In his essay, The Sand Reckoner, Archimedes specified a system of nomenclature for large numbers.  ( The greek title is yammites which is simply, “sand”. Interestingly, Liddel and Scott’s Intermediate Greek Lexicon includes yammakosioi, a “sand-hundred”, stated to be a “comic word”, presumably like “gazillion’, indicating a “countless multitude”. )

The Greeks had names for ten, hundred, thousand, and ten thousand, which was the myriad, and by composition had expressions for numbers up to the myriad myriad, which Archimedes used as the base of his system, and which we designate here by M. He called the numbers from 1 to M the first order of numbers, and using M as a unit of the second order, counted from M to M², which became the unit of the third order. Continuing in this way, he postulated M such orders, counting up to M^M.

He then called this the 1st period of orders, and continued with the 1st order of the second period counting in units of M^M up to M^M+1. Then M orders of the 2nd period count up to M^2M, and finally the Mth period counts to M^M2, which was the limit of his system.

Note that the orders of each period all count from 1 to M, so that the M periods of M orders comprise M² orders all together, accounting for M² factors of M.

This system of Archimedes is considered as something of an oddball, but it is actually familar to us in the form of our number naming system, if we take M to stand for “million” instead of “myriad myriad”.

Note that “million” derives from the Italian, “millione” meaning “grand thousand”,  i.e. a thousand thousand, so our million corresponds directly to the myriad myriad, or “grand myriad”, even though different in value. Then we see that the “long scale” or European system, which uses a million as a multiplier, starts off just like Archimedes’ system, with the numbers of the second order counting by millions up to one billion, the unit of the third order numbers, which count up to one trillion, and so on.

Well then, how well does our nomenclature keep up with Archimedes in this analogy?  The first sign of trouble ahead comes at  the end of the 1000th order, when we hit one milliatillion, as this is where the nomenclature plays its last card. Still, we make out OK up to the end of the first period, where the one millionth order ends at one milliamilliatillion.

Now in the Archimedian system, we “start fresh” at each period, so we can specify an order of each period between one and one million and get any power of a million up to a million million. In the modern “long scale” we are stuck with concatenating an extra “milliamillia” for each period, and the unit of the first order of the 5th period is

one milliamilliamilliamilliamilliamilliamilliamilliamilliamilliatillion

( I’ve been using the site: The English name of a number, for these values. Set “power of ten” and “European ruleset” . )

So I think Archimedes wins the point. But could we extend the modern nomenclature to accommodate the Archimedean scheme? This would mean extending it to one million periods, and in fact this can very easily be done.

Just as the suffix “-tillion” indicates powers of one million, we can introduce the suffix “-zillion” to indicate powers of milliamilliatillion, or one million to the one millionth power. Each period introduces a factor of one zillion, so we can call the largest number of the second period one bazillion, in analogy to one billion as a million million ( in the “long scale” ) . Then we march through trazillion, quatrazillion, etc. in one to one correspondence with the orders of the first period, and the largest number of the millionth period, which is the limit of this scheme, is one miliamilliazillion. So there you are.

RUDY RUCKER’S “SUGGESTION”

Rudy Rucker makes some cursory but pithy comments in his 1983 book, INFINITY AND THE MIND. Here is an excerpt from my copy of the Bantam paperback edition.

Regarding the highlighted recursion rules, there are two mistakes. The first is a trivial error in ii) A(1, j+1 ) = M*A(k,j) which evidently s/b ii) A(1,j+1) = M*A(1,j), noting that rule iv covers values of k > 1, ( although it allows k+1 = M+1, which we shall overlook.)

The second error is not so easily dismissed. This is in iii) A(k+1,1) = A(k,m), which s/b iii) A(k+1,1) = M*A(k,M), since A(k,j) is the GREATEST value of the jth order of the kth period, and A(k,M) is the UNIT of A(k+1,1) . I checked this out with actual recursive definitions in “bc” on cygwin, for low base values such as 3,4, and 5, so don’t doubt me!

But what about this? I think it’s all explained by the author’s  suggested improvement using rule iv*) . The modified rule, with the other rules as stated ( except for the above cited trivial correction ) does produce values of A(M,M) = M^M^M, as claimed, so it would appear that the original formulas were reverse engineered from the “improved” version.

The machinations of the latter can be easily discerned. The “highest values of the orders of the periods” ( inconsistently so called )  progess as follows:

M        M²       … M^M
M^M      M^2M      … M^M2
M^M2     M^2M2     … M^M3
…
M^M(M-1) M^2M(M-1) … M^MM

… but this seems to bear very little relation to Archimedes’ scheme, which is based on simple enumeration. The multiplier value of M comes from counting 1 thru M in each order of each period, and this is the defining principle. It is the unit of the enumeration that changes, and the numbers of all the orders are the same, namely 1 thru M.

But nevertheless ! … there is a connection, and Rucker’s modified formula is contained in the Archimedean scheme. Note the very suggestive sequence of      A(k,M) = M^Mk , where k=1 corresponds to the end of the Mth order, and k=2 corresponds to the end of the Mth period. The suggestion is namely that we consider orders and periods to be the 1st and 2nd levels of a generalized sequence of nth level periods.

Furthermore, each row comprises successive powers of the first element, as specified by rule iv*). So Rucker’s scheme gives a summary of the values reached by the jth element of the kth-level period, j and k passing from 1 to M.

Note that it would require a recursive function with M arguments to specify M levels of periods in a direct continuation of the original scheme.

It seems that Rucker must have used this reasoning, or some form of it, to derive his function, but his remarks that Archimedes “could” and “should” have made a similar specification remain puzzling.

Charles Sanders Peirce, in his essay, LOGIC AS SEMIOTIC: THE THEORY OF SIGNS, presents a diagram displaying the ten types of signs in a triangular array, like bowling pins. These are categorized by three different threefold classifications, his “three trichotomies”, which allow for the specification of 27 types, but only ten are valid. There’s a very simple rule that performs this reduction, but it is not specifically enunciated, that I can notice, even though you can see him applying it.

The Wikipedia article Semiotic elements and classes of signs makes the rule fairly clear with a diagram, but again does not specifically enunciate it, saying simply, ” many co-classifications aren’t found”.

Some years ago, in my own ruminations of the essay I cited, I realized the rule was simply this, that in each label, reduced to a three digit ternary number, abc, where a, b, and c are elements of {0,1,2}, we must have a >= b >= c. Thus when b=0, specifying an Icon, we can only have 000, 100, and 200. The logic of this is that each succesive category can only partake of a quality present in the previous category, so that the Qualisign, 000, is unique, and conversely the Argument, 222, is as well.

Realizing all this, I made a model representing the subset of the 3X3X3 cube:

This crude model is a number of years old, and its survival is no doubt partly due to its simple construction, as it is composed of exactly two identically folded paper pieces, a reflection of the symmetry of this subset of the cube. It is essentially, except for edge conditions, 1/6 of the cube, as partitioned by 3 bisecting cuts along a major diagonal. I felt that all this symmetry impinged, or ought to impinge on the significance of the categories, but I never pursued such thinking. Nevertheless, the construction does serve as a powerful mnemonic, and stands as a rival to Peirce’s own diagram.

Well, just lately, I had the thought to make a direct comparison of the two, and found that there is a direct relationship, and in fact Peirce’s tableaux does embody most of the symmetry of the cubic model, even if it doesn’t directly display the restrictive rule.

So with that preamble, here is Peirce’s tableaux marked to display the groupings of the Icons, Indexes, and Symbols:

Then here is a new model of the cubic representation using a lattice so you can see all the elements, and of course the same color coding. You can see that the Icons, Indexes and Symbols belong to three separate planes perpendicular to their defining axis:
This is suggestive, but how can we relate this lattice directly to the Tableaux? A start is to try a projection, or view along an axis of symmetry, to spread things out a little. Here is such a view of the same model:
This goes pretty far. Note that the ( green ) Indexical categories do in fact correspond to the Tableaux. ( You may believe me that the correspondence is exact. ) However, the other elements are out of whack. The three rows of this projection are not the same as the rows of the Tableaux.

We can almost rectify everything by flattening the lattice.  This only requires “unbending”  the two right angle connections between the blue and green, and the pink and green:
This is almost it! We have retained the connective character of the lattice model, and come very close to Peirce’s original representation. In fact, we could modify the Tableaux by a simple cut and paste of the Argument and Qualisign elements. But let us bow to precedent and esthetics, and simply swing the lattice elements, which have only one connection, into the “correct” place, and we have an exact correspondence without really sacrificing anything: