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Joel Kalvesmaki Expands Arithmetically
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This extended interview covers a broad field of esoteric number-theory in history. We range far and wide through the number-strewn fields of post-Platonic ‘Pythagorean’ mathematical symbolism, taking in many of the sights along the way.
Specific topics of discussion include:
- The contextualisation of psephic metaphysics in Gnosticism and early Christian writers as part of the early Greek quest for world-systems,
- More on the alpha-numeric mysteries of the Marcosian Gnostics,
- Some reflections on the polemics about reading the world as an alphabetic-mathematical text inside and outside the early church,
- The fourth-century debate between the followers of Iamblichus and Theodor of Asine about what kinds of realist positions concerning words and numbers (and even written letter-forms!) were permissible,
- The development of the Greek term logos into a metaphysically-real principle and the possibilites this opened up for speculation in early Christianity,
- The fact that the One (and sometimes the Two) are not considered ‘numbers’ in our Greek sources,
- And the Problem of multiplicity in a monist universe (or of unity in a fundamentally-multiple universe), and how the later mathematical concept of the zero is relevant to this problem, which of course leads to some discussion of early apophatic Christian theology.
Works Discussed in this Episode:
- Fritz, K. v. (1945). ‘The Discovery of Incommensurability by Hippasus of Metapontum’, The Annals of Mathematics, Second Series 46 : 242-64.
- Iamblichus. Dillon, J. M. & Hershbell, J. P. (Ed.), 1991. On the Pythagorean Way of Life. Scholars’ Press, Atlanta, GA.
October 24, 2019
The podcast suggests that Ancients did not have the ‘concept of zero’. I am not sure what is meant by that but this may be repeating a common modern confusion about the mathematical not-one or none.
I doubt we would dispute that the ancients had the concept of none as not-one or non-being or not-being-present. We would not dispute that they had the concept of absence in the sense that if the ten green bottles are no longer sitting on the wall then there are ‘no’ green bottles sitting on the wall, which is to say ‘zero’ green bottle. More likely, by ‘zero’ we might mean notational cipher. The ancients did not have that because they did not need it. The marking of an absence is only needed when calculation was transferred to paper (or other writing surfaces).
Throughout most of civilisation, calculation was done on physical counters (boards or abacus). With set place-value columns or rows, an empty place was an empty place. When the calculation was complete, the result would be announced or written in words or symbols. Whenever in history there was as shift to notational calculation (without set place-value columns) then notional was required for the empty place, often symbolised by an empty circle or square.
Where the Greeks did have a problem was with linear geometry, where their counting of the limit of intervals starts with one. We may find this confusing while still counting notes/intervals in the octave in this way. But it worked for them. They would have had a problem if they tried to do negative numbers as per our BCE/CE timeline. But they didn’t.
As for the Monad-origin-of-the-emanation, this is not the same as the timeline’s ‘1 CE’. Instead, it is an absolute origin of self-generation, which might best be described as the arche-one and the ordinal zero combined (hence unity and its opposite as suggested by Plato in Republic, p524). True, Hellenistic Pythagoreans confused and disputed the sequence from there, but that was not only about zero.
The modern obsession with the ‘invention of zero’ is curious because we have always known that place-value began with physical counters (think, shop ‘counter’) and that the cipher was only to make up for a deficiency in notational calculation. We tend to demean the great utility of ancient mechanical computers and misses the good reasons why the move to paper was resisted in the West. This was not only due to concern about dropping zeros but also that calculation by abacus was much faster.
October 28, 2019
Okay, Bernie, good points, but even if it’s wrong to say the ancients ‘didn’t have a concept of zero’ (and I take your point here), there is no doubt that arithmetic advanced by leaps and bounds once a placeholder for ‘no number’ was introduced. You can’t search for primes or model Reimann geometries or whatever with an abacus, and the ancient Greeks were definitely interested in doing much more than counting stuff, as at a shop counter – things like modelling hyperbolic motions or computing the circumference of the earth. So they could have done with a place-value notation system with a zero.
As for the Platonist/Pythagorean monad: quite so, but I suspect that different debates about whether the one (and even the Two) were numbers or not didn’t have much relevance at the counter. If you are here to buy one apple, one apple you shall have.
Finally, the move to paper in the West (by which I guess you mean roughly the Roman realm?) wasn’t exactly resisted – there just wasn’t any paper! We have tons of papyrus fragments from Egypt (and ostraka from Greece etc.) with calculations jotted on them, so people were indeed doing arithmetic on ‘paper’ in antiquity, but, as you say, an abacus is way quicker for on-the-fly addition/subtraction.
October 29, 2019
Some clarifications: by ‘calculation’ I mean addition, subtraction, multiplication and division, which is where place value is almost indispensable for high level work. The abacus was only for calculation! Of course the surface would be used for geometry and for recording or progressing the work. The main point is that while place value is a big deal, its notational zero is not. The widespread view that the zero was important is not supported by the evidence as presented in our histories. These show that it tended to appear with the transition to notational calculation. I have never seen good evidence that non-place-value notation was in common used where large calculations were required (eg with the Romans administration, astronomy, horoscopes etc). That should be no surprise because it would have been so difficult, and that is why it is so curious that folks imagine this. When paper became more available (though it was still expensive by Newton’s time), there certainly was a case for covering it with those long tails of calculation (eg for tracing your errors), but these days folks don’t seem to appreciate that there could be any competition—viz., just how fast and accurate abacists used to be. Yet in the middle ages, at least from Pope Sylvester, there seems to have been a bit of an ‘algorist’ vs. abacist thing going down. This famous woodcut has a fanciful competition between Boetius and…wait for it…Pythagoras!
October 29, 2019
Nice woodcut! And good points.
June 5, 2023
On the topic of complex numbers, I figured Plotinus might suggest that they are derived from the natural numbers and are therefore have to do with quantity rather than quality. But would be interested to hear what you might have to say on this topic.
June 5, 2023
June 5, 2023
Whoo, Oscar, that’s way above my pay-grade!
Svetla Slaveva-Griffin’s book Plotinus on Number (Oxford University Press, 2009) might be a good first place to look: I have not read it, but she’s worked a lot on Plotty and numbers, and I’ve heard good things about the book, so I feel confident in recommending it.