Andrei Rodin
Theorem and Event
(Proclus, Kant, Deleuze)
Accordingly to Proclus [1] every geometrical theorem and problem has the same structure:
(Euclid Elements I.5)
1) protasis, proposition:
In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another.
2) ekqesis, exposition:
Let ABC be an isosceles triangle having the side AB equal to the side AC;
and let the straight lines BD, CE be produced further in a straight line with AB, AC.
3) diorismos, determination:
I say that the angle ABC is equal to the angle ACB, and the angle CBD to the angle BCE.
4) kataskeuh, construction:
Let a point F be taken at random on BD; from AE the greater let AG be cut off equal to AF the less; and let the straight lines FC, GB be joined.
5) apodeixis, demonstration:
Then, since AF is equal to AG and AB to AC, the two sides FA, AC are equal to the two sides GA, AB, respectively; and they contain a common angle, the angle FAG. Therefore the base FC is equal to the base GB, and the triangle AFC is equal to the triangle AGB, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend, that is, the angle ACF to the angle ABG, and the angle AFC to the angle AGB. And since the whole AF is equal to the whole AG, and these AB is equal to AC, the remainder BF is equal to the remainder CG. But FC was also proved equal to GB; therefore the two sides BF, FC are equal to the two sides CG, GB respectively; and the angle BFC is equal to the angle CGB, while the base BC is common to them; therefore the triangle BFC is also equal to the triangle CGB, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend; therefore the angle FBC is equal to the angle GCB, and the angle BCF to the angle CBG. Accordingly, since the whole angle ABG was proved equal to the angle ACF, and in these the angle CBG is equal to the angle BCF, the remaining angle ABC is equal to the remaining angle ACB; and they are at the base of the triangle ABC. But the angle FBC was also proved equal to the angle GCB; and they are under the base.
6) sumperasma, conclusion:
Therefore, in isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another. QED
(Translation of T.Heath [2])
The only difference between problem and theorem relating this structure is the difference of accent : while in the theorem the (auxiliary) construction is an instrument of proof, in the problem the proof is an instrument of (approval of) construction:
(Euclid Elements I.1)
1) protasis, proposition:
On a given finite straight line to construct an equilateral triangle.
2) ekqesis, exposition:
Let AB be the given finite straight line.
3) diorismos, determination:
Thus it is required to construct an equilateral triangle on the straight line AB.
4) kataskeuh, construction:
With centre A and distance AB let the circle BCD be described; again, with centre B and distance BA let the circle ACE be described; and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined.
5) apodeixis, demonstration:
Now, since the point A is the centre of the circle CDB, AC is equal to AB. Again, since the point B is the centre of the circle CAE, BC is equal to BA. But CA was also proved equal to AB; therefore each of the straight lines CA, CB is equal to AB. And things which are equal to the same thing are also equal to one another; therefore CA is also equal to CB. Therefore the three straight lines CA, AB, BC are equal to one another.
6) sumperasma, conclusion:
Therefore the triangle ABC is equilateral; and it has been constructed on the given finite straight line AB. (Being) what it was required to do.
(translation of T. Heath [2])
We can see that Proclus does not treat the theorem in the binary way as we do it now dividing the theorem into statement and its proof. However Proclus himself moves towards the modern approach, saying that among the six parts of theorem the proposition, the proof and the conclusion are the most important. Identifying then proposition and conclusion we come to the modern point of view. In this article we come back to Proclus’ structure of theorem which allows us to discover a place of meeting of such different thinkers of different ages as Proclus, Kant and Deleuze.
The theorem develops between proposition and conclusion. Conclusion is the literal repetition of proposition ( except added “QED” that means the end of the theorem and separate the theorem from other ones). Thus the theorem has a form of cycle. Being textualy identical the proposition and the conclusion are not identical by sense: the proposition is truth to be proved (true statement) while the conclusion is a proved truth. Is a proved truth a true statement and its proof taken together? In this case the theorem would have a linear, but not a circular form: we begin with a proposition, then add some chain of reasoning and call all this a theorem or a proved truth:
proposition proof
fig.3
But how it is possible to obtain the initial statement proved following this scheme while there is no return to the beginning nor repetition in it? Actually beginning to prove a statement we start from this statement and finishing a proof we come back: to prove a statement means not to go from it to something new, but to give the same statement another epistemological status. Thus to complete the proof we should return to the beginning. Representing the theorem as
proposition
proof
fig.4
we may think as before that to prove a statement means to add something to it, to add some cycle. But to return means to discover the same once again or to repeat. Thus the return and therefore the proof is not a appendix nor addenda. We may represent the return like this:
proposition conclusion
proof
fig.5
We can see that our faculty to repeat the same text is not only a faculty to express the same meaning in different situations but also a faculty to distinguish in the same text its different senses in virtue of different contexts. Repeating the statement of theorem we distinguish what is an unfounded statement (proposition) and what is a proved truth (conclusion). The two obviously differs from each other by their senses but not by their meanings.
It is possible also to argue against the cyclic structure of theorem and therefore against the necessity to distinguish the proposition and the conclusion on the basis of idea of deductive structure of mathematical theory. Let us have a set of theorems proved before and axioms taken without proof (“known”), then applying to this set some procedure we obtain as a result a new proved statement. All the construction is called a theorem:
Known Procedure of inference New statement
fig.6
However strictly speaking there is no proof here. Actually, the proof is a procedure applying to statement that changes its epistemological status (from being unfounded to being proved) but not statement itself. That is why proof is not a procedure of inference which transforms one statement into another. If to interpret the proof as a procedure of inference taken together with “known”:
proof statement
fig.7
the result is curious: a proof begins as a proof of nobody knows what and the goal of enterprise is discovered later. In some cases it may possibly describe a real work of a mathematician (though usually mathematical work is far from formal attempts to derive from known statements a new one) but it is not a epistemological description of theorem: what is proved in the theorem it is always some statement that before proof and after proof is the same but has different sense. Thus the return and repetition in the theorem are irreducible.
In the problem conclusion is not a pure repetition of proposition: in proposition they demand to construct some figure and in conclusion they assert that such a figure is constructed and present the figure. Firstly, conclusion differs in the problem from proposition by modality: proposition expresses a demand while conclusion expresses its execution. There is also a demand in the proposition of theorem, namely a demand for proof, however the difference of modalities of proposition and conclusion in the case of theorem is rather internal and not represented grammatically. Secondly, in the problem the conclusion differs from the proposition by the fact that in the proposition they demand to construct something in general (for example equilateral triangle), while in the problem they present a concrete individual having its proper name (for example triangle ABC). To analyse the difference between general and individual (particular) we now come back to the theorem.
We can see that only outermost elements of theorem (proposition and conclusion) deal with general while all the internal elements deal with such individuals as triangle ABC. A transition from generalities to individuals is done in the second and the third elements of theorem, exposition and determination. From purely formal point of view the individualisation is an ascribing of proper name to the name that designates the general notion as for example they ascribe a proper name ABC to the common name “triangle”. However if to speak not about terms but about their sense it is impossible to understand what is triangle in general without imagination of some particular triangles. To think about triangle in general means to think about particular triangles following the general scheme. This is an approach of Kant [3] who opposes the transcendental logic to formal one. Proclus argues another way: mathematics uses some kind of mental images consisting of special kind of matter that Proclus calls “imagination” and that is a principle of plurality of mathematical things. Accordingly to Proclus mathematics differs from arbitrary play of imagination with the fact that the play submits general rules. As far as there are general rules, there are also individuals submitting and arranging general rules: Law and Individual presuppose each other. However in mathematics where Law and Individual are given purely (while in State Law and Individual always remain relative) the limit of both is found. One can imagine many identical triangles. All the triangles are constructed accordingly their general Law. As far as there is no difference between what every triangle should be and what it actually is, every triangle can be called absolutely law-abiding. The same time every triangle can be called legislator because it gives the law of its perfect form to any actual and possible triangle of the same form. The lack of difference between Due and Actual makes doubtful the very notions of Law and Individual (as Low-abiding and Legislator). Individual differences and therefore the identity of General appear to be illusionary. We can say after Deleuze [4] that here individuality transforms into singularity and its generality transforms into universality. The play of individuals submitted general rules transforms into the play of singularities submitted universal rules or the play of Ideas. We can see it more clearly in the case of numbers which are treated by Proclus following Plato [5] as “closer” to Ideas than geometrical figures. Is a number 5 an individual or a general notion? One can understand many individual five’s under the same general notion of Five, thus 5 has both meanings and can be understood or as a general notion or as an individual. However while in geometry individuals differs from general notions with proper names, there are no such a difference between the two in arithmetic. It is hardly possible to believe that writing down a row of natural numbers we deal with one sort of number (general) and writing down the squares of the numbers we deal with another sort of numbers (individual) because to square a number it is necessary to take it twice. Or that writing one time 5+5 and another time 5*2 we deal with different sorts of five’s. Obviously the logic of general and individual (particular) stops to work here. Unlikely geometrical figures numbers even are not given proper names and definitions (at least in Euclid’s Elements). Number appears to be a kind of individual without proper name and in the same time a kind of general notion without definition or a kind of unwritten law. It means that number is rather singular than individual and rather universal than general. However it is not a mathematical number but Idea that is singular and universal in the strict sense of the words.
Accordingly to Proclus all the movement of Mathematics is a movement from individuals with their general rules to universal ideas. Two ways from individual and general to ideas are possible. The one way starts from individual and the other starts from general. The first way is Plato’s and the second is Aristotle’s. Proclus as Platonic pays more attention to the first way. He wants to overcome a plurality of individuals to find Idea behind the form of generality of Law (but not to remain general laws without their subjects ). However Proclus uses also an Aristotelian critic of general as abstract. He does it in a form of critic of Aristotle himself, explaining that idea is not abstract and that singularity is not particular.
Speaking about a relationship between singular and individual (universal and general) Deleuze [4] use a notion of interiorisation of repetition. Going from a numerical repetition of geometrical points (point A, point B, point C etc.) to repetition of numbers themselves (1,1,1 ...) one ceases to distinguish repeated individuals by their names. And in the case of repeated number one can hardly say if he deals with the same number or not: it is the same number that is repeated but taken in different samples which are not the same though they are not definitely distinguishable by names as geometrical points. The identity of repeated individual disappears but repetition does not. A repeated number makes a series that is not a row of identities. Repetition, says Deleuze, ceases to be a repetition of the Same and becomes a repetition of repetition as a principle of the Same. Repetition ceases to be extensive or developing a raw of repeated identities outside and becomes intensive or focusing a singularity. The extensive repetition replace individuals by others individuals keeping their common place (point in general for A, B and so on) while the intensive repetition repeats non-replaceable singularity and its universe.
Let us return to the structure of theorem. The exposition differs from the determination by the fact that in the former it is a condition of the theorem’s proposition that is translated from the language of general notions to the language of individuals while in the later it is a conclusion of proposition (here we speak about conclusion in the sense of logical implication). In other words a logical distinction between condition and conclusion is made here not for general, but for particular judgements. There are reasons for that. For an individualisation in the exposition differs from that in the determination by the conditions of naming: in the case of exposition names are given arbitrary while in the case of determination names are constructed with the names given in exposition. So individuals established in the determination are elements of “primordial” individuals established in exposition. Thus the difference of exposition and determination is irreducible to that of condition and conclusion of hypothetical judgement.
Going from the proposition to determination trough exposition we go from general notions to individuals. However as we mentioned above the theorem’s conclusion is expressed by general notions again. Hence there is a reversal movement from individuals to general notions. Obviously such a reversal movement is done between the proof and the conclusion: the proof concerns individuals established with the exposition and determination as triangle ABC and angle ABC but the conclusion concerns general notions as isosceles triangle in general and angles at its base. A relation of previously established individuals with general notions also appears in the proof itself when they bring the individuals under propositions of theorems proven before or axioms as in theorem 1.5 Euclid concludes from an equality of elements of triangles AFC and AHB to the equality of the triangles on the basis of theorem 1.4. In the later case they go from general to particular judgement accordingly Aristotle’s “perfect” syllogism, but then to receive a general judgement they have to go from particular to general which does not accord syllogistic. However there are no reasons to say that they do so a wrong conclusion because a syllogistic accordingly to Aristotle concerns only proof but not the theorem in the whole. Meanwhile it does not solve the problem about a ground that makes it possible to transfer what was proved for individuals to general notions.
The Kant’s answer is that human mind deals with individuals following general schemes. The generality of mathematical notions means nothing but existence of such general schemes. That is why following Kant it is not appropriate to speak about any transfer here at all: triangle ABC and triangle in general are the same notion taken once in concreto and the other time in abstracto. A seeming paradox when what was proved for an individual triangle is always valid for any triangle of suitable sort accordingly to Kant should be considered as paralogism based on the wrong premise that triangles are things in themselves. Actually any synthetic judgement and the judgement about an equality of angles at the base of isosceles triangle in particular is always a general form of individual.
Proclus makes accent on elevating moment of mathematical synthesis. Accordingly to Proclus a transfer from an individual triangle to general notion of triangle is an epistrofh of triangle or its return to itself. However as we noted above we should rather speak not about a transition from individual to general but about a movement that forms general as well as individual and in the limit identifies them in idea. So Proclus understands a mathematical synthesis as a continuous movement from concrete imagined individuals and their abstract notions to ideas.
According to Kant idea is an “illegal” notion that has no associated individuals (i.e. no corresponding possible experience). Meanwhile the idea is necessary as a regulator of any activity of mind as for example an idea of world for physicist or idea of infinite row of numbers for mathematician. Notions of Mind (Verstand) including mathematical ones and transcendental ideas of Reason (Vernunft) are distinguished by Kant in radical way: there is a gap between the two. That is why any continuous transition from notions of Mind to ideas of Reason according to Kant seems to be impossible. However Deleuze’s analysis of the concept of difference allows to conciliate Proclus and Kant. What does it mean that Kant distinguishes notions and ideas in radical way? Does Proclus mean only relative differences speaking about the elevation to ideas? The crucial question is: what difference is radical and what is relative? Deleuze answers as follows. The radical or pure difference is difference not submitted to the identity. So all the differences
of identities are relative. What differs radically is different but not identical in itself, it is difference of difference. It is difference itself that differs radically from itself. It means that if individuals and ideas differ radically they can not form two worlds - world of individuals or representations and world of ideas - and gap between the two. Gap goes rather through every point of world. Proclus’ platonic elevation does not fill the gap between the two worlds but rather makes the difference of levels irreducible to any identity, i.e. makes the difference radically. Elevating power is very power of difference or power of idea. Kant says that we should never take an idea for representation but to elevate from representations to ideas does not mean to find a way do it. It means on the contrary to make a difference between the two radical. Note that Kant himself treating later a theme of elevation introduced an intermediate element between ideas and representations - the subject of faculty of judgement [6].
Thus the circle of the theorem (a numerical repetition of proposition and conclusion) goes from generalities to generalities through individuals. Speaking about general and individual before we meant only mathematical objects as triangle in general and individual triangle ABC. However the same is valid for a reasoning mathematical subject. Actually the theorem’s assertion (proposition or conclusion) belongs to the general subject of mathematics i.e. mathematical community in the whole. It is a common knowledge. On the other hand everybody studied mathematics knows that a theorem can be proved only by personal effort, even in the case of reproduction somebody else’s proof. The thinking individual not only participates the mathematical reasoning but is constituted by it as that who follows general rules (student) and gives general rules (scholar).
So the circle of mathematical theorem is not only represented by mathematician but mathematician himself is involved into this circle by his own generality and individuality.
We have to consider the construction. It is clear that the construction is the main synthetic element of the theorem and a synthetic judgement expressed by the proposition is finally stipulated by the construction. But can we call this synthesis individual? What happens in the construction with individuals exposed before? Proclus notes: “omitted” elements are constructed that makes it possible to discover some general properties also “omitted” before. However it is enough to look at the drawing to the theorem 1.5 to reject such a naive platonism: the initial isosceles triangle is obviously more “complete” and “perfect” than the presented configuration. Generally it is always configuration that is constructed but a simple individual object is separated after (as a subject of properties to be established in the case of theorem or as a final task of construction in the case of problem). Configuration is not an individual but a complicated mixture or interlacement of individuals. Note that this is not because borders are erased but on the contrary because a lot of new borders are added and individuals become parts of other individuals. (As we mentioned above it begins even before construction - in the determination.) So in the construction individuals exposed before are deindividualisated or “die” while new individuals constituted in the proof are born. However this non-individual or pre-individual is not general: deindividualisation of individuals exposed before gives no general as well as birth of new individuals in the proof is not their exposition from the general (the general is given here when new individuals are submitted to the propositions of theorems proved before and axioms). To complete its individuality mathematical individual exposed before has to die in the construction (note that in the theorem 1.5 triangle ABC is not mentioned after construction).
In the case of problem the accent is transmitted from death to birth, but initially exposed individual also dies becoming an element of new individual. Thus the construction appears to be an accident or event of death and birth of individuals.
We come to the same conclusion looking to the construction from the side of reasoning subject. It is possible to understand a proof only individually but the understanding itself seems to be something superindividual differing the same time from any general knowledge. In the moment of understanding, of insight an individual loses his identity and returns to the new one. Such an accident always happens with everybody dealing with mathematics. A theorem once understood can not be simply forgotten in the sense of return to the previous condition because there is no third to compare new and old conditions. It is possible to forget that one remembers while the theorem is not remembered but (using Plato’s term) reminisced. Speaking about reminiscence Plato [7] point on the moment of loss and acquisition of individuality: the soul is the place of deaths and births of individuals. Is the immortal soul individual itself? Is its memory and its memory the memory and knowledge of individual? Can the soul be called general for individuals being birth and dying in it. If the soul is individual we have a regress ad infinitum. The classical argument of “third man” has in the case the form as follows: to understand soul should have its soul and so on. But the soul is not a general rule for individuals for general rule is a place of individual while the soul is a place of events - births and deaths of individuals.
general rules proposition
individuals exposition
determination
event construction
proof
conclusion
fig.8
Thus behind the external circle of numerical repetition of assertion in the proposition and conclusion of the theorem there is an internal circle of births and deaths of individuals. (In the case of problem the external circle is incomplete for conclusion is individual here.) Construction appears to be a centre of the whole theorem. Such a concentrated circular motion or concentrated repetition Proclus describes in three terms: Stop (monh), Way Out (proodos), Return (epistrofh). Stop is a centre of the circle, point of internal plenitude from which an external emanation goes out. Way Out is a disturbance of symmetry in virtue of that the Stop deploys. The synthetic moment of the system is Return: the deployed plenitude “reminisces” its stop and returns to itself. When Proclus applies the scheme to Euclid’s definition of the circle he gives the image as follows:
return
stop
way out
fig.9
Return is eternal here as well as stop and way out. Return and way out are principles that make a theorem “to work”, to rotate around the ideal centre.
Thus the construction is not a simple addition to the given. The plenitude appears to be available not trough addition but trough death and new birth. The event of construction appears to be Proclus’ stop - the point of going out and of return. Is this point simple? The very point of return we can not see and can only be found there, but going to it up deal with the increasing complexity: relatively simple figure turns into relatively complex configuration. And the same time the point of return appears simple as an intuitive understanding preceding a complex discursive deployment. So the point of event not only consists complexity but also is a simple limit of the complexity. Is this point single? Generalities differs by genus (as subject and object) and by species (as isosceles and equilateral triangles). Individuals differs also numerically by names (as two points). Speaking on the language of set theory the cardinality of the set of individuals is more than the cardinality of the set of generalities (the infinite number of infinite sequences). With every individual can happen a finite number of events that form its individual history or biography. This is actual events or facts. The cardinality of the set of facts is equal to that of individuals (a finite product of sets). However speaking events that happen not between birth and death but between death and birth i.e. about virtual events it is impossible to distinguish one such an event from the other because there is no somebody to distinguish. So such an event appears to be single. Or rather not single but singular. Singularities can not be counted as individuals and can not form a set comparable with other set by one-to-one correspondence because their differences go further than numerical differences of individuals and are pure or radical differences. A numerical difference, says Deleuze, is submitted to the repeated identity (repetition of The Same). A specific difference is also submitted to the identity, namely to the identity of divided genus. A pure difference appears to be closer to the generic difference if a genus is not considered as a basis for following divisions to species. Singularities differs by their internal cardinality or internal power and there are no external differences between them even by names. (Internal here means the limit of external: internal differences of singularities is something more accidental, more fluent than names of individuals. Specific differences from this point of view are the most external.) That is why not being single a singularity is one. Differing inside themselves singularities repeat. And while individuals and actual events of their lives repeat extensively that makes it possible to name and to count them - in such a way repeats the same theorem being proved again and again by schoolteacher - the virtual singular event repeats internally and intensively - in such a way a theorem repeats being every time again and again “finally really understood” (equally in somebody’s personal experience or in historical development of mathematics). It is impossible not only to count how much times a mathematician returns in his mind to the same theorem he works on but also to define when he stops to think about the same theorem and begins to think about another one.
Actual events repeat birth and death by analogy for birth is acquisition of life, death is loss of life and every intermediate event also is acquisition or loss of something as for example an actual understanding is acquisition of knowledge and loss of superstitions. Life between the birth and death is a measure and basis of analogy of actual events. The virtual singular event repeats without identity and analogy i.e. repeats differently because there is nothing between death and life. The singular event, a stop between death and birth is universal because every individual with its general rules returns to it. This is a non-individual singularity (mone but not monad of Proclus). The One (to en of Proclus) is an universal aspect of singularity.
What differs singular from one (universal)? One is thought through the universalisation of general rule while singular is thought through the singularisation of individual. The first strategy Deleuze calls ironic and the second humoral. Philosophy, says Deleuze, is always critic of general, critic of Low. Two directions of such a critic are possible. Ironical philosopher derides the Law as something only pretending to be the Law but essentially illegal. In other words ironical philosopher goes up to last foundations and looks for true universal Law. Humoral philosopher on the contrary goes down from general statements of Low to the sphere of its practical application and shows a weakness of the Low by different curious cases. Although a platonic strategy is really mostly ironic as says Deleuze, we discover in Proclus the both strategies (en and monh).
Speaking about singular and universal we internally repeat. High and Depth, Paradise and Hell, Sky and Cave are super- and preindividual principles of Event. Birth in problem and Death in theorem differ one doubled event. Proclus makes accent on the theorem, because speaking about death we usually presuppose birth but speaking about birth not always remember about death. However behind the variety of life it is easy to lose not only the last but also the first limit when it is necessary to accent a problem as Deleuze does it. Deleuze reminds us that birth is not identical to life (that often forget already Plato) and the depth of Cave is not identical to the surface of Earth. What is earth distinguished not only from sky but also from cave?
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