“On the shoulders of giant numbers”
http://www.allergrootste.com/big/book/ch1/ch1_7.html
bigΨ
Ψ.1. Natural repetition
chapter 1.7, edit 0.2.9
published: 2010-11-08
updated: 2011-12-30
# 1.7. A new ancient world record number
Then again, on a single point
Buddhas may dwell for untold aeons
And as on one point, so on all points
For the same number of aeons...
– Avatamsaka Sutra 30.
§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
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 lengthy fantasy names instead.
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}
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!
10^(5×2^103) =
1050706024009129176059868128215040
~ 2^2^107.05
(Buddhabhadra skipped s100 term)
10^(7×2^103) =
1070988433612780846483815379501056
~ 2^2^107.54
(Shikshananda past a laksha)
10^(9.6×2^103) ~
1097689818800677646398555859804252.
= 2^2^108
(Asamkhyeya's absoluteya)
10^(4.9×2^104) ~
10100000000000000000000000000000000 = 1E1E32
~ 2^2^108.03
(Père Ubu's googolkhyeya)
10^(5×2^104) =
10101412048018258352119736256430080
~ 2^2^108.05394894
(Novaloka's restoration)
10^(10×2^104) =
10202824096036516704239472512860160
~ 2^2^109.05
(Cleary with errors corrected)
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.
= (10^3)^(1/10×2^(5+103)) ~ (2^10)^(1/10×2^108)
<~ 2^2^108
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
square untold 10^(5×2^123)
will be the currently accepted ancient Indian record.By comparison, any number this size (but just one!) 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 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).
= (10^(25×2^120))^^2 (first new record step)
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 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.
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)~
§1.7.3. 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.
step t: 2^(2^2^7*2^...^7) {2#t2} ~ 2^...^7 {2#t3}
~ 2^^(t+5½) (tetration)
record step s120: 2^^(2^2^124.) ~ 2^^2^^5½
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.
The new ancient world record number
is defined exactly and traditionally
as the great Indian
unspeakable tetration
10^^(10^(5×2^120))
of the Avatamsaka Sutra
thus far unknown in the history of mathematics.
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^).
Everybody's talkin' at me,
I don't hear a word they're sayin',
only the echoes of my mind.
People stop and stare,
I can't see the faces,
only the shadows of their eyes.
I'm goin' where the sun keep shinin'
through the pouring rain,
goin' where the weather suits my clothes.
Banking off on the northest wind,
sailing on a summer breeze,
skippin’ over the ocean like a stone.
And I won't let you leave my love behind.
– Everybody's Talkin'
Fred Neil
in the film:
Midnight Cowboy