Hypostatic Abstraction. Thought of the Day 138.0

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Hypostatic abstraction is linguistically defined as the process of making a noun out of an adjective; logically as making a subject out of a predicate. The idea here is that in order to investigate a predicate – which other predicates it is connected to, which conditions it is subjected to, in short to test its possible consequences using Peirce’s famous pragmatic maxim – it is necessary to posit it as a subject for investigation.

Hypostatic abstraction is supposed to play a crucial role in the reasoning process for several reasons. The first is that by making a thing out of a thought, it facilitates the possibility for thought to reflect critically upon the distinctions with which it operates, to control them, reshape them, combine them. Thought becomes emancipated from the prison of the given, in which abstract properties exist only as Husserlian moments, and even if prescission may isolate those moments and induction may propose regularities between them, the road for thought to the possible establishment of abstract objects and the relations between them seems barred. The object created by a hypostatic abstraction is a thing, but it is of course no actually existing thing, rather it is a scholastic ens rationis, it is a figment of thought. It is a second intention thought about a thought – but this does not, in Peirce’s realism, imply that it is necessarily fictitious. In many cases it may indeed be, but in other cases we may hit upon an abstraction having real existence:

Putting aside precisive abstraction altogether, it is necessary to consider a little what is meant by saying that the product of subjectal abstraction is a creation of thought. (…) That the abstract subject is an ens rationis, or creation of thought does not mean that it is a fiction. The popular ridicule of it is one of the manifestations of that stoical (and Epicurean, but more marked in stoicism) doctrine that existence is the only mode of being which came in shortly before Descartes, in concsequence of the disgust and resentment which progressive minds felt for the Dunces, or Scotists. If one thinks of it, a possibility is a far more important fact than any actuality can be. (…) An abstraction is a creation of thought; but the real fact which is important in this connection is not that actual thinking has caused the predicate to be converted into a subject, but that this is possible. The abstraction, in any important sense, is not an actual thought but a general type to which thought may conform.

The seemingly scepticist pragmatic maxim never ceases to surprise: if we take all possible effects we can conceive an object to have, then our conception of those effects is identical with our conception of that object, the maxim claims – but if we can conceive of abstract properties of the objects to have effects, then they are part of our conception of it, and hence they must possess reality as well. An abstraction is a possible way for an object to behave – and if certain objects do in fact conform to this behavior, then that abstraction is real; it is a ‘real possibility’ or a general object. If not, it may still retain its character of possibility. Peirce’s definitions of hypostatic abstractions now and then confuse this point. When he claims that

An abstraction is a substance whose being consists in the truth of some proposition concerning a more primary substance,

then the abstraction’s existence depends on the truth of some claim concerning a less abstract substance. But if the less abstract substance in question does not exist, and the claim in question consequently will be meaningless or false, then the abstraction will – following that definition – cease to exist. The problem is only that Peirce does not sufficiently clearly distinguish between the really existing substances which abstractive expressions may refer to, on the one hand, and those expressions themselves, on the other. It is the same confusion which may make one shuttle between hypostatic abstraction as a deduction and as an abduction. The first case corresponds to there actually existing a thing with the quality abstracted, and where we consequently may expect the existence of a rational explanation for the quality, and, correlatively, the existence of an abstract substance corresponding to the supposed ens rationis – the second case corresponds to the case – or the phase – where no such rational explanation and corresponding abstract substance has yet been verified. It is of course always possible to make an abstraction symbol, given any predicate – whether that abstraction corresponds to any real possibility is an issue for further investigation to estimate. And Peirce’s scientific realism makes him demand that the connections to actual reality of any abstraction should always be estimated (The Essential Peirce):

every kind of proposition is either meaningless or has a Real Secondness as its object. This is a fact that every reader of philosophy should carefully bear in mind, translating every abstractly expressed proposition into its precise meaning in reference to an individual experience.

This warning is directed, of course, towards empirical abstractions which require the support of particular instances to be pragmatically relevant but could hardly hold for mathematical abstraction. But in any case hypostatic abstraction is necessary for the investigation, be it in pure or empirical scenarios.

The Second Trichotomy. Thought of the Day 120.0

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The second trichotomy (here is the first) is probably the most well-known piece of Peirce’s semiotics: it distinguishes three possible relations between the sign and its (dynamical) object. This relation may be motivated by similarity, by actual connection, or by general habit – giving rise to the sign classes icon, index, and symbol, respectively.

According to the second trichotomy, a Sign may be termed an Icon, an Index, or a Symbol.

An Icon is a sign which refers to the Object that it denotes merely by virtue of characters of its own, and which it possesses, just the same, whether any such Object actually exists or not. It is true that unless there really is such an Object, the Icon does not act as a sign; but this has nothing to do with its character as a sign. Anything whatever, be it quality, existent individual, or law, is an Icon of anything, in so far as it is like that thing and used as a sign of it.

An Index is a sign which refers to the Object that it denotes by virtue of being really affected by that Object. It cannot, therefore, be a Qualisign, because qualities are whatever they are independently of anything else. In so far as the Index is affected by the Object, it necessarily has some Quality in common with the Object, and it is in respect to these that it refers to the Object. It does, therefore, involve a sort of Icon, although an Icon of a peculiar kind; and it is not the mere resemblance of its Object, even in these respects which makes it a sign, but it is the actual modification of it by the Object. 

A Symbol is a sign which refers to the Object that it denotes by virtue of a law, usually an association of general ideas, which operates to cause the Symbol to be interpreted as referring to that Object. It is thus itself a general type or law, that is, a Legisign. As such it acts through a Replica. Not only is it general in itself, but the Object to which it refers is of general nature. Now that which is general has its being in the instances it will determine. There must, therefore, be existent instances of what the Symbol denotes, although we must here understand by ‘existent’, existent in the possibly imaginary universe to which the Symbol refers. The Symbol will indirectly, through the association or other law, be affected by those instances; and thus the Symbol will involve a sort of Index, although an Index of a peculiar kind. It will not, however, be by any means true that the slight effect upon the Symbol of those instances accounts for the significant character of the Symbol.

The icon refers to its object solely by means of its own properties. This implies that an icon potentially refers to an indefinite class of objects, namely all those objects which have, in some respect, a relation of similarity to it. In recent semiotics, it has often been remarked by someone like Nelson Goodman that any phenomenon can be said to be like any other phenomenon in some respect, if the criterion of similarity is chosen sufficiently general, just like the establishment of any convention immediately implies a similarity relation. If Nelson Goodman picks out two otherwise very different objects, then they are immediately similar to the extent that they now have the same relation to Nelson Goodman. Goodman and others have for this reason deemed the similarity relation insignificant – and consequently put the whole burden of semiotics on the shoulders of conventional signs only. But the counterargument against this rejection of the relevance of the icon lies close at hand. Given a tertium comparationis, a measuring stick, it is no longer possible to make anything be like anything else. This lies in Peirce’s observation that ‘It is true that unless there really is such an Object, the Icon does not act as a sign ’ The icon only functions as a sign to the extent that it is, in fact, used to refer to some object – and when it does that, some criterion for similarity, a measuring stick (or, at least, a delimited bundle of possible measuring sticks) are given in and with the comparison. In the quote just given, it is of course the immediate object Peirce refers to – it is no claim that there should in fact exist such an object as the icon refers to. Goodman and others are of course right in claiming that as ‘Anything whatever ( ) is an Icon of anything ’, then the universe is pervaded by a continuum of possible similarity relations back and forth, but as soon as some phenomenon is in fact used as an icon for an object, then a specific bundle of similarity relations are picked out: ‘ in so far as it is like that thing.’

Just like the qualisign, the icon is a limit category. ‘A possibility alone is an Icon purely by virtue of its quality; and its object can only be a Firstness.’ (Charles S. PeirceThe Essential Peirce_ Selected Philosophical Writings). Strictly speaking, a pure icon may only refer one possible Firstness to another. The pure icon would be an identity relation between possibilities. Consequently, the icon must, as soon as it functions as a sign, be more than iconic. The icon is typically an aspect of a more complicated sign, even if very often a most important aspect, because providing the predicative aspect of that sign. This Peirce records by his notion of ‘hypoicon’: ‘But a sign may be iconic, that is, may represent its object mainly by its similarity, no matter what its mode of being. If a substantive is wanted, an iconic representamen may be termed a hypoicon’. Hypoicons are signs which to a large extent makes use of iconical means as meaning-givers: images, paintings, photos, diagrams, etc. But the iconic meaning realized in hypoicons have an immensely fundamental role in Peirce’s semiotics. As icons are the only signs that look-like, then they are at the same time the only signs realizing meaning. Thus any higher sign, index and symbol alike, must contain, or, by association or inference terminate in, an icon. If a symbol can not give an iconic interpretant as a result, it is empty. In that respect, Peirce’s doctrine parallels that of Husserl where merely signitive acts require fulfillment by intuitive (‘anschauliche’) acts. This is actually Peirce’s continuation of Kant’s famous claim that intuitions without concepts are blind, while concepts without intuitions are empty. When Peirce observes that ‘With the exception of knowledge, in the present instant, of the contents of consciousness in that instant (the existence of which knowledge is open to doubt) all our thought and knowledge is by signs’ (Letters to Lady Welby), then these signs necessarily involve iconic components. Peirce has often been attacked for his tendency towards a pan-semiotism which lets all mental and physical processes take place via signs – in the quote just given, he, analogous to Husserl, claims there must be a basic evidence anterior to the sign – just like Husserl this evidence before the sign must be based on a ‘metaphysics of presence’ – the ‘present instant’ provides what is not yet mediated by signs. But icons provide the connection of signs, logic and science to this foundation for Peirce’s phenomenology: the icon is the only sign providing evidence (Charles S. Peirce The New Elements of Mathematics Vol. 4). The icon is, through its timeless similarity, apt to communicate aspects of an experience ‘in the present instant’. Thus, the typical index contains an icon (more or less elaborated, it is true): any symbol intends an iconic interpretant. Continuity is at stake in relation to the icon to the extent that the icon, while not in itself general, is the bearer of a potential generality. The infinitesimal generality is decisive for the higher sign types’ possibility to give rise to thought: the symbol thus contains a bundle of general icons defining its meaning. A special icon providing the condition of possibility for general and rigorous thought is, of course, the diagram.

The index connects the sign directly with its object via connection in space and time; as an actual sign connected to its object, the index is turned towards the past: the action which has left the index as a mark must be located in time earlier than the sign, so that the index presupposes, at least, the continuity of time and space without which an index might occur spontaneously and without any connection to a preceding action. Maybe surprisingly, in the Peircean doctrine, the index falls in two subtypes: designators vs. reagents. Reagents are the simplest – here the sign is caused by its object in one way or another. Designators, on the other hand, are more complex: the index finger as pointing to an object or the demonstrative pronoun as the subject of a proposition are prototypical examples. Here, the index presupposes an intention – the will to point out the object for some receiver. Designators, it must be argued, presuppose reagents: it is only possible to designate an object if you have already been in reagent contact (simulated or not) with it (this forming the rational kernel of causal reference theories of meaning). The closer determination of the object of an index, however, invariably involves selection on the background of continuities.

On the level of the symbol, continuity and generality play a main role – as always when approaching issues defined by Thirdness. The symbol is, in itself a legisign, that is, it is a general object which exists only due to its actual instantiations. The symbol itself is a real and general recipe for the production of similar instantiations in the future. But apart from thus being a legisign, it is connected to its object thanks to a habit, or regularity. Sometimes, this is taken to mean ‘due to a convention’ – in an attempt to distinguish conventional as opposed to motivated sign types. This, however, rests on a misunderstanding of Peirce’s doctrine in which the trichotomies record aspects of sign, not mutually exclusive, independent classes of signs: symbols and icons do not form opposed, autonomous sign classes; rather, the content of the symbol is constructed from indices and general icons. The habit realized by a symbol connects it, as a legisign, to an object which is also general – an object which just like the symbol itself exists in instantiations, be they real or imagined. The symbol is thus a connection between two general objects, each of them being actualized through replicas, tokens – a connection between two continua, that is:

Definition 1. Any Blank is a symbol which could not be vaguer than it is (although it may be so connected with a definite symbol as to form with it, a part of another partially definite symbol), yet which has a purpose.

Axiom 1. It is the nature of every symbol to blank in part. [ ]

Definition 2. Any Sheet would be that element of an entire symbol which is the subject of whatever definiteness it may have, and any such element of an entire symbol would be a Sheet. (‘Sketch of Dichotomic Mathematics’ (The New Elements of Mathematics Vol. 4 Mathematical Philosophy)

The symbol’s generality can be described as it having always blanks having the character of being indefinite parts of its continuous sheet. Thus, the continuity of its blank parts is what grants its generality. The symbol determines its object according to some rule, granting the object satisfies that rule – but leaving the object indeterminate in all other respects. It is tempting to take the typical symbol to be a word, but it should rather be taken as the argument – the predicate and the proposition being degenerate versions of arguments with further continuous blanks inserted by erasure, so to speak, forming the third trichotomy of term, proposition, argument.

|, ||, |||, ||||| . The Non-Metaphysics of Unprediction. Thought of the day 67.1

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The cornerstone of Hilbert’s philosophy of mathematics was the so-called finitary standpoint. This methodological standpoint consists in a restriction of mathematical thought to objects which are “intuitively present as immediate experience prior to all thought,” and to those operations on and methods of reasoning about such objects which do not require the introduction of abstract concepts, in particular, require no appeal to completed infinite totalities.

Hilbert characterized the domain of finitary reasoning in a well-known paragraph:

[A]s a condition for the use of logical inferences and the performance of logical operations, something must already be given to our faculty of representation, certain extra-logical concrete objects that are intuitively present as immediate experience prior to all thought. If logical inference is to be reliable, it must be possible to survey these objects completely in all their parts, and the fact that they occur, that they differ from one another, and that they follow each other, or are concatenated, is immediately given intuitively, together with the objects, as something that can neither be reduced to anything else nor requires reduction. This is the basic philosophical position that I consider requisite for mathematics and, in general, for all scientific thinking, understanding, and communication. [Hilbert in German + DJVU link here in English]

These objects are, for Hilbert, the signs. For the domain of contentual number theory, the signs in question are sequences of strokes (“numerals”) such as

|, ||, |||, ||||| .

The question of how exactly Hilbert understood the numerals is difficult to answer. What is clear in any case is that they are logically primitive, i.e., they are neither concepts (as Frege’s numbers are) nor sets. For Hilbert, the important issue is not primarily their metaphysical status (abstract versus concrete in the current sense of these terms), but that they do not enter into logical relations, e.g., they cannot be predicated of anything.

Sometimes Hilbert’s view is presented as if Hilbert claimed that the numbers are signs on paper. It is important to stress that this is a misrepresentation, that the numerals are not physical objects in the sense that truths of elementary number theory are dependent only on external physical facts or even physical possibilities. Hilbert made too much of the fact that for all we know, neither the infinitely small nor the infinitely large are actualized in physical space and time, yet he certainly held that the number of strokes in a numeral is at least potentially infinite. It is also essential to the conception that the numerals are sequences of one kind of sign, and that they are somehow dependent on being grasped as such a sequence, that they do not exist independently of our intuition of them. Only our seeing or using “||||” as a sequence of 4 strokes as opposed to a sequence of 2 symbols of the form “||” makes “||||” into the numeral that it is. This raises the question of individuation of stroke symbols. An alternative account would have numerals be mental constructions. According to Hilber, the numerals are given in our representation, but they are not merely subjective “mental cartoons”.

One version of this view would be to hold that the numerals are types of stroke-symbols as represented in intuition. At first glance, this seems to be a viable reading of Hilbert. It takes care of the difficulties that the reading of numerals-as-tokens (both physical and mental) faces, and it gives an account of how numerals can be dependent on their intuitive construction while at the same time not being created by thought.

Types are ordinarily considered to be abstract objects and not located in space or time. Taking the numerals as intuitive representations of sign types might commit us to taking these abstract objects as existing independently of their intuitive representation. That numerals are “space- and timeless” is a consequence that already thought could be drawn from Hilbert’s statements. The reason is that a view on which numerals are space- and timeless objects existing independently of us would be committed to them existing simultaneously as a completed totality, and this is exactly what Hilbert is objecting to.

It is by no means compatible, however, with Hilbert’s basic thoughts to introduce the numbers as ideal objects “with quite different determinations from those of sensible objects,” “which exist entirely independent of us.” By this we would go beyond the domain of the immediately certain. In particular, this would be evident in the fact that we would consequently have to assume the numbers as all existing simultaneously. But this would mean to assume at the outset that which Hilbert considers to be problematic.  Another open question in this regard is exactly what Hilbert meant by “concrete.” He very likely did not use it in the same sense as it is used today, i.e., as characteristic of spatio-temporal physical objects in contrast to “abstract” objects. However, sign types certainly are different from full-fledged abstracta like pure sets in that all their tokens are concrete.

Now what is the epistemological status of the finitary objects? In order to carry out the task of providing a secure foundation for infinitary mathematics, access to finitary objects must be immediate and certain. Hilbert’s philosophical background was broadly Kantian. Hilbert’s characterization of finitism often refers to Kantian intuition, and the objects of finitism as objects given intuitively. Indeed, in Kant’s epistemology, immediacy is a defining characteristic of intuitive knowledge. The question is, what kind of intuition is at play? Whereas the intuition involved in Hilbert’s early papers was a kind of perceptual intuition, in later writings it is identified as a form of pure intuition in the Kantian sense. Hilbert later sees the finite mode of thought as a separate source of a priori knowledge in addition to pure intuition (e.g., of space) and reason, claiming that he has “recognized and characterized the third source of knowledge that accompanies experience and logic.” Hilbert justifies finitary knowledge in broadly Kantian terms (without however going so far as to provide a transcendental deduction), characterizing finitary reasoning as the kind of reasoning that underlies all mathematical, and indeed, scientific, thinking, and without which such thought would be impossible.

The simplest finitary propositions are those about equality and inequality of numerals. The finite standpoint moreover allows operations on finitary objects. Here the most basic is that of concatenation. The concatenation of the numerals || and ||| is communicated as “2 + 3,” and the statement that || concatenated with ||| results in the same numeral as ||| concatenated with || by “2 + 3 = 3 + 2.” In actual proof-theoretic practice, as well as explicitly, these basic operations are generalized to operations defined by recursion, paradigmatically, primitive recursion, e.g., multiplication and exponentiation. Roughly, a primitive recursive definition of a numerical operation is one in which the function to be defined, f , is given by two equations

f(0, m) = g(m)

f(n′, m) = h(n, m, f(n, m)),

where g and h are functions already defined, and n′ is the successor numeral to n. For instance, if we accept the function g(m) = m (the constant function) and h(n, m, k) = m + k as finitary, then the equations above define a finitary function, in this case, multiplication f (n, m) = n × m. Similarly, finitary judgments may involve not just equality or inequality but also basic decidable properties, such as “is a prime.” This is finitarily acceptable as long as the characteristic function of such a property is itself finitary: For instance, the operation which transforms a numeral to | if it is prime and to || otherwise can be defined by primitive recursion and is hence finitary. Such finitary propositions may be combined by the usual logical operations of conjunction, disjunction, negation, but also bounded quantification. The problematic finitary propositions are those that express general facts about numerals such as that 1 + n = n + 1 for any given numeral n. It is problematic because, for Hilbert it is from the finitist point of view incapable of being negated. By this he means that the contradictory proposition that there is a numeral n for which 1 + n ≠ n + 1 is not finitarily meaningful. A finitary general proposition is not to be understood as an infinite conjunction but only as a hypothetical judgment that comes to assert something when a numeral is given. Even though they are problematic in this sense, general finitary statements are of particular importance to Hilbert’s proof theory, since the statement of consistency of a formal system T is of such a general form: for any given sequence p of formulas, p is not a derivation of a contradiction in T. Even though in general existential statements are not finitarily meaningful, they may be given finitary meaning if the witness is given by a finitary function. For instance, the finitary content of Euclid’s theorem that for every prime p there is a prime > p, is that given a specific prime p one can produce, by a finitary operation, another prime > p (viz., by testing all numbers between p and p! + 1.).

Sellarsian Intentionality. Thought of the Day 59.0

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Sellars developed a theory of intentionality that seems calculated to so construe intentional phenomena as to make them compatible with developments in the sciences.

Now if thoughts are items which are conceived in terms of the roles they play, then there is no barrier in principle to the identification of conceptual thinking with neurophysiological process. There would be no “qualitative” remainder to be accounted for. The identification, curiously enough, would be even more straightforward than the identification of the physical things in the manifest image with complex systems of physical particles. And in this key, if not decisive, respect, the respect in which both images are concerned with conceptual thinking (which is the distinctive trait of man), the manifest and scientific images could merge without clash in the synoptic view. (Philosophy and the Scientific Image of Man).

The first thing to notice is that Sellars maintains that intentionality is irreducible in the sense that we cannot define in any of the vocabularies of the natural sciences concepts equivalent to the concepts of intentionality. The language of intentionality is introduced as an autonomous explanatory vocabulary tied, of course, to the vocabulary of empirical behavior, but not reducible to that language. The autonomy of mentalistic discourse surely commits us to a new ideology, a new set of basic predicates, above and beyond what can be constructed in the vocabularies of the natural sciences. What we get from the sciences can be the whole truth about the world, including intentional phenomena, then, only if there is some way to construct, using proper scientific methodology, concepts in the scientific image that are legitimate successors to the concepts of intentionality present in the manifest image. That there is such a rigorous construction of successors to the concepts of intentionality is, a clear commitment on Sellars’s part. The only real alternative is some form of eliminativism, an alternative that some of his students adopted and some of his critics thought Sellars was committed to, but which never held any real attraction for Sellars.

The second thing to notice is that the concepts of intentionality, especially the concepts of agency, differ in some significant ways from the normal concepts of the natural sciences. Sellars puts it this way:

To say that a certain person desired to do A, thought it his duty to do B but was forced to do C, is not to describe him as one might describe a scientific specimen. One does, indeed, describe him, but one does something more. And it is this something more which is the irreducible core of the framework of persons.

Here the focus is explicitly on the language of agency, but the point is fundamentally the same as in Sellars’s well-known dictum from Empiricism and Philosophy of Mind:

in characterizing an episode or a state as that of knowing, we are not giving an empirical description of that episode or state; we are placing it in the logical space of reasons, of justifying and being able to justify what one says.

In both epistemic and agential language something extra-descriptive is going on. In order to accommodate this important aspect of such phenomena, Sellars tells us, we must add to the purely descriptive/explanatory vocabulary of the sciences “the language of individual and community intentions”. He points to intentions here because the point is that epistemic and agential language – mentalistic language in general – is ineluctably normative; it always contains a prescriptive, action-oriented dimension and engages in direct or indirect assessment against normative standards. In Sellars’s own theory, norms are grounded in the structure of intentions, particularly community intentions, so any truly complete image must contain the language of intentions.

HumanaMente 

Categorial Logic – Paracompleteness versus Paraconsistency. Thought of the Day 46.2

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The fact that logic is content-dependent opens a new horizon concerning the relationship of logic to ontology (or objectology). Although the classical concepts of a priori and a posteriori propositions (or judgments) has lately become rather blurred, there is an undeniable fact: it is certain that the far origin of mathematics is based on empirical practical knowledge, but nobody can claim that higher mathematics is empirical.

Thanks to category theory, it is an established fact that some sort of very important logical systems: the classical and the intuitionistic (with all its axiomatically enriched subsystems), can be interpreted through topoi. And these possibility permits to consider topoi, be it in a Noneist or in a Platonist way, as universes, that is, as ontologies or as objectologies. Now, the association of a topos with its correspondent ontology (or objectology) is quite different from the association of theoretical terms with empirical concepts. Within the frame of a physical theory, if a new fact is discovered in the laboratory, it must be explained through logical deduction (with the due initial conditions and some other details). If a logical conclusion is inferred from the fundamental hypotheses, it must be corroborated through empirical observation. And if the corroboration fails, the theory must be readjusted or even rejected.

In the case of categorial logic, the situation has some similarity with the former case; but we must be careful not to be influenced by apparent coincidences. If we add, as an axiom, the tertium non datur to the formalized intuitionistic logic we obtain classical logic. That is, we can formally pass from the one to the other, just by adding or suppressing the tertium. This fact could induce us to think that, just as in physics, if a logical theory, let’s say, intuitionistic logic, cannot include a true proposition, then its axioms must be readjusted, to make it possible to include it among its theorems. But there is a radical difference: in the semantics of intuitionistic logic, and of any logic, the point of departure is not a set of hypothetical propositions that must be corroborated through experiment; it is a set of propositions that are true under some interpretation. This set can be axiomatic or it can consist in rules of inference, but the theorems of the system are not submitted to verification. The derived propositions are just true, and nothing more. The logician surely tries to find new true propositions but, when they are found (through some effective method, that can be intuitive, as it is in Gödel’s theorem) there are only three possible cases: they can be formally derivable, they can be formally underivable, they can be formally neither derivable nor underivable, that is, undecidable. But undecidability does not induce the logician to readjust or to reject the theory. Nobody tries to add axioms or to diminish them. In physics, when we are handling a theory T, and a new describable phenomenon is found that cannot be deduced from the axioms (plus initial or some other conditions), T must be readjusted or even rejected. A classical logician will never think of changing the axioms or rules of inference of classical logic because it is undecidable. And an intuitionist logician would not care at all to add the tertium to the axioms of Heyting’s system because it cannot be derived within it.

The foregoing considerations sufficiently show that in logic and mathematics there is something that, with full right, can be called “a priori“. And although, as we have said, we must acknowledge that the concepts of a priori and a posteriori are not clear-cut, in some cases, we can rightly speak of synthetical a priori knowledge. For instance, the Gödel’s proposition that affirms its own underivabilty is synthetical and a priori. But there are other propositions, for instance, mathematical induction, that can also be considered as synthetical and a priori. And a great deal of mathematical definitions, that are not abbreviations, are synthetical. For instance, the definition of a monoid action is synthetical (and, of course, a priori) because the concept of a monoid does not have among its characterizing traits the concept of an action, and vice versa.

Categorial logic is, the deepest knowledge of logic that has ever been achieved. But its scope does not encompass the whole field of logic. There are other kinds of logic that are also important and, if we intend to know, as much as possible, what logic is and how it is related to mathematics and ontology (or objectology), we must pay attention to them. From a mathematical and a philosophical point of view, the most important logical non-paracomplete systems are the paraconsistent ones. These systems are something like a dual to paracomplete logics. They are employed in inconsistent theories without producing triviality (in this sense also relevant logics are paraconsistent). In intuitionist logic there are interpretations that, with respect to some topoi, include two false contradictory propositions; whereas in paraconsistent systems we can find interpretations in which there are two contradictory true propositions.

There is, though, a difference between paracompleteness and paraconsistency. Insofar as mathematics is concerned, paracomplete systems had to be coined to cope with very deep problems. The paraconsistent ones, on the other hand, although they have been applied with success to mathematical theories, were conceived for purely philosophical and, in some cases, even for political and ideological motivations. The common point of them all was the need to construe a logical system able to cope with contradictions. That means: to have at one’s disposal a deductive method which offered the possibility of deducing consistent conclusions from inconsistent premisses. Of course, the inconsistency of the premisses had to comply with some (although very wide) conditions to avoid triviality. But these conditions made it possible to cope with paradoxes or antinomies with precision and mathematical sense.

But, philosophically, paraconsistent logic has another very important property: it is used in a spontaneous way to formalize the naive set theory, that is, the kind of theory that pre-Zermelian mathematicians had always employed. And it is, no doubt, important to try to develop mathematics within the frame of naive, spontaneous, mathematical thought, without falling into the artificiality of modern set theory. The formalization of the naive way of mathematical thinking, although every formalization is unavoidably artificial, has opened the possibility of coping with dialectical thought.

Noneism. Part 2.

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Noneism is a very rigourous and original philosophical doctrine, by and large superior to the classical mathematical philosophies. But there are some problems concerning the different ways of characterizing a universe of objects. It is very easy to understand the way a writer characterizes the protagonists of the novels he writes. But what about the characterization of the universe of natural numbers? Since in most kinds of civilizations the natural numbers are characterized the same way, we have the impression that the subject does not intervene in the forging of the characteristics of natural numbers. These numbers appear to be what they are, with total independence of the creative activity of the cognitive subject. There is, of course, the creation of theorems, but the potentially infinite sequence of natural numbers resists any effort to subjectivize its characteristics. It cannot be changed. A noneist might reply that natural numbers are non-existent, that they have no being, and that, in this respect, they are identical with mythological Objects. Moreover, the formal system of natural numbers can be interpreted in many ways: for instance, with respect to a universe of Skolem numbers. This is correct, but it does not explain why the properties of some universes are independent from subjective creation. It is an undeniable fact that there are two kinds of objectual characteristics. On the one hand, we have the characteristics created by subjective imagination or speculative thought; on the other hand, we find some characteristics that are not created by anybody; their corresponding Objects are, in most cases, non-existent but, at the same time, they are not invented. They are just found. The origin of the former characteristics is very easy to understand; the origin of the last ones is, a mystery.

Now, the subject-independence of a universe, suggests that it belongs to a Platonic realm. And as far as transafinite set theory is concerned, the subject-independence of its characteristics is much less evident than the characteristic subject-independence of the natural numbers. In the realm of the finite, both characteristics are subject-independent and can be reduced to combinatorics. The only difference between both is that, according to the classical Platonistic interpretation of mathematics, there can only be a single mathematical universe and that, to deductively study its properties, one can only employ classical logic. But this position is not at all unobjectionable. Once the subject-independence of the natural numbers system’s characteristics is posited, it becomes easy to overstep the classical phobia concerning the possibility of characterizing non-classical objective worlds. Euclidean geometry is incompatible with elliptical and hyperbolic geometries and, nevertheless, the validity of the first one does not invalidate the other ones. And vice versa, the fact that hyperbolic and other kinds of geometry are consistently characterized, does not invalidate the good old Euclidean geometry. And the fact that we have now several kinds of non-Cantorian set theories, does not invalidate the classical Cantorian set theory.

Of course, an universally non-Platonic point of view that includes classical set theory can also be assumed. But concerning natural numbers it would be quite artificial. It is very difficult not to surrender to the famous Kronecker’s dictum: God created natural numbers, men created all the rest. Anyhow, it is not at all absurd to adopt a whole platonistic conception of mathematics. And it is quite licit to adopt a noneist position. But if we do this, the origin of the natural numbers’ characteristics becomes misty. However, forgetting this cloudiness, the leap from noneist universes to the platonistic ones, and vice versa, becomes like a flip-flop connecting objectological with ontological (ideal) universes, like a kind of rabbit-duck Gestalt or a Sherrington staircase. So, the fundamental question with respect to the subject-dependent or subject-independent mathematical theories, is: are they created, or are they found? Regarding some theories, subject-dependency is far more understandable; and concerning other ones, subject-independency is very difficult, if not impossible, to negate.

From an epistemological point of view, the fact of non-subject dependent characteristic traits of a universe would mean that there is something like intellectual intuition. The properties of natural numbers, the finite properties of sets (or combinatorics), some geometric axioms, for instance, in Euclidean geometry, the axioms of betweenness, etc., would be apprehended in a manner, that pretty well coincides with the (nowadays rather discredited) concept of synthetical a priori knowledge. This aspect of mathematical knowledge shows that the old problem concerning the analytic and the a priori synthetical knowledge, in spite of the prevailing Quinean pragmatic conception, must be radically reset.

Organic and the Orgiastic. Cartography of Ground and Groundlessness in Deleuze and Heidegger. Thought of the Day 43.0

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In his last hermeneutical Erörterung of Leibniz, The Principle of Ground, Heidegger traces back metaphysics to its epochal destiny in the twofold or duplicity (Zwiefalt) of Being and Thought and thus follows the ground in its self-ungrounding (zugrundegehen). Since the foundation of thought is also the foundation of Being, reason and ground are not equal but belong together (zusammenhören) in the Same as the ungrounded yet historical horizon of the metaphysical destiny of Being: On the one hand we say: Being and ground: the Same. On the other hand we say: Being: the abyss (Ab-Grund). What is important is to think the univocity (Einsinnigkeit) of both Sätze, those Sätze that are no longer Sätze. In Difference and Repetition, similarly, Deleuze tells us that sufficient reason is twisted into the groundless. He confirms that the Fold (Pli) is the differenciator of difference engulfed in groundlessness, always folding, unfolding, refolding: to ground is always to bend, to curve and recurve. He thus concludes:

Sufficient reason or ground is strangely bent: on the one hand, it leans towards what it grounds, towards the forms of representation; on the other hand, it turns and plunges into a groundless beyond the ground which resists all forms and cannot be represented.

Despite the fundamental similarity of their conclusions, however, our short overview of Deleuze’s transformation of the Principle of Sufficient Reason has already indicated that his argumentation is very different from Heideggerian hermeneutics. To ground, Deleuze agrees, is always to ground representation. But we should distinguish between two kinds of representation: organic or finite representation and orgiastic or infinite representation. What unites the classicisms of Kant, Descartes and Aristotle is that representation retains organic form as its principle and the finite as its element. Here the logical principle of identity always precedes ontology, such that the ground as element of difference remains undetermined and in itself. It is only with Hegel and Leibniz that representation discovers the ground as its principle and the infinite as its element. It is precisely the Principle of Sufficient Reason that enables thought to determine difference in itself. The ground is like a single and unique total moment, simultaneously the moment of the evanescence and production of difference, of disappearance and appearance. What the attempts at rendering representation infinite reveal, therefore, is that the ground has not only an Apollinian, orderly side, but also a hidden Dionysian, orgiastic side. Representation discovers within itself the limits of the organized; tumult, restlessness and passion underneath apparent calm. It rediscovers monstrosity.

The question then is how to evaluate this ambiguity that is essential to the ground. For Heidegger, the Zwiefalt is either naively interpreted from the perspective of its concave side, following the path of the history of Western thought as the belonging together of Being and thought in a common ground; or it is meditated from its convex side, excavating it from the history of the forgetting of Being the decline of the Fold (Wegfall der Zwiefalt, Vorenthalt der Zwiefalt) as the pivotal point of the Open in its unfolding and following the path that leads from the ground to the abyss. Instead of this all or nothing approach, Deleuze takes up the question in a Nietzschean, i.e. genealogical fashion. The attempt to represent difference in itself cannot be disconnected from its malediction, i.e. the moral representation of groundlessness as a completely undifferentiated abyss. As Bergson already observed, representational reason poses the problem of the ground in terms of the alternative between order and chaos. This goes in particular for the kind of representational reason that seeks to represent the irrepresentable: Representation, especially when it becomes infinite, is imbued with a presentiment of groundlessness. Because it has become infinite in order to include difference within itself, however, it represents groundlessness as a completely undifferentiated abyss, a universal lack of difference, an indifferent black nothingness. Indeed, if Deleuze is so hostile to Hegel, it is because the latter embodies like no other the ultimate illusion inseparable from the Principle of Sufficient Reason insofar as it grounds representation, namely that groundlessness should lack differences, when in fact it swarms with them.

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Deleuzian Grounds. Thought of the Day 42.0

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With difference or intensity instead of identity as the ultimate philosophical one could  arrive at the crux of Deleuze’s use of the Principle of Sufficient Reason in Difference and Repetition. At the beginning of the first chapter, he defines the quadruple yoke of conceptual representation identity, analogy, opposition, resemblance in correspondence with the four principle aspects of the Principle of Sufficient Reason: the form of the undetermined concept, the relation between ultimate determinable concepts, the relation between determinations within concepts, and the determined object of the concept itself. In other words, sufficient reason according to Deleuze is the very medium of representation, the element in which identity is conceptually determined. In itself, however, this medium or element remains different or unformed (albeit not formless): Difference is the state in which one can speak of determination as such, i.e. determination in its occurrent quality of a difference being made, or rather making itself in the sense of a unilateral distinction. It is with the event of difference that what appears to be a breakdown of representational reason is also a breakthrough of the rumbling ground as differential element of determination (or individuation). Deleuze illustrates this with an example borrowed from Nietzsche:

Instead of something distinguished from something else, imagine something which distinguishes itself and yet that from which it distinguishes itself, does not distinguish itself from it. Lightning, for example, distinguishes itself from the black sky but must also trail behind it . It is as if the ground rose to the surface without ceasing to be the ground.

Between the abyss of the indeterminate and the superficiality of the determined, there thus appears an intermediate element, a field potential or intensive depth, which perhaps in a way exceeds sufficient reason itself. This is a depth which Deleuze finds prefigured in Schelling’s and Schopenhauer’s differend conceptualization of the ground (Grund) as both ground (fond) and grounding (fondement). The ground attains an autonomous power that exceeds classical sufficient reason by including the grounding moment of sufficient reason for itself. Because this self-grounding ground remains groundless (sans-fond) in itself, however, Hegel famously ridiculed Schelling’s ground as the indeterminate night in which all cows are black. He opposed it to the surface of determined identities that are only negatively correlated to each other. By contrast, Deleuze interprets the self-grounding ground through Nietzsche’s eternal return of the same. Whereas the passive syntheses of habit (connective series) and memory (conjunctions of connective series) are the processes by which representational reason grounds itself in time, the eternal return (disjunctive synthesis of series) ungrounds (effonde) this ground by introducing the necessity of future becomings, i.e. of difference as ongoing differentiation. Far from being a denial of the Principle of Sufficient Reason, this threefold process of self-(un)grounding constitutes the positive, relational system that brings difference out of the night of the Identical, and with finer, more varied and more terrifying flashes of lightning than those of contradiction: progressivity.

The breakthrough of the ground in the process of ungrounding itself in sheer distinction-production of the multiple against the indistinguishable is what Deleuze calls violence or cruelty, as it determines being or nature in a necessary system of asymmetric relations of intensity by the acausal action of chance, like an ontological game in which the throw of the dice is the only rule or principle. But it is also the vigil, the insomnia of thought, since it is here that reason or thought achieves its highest power of determination. It becomes a pure creativity or virtuality in which no well-founded identity (God, World, Self) remains: [T]hought is that moment in which determination makes itself one, by virtue of maintaining a unilateral and precise relation to the indeterminate. Since it produces differential events without subjective or objective remainder, however, Deleuze argues that thought belongs to the pure and empty form of time, a time that is no longer subordinate to (cosmological, psychological, eternal) movement in space. Time qua form of transcendental synthesis is the ultimate ground of everything that is, reasons and acts. It is the formal element of multiple becoming, no longer in the sense of finite a priori conditioning, but in the sense of a transfinite a posteriori synthesizer: an empt interiority in ongoing formation and materialization. As Deleuze and Guattari define synthesizer in A Thousand Plateaus: The synthesizer, with its operation of consistency, has taken the place of the ground in a priori synthetic judgment: its synthesis is of the molecular and the cosmic, material and force, not form and matter, Grund and territory.

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Meillassoux, Deleuze, and the Ordinal Relation Un-Grounding Hyper-Chaos. Thought of the Day 41.0

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As Heidegger demonstrates in Kant and the Problem of Metaphysics, Kant limits the metaphysical hypostatization of the logical possibility of the absolute by subordinating the latter to a domain of real possibility circumscribed by reason’s relation to sensibility. In this way he turns the necessary temporal becoming of sensible intuition into the sufficient reason of the possible. Instead, the anti-Heideggerian thrust of Meillassoux’s intellectual intuition is that it absolutizes the a priori realm of pure logical possibility and disconnects the domain of mathematical intelligibility from sensibility. (Ray Brassier’s The Enigma of Realism: Robin Mackay – Collapse_ Philosophical Research and Development. Speculative Realism.) Hence the chaotic structure of his absolute time: Anything is possible. Whereas real possibility is bound to correlation and temporal becoming, logical possibility is bound only by non-contradiction. It is a pure or absolute possibility that points to a radical diachronicity of thinking and being: we can think of being without thought, but not of thought without being.

Deleuze clearly situates himself in the camp when he argues with Kant and Heidegger that time as pure auto-affection (folding) is the transcendental structure of thought. Whatever exists, in all its contingency, is grounded by the first two syntheses of time and ungrounded by the third, disjunctive synthesis in the implacable difference between past and future. For Deleuze, it is precisely the eternal return of the ordinal relation between what exists and what may exist that destroys necessity and guarantees contingency. As a transcendental empiricist, he thus agrees with the limitation of logical possibility to real possibility. On the one hand, he thus also agrees with Hume and Meillassoux that [r]eality is not the result of the laws which govern it. The law of entropy or degradation in thermodynamics, for example, is unveiled as nihilistic by Nietzsche s eternal return, since it is based on a transcendental illusion in which difference [of temperature] is the sufficient reason of change only to the extent that the change tends to negate difference. On the other hand, Meillassoux’s absolute capacity-to-be-other relative to the given (Quentin Meillassoux, Ray Brassier, Alain Badiou – After finitude: an essay on the necessity of contingency) falls away in the face of what is actual here and now. This is because although Meillassoux s hyper-chaos may be like time, it also contains a tendency to undermine or even reject the significance of time. Thus one may wonder with Jon Roffe (Time_and_Ground_A_Critique_of_Meillassou) how time, as the sheer possibility of any future or different state of affairs, can provide the (non-)ground for the realization of this state of affairs in actuality. The problem is less that Meillassoux’s contingency is highly improbable than that his ontology includes no account of actual processes of transformation or development. As Peter Hallward (Levi Bryant, Nick Srnicek and Graham Harman (editors) – The Speculative Turn: Continental Materialism and Realism) has noted, the abstract logical possibility of change is an empty and indeterminate postulate, completely abstracted from all experience and worldly or material affairs. For this reason, the difference between Deleuze and Meillassoux seems to come down to what is more important (rather than what is more originary): the ordinal sequences of sensible intuition or the logical lack of reason.

But for Deleuze time as the creatio ex nihilo of pure possibility is not just irrelevant in relation to real processes of chaosmosis, which are both chaotic and probabilistic, molecular and molar. Rather, because it puts the Principle of Sufficient Reason as principle of difference out of real action it is either meaningless with respecting to the real or it can only have a negative or limitative function. This is why Deleuze replaces the possible/real opposition with that of virtual/actual. Whereas conditions of possibility always relate asymmetrically and hierarchically to any real situation, the virtual as sufficient reason is no less real than the actual since it is first of all its unconditioned or unformed potential of becoming-other.

Reza Negarestani’s Ontology as Science of Cruelty and Deleuzean Excavation of the Architectonic. Thought of the Day 40.0

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The problem of the principle of reason/ground is architectonic. As such it is the great theme of modern philosophy: how and where to begin? The two classical answers are provided by romanticism and enlightenment thinking. If there is a romantic side to Heidegger, as Deleuze says, then Meillassoux inherits and continues a long-standing tradition of enlightenment. Whereas the first always looks for a foundation or ground, even if it turns out be an abyss, the critical reason of the latter rabidly dismantles all grounds. Alternatively, Deleuze calls for a third answer which he calls modernism or constructivism and which always begins by the milieu (par le milieu). Instead of rising out of first principles like a tree from its roots, his metaphysics proliferates like a rhizome, never straying far from the events at the surface in a groping experimentation with the conditions of real experience. For Deleuze, the milieu is not the solid ground on which we stand, but neither is it an abyss or a void. Rather it is the fluctuating ground in which we must learn to swim. It is the element of the problematic as such, an element that matters and calls for an ethics of life. To think by the milieu means to think both without reference to a fixed ground yet also without separating thought from the forces it requires to exist. Whereas Meillassoux reinstalls the Kantian tribunal of reason and the generality of its judgments, Deleuze always emphasizes his own conditions of enunciation, i.e. the matters of concern that enable him to learn. While the anti-correlationist position is one of right, Deleuze’s own position is always one of fact.