“On the shoulders of giant numbers”
 http://www.allergrootste.com/big/book/ch1/ch1_7.html  bigΨ

§1.7.1. 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, 1st-7th 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 that foremost innumerable number – the asamkhyeya – is most sacred :o)~
Where to begin? The oldest translation by Buddhabhadra (c.420 AD) starts credibly by squaring a base 100000 (Sanskrit: laksha). Cleary, whose translation follows Shikshananda's (c.700 AD), 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 what a koti is, but he reserves it for the start of a similar list [pp.1229-1230 FOS] which results in the smaller incalculable asamkhya 10^(7×2^96) of the Gandavyuha or Garland chapter.
To make our squaring formula work for Shikshananda, think of his koti as the induction base and consider his list one item shorter than usual, namely n=103 squares before it reaches the uncountable numbers.
From the choice of the base s₀ = 10... {0#a} a general formula for the n-th square s_n 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 lengthy fantasy names instead.

The highest exponent n is the length of the list, and the size of the asamkhyeya depends most on this.
But 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 Empress Wu, 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 reckless. Their number list contains 10 cumulative miscalculations, 6 copying and 2 printing errors, so that Cleary's last sum at n=103 is barely accurate up to 4 decimal digits.

We have reason to assume the precise asamkhyeya is given in the above formula by s₀=10^5 after 104 steps. But because other constructions are also justifiable it is best to think of our number as an authoritative restoration – like an artwork is restored when the paint gets blurred over time.

To compare ours with other versions of the asamkhyeya click here!
For two modern approximations of the incalculable number, click here!

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 3d century India have figured this out? The answer is yes – counting the Archimedian power laws from index 0 and using the obvious facts that 10^3 ~ 2^10 and 2^5 = 32 …beautifully!
George Joseph quotes the Anuyoga Dwara Sutra and affirms that both the power laws and specific logarithms of base two (ardhacheda) where known in ancient Jaina mathematics, so this may very well have happened.

It's important to hide the absolute boundary power 108 in another base 2 than in the base 10 of the system used for writing numbers. So there can be no crossover of numbers in between, given that the scale is so large.
From all this we conclude that the number asamkhyeya of the Avatamsaka Sutra is now properly restored.

With 2^60 TB the asamkhyeya lies barely out of reach, perhaps one day humanity will produce such an amount of digital information. When bit size is reduced to a single atom, random numbers the size of this asamkhyeya can be expressed in 2766 metric ton of silicon _{<which fits inside a big house>}.

§1.7.2. Untold number records

The construction of Big numbers by squaring in the buddhist Avatamsaka Sutra had its precursor in the 1st? century Anuyoga Dwara Sutra of the Jains. Still both sects seem to try to attain infinity with explicitly finite comparisons. So infinity can be watched walking away incessantly – as expected, after one Big number comes another… Though the ancient Indians may have believed the purpose of these numbers was just to ornament the uncountable.

Now the realm of still nameable infinity in Buddhist mathematics mentions ten names with their squares. Starting from the first uncountable number, the asamkhyeya s₁₀₄ incalculable, 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) ~ 2^2^124.05

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 page 833 in Cleary's FOS a similar list reads impure instead of untold, suggesting that a large enough quantity can turn into a quality. There the atomic and the astronomically large become interchangeable steps on the path to enlightening concentration – any order in terms of size a passing stage in the discrimination of a living world.

Taking the group of asamkhyeyas (uncountable 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 poem is no less than a mystic mathematical formula that extends the asamkhyeyas to a stairway of exponents a^b^c^.. or power tower.

The highest number in the Anuyoga Dwara is described by counting mustard seeds contained by cylinders with recursively increasing radius. The parallel passage in the poem in the Avatamsaka starts iterating over the extremes of atoms in universes (buddha-lands) and instants in ages of universes (Sanskrit: kalpas = aeons).

[Generally every item in] an unspeakable s₁₂₀ [quantity] is filled with untold s₁₂₂ numbers of unspeakables, and when this substitution is repeated for endless ages [to arbitrary depth] not a single unspeakable atom can ever be completely explained. [vs.1]

Now if untold buddha-lands are reduced to unspeakable atoms in an instant, where every atom contains untold lands... [1½ iteration, subtotal s120*s122^2 = 10^(45×2^120) lands]
...and this continuous reduction [recursion] moment to moment goes on for untold aeons, then it's hard to tell the final number of lands or atoms... [vs.2-3]

verses 1 ·· 3 · chapter 30 · FOS

It's hard… but let's give it a try! Fill in the known values and let m be the number of moments in an aeon. The larger than untold length of the continuous reduction will be dominant and dwarf the number operation in the recursion step itself. Also if m is less than unspeakable it's hardly significant, as shown in the calculation below.
Following the general principle set out in verse 1 above, we may assume that an aeon (of which there is an untold number) contains an unspeakable s₁₂₀ amount m of instances (atoms of time).

A power tower of almost 2^2^2^2^7 about equals the last number of atoms expressed in verse 3 (or the cumulative total of lands and atoms for that matter).
Here we approximate Big numbers with binary power towers 2^...b {2^#c 7≤b<2^7} which is the notation we prefer for numbers that go unnamed, but are actually described in the buddhist poem at hand.

In our own universe m is negligible (as time is running short), but in higher Buddha worlds it reaches the size of the asamkhyeyas. This interpretation is based on science! and scripture! but has no relevance for our new record!

It's historically unclear what kind of instant is meant here (Luk notes 60 kshana in a finger-snap at 75 snaps per minute), but a lower bound can be given by modern physics, as there are 1.855E42 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. Scriptural explanation of the data use 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 m_r 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 f_r 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.

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 m_r to derive the level r ~ 4E35 of the Buddhas talking from the Avatamsaka platform (a well kept secret ;o)~

§1.7.3. 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 theory of recursion was never in the purview of the translators. What makes the interpretation difficult is to establish the intended order of reasoning, which is to begin with in Sanskrit poetry often the other way round, and then obscured by Chinese grammar which is all too flexible.
Our focus must be on paragraph 4 of the poem in Thomas Cleary's literal version, reinterpreted. For the latter part of the chapter there is a cornucopia of multiplications, widening in a grand ornamented parable, but then lacking the pure speed to race up a power tower, as we see here.

Iterating this way an unspeakable number t=s₁₂₀ of times [or worse: aeons],
while recursively counting aeons by these atoms [by their expanding number].

verse 4 · chapter 30 · FOS

What this verse says is, take the expression from the previous verses as the first step of a formula that continues to raise powers but now to arbitrary height. Each step t of this formula expresses a number of atoms, which will be fed back the next step t1 into the coefficient for the number of aeons, each time adding an exponent on top.
In reality it will cost you at least 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 height neither the value of m nor the other coefficients so precisely defined in the past carry weight any more.

The new record number is not just speculation, for this second iteration is a minimal interpretation. It is possible that two separate, consecutive recursions can be read in verse 4 (the 2nd verse on page 892), which would lead to even higher operations, but we can't support it – the translation is inevitably confused and the evidence is too thin.
So what we have here is probably the first description of a superpower in history. Despite their obviously obscure, perhaps even clumsy, literary formulations we claim that the Indians achieved the superpower operation of tetration already… well before Celtic poets sang of CuChulinn _{<and the Ulster Bull>}.

The first iteration grinds worlds to atoms during a number of ages, counted in the second iteration by the atoms left over from past (minor) ages. This combination, of minor and major kalpas, has the tetrational might to raise a power tower to unspeakable height after stepping it up so many (major) times.
It doesn't matter if an extra exponent was added to the stairway in the beginning – not even if we run up these stairs two steps at a time – which settles the question whether to try to define aeons as in the Avatamsaka chapter 31 or to strictly count the unspeakable moments of chapter 30 what we do here. The answer is that these stations have past and have no significant impact on the resulting record number, which is thus established.

What value the exponents of the power tower actually have is not so significant, given that the iterator over the height of the tower is itself a large enough number.
The value of the generic 10 could historically be put at 10 (decimal base) or standardized as 2 (binary), without changing the result significantly. In fact the constant ê of the power whose derivative increases equally fast, would be the natural choice for such an idealized exponent.

While these buddhist machinations remained hidden behind the proverbial oriental veil, an absolute form of infinity was planted over from Neoplatonist philosophy to the Christian world by St. Augustine (354-430 AD).
The great Indian unspeakable tetration is still less than 4^^^3 so in the context of true infinity it doesn't stand tall. Strange, but the creators of the Avatamsaka Sutra thought they were mapping the uncountable all the way!

Enlightening beings called Universally Good, each praised for having 2^^2227 virtues or more, will return to a point as small as the tip of a fine hair, and occupy it in unspeakable numbers...
The same is true of all points in the universe.

verses 4 · 5 · chapter 30 · FOS

(^_^) The Buddha way appears mixed up, modular, with little large loopholes for buttonholes in the end (^o^).