3rd: Alain Badiou – Logic Course (1982 – 83)

[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

2nd: Alain Badiou – Logic Course (1981 – 82)

[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.

1st: Alain Badiou – Course on Logic (1980 – 81)

[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.

Critchley & Simmons

It is said that Nietzsche, during his fascination with Wagner, sent some of his compositions to Franz Liszt. Liszt’s first response was to believe that it was some kind of joke (which, of course, it wasn’t). With this in mind, I give you Simon Critchley’s artistic expression.

Lucretius Contra Heidegger

It has become de rigueur in the last century, the last philosophical century, to approach thought via a limit-concept, and our philosophies are marked by the tropes of this finitude. In short, thought is tragic; or put slightly differently, thought is Heideggerian. Tragic since the ultimate horizon of thought and being are found in the human. The question that arises in this paper is whether the next philosophical century is possible. Is a non-tragic thought possible? Or put slightly differently, is it possible for a thought to exist that does not simply reject the previous conditions of thought but instead pushes these conditions to their most radical conclusion?

The difficulty that arises is from where a non-tragic thought could find its bearings. Tragic thought does not simply affect the current position of philosophy, but instead transforms the entire history of philosophy, through Heidegger’s appropriation of ancient thought through his studies from Parmenides to Aristotle up through Augustine. What is needed, then, is a position within the stream of philosophy that is not already appropriated by the tragic moment in thought. It is the wager of this paper that that position is the one occupied by Titus Lucretius Carus, and his magnificent De Rerum Natura (On the Nature of Things ). Lucretius is, quite simply, unthinkable from the perspective of tragic thought, from Heidegger’s thought, and thus remains an obscure figure in the history of philosophy, at best. More importantly, Lucretius continually runs ahead of Heidegger and thus he appears to be beyond the tragic era. What Lucretius’ philosophy provides is the possibility to re-think the tropes of thought that have been brought forth by the tragic century and thus, allows for the beginning of a non-tragic thought.

Ultimately, tragic thought operates via a particular relationship between philosophy and science; it is an assertion that science must take philosophy as its condition. This point is made explicitly in Heidegger’s earliest work, whether philosophy is seen as a fundamental ontology versus science’s regional ontology or philosophy as critical science with science as positive science. In his lecture course on Kant’s Critique of Pure Reason, Heidegger states,

…whereas the physicist defines what he understands by motion and circumscribes what place and time mean—whereby he relies in part on ordinary concepts—still, however, he does not make motion’s way of being a theme of his investigation. Rather he examines only certain movements. The physicist does not inquire into the ownmost inner possibility of time, but rather uses time as that with respect to which he measures motion…The scientific methods have been developed precisely in order to explore beings. But they are not suited for examining the being of these beings. If this is to happen, then what we need is not to objectify a being, e.g., the existing nature as a whole, but the ontological constitution of nature or the being of that which exists as historical.

In the course the previous year, Heidegger states, “…the other sciences, mathematics, physics, history, philology, linguistics, do not begin by asking what is mathematics (etc.); instead they just set about their work, they plunge into their subject matter…In the very essence of all these sciences, in the fact that they are positive sciences, versus philosophy, which we call the critical science.” For Heidegger, science is only capable of dealing with beings but is incapable of understandings its own foundations in being. Philosophy, on the other hand, is capable of providing this foundation via its role as fundamental ontology. In short, science remains lacking, existing only as a possible philosophy at best.

But what is the effect on science from this approach? At the surface level, this requirement is little more than a version of the Kantian transcendental, a search for the conditions of possibility for any science. But what must be recognized, and this is the contention of this paper, is that this seemingly innocuous requirement is a poison pill and that science’s acceptance of philosophy as its condition is nothing short of the destruction of science itself, and the casting of philosophy into the sea of idealism. Why is this the case?

Heidegger begins by asserting the necessity of science’s acceptance of philosophy as its condition, pointing to antiquity for proof of this relationship. He then adds that what philosophy, as the “freest possibility of human existence” provides is the “most original and necessary relationship” to being. But since Heidegger’s understanding of being is its openness, its givenness, or its manifestation, then he has affirmed himself within the lineage of Kant, and thus, science, with its condition set by philosophy, loses its ability to touch the real, the reality indifferent to dasein or the human. But such a loss cannot be overcome by science and so results in science’s utter destruction. Being, for Heidegger, cannot be separated from its relationship to the human, imagined as dasein, and thus remains a humanistic idealism masquerading as a radical philosophy.

But what if the relationship is reversed? What if science becomes the condition of a philosophical thought? It is already in Plato where this understanding of philosophy is put into practice. Mathematics and specifically its non-empirical status serves as the student’s, and our own, path from becoming to being. It is mathematics that sets itself against doxa, against the endless assertion of one’s opinion and perspective. Let us look at Plato’s Republic Book 7:
Think a little and you will see that what has preceded will supply the answer; for if simple unity could be adequately perceived by the sight or by any other sense, then, as we were saying in the case of the finger, there would be nothing to attract towards being; but when there is some contradiction always present, and one is the reverse of one and involves the conception of plurality, then thought begins to be aroused within us, and the soul perplexed and wanting to arrive at a decision asks ‘What is absolute unity?’ This is the way in which the study of the one has a power of drawing and converting the mind to the contemplation of true being.

What then of Lucretius, this obscure Latin poet philosopher? The insight is Michel Serres’, found in his extended study on Lucretius, The Birth of the Physics . Serres rejects the notion that ancient atomism, specifically Lucretius’, is naïve or non-mathematical. Instead, he asserts that Lucretius’ most fundamental insight, the swerve of the clinamen, operates via a specific dialogue with the mathematics of Archimedes, specifically the proto-calculus of infinitesimals. “Not only did the atom have to be born by way of the treatment of curved elements, in the irrational and differential, or by way of the indefinitely divisible…This is because the angle of contingency may not be subdivided: it is demonstrably minimal. It is null, but without the lines which form it overlaying one another.” Let us look to Lucretius’ actual text to see this for ourselves.

It begins with the laminar flow, the gravitational descent of atoms in the perfect parallel order, then “at absolutely unpredictable times” the atom swerves off its path, slightly, by “only an infinitesimal degree,” a deviation of the smallest possible angle. And thus the lines of the laminar flow become spirals, at utterly incomprehensible moments since they are moments without witness or givenness, as the clinamen begins to wobble, order is broken, the void is joined by the reality of the atom. Atom meets atom, eventually, as the spiral forms a cone and force of gravity is transferred via an encounter. As Althusser will say in his rediscovery of the secret history of materialism: “it is clear that the encounter creates nothing of the reality of the world, which is nothing but agglomerated atoms, but that it confers their reality upon atoms themselves, which without swerve and encounter, would be nothing but abstract elements…” The swerve of the clinamen, long dismissed as “puerile” or a “most monstrous absurdity” or worse yet, used to introduce free will, should be seen as the most basic mathematized thought of a temporality without givenness.

We should pause here just to note the radical difference between this mathematical beginning and the other ancient model that has received so much attention in our tragic era, namely the one discovered in Plato’s Timeaus. The point is not primarily a question on the value of chaos versus order, although this certainly plays a role, but instead the relationship between the world and the human. In Plato, the world is placed into order by the demiurge because of its moral purity, which is to say, because of the coming human soul that the demiurge is preparing. Whereas in Lucretius, the human, the product of the random combination of atoms, is ill-suited for the world, since it exists without reason, and is incapable of forming itself as dominion. Lucretius’ approach is thoroughly scientific; Plato is unable to break from the doxa of Greek religion. To paraphrase Lautréamont, “he who knows and appreciates (mathematics) no longer wants the goods of the earth and is satisfied with (its) magical delights…(the demiurge) only offers (Plato) illusions and moral phantasmagoria.”

What, then, is the effect of such a reversal of the basic relation between science and philosophy? What becomes of the tropes of finitude that we philosophers have been unable to operate without? Let us look at two of these tropes, death of the mortal man and death of the immortal god, and how these are transformed in Lucretius’ philosophy. Heidegger, in the Beitrage, writes, “The uniqueness of death in human Da-sein belongs to the most originary determination of Da-sein, namely to be en-owned by be-ing itself in order to ground its truth (openness of self-sheltering). What is most non-ordinary in all of beings is opened up within death’s non-ordinariness and uniqueness, namely be-ing itself, which holds sway as estranging.” Or in Being and Time, we are presented with “With death, Dasein stands before itself in its ownmost potentiality-of-being. In this possibility, Da-sein is concerned about its being-in-the-world absolutely…As the end of Da-sein, death is the ownmost nonrelation, certain, and as such, indefinite and not to be bypassed possibility of Da-sein. ” In short, death is an event since it pronounces the highest possibility of Dasein and with this, death displays the ultimately groundedness of Da-sein as the ab-ground, the abyss. Death reveals the truth of being to Dasein; this truth being the original givenness of the world, or what Heidegger comes to call the “other beginning.” This other beginning is not the material conditions for a world but instead the ultimate human centeredness, or Dasein centeredness if you prefer, of being itself.

This being said, there is a remarkably different approach to death in the work of Lucretius. He states, “Death, then, is nothing to us and does not affect us in the least, now that the nature of the mind is understood to be mortal.” The most important difference between the tragic approach and Lucretius’ is that Lucretius robs death of its status of the singularity. Death, according to Lucretius, is a property of the body, as it is a property of all bodies. A property according to Lucretius “is what cannot under any circumstances be severed and separated from a body without the divorce involving destruction.” Death then is a property of all bodies, save one, the atom. Humans die, but so do plants and planets. Heidegger magnifies the importance of death since the revealedness of being is directly correlated to Dasein. No such relationship is presupposed by Lucretius, since his mathematics has presented him with the possibility of a thought without human witnesses. Mathematics is, after all, “an instance of stellar and warlike inhumanity.” We should be wary of making the mistake of supposing that Lucretius’ view on death is an historical curiosity, and mere replication of the zeitgeist of Roman philosophy, a masculine warrior’s response to the inevitable. While it is true the Stoics dismissed death as a concern, they also offered themselves an escape clause via their insistence that the gods would honor their triumphs. Lucretius, as will discuss in the next section has no such clause.

What then of the death of the immortal god? The death of God or, to use Heidegger’s later formulation, the fleeing of the Last God is the condition of possibility for world. “The last god has its essential swaying within the hint, the onset and staying-away of the arrival as well as the flight of the gods who have been…In such essential swaying of the hint, be-ing itself comes to fullness. Fullness is preparedness for becoming a fruit and a gifting…Here the innermost finitude of be-ing reveals itself: in the hint of the last god.” Once again we return to the centrality of the human, Da-sein. The fleeing of the Gods reveals be-ing as gift, whether delivered or delayed, it makes no difference. Heidegger has asserted the impossibility of any thought beyond the manifestation of world to Dasein, and tragic thought receives its trope. This position is not unique to Heidegger and is found both before his explicit creation of tragic thought and in those tragic thinkers who have carried in his wake. We can, for instance, see this beginnings of this approach to God’s absence in the sorrow of Nietzsche’s madman, who wonders what humanity will do now that it as lost its greatest treasure. Also, we have a more recent contribution to this thought in the work of Simon Critchley, specifically his newest book Infinitely Demanding. In this work, Critchley locates the motivating force of philosophy itself in the disappointment that the philosopher feels in losing God. What we get in all three of these thinkers is the arrival of Aristotle’s tragic hero, the great man who has suffered a reversal of fortune.

Lucretius, on the other hand, commits the unforgivable sin of rejecting God at its premises, and therefore God’s death as logically impossible. There is simply no role preserved for the gods in Lucretius’ philosophy, since the beginning of the reality occurs via mathematics or physics rather than divine commandment. “For it is inherent in the very nature of the gods that they should enjoy immortal life in perfect peace, far removed and separated from our world; free from all distress, free from peril, fully self-sufficient, independent of us, they are not influenced by worthy conduct nor touched by anger.” It is simply logically impossible that the gods are here in our world or were ever here. To say otherwise to assert the importance of the human and its perspective, but humanity has never been a hero, is not touched by destiny, and therefore, lacks the fortune that vanity perceives.

What is important is to see that these tropes of tragic thought were not selected at random; instead they are the central coordinates of tragic thought itself. The traditional relationship in western philosophy between humanity and God has focused on the “great chain of being,” with the immortal god seating comfortable at the top, well above mortal humans. But the death of god has brought about a collapse of this chain. What has been missed by tragic thought is effect that this event has on the rest of the chain. We have been told repeatedly the loss of God solidifies our finite position, and has engendered disappointment, etc. But the dyad of immortal/mortal can only be maintained so long as the point of comparison, here infinity, survives. The ultimately effect of the loss of God is not the absolutizing of the finitude of be-ing, but instead the infinitizing of reality itself. Much has been made by Alain Badiou of Georg Cantor’s discovery of infinity as number, but we can already see the idea of numbered infinity at play in Lucretius. From a distance the sheep bleed into one another and appear as a mass of white upon the hill, moving as one, and as it is with the sheep, it is with atoms. Once again, it is Archimedes and his Sand Reckoner, which provide the first hints at the infinite, not as divine, but as number.

One possibility that confronts us is to assume that non-tragic thought is simply the opposite of tragic thought. This would be a mistake. Non-tragic thought is tragic thought taken to its most extreme possibility, which is the full embrace of finitude. Contrary to the approach of contemporary thought, human finitude does not operate as a limit to thought but instead allows thought to get beyond the human perspective; the radical acceptance of finitude makes it possible that, to quote Ray Brassier, “Philosophy should be more than a sop to the pathetic twinge of human self-esteem.”

As we have already discussed, Heidegger sees death as the ultimately potentiality of Dasein. The failure that Heidegger succumbs to is the inability to fully embrace this potentiality for all possible Dasein. Heidegger touches twice on this possibility but abandons the path both times to return to the singular death of a particular Dasein. The analysis of the being-toward-death of Dasein begins as the death of others. This first path is found to be limited since “in dying, it becomes evident that death is ontologically constituted by mineness and existence.” And so, although “every Dasein must itself actually take dying upon itself…”, Heidegger fails to truly grasp the “every” of finitude, and continues to think death only as a singular occurrence of a singular Dasein. When Heidegger does approach the concept of death in the collective, it is only in relation to the “the they.” But the “the they” can only think the death of the generic Dasein, “one dies”; and thus, “the they” fails to grasp the real occurrence of death for themselves.

The problem for Heideggerian thought is that commits a similar error. If “the they” cannot think its own death, Heidegger cannot get beyond his own death, which is the death of a specific Dasein. The proper procedure would be the combination of both approaches to death, that is, the recognition that one does indeed die, but also that this one is always a specific one, a Dasein. Thus we fully accept the possibility that “every Dasein” will die, not someday, but at the specific point of extinction, at the arrival of the stellar explosion. This extinction, which is first properly thought by Lucretius, is the absolute exhaustion of the human given.

Now the aged plowsman shakes his head and time after time sighs that his hard labor has all come to nothing…His gloomy sentiments are echoed by the planter of the old and shriveled vine who deplores the tendency of the times…Only he fails to grasp that all things gradually decay and head for the reef of destruction, exhausted by long lapse of time.

From the thought of the absolute finitude of all possible Dasein we arrive at the possible thought of a thinking beyond the existence of Dasein, a point when “the earth be confounded with the sea, and the sea with the sky.” And as we can now think the beyond of Dasein, we can also think the before of Dasein. Through a radical embracing of finitude, a radicality absent from Heidegger and his followers, thought is severed from the limits of humanity and becomes thoroughly inhuman.

Ultimately, tragic thought, in general, and Heidegger’s thought, in particular, remains little more than elaborate and hyper-stylized astrology. In Quentin Meillassoux’s After Finitude, he calls into question Kant’s relationship to his so-called “Copernican revolution” and instead, asserts that what critical and, most importantly for us, post-critical philosophy represents is a “Ptolemaic counter-revolution.” If Galileo’s great insight was to expel humans from the center of reality, then it was Kant’s greatness that returned us to our place. We can even see in Ptolemy the desire to supplement the science of astronomy with a proper metaphysical outlook, so as to understand the effects of the “ambients.” And thus we today are the inheritors of a thought that places the cosmos back in orbit around us, a thought that can only be tragic since it assumes that the movements of the planets are forever tied to the whims and desires of the human ego. Therefore, non-tragic thought is that thought that is willing to go all the way to the end and accept the utter meaninglessness of our very existence and in so doing re-affirm philosophy’s materialist binding to science.

Deleuze & Meinong: The Prejudice in Favor of the Actual

In 1999, after the Social Text farce, Alan Sokal, along with fellow physicist Jean Bricmont[1], skewered the contemporary French philosophic community by impugning their use of mathematics and science in their theoretical work. In particular, Sokal and Bricmont attack Gilles Deleuze for his repeated discussion of infinitesimals throughout Difference and Repetition[2]. Sokal and Bricmont write,

“After the birth of this branch of mathematics in the seventh-century through the work of Newton and Leibniz, cogent objections were raised against the use of ‘infinitesimal’ quantities such as dx and dy. These problems were solved by the work of d’Alembert around 1760 and Cauchy around 1820, who introduced the rigorous notion of limit—a concept that has been taught in all calculus textbooks since the middle of the nineteenth century. Nevertheless, Deleuze launches into a long and confused meditation on these problems…What is the point of all these mystifications about mathematical objects that have been well understood for over 150 years.”[3]

The question to arise from this is how Deleuze is to be defended (if one thinks that he must)? One possible approach is to claim that although the mathematical problem of infinitesimals, and thus the status of the infinitesimals in mathematical discourse, has been solved, the philosophical problem and the ontological status of infinitesimals remains. After all, infinitesimals are the invention of Gottfried Leibniz, a philosopher, and the problem of the infinitesimals was not so much solved as it was ignored (as infinitesimals are indistinguishable from zero they can be treated as zero and dropped). This is the approach that Lyotard advanced in his television debate with Sokal after the publication of Fashionable Nonsense, that the philosophers in question were playing language games with mathematical lingo.

But is this correct defense? At least in the case of Deleuze and infinitesimals, the answer is a resounding “No.” To assert that philosophy’s use of mathematics is simply metaphorical, or a language game, is to already give up too much ground to the enemy. The mathematical and philosophical status of infinitesimals is not opposed in the work of Deleuze; they ultimately cannot be distinguished. Mathematics plays an essential role in Deleuze’s thought, and to reduce this role to metaphor is to do great damage to his philosophical edifice.

Why then does Deleuze take up the “solved” problem of infinitesimals? The answer has to do with Deleuze’s understanding of the nature of problems. For Deleuze, problems are not solvable in the way that Sokal and Bricmont understand.[4] Following Kant, Deleuze asserts that Ideas are Problems, and that the problem must be understood as “the indispensable condition without which no solution would ever exist.”[5] Whereas for Sokal and Bricmont (and for a peculiar line of thinkers in Western thought), the solution causes the problem to disappear into the solution, like a conditional statement whose antecedent has been met and thus is collapsed into the consequent, thus transforming the consequent into a detachable conclusion.[6] On the other hand, in Deleuze, the problem is understood as the transcendental condition of the solution, whose actualization will result is a particular outcome, a solution, which does not, for all of that, eliminate the problem as idea. The outcome of this shift from the solution to the problem has a major impact in the way in which philosophy and mathematics come to be related to thought. According to Deleuze, “The relation between mathematics and man may thus be conceived in a new way: the question is not that of quantifying or measuring human properties, but rather, on the one hand, that of problematizing human events, and on the other hand, that of developing as various human events the condition of a problem.”[7] As a result infinitesimals are re-established in the place as problem, and thus work remains to be done.

The shift from the solution to the problem has another impact beyond that of infinitesimals, which is Deleuze’s entrance into intellectual fights that are thought of, like that of infinitesimals, as long settled affairs. An important, and related, example being Deleuze’s use of Alexius Meinong in Logic of Sense. It is reported that Gilbert Ryle, famed analytic philosopher, once said that “If meinongianism isn’t dead, nothing is.”[8] But Meinong, although often treated as the loser in fights involving Bertrand Russell[9] and W.V.O. Quine,[10] plays an essential role in Logic of Sense, especially with the use of his concepts of Sein (being) and Sosein (subsistence), the importance of which will be explained in a coming section. Ultimately, the problem of infinitesimals and the problem of being, as displayed in the work of Meinong, are interrelated and it is the purpose of this paper to bring these seemingly disparate areas together.

Section 1: Smooth Infinitesimal Analysis

Infinitesimals make their first appearance in the atomistic philosophy of ancient Greece, are discredited by the likes of Aristotle and Zeno, and are used by Archimedes in his geometry.[11] Lucretius used the concept of infinitesimals as well in his description of the swerve of the clinamen in his excellent On the Nature of Things.[12] But it is not until the creation of differential calculus by Leibniz that the infinitesimal gets its proper mathematical home. But from the very beginning it was not accepted as properly rigorous, and the process of getting past the infinitesimal, to solve the problem of the infinitely small, began.

The most consistent reaction against the infinitesimal was its violation of the law of the excluded middle (given x, x must be true or false). D’Alembert, who Sokal and Bricmont cited as the first to overcome the problem, stated that “A quantity is something or nothing: if it is something, it has not yet vanished; if it is nothing, it has literally vanished. The supposition that there is an intermediate state between these two is a chimera.”[13] Bishop Berkley, who famously referred to infinitesimals as the “ghosts of departed quantities,”[14] makes a similar argument when he asserts the impossibility of the mind conceiving of something infinitely small.[15] And so with an obvious logical quandary brewing with infinitesimals, the concept of limit is introduced, as Sokal & Bricmont said it has been taught in calculus textbook ever since.

Then why does Deleuze attempt to reassert these logically dubious entities? The first part of the answer is that Deleuze is not alone in returning to the infinitesimal, and so the authors of Fashionable Nonsense are simply wrong to assert that the problem of infinitesimals is dead, and that no controversy exists within mathematics. There are, in fact, at least two approaches to the calculus that do not do away with infinitesimals. The first, is the creation of non-standard analysis by Abraham Robinson, of which we will not return to, and second, the invention of Smooth Infinitesimal Analysis (SIA) in the 1970’s by F. W. Lavere. It is the assertions of this work that bare a striking resemblance to Deleuze’s own work with infinitesimals and thus will be referred to ground his mathematical speculation, although it would be false to say that Deleuze is using SIA in any way himself.

The most important detail, if we are to return to infinitesimals, is that whatever approach that is adopted must break with the logical condition of the excluded middle. In fact, it is C.S. Peirce who is the first to see this necessity in relation to infinitesimal calculus. Peirce asserted that “Now if we are to accept the common idea of continuity…we must either say that a continuous line contains no points or…that the principle of excluded middle does not hold of these points. The principle of excluded middle applies only to an individual…but places being mere possibilities without actual existence are not individuals.”[16] There are two important outcomes of this insight by Peirce. The first is what we have already discussed in relation to infinitesimal analysis, but the second is that logic should not be confused with Logos, as an over-arching principle of existence, but instead that it must, to use Deleuze’s terminology, be engendered within a world, what Peirce here calls individuals. This understanding of logic will reappear later in the discussion on Meinong.

So, what then is the status of the law of the excluded middle in SIA? To quote from Bell, “…one is forced to acknowledge that the so-called law of excluded middle – every statement is either definitely true or definitely false – cannot be generally affirmed within smooth worlds.”[17] Bell’s argument in defense of this position is as follows:

Assuming the law of the excluded middle, each real number is either equal to 0 or unequal to 0, so that correlating 1 to 0 and 0 to each nonzero real number defines a function – the ‘blip’ function – on the real line which is obviously discontinuous. So, if the law of excluded middle held in a smooth world S, the discontinuous blip function could be defined there. Thus since all functions in S are continuous, it follows that the law of excluded middle must fail within it. More precisely, this argument shows that the statement

for any real number x, either x = 0 or not x = 0

is false in S.[18]

What should not be missed is how the definition of line in SIA, that a line is made up not of points but of lines of infinitesimal length (the fact that Bell has used to eliminate the law of excluded middle), meshes with Deleuze’s own understanding of Sense as aliquid, the “minimum of being which befits inherence,”[19] a position that Deleuze admits originates, at least in part, in Meinong.

Section 2: Meinong

The importance of Alexius Meinong cannot be overstated in reference to Deleuze’s Logic of Sense. Primarily, what interests Deleuze is Meinong’s “Theory of Objects,” specifically his concepts of Sein (being) and Sosein (subsist). The concept of Sein[20] can be understood as the existential status of the object in reference to a physical reality. On the other hand, the concept of Sosein refers not to this existential status of the object but to the characteristics that the object has. For example, an object such as a “round rock” would have being (Sein) if I am able to discover a rock that would match the description, whereas the characteristics (Sosein) of the “round rock” would be based upon its shape, mass, and composition, not as they appear in reality, but as they appear within the intellect. Furthermore, the concept of Sein and Sosein exist via a principle of independence, which is to say, that they are not dependent upon one another. The round rock does not need to exist, or even be capable of existing, for its Sosein to subsist. Or as Meinong puts it,

Now it would accord very well with the…prejudice in favor of existence to hold that we may speak of a Sosein only if a Sein is presupposed…However the very science from which we were able to obtain the largest number of instances counter to this prejudice shows clearly that any such principle is untenable. As we know, the figures with which geometry is concerned do not exist. Nevertheless, their properties, and hence their Sosein, can be established.[21]

What the principle of independence asserts is that there is sharp divide between discourse and material reality, between Logos, on one hand, and Being, on the other. The result is that a meinongian discourse would be quite capable of dealing with the “golden mountain made of gold” or “the current king of France being bald.”[22] These statements lead to what Deleuze calls “Meinong’s Paradox” since these statements,

…are without signification, that is, they are absurd. Nevertheless, they have a sense, and the two notions of absurdity and nonsense must not be confused. Impossible objects – square circles, matter without extension…- are objects “without a home,” outside of being, but they have a precise and distinct position within this outside: they are of “extra being—pure, ideational events, unable to be realized in a state of affairs.[23]

To return to the principle of independence, the square circle exists virtually, since it exists as a Problem, an Idea; and of course, it was a problem that was particularly productive in Greek mathematics. It is the separation of Sein and Sosein, of being and language, that allows such an impossible object to be comprehended without damage occurring to rationality itself. Furthermore, Meinong’s Sein and Sosein appear in the Logic of Sense as two series united in Deleuze’s conception of sense. Deleuze states,

Sense is never only one of the two terms of the duality which contrasts things and propositions, substantives and verbs, denotations and expressions; it is also the frontier, the cutting edge, or the articulation of the difference between the two terms, since it has at its disposal an impenetrability which is its own and within which it is reflected.[24]

Sense, then for Deleuze, is a boundary that both separates the thing and the proposition, but equally allows for the two to come into contact via its membrane. And thus Meinong, with his principle of independence, a principle mirrored in Deleuze’s Logic of Sense,[25] begins a fight against Hegel and his insistence that Being says itself.[26]

Section 3: Hegel’s Logic

Deleuze’s first work on the philosophy of language is not Logic of Sense, but instead a review of Jean Hyppolite’s Logic and Existence published in 1954[27]. Although short, this article is highly significant for many reasons. First is the singular importance of Hyppolite for the philosophers of Deleuze’s generation in their attempts to overcome Hegel. Michel Foucault has stated that,

If, then, more than one of us is indebted to Jean Hyppolite, it is because he has tirelessly explored, for us, and ahead of us, the path along which we may escape Hegel…For Hyppolite, the relationship with Hegel was the scene of an experiment, of a confrontation in which it was never certain that philosophy would come out on top. He never saw the Hegelian system as a reassuring universe; he saw in it the field in which philosophy took the ultimate risk.

Secondly, the article provides us with the Deleuze’s first explication of philosophy’s relation to sense. Following Hyppolite and his anti-humanist reading of Hegel, Deleuze writes, “Philosophy must be ontology, it cannot be anything else; but there is no ontology of essence, there is only an ontology of sense.”[28] This ontology of sense arrives due to the Hegel’s destruction of the second world, the transcendent world of essences, of God. According to Hegel, predating Nietzsche and his madman in the market, God is dead, since “inquires, for instance, into the into the immateriality of the soul, into efficient and final causes, where should these still arouse any interest? Even the former proofs of the existence of God are cited only for their historical interest…”[29] And so philosophy can no longer concern itself with the question “what is?”, the platonic question par excellence. Instead, philosophy must come to recognize that there is no true world behind the appearance, but only the truth of sublated appearance, what we have been calling sense. This collapse of the apparent world into the true world means that there is no longer any difference between thought and language, as Plato asserted in the distinction between the form and its empirical appearance. “In logic, therefore, there is no longer…what I say on the one hand, and the sense of what I say on the other…my discourse is logically or properly philosophical when I speak the sense of what I say, and when Being this speaks itself.”[30]

Hegel’s logic, then, offers us the equation Being=Thought=Language as the Spirit “is home with itself” in the logic, as there is no longer a distinction to be made between thought and thing thought.[31] This is what Deleuze refers to as the ontology of sense. The question for us is what is the relationship between this review and the Logic of Sense that will take up many of the same themes? Len Lawlor seems to claim[32] that there is a nature slide from the earlier essay into Logic of Sense. But he accomplishes this only by explicitly rejecting the meinongian theme present in the latter work. This is quite clear when he asserts that Logic of Sense is a misnomer, since Deleuze should have entitled the work the Logic of Nonsense. What Lawlor misses is that what has changed from the review to the book, is the transformation of the ontology of sense into the logic of sense. There simply is no ontology of sense in Deleuze’s actual philosophy, anymore than there is logic of nonsense. This is obvious if one notices the conditions for an ontology of sense set by Deleuze. An ontology of sense exists only with the identification of being and language, with Being capable of saying itself. To quote Hyppolite, “Hegelian Logic is the absolute genesis of sense, a sense which, to itself, is its own sense, which is not opposed to the being whose sense it is, but which is sense and being simultaneously.”[33]

Deleuze is quite correct to assert that philosophy has undergone a quite serious transformation after Hegel’s logic and that is marked by an ontology of sense. In fact, this new understanding of philosophy brings together philosophers whose connection was formerly thought to be tangential at best. We are now able to see the connection between philosophers as disparate as Heidegger and Russell, Derrida and Carnap. What these philosophers have in common is the assertion no only that thought is synonymous with language, but that philosophy finds it end in this realization. It matters little if one associates thought with poetry, as in the case of Heidegger and Derrida, or with logic, as with Russell and Carnap, the end result is the same, and both groups, while explicitly attacking Hegel, affirm the insight that being=thought=language dominates.

What then of Deleuze? It should be clear that Deleuze is not part of the above lineage of thinkers coming from Hegel’s Logic. Although, Deleuze does affirm that philosophy must be an ontology of sense, when he has the opportunity to construct such an ontology, he instead calls it a logic of sense. It could be argued that this shift in terminology is meaningless, a point of style or aesthetics. But such a claim does not take into consideration the relationship between the Logic of Sense and Difference and Repetition. We are told by Hyppolite that the Phenomenology of Spirit is meant to present the empirical that is sublated into the Science of Logic, and can therefore be abandoned after this move of the dialectic. This is obvious if one understands that the Logic removes the distinction between thought and object of thought. However, if one rejects that Being can say itself then one must also reject the relationship between ontology and logic that Hegel had presented. And this is precisely what Deleuze does.

According to Deleuze, sense is incapable of saying itself because of the paradox of infinite regress. Let us look at this paradox as Lewis Carroll presents it in “What the Tortoise said to Achilles.”[34] Let us take an argument:

(A) Things that are equal to the same are equal to each other.

(B) The two sides of this Triangle are things that are equal to the same.

Therefore,

(Z) The two sides of this Triangle are equal to each other.

Obviously, the argument is built upon the acceptance of A and B, and their acceptance leads to Z. However, this formulates another proposition, let us call it C.

(C) If A and B are true, Z must be true.

This leads to the assertion the following:

(A) Things that are equal to the same are equal to each other.

(B) The two sides of this Triangle are things that are equal to the same.

(C) If A and B are true, Z must be true.

Therefore,

(Z) The two sides of this Triangle are equal to each other.

However, this leads formulates another proposition, let us call it D.

(D) If A, B, and C are true then Z must be true.

Etc.

The paradox of infinite regress asserts that the sense of any proposition is always another proposition. “I never state the sense of what I am saying…This regress testifies both to the great impotence of the speaker and to the highest power of language: my impotence to state the sense of what I say, to say at the same time something and its meaning; but also the infinite power of language to speak about words.”[35]

As we have already seen, for Deleuze, following Meinong, sense operates as a duality, a series made up of the states of affairs, where sense exists, and the proposition, where sense subsists. But sense itself is one part of a duality, with the other series being nonsense. Ultimately, this duality is made of ontology and logic. These separate series then allow us to see the relationship between the Logic of Sense and Difference and Repetition. Unlike, Hegel who wishes the ontology (Phenomenology of Spirit) to be overcome by the logic (Science of Logic), Deleuze inverts the relationship, while asserting the principle of independence between the two. Thus one cannot overcome the logically in favor of the ontological in Deleuze, instead they must be understood as two sides of a duality. This leads to the necessity of reading the Logic of Sense with Difference and Repetition.

In fact, we can see Deleuze construction of this duality in Series Sixteen and Seventeen in the Logic of Sense. In Series Sixteen, entitled “Static Ontological Genesis,” we find Deleuze laying out the production of the world around the actualization and convergence of singularities. Furthermore, in Series Seventeen, entitled “Static Logical Genesis,” we are given an analysis of the logical boundaries of worlds, both in there own construction (the rule of contradiction) but also in their relationship to each other (the law of incompossibles). What we get then with these two series, is a split discourse, where worlds are approached through the discourse of logic, but where the individuation of these worlds is achieved through the discourse of mathematics. Thus ontology for Deleuze, in a way similar to the work of Alain Badiou, is mathematics, specifically the mathematics of infinitesimal calculus.

Section 4: Conclusion

What ground have we covered? As we have seen the problem of the infinitesimals, far from being solved, operates as an essential feature of Deleuze’s ontology. And it is this ontology that allows us to come to an understanding of Deleuze’s placement of logic within his philosophical work. Deleuze, unlike the other major figures in continental philosophy, does not dismiss logic as a metaphysical trap, whether one sees it as technological thinking at its purest (a la Heidegger) or whether one sees it has the culmination of the thinking of presence (a la Derrida). But neither does Deleuze assert its absolute superiority in a way that anglo-american philosophy has. What Deleuze offers is the possibility to understand how logic comes to work, which is to say how a logical world could be engendered, individuated. But he accomplishes this via a turn to mathematics, a turn that I have only been able to hint at here. And so mathematics is ontology for Deleuze and it is through infinitesimal calculus that we are able to understand the ontological conditions that allow for logical worlds.

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[1] Sokal, Alan and Jean Bricmont. Fashionable Nonsense. New York: Picador. 1999, 154-168.

[2] Deleuze, Gilles. Difference and Repetition. translated by Paul Patton. New York: Columbia University Press, 1994.

[3] Sokal and Bricmont, 160-165.

[4] For a detailed explication of the Deleuze’s problematics, see Smith, Daniel. “Axiomatics and problematics as two modes of formalization: Deleuze’s epistemology of mathematics.” in Virtual Mathematics: the Logic of Difference. edited by Simon Duffy. London: Clinamen, 2006.

[5] Deleuze 1994, 168.

[6] See Deleuze, Gilles. Logic of Sense. translated by Charles Stivale. New York: Columbia University Press. 1990, 3^rd Series of the Proposition.

[7] Ibid., 55.

[8] Quoted in Priest, Graham. Towards Non-Being. Oxford: Clarendon Press. 2005, vi.

[9] Russell, Bertrand. “On Denoting.” Mind 14 (56): October 1905, 479-493.

[10] Quine, W.V.O. “On What There Is.” Review of Metaphysics 2(5): 1948, 21-38.

[11] See Boyer, Carl. The History of the Calculus and its Conceptual Development. New York: Dover, 1949.

[12] Lucretius. On the Nature of Things. translated by Martin Ferguson Smith. New York: Hackett Publications, 2001.

On the relation between Lucretius and Archimedian mathematics, see Serres, Michel. The Birth of Physics. London: Clinamen Press, 2001.

[13] Quoted in Boyer 1948, 248.

[14] Quoted in Smith 2006, 152.

[15] Berkeley, George. “On Infinites” in From Kant to Hilbert: A Sourcebook in the Foundations of Mathematics, Vol. 1. edited by William Ewald. Oxford: Clarendon Press. 1996, 17.

[16] Quoted in Bell, J.L. A Primer of Infinitesimal Analysis. London: Cambridge University Press. 2008, 5.

[17] Ibid.

[18] Ibid.

[19] Deleuze 1990, 22.

[20] I am really on Preist 2005 for this understanding of Sein as existential status.

[21] Meinong, Alexius. “Kinds of Being” in Logic and Philosophy. edited by G. Iseminger. New York: Appleton-Century Crofts. 1968, 122.

[22] Both of these statements cause a great deal of stress for the likes of Bertrand Russell, see Russell 1905.

[23] Deleuze 1990, 35.

[24] Deleuze 1990, 28.

[25] Deleuze says as much in Logic of Sense when he states, “Undoubtedly there are reasons for these moments (the discovery and rediscovery of sense as expressed of the proposition): we have seen that the Stoic discovery presupposed a reversal of Platonism; similarly Ockham’s logic reacted against the problem of Universals, and Meinong against the Hegelian logic and its lineage.” (emphasis added). Deleuze 1990, 19.

[26] This is the conclusion that Hyppolite claims for Hegel’s logic. See Hyppolite, Jean. Logic and Existence. translated by Len Lawler and Amit Sen. New York: State University of New York Press, 1997.

[27] Deleuze, Gilles. “Jean Hyppolite’s Logic and Existence.” translated by Michael Taormina. In Desert Island and Other Texts 1953-1974. edited by David Lapoujade. New York: Semiotext(e). 2002, 15-18.

[28] Deleuze 2002, 15.

[29] Hegel, G.W.F. Science of Logic. translated by A.V. Miller. New York: Humanity Books. 1969, 25.

[30] Deleuze 2002, 17.

[31] Hegel, G.W.F. The Encyclopaedia Logic. translated by T.F. Geraets, et al. New York: Hackett Publishing. 1991, § 24.

[32] Lawlor, Leonard. “Translator’s Preface.” in Hyppolite, Jean. Logic and Existence. translated by Len Lawler and Amit Sen. New York: State University of New York Press, 1997.

[33] Hyppolite 1997, 161.

[34] Carroll, Lewis. “What the Tortoise said to Achilles.” Mind 104 :416 (October 1995), 691-693

[35] Deleuze 1990, 28-9.