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Translation by AB – January 28, 2023
As soon as the child perceives the inexhaustible dimensions of language, he seizes it and plays with it. Faced with the unknown, he instinctively creates new words. On the other hand, the adult does not know any more the brightness of the discoveries, which tarnished by automatic subjection of his representations to a lexicon become a second nature (like these monster-AIs, adults in essence). There are, however, some “professional” exceptions. Some authors know how to maintain their propensity for language games. Some scientists, philosophers or mathematicians still discover territories that cannot be explored or conquered with the words available. So, if very often ancient Greek or dead languages are used as a reservoir, some feel the need to act “as a child” again.
This was the case of the American mathematician Edward Kasner. In his famous book “Mathematics and the Imagination”, co-written with James Newman and published in 1940, Kasner imagined a number big enough to, he thought, baffle the mind and give a vague idea of infinity. The story of the name of this number, as told by James Newman, begins as follows1:
The name “googol” was invented by a child (Dr. Kasner’s nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name.
This number is written 10100 in modern notation. It is not strictly speaking a discovery, like a new particle in physics, but this number belongs to a numerical territory located well beyond the “current affairs” in the middle of the 20th century. This territory remains in principle accessible and the numbers that occupy it must therefore be able to be named, which did not escape the instinct of Kasner’s nephew.
But googol is today, so to speak, “smaller” than it was back then. In 2023, it does not baffle our minds so much because we sometimes meet it. For example, the estimated mass of the universe is about 1080 to 1090 times that of the electron. These measures are gigantic but, since they come from our representations of the world, we must acknowledge their existence. The evaporation time of a supermassive “black hole” (still a childish name) is of the order of 10100 years. It would take 10128 neutrons to fill the universe, etc. etc. Very real numbers of the order of googol thus appear on the horizon.
However, we know how to write much larger numbers, such as 1010000, but they never occur in our representations of the world. Googol thus seems to determine an order of magnitude which separates two kinds of numbers: those which “account for” the world and those which, well beyond googol, belong only to mathematicians and to their language games (which are also, sometimes, “child’s games”2).
This comments have given rise to two separate articles. This one, subtitled “Numbers and Progress” looks after numbers smaller than googol. We will see that they are spreading rapidly, escorting what we call “progress”. The second article, Numbers and Mathematic, is devoted to numbers much larger than googol. Although they can, to a certain extent, be named or written like the others, they belong to a fundamentally different species. We will try to better describe this curious phenomenon of “phase shift”.
However, there will remain a mystery that we will leave unexplored: why does an “edge” appear around googol, 10100, rather than around any other order of magnitude?
Very small numbers
The perception of the number of objects, or “numerosity”, especially in humans not educated in modern mathematics (children, indigenous or prehistoric peoples…), has long been studied by psychologists, ethnologists and neuroscientists. However, these works produce little certainty about the mechanisms of estimation and manipulation of numbers. But at least one skill is generally confirmed: “subitizing”. It is the rapid, accurate and confident judgment of the number of elements in a scene. It is limited to a very small number3 :
Judgments made for displays composed of around one to four items are rapid, accurate and confident. However, once there are more than four items to count, judgments are made with decreasing accuracy and confidence. In addition, response times rise in a dramatic fashion, with an extra 250–350 ms added for each additional item within the display beyond about four.
Thus, prior to any concept of number, subitizing seems to be a biological phenomenon shared by many species, a “sense” of numerosity. For humans, around 4 and beyond, the evaluation seems on the other hand to call upon different cognitive processes and its precision depends on various tools, whether it is the body itself or artifacts such as language. Around 4 and beyond, it is thus no longer a question of direct, biological perception, but already of technique.
Many ethnographic surveys confirm this phenomenon. Aparecida Vilaça, Professor of Social Anthropology at Brazil’s Museu Nacional, narrates some of these observations about the Wari’ Indians of the Amazonian southwest (3,000 individuals)4:
After “1” and “2”, Wari’ students could only use quantifiers in a random way, except for the word for “little” which most of them translated as “3”. Beyond that, numbers from 4 to 10 were designated by various terms meaning “a lot” to which, to reach higher numbers, they added, for example, the adverb “very” and the qualifier “really” which gave rise to expressions like “really a lot”.
Aparecida Vilaça also recalls that Claude Lévi-Strauss had studied the North American peoples of Eskimo, Athapaskan and Penutian language and noticed that they use distinct terms for the numbers from 1 to 6. But “strange as it may seem, [they] form 7 by derivation of 6 + 2, 8 by derivation of 6 + 3, and 9 by derivation of 6 + 4”. This pre-arithmetic system is quite accurate, although wrong according to our mathematics, and involves only a few very small numbers.
Another well-known survey, conducted in the early 2000s among the Mundurukú-speaking Indians (about 10,000 individuals), shows the same phenomenon of rapid dispersion of terms designating numbers beyond the subitizing range. The Mundurukú language thus has fixed expressions for numbers from 1 to 5, although long and polysemous beyond 2, and this study concludes5:
What the Mundurukú appear to lack, however, is a procedure for fast apprehension of exact numbers beyond 3 or 4. […] It is noteworthy that the Mundurukú have number names up to 5, and yet use them approximately in naming.
For example, of the Mundurukú speakers who were presented with five “things”, less than 30% used the correct term “pūg pōgbi” which literally means “one hand”, with the remainder using “adesū” which means “not much”, or “ebaddipdip”, literally “2+1+1” i.e., 4. Younger children, in particular, consistently ignored the use of “pūg pōgbi”. The quantities “5” and beyond did not seem to serve any purpose to the point of needing to be named precisely.
If the biological mechanisms of subitizing seem to be universal and confirmed by numerous ethnographies, we see that these very small numbers have diverse fates. But in all cases the language, in particular the shaping of the names of numbers, is critical for the arithmetical “progress”:
Around the age of 3, Western children exhibit an abrupt change in number processing as they suddenly realize that each count word refers to a precise quantity. This “’crystallization” of discrete numbers out of an initially approximate continuum of numerical magnitudes does not seem to occur in the Mundurukú.
The Mundurukú speakers do not exceed, as far as the evaluation of the numbers is concerned, the capacities of a “western child” of less than 3 years old. This is obviously not a biological question but a question of technical and linguistic environment: “progress” seems to have spared these speakers.
The tiny range of subitizing is sometimes extended by bodily markers (limbs, fingers…), then followed by a blur where numbers are barely distinguishable, often summing up to “little” and “much”. These forms have no relation, even “genetic”, with the vastness implacably ordered by mathematics. This has not prevented the Wari’ and Mundurukú peoples from elaborating harmonious and complex relationships between themselves and with their environment. If they did not invent the “number” it is because they never needed it, any more than they needed trowels or knitting needles. Indeed, there were only a few thousand of them and, as far as we know, they had nothing. So why count because what is there to count?
A clue comes forth: the “progress” of the evaluation of numerosities (as well as that of language – see Following Julian Jaynes – comeback of the Bicameral Mind?) would be correlative to the growth of human groups that must identify and “share” more and more things. More precisely, counting becomes a necessity when, in everyday life, there are more beings (fellows, sheeps, paths…), simultaneous and de-individualized, no longer considered in their singularity (this fellow who is my brother, this sheep which has a black spot, this path that leads to the river…) but as objects pertaining to prototypes (an individual in general, a head of cattle in general, a path in general).
A necessary and typically human relation to these prototypical beings could be that of “possession”, in the very broad sense of a material and/or symbolic appropriation (let us remember how the children appropriate the new world by having to invent words). Under this hypothesis, “there are more beings” means: there are more beings for such and such a use (“appropriation” designates indeed “the fact of adapting something for a specific use”).
This, perhaps, is why humans must count, and count more and more as they gather and concentrate their power.
Beyond 4, two other cognitive processes emerge: “estimation”, which allows one to approximate the numerosity of a set of arbitrary size, and especially “counting” which6 …
… allows to enumerate with precision any set. It consists in matching, one by one, each of the enumerated objects with a reference list that can be verbal (names of numbers) or non-verbal (fingers, body parts).
The action of “matching one by one” should be seen here as a series of concrete and precise gestures in order to make a “form”, like the mason builds a wall or the knitter a sweater. Counting is a real activity that allows one to rise gradually, with accuracy, above the tiny subitizing range. We can see that this equipped cognitive process is distinct from simple recollection, which does not make any number (“a die has 6 sides” or “the Dalmatians are 101”). Counting always presupposes a situation, an intention, and requires a certain duration. Incidentally, each number produced by counting thus has, in a way, a history.
Counting would thus be an activity of appropriation that produces “possession” represented by the number. Like any human activity, it is repeated, perfected, transmitted, and thus becomes technical.
The counting techniques, of which the first known traces go back to a few tens of thousands of years, were quite diverse. However, the principle always remained to match one by one “by hand” the beings to be counted (fellow creatures, sheep…) with other conventional “beings”, stripped of their own essence and small enough to be easily transported. These were for example the multiple ways of using fingers (dactylonomy7), notches on bones or sticks (the shepherd makes his herd pass in front of him and shifts his fingernail from a notch each time a beast passes in front of him8), pebbles or knotted strings (Inca quipus9) … These multiple counting techniques are attested in all human groups, and even until very recently in illiterate environments. Thus, in France, the Napoleonic Code provides that “The notches correlative to their samples are evidence between persons who are in the habit of so ascertaining the supplies they make or receive in detail”. This article was still in the French Civil Code until 2016.
One can thus count without any concept of number, by simply matching the beings that one appropriates for their use with conventional beings. The latter are not yet numbers, but they are already representations of representations, that is to say material representations (notches, knots, pebbles…) of mental representations (numerosity). This particular phase of the technical evaluation of numbering is quite remarkable. Indeed, if the representation of a “natural thing”, for example a bird, remains attached to the thing, on the other hand the material representation of a mental representation can acquire a proper existence and “travel”. It will take several millennia before it returns to us, crafted by mathematics, in the form of a pure concept.
It could be that writing itself, at least in Mesopotamia, comes from successive technical fine-tunings of the representation of numbers. The tokens or calculi (small stone or clay objects), used to count the beings of interest, are enclosed in clay envelopes on which are engraved sizes corresponding to their number and their “species” (the prototypes of beings that they themselves represent). This double counting will experience a remarkable evolution10:
The tokens would be accounting instruments used for several millennia, which in the Uruk period become more complex in their form as well as in their use, in particular by being integrated into bubbles and printed on them. The fact of printing the tokens made their inclusion in a bubble progressively useless, so they disappeared and were replaced by signs representing them. As for the bubble, it flattened out into a more convenient tablet that nevertheless contained as much information.
There is no more “craft” gesture of matching nor, strictly speaking, technical activity of appropriation, but signs representing the results of hypothetical enumerations, i.e., numbers. It is necessary to insist: the number, that is to say the writing, appears as the result of the technical evolution, not of an object but of an activity. With an approximate formulation, and by reference to the Simondonian lexicon, we could say that the number is a product of the “concretization” of the activity of appropriation by counting (Gilbert Simondon, “philosopher of information”?).
An even more telling example is that of Roman writing, in which numbers are represented more or less by the drawing of notches on the counting sticks (“III” for example). In this Roman case, as in the Mesopotamian case, the matching gestures are progressively replaced by autonomous symbols. We refer here to another exploration of numbers showing the capture of a numerical “monster” by its naming (Liu Hui overcomes a monster), and where we recalled these words of Ernst Cassirer11:
The function of signification attains pure autonomy. The less the linguistic form still aspires to offer a copy, be it direct or indirect, of the world of objects, the less it identifies with the being of this world and the better it accesses its role and its proper meaning.
In this case, the linguistic form of number does not offer a copy of the “world of objects” but of the world of our pure senses. The linguistic form can break away and “travel”, as we suggested above. But now we are getting closer to this edge, which we will only cross in the second part of this “Proof by Googol”.
Here, the number accelerates!
“Progress” in general, as seen elsewhere where we reduced it to a few aphorisms through the prism of photography (The “progress” unveiled by Photography (with Henri Van Lier)), is correlative of techniques that allow us to master increasing orders of magnitude, in particular those of human groups. Pliny the Elder declared in his Natural History (around 77 AD)12:
The ancients had no number whereby to express a larger sum than one hundred thousand; and hence it is that, at the present day, we reckon by multiples of that number, as, for instance, ten times one hundred thousand, and so on.
This was not so bad. Being able to count up to a hundred thousand had apparently become necessary to manage the Empire, but was no longer concretely possible with stones, knots or notches. The domain of things to be “appropriated” had already been considerably extended while remaining, let’s say, terrestrial (herds, land, population…) and thus still falling under artisanal techniques and a social organization around the number (“calculator”, “dispensator”, etc.).
These big Roman numbers are nothing compared to those produced by the fascinating Jain cosmology. Jainism appeared in India around the 6th century BC. Its philosophy consists of a kind of “ethical materialism” which does not consider our world to be bounded by a divine, inaccessible and therefore immeasurable domain. For the Jains (additions and modifications in brackets)13:
Time is thought of as eternal and without form. The world is infinite, it was never created and has always existed. […] This cosmology has strongly influenced Jaina mathematics in many ways and has been a motivating factor in the development of mathematical ideas of the infinite which were not considered again until the time of Cantor [ at the end of the 19th century ]. The Jaina cosmology contained a time period of 2588 years [ or 10177 ]. Note that 2588 is a very large number!
This (almost) limitless world offers itself to any measurement. The human mind, driven by curiosity, does not fail to count everything that the cosmogony allows. A last well-known example of “ultimate” counting work, at the cosmogonic limits, is that of Archimedes in his “Sand Reckoner” (230 BC) where “he set out to determine an upper bound for the number of grains of sand that fit into the universe. In order to do this, he had to estimate the size of the universe according to the contemporary model, and invent a way to talk about extremely large numbers”14. This way of talking about large numbers consists of a sort of tower of successive powers starting from the largest number that had a name at the time: the “myriad”, or 10,000. Archimedes then tries to commensurate a very large thing (the universe, which he estimates to be finite and smaller than 1014 stadia, or 2 light-years) with a very small thing. He thus obtains an estimate of 1063 grains of sand to fill the universe. This number is in some way the largest number that the Greek cosmogony “allows”.
In any case, the cosmogony leads to this edge marked by googol.
These examples allow us to see three successive orders among the accessible, pre-mathematical numbers.
First, even to this day, indigenous groups of a few thousand people do not really need numbers. Their environment is limited and they are part of it: there is almost nothing to appropriate, neither physically nor conceptually. To evoke the first few numbers that these groups sometimes handle, we can speak of “Numbers for Body”, a few units, no more than tens.
Then, the Roman example shows us that it is not necessary to have large numbers nor a very efficient arithmetic system to build an empire. The Romans were satisfied with what we will call the “Numbers for Technique”, those that allowed them to control the affairs of the empire such as money, population, herds or economic time… One million (106) seems an order of magnitude limit of these numbers.
Finally, the cosmogonic measurement of the universe (time, space, cycles…) produces limit numbers, which have posed very tough writing problems.
For a long time, the orders of magnitude remained more or less as follows:
Let there be Machine…
All the techniques, processes and social organizations around counting have proliferated in human affairs (agriculture, commerce, administration…). Numbers have flooded in, gradually invading the unlimited space of orders of magnitude. Like any activity, this numerical production became mechanized and, with the computer, humankind passed in the 20th century from the craftmanship to the industrial era of numbers, to their automated mass production. A number in a computer remains however a trace as material as a notch on a tally stick. Far from being an abstraction, it results from an activity that requires time and energy (in French only: Données et traces numériques (sous rature)).
Today, an ordinary computer (64 bits) can theoretically designate each number up to 264, that is to say 1019 approximately; let us round up to 1020. The order of magnitude of the “Numbers for Technique” has thus experienced, in only a few decades, an extraordinary inflation according to a generalized “Moore’s Law”:
The machine can also seize with a good precision the numerosity of cosmogonic descriptions produced by science, in particular by physics and astrophysics in charge of measuring the universe. This science remains surprisingly confined behind the edge around googol, stable since the Jains and Archimedes, more than 2000 years ago.
Today, here are the orders of magnitude of Mundus Numericus:
A thousand things go forward, nine hundred and ninety-eight go backward; that is the progress.15
Humankind has globally progressed in correlation with the orders of magnitude of the numbers it has produced. We do not only say that numbers grow with scientific and technical progress, which is rather obvious, but we also hold to the second term of the correlate: the technical capacity to represent and manipulate larger and larger numbers (sticks, quipus, calculi, languages, machines…) is a condition of progress in general, for which there is still a vast terrain to conquer between 1020 and googol.
These speculations may be risky, but the number undoubtedly structures our conception of the world and represents our ambitions, such as Google’s to index the planet (to begin with), and whose name, registered in 1997, obviously comes from googol16. What could be better than this number to carry the objective of a progress defined by the digital conglomerates: to appropriate the world?
However, there remains a vastness beyond googol, populated by digital beings that have no comparison with those which escort this “progress”. We will meet them in the next article: Numbers and Mathematic.
1. ↑ Edward Kasner, James Newman / Simon & Schuster – 1940 – Mathematics and the Imagination
2. ↑ Didn’t the French mathematician Alexandre Grothendieck call “child’s drawings” mathematical objects of his invention, whose name should not deceive us: they are inaccessible to non-mathematicians.
3. ↑ Wikipédia – Subitizing
4. ↑ (in French) Aparecida Vilaça / Conférence Lévi-Strauss – 2018 – Le diable et la vie cachée des nombres – “Après le « 1 » et le « 2 », les étudiants wari’ ne purent employer que des quantificateurs de façon aléatoire, exception faite du mot utilisé pour « peu » que la plupart d’entre eux traduisaient par « 3 ». Au-delà, les nombres de 4 à 10 furent désignés par divers termes signifiant « beaucoup », auxquels, pour atteindre des nombres supérieurs, ils ajoutaient, par exemple, l’adverbe « très » et le qualificatif « véritable », ce qui donnait lieu à des expressions du type « vraiment beaucoup »”.
5. ↑ Pierre Pica, Cathy Lemer, Véronique Izard, Stanislas Dehaene / Science, vol. 306, p.499-503 – October 15, 2004 – Exact and Approximate Arithmetic in an Amazonian Indigene Group
6. ↑ (in French) Collège de France – 2008 – Le concept de nombre
7. ↑ Wikipedia – Dactylonomy
8. ↑ Wikipedia – Tally stick
9. ↑ Wikipedia – Quipu
10. ↑ (in French) Wikipédia – Débuts de l’écriture en Mésopotamie
11. ↑ Ernst Cassirer – 1997 – Trois essais sur le symbolique – Œuvres VI
12. ↑ Pliny the Elder – The Natural History – Book XXXIII. The Natural History of Metals
13. ↑ J J O’Connor and E F Robertson/ MacTutor – November 2000 – Jaina mathematics
14. ↑ Wikipedia – The Sand Reckoner
15. ↑ Henri-Frédéric Amiel – December 30, 1874 – Journal intime
16. ↑ David Koller / Stanford – 2004 – Origin of the name “Google”