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November 3, 2003

Radcliffe Edmonds (Classics)
"The Theology of Arithmetic"

Prepared by Anne Dalke
Additions, revisions, extensions are encouraged in the Forum

Drawing on a range of classical texts (for which see The Theology of Arithmetic, a PDF file, 134 KB), Rad traced the impulse among the Greeks to describe the nature of the world not in terms of a substance at the center of the universe, but as a pattern (the word "cosmos" actually means order, or beauty). With the phenomenal world in flux, only the patterns and the numbers in which it is represented were thought to be meaningful and truly significant. Rad traced a range of ways in which the followers of Pythagoras (569 BCE- 475 BCE) described the generation of all of the cosmos from numbers, the place where the "unlimited meets the limited": starting w/ a point (#1), going to a line (2), a plane (3), solids (4), one could arrive at the motion of the perceptible universe. The five regular solids (see again The Theology of Arithmetic) were associated with the five essences/elements; numbers were used to talk both about physical phenomena and religious ideas--for instance, to evoke the generation of the gods. According to Plato, phenomena were fallible and changeable; only the world of numbers was truly real. Emphasis was placed on the "movement of number into phenomena," as in music (consider the calculation of the ratio of an arithmetic or a harmonic mean). Rad closed his presentation with a series of questions:

  • Why tell a story rather than give an equation?
  • Why do we value numbers?
  • Why do we make this move to the abstract?
  • What are the problems involved in doing so?
  • What are the specific problems of attaching numbers to phenomena?
  • Is there "something special" about numbers?

During discussion, it was observed that Rad had given us a "useful reminder" that the Pythagoreans had "made the same mistake" we are making in the present day: they, like us, were interested in numbers as a way of seeking fixed invariants in a universe that is full of change. Finding flux undesirable, needing transcendence, they did then what we do now (see earlier discussions in this series on student assessment and college rankings): fight the suspicion that there is no stability in the world by inventing something that is fixed. An alternative to the Pythagorean gesture is to decide that flux is not an imperfection, not an expressed variation on the perfect real, but real itself. Heraclitus (of "never step into same river twice" fame) was a critic of Pythagoras who understood flux as the basic principle of life. But why did Heraclitus lose in this battle of ideas? He was famous for his obscurity, was unable to provide a way of describing the flux. It's difficult to study the noise; more do-able (and perhaps more worth our attention?) is finding the patterns that underlie changing phenomena. Numbers were invented to make sense of the world; counting enabled our ancestors, for instance, to know when to plant their crops; it gave them a sense of predictability. Invented for practical use, numbers became later an object of study venerated for their beauty.

But what of irrational numbers? Why did the Pythagoreans keep them a secret? The public account they provided was that every pair of numbers is commensurable, expressible in a ratio of two integers; but many are not (consider the square root). The Pythagoreans made a choice to believe the proof rather than the experience of measurement. Also significant is that Pythagoras, like Jesus and Socrates, recorded none of his thinking in writing; what we have are the fragmentary accounts of his followers, who have left us a range of solutions.

Of real interest to the group was the matter of numbers used by contemporary physicists to describe the world. Are such numbers "real" or "invented"? What of the mystery of the "fine structure constant" (1/137)? Is this number real, or the representation of some interactive strength? Is Planck's constant an observable number? Which of such numbers were known to the Greeks? Pi--but this not an observable number: it is an infinite sequence of digits measurable only to an arbitrary precision. But what is measurement? What can we measure? (The fingers on a hand?) Measurement is always explained in terms of relations between two numbers. One way mathematicians will handle uncertainty is to identify one constant they can know with infinite precision, and then put all the uncertainties into another number.

We moved then to a different level of question, thinking about what use we could make today of Pythagoras's notions:

  • We can choose only to look at our observations, and end up in chaos.
  • Or we can fall in love with the theology/theory of numbers.
  • Or we can claim that theory and practice work together, without privileging one over the other.
The study of numbers only gets us into trouble when we attempt to apply them to the phenomenal world; a pure mathematician will never bring the two together. Is it only possible to make judgments if we have an outside standard for comparison and meaning? Conventionally, it has been thought that if you cannot explain phenomena, then there's something wrong either with
  • the phenomenon,
  • the powers of observation, or
  • the explanatory theory.
The assumption has been that, if you can find a better theory, the phenenomen will make sense. This is what the Pythagoreans did: they wanted to go beyond chaos, to a pure world.

There was considerable discussion about whether the desire to do so was "escapist," a "narcotic" (and some further discussion about whether "narcotic" was good or bad: an attempt to "find meaning" or "go to sleep"). On the one hand, there is a strong tradition that the world is full of beauty and order, that our job is to perceive it, and so align ourselves with the divine. On the other hand, there exists the claim that, in the real world, things change, that the charge of "escapism" describes the retreat of both economists and psychologists into mathematical explanations, that such a gesture is the traditional academic one of all "lumpers," searching to impose patterns on the confusion of things. Is that gesture an idealizing one? A controlling one? Or an illusion that one can possess such power? Is such a move an attempt to avoid the messiness of real life, or a useful, clarifying step needed to understand and then apply that understanding to alter the world?

The claim was made that there are two different kinds of knowledge/numbers/"narcotics": the work of pure mathematicians, for whom numbers operate in a totally self-contained system, and who play with them there; and the work of experimental physicists, for whom numbers are representations of their observations of world. These are two kinds of connectivity; the latter is a shorthand way to express experience. Caution was voiced about the "hyper-theoretical" stance, which has no concern with the real world; mention was made, in this context, of the work of cleometricians, who used the science of numbers to reduce the horrors of slavery and the subjectivities of those who were enslaved.

This conversation continues online and will pick up again in person next Monday, November 10, when Liz McCormack will discuss "Concepts of Measurement in Physics."

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