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Early evidence of tetration
The ancient Indian record number of tetration with an "unspeakable" super-exponent 10^^(10^(5*2^120)) can be found in the Avatamsaka Sutra, chapter 30, the first verses of the poem.

Page contains notes, click the *
My comments are pasted below from:

§1.6.2. Measuring the asamkhyeya

To give the most definitive definition of an asamkhyeya – the traditional Indian number where infinity starts – we turn our eyes to the most grandiose work of buddhist literature. The Flower Ornament Scripture (Sanskrit: Avatamsaka Sutra, India, 2nd-5th century AD), in the West translated by the perspicacious linguist Thomas Cleary, hail!
In chapter 30 The Incalculable (Sanskrit: Asamkhyeyas) a list is constructed of the largest numbers in ancient history, even surpassing the asamkhyeya itself, but experts disagree on the exact size of these number records.

The problem with the primary list of squares is that the two earliest Chinese translators sum their exponents up differently, resulting in somewhat different incalculable numbers. To confuse the issue, Cleary takes his own course, (but our definition of the foremost innumerable number – the asamkhyeya – is most sacred :o)~
Where to begin? The oldest translation by Buddhabhadra (c.400 AD) starts credibly by squaring a base 100000 (Sanskrit: laksha). Cleary, whose translation generally follows Shikshananda's, places his induction base 10^10 one square higher than Buddhabhadra, but Shikshananda starts squaring only after he counts 100 lakshas 10^7 (Sanskrit: koti).
Cleary knows perfectly well how to begin, which he reserves for a similar, later list [pp.1229-1230 FOS] resulting in the much smaller incalculable 10^(7*2^96) asamkhya of the Gandavyuha Sutra chapter.

To make our squaring formula work for Shikshananda, think of his koti as the induction base and consider his list of numbers one item shorter than usual, namely n=103 countables before it reaches the uncountables.
From the choice of the base s0 = 10... [0#a] a general formula for the n-th square sn in the Sutra's list of numbers follows. We ourselves take the initial lakh exponent 5 from Buddhabhadra as our base. Bear in mind that the Indians (although inventors of the decimal system) assemble very lengthy words instead, or use fantasy names.

s0 = 10^a & sn+1 = sn*sn <=>

sn = 10^(a*2^n) ~ 2^2^(n+log(a*3.32)/log(2))
= 10^(5*2^n) ~ 2^2^(n+4.05) [a=5] (Buddhabhadra)

The highest exponent n is the length of the list, and the size of the asamkhyeya depends most on this.
So when to end? Bhikshu Jin Yong (our intermediary source) notes that the list of Buddhabhadra misses a term before it gets to the asamkhyeya. Thomas Cleary considers that Shikshananda, who did his work for the Chinese empress, offers the most complete text. But Cleary places his incalculable one large step ahead of Shikshananda.
Also Cleary and/or his printing devil were a bit wreckless. Their number list contains 10 cumulative miscalculations, 6 copying and 2 printing errors, so that Cleary's final sum at n=103 is barely accurate up to 4 decimals.

Van Novaloka assumes the true asamkhyeya is given by s104 in the above formula with s0=10^5
To compare ours with the other propositions based on scroll and chapter, click here!
For two modern approximations of this incalculable number, click here!

10^(5*2^104) = 10101412048018258352119736256430080
~ 2^2^108.05394894 (Novaloka)

The question why s104 = 2^2^108.05 should lie just out of reach in buddhism is of special interest.
A likely answer is that during the traditional practice of mantra meditation a monk keeps count of his prayers with rounds of 108 beads on his rosary. Because past 108 there is no counting possible – round we go!
Our own asamkhyeya and the modern and preciser estimates have a hidden property. For when the binary power tower exponent 108 has just passed by, the great gate to Indian infinity is officially opened.

Could a mathematician in the 3d century have figured this out, without knowing logarithms? The answer is yes, using the Archimedian power laws and the obvious facts that 10^3 ~ 2^10 and 2^5 = 32 …beautifully.

2^2^108 ~ (2^10)^(1/10*2^108 )
~ (10^3)^(32/10*2^103) <~ 10^(10*2^103) = s104
It's important to express the boundary asamkhyeya in another power (2) than the system (10) one uses to write numbers in, so the boundary is not a precise common number, given that it's so large.
With 2^60 TB the asamkhyeya comes within reach nowadays, perhaps one day humanity will be able to produce that amount of digital information. When bit size is reduced to a single atom, 2766 metric ton of silicon expresses random numbers the size of this asamkhyeya, which fits inside a large house.

§1.6.3. Untold number records

The realm of still nameable infinity in Buddhist mathematics mentions ten names with their squares. Starting from the first uncountable number, the asamkhyeya s104 uncalculable, the Avatamsaka Sutra continues with a series of fourth powers to further name s106 measureless, s108 boundless, s110 incomparable, s112 innumerable, s114 unaccountable, s116 unthinkable, s118 immeasurable, s120 unspeakable, s122 untold and finally s123 square untold. You can use our squaring formula to calculate the exponents of this group of asamkhyeyas, for example the number unspeakable s120 = 10^(5*2^120) ~ 2^2^124.

Then square untold 10^(5*2^123) will be the currently accepted (yet ignored) ancient Indian record. Any number this size – but not all! – can be expressed, on a fine future day, by atom sized bits in a solid iron cube with sides measuring 452 meter.
With the untolds we've arrived at the end of the Sutra's long list of number names. What follows may be called unnameables or unmentionables and then the uncallables and unlistables (all fine Buddhist paradoxes ;-)
On pages 832-834 of Cleary's FOS a similar, earlier list reads impure instead of untold, suggesting that a large enough quantity can turn into a quality. But the passage is philosophical – it declares the atomic and the astronomically large as interchangeable steps on the path of enlightening concentration – any order in terms of size a passing stage in the discrimination of living bodies in this material world.

Taking the group of asamkhyeyas numbers as fuel for an enlightenment rocket that leaves our petty little universe, the Sutra chapter of which we've thus far studied the prose, concludes with a long poem. Highly elevated concepts come into play, first still squares by nature, then recursively expanding… This is no less than an attempt to extend the uncountables to a stairway of exponents a^b^c^.. or power tower.

From now on Big numbers are approximated with binary power towers 2^...b [2^#c 7≤b<2^7] which is our preferred notation for numbers that go unnamed, but are actually described in the Avatamsaka Sutra, we believe.

Let every item in an unspeakable s120 quantity be filled with untold s122 quantities of unspeakables, and allow this substitution to repeat arbitrarily many times. [vs.1]
If untold Buddha lands are reduced to unspeakable atoms in an instant, and in every single atom are untold lands... [1½ reduction, subtotal 10^(45*2^120) lands]
...and this continuous reduction [recursion] goes on moment after moment for untold aeons, then it's hard to tell how many atoms... [vs.2-3]

verses 1 ·· 3 · chapter 30 · FOS

But we can try! Fill in the values that are known and let m be the number of moments in an aeon. The untold length of the continuous reduction will be dominant and dwarf the specification of the reduction step itself. Also if m is less than unspeakable it's hardly significant, as shown in the next calculation.
Following the principle set out in verse 1 above, we may safely assume that an aeon (of which there is an untold quantity) contains an unspeakable number m of instances (atoms of time).

(2^2^2^7)^(s122*m) ~ 2^(2^2^126*m)
~ 2^2^2^126.4 [m=s120] (first new record step)
For our universe m is negligible (as time is running short), but in higher Buddha worlds it may reach the size of the asamkhyeyas. This is based on science and scripture and more than speculation!

Though it's historically unclear what an instant is meant to be, a lower bound can be given by modern physics, as there are 1.855E+42 shortest instants of Planck Time in a second, about 10^47 per day or 10^50 per year.
Define an aeon as the period that life (or consciousness) exists in a universe. Everyone will probably be dead when star formation stops after a hundred trillion years – which sets a maximum of 10^17 days and 10^64 moments for the aeon of our cosmos. Data used in the formula below.

In chapter 31 of the Avatamsaka Sutra called Life Span, every aeon in the field of a Buddha equals a day (and night) in a higher Buddha world. So the number of moments mr in a Buddha's aeon is increasing exponentially against the number of reduction levels r = s122 of Buddha lands or fields.
Our instant land formula calculates the total number fr of level r fields issuing from a higher level r+1 field, where f0 is the field of Shakyamuni Buddha, our stellar universe. The constant c is the number of fields resulting from the reduction of all the atoms in a single Buddha land, which was fixed in the 2nd verse.

c = 10^(25*2^120) ~ 2^2^126 (untold unspeakables)
mr ~ 10^(64+17*r) ~ 2^(212+55*r)
fr = c^mr ~ 2^2^(338+55*r) (instant land formula)
The exact values of the physical coefficients in this instant land formula don't matter much, and the effect of the constant c is negligible. Important is that when, as argued above, the aeon consists of s120 instants, we can use mr to derive the level r ~ 4E35 of the Buddhas talking from the Avatamsaka platform (a well kept secret ;o)~

§1.6.4. Early evidence of tetration

The numbers that follow in the poem (in chapter 30 of the Avatamsaka Sutra) are certainly larger, but exactly how large is uncertain. It's a pity the mathematical concept of recursion was never in the purview of the translators. What makes the interpretation difficult is that the order of sentences in a Sanskrit poem is often the other way round.

Counting the aeons [recursively] by their various [expanding] numbers of atoms...
Count [iterate] this way an unspeakable number t=s120 of times [or worse: aeons].

verse 4 · chapter 30 · FOS

What this verse says is – take the old expression from vs.1-3 as the first step of a new formula, that continues to build power towers but now to arbitrary height. Each step t of this formula expresses a number of atoms, which will be fed back the next step t+1 into the coefficient for the number of aeons, each time putting an exponent on top. Of course it will cost you an aeon to take one such step, which is why we chose the word times (to stay safe).
Finally, after counting an unspeakable number s120 of recursive steps t the true tetrational record is set. From this new height the value of m nor the other coefficients so precisely defined in the past carry weight anymore.

step 2: (2^2^2^7)^2^2^2^2^7 ~ 2^2^2^2^2^7
step t: 2^(2^2^7*2^...^7) [2#t+2] ~ 2^...^7 [2#t+3]
~ 2^^(t+5) (tetration)
record step s120: 2^^(2^2^124) ~ 2^^2^^5½
Despite their obviously obscure, perhaps even clumsy, literary formulations we claim the Indians already achieved the super-exponential operation of tetration some time around the 3d century.
It is possible to read two separate, consecutive recursions in verse 4 (the 2nd verse on page 892), which would lead to the operation ^^^ of pentation, but we can't support it – the passage is confused and the evidence is too thin.
In any case, what we have here is probably the first description of a super-power in history.

Note that, because the last iteration has the tetrational might to raise a power tower to unspeakable height, it doesn't matter if an extra exponent was added to the stairway in the beginning – not even if we could run up these stairs two steps at a time. This settles the question whether to define aeons as in the Avatamsaka chapter 31, or to strictly apply the iteration systems of chapter 30 like we've done here. The answer is that these stations have past and have no impact on the resulting number, which is thus established.

The new ancient Indian world record number, with hindsight more suitably defined.
We hope the unspeakable tetration 10^^(10^(5*2^120)) <~ 4^^^3
from the Avatamsaka Sutra will find its place in some future History of Mathematics.

While these machinations remained hidden behind the veil for more than a millennium, a more fundamental type of infinity planted over from Greek philosophy to the Christian world. Strange, but the creators of the Avatamsaka Sutra (export product ® made in India) imagined they were mapping the uncountable all the way.

Even a being as superior as Universally Good, who holds more than 2^^(2^2^2^7) virtues, will return to a point the size of the tip of a hair, and occupy it in unspeakably many forms...
The same applies to all points in the cosmos.

verses 4 · 5 · chapter 30 · FOS
(10/01/2010, 10:46 PM)Giga Gerard Wrote: The ancient Indian record number of tetration with an "unspeakable" super-exponent 10^^(10^(5*2^120)) can be found in the Avatamsaka

Very nice - thank you for this discussion! Although the teaching of the Buddha concerns especially the constructions of the mind, or more precisely, not so much the contents of it but the way how they come into existence. So I'm always much interested in the discussion about "infinity and mind", of which "large numbers and mind", "iteration and mind" and as such also ancient "tetration" may be sub-chapters ... :-)

Gottfried Helms
Gottfried Helms, Kassel

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