are quite persuasive reasons to employ other methods in certain contexts, as a result of the complexity therein exposed.

To my eye this is a mistaken view : the sum as used by the righteous is not only perfectly fine, but quite a necessity on the basis of their definition of numbers. Because you see, people who take no heretical interest in math, people who live their entire lives on the precarious basis of Euclidean geometry and basic arithmetics understand and define the number as the cardinal of a finite set. Six simply means "the number of ducks in a certain well chosen, given and definite set of ducks". That's how kids are introduced to numbers, they're shown a box with four severed nipples or a bag with three stretcher dildoes next to the abstract notation of 4 or 3 or what have you.

Obviously there's no particular reason a number should mean this rather than any other thing, such as the entirely different type of number that the 12 in -1/12 represents. Sure, the differences may be minute. Sure, they may not even be discernible, unless one is heretically equipped, leaving the righteous in the unfortunate state of confusing unlike things because nobody told them there's a space on the paper between the tray of cocks and the number 6. The space was there, but nobody bothered to point it out, as it were. Sure, defining numbers in the pedestrian manner is uniquely "stupid", in that it is quite uncomplex, so uncomplex in fact as to render the infinity symbol liberally sprinkled about eminently meaningless. And sure, problems of cardinality of infinite sets confound even otherwise very bright minds, for as long as they stay righteous, which is to say naive, or uncomplex, or untainted, or whatever you'd call it. In touch with their inner gerbil.

All that notwithstanding, for as long as what we mean by number is "the cardinal of a finite set" as depicted in first grade textbooks it is absolutely the case that the only possible sum of all positive integers will yield a result that is both an integer and positive. In fact, on that precarious basis no other summation could be defined, and the numerous available alternatives listed are, rightfully, quite as much hogwash in this perspective.

Or in other words, while the sauce for the goose works just fine for the gander, this point is well wasted on vegetarians : the stuff they eat doesn't even have flavor colour gender.

1. Because that's how the Internet works, someone somewhere says your name and something stupid in the same sentence and you're stuck.

   This may seem absurd as plainly stated above, but it is nevertheless exactly true, and a lot further reaching than people are generally willing to even consider. Take the situation of all the people Bitcoin butthurt : they live in a "classical" world in which they imagine themselves entitled to self-determination, which is to say they view themselves as the ultimate arbiters of what they are. They also imagine themselves entitled to version control, which is to say that if they do not sign off, if they don't approve to any changes to what they are, then no such changes can supposedly take place.

   This naive worldview belies very little reading, as Foucault among others plainly if extensively points out the problems with the approach. In short, a convention between some Dutch merchants in Antwerp and some German-ish land owners in Missouri can easily, factually and quite commonly result in the change of what black people all over the African coast are. From free ranged aborigines to slaves, as the case may be. A "proper" writ, for whatever meaning of "propriety" as may be arbitrarily allocated by other people can change one from ζῷον to βίος and - fearfully - back again. Or whatever you may wish to call these, the magical transformation from "freedom fighter" to "terrorist" and vice-versa.

   So, in point of fact, someone you've never met writing about things you never heard of somewhere you don't know exists can, in fact does, in fact already has and almost certainly will at some future point alter what you are. More or less sensibly - perhaps you're not even equipped to perceive the manner in which Bernanke printing money alters what you personally are in the grander scheme of things - but nevertheless quite factually. Any attempt at a subjective construction of identity outside of outright control of the environment - which necessarily means sovereignity, in its invariant form from Antiquity to present times - is not much unlike an attempt to write software out of the contents of pointers. While 2+2 may appear quite meaningful, and ^2 + ^2 may appear synonymous, the fact of the matter is that in the second formulation it is he who handles pointer allocation that actually writes the code.

   And so it goes. [↩]

2. Yes, I'm aware that Glenn Beck&co are not particularly popular among the more educated and thus further seeing folk. This is fine. I'm also aware that Foucault &co deeply hold the belief that in any comparison of anything, the more complex that still maintains coherence wins. This is also fine. As there's in point of fact very little common ground if at all, I find it difficult to register an actual dispute anywhere in there. So French people speak stupidlanguage and German people speak dumblanguage. What of it ? No, there's no requirement for all constructs of dumblanguage to be translatable in stupidlanguage, nor is there a converse requirement. Am I perhaps expected to wax melodical over the "right" way a thing should sound ? I should hope not. The case is therefore dismissed for failure to state a claim. [↩]