Reading time: 50 minutes
Translation by AB – September 28, 2024
This long article is part of the “Puissance & Raison” fresco questioning the Informatization Age and, in particular, the hypothesis of the “intelligent” or even “conscious” machine.
The German mathematician Georg Cantor (1845-1918) published in 1883 a major text entitled “Grundlagen einer allgemeinen Mannigfaltigkeitslehre”, which makes a singular contribution to this questioning. Cantor was obviously not thinking of computing machines when he wrote this essay, but our somewhat oblique reading reveals some interesting hints about the “proper characteristic of man”, as François Rabelais put it, and consequently the problems posed to a machine that would like to “resemble” him. These new elements will be developed in a second article (coming soon), where we’ll talk about “levels of intelligence”, shedding new light on the boundary between human and machine.
This exploration is not, therefore, aimed directly at Georg Cantor or his work, but at those clues we have identified in Grundlagen, in particular the curious motives behind Georg Cantor’s development of his “theory of multiplicities”, later to be called “set theory”, and the “monster” that accompanies it: actual infinity. We have drawn on numerous sources, and in particular on a study by José Ferreirós, Doctor of Philosophy at the University of Seville, which explores Cantor’s main extra-mathematical motivations[1].
To the mathematician reader
This article is not mathematical, and mentions nothing, for example, about transfinite arithmetic, axiomatic set theory and its paradoxes, or the continuum hypothesis… On the other hand, the mathematical philosopher (or mathematical philosopher) may be interested in our full retranslation of Cantor’s text (in French!), made necessary for the purposes of this particular reading. This translation and the original German text are available on this page. The excerpts from Grundlagen quoted in English for ease of reading are doubled in footnote by our own French translation.
A reading of Grundlagen
Infinities
The German-Russian mathematician Georg Cantor (1845-1918) laid the foundations of the mathematical theory known as “set theory”. This theory may stir memories for some, particularly in France, where it was the basis for the famous “New Mathematics” that appeared in the 1960s-1970s and was taught from elementary school onwards[2]. But when Cantor developed the core of his ideas in the 1870s and 1880s, he met with a particular hostility. In Germany, his former teacher, Leopold Kronecker, turned into a character in a tragedy and barred him from the prestigious University of Berlin, calling him a “scientific charlatan”, a “renegade” and a “corrupter of youth”[3]. Cantor would later suffer other, more measured but no less indignant critics, from mathematicians as renowned as his contemporary Henri Poincaré and philosophers such as Ludwig Wittgenstein, who claimed that « [m]athematics is ridden through and through with the pernicious idioms of set theory »[4].
Despite these august enmities, set theory eventually came to dominate and structure all of mathematics. This theory was even endorsed by David Hilbert, the great organizer of the mathematical kingdom at the beginning of the 20th century, who declared in 1925[5]:
No one shall drive us out of the paradise which Cantor has created for us.
What is it about this theory that has aroused such great excitement, both ecstatic and horrified? It’s quite simple: Georg Cantor dared to cross a divide that has been present in the West at least since Aristotle, by asserting the existence of an actual, or “authentic” [Eigentlich] infinity, and not just a speculative or ideal one. Thus, Cantor argues, there are truly infinite things, and to “measure” them there must be infinite numbers just as palpable as the usual ones. Cantor succeeded in giving them a mathematical form and subjecting them to traditional arithmetic operations (addition, multiplication…). So set theory was first and foremost about establishing a ratio, in this case a mathematical one, for this authentic infinity that Cantor saw all around him.
Tensions
But infinity as such has always been the prerogative of the religious, and Cantor’s mathematical secularization came at a time when the Catholic Church was seeking to preserve its domain from the increasingly audacious intrusions of scientific rationalism[6]. Mathematicians, perhaps because of this, have come to accept infinity as a horizon towards which mathematical constructs tend: it is not an object in itself. So for them – as for most of us – there is only one kind of infinity, absolute and unattainable, symbolized since the 17th century by the famous sign “∞”[7] which does not designate a mathematical object, but rather means something like “etc.” or “on the horizon”.
But since the Enlightenment, there has been a compelling impulse from an unfettered, rationalist and conquering human spirit, which developed a victorious scientific, technical and industrial system in the 19th century. Cantor is of course part of this impulse, but as a rather pious man himself, he is bound by the double constraint of the respect due to the Divine, which requires him to set aside the principle of an earthly infinity, and the impulse of reason, which he experiences as self-evident: there’s infinity right there, in front of his eyes, which lends itself to manipulation. « I see it, but I don’t believe it »[8] he wrote to mathematician Richard Dedekind, like Galileo in the face of the evidence: « and yet it moves »! Cantor even paid for his own transgressions, based on firm convictions, with increasingly severe bouts of depression.
Grundlagen
In 1883, a year before his first depressive episode, Cantor published the founding essay of the future theory of sets, entitled “Grundlagen einer allgemeinen Mannigfaltigkeitslehre”, which can be translated as “Foundations of a general theory of multiplicities”[9]. These “multiplicities” correspond more or less to the “sets” of the theory, but the term here insists on “things” insofar as they are immediately perceived as composed and therefore measurable. This text, which we will abbreviate “Grundlagen” here, in which Cantor mixes mathematics and philosophy, strives to make room for his strange infinite numbers while at the same time sparing religious doctrines. He opens up a narrow passageway, based mainly on the philosophical insights of Spinoza and Leibniz. These new numbers find their place between the finite (1, 2, 3, etc.) and the unknowable, “the true infinite or Absolute, which resides in God”. Cantor calls this in-between place the “Transfinitum”, a new mathematical territory where reason regulates a kind of terrestrial infinity that is absolutely respectful of the Divine because it is absolutely distant from it.
The reading of Grundlagen that we are proposing hardly focuses on infinity itself (a concept that is certainly fascinating, but not very fruitful, given how difficult it is to say anything about it), but rather on Cantor’s central argument, which attempts to establish its necessity or “naturalness” in order to force the assent of mathematical and religious doxas. Looking at this argumentation, we come to a question that will guide this reading: Considering that the “invention” of terrestrial infinity is an achievement of the human being, could this invention have been born of a machine? This question, addressed to transhumanists, is far from incidental. It seems to us that this seemingly purely intellectual and rational accomplishment is unique to human beings, and thus sheds light on a natural boundary between human and algorithmic intelligence. The Cantorian development of infinity, as it appears in Grundlagen, seems to us, to use the contemporary vocabulary, to be a “non-replaceable” activity. At a time when so-called “generative” AIs (ChatGPT et al.) seem capable of replacing all the “generative” activities thought to be reserved for human beings (writing text, composing music, drawing, arguing…), here we have a serious counter-example.
But why go looking for Cantor’s transfinite mathematics? Isn’t art, for example, obviously non-replaceable? The problem is that it’s virtually impossible to say how a human work of art differs from an algorithmic one (Artificial Intelligence-Art in its infancy). This does not mean that there is no difference, only that it cannot be described (with words). In mathematical practice, on the other hand, the difference is obvious: human intelligence is characterized by its exclusive ability to create concepts, and not simply to manipulate existing ones[10]. It is then possible to say why Cantor’s creation of the digital transfinite is a specifically human work. Such a theory can only find support in the dimensional affections of the human body, which in turn trigger the “feeling” of dimension. In a way, infinity must be perceived before it can be invented. Cantor himself was well aware of this, and found essential support in Spinoza, who asserted[11].
[…] the dictates of the mind are but another name for the appetites, and therefore vary according to the varying state of the body.
Some passages in Grundlagen bear witness to the affects of “Cantor’s body”, and even reveal what a human body and intellect specifically “do” in relation to the world. So here we are, with this analysis, in touch with our general questioning of the human being and the machine.
Text outline
This text is in three parts.
In the first part (Excess, transgression, meditation), we ask why the human being can’t help but “build theories”. In particular, his relentless active brain has learned to regulate its relationships with the world through the intermediary of “numbers” and, in short, to satisfy its desires with ever larger numbers (The Proof by Googol (1) Numbers and Progress). Grundlagen is a theory both born of this movement and, more to the point, about this movement.
In the second part (Cantor’s body), we explore the inspirations behind Cantor’s theory of multiplicities. Though the human being grasps reality “dimensionally”, reality does not naturally take the form of numbers: Cantor establishes a kind of correspondence between the “things” in the world and the numbers that represent them. As some things are truly infinite, so are their “transfinite” numbers. But we need Cantor’s “body” to be convinced…
In the third section (Principles of generation), we look at Cantor’s own astonishing dismantling of the deployment mechanism of his transfinite numbers. This mechanism seems natural and therefore valid for all forms of human activity, such as art.
Finally, we will summarize (In a nutshell).
1. Excess, transgression, meditation
Desire comes first, so let’s not wait for life to be easy and happy before we start loving it…[12].
Building theories
Cantor introduces his mathematical essay in a rather unusual way, by invoking philosophy from the outset (Gr. Foreword)[13]:
In delivering these pages to the public, I do not want to forget to point out that I have written them with two types of reader in mind above all: philosophers who have followed the development of mathematics up to the present day, and mathematicians who are familiar with the main developments, ancient and recent, in the philosophy of mathematics.
Cantor shows in Grundlagen that, once the existence of transfinite numbers that measure infinite multiplicities has been accepted, they comply easily to the rules of mathematical language. This language has no problem with infinity and, more generally, seems neutral and indifferent to what it represents, whether circles, curves, irrational numbers or infinity: it “works”. But the human mathematician obviously loads this language with his affects (just as we load our technological prostheses with meaning). In this case, the existence of transfinite numbers is indeed an “affect” of Cantor’s body that was not born within the mathematical game (a difference between the human being and the machine is already present). That’s why Cantor has to tie it to that great philosophical network that can be said to institute in language the common “norm” of our affects. In this sense, Grundlagen is also a work about philosophical normalization of infinite multiplicities. However, it’s neither the mathematical tricks nor the philosophical arguments that make Grundlagen noteworthy. Cantor seems to be practicing another, more organic and in some ways “preverbal” activity, to which he endeavors to give mathematical and philosophical form. This endeavor, betrayed by some lexical hesitations that we’ll look at later, leads directly to this first question: why do human beings feel such a compelling need to develop theories?
Like Kronecker, Wittgenstein and others, we may rightly feel that the theory of infinite multiplicities is unnecessary, or even fallacious. Cantor therefore uses all kinds of mathematical and philosophical arguments to make it necessary. But let’s put these arguments to one side and ask ourselves: why does he need it so much he’s sick of it? A somewhat oblique reading of Grundlagen would therefore be of a quasi-anthropological nature: isn’t Grundlagen a kind of explanation (or at least a description) of this typically human propensity to build theories “against all odds”? In an astonishing mise en abyme, wouldn’t Cantor be building an extreme theory, because it deals with infinity, which would explain (or at least describe) how and why human beings, and only human beings, need to build theories?
The “proper characteristic of human”
However, when it comes to human beings as such, we’re always in doubt. During this exploration, an article recounting the death of primatologist Frans de Waal was published, in which it was recalled that “if there was one question that annoyed Frans de Waal to no end, it was the ‘proper characteristic of man’”[14]:
In my life, I must have seen twenty-five proposals on the proper characteristic of man […]. They’ve all fallen by the wayside. We’re wasting our time (…) Why are we always looking for what’s unique to us?
But suppose a scientific extraterrestrial (“SE”) discovered planet earth and set about studying its biotope. Among the living species he or she would spot, all of which are of course identically characterized by their “intensiveness” as living beings, the human species would immediately stand out from all the others: macroscopic traces (roads, buildings, trails…), curious destructive eruptions of fellow creatures, devastation of its own environment, etc. The human species would also stand out from all the others. For the SE, there would obviously be a nature peculiar to this singular species and, as a good scientist, he would carry out some “dissections” to characterize it. Yet Cantor appears to us precisely in the position of the SE who studied the human species and endeavored to dissect the strange and very specific “mechanisms” by which human beings see numbers everywhere, appearing in retrospect to be the only living being to regulate its social functioning entirely around numbers and measurement (Incidentally, the fact that this functioning can lead to disorders is perhaps only an indication of the associated pathology, namely excess[15]).
In this respect, building theories always seems to consist in disciplining this “dimensional” relationship between human beings and the outside world. This is what Cantor devotes himself to in Grundlagen, driven by his bodily affections for the authentic infinite. But in addition, in response to his detractors, he endeavors to establish the “naturalness” of his own dimensional relationship with the outside world, offering us, as it were, an account of a “dissection” of the human being that is neither philosophical nor mathematical.
Excess
We have hardly escaped the mixture of fascination and rejection that set theory has aroused. We ourselves have never really tasted this theory and its procession of transfinite numbers. For far from being a “paradise”, Cantor’s creation, virtuoso though it may be, is an extreme example of the compulsive “surpassing-oneself” (or “being-in-front-of-oneself”) that drives human beings, and which we might simply call “excess”. Every SE can see this excess from space. Performing a real dissection this time, it doesn’t take long for him to discover its biological origin: an absurdly powerful brain. Perhaps all human civilizations are simply different versions – sometimes sublime, sometimes dramatic – of this particular gift of nature.f
Wittgenstein, an outspoken critic of set theory, concluded his “Tractatus Logico-Philosophicus” with this high moral, rather ignored these days: “Whereof one cannot speak, thereof one must be silent”. But human beings seem incapable of doing this, as the only organism on earth forced by its “hyperbrain” to represent the totality of reality in its language, without any apparent end or purpose – in other words, to build theories. But why and how? Once again, dissection operates on and with numbers.
Let’s start with this simple observation: the common numbers we use every day measure what we “have” and the mathematical numbers, with no real limit (even before Cantor), measure what we “want” (The Proof by Googol (2) Numbers and Mathematic). This is why there is literally no common ground between these two expanses of numbers: the former are always annihilated by the latter. If mathematics seems to civilize this “hubris of desire”, it is at the same time sheathing the weapons of excess, or at least of the perpetual surpassing of having. Despite appearances, the implacable regulation of their language is no temperance. Rather, mathematical practice takes the form of an “ordered compulsion” (rules, theorems, logic…), a form that is spread throughout Mundus Numericus by the algorithm.
In Cantor’s case, the onslaught is clearly maximal, for what more can we “desire” than infinity? This maximality has a fascinating consequence: Grundlagen certainly already reveals the excess that characterizes all human practice, but “desire” pushed to the limit here is somehow summoned, in the face of embarrassed or even annoyed mathematicians, philosophers and religious figures, to explain itself or give an account. Cantor must therefore proceed, at least according to our reading, to a rational, mathematical and therefore indisputable dismantling of this “mechanism of excess”, i.e. of the primitive-compulsive form of the “numerical” articulation of body and intellect in the human being.
This dismantling, which (at least) explains infinity, is simple and fascinating. But before we get to that, we want to better understand what Cantor is doing when he does mathematics, i.e. when he builds theories.
Festivals
A festival is a permitted, or rather an obligatory, excess, a solemn breach of a prohibition (Freud – Totem and Taboo[16])
If excess permeates all possible forms of human activity (science, art, war, technology, consumerism… mathematics), the festival is one of its most typical manifestations, and it seems to us that mathematical invention is akin to it, moving like it in the “space of ambivalence” identified by Freud[17]:
[The festival ] is a negation of the Law and at the same time an affirmation of its indispensable presence. […] In these festivities, the “unleashing of instincts” is circumscribed within a spatio-temporal framework pre-established by society or the group. Transgression thus leads to a reinforcement of submission and dependence, since, in the final analysis, its ordering is a matter for the Law.
But if we can discern this “unleashing of instincts” in Cantor’s work – an unleashing that is civilized because it is “a matter for the Law” of mathematics (there is indeed some quite rigorous mathematics in Grundlagen) – the existing Law, stipulating, for example, that the actual infinite is not a mathematical concept, will ultimately be amended: after Cantor’s “festival”, nothing will be the same, and set theory will be imposed on all.
Has mathematics thus progressed? Of course, but the “Law of Excess”, by which every desire must be sublimated into a new, numerically attested possession, remains unchanged. In a way, mathematics only transgresses itself, and that’s how it progresses. Technology proceeds in the same way with “innovation”, which stages transgressions and “disruptions” of an existing world, but perpetuates the same Law of Excess, in obedience to a well-established normative discourse. In the words of Tancredi Falconeri, the character in Giuseppe Tomasi di Lampedusa’s “The Leopard”: “If we want things to stay as they are, things will have to change”. This could be the very law of progress, to which religious “common sense” seems to respond with a counter-maximus: If we want things to stay as they are, nothing must change!
Cortex
The human species thus seems subject to this law of excess and, like the SE, we suspect a cerebral specificity which should be clarified a little. While our brain is relatively large compared to that of other species (albeit three times smaller than that of the elephant) and dense in neurons (albeit ten times smaller than that of the bird), it is above all characterized by a “secondary altriciality”, i.e. a speed of growth and maturation after birth that is unique on earth: the brain of a newborn weighs a quarter of that of an adult at birth and almost half six months later! It is hardly more vulnerable or more fragile than the human newborn, but this physiological disposition also explains…[18]
[…] the enormous, terrifying potential of a creature that spends 75% of its cognitive growth interacting with adults and the environment.
Thus, ingenuity, intelligence, creativity… which most of us develop after a decade or so of maturing and interacting with our fellow creatures, are culturally valued advantages (and rightly so, since they have enabled us to “numerically” dominate all other species). But at the same time, this poses a huge problem, because this essentially “post-partum” civilized brain also deploys activities that we, as living beings, have no obvious need for, and which deplete a biosphere not designed for it. This organ, which is too large and complex, has to work constantly. We all experience this at every moment.
In this direction, we note the theses of philosopher Daniel S. Milo in his latest book, “Good Enough”. The Inquisitive Biologist has commented on this work in an article, of which the following is an extract[19]:
We have achieved this now, but the drive persists, “our bored neurons crave action”. So we construct problems for ourselves to solve, endless diversions to lose ourselves in. Politics, sports, cuisine, art, fashion, science, work – all our culture, all our questing for athletic, emotional, and spiritual goals. They are all but exercises, endless loops, to give our lives meaning. Even though we have never had it so good, we continue to compete as if our lives depend on it, for without it “we would succumb to boredom and despair”.
Meditation
With its hyperbrain, the human being is condemned to a constant search for “existential fulfillment”, a never-ending quest that, if modern coaches are to be believed, should lead the luckiest of us to… ourselves, that is, not far away. We therefore wondered about the opposite human prototype, located at the other end of the brain activity spectrum, where excess would be almost totally contained and Cantorian theory impossible. Is it the “mediocre” who, in Milo’s sense of the word, is precisely “good enough” to survive without having to go through the sieve of ruthless Darwinian selection? Is it even the simpleton whose brain seems to have been abandoned, or the sportsman whose body seems to rule…? For the sensitive souls of the time, we would point out that this spectrum is not that of “intelligence”, with imbeciles on one side (bad) and geniuses on the other (good), but that of brain activity as such. This activity can only really slow down, or even shut down just enough without perishing, at the price of extremely rigorous training, the prototype of which is probably meditation. This cerebrally opposed individual (or state) to the “mathematician”, and therefore to the human being who compulsively builds theories, seems to be the meditator.
Vipassanā meditation[20] for example, one of the most ancient of Buddhist techniques, aims through intense mental training to perceive things as they really are, and to observe the sensations that pass through us in order to better detach ourselves from them. What the American medical professor Jon Kabat-Zinn calls “mindfulness” meditation is a similar quest for authentic mental silence[21]. A more extreme example: sokushinbutsu refers to one of the very few Japanese meditators who are supposed to reach a state beyond life and death. Biologically, these Buddhists have died and, in a way, self-mummified[22]. In all cases, the meditator can only achieve his or her practice of cerebral rest, whether temporary or definitive, at the price of complete social withdrawal. Meditation cannot therefore be a “species” practice, but only a perfectly deliberate individual state of indifference and equanimity. The meditator must in some way want to withdraw from the human condition (here’s a second clue: if a machine with intelligence comparable to that of a human being ever exists, it won’t meditate).
Despite appearances, the mathematician is therefore the opposite of a meditator: he is the human being “naked”. He or she has the same brain, but gives in to all its excesses, festivals, transgresses, disciplining his outbursts as best he can through a logical, rational language that seeks to represent and tell everything, and through a compulsive conceptualization that often has no object other than itself.
This is how the numbers of the Transfinitum appear, in the absolute opposite of meditation: Cantor observes the sensations that pass through his body, not to detach himself from them but to seize them in a theory.
2. Cantor’s body
Necessity dictates Law
Cantor’s works have been the subject of many readings, both mathematical and philosophical, but these readings have never gone beyond a fairly restricted circle of initiates. It is also necessary to emphasize the formidable gap between Cantor’s brilliant achievement, not devoid of a certain aesthetic, and his practical uselessness. The classic argument of “pure” research, which glorifies human reason but above all finds its social justification in the possibility of future spin-offs, doesn’t hold for the arithmetic theory of infinity. This theory never has the slightest chance of concerning us, for the simple reason that infinity is only a “limit concept”, a horizon that we never encounter in practice. At most, it can occupy our hyperbrain as an enigma or paradox.
Since Aristotle, the mathematical and philosophical tradition has cautiously held infinity in a position of potentiality, and for a long time only religion was able to maintain a direct and legitimate relationship with this concept[23]. For the legalist Cantor, welcoming the infinite into the conceptual field of mathematics was a real transgression, a tour de force requiring great precautions and extreme personal tension. This strain could only have been ordered by a compelling necessity linked, for our purposes here, to the natural articulation between body and brain, between what Cantor calls the outside world [Aussenwelt] and the intellect [Verstand]. If this articulation characterizes the “proper characteristic of man”, it must manifest itself in every form of activity (sport, art, cooking, philosophy, literature…), but mathematics seems to have the particular ability to sublimate it in the form of Laws. So we need to interpret these mathematical laws, and above all their reasons for being, as “anthropological” traces of this natural articulation.
Number-as-a-tool
Reality can only be grasped by the intellect and the representational compulsion of our hyperbrain by means of numbers. Admittedly, this statement may come as a surprise, and will be difficult to accept for an artist, philosopher or constitutionalist… who may only see numbers on the horizon of their efforts. But between the conscious grasp of reality and numbers lie a multitude of intermediate concepts, such as language, logic, ideas, forms… that conceal it. Let us observe, for example, that the contemporary technical system which, thanks to the techniques of “artificial intelligence”, now drives our videos, our photos, our music, our language… is based exclusively on numbers, and more precisely on the tiny, invisible bit, the ultimate reality as elusive to our senses as the atom.
Ultimately, Cantor can only proceed to an intellectual grasp of the infinite by means of numbers that he describes as “transfinite” (Gr. §5)[24]:
What I assert and believe I have proved through this work, as well as by my previous research, is that there is, after the finite, a Transfinitum (which might also be called Suprafinitum), i.e. an unlimited scale of determinate Modes which, by their nature, are not finite, but infinite, but which, like the finite, can be determined by specific numbers, well defined and distinct from one another.
But isn’t this statement counter-intuitive? Doesn’t infinity, on the contrary, qualify what is beyond all possibility of measurement? An impotent “etc.”? And yet, Cantor achieved the feat of opening up the intellect to an infinite number of transfinite numbers. This success was only possible because Cantor recognized, implicitly or explicitly, that the concrete appropriation of infinity could only be achieved by means of number rather than any other concept, perhaps because only number, as a representamen, has a direct relationship of truth with reality for human beings.
Sets
But this poses a problem: not all reality presents itself naturally in the form of numbers – a fair objection of our artists, philosophers and other constitutionalists… – an observation by which we now approach Cantor’s “body”. To understand this essential point, we must approach Grundlagen more closely.
Mathematical activity is based in part on a rather imprecise mental operation, for which we have yet to find a clear description, consisting roughly in bringing together “beings” and constituting this gathering into a new being with its own “modes”, to use Cantor’s term above, which he borrows from Spinoza and the Scholastics. It was perhaps the incessant repetition of this elementary mental operation, as well as the precise identification of this operation in other mathematical authors, that led Cantor to have to translate it mathematically. In the end, Cantor sees reality as made up of “things” that the intellect pre-consciously analyzes as “gatherings” of other “things”. As these gatherings become mathematical objects in their own right, Cantor sets out to define them and, above all, to give them a name. Four different terms appear in the following short passage (Gr. Notes to §1)[25]:
I understand […] by multiplicity [Mannichfaltigkeit] or set [Menge], in a general way, any multitude [Viele] which can be conceived as one, i.e. any collection [Inbegriff] of determined elements which can be connected into a whole by a rule.
This remarkable linguistic hesitation governs the birth of a fundamental concept that future set theory would fix once and for all with the word “set” (“Menge” in German, “ensemble” in French…). This concept still seems a little confused and barely nameable, but it undeniably appears in Cantor’s inner vision [Anschauung] after a long mathematical practice.
We can be fooled by the apparent banality of this extract. Common vocabulary would simply speak of “thing”, which the dictionary defines circularly as “an object, fact, affair, circumstance, or concept considered as being a separate entity”[26]. So, whether it is a cloud, rain, an animal, a human being, a rocket, etc., each of these things appears to us pre-consciously as a whole, a “mass”, a “heap”. Either linguistic practice and convention have fixed these perceptions in this way (any “heap” that already has a name is perceived as a thing) or, conversely, the immediate “hardness” or “compactness” of the perception – a bodily affect – indicates to us from the outset that it is a thing that must be given a name. Thus, it seems to us, Cantor perceives “the outside world facing the intellect” (Gr. §8), as elementally constituted of things, both concretely and in the intellect, and which mathematics will take hold of under the concept of “set”. Everything is a “set” and perceived as such, such as this line segment:
Given the universal nature of the concept, it’s not surprising that after Cantor, all mathematics could be rebuilt on the single concept of set, since everything mathematical, i.e. everything that mathematics can talk about, is ultimately a set. Thus, this line segment would ultimately be an infinite set of “points” rather than an indivisible and unanalyzable mass:
This vision may seem natural, but it’s nothing more than an indemonstrable petitio principis. Note in passing Wittgenstein’s objection to “the way we speak of a line as composed of points”, when in fact “a line is a rule and is not composed of anything”. We could say that the hyperbrain is at work here, trying to divide and conquer, to count and measure everything, instead of taking, as the meditator does, reality as it comes forth.
By means of the language game of sets, mathematics seems to have succeeded in breaking away from reality, which it has no use for itself, while remaining as close as possible to the boundary that connects it to us, so that it retains an imprint of it, a model as faithful as possible. But for Cantor, mathematics is more than modeling: it remains firmly anchored to reality thanks to this troubling set-theoretic connection. This is probably the main reason why he is so keen to defend his theory. This is how he justifies himself in his foreword (Gr. Foreword)[27]:
I am well aware that the subject I am dealing with has always given rise to the most diverse opinions and conceptions, and that neither mathematicians nor philosophers have reached universal agreement on this point. I am therefore far from believing that I can have the last word on a subject as tricky, as demanding and as vast as that presented by infinity; but as I had arrived by long research at certain convictions on this subject, and as they have not weakened throughout my investigations, but have become fully consolidated, I felt obliged to put them in order and to make them known.
When, by dint of hard work, experience or indoctrination, we come to see the world in a certain, intrinsically coherent way, i.e. when the dynamic coupling of body and mind (Aussenwelt/Verstand) reaches an “energetic” optimum, nothing can change our mind; this optimum is in fact a possible state of “existential plenitude”, the self that some people seek.
Number-as-a-set
The concept of set, then, is not the one Cantor started from, but rather the one he arrived at in a vast inductive movement. That’s why he won’t be building a somewhat abstract mereology, but a genuine arithmetic of sets. Indeed, since every existing being is a collection of more elementary existing beings, one of the essential modes of any “thing” is the number of things that constitute it, which we can represent elementally as its dimension or internal power. This is the only mode that mathematics can really take hold of and legislate, since all the rest is just “logic”, i.e. ways of talking.
Not only does every set (every “thing”) have a number that is its number of elements but, in Cantor’s vision, reciprocally every number is a set (a thing), or rather is identical to the class of all sets (all things) that are measured by this number (Cantor is in fact not at all interested in other Modes of sets, especially those that might depend on the nature of their elements). The outside world is thus vaguely perceived as made up of sets, disciplined in the intellect under the mode of number. Here we are in the midst of dissection:
The mathematician has thus wrested from the “outside world” the most elementary and universal conceptual unit that human beings can speak about, the set, and has seized upon it in his arithmetical language game, thanks to his universal numerical Mode[28]. Cantor can now examine the outside world from a mathematical point of view, with rather strange questions such as: what is the “number” of this line-segment-set? In other words, how many “points” are there in this set?
It will, of course, be a transfinite number. Infinity is no longer just a religious or limiting concept, but the numerical mode of very concrete sets. The proof? We’ve just drawn a real infinite set! So, like Cantor, all you have to do is look around to see that infinity is everywhere.
Cantor’s ratio / Intrasubjectivity
For his transfinite number-sets, Cantor has established arithmetical Laws [Gesetz] that are as rigorous and consistent as those everyone knows for the usual finite numbers (Laws of addition, multiplication, etc.). We can indeed easily imagine that when two line-segments are juxtaposed, the number associated with the set formed by the resulting segment is the “sum” of the transfinite numbers associated with each of the two segments. The sum of two transfinite (infinite) numbers thus arises “naturally”[29].
In spite of this evidence and the rigor of the Laws highlighted in Grundlagen, Cantor had to face firm opposition from institutional Mathematics, in the person of Leopold Kronecker, a kind of incarnation of the intimate enemy, as well as extreme mistrust from the religious circles whose recognition was so important to him. This is why, throughout Grundlagen, Cantor takes great care to link his work to all possible branches, both philosophical (Plato, Leibniz, Spinoza…) and mathematical (Bolzano, Weierstrass, Dedekind, Cauchy… and even Kronecker, whom he strives to flatter). Mathematical transgression must be a genuine festival, not a tabula rasa. The essence of his argument is set out in the few more philosophical pages of chapter §8 of Grundlagen (which can be read almost independently), and it rests entirely on this proposition of Spinoza’s[30].
The order and connection of the thought is identical to with the order and connection of the things.
Hence, Cantor tells us, the existence or reality of numbers, both finite and infinite-transfinite, can be understood in two ways.
Firstly, everyone has made room for the idea of number, and intuitively “feels” how this idea distinguishes itself from all the others, while at the same time combining with some of them. Numbers thus occupy a specific position in the structure of our intellect, i.e. in the “order and connection of the thought”. Cantor calls this kind of reality “intrasubjective” or “immanent” [intrasubjective oder immanente Realität]; it seems to owe nothing to the outside world, and depends only on the “truth” conditions of a language game that conforms to the structure of the intellect. Thus, seemingly, nothing prevents the mathematician from admitting the intrasubjective reality of any idea, including that of a truly infinite set and therefore that of a transfinite number, as long as the language game remains coherent (Gr. §8 – emphasis added)[31]:
Mathematics is entirely free in its development, and is bound only by the obvious consideration that its concepts are both non-contradictory in themselves, and that they maintain fixed relationships, ordered by definitions, with previously constituted concepts, already available and proven. […] The essence of mathematics lies precisely in its freedom.
There would be no reason to oppose its developments, since mathematics would be free in the realm of intrasubjectivity; so it could accommodate with infinity without bothering anyone. But the argument is lacking, because what Cantor calls “its concepts” propagate wavefronts throughout intrasubjectivity, i.e. in all domains, and can therefore modify the very structure of consciousness, that “proxy” necessary for what we call, without judgment here, the “domination of human by human”. This is said a little sharply, but we can see the threat to the religious, the philosophic or the political, who can’t totally concede a little potentially subversive territory to the mathematician alone, even if it is meticulously organized and intrinsically coherent.
Cantor’s ratio / Transsubjectivity
Cantor knows all this, but his “body” resists and “physio-logically” establishes a link of mutual existence and conditioning between number and set, cluster, multiplicity or thing… a phenomenological link that clearly goes beyond mathematics. Cantor thus understands the existence or reality of numbers, including transfinite numbers, in a second way, which he calls “transsubjective” or “transcendent” (Gr. §8)[32]:
But we can also attribute a reality to numbers insofar as they must be considered as the expression or image of events and relations in the outside world facing the intellect, and where the different classes of numbers (I), (II), (III), etc. are the representatives of powers that actually occur in physical and spiritual nature. I call this second kind of Reality the transsubjective or transcendent Reality of numbers […].
But how does the intrasubjectivity that unfolds within free mathematics, developing its ideas in its own corner, always “just so happen” to accord with this transsubjective reality? Since the order and connection of the ideas is identical to with the order and connection of the things, there comes an argument based on utter faith (Gr. §8)[33]:
This link between the two Realities finds its authentic foundation in the unity of the Whole, of which we ourselves are a part.
Mathematical reality would thus necessarily fit the transcendent reality of things:
However, the so-called “freedom” of mathematics would be difficult to apply if immanent reality were somehow constrained by transcendent reality. But Cantor, as a good Platonist, considers that through the “unity of the Whole”, reality presents itself from the outset in these two aspects, without the need to “test” or “align” them a posteriori. If it appears to the intellect in the form of numbers, for example, this appearance has necessarily a transsubjective reality, and there is therefore no rationale to reject it a priori (Gr. §8)[34]:
It is not necessary, I believe, to fear in these precepts, as often happens, any danger for science; on the one hand, the conditions indicated in which the freedom to form numbers can only be exercised are such that they leave only an extremely reduced margin for arbitrariness; but moreover each mathematical concept carries within it the necessary corrective; if it is sterile or inappropriate, it very quickly reveals its uselessness and it is then abandoned, for lack of success.
This is where the freedom of mathematics lies: because it carries within it its own regulatory principles, it must not restrict itself in any way. Using an argument that Cantor lent to his critics, and which he ultimately turns around to support his own theory, these regulatory principles (Gr. §4) …[35]
…should serve to indicate the true limits of the speculative and conceptual flight of mathematics, where it does not risk leading to the abyss of the “transcendent”, where, as we say with fear and salutary dread, “anything might be possible”.
So, if the concept of the actual infinite finds its natural place in the intellect among the other concepts, a place clearly regulated by arithmetic, it’s because the (set-theoretic) infinite really exists, transsubjectively, and indeed for Cantor it has come to take shape through practice and exhausting observations[36]. Cantor may be exhausted, but he remains confident, rightly pointing out that the transgression that led him to transfinite numbers is no more out of place than those that led his illustrious predecessors to negative, complex or irrational numbers. These adventures have seen the same hesitations, the same rejections, and ultimately the same glories.
Final dissection
The “physio-logical” perception of the outside world in the form of sets thus led Cantor to form in his intellect the idea of transfinite numbers. “Infinite” sets combine before his eyes, leading to the arithmetical waltz of the numbers that measure them.
Now we need to study how the infinity of transfinite numbers is organized and what its structure is. Cantor goes to great lengths to explain this to us; Grundlagen then takes on a more mathematical tone, but at the same time comes in search of explanations, even descriptions, as fragile and delicate as arabesques, leaving us to think that Cantor is touching on that mysterious “articulation” of the human body with the world in front of it. He then describes, rather than explains, the deployment of transfinite numbers through the operation in the intellect of two “principles of generation” that we assume belong, in their full development, to that famous “proper characteristic of human”.
3. Principles of generation
The two principles of generation by means of which […] the newly determined infinite numbers are defined, are of such a kind that, through their combined action, there are no longer any barriers to the conceptualization of the actual integers. (Gr. §1)[37]
Once the imprint of the outside world on mathematics in the form of sets has been accepted, the numbers that accompany them and the arithmetical laws they obey are presented as derived but somehow natural concepts, so that Cantor’s arguments on the structure of transfinite numbers are based on a quasi-anthropological analysis of the articulation between the intellect (immanent reality) and the outside world (transcendent reality) in human beings. The main result of this analysis/dissection is the presentation of two “principles of generation” (associated with a third, so-called “principle of limitation”, which we won’t go into here).
First principle of generation
What Cantor calls the “first principle of generation” is what we called an “etc.-algorithm” in Body and Language Games, where “etc.” is the name of the stable, repeated algorithm by which a form is revealed in the intellect: extending a line segment or, like Cantor, adding “1” over and over to the previous number. A first section of the “transfinite structure” appears as follows (Gr. §11)[38]:
The series (I) of positive actual integers 1,2,3,…,ν,… finds its raison d’être in the repeated actualization and gathering of units considered identical; the number ν is the expression both of a determined finite quantity of these successive actualizations and of the gathering of the accumulated units into a whole. The formation of finite actual integers thus rests on the principle of adding a unit to an existing number already formed; I call this moment […] the first principle of generation.
Note the meticulousness used to describe a very elementary operation: the addition of “1” to any “determined” number. But if Cantor develops certain details in this meticulous way, it’s because he’s undertaking this “SE work” of methodically teasing out a natural basis for the human approach to reality. So we could read this important passage as a description of an “anthropological” fact rather than a mathematical statement. Three important propositions arise here.
Firstly, Cantor speaks of a “moment” [Moment] and even, later in the text, of a “logical moment”. Remember that, unless it becomes “sterile or inappropriate”, every idea corresponds to a transsubjective reality. This moment is therefore less a moment of “generation” than a further step along one of the paths that lead to this reality[39]. Let’s look at the deep analogy between this kind of “moment”, or principle, at work in the intellect and the body’s forward movement. Cantor’s first principle of generation is thus the mathematical representation of the natural and compelling forward (or forward-looking) movement of the dynamic body/intellect couple, a movement to be understood here in the broadest sense.
Secondly, Cantor breaks down this “extra step” into two successive elementary “gestures”: “actualization” [Setzung], which designates the actual realization of the extra step, such as adding 1 to an existing number, and “gathering” [Vereinigung] which consists in considering the result as a completed whole, i.e. a set, a heap, or an aggregate from which a new step will be taken, since we have to start from somewhere at every moment. In this way, number represents not just the “quantity” of completed steps or “repeated actualizations”, but also their “gathering”, i.e. the totality of these actualizations, a new point of departure[40]. In a way, actualization designates the bodily and actual gesture of forward movement, which any “thing” could be satisfied with in order to move forward (by simple inertia, for example), while gathering designates the intellectual and reflexive gesture of the same movement consisting in taking as such, as a “whole”, the “point where we are”[41].
Thirdly, it is a “repeated” movement, since it involves “units” [Einheiten] “considered identical”. But what are these “units”, and how do we go about treating them as identical? These units are not mathematical units such as “1”, but things distinguished from transsubjective reality (sheep, for example) and “considered”, i.e. envisaged intrasubjectively, as “identical” (in our eyes, every sheep is a sheep). There is therefore an operation by which the thing identified (the sheep) is reduced in the intellect to an elementary identity that mathematicians call “1”:
Problems
Cantor, the “anthropologist”, goes on to detail the general foundations of the raison d’être of numbers, foundations on which he will base his legal justification of transfinite numbers: (1) the logical moment as the expression of the natural and imperative forward movement, (2) the decomposition of this movement into an actual and reflexive double gesture, (3) the repetition of this movement based on the identification of units considered to be identical. This elegant and convincing theory does not apply identically to all beings: everything depends on their “body”.
Remember that for the Cantorian being, everything is a priori a set (a thing) and perceived as such. There is nothing to add to this: we are on the edge of the speakable. Indeed, we can always say that a sheep, a mountain, the wind… everything for which we have invented a noun is first presented as a “heap”, without much risk of being contradicted. But then, how does the inevitable “counting” take place, i.e. where and how do we see identical units? There are at least two ways of proceeding: synthetically or analytically.
Synthetically, we can “count sheeps” to obtain a “flock”. The actual gesture consists of passing the sheep in front of you and, if necessary, passing your finger over a notch on your counting stick (The Proof by Googol (1) Numbers and Progress). The reflexive gesture consists in considering all the sheep already passed as a provisional flock and, where appropriate, the counting stick as the “numerical mode” of this flock. This first way of proceeding depends on the Law by which two “things” are considered to belong to the same “set”.
Analytically, we can consider the sheep itself as a completed aggregate and try to determine its number, i.e. “count” it. We then realize that there are as many ways of doing this as there are ways of “cutting up” the sheep into identical units. We can, for example, weigh it, count its atoms, its bones, its age… and each time the method, the “path” and the result are different. Once again, a Law is needed.
These two methods of counting reality show that the result depends 1) on a Law that must match the concrete possibilities of the being counting and 2) on the intention behind (so it is not certain that a machine could count in the same way as a human being). We can agree that everything is intrinsically a set, a cluster or an aggregate (in any case, this can hardly be said, and therefore contradicted). But to consider, as Cantor seems to assert, that every “thing” has, on the pretext that it is a set, a numerical mode in itself is, in our view, hardly admissible. This particular mode only exists for a given organizing organism, under certain conditions and for a certain purpose. Thus, to count a set (a thing) always requires a path, or a method, which depends to some extent on the possibilities of body movement (and its technical prostheses) available to the measuring being[42]. Returning for a moment to Wittgenstein, for whom “a line is a law and is not composed of anything” and is therefore not countable, the fact remains that this law does indeed depend on a legislator: the being who considers this line as a thing.
This is why, if the first principle of generation is not specific to human and could well apply to other organisms, the “trajectory” it determines depends on a way of moving specific to a certain body articulated to a certain intellect. This reading of Cantor’s first principle of generation brings us back to the concept of “enaction” introduced in Francisco Varela, the heterodox[43]:
Enaction faces the problem of understanding how our existence, the practice of our life, is coupled with a surrounding world which appears filled with regularities which at every moment are the result of our biological and social history… to find a medium way to understand the regularity of the lived world which we experience at every moment, but without any other point of reference than ourselves which would give certainty to our descriptions and affirmations […]
Georg Cantor is convinced that the harmony of the world and of ideas rests on a transcendent “Unity of the Whole”, but he has no point of reference other than himself. Mathematical language seems to put his “I” at a distance (the number is this or that…), but it must be recognized that the first principle of generation only works to the full on the “proper characteristic” of an organism capable of a (reflexive) experience of every moment.
Second principle of generation
The second generating principle, both simpler and more mysterious, is introduced rather abruptly by Cantor. Far from corresponding to this repeated and somewhat mechanical movement of the first principle, it is at the origin of a “conceptual leap” of the same nature as that accomplished by the Chinese mathematician Liu Hui 1700 years ago[44]:
Confronted with the impossibility of extracting the root of a number according to an algorithm, Liu Hui still gives a result to the operation. But this one is different in nature from what a calculation would have given. We refer to it, not by stating it explicitly, but by giving it a name, more precisely by giving the name of “root” to the number on which we operate. It will be “root of”.
Cantor’s first transfinite number arises from an analogous conceptual leap (Gr. §11 – in the following passage, we can think of “class (I)” as the set of integers: 1, 2, 3…)[45]:
The quantity of numbers of class (I) to be constructed in this way is infinite, and there is no larger of them. If it is therefore contradictory to speak of a greater number of the class (I), there is nothing shocking, on the other hand, in conceiving a new number, which we will call ω, and which would be the expression of the fact that the whole collection (I) is given, according to the law, in its natural succession.
Among the integers 1,2,3…, there is indeed none that is the greatest, and it is therefore quite “contradictory” [widerspruchsvoll] to want to designate one of them. But, Cantor first tells us, there’s nothing “shocking” [Anstößig] about naming “ω” the set of these numbers (this thing) and, he adds further on, “creating [in this way] a new number that is conceived as the limit of these numbers, i.e. that is defined as the number immediately greater than all of them”. We see here, in all its clarity, the reason for the necessity of the existence of a first transfinite number “ω”: a number must be a set, i.e. correspond to a thing. The fact that every transsubjective set has a numerical mode (is “measurable”) gives rise to the idea that there also exists, transsubjectively, the pure number-set. “ω” is thus both the name of the infinite set 1,2,3… and the first number that follows them all. Just as Liu Hui authoritatively completes the infinite calculation of an irrational root, Cantor completes the infinite iteration of the etc.-algorithm “+1”. These completions are always sealed by names. It is then, as the German philosopher Ernst Cassirer put it, that “the function of signification attains pure autonomy”[46]. Starting with “ω”, the smallest infinite set, Cantor becomes free to develop the extraordinary series of transfinite numbers in the sole signifying territory of mathematics.
All Cantor had to do then was convince his detractors of the existence of “ω” alone as a number and not, like “∞”, as an indeterminate horizon for the potentially endless application of the first principle of generation. His argument that there’s nothing “shocking” here remains short-lived, perhaps because this abrupt manner comes not from mathematics but from the depths, from some kind of inexplicable vital impulse. The only real intellectual argument for the existence of “ω”, the first transfinite number, remains that of the Unity of the Whole introduced in §8 of Grundlagen, which grants that if “ω” resists (in the intellect), then it exists (in the world). But this argument itself, steeped in mysticism, remains flimsy and, above all, supremely human.
A step sideways with art
In all fields, concepts (names for things) have eventually joined the common corpus after long theoretical gestations. Mathematicians, for example, have come to admit the long-problematic existence of the numbers “-1” (negative), “π” (irrational) or “i” (complex)… These concepts are more or less stable, but they constitute a structure that only language in the broadest sense, a concrete artificial form, can represent and manipulate.
To better grasp the scope of the two principles of generation, let’s take an example from artistic practice, based on the principle that an (authentic) work of art is like a “name of thing” seeking to “join the common corpus after a long gestation period”. Each work of art is more or less easily received, and sometimes violently rejected, before eventually achieving glory. A work of art is, in truth, a theory.
The painter “develops” his painting by means of tools and gestures repeated over a long period of time. Tools and gestures form what we might call a technique. Having said this, let’s consider the final result, the finished painting. Did this painting exist a priori in the painter’s mind, and is it therefore merely the reproduction of an existing idea, or is it a surprise to him, owing its “discovery” only to the instinctive use of his technique and his endurance? Certainly a little of both. In any case, the painting did not exist in the outside world before his intervention, and in this sense, painting and theory-building (an activity to which human beings compulsively devote themselves) are one and the same thing. It could be, then, that the activity of painting stems from the same principles of generation, with the technicized body enabling concrete movement governed by these principles.
Let’s start with the first principle, whose characteristics we recall: (1) the logical moment as the expression of a natural and imperative forward movement – in this case, the brushstroke; (2) the decomposition of this movement, for the human being, into an actual and reflexive double gesture – the brushstroke and the result as a provisional painting; (3) the repetition of this movement based on the identification of units considered identical – the identity of the brushstrokes insofar as they participate in the same painting. In this way, the painting is revealed after each brushstroke, just as the set of numbers is revealed after each iteration. But the application of the first generating principle alone, a kind of “etc.-algorithm” for painting, never completes the painting, just as we never finish counting whole numbers. The painter must therefore decide when the painting is finished. This decision corresponds to the moment of the second principle of generation, that abrupt “conceptual leap” from the depths that overcomes a perpetual movement to symbolically complete it (since in concrete terms it’s impossible): the painting finds a name, a place, a frame, a buyer…
The second principle of generation therefore seems to be the key principle of creation. It corresponds to the moment when a concept unveiled by the incessant forward march of the first principle finds its place in the intellect. Each concept born of the second principle is thus the trace of a repetition likely to be taken up again each time it is evoked. Every time a painter contemplates his work, he remembers the brushstrokes and imagines new ones…
In a nutshell
This reading has opened up a number of new avenues.
Let’s start by recalling that Georg Cantor developed his “theory of multiplicities”, which is in fact a mathematization of the actuel or “authentic” infinite, against the mathematical dogmas of the time and, above all, by seeking to tune his true religious feeling to this secularized and potentially “heretical” infinite. The compelling need to build such a theory, to the point of literally making himself sick of it, seems to stem from that “proper characteristic of human”: the need to constantly exceed what he already has.
The intellect has an imprint of the world (“The brain takes on the ‘shape’ of the body’s experiences”, as we said in The Brain is an Edge (not a Center)), and language then plays a dual function: a function of representation, the collective memory of this imprint, and a function of action, i.e. of effective articulation between the intellect and the body in contact with the outside world. All animals probably have such a “conceptual structure”, however elementary, but only human beings have a hyperbrain whose cognitive development takes place mainly post-partum, in interaction with other animals and the environment. Human beings don’t come with a ready-made “conceptual structure”, but they do get used to continually adapting it to their affections, and even develop a need for it. His hyperbrain never stops building theories: inventing transfinite numbers, painting works of art, even brooding over conspiracy theories without any contact with the real world…
In Grundlagen, Georg Cantor grapples with this particular disposition to “excess” on two levels: firstly, he succumbs to it by deploying his theory of transfinite numbers, a kind of compulsive conquest of numerical territory; secondly, he dissects the “Law of Excess” itself, which has become totally transparent with the vertiginous deployment of transfinite numbers, a dissection based on the theory of the two “principles of generation”. In seeking to base actual infinity on a “natural” cause, Cantor incidentally uncovers principles that are probably valid, at least on a descriptive level, for all forms of cognitive activity, whether human, animal or mechanical. These principles of generation, combined with some of Cantor’s more anagogical arguments, simultaneously outline a scale of cognitive capacities (or, as we now say, “intelligence”) that we will explore in the second part of this Georg Cantor-inspired exploration (coming soon).
1. ↑ José Ferreirós / Science in Context 17(1/2), 49–83 (2004) – 2004 – The Motives behind Cantor’s Set Theory – Physical, Biological, and Philosophical Questions
2. ↑ Wikipedia – New Mathematics – This pedagogical byproduct, however, has little to do with Cantor’s work. See also Naive set theory
3. ↑ Wikipedia – Georg Cantor
4. ↑ Stanford Encyclopedia of Philosophy – 2018 – Wittgenstein’s Philosophy of Mathematics
5. ↑ David Hilbert / Math. Ann., t.95, pp 161-190 – 1925 – On the infinite
6. ↑ Pope Pius IX’s “Syllabus of Errors”, published in 1864, some twenty years before Grundlagen was published, is a vigorous critique of pantheism, naturalism and absolute rationalism.
7. ↑ Created by mathematician John Wallis in 1655 and set out in his book De sectionibus conicis
8. ↑ Not so much about infinity itself as about one of its particular manifestations. But astonishment seems to accompany all his work.
9. ↑ Or even “Foundations of a general theory of sets” as for example in Logic and Language by James Meyer. But, at least for a French reader, the word “set” (“ensemble”) is problematic in the context of Grundlagen.
10. ↑ It should be noted, however, that this test is based on an uncertain premise: the permanence of “reason” as a characteristic of the human species…
11. ↑ Baruch Spinoza – 1677 – Ethics III. Note of Proposition II: “Body cannot determine mind to think, neither can mind determine body to motion or rest or any state different from these, if such there be.”
12. ↑ (in French) Le Monde – April 7, 2024 – André Comte-Sponville, philosophe : « La mort ne peut plus me prendre qu’une partie de ma vieillesse, et sans doute pas la plus intéressante » – “Le désir est premier, donc n’attendons pas que la vie soit facile, heureuse pour commencer à l’aimer…”
13. ↑ “En livrant ces pages au public, je ne veux pas oublier d’indiquer que je les ai écrites en pensant avant tout à deux types de lecteurs : les philosophes qui ont suivi le développement des mathématiques jusqu’à nos jours et les mathématiciens qui sont familiarisés avec les principaux développements, anciens et récents, de la philosophie.”
14. ↑ Nathaniel Herzberg / Le Monde – March 18, 2024 – La mort de Frans de Waal, le primatologue qui voulait remettre « sapiens » à sa place – “Dans ma vie, j’ai dû voir vingt-cinq propositions sur le propre de l’homme […]. Toutes sont tombées. On perd notre temps. (…) Pourquoi toujours chercher ce qui nous est unique, à nous ?”
15. ↑ The French pun is unfortunately impossible to render in English. “Démesure”, translated here as “Excess”, literally means “that which exceeds the capacity to be measured”.
16. ↑ Sigmund Freud – 1913 – Totem and Taboo – p.163 (translation by James Strachey)
17. ↑ Amadeo López / Cahiers du CRICCAL, année 2001, 27, pp. 5-9 – 2001 – La fête. Solennité, transgression, identité [liminaire] – “[ La fête ] est négation de la Loi et en même temps affirmation de son indispensable présence. […] Dans ces fêtes, le « déchaînement des instincts » est circonscrit dans un cadre spatio-temporel préétabli par la société ou le groupe. La transgression aboutit ainsi à un renforcement de la soumission et de la dépendance puisque, en définitive, son ordonnancement relève de la Loi”.
18. ↑ (in French) Daniel S. Milo / Annuaire de l’EHESS 2007-2008 – Philosophie naturelle
19. ↑ The Inquisitive Biologist – August 2019 – BOOK REVIEW – GOOD ENOUGH: THE TOLERANCE FOR MEDIOCRITY IN NATURE AND SOCIETY
20. ↑ Wikipedia – Vipassana movement
21. ↑ Inevitably, this technique has ended up in the clutches of therapeutic recycling, and therefore of a lucrative market, along with “cognitive therapies” in general, which are controversial by the way. Once again, these therapies are symptomatic of the cerebral excess that sometimes overwhelms us.
22. ↑ Julia Shiota / National Geographic – January 19, 2024 – Why did these monks in Japan choose to mummify themselves?
23. ↑ For example: Adam Drozdek / Laval théologique et philosophique, Volume 51, numéro 1 – février 1995 – Beyond Infinity: Augustine and Cantor
24. ↑ “Ce que j’affirme et que je crois avoir prouvé par ce travail, ainsi que par mes recherches antérieures, c’est qu’il y a, après le fini, un Transfinitum (qu’on pourrait aussi appeler Suprafinitum), c’est-à-dire une échelle illimitée de Modes déterminés qui, par leur nature, ne sont pas finis, mais infinis, mais qui, comme le fini, peuvent être déterminés par des nombres spécifiques, bien définis et distincts les uns des autres.”
25. ↑ “J’entends […] par multiplicité [Mannichfaltigkeit] ou ensemble [Menge], d’une manière générale, toute multitude [Viele] qui peut être conçue comme une, c’est-à-dire toute collection [Inbegriff] d’éléments déterminés qui peuvent être reliés en un tout par une loi […]”
26. ↑ Dictionary.com – Thing
27. ↑ “Je suis bien conscient que le sujet que je traite a de tout temps fait l’objet des opinions et des conceptions les plus diverses et que ni les mathématiciens ni les philosophes ne sont parvenus à un accord universel sur ce point. Je suis donc très loin de penser que je puisse avoir le dernier mot dans une matière aussi délicate, aussi exigeante et aussi vaste que celle que présente l’infini ; mais comme j’étais parvenu par de longues recherches à des convictions déterminées sur ce sujet, et qu’elles n’ont pas faibli tout au long de mes investigations, mais qu’elles se sont pleinement raffermies, je me suis senti obligé de les mettre en ordre et de les faire connaître.”
28. ↑ We thus echo to the note about Cantor in the exploration The Proof by Googol (2) Numbers and Mathematic, observing that if the number is the historical/anthropological figure of “ownership”, it is indeed the Cantorian set that “gathers” that is the source concept.
29. ↑ Even so, this result may seem paradoxical, since two joined segments produce a new segment that appears to be measured by the same “infinite” number. But the mathematical developments in Grundlagen, and in the texts that preceded it, have precisely highlighted the importance of the “way of counting”, i.e. the order of the elements, in the final result.
30. ↑ Baruch Spinoza – 1661-1675 – Ethics, II, proposition VII – “Ordo et connexio idearum idem est ac ordo et connexio rerum”
31. ↑ “Les mathématiques sont entièrement libres dans leur développement et ne sont liées qu’à la considération évidente que leurs concepts sont à la fois non contradictoires en eux-mêmes et qu’ils entretiennent des relations fixées, ordonnées par des définitions, avec les concepts précédemment constitués, déjà disponibles et éprouvés. […] l’essence des mathématiques réside précisément dans leur liberté.”
32. ↑ “Mais on peut aussi attribuer une réalité aux nombres dans la mesure où ils doivent être considérés comme l’expression ou l’image d’événements et de relations dans le monde extérieur faisant face à l’intellect, et où les différentes classes de nombres (I), (II), (III), etc. sont les Représentants des puissances qui adviennent effectivement dans la nature physique et spirituelle. J’appelle cette deuxième espèce de Réalité la Réalité transsubjective ou encore transcendante des nombres […]”
33. ↑ “Ce lien entre les deux Réalités trouve son fondement authentique dans l’unité du Tout, dont nous-mêmes faisons partie.”
34. ↑ “Il n’est pas nécessaire, je crois, de craindre dans ces préceptes, comme cela arrive souvent, un quelconque danger pour la science ; d’une part, les conditions indiquées dans lesquelles la liberté de former des nombres peut seulement s’exercer sont telles qu’elles ne laissent qu’une marge extrêmement réduite à l’arbitraire ; mais de plus chaque concept mathématique porte en lui le correctif nécessaire ; s’il est stérile ou inapproprié, il se manifeste très vite par son inutilité et il est alors abandonné, faute de succès.”
35. ↑ “… doivent servir à indiquer les vraies limites de l’envol spéculatif et conceptuel des mathématiques, là où il ne risque pas de mener vers l’abîme du « transcendant », là où, comme on le dit avec crainte et dans un effroi salutaire, « tout est possible ».”
36. ↑ Similarly, to use a mathematical argument, it is because a number exists, in both ways, that a numerical series can admit it as a limit, and not vice versa. So, in a way, the “experience” of the convergence of a series – an experience that takes time – awakens the idea of an irrational number and brings it to consciousness (see also Liu Hui overcomes a monster). This remark also ties in with the idea of the concept as singularity, i.e. the result of a history (The Body of René Thom (singularities)).
37. ↑ “Les deux principes générateurs à l’aide desquels […] les nouveaux nombres infinis déterminés sont définis, sont d’une espèce telle que, par leur action combinée, il n’existe plus aucune barrière à la conceptualisation des nombres entiers actuels.”
38. ↑ “La série (I) des nombres entiers actuels positifs 1,2,3,…,ν,… trouve sa raison d’être dans l’actualisation et le regroupement répétés d’unités considérées comme identiques ; le nombre ν est l’expression aussi bien d’une quantité finie déterminée de ces actualisations successives que du regroupement des unités accumulées en un tout. La formation des nombres entiers actuels finis repose donc sur le principe de l’ajout d’une unité à un nombre existant déjà constitué ; j’appelle ce moment […] le premier principe générateur.”
39. ↑ Series must therefore lead to existing limits; it is the existence of these limits that, according to Cantor, reversing the usual reasoning of mathematicians, justifies the “asymptotic” behavior of series.
40. ↑ The formalism of axiomatic set theory gives substance to these numbers (integers) seen as completed sets.
41. ↑ In The Body of René Thom (singularities), we identified this imaginary point as a “singularity”. The unbroken thread of our consciousness is woven from these singularities.
42. ↑ Cantor is thus obliged to impose a purely intrasubjective condition (in a way specific to human) on these sets, which would later be called “ordinals”: that of being “well-ordered”. Without going into the details of this subtle and very precise mathematical notion, this condition imposes the presence of an exogenous rule for the path of the set, i.e. a path that the human “body” can take. But whether anything in the outside world can be “well-ordered” by ourselves, nothing is less certain.
43. ↑ Wikipedia – Enactivism
44. ↑ Quoted in Liu Hui overcomes a Monster: (in French) Chemla Karine / Revue d’histoire des sciences, tome 45, n°1, 1992. pp. 135-140 – 1992 – Des nombres irrationnels en Chine entre le premier et le troisième siècle
45. ↑ “La quantité des nombres ν de la classe (I) à construire ainsi est infinie et il n’y a pas de plus grand d’entre eux. S’il est donc contradictoire de parler d’un plus grand nombre de la classe (I), il n’y a rien de choquant, d’autre part, à concevoir un nouveau nombre, que nous appellerons ω, et qui serait l’expression du fait que toute la collection (I) est donnée, suivant la loi, dans sa succession naturelle.”
46. ↑ (in French) Ernst Cassirer – 1997 – Trois essais sur le symbolique – Œuvres VI