The Attic Numerals

1.2.6?

The Attic Numerals

Introduction

Culturally we have inherited much of the rich mythology and history of the greeks. From the myth of narcissus, the beautiful youth who fell in love with his own reflection and drowned in an effort to embrace it, we get the word narcissist. From the Pyrrhic War in which King Pyrrhus of Epirus suffered heavy losses to defeat the romans, we get the phrase pyrrhic victory, for a victory that inflicts such a heavy toll on the victor that it is tantamount to defeat. From the myth of King Sisyphus who was punished by having to repeatedly roll a heavy stone up a hill only to have it roll down again, we have the phrase sisyphean task, for any laborious and futile task. This last one could be said to have a special significance here as this very book could be called a sisyphean task, as well as the unending quest of trying to come up with larger and larger numbers...

The word philosophy is itself of greek origin. It is composed of two parts, philo, meaning love, and sophia, meaning wisdom. So philosophy is literally the "love of wisdom", and philosophers are "lovers of wisdom". From Socrates we get the socratic method, which asks us to question everything in the hopes of getting to our most central and basic unquestioned assumptions. From Plato we get platonism, the philosophy that gives ideas primacy over form. Mathematics is often thought in platonic terms as an abstract realm which we try to express and manipulate through mathematical symbolism, but whose symbols represent a deeper mathematical reality. From Antisthenes we get the ancient greek philosophy of cynicism, which inherently mistrusts both human motives and conventional pursuits of happiness. From Aristotle we get such seemingly theological notions as the prime mover, and the uncaused cause. These philosophical notions were borrowed by medieval theologians and had a powerful impact on judeo-christian understandings of the nature of God.

From the greeks we get our natural mistrust of the governing as susceptible to collusion by the rich and powerful. The term oligarchy was originally used by the greeks to describe rule by a small number of elites and was considered an improvement of a monarchy, rule by a single person. However the term eventually gained a negative connotation for them as well as a monopoly of power held and maintained by rich families. To correct for this the greeks of Athens championed the ideals of democracy, in which policies were decided by a vote from the populace. They had elected officials with limited terms of office. From the greeks of Athens we also get the notion of being judged by a jury of our peers. This is a system of justice which is admittedly flawed, and it lead to the death sentence for Socrates for purportedly corrupting the youth, and rejection of the existence of the greek gods, though his vilification is most likely due to him being a nuisance to people in power for which the status quo being questioned was a problem, particularly religious, moral, and political authorities.

From the greeks we get some of the earliest examples of the rejection of theistic explanations for naturalistic ones. Basic skepticism for confirmation bias was recognized, and superstitions questioned. The greeks gave us some of the earliest hypotheses about the material world, such as it being composed of various fundamental elements: fire, earth, water, and air. Democritus gave us the earliest example of atomic theory, the idea that there are indivisible particles of matter which can not be split. Eratosthenes was the first to make observational measurements for the circumference of the earth, based on the length of shadows at two distant locations and the assumption of a spherical earth. Aristarchus proposed a model of the universe in which the sun was the center of the cosmos, not the earth. Many astrological concepts, such as the names of constellations comes to us from the greek astronomers nomenclature.

In mathematics the greeks gave us the earliest examples of mathematical proofs and axiomatic systems, exemplified most prominently by 'The Elements', a thirteen part treatise on geometry written by Euclid. The so called "Pythagorean Theorem", that the sum of the square of the legs of a right triangle is equal to the square of the hypotenuse, was known to the egyptians, and they were familiar with the simplest of pythagorean triples, the 3-4-5 triangle, which they used in their surveying to create right angles. Pythagoras however is credited with the earliest mathematical proof of this property. Pythagoras is also credited with coining the word mathematics, which means "that which is learned". The Pythagorean's, followers of Pythagoras, believed that all numbers were rational, were the first to discover the irrationality of the square root of two, when the length of the hypotenuse of a right triangle where both legs equal one was considered. From the greeks we also get the earliest proof of the infinitude of prime numbers, as well as a proof that there exists arbitrary stretches of integers with no prime numbers among them. Archimedes anticipated the calculus with his method of exhaustion, and derived many geometric relations and formulas such as that the volume of a sphere is 2/3 that of it's circumscribing cylinder. Of particular importance to googologist's is Archimedes writing of the sand reckoner, which features a system of naming extremely large numbers, just for the sake of taking number systems to their logical extreme and showing how they vastly exceed anything we physically observe in the natural world. This is considered to be the earliest example of a work of a purely googological nature.

Suffice it to say that the influences and contributions of the greeks are far too numerous to exhaust here. It turns out that they also made important contributions to number notation, that would eventually influence other civilizations. They have one of the earliest examples of a rank-and-file system, which went on to influence other rank-and-file systems as we'll see. For this reason alone, the greek numeration system is of interest to us as part of our unfolding story of continuing numeric progression through the use of better and better numeration systems. But even more importantly, in a later article I'll show how the classic greek system was eventually expanded to heights not seen before!

We will begin with the earliest systems of greek numeration and writing and then discuss the eventual development of their later rank-and-file system.

Phonetics, Alphabets and Abjads

Nestled in the heart of the Mediterranean Sea, resting just above the center line of the continent of Africa, lies Greece, the birth place of so called western civilization. Although seemingly small as seen on a world map, Greece has an area exceeding 50,000 square miles. Some 2000 Islands sprinkle it's native Aegean Sea though only a little over a hundred are actually inhabited.

For an age lasting a thousand years before the birth of Christianity, a civilization flourished here that would form the basis of the cultural, philosophical, political, scientific, and mathematical heritage that still exists today in the United States and much of Europe.

The Phoenician Alphabet, circa 1050 B.C.E , is the oldest verified alphabet. Forms of written language existed before the phoenicians, but in these the units of the language were symbols represented whole concepts, what we would recognize as words. As such, to learn any language required the memorization of thousands upon thousands of symbols and their associated spoken counterpart, with no clue as to how a particular symbol was to be spoken. With the phoenicians we get the first phonetic written langauge. That is to say, the symbols here represent spoken sounds, and have no inherent meaning in isolation. However by combining groups of sounds one could form whole words with meaning. You may notice that the Phoenician alphabet contains no vowels, only consonants. The vowel sounds had to be supplied based on the familiarity of how words were meant to be spoken, but such vowels were not notated in written form in any way. An alphabet without vowels is called an Abjad. Despite this deficiency the idea of a phonetic system would go on to have a great influence on my proceeding civilizations and languages.

The greeks inherited and adopted the phoenician alphabet for their own language with one important innovation: the inclusion of vowels. Thus the greeks system of writing is considered to be the earliest example of a true alphabet: a phonetic system which includes letters for consonant and vowel sounds. The table below shows the standard greek letters, in upper and lower case form, in their conventional order along with their original names and their english equivalents:

You may notice the similarity of many of these to our own english alphabet. This is no coincidence. The greeks had a strong influence on their roman conquerors, and when rome itself fell it's own influence could be felt through the spread of the so called romance languages of which english is but one example. You may also notice that these symbols are common in mathematics. This is no doubt a tribute to the greek mathematicians. Of particular interest to us here is that many of these symbols were used in the common ordinal notations. We will learn about this much much later. Because we will be revisiting these symbols over and over again later on, it may be a good idea to take a good look at the above list and memorize them. We will need them later.

For now our main interest in this alphabet is in how it was used to form the earliest greek system of numeration: the attic numerals.

The Attic Numerals

The Attic Numerals were used by the ancient greeks as early as the 7th century B.C.E. They are a Denominational System, the same type as the Egyptian Numerals we saw earlier. As you will see however they have a few important innovations that make them slightly more useful and easy to work with. It is believed that the Egyptian Numerals did have some influence on the Attic Numerals, as the greeks took much of the egyptian learning into their own country. It formed the basis of their mathematics, though they later made important contributions and improvements over it.

We haven't discussed much about the names of numbers up until this point, but the names of the greek numbers played an important part in their number notation here, so we make note of it here. For the singleton, the Attic system used the single vertical stroke:

Ι

So far nothing surprising, this is a typical choice for the singleton, or unit. This has nothing to do with their word for the singleton, ena , or in their native greek, ένα. For the next denomination, representing the number of fingers on a single hand, the greeks used the word pente, or πέντε in greek. From this word we get the english word pentagon. The greeks took the capital of the first letter of this word, pi, and used this as their next denomination:

Π

We can define this symbol explicitly as equal to the same as this many strokes: Ι Ι Ι Ι Ι.

For the next denomination we can look at the greek word for the number of all fingers on both hands, deka or δέκα. The capital of δ is Δ. So the denomination symbol for δέκα is predictably:

Δ

Which is the same as Π Π

Next the original greek word for having a δέκα of δέκα's, was hekaton or ηεκατον. The Capital of η is Η, and so we have:

Η

For

Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ

Next up is the greek name for a δέκα of ηεκατον's, which is chilia or χιλιά. Again the capital of χ is Χ, and so we have:

Χ

for

Η Η Η Η Η Η Η Η Η Η

The greeks had one more denomination with a special unique name. Instead of δέκα χιλιά for a deka of chilia's as we might do, this too had a special designation of a myrion or μυριον. The capital of μ is Μ, and so we have:

Μ

for

Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ

So in summary we have the denominations: Ι Π Δ Η Χ Μ

A nice upshot of this system is that if you forget any of the denominations you can just remember the first letter in the name of that denomination. For this reason the attic numerals may be described as acrophonic. This means that the symbols for the denominations are also the first letter in their names. For this reason the Attic Numerals are also sometimes called the Acrophonic Numerals. For memorization purposes this alone is an improvement over the egyptian denomination system.

The table below summarizes the results so far:

Just like the Egyptian Numerals, the Attic Numerals worked on the additive principle that numeric symbols next to each other were to be understood as their sum.

This however is not the system in total. ΙΙΙΙ more symbols were derived from deka, hekaton, chilia, and myrion. It begins with the pi symbol for pente. The second pillar is shortened and a small denomination is then written just underneath it's top line. The result is the product of pente and that denomination symbol. So For example:

With the small deka sign in the middle would mean Π times Δ. That is to say it meant pente deka's or Δ Δ Δ Δ Δ, analogous to how Π means Ι Ι Ι Ι Ι. Nifty solution. ΙΙΙ further such "compound" symbols were used. These were:

For Η Η Η Η Η

For Χ Χ Χ Χ Χ.

and

For Μ Μ Μ Μ Μ.

These were all the official denominations used by the ancient greeks. With the other symbols this makes for a total of Δ denominations used, more than the egyptians which used only Π Ι Ι, which is Ι Ι less.

Denominations were ordered from greatest to least from left to right, using the same additive principle as the egyptians. One important advantage of this system which was unlike the egyptian system which could have a maximum of Ι Ι Ι Ι Ι Ι Ι Ι Ι copies of the same denomination, the Attic numerals never require more than Ι Ι Ι Ι for any canonical representation. That's a considerable improvement. It's worth noting that Ι Ι Ι Ι is also the limit of our in-born number-sense, so this system lends itself to easy reading with no "counting" required.

With these symbols, the additive principle, and the ordering principle that says symbols must be ordered from greatest to least, it is still possible to express some numbers in multiple ways. For example we may express ΔΠ Ι Ι as Π Π Π Ι Ι or Δ Ι Ι Ι Ι Ι Ι Ι or Π Π Ι Ι Ι Ι Ι Ι Ι etc. In order to force a unique canonical form the implicit rule is that you always use the least number of symbols possible. This can always be accomplished by taking from the total the largest denomination possible at every stage. So for example if we have:

Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι

We can count to show that it is more than Δ but less than

, so the first symbol must be Δ. Next we deduct this many units from the total and repeat the process:

Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι Ι

ΔΙ Ι Ι Ι Ι Ι Ι

For the remaining units we can find that it is more than Π but less than Δ. So we next use a Π and replace the appropriate number of units.

ΔΙ Ι Ι Ι Ι Ι Ι

Δ Π Ι Ι

Since the remaining units are less than any of the denominations past units, we leave them alone. This unambiguously tells us that Δ Π Ι Ι is the canonical form for this number. We can call this the efficiency principle. So with these Δ symbols along with the I I I principles, the additive principle, the ordering principle, and the efficiency principle, we have a unique way to name some of the numbers.

To get a good feel for this works in practice I have listed out the first hekaton attic numerals:

Ι

Ι Ι

Ι Ι Ι

Ι Ι Ι Ι

Π

Π Ι

Π Ι Ι

Π Ι Ι Ι

Π Ι Ι Ι Ι

Δ

ΔΙ

ΔΙ Ι

ΔΙ Ι Ι

ΔΙ Ι Ι Ι

ΔΠ

ΔΠ Ι

ΔΠ Ι Ι

ΔΠ Ι Ι Ι

ΔΠ Ι Ι Ι Ι

ΔΔ

ΔΔΙ

ΔΔΙ Ι

ΔΔΙ Ι Ι

ΔΔΙ Ι Ι Ι

ΔΔΠ

ΔΔΠ Ι

ΔΔΠ Ι Ι

ΔΔΠ Ι Ι Ι

ΔΔΠ Ι Ι Ι Ι

ΔΔΔ

ΔΔΔΙ

ΔΔΔΙ Ι

ΔΔΔΙ Ι Ι

ΔΔΔΙ Ι Ι Ι

ΔΔΔΠ

ΔΔΔΠ Ι

ΔΔΔΠ Ι Ι

ΔΔΔΠ Ι Ι Ι

ΔΔΔΠ Ι Ι Ι Ι

ΔΔΔΔ

ΔΔΔΔΙ

ΔΔΔΔΙ Ι

ΔΔΔΔΙ Ι Ι

ΔΔΔΔΙ Ι Ι Ι

ΔΔΔΔΠ

ΔΔΔΔΠ Ι

ΔΔΔΔΠ Ι Ι

ΔΔΔΔΠ Ι Ι Ι

ΔΔΔΔΠ Ι Ι Ι Ι

Ι

Ι Ι

Ι Ι Ι

Ι Ι Ι Ι

Π

Π Ι

Π Ι Ι

Π Ι Ι Ι

Π Ι Ι Ι Ι

Δ

ΔΙ

ΔΙ Ι

ΔΙ Ι Ι

ΔΙ Ι Ι Ι

ΔΠ

ΔΠ Ι

ΔΠ Ι Ι

ΔΠ Ι Ι Ι

ΔΠ Ι Ι Ι Ι

ΔΔ

ΔΔΙ

ΔΔΙ Ι

ΔΔΙ Ι Ι

ΔΔΙ Ι Ι Ι

ΔΔΠ

ΔΔΠ Ι

ΔΔΠ Ι Ι

ΔΔΠ Ι Ι Ι

ΔΔΠ Ι Ι Ι Ι

ΔΔΔ

ΔΔΔΙ

ΔΔΔΙ Ι

ΔΔΔΙ Ι Ι

ΔΔΔΙ Ι Ι Ι

ΔΔΔΠ

ΔΔΔΠ Ι

ΔΔΔΠ Ι Ι

ΔΔΔΠ Ι Ι Ι

ΔΔΔΠ Ι Ι Ι Ι

ΔΔΔΔ

ΔΔΔΔΙ

ΔΔΔΔΙ Ι

ΔΔΔΔΙ Ι Ι

ΔΔΔΔΙ Ι Ι Ι

ΔΔΔΔΠ

ΔΔΔΔΠ Ι

ΔΔΔΔΠ Ι Ι

ΔΔΔΔΠ Ι Ι Ι

ΔΔΔΔΠ Ι Ι Ι Ι

Η

The Attic Numerals appear to be aesthetically pleasing, simple, and easy to read. By limiting the notation to no more than I I I I copies of any denomination it is much more conducive to almost instant recognition as our natural in-built number sense can handle repetitions up to I I I I. This gives them a strong advantage over the egyptian numerals where having up to I I I I I I I I I of the same denomination forcing one to actually count out the symbols individually to fully "read" the number. Lastly the denominations themselves are streamlined and simplified so that they are easier on the eyes and easier to identify quickly than the more elaborate and pictorial egyptian numerals.

We can express any arbitrary number with ease by simply breaking it down into it's appropriate groups. So for example we can write the number of dots:

..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .....

..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .....

..... ....

ΗΗΔΔΔΠ Ι Ι Ι Ι

This numeral itself uses ΔΙ Ι symbols. Certainly a vast improvement over having to write out as many tally marks as there are dots. The number of dots itself seems to be a modestly large number, but still quite terrestrial. Let us now consider the limits of our nifty little attic numerals, and see if they have aided us in the furtherance of our quest for larger and larger numbers. Perhaps more importantly, do they even manage to be an improvement over the egyptian numerals in terms of range? Let's find out...

Attic Numerals as a "Large Number Notation"

Fundamentally the Attic numerals have the same draw back as the egyptian numerals, and all denominational systems for that matter: they have a pre-built in limitation because there is only a small number of denominations and necessarily a largest one. We can show a direct comparison between the denominations of the egyptians and the attic numerals. We have that:

Ι is equivalent to

Δ is equivalent to

Η is equivalent to

Χ is equivalent to

and lastly...

Μ is equivalent to

The myrion is the last denomination for the attic numerals but the egyptian numerals have two additional denominations beyond this. So with this consideration alone we can see that the Attic system is actually more modest than the egyptian system, despite the fact that it has more denominations and seems to be an improvement in terms of compactification.

But just how sizable are these denomination's in physical reality? Perhaps a myrion is still a pretty big number. Well it is ... just try counting to it, but it is nothing really out of this world.

Δ can be represented directly by a mere: ..... .....

Η is the more sizable:

..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .....

Χ is the appreciable:

..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .....

Not bad,...

Μ is the somewhat large:

..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... .....

It's certainly a vast improvement to contain all this in a single nifty symbol of Μ but it's certainly not the best we can do. As we saw earlier our sheer imagination can soar much much higher, even if only vaguely. But before we give up on the attic numerals, let's remember that Μ is NOT the largest number it can canonically represent. Surely we can have ΜΙ and ΜΙΙ and ΜΙΙΙ and so on ...

But just how big is "and so on..." in this case. In mathematical circles "and so on..." is just a shorthand of saying the continuation is trivial and not worth considering further. In essence ... if we needed to continue we could indefinitely without any real unforeseen difficulties and so it's not worth any real effort. Googology teaches us however that such a notion is naive at best. It teaches us never to take "and so on..." for granted. So again, just how big is this particular "and so on...".

Well in this case we see that we can follow the forms without breaking any convention until we fill up all the lower denominations and thus we could eventually reach

Μ

ΧΧΧΧΗΗΗΗΔΔΔΔΠ Ι Ι Ι Ι

That expression might feel like a "big" number but remember that the size of a representation of a number is not directly proportional to the number itself. In fact this is just slightly less than I I times Μ. In fact in short order we can see that the next canonical numeral would be ΜΜ without violating any of the notations principles. In fact we could continue to ΜΜΜ and even ΜΜΜΜ. Finally we reach:

ΜΜΜΜ

ΧΧΧΧΗΗΗΗΔΔΔΔΠ Ι Ι Ι Ι

What's next? Well we have exhausted all the denominations except for the last:

.This is, of course, the next canonical numeral. We can then continue logically with:

Ι

Ι Ι

Ι Ι Ι

etc.

Well that's all well and good, but we have no intention of actually counting. What we want is to see how much further this will eventually get us. Eventually we reach

Μ, and then ΜΜ and then ΜΜΜ and then ΜΜΜΜ. The best we can do at this point is fill out all the denominations with the maximum number of allowed copies to obtain:

ΜΜΜΜΧΧΧΧΗΗΗΗΔΔΔΔΠ Ι Ι Ι Ι

This is the largest canonical numeral possible in the Attic System. It is just a unit less than the egyptian "falcon" or "frog". This means that actually the Attic system is less expressive as the egyptian system, this despite the fact that it has III more symbols than the egyptian system. So why is it that we have achieved less with apparently more? This is because the Attic system uses intermediary denominations whereas the egyptian system does not. So what the Attic system gains in conciseness it loses in range. We need roughly II the number of symbols in the Attic system to cover the same range in the egyptian system.

But why is this the largest number we can legally express in the Attic System? Well firstly, we have already exhausted all Δ official denomination of the Attic System. There simply are no new symbols after this. Why do we need a new symbol? It is implied by the structure of the system itself. The Attic system can be thought of as a mixed radix system. The egyptian system by contrast is a fixed radix system. This means that every new denomination is always the same multiple larger than the previous denomination. But in the Attic system the sequence of denominators alternate from being Π times larger than the previous denominator and I I times larger than the previous denominator. This means the next denomination has to be I I times larger than

. That is to say, we need a new denomination for

. But it doesn't exist. Why can't we just use ? Because this breaks the efficiency principle which prevents us from having I I of the symbols Π ,

, , , or , since they can all be described more efficiently by I copy of the next symbol. The problem for is that there is no next symbol to use, and thus we can't move on without violating some previously established principle or inventing something new to continue!

Why am I going into detail on this point? Because I want to use it to illustrate an important feature of large number notations that is going to come up again and again. The point is that a system will always have either an explicit or implicit limit in what it can express; a theoretical or practical limit to what it can express. Once we have hit such a limit we can't continue ... without inventing something new that is actually outside the system. You may think, well fine just add something outside the system and there you go ... no real impasse. But this addition will just lead to a larger system which itself has a limit, requiring you to add something new again, which itself is not a simple consequence of the previous system, and so there is no way to travel unimpeded towards infinity. All we can do is try to make each new system as efficient as possible so that it can reach as far as possible, so that we don't have to keep making as many ad hoc extensions along the way.

So how might we proceed past this first impasse in the attic numerals?

Well there is more than one way but no unique or predetermined path. Whenever this happens it is the sign that you have reached the limit of your current system. However all paths forward are not created equal. While we can't continue without inventing something new or violating something old, we can at very least chose a continuation which preserves at least some of the previous principles.

The first thing we might consider, is inventing a new denomination for

. Continuing with the acrophonic principle, we might be tempted to say that the next symbol should be the capital of the first letter for the greek name of this number. The problem is, that this number does not have a special name is greek, but is instead given the compound name deka myrion, essentially meaning Δ Μ's. So we don't have a unique name, nor a unique letter at the start of this numbers official "name". We could just pick any of the greek letters not yet used, but such a choice would by necessity be arbitrary. So to continue with the denomination symbols we would have to violate the acrophonic principle. Oh well :/

What's another option? Well we can just say there are no more symbols, but that the efficiency and additive principles are still true. So when we want to have a numeral for deka myrion, we can just say it's

which employs the additive principle and is the most efficient expression possible assuming is our largest denomination. With that decided we do indeed have a canonical form for every number.

So there, we now have an unimpeded path towards infinity!

Not so fast!

Here it is worth bringing out another important point about large number notations. If we stay within the strictures of the original system there is indeed a largest canonical numeral. This system has an explicit and theoretical glass ceiling we can't escape. This is what we can call a closed form system. But it is also possible to create an open formed system that gives us a canonical representation for every member of a strictly infinite collection, but ... there is a catch. The catch is that, while in theory every member can be expressed, in practice only a finite number will be able to be expressed. Such systems have an implicit practical limit, determined by the longest expressions we deem feasible to actually write out.

First an foremost we can continue our extended attic numerals all the way up to

ΜΜΜΜΧΧΧΧΗΗΗΗΔΔΔΔΠ Ι Ι Ι Ι

Without any appreciable difficultly, but the next numeral would be

, which contains more than I I I I of a single symbol, violating an earlier established principle. More importantly for any number equal to or greater than this, we can no longer easily "read" it because it will exceed our innate number-sense capabilities. We would be forced to count the number of occurrences of the

symbol. So the limit of our extended system is ultimately a tally system that is just a scalar multiple of a simple unit tally system, like we saw at the beginning of this chapter. This means the same limits that we discussed for writing out and reading tally marks applies here, except that each mark now represents a large multiple instead. But as far as infinity is concerned this system is no better than a tally system. Eventually we have so many

's that we can neither read nor even write them all out! Even if an ancient greek scribe were to write

's for their entire lives, filing up book after book with the symbol, they would likely pass away long before reaching even the grains of sand in the sea, which is but a small number compare to the stars in the heavens, or even the atoms in a single drop of water!

Hmm. So now it seems quite reasonable that those of antiquity said that the grains of sand in the sea, are without number. That is to say, if by without number you mean without a numeral small enough for you to actually express. But as Archimedes showed in the sand reckoner, a work we will visit a little later in this chapter, this assumption is dead wrong. In fact, it takes no great cleverness to imagine much bigger numbers than our hapless scribe, and at an infinitesimal fraction of the effort. What if, instead of actually writing out all those

's we simply imagined writing them out. It would not have taken a clever greek to imagine the magnitude expressed by having

copies of . This might have seemed mind bogglingly big to the average person at the time. This does work out to be a pretty big number by ordinary standards. It's about on the scale of the number of humans on the earth today, but it would still be dwarfed by the number of fish on the earth, or insects. If our common day dreamer wanted yet a bigger number it might occur to him to have instead

copies of . But after a little contemplation perhaps they would quickly realize that this is merely twice the size of the previous number, which isn't very impressive. And then we have the 'eureka!' moment that really marks the beginning of googology. We now have this idea of 'some number' copies of

. So what we want to do is replace 'some number' with the largest number we can come up with so far, which brings to mind

copies of . And so now our common daydreamer becomes very excited at the prospect of having '

copies of ' copies of . This now is a suitably monsterous number, but our intrepid daydreamer nearly passes out when we considers now the continuation of the sequence in which every new member has as many

's as the value of the old number! If they can count the members of this sequence, they can reach truly transcendent numbers that exceed as the atoms in the observable universe and beyond!

It is just such crazy heights that Archimedes himself reaches, envisioning a similar system as described here, but with far better clarity. But for now let us return to the earth once again. We have a few other things to discuss before we get to Archimedes. Rest assured that we will soon return to such heights, but with better clarity, enabling us to think even further still!

Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω Ϊ Ϋ ά έ ή ί ΰ α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ ς σ τ υ φ χ ψ ω U+10140