The Proof by Googol (2) Numbers and Mathematic

Reading time: 29 minutes

Translation by AB – March 14, 2023

Translation note:

“Mathematics”, in the plural, is the usual designation for the mathematical sciences, and in this article, we will say “mathematics are”, not “mathematics is”. Indeed, we will focus on the unification of these mathematics into a “Mathematic”, in the singular, which corresponds to the discipline we know today. This singular is not common (and may be incorrect). Therefore, when referring to this unified discipline, we will refer to it as “Mathematic” with a capital “M”.

This reading requires no knowledge of mathematical techniques other than the exponential notation “a^b”, meaning “a ✕ a … ✕ a” b times.

Googol

Thus is the story of big numbers a story of human progress.

Scott Aaronson

After more than two millennia of counting and measuring by various means, the computer has finally put everyone in agreement, from North to South, from East to West. The machine produces numbers in huge amounts, the same for all, according to the same processes and for the same uses. Technological progress has thus made numbers a “transcultural material”: no one claims, interprets or questions “2”, “250” or “10¹⁹”; no one questions the universality of numbers and, consequently, their own existence, independent of the human being.

But our technical and scientific devices may produce more and more numbers, and bigger and bigger, but none of them ever goes beyond this border around the number googol (10¹⁰⁰), as if the numbers were squeezed behind a locked door. If sciences and techniques are thus contained by the “real”, mathematics on the other hand knows no borders: beyond googol, there is a somewhat nebulous “infinity” of numbers like 10¹⁰⁰⁰, 10¹⁰⁰⁰⁰, 10¹⁰⁰⁰⁰ + 1… What should we understand about our capacity to write, designate or simply imagine numbers that nature will never “produce”? This apparently uninteresting question, quite irrelevant for our practical lives, can nevertheless teach us a lot about the human as human. The number is somehow excellent “ethological” evidence.

Then, after Numbers and Progress, the first part of our “Proof by Googol”, we are going tackle these extreme numbers by starting, not from real phenomena which seem to ignore them, but from the only language which allows us to talk about them: mathematics.

This text is made of three parts. The first part (To Infinity and Beyond) examines some selected examples from the history of mathematics and its unification, where the number loses all contact with reality. The second part (To Finity?) discusses some issues about this loss of contact. The third part (Impossible Return) concedes the infinitary nature of mathematics and of language in general.

1- To Infinity and Beyond

Music!

Music charms us, although its beauty consists only in the agreement of numbers and in the counting, which we do not perceive but which the soul nevertheless continues to carry out, of the beats or vibrations of sounding bodies which coincide at certain intervals.

Leibniz¹

Leibniz thus presented music as the phenomenon by which the soul immediately and instinctively recognizes numbers and their “agreement”, their essential and harmonious proportions. Nearly 2000 years earlier, music had already aroused this “sense of numerosity” for the Pythagoreans, and Archytas of Tarentum, of whom Ptolemy made “the reputation of being the most musical of the Pythagoreans”², persistently studied musical intervals and their numerical proportions. According to these thinkers, music hits the soul directly because “the soul nevertheless continues to carry out” and instinctively attunes itself to the number as it occurs in nature. We feel the number as the caress of the air or the heat of the sun. Numbers then rise among the Pythagoreans to reach an “ontological” status leading to a grandiose vision of the cosmos³:

Spheric and harmonic combined to produce a descriptive theory of the cosmos. […] Some verses of Horace describe thus “Archytas as having explored the aerial dwellings and browsed in thought the celestial vault”.

This “emancipation” of the number does not go without causing some misunderstandings, as any concept born in minds still not secure⁴ :

For the Pythagoreans, things are numbers, or things consist of numbers, or things imitate numbers (which would be principles), or things have numbers: a certain vagueness remains.

History will retain the famous motto of the Pythagorean school: “all things are number”⁵. This quasi-religious dogma was taken up by Plato, friend of Archytas, who, according to Aristotle, rectified it as follows⁶:

[ Plato ] is also peculiar in regarding the numbers as distinct from sensible things, whereas [Pythagoreans ] hold that things themselves are numbers, nor do they posit an intermediate class of mathematical objects. His distinction of the One and the numbers from ordinary things (in which he differed from the Pythagoreans) and his introduction of the Forms were due to his investigation of logic (the earlier thinkers were strangers to Dialectic).

By anchoring all “mathematical objects” in the “midwater” between the sensible world and the Ideas, Plato prepares the position of mathematics for centuries to come: mediators between the real and the conceptual, constantly oscillating from one to the other. Consequently, concerning this “object” that is the number, the soul will not only be subjugated by its harmonic displays: it will finally be able to “make an Idea” of it.

Mathematical arts

Like any discipline in search for “truth”, mathematics develop within a highly codified and organized social framework. Nothing Pythagoras taught was to be written down or divulged to the uninitiated, and even the disciples were divided into two classes. The “acousmaticians” (άκουσματικοί), the mere listeners, were allowed to know a little of his teaching. It is said that Pythagoras gave them lectures while standing behind a curtain and did not give them any explanation or demonstration, asking them to know his precepts by heart⁷. The “mathematicians” (μαθηματικοί) were students privileged to see the Master and to know his thoughts regarding the workings of the world and of the numbers.

On these ubiquitous numbers relied for Pythagoreans these four fields still gathered in the Middle Ages under the name of “quadrivium”: arithmetic, geometry, astronomy and music. The American mathematician Morris Kline (1908-1992) commented in his book “Mathematics in Western Culture” on this Numerical Brotherhood of these “mathematical arts”⁸:

From the time of Pythagoras, the study of music was regarded as mathematical in nature and grouped with mathematics. This association was formalized in the curriculum of the medieval system of education wherein arithmetic, geometry, spherics (astronomy), and music comprised the famous quadrivium. The four subjects were linked further by being described as pure, stationary, moving, and applied number, respectively.

In the quadrivium, space and time are somehow “shaped” by the number⁹: arithmetic concerns only the number in the abstract (the “pure” number), geometry the number in space, music the number in time, and astronomy the number in space and time. The work of unifying these mathematical arts, considered as different points of view on number, will lead to the emergence, at the turn of the 17th century, of a field that we will call here, not mathematics (i.e., mathematical arts), but Mathematic. It is it and only it that will unfold the shape of the Ideal number, ignoring the “googol edge”, far from the shores of the sensible world, counting sticks and other calculi…

Mathematic!

If the time of the quadrivium is well past, modern mathematics still include several “arts” (arithmetic, geometry, analysis, topology…) which still seem to justify its plural form. But these arts are now more unified than ever, not in the sense that they are all based on the same numerical foundations, but because they all share the same “shape”, the same language carefully elaborated since the time of François Viète in the 16th century. Mathematic is thus enthroned as a kind of Constitutional Council of a society of concepts intended primarily for the representation of sensible phenomena and for the conduct of earthly affairs (we shall return to this constitutional metaphor in the conclusion). Galileo is the one who best embodied this “Darwinian” (r)evolution of the medieval-Pythagorean doctrine, saying (emphasis added)¹⁰:

Philosophy is written in this grand book, the Universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.

Mathematical concepts are no longer essences as with the Pythagoreans (“all things are number”), nor “objects” cautiously installed by Plato halfway between the sensible world and that of the Ideas, but they are determined by a language with which the Universe tells itself to the human. He thus sets here a truly scientific project in which the mathematical language, always bound to reality, plays for the human the role of descriptive means.

If this position seems clear, there is still a gap in Galileo’s formula through which mathematical concepts can always appear as eternal “essences”: the Universe / Philosophy is “written” and not merely “described”, thus indicating a tenuous but direct relationship, without human mediation, between a Universe and a Mathematic that would have the same “shape”, so to speak. This hypothesis still remains today. In a well-known article entitled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, physicist Eugene Wigner notes¹¹:

[…] mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections. […] the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.

The Universe and the Mathematic seem to be “mysteriously” connected and the mathematical concepts adjust themselves “secretly” to the phenomena. Therefore, why numbers like 10¹⁰⁰⁰, 10¹⁰⁰⁰⁰, 10¹⁰⁰⁰⁰ + 1… would they not be already present in the Universe but still unobserved for lack, for example, of adequate measuring instruments or sufficient theories? Does mathematics invent or discover? This is the classical epistemological dilemma that emerges from the Platonic “midwaters”.

A mathematical language developed by an extraterrestrial species or, a more uncertain hypothesis, by a machine, would certainly have a different expression from the one we know (the numbers, for example, would not be written in the same way). But, being by hypothesis isomorphic to the sensible Universe, this language would have a mechanical translation in our Mathematic. Therefore, neither the human, nor such extraterrestrial, nor such machine, are necessary conditions of Mathematic in general but only of its “local” forms. Any concept coherent with terrestrial Mathematic is likely to correspond to something in the Universe, and simply waits for its confirmation by reality, including therefore the numbers beyond googol, including, why not, the infinite itself.

Thus, humans inevitably believe in some transcendence of the concepts that their language uses. In this matter, mathematical language has reached the highest levels.

Googolplex

Kasner’s nephew (see Numbers and Progress) had well understood the recursive character of the language by inventing this other number called “googolplex”, which would be written 1 followed by googol zeros. We write it very easily as 10^gooogol but, for this young boy who did not know exponential notation, the largest number one could imagine had to be written with the largest possible number of “0”’s after the “1”. Very cleverly, he defined googolplex as “one, followed by writing zeroes until you get tired”. He perceived a physical limit to a mathematical exercise that was otherwise purely ideal. Everything may be a number, but does any number correspond to something if nothing can ever write it down?

In the example of googolplex, even equipped with a very competitive notation, mathematicians would quickly lose patience and the placid machines would give up for lack of time or energy. More generally, not all finite numbers can be named since the mathematical language has a finite number of symbols (“0”, “1”, “2” …) and we have only a finite amount of time to name or write anything. The set of numbers that we can name or write is therefore finite¹². Googol and googolplex are part of this big family. Since it is not infinite, there is necessarily among its members a number and only one that is the largest. Let’s call it “googolmax”. Beyond googolmax there is no instance of the concept of number (no number that we can designate): there remains only the Platonic Idea of “number”, and nothing else comes to us to talk about it than the expression “really a lot” of the Wari’ Indians. Logic however forbids googolmax to exist because, to use a simple argument on this ground filled with traps, if we can designate googolmax, then “googolmax + 1” comes immediately and googolmax is thus not the greatest of all.

Logic goes astray with very large numbers but, curiously enough, the landscape lights up when we jump over them to reach infinity and beyond: infinity solves a sort of “anguish”.

Cantor

All trust in the world begins with names, which allow stories to be told. This is what the biblical account of the gift of names in paradise testifies to, but also the belief that is at the foundation of all magic, and that remains determining in the beginnings of all science, that the relevant naming of things will remove the enmity between them and man to allow him to use them. The terror which found the way of the language is already borne.¹³

The drifts of a society of concepts have no other limits than those of language when nothing real comes to moor them. The language games of the German mathematician Georg Cantor, at the end of the 19th century, have pushed these drifts to the extreme. Cantor revealed (or invented) several kinds of infinities, cheerfully stepping over the difficulties of designating the finite numbers themselves¹⁴.

Even if it is not excluded that these strange species will one day agree with the phenomena (benefit of the doubt), it must be recognized that the mathematician has withdrawn behind his curtain and seems to speak only to a few “fanatical” mathematicians. Cantor calls “ℵ₀” the infinite number of integers, the numerosity of an unimaginable collection. It is indeed a number to which we can add “1” by writing “ℵ₀ + 1” and to which other infinities (called “transfinite cardinals”) follow, such as “ℵ₁”, “ℵ₂”… “ℵ_googol”, etc. But Cantor was not crazy. Noticing, for example, during his work that there are “as many” elements (points) in a line segment when in a square, in this case “ℵ₁”, he did not believe it, and he wrote to his colleague Richard Dedekind “I see it, but I don’t believe it!”¹⁵.

Does this drift of numbers make sense? The power of mathematical language is extraordinary, but this power seems to be like our own: caught up in hubris, the vertigo entailed by a too permanent success to which the Greeks opposed temperance and moderation, which is first of all knowledge of oneself and of one’s limits¹⁶. However, hubris should be confronted by civilization, by reason (logos), and therefore by mathematics. But the “disproportion” seems to have to penetrate each of our creations, making like an echo to our anguished vitality, and at the same time desiring, in front of the future (i.e., of “+ 1”)

2- To Finity?

Émile Borel and the strange Law of Chance

Some mathematicians remained wary in front of these conceptual and numerical assaults. This was the case of Émile Borel (1871-1956), a graduate of the French Ecole Normale Supérieure and a brilliant precocious mathematician, who at the same time led a political career marked by his republican intellectualism, far from the solitary getaways in the world of Ideas. Borel was interested in a very particular mathematical art: the calculus of probabilities, or “science of randomness”, which development had only really begun with Fermat and Pascal in the middle of the 17th century. A probability is a measure of “chance” and therefore takes the form of a number. Indeed, although we say for example “1 chance in 10,000” (written “1 / 10,000”), the whole number “10,000”, measuring the number of exclusive “possibilities”, is enough to describe the phenomenon.

The French epistemologist Ernest Coumet writes¹⁷ :

[…] the same inspiration had guided Jacques Bernoulli, Buffon, Condorcet…, and many other probabilists who had seen in the calculus of probabilities the ideal instrument of reason when it is applied to the “affairs of civil life” and that it must help men to choose better where the intervention of chance embarrasses them.

Mathematic thus takes charge, with this calculation, not only of providing to the other sciences the descriptive forms of nature and its laws, but also of those of the “affairs of civil life” which must be regulated by the “reason”: economics, finance, business administration, decision-making… in short, praxeology in the broad sense¹⁸. Probabilistic “art” was thus born from very concrete topics, responding to a need for mastering our confused and often biased evaluations by “measuring” these situations of chance that abound in everyday life. Émile Borel thus seized a mathematical art which questions de facto the morals¹⁹:

On the eve of the First World War, his first synthesis on the question (Le Hasard) allowed him to note that the “science of randomness” had taken on universal importance: in particular, it obliges us to reconsider the “practical value” of calculus, i.e. its importance for the daily decision-making of everyone, as well as for the social conscience of individuals.

Borel thus wondered, from a mathematical point of view, about the number of exclusive sets of possibilities that can be encountered in practice and, beyond, in the universe. We must go here into some more details.

Borel was, with Henri Lebesgue and René-Louis Baire, a pioneer of the “Measure Theory” consisting in a vast generalization of the concept of “measure” (such as surface, volume, mass or even… probability). Intuitively, measuring consists in folding or projecting any conceivable “form”, however complex, (such as a computer) on a single number (1.2 kg)²⁰. In the process, we lose almost all the information about this form – its structure, its topology… – but we gain by being able to “appropriate” it in one go (the miracle of numbers!). This Measure Theory applied to probabilities i.e., the measure of probabilistic forms or configurations of chance, led Emile Borel to wonder about the numbers “accessible” by the measure of probabilities concerning events likely to occur in reality. This is how he stated what he called the “Single Law of Chance”²¹:

Events whose probability is sufficiently small never occur.

This Law, between the Idea and the Sensible, the finite (“small”) and the infinite (“never”), is quite extraordinary. If an event of very small probability is not only very unlikely but will never actually occur, there would thus be a numerical threshold beyond which reality never transpires. But what is the value of this threshold? Borel suggests an intuitive estimate based on the scale of the phenomenon²²:

Borel proceeds to a numerical estimation of negligible probabilities by distinguishing 3 different scales: human, terrestrial, cosmic. Of course, his goal is not to provide an exact measure of the limit at which a probability becomes negligible, but only to indicate the order of magnitude. On the human scale, i.e., at the individual level, a probability can be considered negligible if it is less than 10^-6. On the terrestrial scale, i.e., for the whole of humanity, this limit could be lowered to 10^-15 if we estimate this population, as Borel does, at about 1 billion individuals, or 10⁹. Finally, it will be, he says, 10^-50 on a cosmic scale.

Bringing the probability (“1 / 10 000”) to the number of complexions (“10 000”), we get this:

We observe a certain similarity with the scale of numbers at the time of Borel, before the rise of the computer (see Numbers and Progress):

The number of complexions corresponding to all the possibilities for an event to really occur, there is no reason for this number to exceed this “googol edge” beyond which only instances of the Ideal number exist.

Reason

Émile Borel was not isolated. The mathematical growth was indeed in introspective suspension at the turn of the 20th century. This pause paralleled the suspicions expressed about languages in general as foundations of universal truths. It is because, we think, technical power was beginning to work at full speed, allowing the production and confrontation of more and more “linguistic inputs and outputs” (books, press, letters and now e-mails, posts, blogs…). Our inclination to “deify” language makes potentially true everything it allows us to tell. However, we do set a condition: no contradiction should appear; everything should be consistent.

As all the innumerable linguistic productions cannot be coherent between themselves and by themselves, the “truth-telling” devices from which they come are questioned and sometimes stiffen. To each his own way. A social network, for example, whose truth is not governed by any Constitutional Council, fragments into more or less watertight and coherent sub-systems (“filter bubbles”, “groups of friends” …). A (totalitarian) ideology cannot admit fragmentation and proceeds otherwise: contradictions are crushed by force. If necessary, reality itself is made to conform to language. Finally, the more civilized Mathematic is a device whose inconsistencies are repaired, sometimes with difficulty, by the supreme authority of reason.

The Cantorian creations or the very large Borelian numbers have given Mathematic a hard time. But it is its very nature: reason resists, repairs, and continues to work on our representations, to concretize them in order to make them ever more effective. This effectiveness is both the cause and the condition of its universal adoption and of peaceful trade between humans. It is reason that assigns mathematics to some coherence with reality and that led Emile Borel when he declared²³:

In the ordinary conduct of his life, every man usually neglects the probabilities whose order of magnitude is less than 10^-6, that is to say, one millionth, and we will even note that a man who would constantly take into account such unlikely possibilities would quickly become a maniac or even a fool.

Ultrafinitism

This return to reality found a way to express itself within Mathematics with the “intuitionistic” current initiated by the Dutch mathematician Luitzen Egbertus Jan Brouwer at the beginning of the 20th century. His singular philosophy opened a schism between the proponents of classical formalism and the contemptuous of this formalist epidemic which broke the moorings of reality and ignored the “googol edge”. Intuitionism mainly seeks to re-establish criteria of truthfulness, not internal to language (everything that can be expressed correctly in language is, according to the “dogma of the modal uniformity of mathematics”, either true or false²⁴), but relative to the possibility of accessing actually to what is “true”. Truth is thus subjective, and more precisely relative to the subject’s ability to access it by construction. To give just one consequence of this razor’s edge theory, Brouwer rejected the idea of an actual infinite such as “the set ℕ of all integers”. This set is potentially conceivable as something whose path can never be completed (constantly created), but it does not exist as a completed object, even if it has a proper name. “ℕ” is thus not the name of a thing but, if one wishes, the name of a procedure whose stopping conditions are not specified, or of an intention, like the number googolplex whose writing stops only when one “gets tired”, according to one’s appreciation and endurance.

An extreme interpretation of intuitionism, called “ultrafinitism”, radically raises the question of the very existence of large numbers and therefore of the necessity for Mathematic to take them in charge. Ultrafinitism was developed in the early 1960s by the Russian mathematician and dissident Alexander Esenin-Volpin. He dismissed not only the existence of infinite sets but also of “very large” integers, although as “small” as googol, for example. It is probably no coincidence that ultrafinitism was invented by a man, living under a totalitarian regime, who loved freedom and justice, for in the end intuitionism, ultrafinitism and, more generally, all constructivist approaches ask reality to validate what others claim, using only language, to be the truth. In a certain way, these approaches respond to this “moral” posture seen with Borel²⁵ :

[ Esenin-Volpin ] believed that the liberation should emerge through authenticity and precision of language, understood ideally as mathematically-inspired formalisation of the language of areas closest to the practical and social life: ethics and jurisprudence. Without a language that is transparent and unambiguous we will not be able, he believed, to trust our thoughts.

Ultrafinitism rejects both actual and potential infinity (“totalitarian” mirages?) and even disputes the existence of numbers too large to ever emerge from the sensible universe. It advocates the return of mathematics to a kind of “natural state” and of the numbers to concrete numbers. Let’s see about this nice anecdote told by the mathematician Harvey Friedman and related in the above-mentioned article. We reproduce it in extenso²⁶ :

I have seen some ultrafinitists go so far as to challenge the existence of 2¹⁰⁰ as a natural number, in the sense of there being a series of “points” of that length. There is the obvious “draw the line” objection, asking where in 2¹, 2², 2³, …, 2¹⁰⁰ do we stop having “Platonistic reality”? Here this is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas. I raised just this objection with the (extreme) ultrafinitist Yessenin-Volpin during a lecture of his. He asked me to be more specific. I then proceeded to start with 2¹ and asked him whether this is “real” or something to that effect. He virtually immediately said yes. Then I asked about 2², and he again said yes, but with a perceptible delay. Then 2³, and yes, but with more delay. This continued for a couple of more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take 2¹⁰⁰ times as long to answer yes to 2¹⁰⁰ then he would to answering 2¹. There is no way that I could get very far with this.

A fine illustration of radical constructivism! Of course, Harvey Friedman has good reasons to judge this attitude as far-fetched and, above all, mathematically pointless. But practically and ethically, this ultrafinitistic questioning seems to us absolutely valid: why indeed accept without discussion the existence a priori of finite instances of the Number like 2¹⁰⁰ or googol? The mere observation that a machine would answer Harvey Friedman’s “challenge” much faster than Esenin-Volpin shows that it is necessary to keep an eye on this kind of question.

Etc.

I looked for (without specifying it too much) the connection of the definitions – postulates – of mathematics with the (special) sensibility and the motricity – that is to say with the constituents of the acts. – What is natural – since mathematics is, in the last analysis, only a prescription of acts leading to a number or to a tracing, by way of successive I-can.

Paul Valéry²⁷

Archytas of Tarentum was the first to state a paradox aiming at demonstrating the absurdity of the existence of a material edge of the world, a bit like Friedman who wanted to show Esenin-Volpin the absurdity that a particular element of his list is the last to exist²⁸:

If I arrived at the outermost edge of the heaven, could I extend my hand or staff into what is outside or not? It would be paradoxical [given our normal assumptions about the nature of space] not to be able to extend it.

The strange peculiarity of these arguments is to use the material, psychological or imaginary position of the subject (“I”) as the place from which it is always possible to extrapolate (see also Body and Language Games), and then to turn this solipsistic reasoning around to conclude something about an a priori Universe, without any “I”. What problem would arise if we posed that the Universe cannot accommodate “I” everywhere²⁹? Well, we should follow Ludwig Wittgenstein’s aphorism and keep silent since speaking is only possible from “I”³⁰. But it is not so simple to be really silent, that is to say, not only to forbid ourselves the use of certain words or numbers, but better still: to forge a language that never produces them.

Let’s take Googolplex. This word is available, but without any “I” to really reach anything that can correspond to it. So, how to keep a “mathematical silence” on googolplex and on all other numbers beyond googol? This is what these ultrafinitist mathematicians are trying to do, but isn’t it, in a way, already too late?

3- Impossible Return

Constitution

A Constitution is a fundamental law that legitimizes all lower norms. According to us, it must obey, like any language, in particular mathematical, four “commandments”: consistency, aesthetics, truthfulness and effectiveness. The first two are internal (the Ideal), the other two govern the relationship of the constitutional game to reality (the Sensible). Let us review them by identifying each time where ultrafinitism fails.

Consistency is an absolute imperative and “the constitution [ must enjoy ] an intrinsic unity, and each of its elements has a meaning related to that of the other provisions”³¹. But the existence of a larger number, say googol, is at first sight incoherent. Indeed, if googol is a number, then an article of the “Mathematical Constitution” says that one can “make” the number “googol + 1”. But since googol is the largest number, then we must have “googol + 1 = googol” and therefore by subtraction “1 = 0”, which another article of the Constitution forbids. Ultrafinitism must thus write mathematical articles consistent with each other and with the existence of “a” greater number. Some ultrafinitists like the Russian mathematician Vladimir Sazonov pose for example that “What is known definitely is the “fact” that 2¹⁰⁰⁰ (and even 2¹⁰⁰) is not feasible”³². There should therefore be a limit, a “googolmax”. But in our opinion this project must fail for one essential reason: beyond the immense difficulty of the task, “this” greater number has no logical reason to emerge from mathematics in a purely internal way. It is as if the Constitution had to set a limit, for example on the number of citizens, and ensure its internal consistency with this quantity coming from elsewhere.

Aesthetics is also an absolute imperative because it determines the sensible form of the Idea in order to make it perceptible by intuition. The constitutional form must obviously be simple, clear and simply intelligible. This applies both to its acceptance and to its concrete application. This is why the constitutionalist has an aesthetic duty, and the mathematician for the same reasons. The latter will thus look for economical and symmetrical forms; he will hunt for hypertelias, erase roughness and exceptions… But Sazonov’s “fact” of a greater number has nothing aesthetic: it introduces an exception, de-symmetrizes the timeless succession of numbers. On the contrary, infinity aestheticizes by simplifying and harmonizing the whole mathematical edifice, and even gratifies it with a certain beauty. On the other hand, we must recognize that a powerful machine could very well accommodate a great complexity and the ugliness of a tricky mathematical Constitution.

Third, a Constitution must be legitimate, and thus establish a close relationship with “truth”. It should be remembered that constitutionalism is not self-evident: it is a recent movement, which emerged in Europe during the Enlightenment in order to limit the arbitrariness and despotism of monarchs. The “truth” that the Constitution must reflect is that of the people and no longer that of an absolute power, which hardly needs texts or language but only force to impose its truth. Traditional Mathematic certainly offers an already exceptional “truth-telling” framework, but the rise of computers (mathematical machines that are finite by nature) and the collective awareness, now established (and anguished), of the finiteness of our environment work in depth on the criteria of mathematical truth: is the sensible world finally representable by an infinitary language? Let us say it simply: it is not this kind of practical truth that mathematics tries to reach. This is why, in a certain way, the ultrafinitist is an ideologist, a praxeologist, even a constitutionalist. But is he still a mathematician?

Finally, a Constitution must be effective insofar as it applies to solving the problems related to the political system it supports³³. Mathematics does not escape this requirement if by “political system” we mean “way of considering the universe”. The “unreasonable effectiveness” of the Mathematic is thus understood for a “political system” determined by the science, the naturalism and the technical corollary aiming at the conquest of the things of the nature. Our way of looking at the universe must therefore be based on an infinitary Constitution that sets no limit to this conquest. Ultrafinitism supports, consciously or not, another political system of deconstruction of progress that nobody wants and whose time may never come. A bit like blockchain (Some political dimensions of blockchains), ultrafinitism is confronted with a problem of “bootstrapping” by the milieu of which it would be the efficient tool.

These few arguments deserve to be reviewed and developed, but the exploration of numbers leads us in any case to this: Mathematic, like any language, is, by nature, infinitary. There is, finally, one last argument.

Desire

Quand il y a de moins en moins d’espace entre l’infini et nous, entre le soleil libertaire et le soleil procureur, nous sommes sur le banc de la nuit.

René Char, Tables de longévité (left untranslated)

Mathematic exceeds “really a lot”, without any finitary recourse, our small cantonment between zero and googol:

We always share, in the end, the same appreciation of numbers as the Mundurukú or wari’ Indians: beyond a certain threshold, certainly considerably higher for us who are equipped with technological prostheses, gathers a fog of unspeakable numbers collectively considered under the signifier “many” (or the indeterminate Greek “murios” which means more or less the same thing³⁴). This fog is “structured” only by mathematicians and for their own account.

This being recalled, we can now better qualify what we have named a “phase shift” in Numbers and Progress between “earthly” numbers and mathematical numbers: earthly numbers measure what we possess and mathematical numbers what we desire. This is why the first ones accompany the “progress” and the second ones testify of an impossible temperance.

Jean-Pierre Cléro, evoking the work of the French psychoanalyst Jacques Lacan, formulates a possibility as to the “office” of mathematical signifiers (including numbers)³⁵:

The signifiers of desire are not an expression in the strict sense. They are its structure, its mode of operation. Mathematics offer the best example of this type of discourse that progresses without thinking, if not symbolically. It is mathematics that best says desire in its ultimate reality.

It remains to be understood if this remark is only a psychoanalytical or even poetic whim, or if it does not open one of the paths of the “proof” of the human singularity by Homo Mathematicus (Ecce Homo Mathematicus).

1. ↑ Gottfried Wilhelm Leibniz – 1714 – Principes de la nature et de la grâce fondés en raison
2. ↑ (in French) Bernard Mathieu / Bulletin de l’Association Guillaume Budé, Année 1987, 3, pp. 239-255 – 1987 – Archytas de Tarente, pythagoricien et ami de Platon
3. ↑ Ibid. 2, p. 243 – “Sphérique et harmonique se combinaient pour produire une théorie descriptive du cosmos. […] Certains vers d’Horace décrivent ainsi « Archytas comme ayant exploré les demeures aériennes et parcouru en pensée la voûte céleste »”.
4. ↑ (in French) Wikipédia – Pythagore
5. ↑ Ambiguous translation of “τὰ ὄντα πάντα ἀριθμοὺς”, words of an anonymous commentator on the life of Pythagoras (in French: Bibliothèque de Photius)
6. ↑ Aristotle – Metaphysics, I
7. ↑ By the way, these acousmaticians are exactly in the position of Searle in the “Chinese room” experiment reported in GPT-3, LaMDA, Wu Dao… The blooming of “monster” AIs. In Mundus Numericus, most of us reach, at best, the “grade” of acousmatician…
8. ↑ Morris Kline – 1953 – Mathematics in Western Culture (p. 287)
9. ↑ Wikipedia – Quadrivium
10. ↑ Galileo – 1623 – The Assayer (in Discoveries and Opinions of Galileo, p.237)
11. ↑ Eugene Wigner / Communications on Pure and Applied Mathematics, vol. 13, no 1,‎ 1960, p. 1–14. – 1960 – The unreasonable effectiveness of mathematics in the natural sciences
12. ↑ It would of course be necessary to specify the conditions of writing (duration, number of authorized characters, dimensions of the sheet of paper…) and the features of the writer (human, machine…).
13. ↑ Hans Blumenberg / trad. fr. Denis Trierweiler – 1979 – Arbeit am Mythos – In French: “Toute confiance dans le monde débute par les noms, qui permettent de raconter des histoires. C’est ce dont témoigne le récit biblique de la donation des noms au paradis, mais aussi la croyance qui est au fondement de toute magie, et qui reste déterminante dans les commencements de toute science, que la nomination pertinente des choses supprimera l’inimitié entre elles et l’homme pour lui permettre de les utiliser. La terreur qui a retrouvé la voie du langage est déjà supportée”.
14. ↑ Georg Cantor (1845-1918) is a prominent character in the history of ideas, especially in mathematics. The “Set theory” of which he is the instigator posits roughly that “everything is a Set” or rather that we can imagine everything as a Set. In this theory, the fundamental relation is that of the membership of an Element “x” to a Set “E”, which is written “x∈E”. We note the strange similarity with what we say about number as a tool of appropriation. The set and the number are two historically distinct figures of the “property”. It is no surprise that the set has ended up subsuming the number.
15. ↑ (in French) Thierry Berkover – October 1st, 2006 – Digressions sur « Le drame subjectif de Cantor »
16. ↑ Wikipedia – Hubris
17. ↑ (in French) Ernest Coumet / Annales. Économies, sociétés, civilisations. 25ᵉ année, N.3, 1970. pp. 574-598 – 1970 – La théorie du hasard est-elle née par hasard ?
18. ↑ Today, all the autonomous or semi-autonomous devices of the technological system, in particular the “smart” ones, operating in an open environment have to do with chance and thus depend in their conception and functioning on these mathematics of chance.
19. ↑ (in French) Alain Bernard / Cahiers philosophiques 2018/4 (N° 155), pages 81 à 95 – 2018 – Émile Borel (1908) : le calcul des probabilités et la mentalité individualiste
20. ↑ This is also what a neuromimetic network does.
21. ↑ This is a stronger version of what some have called “Cournot’s principle”, named after the 19th century French mathematician who stated that “The physically impossible event is the one whose mathematical probability is infinitely small”. Cournot used a striking example: it is impossible to physically trace the center of a circle, since this center corresponds to a single point among an infinite number of points located right around it (which Cantor would measure by the number “ℵ₁”). Therefore, there is no chance of getting there since this event constitutes the only favorable case of an infinity of possible cases (i.e., “1 / ℵ₁” which is “worth” zero in the mathematical language). But Borel somehow considered that the tracer itself had a minimal “thickness” and that it was therefore possible in practice to cover the exact theoretical center of the circle, with a probability that gets smaller and smaller as the tracer becomes thinner. However, Borel never referred to Cournot or to this thought experiment.
22. ↑ (in French) Thierry Martin / Images des mathématiques – CNRS – February 14, 2018 – Les probabilités négligeables selon Émile Borel
23. ↑ Ibid. 22 (Borel as quoted in the article)
24. ↑ (in French) Ivahn Smajda / Les Études philosophiques 2008/1 (n° 84), pages 49 à 69 – 2008 – Mathématiques, réalisme et modalités
25. ↑ Jan Gronwald / cantorsparadise.com – December 16, 2020 – The Beautiful Consistency of Mathematics — Alexander Yessenin-Volpin
26. ↑ Ibid. 25
27. ↑ Paul Valéry was a French poet, essayist and philosopher. Quoted by Norbert Schappacher – 2016 – Paul Valéry et la potentialité des mathématiques – “J’ai cherché (sans trop me la préciser) la liaison des définitions – postulats – des math[ématiques] avec la sensibilité (spéciale) et la motricité – c’est-à-dire avec les constituants des actes. – Ce qui est naturel – puisque les math[ématiques] ne sont, en dernière analyse, qu’une prescription d’actes aboutissants à un nombre ou à un tracement, par voie de Je puis successifs”.
28. ↑ Quoted here: Stanford Encyclopedia of Philosophy – Archytas
29. ↑ Let us specify that the “I” of which we speak is the “I” extended by the technique and having, for example, a photographic device (The “progress” unveiled by Photography (with Henri Van Lier)).
30. ↑ Ludwig Wittgenstein – 1921 – Tractatus logico-philosophicus – Aphorism #7 : « What we cannot speak about we must pass over in silence » (Pears/McGuinness translation).
31. ↑ (in French) Olivier Pluen / Actu-juridique.fr – July 9, 2018 – La cohérence de l’écriture constitutionnelle – “la constitution [ doit jouir ] d’une unité intrinsèque, et chacun de ses éléments revêt une signification liée à celle des autres dispositions”.
32. ↑ Vladimir Sazonov (email exchange) – 1998 – FOM: ultrafinitism; objective vs. subjective
33. ↑ See for example (in French): Alain Marciano / Revue interdisciplinaire d’études juridiques 2017/1 (Volume 78), pages 59 à 65 – 2017 – Alain Marciano / Revue interdisciplinaire d’études juridiques 2017/1 (Volume 78), pages 59 à 65 – 2017 – Constitution, économie et efficacité
34. ↑ Mirco A. Mannucci, Rose M. Cherubin – February 1st, 2008 – Model Theory of Ultrafinitism I: Fuzzy Initial Segments of Arithmetic (Preliminary Draft)
35. ↑ Words reported here