Alain Badiou (1937-) is a French philosopher. At one point he was the chair of Philosophy at the École normale supérieure (ENS) and founder (with Michel Foucault, Gilles Deleuze and Jean-François Lyotard) of the Faculty of Philosophy at the Université de Paris VIII. He’s now René Descartes Chair and Professor of Philosophy at The European Graduate School. Badiou has also been involved in a politics and political organisations since early in his life. Indeed he has often commented on both French and global political affairs.

More relevantly to this piece, Badiou has a strong mathematical background. He’s the son of the mathematician Raymond Badiou (1905–1996). And, according to Badiou himself, by 1967 he “already had a solid grounding in mathematics and logic”. Badiou again describes his own history when he tells us that he studied “contemporary mathematics in greater depths by taking the first two years of university math”. He then goes on to say that “[t]his was from 1956 to 1958, my first two years at at the École Normale Supérieure”.

Introduction

Despite writing this piece, much — or at least some — of the philosophical things Badiou says about mathematics I (to be honest) find incomprehensible. Either that or plain pretentious. And many of the explanations (or “textual analyses”) of Badiou’s ideas by other academics are even worse than his own. I’m not alone here. Various critics have a problem with Badiou’s philosophy of mathematics. For example, the English philosopher Roger Scruton (who died in January this year) questioned Badiou’s knowledge of the foundations of mathematics. He once wrote that Badiou doesn’t

“Really understand [philosophically or mathematically?] what he is talking about when he invokes (as he constantly does) Georg Cantor’s theory of transfinite cardinals, the axioms of set theory, Gödel’s incompleteness proof or Paul Cohen’s proof of the independence of the continuum hypothesis”.

Added to all that, the mathematician Alan Sokal and physicist Jean Bricmont can be quoted stating the following words:

“Badiou happily throws together politics, Lacanian psychoanalysis and mathematical set theory […] After a brief discussion on the situation of immigrant workers, Badious refers to the continuum hypothesis.”

All the above is odd if one considers Badiou’s background and grounding in mathematics (which has just been mentioned). Then again, a grounding and background in mathematics doesn’t guarantee Badiou — or anyone else — anything. It certainly doesn’t guarantee that Badiou’s philosophy of mathematics will offer us anything worthwhile — or, in his case, comprehensible.

Having said all that, some of what Badiou says about mathematics is both crystal clear and informative. Indeed obviously this piece wouldn’t have been written if that weren’t the case. However, it’s interesting to note that most of the comprehensible quotes in the following piece are from Badiou’s book In Praise of Mathematics. Now that book is actually a translation (by Susan Spitzer) of Badiou’s “dialogue” with Gilles Haéri (who’s the Director General of the publishing house Éditions Flammarion). And one can only assume that the fact that it’s both an introductory dialogue and a (perhaps liberal) English translation may help explain its comprehensibility.

Alain Badiou’s Platonism

Alain Badiou is a Platonist. Or, more accurately, he’s a mathematical Platonist. This isn’t to say that Badiou expresses his Platonism in precisely the same way in which other Platonists have done so in the past. And Badiou is certainly a considerable distance from Plato himself on many matters (which is hardly a surprise).

In any case, Badiou says that Plato is “my old master”. He also tells us about Plato’s own position on the importance of mathematics. Badiou writes:

“For Plato… mathematics was the very foundation of universal rational knowledge: the philosopher absolutely had to begin with mathematics. Even if he ultimately went beyond it, he had to learn mathematics first. Plato thought that political leaders, for example, would be well advised to study higher mathematics for at least ten years. He indicated that they were not to be satisfied with just the minimum, since they had to do geometry in space in particular.”

Badiou also explains the “Platonic” position on mathematics. He states that the

“realist (or Platonic) conception, which holds that the object of mathematics exists outside of us”.

More specifically on mathematics and its relation to the world (or to nature), Badiou says that the “Platonic” vision has it that “mathematics is part of the thinking of what there is, of what is”.

All this can be turned into a specific position on the nature of physics.

Badiou expresses a view of mathematics that’s fairly standard among the more philosophical of physicists. That is, he believes that maths “is the science of everything that is”. Indeed with mathematics everything is “grasped as its absolutely formal level”. And Badious certainly believes that mathematics can be used to describe and explain “everything that is”.

Platonism is also tied to a belief in what Badiou himself calls “absolute truths”. Having said that, Badiou does offer a very 21st-century take on such absolutetruths — as can be seen in the following passage:

“I am convinced that there are absolute truths, which, although extracted at the time of their creation from a particular soil (a moment in history, a country, a language, and so on), are nevertheless constructed in such a way that their value becomes universalized.”

I’ve just used the modifying phrase “a very 21st-century take” (i.e., on absolute truth), and that certainly applies to Badiou’s sociological and historical claim that mathematical truths are

“extracted at the time of their creation from a particular soil (a moment in history, a country, a language, and so on)”.

This means that Badiou’s position can be seen as a sociological and historical account of absolute truth. And that account certainly goes against many strong — indeed Platonic — philosophical stances on absolute truth. Or, at the very least, Badiou’s account muddies the philosophical water in that although he accepts absolute truths, he still places them within various contingent contexts. (This may seem — at least to some — to be an oxymoronic position.) The basic point is that although absolute truths do indeed exist, they’re still “discovered” within very specific historical and sociological contexts. And therefore their constructions(if that’s an appropriate word here) display the contingencies of these domains.

Is this, then, simply an ornate way of saying that the equation 2 + 2 = 4 has been expressed by many different symbols and in many different languages? Indeed the same — almost exactly the same — kind of argument is made by philosophers when they talk about propositions. That is, the very same proposition can be expressed in many different languages and in many different ways (see here).

Whatever kind of Platonism Badiou’s philosophy of mathematics actually is, it’s still clear that it can be strongly distinguished from mathematical empiricism — or, indeed, from any kind of empiricism. Badiou is explicit about this when he says that he “reject[s] the theory that mathematics derives from sensory experience”. He then explains his position in a semi-Kantian manner in the following:

“[T]he real of sensory experience is thinkable only because mathematical formalism thinks, ‘ahead of time,’ the possible forms of everything that is.”

Badiou’s Pythagoreanism or Platonism?

Badiou defends his Platonic position by citing the case of the “complex numbers,the imaginaries”. These were initially “invented as a pure game”. (See the later section on formalism.) That’s why they were called “imaginaries”: in order “to make it clear that they didn’t exist”. However, later they “became a basic tool used in electromagnetism in the nineteenth century, something that no one could have foreseen”.

What point is Badiou making here?

Basically, Badiou’s point is that mathematics is always found to be instantiated in the actual world (or “the real” as he puts it). Perhaps Badiou means “nearly always” as it’s certainly the case that some mathematics (or some mathematical ideas) are certainly not instantiated in nature. The physicist Roger Penrose, for example, cites various examples of this. The argument is that if mathematics and the world are one, then why are there (to use Penrose’s words) “bodies of maths with no discernible relations to the physical world”? In terms of Penrose’s actual examples, we have the following:

“Cantor’s theory of the infinite is one noteworthy example… extraordinary little of it seems to have relevance to the workings of the physical world as we know it… The same issue arises in relation to… Gödel’s famous incompleteness theorem. Also, there are the wide-ranging and deep ideas of category theory [mentioned by Badiou too — see later section]that have yet seen rather little connection with physics.”

However, is saying that mathematics is instantiated in nature (or even that maths can describe nature) the same as saying that (as Badiou does) “everything” literally is mathematical? Perhaps, then, this is (almost) a definitional truth. That is:

1. If everything can be described by mathematics,
2. Then everything must be mathematical.

This takes us on to the subject of Pythagoreanism.

It’s often difficult to tell whether a philosopher of mathematics is taking a Pythagorean or a Platonic position on the relation between mathematics and the world (or nature). Indeed it is sometimes difficult to distinguish the two regardless of the philosopher — or physicist! — being commented upon.

Badiou himself tells us that “mathematics touches the real”. Mathematics does so “in a way that is not experimental”. Badiou goes on to say that mathematics can’tbe experimental because it is “presupposed in experience”. The ultimate reason for this is that “the real” is mathematical — or, perhaps more tellingly, already mathematical.

Yet surely if mathematics “touches the real”, then it can’t actually be “the real”. That is, if a finger touches a flower, then the finger isn’t the actual flower it touches. Perhaps this is simply a problem brought about by Badiou’s metaphor “touches”. Yet that metaphor (if it is a metaphor) does at least hint at a separation between mathematics and the world (or the real). That’s unless mathematics is touching itself! In that case, do we have the following Pythagorean identity? -

The Real = mathematics
(The Slovenian philosopher Slavoj Žižek always capitalises the word “real” — à la Hegel — and puts the definite article before it.)

Badiou then goes on to argue that

“Even the great instruments that are used in experiments, from telescopes to giant particle accelerators, are ‘materialized theory,’ and presuppose, even in the way they’re constructed, extremely complex mathematical formalisms”.

This reads like a reworking of the Pythagorean phrase “all things are numbers”. That is, mathematics is immanent in the world. Indeed mathematics is even immanent in the instruments which manipulate that world. To put it plainly, this isn’t the case of mathematics describing the world. This is a case of mathematics — or numbers — literally being instantiated in the world (or, in this particular case, in instruments).

Yet even here we can question any commitment to the world actually beingmathematical. After all, Badiou uses the phrase “materialized theory”. Doesn’t that hint at a separation between mathematical theory and its materialization? That is, firstly there is the (mathematical) theory, and only then is that theory materialized. Thus, the mathematics must surely antedate its materialization (in this case, in instruments).

This situation is not unlike Plato’s ante rem position on universals in which universals exist before they are instantiated by (or in) particulars. Aristotle, on the other hand, believed that universals exist post rem — i.e., only after they’re instantiated by (or in) particulars. So now it can now be said that just as there’s a separation between a universal and its instantiation (or exemplification), so there’s also a separation between a mathematical theory and its instantiation in the guise of an instrument — or its instantiation in anything else in the world. Thus, if the mathematics antedates (“its”) instantiations (or concretisations), then doesn’t that call into question the idea that the world (or its parts) is (or are) literally mathematical? The most we can say is that the world can embodymathematical (to use Badiou’s term) “formalisms” — not that it is mathematics.

It’s true that instruments (to stick with Badiou’s own example) “presuppose” the mathematics in that they (metaphorically at least) abide by the mathematics. Or, more correctly, mathematics describes the instruments and the instruments must also adhere to mathematical formalisms. But simply because these instruments — and all the parts of the natural world — can be described by mathematics, that doesn’t mean that the world literally is mathematics. And the metaphorical word “abide” (or “adhere”) may not help in that the instruments must have mathematical properties (or properties that can be described by mathematics) in order for them to be what they are. However, almost all things can be described by maths — even random or chaotic events. Yet that still doesn’t mean that they actually are mathematical.

Platonic Structuralism

Alain Badiou is explicit about his structuralism — specifically when it comes to mathematics. He writes:

“Structures are first and foremost the business of mathematicians.”

He then cites his influence which not many mathematical structuralists will recognise. In more direct terms, Badiou appears to suggest that his mathematical structuralism came via his interest in — and knowledge of — Claude Lévi-Strauss’s anthropological structuralism. Badiou tells us that

“at the very end of [Levi-Strauss’s] seminal book, The Elementary Structures of Kinship, the great anthropologist [] referred to the mathematician Weil to show the exchange of women could be understood by using the algebraic theory of groups”.

So the interesting thing about Badiou’s mathematical Platonism is that it’s a Platonism about structures, not numbers.

In terms of what Badiou calls “being”, the only way to to capture “being” is “to think purely formal structures”.

So what are “formal structures”?

They’re “structures indeterminate as their physical characteristics”. How does this connect to mathematics? It does so because “the science of these structures is mathematics”.

Badiou also cites a specific example of mathematical structuralism from the world of mathematics: category theory. Badiou tells us category theory is

“roughly the theory of relations ‘in general,’ with no pre-specification of given objects”.

Badiou then goes into more detail when he refers to the nature of category theory. Here he emphasises relations rather than structures. Indeed it can be seen that relations and structures are intimately tied together. Badiou tells us that a

“structural edifice is gradually built up in which relations seem to prevail over entities, or objects, or even to determine their nature and properties”.

Badiou then makes the obvious conclusion:

“So it is tempting to reduce all the so-called ‘intuitive’ objects to structural, or formal, manipulations whose principles only objects the mathematician’s decisions or choices. What then ‘exists’ are structured domains, which are accountable only to the formalism by which they are exhibited.”

The problem here is that Badiou is against what he calls “formalism”. Yet it’s the formalism that brings about the structures and relations which Badiou believes trump mathematical “entities, or objects”. That is, numbers are the children of the structures and relations and therefore of the formalisms which bring about the structures and relations.

Badiou then gives a more concrete and everyday example of mathematical structuralism.

He concentrates on the word “successor”. Firstly, Badiou tells us that“[m]athematical thinking makes a tentative appearance if you say that 235,678,982 is the ‘successor’ of the number 235,678,981”. More relevantly:

“But you can then see that what really matters is the word ‘successor,’ which actually denotes an operation and therefore, ultimately, a structure, in this case addition: if the number n exists, whatever n may be, then there also exists the number n + 1, which will be called the successor of n.”

The English cosmologist and mathematician John D. Barrow also makes similar points (though he stresses operation, not structure). He writes:

“This is done by focusing attention upon the operation by which numbers are changed rather than upon the numbers themselves. Thus, a simple counting process like 1, 2, 3 … is seen not as a list of particular numbers but as the result of carrying out a particular operation of change upon a number, thereby generating its successor.”

Both the quoted passages directly above are slightly problematic in that even ifnumbers are generated by a structure (in this case addition), then this very structure also begins with a number — even if only an unspecified n. In other words, the operations and nature of the successors — and therefore the structure itself — seem to be dependent on the “existence” of n (which is an unspecified number). Even though n is unspecified and Badiou uses the words “carrying out a particular operation of change upon a number”, it’s still a number (or something which is taken to be a number) that begins the show: that is, not a structure or an operation.

It can of course be argued that n itself is a product of prior structures. But that simply replicates the problem in that these prior structures might also have depended on the existence of prior numbers. This means that the numbers (on this reading at least) generate the structures, rather than vice versa.

It may be concluded, then, that the reality of succession and addition (as well as of relations and structures) are distinguishable from the reality of numbers. That means that the successor relation and the operation of addition (as well as structures and relations — at least in mathematics) all depend on the prior existence of numbers. In other words, the prior nature of numbers generates the nature of the structures. Consequently, why can’t we make, say, the number 2 have a prior nature which is completely separate from it having a place in a structure — or, indeed, from it having any relations at all to other numbers?… Having said all that, this position hardly makes sense either!

Against Formalism

As many Platonists have done, Badiou pits himself against what he and many others have called “formalist” accounts of mathematics. Indeed his account of mathematical formalism is also — at least in part — an account of Ludwig Wittgenstein’s ostensibly extreme constructivism when it came to mathematics (see here).

Badiou’s fundamental point is that mathematics isn’t “purely and simply a formal, arbitrary game”. He claims that the formalist simply desires a

“codification of a language that is of course formally rigorous, since the concepts of deduction and proof are normative and formalized in it, but whose rigor cannot claim to have an ongoing relationship with empirical reality”.

In other words, maths in neither about the world nor derived from the world. One reason for this is that formalists believe (at least according to Badiou) that “[m]athematical axioms can be changed, after all”. It is, therefore, the axioms (along with the logical rules) that (as it were) create their own formal worlds.

Badiou, on the other hand, believed that maths has “content”. He says that “there is a real ‘content’ to mathematical thought”. Badiou continues:

“[Maths is] neither a language game — even if complex formalisms are required — nor is it an offshoot of pure logic.”

Indeed Badiou believes that “the majority of mathematicians” take this position too.

The mathematical physicist and mathematician Roger Penrose also replicates Badiou’s position on mathematical formalism. Penrose writes:

“The point of view that one can dispense with the meanings of mathematical statements, regarding them as nothing but strings of symbols in some mathematical system, is the mathematical standpoint of formalism.”

Penrose has a serious problem with the “point of view”. He goes on to say that “[s]ome people like this idea, whereby mathematics becomes a kind of ‘meaningless game’”. Penrose concludes:

“It is not an idea that appeals to me, however. It is indeed ‘meaning’ — not blind algorithmic computation — that gives mathematics its substance. Fortunately, Gödel dealt formalism a devastating blow!”

Perhaps Penrose’s quote above sums up what it is that Alain Badiou’s own mathematical Platonism is all about.