Number and Numbers

by Alain Badiou

The political regime of global capitalism reduces the world to an endless network of numbers within numbers, but how many of us really understand what numbers are? Without such an understanding, how can we challenge the regime of number? In Number and Numbers Alain Badiou offers an philosophically penetrating account with a powerful political subtext of the attempts that have been made over the last century to define the special status of number. Badiou argues that number cannot be defined by the multiform calculative uses to which numbers are put, nor is it exhausted by the various species described by number theory. Drawing on the mathematical theory of surreal numbers, he develops a unified theory of Number as a particular form of being, an infinite expanse to which our access remains limited. This understanding of Number as being harbours important philosophical truths about the structure of the world in which we live. In Badiou's view, only by rigorously thinking through Number can philosophy offer us some hope of breaking through the dense and apparently impenetrable capitalist fabric of numerical relations. For this will finally allow us to point to that which cannot be numbered: the possibility of an event that would deliver us from our unthinking subordination of number.

Genres: Philosophy, Mathematics

240 pages, Paperback

First published January 1, 1990

About the author

Alain Badiou, Ph.D., born in Rabat, Morocco in 1937, holds the Rene Descartes Chair at the European Graduate School EGS. Alain Badiou was a student at the École Normale Supérieure in the 1950s. He taught at the University of Paris VIII (Vincennes-Saint Denis) from 1969 until 1999, when he returned to ENS as the Chair of the philosophy department. He continues to teach a popular seminar at the Collège International de Philosophie, on topics ranging from the great 'antiphilosophers' (Saint-Paul, Nietzsche, Wittgenstein, Lacan) to the major conceptual innovations of the twentieth century. Much of Badiou's life has been shaped by his dedication to the consequences of the May 1968 revolt in Paris. Long a leading member of Union des jeunesses communistes de France (marxistes-léninistes), he remains with Sylvain Lazarus and Natacha Michel at the center of L'Organisation Politique, a post-party organization concerned with direct popular intervention in a wide range of issues (including immigration, labor, and housing). He is the author of several successful novels and plays as well as more than a dozen philosophical works.

Trained as a mathematician, Alain Badiou is one of the most original French philosophers today. Influenced by Plato, Georg Wilhelm Friedrich Hegel, Jacques Lacan and Gilles Deleuze, he is an outspoken critic of both the analytic as well as the postmodern schools of thoughts. His philosophy seeks to expose and make sense of the potential of radical innovation (revolution, invention, transfiguration) in every situation.

Reviews

[BlackOxford · Rating 3 out of 5]

French Arithmetic

There is always something of the tongue-in-cheek about French philosophy. An ironical tone as if communicating to a colleague over your shoulder about whom the reader should have known but doesn’t. As if the writer is making history, perhaps, and giving the reader a privileged seat before the prizes are handed out.

What seems like hot air may only be heavy breathing.

Badiou considers that numbers dominate our lives. Not just in economics where numbers are obviously important, but in everything from medicine to culture. They constitute our practical existence. What is counted counts. And yet most of us feel alienated from numbers. Mathematics is arcane wisdom and aside from making change for the purchase of a newspaper or a restaurant tip, we have no interest in these numbers. But Badiou thinks we should: “if we don’t know what a number is,... we don’t know what we are.”

Fair enough.

So, Badiou is on a quest - to overcome “the despotism of number,” to correct the condition that “we have at our disposal no recent, active idea of what number is,” to create “a second modernity,” the route to which “... constrains thought to return to zero, the infinite, and the One. A total dissipation of the One, an ontological decision as to the being of the void and that which marks it, proliferation without measure of infinities: these are the parameters of such a passage. The amputation of the One delivers us to the unicity of the void and to the dissemination of the infinite.”

Who knew?

What motivates Badiou is that the leading lights of 19th century mathematics - Dedekind, Cantor, Frege and Peano - couldn’t come up with a theory that included all the various kinds of numbers - natural, real, integers, rational, ordinal, etc. Each used a distinct but incomplete method - axiomatics, set theory, logic, etc. But the result has been disappointing: “The thinkers of number have only in fact been able to demonstrate how the intellectual procedure that conducts us to each species of ‘number’ leaves number per se languishing in the shadow of its name.”

I can’t help but feel deep sympathy for such numerical languish. One is reminded daily of the numbers who suffer but never of the suffering numbers!

Some of the ideas of these mathematicians are arresting. For example, Frege’s definition of Zero as that which “is not identical with itself.” Since by definition something not identical to itself does not exist, voila Zero appears. Even more remarkable, Frege goes on to define "One as the number that belongs to the concept ‘identical to Zero’". Which makes perfect sense since Zero is not identical to itself.

Confused yet?

Nevertheless there are also some enlightening observations. For example, Badiou recognises that the Russell Paradox* which blew Frege’s logical boat out of the water has a profound implication: “It is impossible, says the ‘paradox,’ to accord to language and to the concept the right to legislate without limit over existence.” In other words, the map of language is not the territory of reality, even when the language is extremely precise.

Dedekind takes a very different approach. For him the issue is not building up a number system but selecting from an already established infinite set so that the unit of analysis, as it were, is “not ‘a’ number, but N, the simply infinite ‘system’ of numbers.” **Using this tack, Dedekind is able to show that when it comes to infinite numbers, the Euclidean maxim that the whole is always greater than the sum of its parts is in fact bunk. Unfortunately for Dedekind, he uses a variant of the Cartesian ‘cogito’ or ego which is outside the infinite system it hypothesises and so ends up in a paradox similar to that of Frege.

It is in discussing Peano that Badiou gets up a head of intellectual steam and plies into his real foe - language-based philosophy: “We see here, as if in the pangs of its birth, the real origin of that which Lyotard calls the ‘linguistic turn’ in western philosophy, and which I call the reign of the great modern sophistry: if it is true that mathematics, the highest expression of pure thought, in the final analysis consists of nothing but syntactical apparatuses, grammars of signs, then a fortiori all thought is under the constitutive rule of language.”

Yup. And if it’s the case with mathematics, what chance does even the French language have for claiming precision in representing reality? I feel Badiou’s pain. And I understand why he quickly exits that field of battle in order to engage his remaining Mathematicians, who are at least gentlemen in their ignorance of linguistics.

Cantor was also concerned with the infinite, but with the infinite in a grain of sand rather than the whole beach. His insight was that any subset of real numbers, no matter how limited, was also infinite. Put another way, no matter how precisely two numbers might be expressed, there is always a number between them. This leads to the counter-intuitive conclusion that there are more parts to any set than there are elements. I can’t follow Badiou’s digressive criticism of Cantor but it appears to be summarised in the phrase “We do not want to count; we want to think counting,” and somehow this has to do has to do with cats.

Eventually Badiou comes to rest on a concept he believes addresses all the various issues of what numbers really are. They are, he believes ‘surreal.’ Not being a mathematician, Surreal Numbers are new to me despite the fact they have been around for some time. From what I understand in Badiou’s presentation (not that much), I am sympathetic to the general idea. The concept is presented, appropriately enough, not in a professional paper but in a novel from 1974, Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness, which begins “In the beginning everything was void, and J.H.W.H. Conway began to create numbers. Conway said, ‘Let there be two rules which bring forth all numbers large and small...’”

And on it goes. The two rules, the syntax of/for surreal numbers, are enough to generate almost all the sorts of numbers Badiou is worried about. There is of course one small problem: these two rules are the apotheosis of Lyotard’s ‘linguistic turn;’ they are the fundamental grammar of what Badiou considers a coherent mathematics. And indeed they do mean that all thought is ‘under the constitutive rule of language.’ It seems we had already arrived at the destination before we departed on the journey. As I said: French philosophical irony.

See here for a more or less parallel argument: https://www.goodreads.com/review/show...

*Regarding the membership of the set of all sets with itself.

** This he calls “secularisation of the infinite,” a lovely Badiouan flourish.

[Alexander · Rating 4 out of 5]

Somehow, Alain Badiou has managed to write both an incredibly straightforward and very conventional account of Number, while at the same time having written an wildly idiosyncratic and very, well, Badiou-specific exposition of it too. How? Well, on the one hand, the answer to 'what is Number?' for Badiou is torn straight out of what might be any upper grade math textbook on number systems: to be a Number is to be a species of what are called 'Surreal Numbers', a kind of super-category of Number which contains, within itself, almost all other kinds of numbers that we are familiar with: whole numbers (1,2,3...), rational numbers (1/2, 3/4), irrational numbers (pi, the square root of 2), and so on. Surreal numbers really are a thing - they were invented/discovered/elaborated by John Conway and Donald Knuth back in the 70s - and for the most part, Badiou simply reproduces their results, albeit in his own, inimitable style. As he writes here, any disagreement he has with them is merely 'poetical'.

On the other hand, as has been his position for some time, for Badiou, it is also the case that the mathematicians know not of the (philosophical) significance of what they speak! Surreal numbers are not just another kind of number in a long line of different kind of numbers stumbled upon by mathematicians - they are the very Stuff of Number itself, they are that from which every other number is made from and are cut out of. Thus in this philosophical mode does Badiou offer his definition of number: Number is a form of Being. The idea being that 'Being' shows up in many forms, and Number is one among them. As for what this means, suffice to say that Number - understood as surreal numbers - shares the same 'properties' (understood loosely) as all things considered under the sign of Being. In particular, the property of being composed of what Badiou calls 'multiple-being' - a kind of base-level of Being out of which everything - Number among them - is similarly composed.

It's hard to say more than this without getting into the weeds of technicality, but, where Badiou's full exposition of 'Being' takes place in the book is he better known for - Being and Event - what is interesting in Number and Numbers is the way he finds, in surreal numbers, a kind of mirror of Being in Numbers themselves. Just as, in Being and Event, Being is what he calls 'inconsistent' - full of holes, as it were, sites in which potential Events which upend the order of Being can take place - so too is Number (again, thought of in terms of surreal number) also full of sites in which new numbers can themselves be created (by means of a technical operation called a 'cut'). Of the two great categories under which all of reality can be thought - Being on the one hand, and 'the Event' on the other - Number is that which is coextensive with Being: "It [Number] inconsists, is disseminated and profused just like the pure multiple, the general form of being qua being".

But these are the high-altitude stakes of Number and Numbers. As a reading experience, 'on the ground', as it were, most of the book is in fact given over to Badiou's calm and patient exposition of the ins-and-outs of the fiddly technicalities of how number 'works'. In fact the book is practically split down the middle, with the first half detailing both the history of attempts to reckon with number (in particular by Frege, Dedekind, Peano, and Cantor), as well as a preliminary exposition of the ordinals understood in terms of set theory, while the second half - a full hundred pages - lays out, piece by piece, the way in which surreal numbers function. It's a real labour of love, and while it isn't easy reading by any means, Badiou has a charming pedagogical voice that shines through his writing. Even if one were to disagree with his philosophical glosses, reading this book would nonetheless be an education in Number, a real adventure of thought with Badiou as supple guide through the thickets of Number in all their variegated diversity.

NB: For whatever scraps it's worth, I do (ultimately) disagree with them glosses, largely because of Badiou's relegation of complex numbers and hypercomplex numbers to the shadow realm of Not Good Enough to be Numbers. As it happens I reckon it's just in these kinds of numbers that the key to understanding wtf is number is to be found. But that's for another day.

[Mark Moon · Rating 3 out of 5]

Not a math textbook, but contains a lot of very clear arguments. I learned some of the history of how mathematicians have thought about numbers in the past, and about what mathematicians call the "sign expansion" or "sign sequence" presentation of the surreal numbers.

[Mark · Rating 3 out of 5]

I don't tend to read philosophy, but took this on partially as a challenge to myself and partially to better understand some ideas a friend has been sharing. A challenge it was, but not because of the style or lack of momentum but because the theory itself grows quite complex. It's all very interesting, though, and requires little background in philosophy, and even the path is well contained. I will probably read a little more badiou, though, as this book doesn't really give a broad understanding of his philosophy.

[Michael · Rating 5 out of 5]

Excellent read, take notes though. The dialect and syntax are a little hard to follow at first.

[David · Rating 5 out of 5]

Marvelous and fascinating. But strap on your math shoes. You'll need them.

[Michael Greer · Rating 5 out of 5]

Philosophy is always the opponent of tyranny. The paradigm case is Socrates and his opponent Thrasymachus in the Republic. This is precisely the reason that few major in or bother to study philosophy. Philosophical interests are always the interests of free minds. But the vast majority of people favor slavery: slavery to religion, slavery to the economy, slavery to social convention. One slavery that philosophy "calls out" is slavery to number. You might ask, why would the philosopher Alain Badiou contribute to number theory? Is it because he's a displaced mathematician? No, not at all. It's because he is a philosopher who "calls out" our slavery to number.

1. "We live in the era of number's despotism." This remark, a remark which launches a thousand criticisms both from its author and at the author, will provide the context for the reader's journey. The point of Badiou's exercise, throughout the course material, is to reveal exactly what number is. By attaining this conceptual clarification, the tyranny is diminished if not expelled.

2. "The factual impact of number 'escorts' the silence of the concept." The custom is to respect number, to defer to those 'with the numbers,' to have a look 'at the numbers.' Yet this custom is the key chain that binds the mind to a pointless cognitive existence. In fact to reduce the human to mere calculation is a degradation not witnessed since the Holocaust some seventy years ago.

3. "Number currently performs the function of uniting the All." Does anyone do anything any more before counting? Don't count your chickens before they hatch. Don't drive unless you've counted out the number of miles you need to traverse. How many years have you been married? Ah, I see, a successful marriage is upward of thirty years, a failed marriage less than five.

4. "Number governs our conception of the political, with the currency of suffrage, opinion polls, and the majority. Notice how recently the Democrats have a "majority." Only if you factor in that a third of eligible voters didn't vote and then half of those who did favored the opponent do you see what the term "majority" actually means today. Tyranny at all levels.

5. "Numbers govern all the human sciences." Statistics is the course de jour. The rise of bureaucratic practices is the possibility condition of endless parades of numbers. When we visit the doctor, our first question is, "how do the numbers look on the blood test?"

6. "All cultural facts now yield to numbers." Netflix can't provide entertainment before you become aware of the top ten movies on view this week.

I could go on, but Goodreads is not foolish enough to allow unlimited space. They too are bound by 'number.'

[Jeremy Savage · Rating 3 out of 5]

What a fascinating book that I probably didn't understand at all!!