I just saw that Deontologistics is now using the the term deflationary in reference to Graham Harman’s realism. There is a slight difference as he as attached it specifically to the question of realism. Anyway… as usual it is a good post on Harman’s philosophy.

Mikhail over at Perverse Egalitarianism is putting on a show this summer. Much as he did last summer with Lee Braver’s book, A Thing of this World, Mikhail will be putting on a reading group over Saloman Maimon’s Essay on Transcendental Philosophy. I got my book today and am looking forward to the start of this thing.

Here is my contribution to “Material Subjects or Real Objects” in Dundee

The Parmenidian equation Being = Thought has dominated philosophy from its inception. Nearly every philosophy since Parmenides, maybe every philosophy, has asserted and re-asserted this equation, this happy correlation, at the heart of their formulations. This equation naturalized the philosophical task and allowed philosophy to recognize itself within God’s plan, with Man at the center. Being = Thought is the doxa of philosophy; it is, to borrow a phrase from Deleuze, philosophy’s “image of thought.” And it is in this function that the equation acts as a veil that confuses and distorts, causing philosophy to become stagnant and too sure of itself. It is Being = Thought that prevented philosophers from recognizing what were the actual implications for Kant and his so-called “Copernican Revolution.” But even further, it allowed philosophy to commit a suspicious double move where it simultaneously sneered at the common sense that tells us that the world is exactly as it appears to us while asserting a much more fundamental version of the same story, where the world is exactly as we philosophers think it.

However, we are currently experiencing a crack in this correlationalist façade and new equations are being written. The relationship between Thought and Being can be formulated in four ways. Philosophy has experienced the first formulation for its entire existence. Of the three remaining possibilities, two are obvious. First, there is the path that I have chosen to call inflationary metaphysics. This formulation asserts the inequality of Thought and Being, but does so by asserting the primacy of Thought over Being. The primacy of Thought arriving via the creation of a special form of Thought that is not imprisoned by the empirical (e.g. mathematics). The second possibility, which I will call deflationary metaphysics, reverses this inequality by asserting the primacy of Being over Thought, where Being arrives as specific entities or objects that escape from Thought itself. The third possibility, which I will only be able to touch on, would then assert that Thought and Being extend beyond each other in a game of give and take, combining aspects from both of the other ways of formulating the relationship.

What this paper will attempt to do is to look into the two obvious ways of re-formulating the relationship by specifically looking at a representative from each camp: Quentin Meillassoux for the inflationists and Graham Harman for the deflationists. Specifically, we will be looking at the theories of causality that each of these thinkers puts forward. The issue of causality allows for a framing of precisely what Meillassoux and Harman do and do not share. But first a quick note: The terms inflationary and deflationary are meant to refer to the treatment of Thought under each of the new philosophies. If Parmenides functions through an act of equating Thought and Being, both Meillassoux and Harman undermine the equation by selecting one of the sides to privilege. At first sight, this terminology might be said to display a bias towards thought in that describing Meillassoux’s philosophy as inflationary metaphysics, the word inflationary bringing forth positive connotations, whereas Harman’s philosophy being assigned the term deflationary metaphysics might bring forth a negative connotation. This is not the case as these terms are relative to a particular side of the equation, meaning that if the philosophies under discussion are said to be relative to Being, instead of Thought, then the terms would necessary switch.

Curiously, Meillassoux and Harman’s theories of causality represent a radical re-evaluation of the Ash’arite School in Islamic Philosophy. Harman’s relationship with occasionalism, their theory of causality, is well-known, as he has specifically placed his own work within this history. But Meillassoux is no less influenced, although the influence is of the mediated variety. In The Incoherence of Philosophers, Al-Ghazali argues that cause and effect are related only through the contingency of God’s will and not through any form of necessity. The assertion that there exists a natural and thereby necessary relation between cause and effect, as argued by the Aristotelian philosophers, would eliminate the possibility of miracles and would thereby limit the power of God’s will. Such an encroachment on God’s powers by philosophy is illegitimate, and represents an example of demonstration exceeding what it is capable of understanding. Instead, Al-Ghazali argues that the relationship between cause and effect is contingent upon the will of God, meaning that any perceived necessity arises from both the habit of the observer, who has seen the effect follow the cause over and over before, and the habit of God, who has allowed a pattern to be formed so that humanity may recognize a miracle when it occurs. Harman has shown in his own work that the influence that occasionalism has had on 17^th-century philosophy is significant. But the re-interpretation offered by Meillassoux and Harman is radical in its scope. We can understand Hume and Kant, central figures for the new metaphysics, as offering a form of occasionalism that only changes who sits in for God. Hume rejects God and therefore losses one of the strands of habitual action from Al-Ghazali’s own formulation. Without God, it is only humanity’s habit that truly matters. On the other hand, Kant awakes from his “dogmatic slumber” only to collapse the role of God and humanity into one figure, where cause and effect reside within us in the form of the categories. But Meillassoux and Harman both take a more radical line in the formation of their philosophies. Meillassoux’s assertion that the only necessity is contingency mirrors Al-Ghazali’s view of nature, but with only one caveat: There is no God and without God there is no causal relationship. Harman’s own appropriation of Al-Ghazali comes down to his assertion that two objects are never fully available to each other, and thus can only have a mediated relationship. Unlike Al-Ghazali, however, this mediation is not affected by God or his angels but the objects within the relationship only ever approach each other through an intermediary or sensual vicar.

To begin I would like to very quickly run through Meillassoux’s argumentation against correlationism. First, Meillassoux introduces what he calls the arche-fossil, a material “indicating the existence of an ancestral reality…anterior to terrestrial life.” The arche-fossil allows for the formulation of ancestral statements, which as statements about the non-being of Thought are not permissible under correlationism. The ancestral statement is then absolutized by Meillassoux into facticity which he describes as the necessity of contingency. Facticity provides Meillassoux with the principle of unreason, the idea that nothing exists for any reason whatsoever. The principle of unreason is the location of Thoughts ability to extend over reason. As Meillassoux will state, “In a rational world everything would be devoid of any reason to be as it is. A world which was entirely governed by logic, would in fact be governed only by logic, and consequentially would be a world where nothing has a reason to be as it is rather otherwise since nothing contradictory can be perceived in the possibility of such a being-otherwise.” In this world described by Meillassoux the laws of physics are reduced to mere facts and like all facts they can change with time.

What, then, is the problem? The first problem may appear as a rather shallow complaint against such a provocative philosophy. The question is what is the status of science in Meillassoux’s work. In the final chapter of After Finitude entitled “Ptolemy’s Revenge”, he establishes the great invention of modern science was a series of “cognitive processes…no longer of the order of myths, theogonies, and fabulations, and instead become hypotheses susceptible to corroboration or refutation by actual experiments.” So where does the problem lie? Gabriel Catren has pointed out that Meillassoux’s assertions on the principles of unreason are based on his dedication to the event named Cantor and the existence of the non-All. But the absence of a totality should not preclude the existence of localized order or necessity. As Catren writes, “the project of understanding the rational necessity of physical theories defines local problems of scientific knowledge…It is difficult, then, to understand why the supposed impossibility of providing a satisfactory rational global model for the ‘topology’ of absolute knowledge would imply the futility of such a local project.” Worse yet, Catren sums up Meillassoux’s argumentation in After Finitude as follows, “…we can say that Meillassoux’s ‘proof’ begins with an unquestioned (and probably false) presupposition (namely, that physics cannot discover any rational necessity in physical laws), and proceeds by means of an illegitimate ‘deductive’ inference (namely, that of absolutising a supposed limitation).” Meillassoux’s mistake is conflating the absence of a First and Final cause (those that would deliver an absolute necessity) with the absence of an efficient cause. This allows him to dismiss the laws of nature as merely factual arrangements that could change at anytime. But if this were true this would invalidate any experimental data that could corroborate or refute the “cognitive processes” that have replaces “myths and fabulations.” But things are worse for Meillassoux.

As I already said, Meillassoux should be understood as participating in the continuation of the legacy of Al-Ghazali and as such it would be enlightening to look a similar critique that Al-Ghazali himself invokes to critique his own position. “And if someone leave a book in the house, let him allow as possible its change on his returning home into a beardless slave boy—intelligent, busy with his tasks—or into an animal…If asked about any of this, he ought to say: “I do not know what is at the house at present. All I know is that I have left a book in the house, which is perhaps now a horse that has defiled the library with its urine and its dung, and that I have left in the house a jar of water, which may well have turned into an apple tree.” Now, Meillassoux has already dealt with part of this critique in his differentiating of chance and contingency. He defines chance as being the idea that transformation functions via a probability that would produce continual change, thereby resulting in the absolute instability of the world. On the other hand, he puts forward the concept of contingency as a process of potential transformation that is wholly without necessity and therefore, one transformation (or even no transformations) are as likely as any other. In forming the concept of contingency Meillassoux accepts as obvious the stability of the world, but this assumption is without reason. If it is true that the laws of physics are merely factual (i.e. contingent) then even the supposed fixity of the world would have to be called into question. Contingency would allow for the possibility that ever moment before this moment is merely the by-product of a mass formulation of the quanta of memory for all humans resulting in a collective belief in that all previous moments had actually occurred when in fact all of existence began with this very moment. This formulation would include supposed arche-fossils which did not exist the moment before. And without the arche-fossil, correlationism remains untouched.

On top of this Meillassoux’s split between chance and contingency ignores the real advancements in the formulation of probabilities in contemporary physics. In doing this he commits what we could call the Mallarme Fallacy. This fallacy is committed when one absolutizes our own ignorance of the outcome of a throw of the dice into a concept of absolute chaos. Such an approach ignores the other possibility involving probability, which is the assertion that probability is part of the very fabricate of the cosmos. The probability in quantum mechanics is not the same as that of dice or coins. The probability of throw or a flip is produced by out of our own ignorance of conditions effecting the movement of the object, whereas the probability of an electron’s location is part of the structure of the electron. Thus we can see that Meillassoux’s attempts to treat Thought as primary result in the construction of a world that is not only without reason but wholly irrational.

Harman’s philosophy aims to move from a philosophy of access, which is roughly equivalent to Meillassoux’s correlationism, to an object-oriented philosophy. Harman writes, “The drama of the world is never confined to that single layer where human consciousness happens to be located at any given moment. The phenomenal sphere fails to exhaust the riches of reality itself, and for this reason falls short of defining the full scope of philosophy…” For Harman, the move away from the questions of human access or the condition of possibility of human experience, is a move toward autonomous objects that are variously described as “vacuum sealed”, “receding”, “black boxes”, “black holes”, “having a molten core”, etc,; signifying that the inner life of objects are sealed off from everything else, all of these terms are meant to convey the return of the once discredited concept of substance, here revitalized into meaning that which escapes relation.

Harman’s argumentation begins with a re-working of Heidegger’s tool-analysis, which he locates as the central thesis of the Heidegger’s work (the greatest insight of the greatest philosopher of the 20^th century). The failure of the hammer, for example, points to a hammer beyond my experience of banging nails or hit my thumb. Harman takes this notion as not just describing how humans interact or caricature their surroundings nor does this become a condition for all animal experience, instead he asserts that this is the condition of relation as such. All relations, whether between animate or inanimate substances or a combination of both, function through a process of caricature where the components of the relation reduce the other members to ridiculous versions of themselves. Whether it’s a baseball striking a window or a dodge ball striking the face of helpless child, the ball does not exhaust the possibilities of either object that it strikes. The result is a view of the world filled with objects, distributed along the vertical and horizontal axis and at all levels of zooming, that retreat from any and everything. The problem that follows is obvious and provides Harman the opportunity to present the most provocative aspect of his philosophy, namely vicarious causation.

Vicarious causation is a process that allows retreating objects to nonetheless come into contact with each other, although it is a contact that is always mediated, never direct. As Harman explains it, this occurs through the creation of a sensual object, the object of perception. For example, when I stand before a tree, the real tree withdraws from my perceptual presence but the sensual tree lies before me allowing for the creation of the intentional object within which perception occurs. Although this example deals with human experience, for Harman perception by a living creature and the causal relationship of inanimate objects are structured the same.

There are, I think, two related problems with Harman’s description of reality: one methodological, the other with the actual formulations. Harman’s horizon remains phenomenological, even if he criticizes it as being a theory of access. My criticism can be seen by returning to Heidegger’s tool-analysis. As Harman has already noted, Heidegger’s insight into the tool is not one insight among many but the central concept of his philosophy, namely the withdrawal of Being (even if in Heidegger it always remains a question for Dasein). The problem, and I believe that this is a problem with phenomenology per se, is that Heidegger moves from the particular, the hammer, to the universal, Being, but the move itself is illegitimate for phenomenology. The “withdrawal” of the hammer could be understood as a by-product of the finitude of a specific individual user of the tool and does not touch upon the structure of Being nor the condition of relation. In this re-formulation, tool-analysis becomes an epistemological issue not an ontological one, as it deals with the ignorance of the user and not the structure of use. In other words, the problem of the withdrawal of the hammer could be solved by a more dedicated individual who, ignoring all other aspects of his life, comes to function as an obsessive attempting to “complete” the hammer-knowledge.

This problem manifests itself in Harman through a conflation of perception and causation. It must be remembered that Harman wishes to move Heidegger’s tool-analysis out of the philosophy of access and the human and make it the condition of all relations no matter the individuals make up this relation. But to do this he must assert the equality of perception and causation. He writes, “All objects relate only on the inside of another object; all perception occurs on the inside of an object. Hence, causation and perception are equivalent to objects and the interior of objects. To say that the entire universe is made up causation and perception is to say it is made up solely of objects and their molten interiors…” But it should be noted that this action of equivalency occurs too quickly in Harman and thus it is not really clear how the quite brilliant descriptions of human experience that one finds in Harman’s work are actually related to the interaction between inanimate objects. And this brings us to the second problem.

The transference of vocabulary between the descriptions of human experience to that of inanimate interaction seems to confuse matters more than clarify. For example, we could return to the classic example from Al-Ghazali that Harman himself uses throughout his work: namely, fire burning cotton. Under Harman’s description there are five objects functioning within the relation of fire and cotton. There is the real fire, real cotton, sensual fire, sensual cotton and the intention as a whole. The fire that burns the cotton is the real fire as it is the active participant, but the cotton that it burns is the sensual cotton, as the real cotton has withdrawn in the subterranean or obscure cavernous world, away from all relation. On the other hand, one could describe the burning of cotton by saying that the presence of fire causes an increase in temperature resulting in the chemical transformation of the cotton into fuel and remnants thus combining with the oxygen in the air bringing about the burning of cotton. I could point out that in the second description that fire never touches the cotton either in that fire is always separated from the object that it burns through what is called the “flame interface.” One could complicate the second description through the introduction of mathematical formulas describing the chemical changes, etc. The question is what is the difference between these two descriptions. I am willing to admit that they may be completely compatible, with the first description serving as a philosophical version and the second roughly approximating a scientific description. But if this is so, then what is point? There are, however, problems that one would have to deal in make these two different perspectives of the same event. First, the question of active and passive is wholly absence in the second description as the absence of the heat, oxygen or the fuel would result in the absence of fire. Second, it is not entirely clear how the destruction of any object is possible under Harman’s description. If it is true that “Real objects can never be brought to presence even partially…”, then how can we understand what fire actually does to the cotton, an action that is different in kind and not degrees from the perceptual relationship. Harman himself has framed his attempt to formulate a new form of causality on the absence of such a theory in science and philosophy’s own rather pathetic position in contemporary society. But such a view of the relation between philosophy and science is return to Al-Ghazali and his assertion that demonstration must not trespass on the realm of metaphysics, a view that itself necessitates Al-Ghazali’s occasionalism.

It is the very inequality between Thought and Being that makes philosophy possible. This is already understood, although perhaps unconsciously, by nearly all the philosopher of the correlation. Strictly speaking, a true congruence between Thought and Being would result in the absolute presence of the world to our thinking selves. But this obviously does not occur. This explains the need in so many philosophers to tell a tale about the difficulty of thought due to some human failing, whether that failing be a stubborn horse, a girl and an apple or the dreaded boogie-boo of human nature. It is only through the incongruence of Thought and Being that the catastrophe of human consciousness can be understood by necessitating a whole series of rational discourses that are broadly named philosophy and science. It is in Meillassoux that the necessity of mathematical thought is proclaimed so as to be able to speak of the nature of reality independent of the human. Furthermore, Meillassoux presents us with a description of philosophy that should not be abandoned. “Philosophy is the invention of strange forms argumentation…To philosophize is always to develop an idea whose elaboration and defense requires a novel kind of argumentation, the model for which lies neither in positive science…nor in some supposedly innate faculty for proper reasoning. Thus it is essential that a philosophy produce internal mechanisms for regulating its own inferences.” Philosophy must be formulated under an epistemological condition so as to prevent it from running too far afield in its speculation. We should eye with suspicion any philosophy that claims to operate above or before epistemology, since the severing of Being and Thought provides Thought with an entire menagerie of mythical creature to become fascinated by. Meillassoux seems to have forgotten his own lesson and creates a world that is at least untenable, if it is no longer unthinkable. Ultimately, Meillassoux too readily abandons Being for Thought.

But this brings forward the other requirement of philosophy. We should eye with suspicion any philosophy that abandons its other condition of ontology. The delving into Being, in both the way it relates to us and in that with no relation, is the other condition of philosophy. This is the lesson that we must take from Harman’s philosophy. The severing of the congruence between Thought and Being must result in Being stretching beyond the reach of Thought. And Harman is quite right to reject materiality as the stuff of which Being is made. Materialism remains tied to a science that really no longer exists and it is time for philosophy to move on. But Harman is mistaken to allow the all too human experience of objects is to be the ultimate horizon of reality. It is highly probable that human experience simply misperceives the world as it is. But again this is to be expected when Being and Thought are no longer so intimately tied together.

“All science is either physics or stamp collecting.” (Ernest Rutherford)

“According to Meillassoux we must accept that even physics is mere (mathematised) stamp collecting.” (Gabriel Catren)

In the earlier version of Object-Oriented Philosophy, Graham Harman puts forward the thought experiment of trying to image a time when “Speculative Realism/OOP/Speculative Materialism” would stop beating correlationalism, turn its knives inward and begin hacking at itself. The premise being that something like “Specultive Realism” is a generic term that only finds specificity with a specific enemy (i.e. Kant and his ilk) and that once this enemy is dispatch, the SP will undergo a process of individuation that will create “camps” in the movement. Graham lays it out like this:

“There would be the eliminativist wing with a heavy cog.-sci. bent. There would be a Meillassouxian wing generating fascinating philosophical proofs out of a radicalized correlate (and with Zizekians, Lacanians, and Badiouians in the vicnity). There would be a Grantian wing with a more vitalist/materialist approach and more of a Deleuzian flavor than the others. And then there would be an object-oriented wing, with Latour as a key patron saint and a flat ontology as the price of admission.”

This themes was then picked-up on over at Naught Thought, with the concept of “dark vitalism”. I am sure others that I missed or haven’t formulated their version of the fight is out there. It is in relation to this that I read Gabriel Catren’s “A Throw of the Quantum Dice Will Never Abolish the Copernican Revolution.” This essay is remarkable, both in its positive construction of a theory of objects but also a negative element, constructed as an attack on Meillassoux’s own construction of Speculative Materialism. As I have hinted above, Catren accepts the general outline of Meillassoux’s attack on correlationalism, as he states:

“…the ultimate sense of the Copernican Revolution was, as Meillassoux clearly shows in After Finitude, completely distorted. A narcissistic reaction aims to counteract the Copernicum decentering of the planet earth – and tries to heal what Freud called the ‘cosmological humiliation’ – by re-situating human existence on a transcendental ‘unmoving Ur-earth’ (Husserl)”

But this initial affirmation turns, however, as Catren hits Meillassoux where he lives, so to speak. Part of Meillassoux’s argument against correlationialism is of a emotive variety, in that it attempts to draw a clear line connecting today’s philosophers (i.e. correlationialists) and the creationists that currently populate some areas in the US (such as the area in which I write this). Obviously, this an argument that functions along the “do you know who you are now agree with? Them!” While I don’t really have a problem with this, according to Catren, Meillassoux’s polemics against physics, as that science that wants to talk about the non-existent laws of nature, does not put him in a much better group of friends. Lets have a look.

When Meillassoux turns away from his critical section and attempts to formulate a positive construction of reality he decides to operate on a logical level. In a move that I assume anyone reading this will already know, Meillassoux denies that there are no non-logical rules that govern existence, meaning that the laws of science (e.g. Physics, genetics, etc.) that would provide consistency and permance are illusions, produced from habit (following Hume). One can easily see this as the product of a reading of Hume’s discussion on the “sun rising tomorrow”. What laws do govern reality? Well, the same that govern our own thinking. So, the law of non-contradiction is the only law governing reality. Meillssoux writes, “…the principle of unreason teaches us that it is because the principle of reason is absolutely false that the principle of non-contradiction is absolutely true…As for the principle of non-contradiction, it allows us to establish a priori, and independently of any recourse to experience, that a contradictory event is impossible, that it cannot occur either today or tomorrow. But for Hume there is nothing contradictory in thinking that the same causes could produce different effects tomorrow.” We can see then that causal effects that populate science have no basis in reality, whereas the laws of thought (i.e. logical rules) are at play. But where is the real source of this thinking, as it is a radicaliztion of Hume, and not Hume himself. Of course, it is found in Cantor and the transfinite.

As those of us familar with Badiou know, Cantor provides the ultimate proof for the force of thought in the twentieth century, namely “The One is not.” For Meillassoux, the “One is not” is translated into the impossibility for providing the global laws of nature, i.e. physics is forclosed.

But it is precisely this totalization  of the thinkable which can no longer be guaranteed a priori. For we know – indeed, we have known it at least since Cantor’s revolutionary set-theory – that we have no grounds for maintaining that the conceivable is necessarily totalizable. For one  of the fundamental components of this revolution was the detotalization of number, a detotalization also known as the ‘transfinite.’

This is precisely where Catren’s critique begins. As Catren points out “the project of understanding the rational necessity of physical theories defines local problems of scientific knowledge…It is difficult, then, to understand why the supposed impossibility of providing a satisfactory rational global model for the ‘topology’ of absolute knowledge would imply the futility of such a local project.” In other words, whatever the status of a global (i.e. universal) order, the local projects of science would not be effected. One could equal attempt Meillassoux’s move with Godel’s incompleteness theorems instead of Cantor’s theory “transfinite”, but only the worse reading of Godel would lead one to say that because there is no system that can be shown to be consistent and complete that therefore, all science fails.

Furthermore, Catren even rejects Meillassoux’s image of how science works. The failure seems to revolve around axiomatics. Meillassoux views science as an axiomatic choice made between competing, but equal, systems. This leads Catren to make his statement quoted above about physics and stamp collecting. Ultimately, Catren’s criticism comes down to Meillassoux’s refusal to dirty his hands with the lowly ontic investigations of science and instead remained locked within an ontological framework which will refuse any evidence to the contrary. Catren writes, ” (Meillassoux’s argumentation) is consistingly done without any consideration of physics itself – after all, why should the philosopher consider in detail scientific (ontical) descriptions lacking any rational necessity if he can produce philosophical (ontological) demonstrations? Why would he analyse in accurate terms that of which he speaks -  namely, physical theories – if he knows in advance that physics is only a collection of contigent laws that can change without any reason?”

I believe that Catren’s criticism is quite damning, maybe not completely, but it certainly sweeps most of the legs of support out from under Meillassoux’s edifice. A final quote from Catren:

Even if we can decide legitimately to explore the hypothesis according to which the laws of nature lack any rational a priori necessity, we cannot pretend that we are rationally forced to accept the validity of such an hypothesis, nor that the principle of reason must be abandoned

More later.

Proposition 1: The Event (whether named Event, Other, Difference, Differance, Ereignis, etc.) is the necessary and sufficient condition for philosophy today.

Proposition 2: The Event (whether named Event, Other, Difference, Differance, Ereignis, etc.) provides for the construction of the philosophical world.

Proposition 3: The philosophic world is delineated by a normative factor, i.e. a political or ethical project made obvious by the Event (whether named Event, Other, Difference, Differance, Ereignis, etc.).

Proposition 4: The Evental Fallacy is operative within all contemporary philosophical worlds (i.e. worlds conjured by the event).

Proposition 5: The Evental Fallacy erases the operation of philosophical thought (i.e. the thought conditioned by the Event).

Over at Object-Oriented Philosophy, Graham Harman and (at Larval Subjects) Levi Bryant have been carrying on a discussion of Badiou’s idea of the count (and its related concepts, i.e. inconsistent & consistent multiple etc.) and what effect it has on the status of objects (real or subjective, etc.).Here is a taste:

if I were to start saying: “I’m a Badiouian, but I think that rocks and earthworms are also capable of invoking the generic through art, politics, science and love,” what do you honestly think Badiouians would say in this case? Would they say: “Cool. Badiou never specifies that it has to be a human”? You know full well that they would dismiss such a position as vitalist crap. The whole spirit of Badiou’s philosophy is of a militant human subject disrupting given states-of-situations in truth events.

The discussion strikes something that I have been bouncing around in my readings of Badiou. First, I must admit that there is something that I like about the language of the count, which is to say that I like it better than say Deleuze & Guattari’s discussion of infinite speeds and slowing down. But, second, I have found myself troubled by it in a way that is similiar to Graham and Levi’s discussions of it. It strikes me that Badiou and his discussion of the the count-as-one sounds remarkable close to Plato’s description of the demiure in the Timeaus. Specifically, the demiurge (the rational ordering God) is confronted by the choas of pure nothingness; a nothingness without void, but excessive becoming without order. The demiurge then “orders” the nothing under the function of same and different (following Plato’s understanding of order). Doesn’t this sound an awful lot like Badiou’s conception of the movement from the inconsistent multiple to the consistent multiple via the count. Badiou’s willingness to accept his “platonism” aside, this can be quite troubling for a materialist.

This brings us to the second point. Taylor Adkins (Speculative Heresy) and myself will soon publish a translation of Francois Laruelle’s “review” of Deleuze & Guattari’s What is Philosophy? in the next issue of PLI. In this article, Laruelle states that what is needed is a “Non-Cantorian Theory of the Multiple.” This is a curious concept indeed. Since finishing the translation, I have been trying to figure out what this would entail. Obviously, it would be an understanding of the multiple without count, a given without givenness. We have here a first step in a non-philosophical critique of Badiou (with Meillassoux being guilty by association). I wish I had more to say but I have quite got it down yet. It will come. Eventually.

(Here is a translation of Albert Lautman’s “Mathematiques et Réalité”. Its a bit rough but I will try to clean it up over the next couple of days.)

“Mathematics and Reality”
Albert Lautman

The logicians of the Vienna School maintain that the formal study of scientific language must be the only object of the philosophy of science. It is a difficult thesis to accept for those philosophies that consider it their sole task to establish a coherent theory of the relationship between logic and the real. There is physical reality and there is the miracle of explanation, it is only that it needs more developed mathematical theories in order to interpret it. There is in the same way mathematical reality and it is similar object of admiration to see certain domains resist their exploration in a way, so that one must approach them with new methods. It is just as when analysis is introduced into arithmetic or topology into the theory of functions. A philosophy of science that would be unable to carry the entirety of the study into this solidarity between domains of reality and the methods of investigation would be singularly without interest. Philosophy is not the effect of mathematical nature. If logico-mathematical rigor can seduce it, it is certainly not because it permits the establishment of a system of tautological propositions, but because it shines a light on the tie between rules and their domain. It, at the same time, produces a curious event that is for the logicians an obstacle to eliminate, becoming for the philosopher the highest object of interest. It is a question of all the “material” or “realist” implications that the logicist is obliged to accept: this is the well-known axioms of Russell, the axiom of infinity and the axiom of reducibility. It is above all, following Wittgenstein, the affirmation that all true propositions correspond to an event of the world, which involves the entire procession of restrictions and precautions for logic. In particular, any proposition relating to the set of propositons, every logical syntax, in the sense of Carnap, is impossible since it would need for itself the power to consider, correlatively, the world as a totality, which is impossible.

The logicistian of the Vienna School continuously affirm their complete accord with the school of Hilbert. Yet nothing could be more questionable. In the logicist school, in the style of Russell, one strives to find the atomic constituents of every mathematical proposition. The arithmetic operations are defined starting from the primitive notions of elements and classes and the concepts of analysis are defined by extension starting from arithmetic.  The concept of number plays here a central role and this role is again augmented by the arithmetization of logic in the style of the work of Gödel and Carnap. The primacy of the concept of number, nevertheless, seems not to be confirmed by the development of modern mathematics. Poincaré had already indicated in relation to the theory of dimension that the arithmetization of mathematics does not always correspond to the true nature of things. Herman Weyl has, in the introduction to his work Gruppentheorie und Quantenmechanik, established a distinction that appears fundamental to us and will have to be accounted for in all future philosophies of mathematcs. It distinguishes two currents in mathematics. The first, issuing from Indian and Arab thinkers, puts the concept of number into relief and succeeds in the theory of functions of complex variables. The second is the Greek point of view in which every domain carries with it a system of characteristic numbers. It is primacy of the geometric idea of the domain over that of whole numbers.

The axiomatics of Hilbert and his students, far from wanting to restore the mathematics of sets as only a promotion of arithmetic, tend, on the contrary, to free for every domain of study a system of axioms as such, the union of conditions implied by the axioms emerging at the same time as the domain and the valid operations within this domain. It is also that one is axiomatically constituted in the modern algebraic theory of groups, of ideas, of the system of hypercomplex numbers, etc.

The consideration of a purely formal mathematics must therefore give order to a dualism of a topological structure and the functional properties in relation to this structure. In a similar way, the formalist presentation of theories: axiomatization is only a question of greater rigor. The object studied is not the set of propositions derived from the axioms, but organized being, structures, totality, having an anatomy and physiology proper. Citing as an example Hilbert’s space defined by the axioms that confer on it a structure appropriate to the solution of integral equations. The significant point of view here is the synthesis of the necessary conditions and not this analysis of first notions.

This same synthesis of the domain and the operation finds itself in physics under a slightly different point of view. Carnap, sometimes, seems to consider the relationship between mathematics and physics as that between the form and the matter. Mathematics would provide the system of coordinates in their inscription of physical givens. This conception hardly appears defendable seeing that modern physics, far from maintaining the distinction of a geometric form and a physical matter, combines, on the contrary, spation-temporal and material givens in the common framework of a mode of synthetic representation of phenomena; that this were by the tensorial representation of the theory of relativity or by the Hamiltonian equation in mechanics. Thus, one witnesses for every system with a simultaneous and reciprocal determination of container and contents. It is again a determination proper to every domain, the interior of which no longer subsists for the distinction between matter and form. Carnap seems, it is true, to have another theory of the relationship between mathematics and physics, conforming much more closely to his logistic tendancy. It considers physics, not as the science of real facts, but as a language in which one expresses experimentally verifiable statements. This language is submissive to the rules of syntax, of mathematical nature when it is uniformly valid in all their defined domains, of physical nature when their determinations vary with experience. There is the new affirmation of the universality of mathematical laws by opposition to the variation of physical givens. It seem to us that this conception does not reflect that the variation of physical givens only make sense in relation to the presumed variety of varying sizes, and this variety is physics. Carnap’s example from page 131 of his book The Logsche Syntax der Sprache is characteristic. When, we say, the components of the fundamental metric tensor are constants, it is a mathematical law; when they change they observe laws of physics. The real philosophical problem would rather be of the knowledge of how a differential geometry could become the theory of gravitation. This relation between geometry and physics is the proof of an intelligibility of the universe. It results in the development of the mind, in a way, to structure the universe in profound harmony with the nature of this universe. One conceives that this penetration of the real by human intelligence has no meaning for those excessively formalistic. These, in effect, would rather see in these pretensions of the mind to know nature a relevant approach to the studies of Lévy-Bruhl. Understanding would be for them a mystical belief analogous to the participation of the subject to the object in the primitive soul. The term of participation has in philosophy another origin more noble and Brunschvieg has justly denounced the confusion of the two senses. The participation of the sensible in the intelligible in Plato can be identified. If the first contacts with the sensible are not sensations and emotions, the constitution of mathematical physics gives us access to the real via the knowledge of the structure it has been given. Similarly, it is impossible to speak of the independent reality of the modes of thought according to which it leaves it to dread and disparage the merely mathematical language indifferent to the reality that it decries, philosophy engages itself through an attitude of meditation where it should achieve the secrets of nature. There is, therefore, no reason to maintain the distinction, made by the Vienna School, between rational knowledge and intuitive experience, between Erkennen and Erleben. In wanting to remove the relationship between thought and the real, as in refusing to give science the value of a spiritual experience, one risks only having a shadow of science, rejecting the spirit in the search for the real towards the violent attitudes where raison plays no part. This is a resignation that the philosophy of science must not accept.

[Here is the final part of Badiou's logic course from 1982 - 83. I find the brief discussion of the infinite and infinitesimal interesting. Enjoy.]

Alain Badiou – Logic Course (1982 – 1983)

Kant: What can I know? → the phenomena and the thing-in-itself. What must I do? → conform my action to the categorical imperative and ensure that my maxim is universalizable without contradiction. What must I hope for? → The immortality of the soul.

Gödel: the not-all of the calculable in relation to the rational.
Lowenhein-Skolem: the not-one of the rational in relation to the calculable.
Tarski: the impossibility of restoring to the calculable the crucial predicate of the rational that is said of this truth. The rational concept of truth is not representable in calculation: what is not representable are true statements that should be true.

Metatheorem of equivalence

Let C be a theorem of L. Let A be a part of a correct formula C. Suppose that A/B. Let C′ be where A is replaced by B. Then C′ is also a theorem of L.

Language LP of propositional calculus

1) alphabet =
• propositional variables: A, A′, B, B′ …
• logical operators: negation (¬), implication (→) [ The operators ‘or’, ‘and’, ‘equivalence’, are logically deduced from these two operators]
• punctuation signs: ‘(‘ ‘,’ ‘[‘ …

2) grammar

3) three axioms
• p → (q→p)
• [p → (q → r)] → [(p → q) → (p → r)]
• (¬p → ¬ q) → (q → p) = transposition

4) rule of deduction
• Modus ponens (MP): (p→q / p // q)

5) Theorem of deduction
• Theorem of Kalmer (1922): theorem of the completeness of propositional calculus that is said of all tautologies/contradictions: syntacto(decidable)-semantic(being a tautology). Where the language LP is coherent, complete, and decidable.

Language L of the predicate calculus of the 1st order

In addition to LP:

1) alphabet =
• individual constants or proper names: a, a′, b, b′ …
• individual variables: x, y, x′, y′ …
• unary predicates: P, P′, Q, Q′ …
• binary (R, R’ …) and tertiary (T, T′) relations
• Logical connectors:
• Universal quantifier (∀) [ Cf. ∃ is logically deduced from ∀]
2) grammar

3) two axioms
• [∀a (A→B)] → [A→(∀a) B] if ‘a’ is not free in A
• (∀x)A → Subst(y/x)A: one can replace in A x with y if it does not have a particular role in A
4) rule of deduction
• rule of generalization: If A then (∀a)A (one can universalize a statement) [The rule of existentialization is logically deduced from the precedent as ∃ is logically deduced from ∀]

→ (Meta)theorem of equivalence
→ Lowenheim-Skolem’s Theorem (1st)
→ Lowenheim-Skolem’s Theorem (2nd)

Gödel’s Theorem (1930): says the completeness of 1st order logic which is introduced in universally valid statements. Where the language L is coherent, complete but decidable. Cf.:
• Church’s Theorem: 1st order logic is not decidable
• Zermelo’s Axiom of Choice is formulable.
• Herbrand, then Henkin, have also worked on the completeness of 1st order logic for universally valid statements.

Logic of arithmetic by Peano = theory of identity + theory of successor + axiomatic schema for demonstration by repetition: {A[0] and (∀x)(∀y)[S(x,y) → A[x] → A[y]} → (∀x)A[x]

Properties of a Theory

a) consistency and coherence: one cannot deduce all at once A and ¬A; one cannot, therefore, deduce, only introduce. Cf. if I am → A and →¬A then:
• ¬A → (A → B): cf. “ex facto”
• ¬A and ¬A → (A → B) brings about (A → B) by Modus Ponens
• then A and (A → B) brings about (by MP) B!
• An inconsistent theory is a theory where at least one statement is not deducible.
• Lowenheim-Skolem’s Theorem (1st): If (closed formula) A is not deducible in (consistent theory) T the T + ¬A is consistent.
• Lowenheim-Skolem’s Theorem (2nd): If T is coherent, the theory obtaining an additional list of individual constants (= proper names), eventually becoming infinite, is equally coherent.
b) completeness (syntax): all statements are deducible. [semantic completeness is where all true statements are demonstrable. This forbids Gödel’s Theorem.]
• more precisely: a theory is syntactical complete if all of the closed universal formulas are decided. A complete theory is maximal: one cannot add them without rendering them incoherent.
• Cf. Closing the universe of open formulas: one adds to a formula a ∀ for every free variable. An open statement can be true or false depending on this or that interpretation. A closed statement, however, is constrained by a single interpretation. Cf. the open statement: (∃x)P(x) → (∀x)P(x) is indecidable. The open statement P(x) → P(x) is transformed into universally closed statement: (∀x)[P(x) → P(x)]
• A universally closed statement:
ß does not have a free variable
ß is closed by the universal quantifier (≠ existential closure)
• Gödel’s Theorem → incomplete in this sense
• Lindenbaum’s Theorem → ∃ extension completes all theory
• One does not have a complete theory of completeness. One cannot know why a theory is complete or not.
• Cf. Complete theories:
ß Additive arithmetic
ß Boole’s non-atomistic algebra
ß Naïve set-theory
• Most of the grand theories are incomplete.
• Lindenbaum’s Theorem: every theory admits a syntaxically complete extension
• Remarks: this extension is monstrous in three ways:
ß This extension is entirely ineffective: one cannot know whether the given statement is or is not.
ß This extension is not given by axioms but in a single package. It is very difficult to axiomatize. Cf. there exists a tension between completeness and axiomatic presentation.
ß One cannot know at all models of extension taken into consideration by the initial model
ß Extension is also ineffective, axiomatically inaccessible and pathological in terms of the model.

c) decidability = There exists a mechanical procedure that permits the verification for all formulas if it is a theorem.
• Cf. idea of a mechanical procedure that is a decidability machine. Cf. concept of recursive function, Turing machine, Markov algorithm…for these verifiable procedures (and not demonstrable)
• It is a calculable on the calculable.
• Example of decidable theory: the formal theory of commutative groups.
• Why a theory is decidable? One cannot know. Its not a question of simplicity.
• One shows that there is no decidability as decidability!
• One shows that in common decidable theories (cf. naïve arithmetic is additive arithmetic) there always exists formulas where numbers approach the gigantic.

Concept of the Model

Semantic: that yields 5 things, minimally:
• a domain of objects
• proper names distinguishing these objects
• there can exist properties of these objects
• there can exist relations between these objects
ß Intensive relations: given in their definition
ß Extensive relations: given in a list of these n-uplets
• rule of the true and the false

The interpretation of a theory T according to a model M makes:
• variable of T → object of M
• individual constant T → fixed element of M
• predicate in place of T → property of M
• relation of T → relation of M
One translated in this way all elementary formulas of T and one cuts on true/false.

non p        p and q   v     f          p or q    v   f        p → q    v    f          p ↔ q    v    f
v       f             v      v    f              v          v   v            v        v    f             v         v     f
f       v             f          f     f              f         v   f               f          v    v             f          f     v

The logical operators are not translated but one has directly the rules of evaluation. The logical operators are impossible to translate. Towards them, one has a disposition of evaluation, not translation.
Critique of modal logic (with possible, necessity, etc.): one can formalize the theories of probability with classical logic.

N.B. Formalization is the experimental moment of mathematics. It is not true to say, from this point of view, that mathematics is a formal science.

Mathematics exemplifies, in the simplest way, the fact that reality is an impasse. And all experience of the reality is a tense experience, extreme,

History

Boole (1850): 1st version of propositional calculus (with ¬, ø; but without quantification)
Frege, Peirce (1870): idea of Frege that everything returns ideography, that is to say, a writing where the concept would be transparent.
Whitehead and Russell: cf. criticize an inconsistency in Frege’s system holding to the possibility of applying the predicates to the themselves, holding quantifications over properties. Where stratification of syntax and the theory of types: what is to the left and to the right of the sign ∈ is not the same type. Cumbersome and useless (One needed an axiom of reducibility to destratify.
Hilbert: project of the auto-foundation of mathematics over calculus; to seek that everything true be demonstrable = reinstate without rest the truth of demonstrability. Auto-founding the demonstrable.
1915: Lowenheim’s 1st version
1920: Skolem’s recaptures this
1930: Gödel’s theorem

Universally valid formula = valid formula in all interpretive domains.
In logical language, all theorems are universally valid.

Kalmer’s theorem (1920) = theorem of completeness of propositional calculus is said of all tautologies/antilogies. Where language LP is coherent, complete, decidable.

Gödel’s Theorem (1930): says of the completeness of 1st order logic that introduced into universally valid statements. Where the language L is coherent, complete, but not decidable. Cf.:
Church’s Theorem (1937): 1st order logic is not decidable.
Cf. in logic, one cannot know what one says. Cf. mathematics ≠ logic. In mathematics, contrary to Russell, one knows of what they speak.

Logicist School (cf. Russell): logic and mathematics indistinguishable, under the order of a single syntax and argument that if two things are axiomatizable, they are the same.

Therefore:
• Propositional calculus → tautologies → language LP with “¬” and “→” → three axioms and rules of deduction.
• 1st order logic → universally valid formulas → Language L with another ∀, an axiom and rule of deduction.

All coherent theories have at least one model (theorem of Léon Henkins) = anti-idealist position: coherence allows the reference. There is not uninterpretable coherence. All coherence makes sense. Cf. Hegel: “all rationality is real”
All theory that have a model are coherent.

Lowenheim-Skolem’s (1st): all coherent theory has a numerable model
Lowenheim-Skolem’s (2nd): all coherent theory has a model of poor infinity

Categoricity of a theory:

The models of a type infinity given are isomorphs? If yes for a given infinity α the theory is said α-categoricity. It is not of an absolute categoricity since two different models of infinity are not isomorphs.

Examples:
• theory categorical (and therefore univocal) in numeration but not categorical for the superior types of infinity.
• theories α-catergorical for α > numerable but not for the numerable; ex. group theory communitive to unique divisor.
• theories categorical for all infinites separately; ex. theory of vectorial spaces of the entire modulo 2.
• theories never categorical; ex. group theory
Conjecture of Los: it would not have to have these 4 cases.

Morley’s Theorem: has demonstrated this conjecture! It is the longest and most difficult theory of all of logic.

In this way, categoricity does not distinguish the two types of infinity: the numerable, all that is beyond and which makes for categoricity. It is close to the numerability of writing.

Infinites, transcendence

Indefinite ≠ infinite, as potential infinite ≠ actual infinite

The infinite is not connected to the one in mathematics for Cantor. Before this connection was theological. (Uncreated) God was infinite and the (created) world was finite. Their one takes precedence over their infinity: It is the united transcendence of God that brings about their infinity.

One has therefore, before the 17th-century:
• a theological infinite (God)
• a physical finite (world)
• a purely potential logico-mathematical infinite.

Shaking the 17th-century: introduce the actual infinite in the physical world (cf. dialectical dramatization of this point by Pascal) but still not the concept of the actual infinite in mathematics.

The mathematical shaking comes instead from infinitesimal quantities. One progressively eliminates (between Newton = end of 17th and Cauchy = beginning of the 19th) these quantities by the introduction of a calculus of limits: “tightening toward” restoring the potential infinite. (Cauchy ≡ Russell more than Brouwer in the crises of sets)
Cantor introduces the radical revolution in secularizing the infinite by calculating the actual infinite. The introduction of the Hebrew alphabet!
The mathematical concept of infinitesimals is not far behind. Cf. Abraham Robinson (1960) and non-standards analysis. Nevertheless this has little repercussions: one redeems these known results for a long tome to come!
In this way, the mathematical revolution of the infinite has more than a century whereas the infinitesimal only has 25 years.

Pascal makes these two infinites symmetrical. Cantor does not believe in the infinitesimal.

Large Cardinals
• inaccessible
• ineffables
• of Ramsey
• low compacts [faiblement compacts]
• compacts
• super compacts
• enormous
• measurable

[Here is the second translation that I promised. It is a little more interesting than the first, but the two do flow together quite well. Enjoy!]

Alain Badiou – Logic Course (1981 – 1982)

Lacan: “the real is the impasse of formalization” as well as the symbolic. Its nature pierces
the symbolic.
Theorems of limitation: theorems of the formal real
Gödel: impossibility of nomology, of completeness, of the decidable. Every formal
system is pierced. There is an excess of the true in the calculable.
Tarski: impossibility to formalize the notion of truth (of a statement): The true is
unrepresentable in arithmetic.
Beth: impossibility to demonstrate that such a notion is definable in a given system.
Lowenheim-Skolem: superiority of the infinite in relation to the numerable. There is an
excess of the calculable in the true.
However, this theorem = theorems also of excess! There is something of the truth that
exceeds the formal.

General Crises

Greeks: Crisis of irrationals. The idea that being is given in numbers, that all are counted,
is called into question by a “being” that is no longer a “rational” number.
17th – 18th century: differential and integral calculus. They can have a mathematics of
infinity and no longer only of finitude. They can therefore have a mathematics of movement.
Beginning of the 19th century: Non-Euclidian geometry. Doubt over the absolutivity of
Euclidian space taken by Kant as a structural category of experience.
End of 19th century: Paradoxes of set-theory. Attained through the reliability of the
intuition of the multiple.

Gradual Intersecting

Complete result of signs → correct formulas → true formulas → demonstrable formulas
The indecidable that limits calculation is wandering and cannot be assigned.

Theorem of Lowenheim-Skolem (TLS or LS):
if a formal system has an infinite model, it necessarily has a demonstrable (pathological)
model. All formal theories necessarily have an infinite demonstrable model. The infinite non-demonstrable does not have a specific formalization! All formalization is in excess in relation to possible interpretations. There is an over-powering logic in regards to that which one might formalize.
Gödel: true (and rational) > calculable; or: the semantic exceeds the syntax.
LS: the contrary: There is a point that flees formalization.
Torsion = this double excess of something at an indecidable and non-conforming
(pathological) time.

Deviations:
• true → calculable → the only indecidable → reversal in the irrational!
• purely productive insight by overlooking the point fleeing formalization → reversal by ruining the rigor of calculus at the same time

Metatheorem of deduction:
1. ((S+A)→B)→(S→(A→B)) ≡ deduce B from S+A again by deducing (A→B) from S ≡ joining A to S and in deducing B again from by deducing B from A in S alone.
2. This demonstrates a connection between the rational notion of the hypothesis (cf. S+A) and this deduction of implication! It’s a connection between the rational and the calculable.
3. This demonstrates this by recurrence along the length of deductions.
4. The calculable → the finite law where the infinite is on this side of reality. Theology posits the contrary: the world is finite but the law of God is infinte.
5. Grand mathematical thought is not secondarily calculating: it is conceptual. Hegel despised mathematics because he believed it was reducible to calculation. He criticized mathematics because the negative was not operative (because the negation of negation = affirmation).
6. Calculation is not the absence of the subject but its withdrawal. In calculation the subject does not lack but comes to be missing.
7. Formalism is mechanisable. The machine, if it deals with the real, has no relation to the real. Only the subject has a relationship.
8. Axiomatic schema: in a deduction, one can replace any variable by any concrete expression.
9. The logic of predicates of the 1st order: one only quantifies the individuals and not the predicates or properties. There is not, therefore, properties of properties.
10. Most of the algebraic theories are axiomatizable in this logic. On the contrary, topological structure necessitate, in general, a superior order logic.
11. Existence defined by double negation. But ∃ and ¬¬; are they equivalent? Existence is then posed as a limit to the total power of negation. And yet the universal does not have the ontological range that the existential does. Example: All gods are infinite → There exists an infinite God. This last statement commits me, but not the first where the verb has no existential value. The equivalence of ∃ and∀ is purely logical and would not know to infer on the ontological.

3 transformations for substituting ∃ and ∀:
1. replace (A → B) with ¬(B→A)
2.  substitute ∃ and ∀
3. put a negation (¬) in front

An erasure is always a marker, but on its borders.
TLS shows that the homogeneity of the inscription does not arrive to homogenize the
semantic.
Univocity does not exist, therefore, more for artificial languages (langues) than it does for
natural languages. It does not exist as an absolutely univocal language (cf. the failed dream of Leibniz). All languages are poorly made. The subject is effaceable from all language.
TLS is a massive return of the repressed.

Formal theory:
0. underlying logic
1. list of individual constants
2. list of individual predicates
3. additional specious axioms

Theory of identity:
1. no individual constants
2. predicative constants at 2 places: I
3. Three axioms:
∀xI(x,x) =  reflexivity
(∀x)(∀y)I(x,y) → I(y,x) = commutativity or symmetry
(∀x)(∀y)(∀z)I(x,y) and I(y,z) → I(x,z) = transivity
This theory gives 1st-order predicate calculus with equality. One can sometimes consider it not even being mathematical and remaining purely logical.

Theory of the strict partial order:
1. not individual constants
2. predicative constants at 2 places <(x,y) or x<y
3. three axioms: irreflexivity, anti-symmetry and transivity

Theory of the strict total order:
Supplementary axiom: ∀x∀y x<y or y>x

Theory of successor:
1. no individual constants
2. binary relation S(x,y) = y is the successor of x
3. four axioms:
¬∃x S(x,0) ≡ 0 is not succeeded
S(x,y) → [S(x,z) → y = z] ≡ there is a single successor.
S(x,y) → [S(z,y) → x = z] ≡ there is a single predecessor.
All numbers have a successor.

[What follows is a translation of lecture notes from a course on Logic delivered by Alain Badiou in 1980 - 81. It is rather short and deals specifically with Gödel and his theorem of incompleteness. I am not sure who would be interested, but here is it. There are two more course notes (1981 - 82 and 1982 - 83) that I will be posting in the realitively near future. Comments would be greatly appreciated. Enjoy!]

Course on Logic (1980 – 1983)
Alain Badiou

1980 – 1981

A formal system S is composed of 4 things:

1) Alphabet = stock of signs, either
• individual constants or proper names (0, for example)
• individual variables: x, y, x’, y’ …
• predicate  constants (“=”, etc.)
• function constants (“+”, “*”, “succession”…etc.)
• statement of propositional variables: p, q, p’, q’ (Not predicate variables)
• logical signs (negation, implication, and, or…and quantifiers ∃ (there is at least one), ∀ (all)); cf.:
i. operator “or”: A or B ≡ ¬A→B
ii. operator “and”: A and B ≡ ¬(A→¬B)
iii. operator “equivalence”: A/B ≡ (A → B) and (B → A)
• signs of punctuation: “(“ “,” “[“ …

2) Grammar = rules used to form the correct     formulas

3) Axioms
Cf. Three axioms of the calculus of propositions (= axiomatic schema)
• p → (q→p)
• (p → (p → r)) → ((p → q) → (p → r))
• (¬p → ¬q) → (q → p) = transposition
Cf. also axioms of calculus of predicates
• [∀a (A → B)] → [A → (∀a)B]
• (∀a)A → Subst(x/y)A: one can  replace Ax for y if y does not play a particular role in A (Universal instantiation)

4) Rules of deduction
• rules of substitution: if A ↔ B, on can replace A by B in a formula.
• Modus Ponens [p →q/p//q]
• Existential Instantiation: if p then (∃x)p
• Universal Generalization: if A then (∀x)A (One can universalize a statement)

Model M constitutes an interpretive domain: all systems of formulas are interpreted as true/false.

In the system, there is not true/false.
In the model, there is not demonstration.
Where the relation of to truth to demonstration (= as in proof)
S → M ≡ Syntax → Semantics = interpretation
M → S ≡ Semantics → Syntax = formalization

Gödel’s Theorem: There is always a formal statement p that is not demonstrable or refutable in S and where it is translated in M it is true.

There are three limitations where one demonstrates that there are equivalents:

1) incompleteness (semantics): ∃ a true statement not demonstrable in S.
2) indecidable (or incomplete syntax): ∃ an indecidable statement (neither demonstrable nor refutable)
3) irreflexibility as in coherence: impossibility to demonstrate in S its coherence (non-nomology)

1+5 conditions authorizing Gödel’s Theorem:
0) Countable alphabet (discreets!→writing) and final statements (= one can number the statements!)
1) The system must receive as a possible model the integers (the system must have an arithmetic capacity): the translation of the symbols must be capable of being the integers (condition for the function crossing translation and numerization).
2) The system must receive a diagonalized numerization: ∀P ∃a as Num(P)=Tran(a): “a” is translated as the number “n” at the same time as the statement P(a) has “a” for a number; “n” is then in a diagonal position for the predicate P (double function!).
3) The system must receive a predicate D support under the demonstrability: D(i) interpreted as Tran(i)∈Dem that is true if n=Tran(i) is the number of a demonstrable statement (= reflexive condition of the system or expressive condition as far as the question of demonstrability).

These three conditions converge ≡ a single condition: that the system is capable of representing all of the recursive functions. For any elementary arithmetic model, it has these properties.
4) The system must have a symbol translating the negative.
5) Being given a predicate P, ∃ the predicate Opposed-P(i) that corresponds to not-P(i) = condition of complementation.

In fact, three conditions arise:
1. the language is finite or numberable (not the design!)
2. the system comprises a symbol for negation
3. the system admits an elementary arithmetic model (whole numbers)

Principles of demonstration

1) Let D be a predicate demonstrably defined by:
Condition 3 = ∃ Opposed-D contrary of D (cf. Gödel Theorem 5); or
Condition 2 = ∃a such that Num[Opposed-D(a)]=Tran[D(a)]=n (It states that a statement “a” is indemonstrable.
2) One can demonstrate then that Opposed-D(a) (i.e. that “a” is not demonstrable in S) is not demonstrable (→incomplete):
If it was demonstrable, on could have D[opposed-D] and therefore Tran[Opposed-D(a)]∈Dem; or the translation is precisely “ ‘a’ is not translatable” therefore, it would not know it belongs to Dem! → Contradiction. Where incomplete: a true statement (in case the statement “ ‘a’ is indemonstrable”) is indemonstrable.
3) One demonstrates that more than that it is not refutable (→ indecidability)
(a) If it was refutable, then the model would no longer be a model because one could have decided that it would be false!
(b) If not-[Opposed-D(a)] is demonstrable, its translation is not-not-Tran[D(a)]∈Dem if Tran[D(a)]∈Dem. Therefore, n=Tran[D(a)] translates demonstrable statement; however n=Num[Opposed-D(a)]. Therefore, one could in a system demonstrate a statement Opposed-D(a) and its contrary not-[Opposed-D(a)]! Inconsistence of the system.
4) One then deduces the irreflexivity of the system:
If one could have demonstrated in the system a predicate affirming the consistency of the system, then one could demonstrate in the system that one could not have at the same time D(a) and Opposed-D(a); therefore one could deduce in the system Opposed-D(a) since one knows that D(a) is true; but then, by modus ponens, the indecidable statement would be decidable in the system and the system must be incoherent!

In this way, the formal does not exhaust the true.

Double function. Canonical example: a[P(a)] if, for example, “ ‘a’ says that it has a property P”
Husserl → nomological theories say that it is its own law.
Program of work defined by Hilbert.
Gödel indicates the failure of the nomological program.