# The "slow and gradual" reduction of numbers from qualitative elements to pure quantities

In a well-known book of classical scholarship, Jaeger's Paideia, Vol.1, there is the claim that "it has been justly observed that the Greek conception of number originally contained a qualitative element, and the process by which numbers were reduced to pure quantities was slow and gradual".
A reference provided therein is: J.Stenzel (1933), "Zahl und Gestalt bei Platon und Aristoteles" (in German), with the remark that this book pays no attention to the Pythagoreans.
If one does not immediately dismiss the above claim as an outmoded attraction to nebulous mystical musings, what other sources are available for its further exploration?
I'm not interested in studies of the general Pythagorean world-view, but on the specific slow and gradual transition from the qualitative to the quantitative view of number(s).

• Do they mean something like the transition from alchemy to chemistry, except here we went from numerology to algebra (or whatever)? Or, from abstract numerology to some other abstraction (I suppose one could distinguish claims like, "All is One," from, "Seven is a lucky number," even though both might be construed numerologically). Commented May 20, 2023 at 12:52
• This question could be improved by a brief explanation of what a qualitative element in numbers is. I can't think of what that could possibly mean unless they are talking about numerology. Commented May 20, 2023 at 12:53
• Good post! Opens a door ... divine!! I'm prone to hallucinations, cum grano salis. Commented May 20, 2023 at 14:59
• @Kristian Berry Plato mentions eidetic, or ideal, numbers, as beings perhaps identical to the forms (ideas) of things and different from mathematical, or monadic, numbers. One difference being that ideal numbers are not sums of monads (ones), so e.g. five-ness is not produced by four-ness by the addition of one-ness. This is where a qualitative element appears. Perhaps it concerns the structural properties of collections of things or even the forces that shape the structure of things. It seems to be a very difficult subject with many publications trying to make sense of it. Commented May 20, 2023 at 19:32
• Ancient magnitudes were mostly qualitative (not "purely"), their ratios more quantitative, and it is the transition from the latter to numbers that was slow and gradual. See Did Euclid consider circle segments as another magnitude?; What did the ratio of two magnitudes mean to ancient Greek mathematicians? and Were ratios of incommensurable magnitudes interpreted as irrational numbers prior to Fibonacci? on HSM SE, and references therein. Commented May 20, 2023 at 20:08

Assume that mathematics has a lot of objective content or structural value, despite its many changes over the ages (and the surprising level of vehement disagreement that attended or even to this day often attends talk of negative, imaginary, infinitesimal, transfinite, nonstandard, etc. numbers).

In particular, assume that the ordinal/cardinal distinction is fundamental and everlasting. I assume anyway that ancient Greek languages had room for distinguishing, "How many?" from, "Which one?" and this can be indirectly corroborated in Latin, where quod (for which) and quot/quam multi (for how (many)), alongside quantitas, seem like reflections of the quality/quantity distinction, which is then the ordinal/cardinal distinction.

And so in fact, down to Cantor's day, it was not always first just the word ordinal that was used as such, but the phrase order type (to be finicky: Cantor himself also used the word Anzahl, which could be glossed as enumeral as in an enumerative number, but I digress). Types, which figure in extension-facts but which have an intensional action of their own, seem qualitative, or to play into qualitative representation, or however you'd want to put it.

Now, per the use of the word type coinciding with set theory more when Russell developed his specifically-named type theory, there was a long period beforehand where the academic distinction was often not so much between types and tokens or types and terms but types and antitypes, which was actually a religiously-themed pairing, and so extremely qualitative in meaning. (C.f. Carl Jung's archetypes, which do go back terminologically to Kant at least, no less.) So there is some sense in which focusing on numerical types is to focus on them in a qualitative way, whereas using numbers cardinally is then less typically. Note that there can be well-ordered cardinals but there can also be amorphous cardinals (the cardinalities of amorphous sets), which are definitively ill-orderly. And although amorphousness is, after all, something of a quality of these sets, it is a quality whose primary value is read off the obscure quantitative facts that can be represented hereby (i.e., let K be the cardinality of some amorphous set; then it is possible to say of some kind of material object x, "There are K-many examples of x").

Plato's Form of the Good: some interpretations of the ultimate Form are that it is some kind of unity (but not necessarily, as neo-Platonism often is read as saying, a monad). Maybe the most fanciful description Plato offers is that the Good is "greater in dignity and might" than even Truth or Knowledge. Peculiarly, and much later, Peano and Cantor would offer that 1 was the "real" first ordinal, although for the most part 0 has been taken for this since; and metaphysical acceptance of 0 as a "real number" was a long time in coming in many places around the world anyway. Taoism seems to be interpretable as saying that something like 0 is the "first ordinal," however ("From the Tao came the One, from the One came the Two, from the Two came the Three, and from the Three came all other things...").

At any rate, the Good has a sort of ultimate priority in the order on the Forms. If Plato thought that there were numerical types, especially as order types (even if he used subtly distinct cognates for those terms in his language), perhaps he thought that ordinal numbers, or the ordinal use of numbers, itself had some kind of priority over cardinal usage. But the back-and-forth in the progress of mathematics from that time on,N somewhat independently of Pythagorean influences (I don't want to claim a strong degree of independence, I don't know enough about the Pythagoreans to say so), could then yet have gone from a metaphysical preoccupation with the priority of the Good, and become a reflection upon the infinite magnitude of the number of numbers (the quantity of all numbers), which might no longer have been kept in view as the Form of the Good (the "number of the Good"); or which is to say, if numbers were no longer being traced back to some ultimate foundation, were not being emanated from an absolute unity or monadicity, then this would be one way to look at them as being used for quantitative purposes more than qualitative ones, as having a primary existence as quantities (cardinals) and not qualities (ordinals).

NI just found out that Euclid probably lived in Alexandria and perhaps even taught at a sort of "museum" there (the word just about coincides but the description of the facility doesn't clearly match contemporary uses of museum). Now Alexandria also featured the monotheistically-oriented Philo, not at the same time as Euclid, but arguably in a nearby-enough "segment of history." So, if I remember correctly, Philo was sometimes appreciated by Christian analysts as having some forevision of God being multiple in personality. I would suspect, then, that type-talk in those days, even as based on Christian theories of Jewish prophecies, could be found to resonate with talk of qualitative mathematics. More esoteric Jewish researchers were prone to much gematria, a kind of numerology, but which might have nevertheless involved serious comprehension of various arithmetical patterns. I know you were trying to point away from "nebulous mysticism" but we might not be able to avoid greater consideration of how various religious and philosophical subcultures overlapped and interwove back then, to characterize the movement from emphasis on quality to emphasis on quantity (which is still not entirely obviously what happened; infinitesimals might be taken for indicators of "size" or "magnitude," but e.g. the imaginary unit occurs first as a sort of placeholder/abstracted solution to a function, and which is comprehended more substantively through the elaborate interpolation of the real line and the complex plane; so perhaps there hasn't been much of a clear "quantity over quality" or "quality over quantity" outcome, ever, in the end, here).

I should add that I doubt that Plato or almost anyone back then had a vivid understanding of the ordinal/cardinal distinction, because as far as they all knew, every ordinal was otherwise "the same thing as" some cardinal: (zeroth, zero), (first, one), (second, two), etc. and so it wasn't really until Cantor that we had a strong method for identifying multiple ordinals corresponding to one cardinal (which however then becomes something of a type for those ordinals!).

• I believe you've hit the bullseye there!! Keep it coming! "Not everything that counts can be counted, and not everything that can be counted counts." ~ Albert Einstein Commented May 21, 2023 at 1:03
• Crucial is, as you say, that "if numbers were no longer being traced back to some ultimate foundation (...) then this would be one way to look at them as being used for quantitative purposes more than qualitative ones". And I think that to trace numbers back to some ultimate foundation involves embracing a realm broader than mathematics in a way that is both qualitative and precise. A lost art of analogical reasoning, physics & psychology with direct intuition about numbers, perceived as forces of nature. Qualitative examples within mathematics (e.g. topology) are welcome but inadequate. Commented May 21, 2023 at 15:47