Zero-Centric History

Reaction – even Neoreaction – tends to be hard on Modernity. God knows (so to speak) there are innumerable reasons for that.

If the criterion of judgment is set by the Occident, whether determined through its once dominant faith or its once dominant people, the case against Modernity is perhaps unanswerable. The Western civilization in which Modernity ignited was ultimately combusted by it. From an Occidental Traditionalist perspective, Modernity is a complex and prolonged suicide.

An Ultra-Modernist, who affirms the creative destruction of anything in modernization’s path, assumes an alternative criterion, inherent to Modernity itself. It asks: What had to happen to the West for it to become modern? What was the essential event? The answer (and our basic postulate): Zero arrived.

We know that arithmetical zero does not make capitalism on its own, because it pre-existed the catalysis of Modernity by several centuries (although less than a millennium). Europe was needed, as a matrix, for its explosive historical activation. Outside in is persuaded that the critical conditions encountered by zero-based numeracy in the pre-Renaissance northern Mediterranean world decisively included extreme socio-political fragmentation, accompanied by cultural susceptibility to dynamic spontaneous order. (This is a topic for another occasion.)

In Europe, zero was an alien, and from the perspective of parochial tradition, an infection. Cultural resistance was explicit, on theological grounds, among others. Implicit in the Ontological Argument for the existence of God was the definition of non-being as an ultimate imperfection, and ‘cipher’ – whose name was Legion – evoked it. The cryptic Eastern ‘algorism’ was an unwelcome stranger.

Zero latched, because the emergence of capitalism was inseparable from it. The calculations it facilitated, through the gateway of double-entry book-keeping, proved indispensable to sophisticated commercial and scientific undertakings, locking the incentives of profit and power on the side of its adoption. The practical advantage of its notational technique overrode all theoretical objections, and no authority in Europe’s shattered jig-saw was positioned to suppress it. The world had found its dead center, or been found by it.

Robert Kaplan’s The Nothing That Is: A Natural History of Zero is an excellent guide to these developments. He notes that, at the dawn of the Renaissance:

Just as pictorial space, which had been ordered hierarchically (size of figure corresponded to importance), was soon to be put in perspective through the device of a vanishing-point, a visual zero; so the zero of positional notation was the harbinger of a reordering of social and political space.

Capitalism – or techno-commercial explosion – massively promoted calculation, which normalized zero as a number. Kaplan explains:

[The growth of] a language for arithmetic and algebra … was to have far-reaching consequences. The uncomfortable gap between numbers, which stood for things, and zero, which didn’t, would narrow as the focus shifted from what they were to how they behaved. Such behavior took place in equations – and the solution of an equation, the number which made it balance, was as likely to be zero as anything else. Since the values x concealed were all of a kind, this meant the gap between zero and other numbers narrowed even more.

That is how zero, as a number rather than a mere syntactic marker, crept in. In three of the elementary arithmetical operations the behavior of zero is regular, and soon accepted as ordinary. It is of course an extreme number, perfectly elusive in the operations of addition and subtraction, whilst demonstrating an annihilating sovereignty in multiplication, but in none of these cases does it perturb calculation. Division by zero is different.

Zero denotes dynamization from the Outside. It is a boundary sign, marking the edge, where the calculable crosses the insoluble. Consolidated within Modernity as an indispensable quantity, it retains a liminal quality, which would eventually be exploited (although not resolved) by the calculus.

The pure conception of zero suggests strict reciprocity with infinity, so compellingly that the greatest mathematicians of ancient India were altogether seduced by it. Bhaskara II (1114–1185) confidently asserted that n/0 = infinity, and in the West Leonhard Euler concurred. (The seduction persists, with John D. Barrow writing in 2001: “Divide any number by zero and we get infinity.”)

Yet this equation, appearing as the most profound conclusion accessible to rigorous intelligence, is not obtainable without contradiction. “Why?” [Kaplan again]

Our Indian mathematicians help us here: any number times zero is zero — so that 6×0 and 17×0 = 0. Hence 6×0 = 17×0. If you could divide by zero, you’d get (6×0)/0 = (17×0)/0, the zeroes would cancel out and 6 would equal 17. … This sort of proof by contradiction was known since ancient Greece. Why hadn’t anyone in India hit on it at this moment, when it was needed?

Kaplan’s proof demonstrates that for zero, peculiarly, multiplication and division are not reciprocal operations. They occupy an axis that transects an absolute limit, neatly soluble on one side, problematical on the other. Zero is revealed as an obscure door, a junction connecting arithmetical precision with philosophical (or religious) predicaments, intractable to established procedures. When attempting to reverse normally out of a mundane arithmetical operation, a liminal signal is triggered: access denied.

May 7, 2013admin 32 Comments »
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32 Responses to this entry

  • Handle Says:

    Isn’t it remarkable how it took thousands of years for the best minds to work out a concept that we now take for granted as being child’s play? I show my youngster a little string of a number line with the numbers -10 to 10, and there is 0, right in the middle, and in the last decimal of both ends. He grasps the notion right away, as if it were commonsensical and obvious.

    [Reply]

    Thales Reply:

    What is simultaneously amazing is how much people take for granted and how many of their common assumptions are just plain wrong. When you see how much work there still remains to be done and yet how hard it is to do it, the apparent paradox resolves itself.

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    admin Reply:

    Zero is utterly absorbing in so many ways. The history is rich and fascinating, the operation is weirder than it seems, and the arithmetical consequences are both vast and deeply problematic — even when accessed at a low level of technical competence. That presently division by zero is simply forbidden (as ‘meaningless’) surely cannot be a stable situation. How can an ‘ordinary’ number (that we “take for granted”) be excluded from the quadrate of elementary arithmetical operations? To have a piece of basic notation — used quadrillions of times every hour — compounded with seemingly intractable mysteries is mind-meltingly odd (although zero is even).

    Exponentiation of zero by zero also jolts quickly into very strange places …

    [Reply]

    Nick B. Steves Reply:

    That presently division by zero is simply forbidden (as ‘meaningless’) surely cannot be a stable situation.

    Not only do we not divide by zero, we try not dividing even by numbers close to zero, which is what we get with poorly conditioned matrices: they are unstable and results cannot be considered trustworthy (in limited precision computers).

    Division is multiplication by the inverse, so really “not dividing by zero” is really a way of saying “0 is not invertible”… well, it isn’t…. isn’t in a rather stable way.

    [Reply]

    admin Reply:

    “0 is not invertible”– OK, I’m hooked. It sounds to my inexpert ears as if something truly thought provoking and extreme is being said here.

    [“we try not dividing even by numbers close to zero” — infinitesimals were already jostling towards the front of the zero chat queue, and you seem to be pushing the perplexity out further into mundane number space, but I’m not going to beg until I’ve done some preliminary googling]

    Erik Reply:

    As I understood it from studying number theory, the problem is only partly with zero – there’s also a problem with division, which is an intuitively obvious operator when dividing six by three, but the formalization is hardly intuitive, involving the inversion that Nick mentions.

    One method for formal construction of elementary mathematics from near-scratch starts out with the empty set, denoted {} or Ø. The empty set is unique, and the count of its elements is zero. Then we consider the set containing the empty set, {Ø}, whose element count is 1. And so forth: 2 = count( {Ø, {Ø}} ), and the natural numbers are constructed – defined in terms of the element count of a set of all previously defined sets.

    We don’t yet have addition. We have simple counting, a successor function that gives “the next bigger number”. Rudimentary addition is invented by calling this function +1, and successor-of-successor +2, and successor-of-successor-of-successor +3, et cetera. Then there’s a proof that applying the +2 function to 5 is the same as applying the +5 function to 2: addition is commutative.

    Successor(5) = 6, is there an inverse-successor function f so that f(6)=5 and more generally f(successor(x)) = x = successor(f(x))?
    Maybe. But… what’s f(0)?
    We’ve only constructed the natural numbers so far, and f(0) is as yet undefined. We can define it by inventing the negative numbers, giving us the integers, allowing us to perform new and fascinating subtractions. f(0) = -1, and all seems well.

    Next, invention of multiplication happens, then we start looking for an inverse function to multiplication. Consider a function for multiplication by three. Triple(3) = 9, and we posit that Triple(InverseTriple(9)) shall equal 9, so InverseTriple(9) = 3. But what the hell is InverseTriple(10)?

    Again, something undefined, since we only have the integers so far, and there exists no integer i such that Triple(i) = 10. Thus the rational numbers are invented, allowing us to perform new and fascinating divisions. It also means we can write 10/3 without it being meaningless gibberish like 10/banana.

    So, getting to zero. Following the naming pattern of Triple(x), Quadruple(x), Octuple(x) and so forth, we have a multiplication function Zerople(x). Now we consider constructing an inverse function. It must satisfy the constraint Zerople(InverseZerople(5)) = 5. But Zerople(x) = 0 for all x, so Zerople(InverseZerople(5)) = 0, so the InverseZerople function cannot exist.

    [Various proofs have been skipped and other simplifications made.]

    admin Reply:

    @ Erik
    By introducing the Von Neumann Ordinals you’ve jumped at least three episodes into the future — I’ll try to meet you there on schedule.

    Your final paragraph restates the problem admirably, but doesn’t it still leave us with the problem? Division doesn’t really pose much difficulty on the Real number line until zero demands to be treated ‘normally’ — and the Greek horror of infinities as intrinsic absurdities arises. If zero is already tacitly unusable (in this elementary respect), isn’t it peculiar that it has already been adopted and massively utilized as a basic component of modern calculative competence?

    Erik Reply:

    I didn’t mean to introduce the Von Neumann Ordinals – from what I understand, those make extra assumptions like ordering (where you can look at any two numbers and determine which is smaller), and treating “3” as a lower-level block to be used in the definition of “4”. I just have the successor function from one number to the next, and the set consisting of Ø, {Ø} and {Ø, {Ø}}.

    As I see it, ordering is something I could derive from the successor function, and 3 is a count of the elements in that set, not an intermediate step in generating the next set. I don’t use ordering at all, while Von Neumann Ordinals are dependent on ordering for the definition of numbers.

    I’m not sure what your problem is, and I wonder if it may be a confused question. “Division by zero” is an undefined operation much like “Division by banana”. Is there a serious problem with being unable to divide by banana?

    Why should zero be treated normally in division? It’s the identity element for addition (a fancy way of saying “adding zero to a number doesn’t change the number”), among other things. Zero already has unique behavior for addition, subtraction and multiplication.

    Nick B. Steves Reply:

    Don’t get me wrong, Nick. Zero is a fascinating subject. I read that book a few years ago and now I shall have to read it again. It is quite strange that we have this special point on the number line where ordinary operations break down. Divide by ± &epslilon; and you’re fine, but go to zero and you’re not. But zero is a really, REALLY special place on the line. I don’t think we’re suddenly going to start dividing by it and getting actual answers.

    The irrationality of √2 was apparently known to the Greeks. That deserves some ‘splainin’ too.

    Nick B. Steves Reply:

    oops… that was supposed to be ε which would have rendered ε

    nydwracu Reply:

    Zero demonstrates that, in at least one important way, Europe was not the cradle of all civilization. If that’s true in one way, it could be true in others. Chinese philosophy makes Plato and Aristotle, with their autistically rationalist babble about the irrelevant, look retarded — and ‘philosophy’ in academia still means Western philosophy, which has spent the last five hundred years or so trying to either save itself from or get worse than its eminently stupid canon. When non-Western philosophies are taught, at least as far as I’ve seen, they’re commonly mixed in with the mysticism of groups that have never even had anything like a philosophical tradition, and thereby marked off as not worth taking seriously… meanwhile, Kant and Plato are taught with a straight face.

    [Reply]

    admin Reply:

    Yes … but, do Zero, Philosophy, and Civilization slot together neatly under any circumstances?

    [Reply]

    Posted on May 7th, 2013 at 11:57 pm Reply | Quote
  • Scharlach Says:

    The practical advantage of its notational technique overrode all theoretical objections, and no authority in Europe’s shattered jig-saw was positioned to suppress it.

    Hence the danger of centralized and all-powerful authority . . . Unless power is jig-sawed, things can’t creep in.

    [Reply]

    admin Reply:

    “Unless power is jig-sawed, things can’t creep in.” — a delicious formulation in my warped opinion. I was expecting more pushback from certain quarters against this line of reasoning, but it’s early days.

    [Reply]

    nydwracu Reply:

    Even Moldbug admitted it:

    The periods in which human civilization has flowered are the periods in which it has been most politically divided. Ancient Greece, medieval Italy, Europe until 1914, China in the Spring and Autumn Period, and so on. Burckhardt once observed that Europe was safe so long as she was not unified, and now that she is we can see exactly what he meant.

    I’ve always gotten the impression that the average government in a Moldbuggian patchwork, in terms of land and people ruled, would make France or Texas look like a hulking behemoth.

    [Reply]

    admin Reply:

    Yes (without ‘… but’).

    Nick B. Steves Reply:

    And which (whose?) “certain quarters” are those?

    [Reply]

    admin Reply:

    Strong state ethno-nationalists, to start with, but when this line is followed doggedly enough it has the capacity to alienate and outrage just about everybody … after all, this Thing devoured Europe over a long breakfast, then spat out the socio-political gristle to die in the gutter, and it hasn’t even finished assembling itself yet. That’s surely got to put plenty of people in a bad mood.

    Posted on May 8th, 2013 at 1:33 pm Reply | Quote
  • Erik Says:

    “…any number times zero is zero — so that 6×0 and 17×0 = 0. Hence 6×0 = 17×0. If you could divide by zero, you’d get (6×0)/0 = (17×0)/0, the zeroes would cancel out and 6 would equal 17. … This sort of proof by contradiction…”

    This is only a proof by contradiction if you assume that 6 is not equal to 17.

    There exists a mathematics where you can divide by zero, much like there exists a geometry where Euclid’s parallel postulate is wrong. In this mathematics, 6 is equal to 17. More generally, all numbers are equal to 0 in this mathematics.

    This may seem obviously wrong, but then some of the equivalent assertions to Euclid seem obviously true. (There exists a triangle whose angles add up to 180°.)

    [Reply]

    admin Reply:

    “This is only a proof by contradiction if you assume that 6 is not equal to 17.” — I was tempted to go there, on the basis that if you run this whole problem in the other direction, out of Cantorian transfinite arithmetic, very similar problems arise, and the Cantorian position seems to be that from the perspective of infinity, the differences between finite numbers simply don’t matter. Hell, the differences between countable infinities don’t matter (multiply aleph-0 by googleplex and it makes no difference). What makes me hesitate is that the destruction of ordinary arithmetical functionality de-activates the practical machinery which has driven the whole process to this point.

    [Reply]

    Scharlach Reply:

    Yes, and you’re right not to go there and you’re right for the reasons. Even creative destruction must have its limits; otherwise, we’ll find ourselves huddled naked in a cave needing to reinvent fire.

    [Reply]

    admin Reply:

    Setting limits isn’t easy, especially against stuff adept at ‘creeping in.’

    Posted on May 8th, 2013 at 4:06 pm Reply | Quote
  • Nick B. Steves Says:

    Actually, now that you mention it: The Nothing that Is: A Brief History of Zero is, I think (if I’ve got this right) called a micro-history. I’ve also read Who Cut the Cheese? (A Cultural History of the Fart). There are zillions of micro-histories out there. And because they’re “micro” there is room for zillions more.

    Isn’t this a potentially profitable avenue for reaction?

    Micro-histories, by their very nature, are apolitical. They are therefore an excellent vehicle for revealing truths about very specific subjects, whether innately interesting or made interesting by excellent writing. And because of the inherent limitations of the micro-history genre (e.g., size and reader attention span), these specific subjects can be dealt with in a way that isolates them from the Cathedral Narrative (totalizing and politicizing) which would ordinarily be strictly enforced on more comprehensive historical works.

    Marxist Journalist: So how did the development and deployment of artificial turf further the financial interests of the global banking elite?

    (Crypto)Reactionary Micro-History Author: Umm… I dunno… I just thought the history of artificial turf was kinda cool.

    [Reply]

    Posted on May 8th, 2013 at 5:38 pm Reply | Quote
  • admin Says:

    “‘Division by zero’ is an undefined operation much like ‘Division by banana’.” — but ‘banana’ hasn’t infiltrated the global (modern) number system.

    “Zero already has unique behavior for addition, subtraction and multiplication.” — unique but perfectly tractable (and — for moderns — entrenched in intuition).

    These aren’t intended as definitive rejoinders — I’m only just escaping the grief stage over the loss of 1/0 = infinity, and your remarks are hugely helpful for re-clarifying the topic.

    [Reply]

    Posted on May 8th, 2013 at 11:43 pm Reply | Quote
  • admin Says:

    “Oops …” — The symbol handling status quo in intolerable (I’m not even permitted an infinity sign. I’ll try to find the time to deal with it (i.e. install LaTeX), but it might have to wait until after my return from Urumqi).

    [Reply]

    Nick B. Steves Reply:

    You mean this: ∞ ?

    [Reply]

    Nick B. Steves Reply:

    It’s just 1/0… but 2/0 would work equally well. Or, for that matter, any member of the set of real numbers (other than 0) over 0… For 0/0, us L’Hopital’s rule…

    [Reply]

    Nick B. Steves Reply:

    … or for that matter any member of the set of complex numbers (other than 0+i0) divided by 0… although I suppose in that case you’d get ±∞ ±i∞ which doesn’t seem quite as useful… not least because it is difficult impossible to convert to polar form….

    admin Reply:

    Thanks ∞

    [Reply]

    Posted on May 8th, 2013 at 11:46 pm Reply | Quote
  • admin Says:

    Jeepers guys, the history of arithmetic has hard-docked us upon the Vast Abrupt, and you’re trying to get us to talk about banana farts …

    [Reply]

    Posted on May 9th, 2013 at 12:00 am Reply | Quote
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