(Tomorrow is likely to be socio-technically challenging, but I’m hoping to sleaze back towards functionality from the start of the new year.)

There are innumerable niceties concerning notions, relations, instants, formalities, quiddities, and haecceities, which no-one can pry into, unless he has eyes that can penetrate the thickest darkness, and there can see things that have no existence whatever.

Appealing enough, already, in its light-footed philosophical modernism, it becomes utterly sublime when tackled — inversely — by the method of ‘hyper-literal anagogy’. It then suggests a Miltonic recovery of ancient philosophy, undertaken — with blind irony — by modernity itself.

It thus naturally tends to seize semantically on the substantiality of the negative and on what might have been said otherwise but was not — a not that is felt to contain the secret of everything. For example, Meister Eckhart’s exegesis of Paul’s blinding vision on the road to Damascus entirely ignores the ordinary, regular sense of “and when his eyes were opened he saw nothing” (Acts 9:8) [apertisque oculis nihil videbat] in favor of a mystically literal plenitude of possibilities: “I think this text has a fourfold sense. One is that when he rose up from the ground with open eyes he saw Nothing, and the Nothing was God; for when he saw God he calls that Nothing. The second: when he got up he saw nothing but God. The third: in all things he saw nothing but God. The fourth: when he saw God, he saw all things as nothing.”[2] Similarly, Augustine’s well-known statement as to the unknowable knowability of time — “What therefore is time? If no one [nemo] asks me, I know; if I want to explain it to someone questioning me, I do not know”[3] — may be (im)properly read as saying that time is known in the positively negative presence of a nemo, a not-man (ne+homo) who asks about time, a pure question posed by nobody. The presence of this no-one who is still there, a senseless letter-spirit and sudden negative indication upon which superlative understanding depends, provides a fitting structural figure for this method and an image of its divinatory, daimonic form, its sortilegic reading of received signs.

[2] Meister Eckhart, The Complete Mystical Works, trans. Maurice O’C Walshe (New York: Crossroad Publishing, 2009), Sermon 19, p. 142.
[3] “Quid est ergo tempus? Si nemo ex me quaerat, scio; si quaerenti explicare velim, nescio” (Augustine, Confessions, 11.14).

Between The Nothing and Abstract Horror there is no difference. Some related hints (and others). Eventually we reach the Vast Abrupt.

For instance (grabbing what’s immediately to hand), John D Barrow’s The Book of Nothing organizes its discussion of ‘the Origin of Zero’ by relating how

… the zero sign and a positional significance when reading the value of a symbol, are features that lie at the heart of the development of efficient human counting systems.

Robert Kaplan, when discussing the retardation of Greek arithmetical notation, explains:

… the continuing lack of positional notation meant that [the Greeks] still had no symbol for zero.

As everyone ‘knows’, the Babylonians, and later the Indians, got it right: discovering or inventing a sign for zero to mark the empty place required for unambiguous positional-numerical values. Zero arose, and spread, because it allowed modular number systems to develop. Except that, conceptually, there is no basis to this story at all.

Counting is primarily practical, so that no argument counts for much besides a demonstration. In this case, demonstration is peculiarly simple, especially when it is noted that nobody seems to think it possible.

Modulus-2 is convenient, but there is nothing magical about it in this regard. A decimal demonstration, for instance, would be no more intellectually taxing, although it would be considerably more cumbersome. Any modulus works.

Start with the basics. The positions or places of a modular notational systems represent powers. If we count from zero, the number of each successive place (ascending to the left by our established convention) corresponds to the modular exponent. The zeroth power for a single digit number, the first and then zeroth power for two digits, the second, first and zeroth power for three digits, and so on.

As the accepted story goes, each place must be filled, if only by a marked nothing (zero), if the proper places, and their corresponding (modular exponential) values, are to be read. The places must indeed be filled. There is no need whatsoever for a zero sign to do this.

The demonstration, then. Our non-zero modulus-2 positional system has two signs, 1 and 2, each bearing its familiar values. The places also have their mod-2 values, counting in sixteens, eights, fours, twos, and units as they decline to the right. Here we go, counting from 1 to 31 (watch carefully for the point at which the supposedly indispensable zero sign is needed):

1, 2, 11, 12, 21, 22, 111, 112, 121, 122, 211, 212, 221, 222, 1111, 1112, 1121, 1122, 1211, 1212, 1221, 1222, 2111, 2112, 2121, 2122, 2211, 2212, 2221, 2222, 1111 …

Conclusion: the positional function of zero is wholly superfluous. The Greeks, or anybody else, could have instantiated a simple, fully-functional positional-numerical notation without any need to accommodate themselves to the trauma of zero. In regard to this matter, the history of numeracy is utterly diversionary (not just the historiography, but the substantial history — the facts).

Perhaps this won’t seem puzzling to people, but it puzzles the hell out of me.

ADDED: Mathematical lucidity on the topic from Alan Liddell. Part 2.

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.