Valencies of Predicates. Thought of the Day 125.0

Naturalizing semiotics - The triadic sign of Charles Sanders Pei

Since icons are the means of representing qualities, they generally constitute the predicative side of more complicated signs:

The only way of directly communicating an idea is by means of an icon; and every indirect method of communicating an idea must depend for its establishment upon the use of an icon. Hence, every assertion must contain an icon or set of icons, or else must contain signs whose meaning is only explicable by icons. The idea which the set of icons (or the equivalent of a set of icons) contained in an assertion signifies may be termed the predicate of the assertion. (Collected Papers of Charles Sanders Peirce)

Thus, the predicate in logic as well as ordinary language is essentially iconic. It is important to remember here Peirce’s generalization of the predicate from the traditional subject-copula-predicate structure. Predicates exist with more than one subject slot; this is the basis for Peirce’s logic of relatives and permits at the same time enlarging the scope of logic considerably and approaching it to ordinary language where several-slot-predicates prevail, for instance in all verbs with a valency larger than one. In his definition of these predicates by means of valency, that is, number of empty slots in which subjects or more generally indices may be inserted, Peirce is actually the founder of valency grammar in the tradition of Tesnière. So, for instance, the structure ‘_ gives _ to _’ where the underlinings refer to slots, is a trivalent predicate. Thus, the word classes associated with predicates are not only adjectives, but verbs and common nouns; in short all descriptive features in language are predicative.

This entails the fact that the similarity charted in icons covers more complicated cases than does the ordinary use of the word. Thus,

where ordinary logic considers only a single, special kind of relation, that of similarity, – a relation, too, of a particularly featureless and insignificant kind, the logic of relatives imagines a relation in general to be placed. Consequently, in place of the class, which is composed of a number of individual objects or facts brought together by means of their relation of similarity, the logic of relatives considers the system, which is composed of objects brought together by any kind of relations whatsoever. (The New Elements of Mathematics)

This allows for abstract similarity because one phenomenon may be similar to another in so far as both of them partake in the same relation, or more generally, in the same system – relations and systems being complicated predicates.

But not only more abstract features may thus act as the qualities invoked in an icon; these qualities may be of widely varying generality:

But instead of a single icon, or sign by resemblance of a familiar image or ‘dream’, evocable at will, there may be a complexus of such icons, forming a composite image of which the whole is not familiar. But though the whole is not familiar, yet not only are the parts familiar images, but there will also be a familiar image in its mode of composition. ( ) The sort of idea which an icon embodies, if it be such that it can convey any positive information, being applicable to some things but not to others, is called a first intention. The idea embodied by an icon, which cannot of itself convey any information, being applicable to everything or nothing, but which may, nevertheless, be useful in modifying other icons, is called a second intention. 

What Peirce distinguishes in these scholastic standard notions borrowed from Aquinas via Scotus, is, in fact, the difference between Husserlian formal and material ontology. Formal qualities like genus, species, dependencies, quantities, spatial and temporal extension, and so on are of course attributable to any phenomenon and do not as such, in themselves, convey any information in so far as they are always instantiated in and thus, like other Second Intentions, in the Husserlian manner dependent upon First Intentions, but they are nevertheless indispensable in the composition of first intentional descriptions. The fact that a certain phenomenon is composed of parts, has a form, belongs to a species, has an extension, has been mentioned in a sentence etc. does not convey the slightest information of it until it by means of first intentional icons is specified which parts in which composition, which species, which form, etc. Thus, here Peirce makes a hierarchy of icons which we could call material and formal, respectively, in which the latter are dependent on the former. One may note in passing that the distinctions in Peirce’s semiotics are themselves built upon such Second Intentions; thus it is no wonder that every sign must possess some Iconic element. Furthermore, the very anatomy of the proposition becomes just like in Husserlian rational grammar a question of formal, synthetic a priori regularities.

Among Peirce’s forms of inference, similarity plays a certain role within abduction, his notion for a ‘qualified guess’ in which a particular fact gives rise to the formation of a hypothesis which would have the fact in question as a consequence. Many such different hypotheses are of course possible for a given fact, and this inference is not necessary, but merely possible, suggestive. Precisely for this reason, similarity plays a seminal role here: an

originary Argument, or Abduction, is an argument which presents facts in its Premiss which presents a similarity to the fact stated in the conclusion but which could perfectly be true without the latter being so.

The hypothesis proposed is abducted by some sort of iconic relation to the fact to be explained. Thus, similarity is the very source of new ideas – which must subsequently be controlled deductively and inductively, to be sure. But iconicity does not only play this role in the contents of abductive inference, it plays an even more important role in the very form of logical inference in general:

Given a conventional or other general sign of an object, to deduce any other truth than that which it explicitly signifies, it is necessary, in all cases, to replace that sign by an icon. This capacity of revealing unexpected truth is precisely that wherein the utility of algebraic formulae consists, so that the iconic character is the prevailing one.

The very form of inferences depends on it being an icon; thus for Peirce the syllogistic schema inherent in reasoning has an iconic character:

‘Whenever one thing suggests another, both are together in the mind for an instant. [ ] every proposition like the premiss, that is having an icon like it, would involve [ ] a proposition related to it as the conclusion [ ]’. Thus, first and foremost deduction is an icon: ‘I suppose it would be the general opinion of logicians, as it certainly was long mine, that the Syllogism is a Symbol, because of its Generality.’ …. The truth, however, appears to be that all deductive reasoning, even simple syllogism, involves an element of observation; namely deduction consists in constructing an icon or diagram the relation of whose parts shall present a complete analogy with those of the parts of the objects of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts. 

It then is no wonder that synthetic a priori truths exist – even if Peirce prefers notions like ‘observable, universal truths’ – the result of a deduction may contain more than what is immediately present in the premises, due to the iconic quality of the inference.

Is There a Philosophy of Bundles and Fields? Drunken Risibility.

The bundle formulation of field theory is not at all motivated by just seeking a full mathematical generality; on the contrary it is just an empirical consequence of physical situations that concretely happen in Nature. One among the simplest of these situations may be that of a particle constrained to move on a sphere, denoted by S2; the physical state of such a dynamical system is described by providing both the position of the particle and its momentum, which is a tangent vector to the sphere. In other words, the state of this system is described by a point of the so-called tangent bundle TS2 of the sphere, which is non-trivial, i.e. it has a global topology which differs from the (trivial) product topology of S2 x R2. When one seeks for solutions of the relevant equations of motion some local coordinates have to be chosen on the sphere, e.g. stereographic coordinates covering the whole sphere but a point (let us say the north pole). On such a coordinate neighbourhood (which is contractible to a point being a diffeomorphic copy of R2) there exists a trivialization of the corresponding portion of the tangent bundle of the sphere, so that the relevant equations of motion can be locally written in R2 x R2. At the global level, however, together with the equations, one should give some boundary conditions which will ensure regularity in the north pole. As is well known, different inequivalent choices are possible; these boundary conditions may be considered as what is left in the local theory out of the non-triviality of the configuration bundle TS2.

Moreover, much before modem gauge theories or even more complicated new field theories, the theory of General Relativity is the ultimate proof of the need of a bundle framework to describe physical situations. Among other things, in fact, General Relativity assumes that spacetime is not the “simple” Minkowski space introduced for Special Relativity, which has the topology of R4. In general it is a Lorentzian four-dimensional manifold possibly endowed with a complicated global topology. On such a manifold, the choice of a trivial bundle M x F as the configuration bundle for a field theory is mathematically unjustified as well as physically wrong in general. In fact, as long as spacetime is a contractible manifold, as Minkowski space is, all bundles on it are forced to be trivial; however, if spacetime is allowed to be topologically non-trivial, then trivial bundles on it are just a small subclass of all possible bundles among which the configuration bundle can be chosen. Again, given the base M and the fiber F, the non-unique choice of the topology of the configuration bundle corresponds to different global requirements.

A simple purely geometrical example can be considered to sustain this claim. Let us consider M = S1 and F = (-1, 1), an interval of the real line R; then ∃ (at least) countably many “inequivalent” bundles other than the trivial one Mö0 = S1 X F , i.e. the cylinder, as shown

Untitled

Furthermore the word “inequivalent” can be endowed with different meanings. The bundles shown in the figure are all inequivalent as embedded bundles (i.e. there is no diffeomorphism of the ambient space transforming one into the other) but the even ones (as well as the odd ones) are all equivalent among each other as abstract (i.e. not embedded) bundles (since they have the same transition functions).

The bundles Mön (n being any positive integer) can be obtained from the trivial bundle Mö0 by cutting it along a fiber, twisting n-times and then glueing again together. The bundle Mö1 is called the Moebius band (or strip). All bundles Mön are canonically fibered on S1, but just Mö0 is trivial. Differences among such bundles are global properties, which for example imply that the even ones Mö2k allow never-vanishing sections (i.e. field configurations) while the odd ones Mö2k+1 do not.

Platonist Assertory Mathematics. Thought of the Day 88.0

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Traditional Platonism, according to which our mathematical theories are bodies of truths about a realm of mathematical objects, assumes that only some amongst consistent theory candidates succeed in correctly describing the mathematical realm. For platonists, while mathematicians may contemplate alternative consistent extensions of the axioms for ZF (Zermelo–Fraenkel) set theory, for example, at most one such extension can correctly describe how things really are with the universe of sets. Thus, according to Platonists such as Kurt Gödel, intuition together with quasi-empirical methods (such as the justification of axioms by appeal to their intuitively acceptable consequences) can guide us in discovering which amongst alternative axiom candidates for set theory has things right about set theoretic reality. Alternatively, according to empiricists such as Quine, who hold that our belief in the truth of mathematical theories is justified by their role in empirical science, empirical evidence can choose between alternative consistent set theories. In Quine’s view, we are justified in believing the truth of the minimal amount of set theory required by our most attractive scientific account of the world.

Despite their differences at the level of detail, both of these versions of Platonism share the assumption that mere consistency is not enough for a mathematical theory: For such a theory to be true, it must correctly describe a realm of objects, where the existence of these objects is not guaranteed by consistency alone. Such a view of mathematical theories requires that we must have some grasp of the intended interpretation of an axiomatic theory that is independent of our axiomatization – otherwise inquiry into whether our axioms “get things right” about this intended interpretation would be futile. Hence, it is natural to see these Platonist views of mathematics as following Frege in holding that axioms

. . . must not contain a word or sign whose sense and meaning, or whose contribution to the expression of a thought, was not already completely laid down, so that there is no doubt about the sense of the proposition and the thought it expresses. The only question can be whether this thought is true and what its truth rests on. (Frege to Hilbert Gottlob Frege The Philosophical and Mathematical Correspondence)

On such an account, our mathematical axioms express genuine assertions (thoughts), which may or may not succeed in asserting truths about their subject matter. These Platonist views are “assertory” views of mathematics. Assertory views of mathematics make room for a gap between our mathematical theories and their intended subject matter, and the possibility of such a gap leads to at least two difficulties for traditional Platonism. These difficulties are articulated by Paul Benacerraf (here and here) in his aforementioned papers. The first difficulty comes from the realization that our mathematical theories, even when axioms are supplemented with less formal characterizations of their subject matter, may be insufficient to choose between alternative interpretations. For example, assertory views hold that the Peano axioms for arithmetic aim to assert truths about the natural numbers. But there are many candidate interpretations of these axioms, and nothing in the axioms, or in our wider mathematical practices, seems to suffice to pin down one interpretation over any other as the correct one. The view of mathematical theories as assertions about a specific realm of objects seems to force there to be facts about the correct interpretation of our theories even if, so far as our mathematical practice goes (for example, in the case of arithmetic), any ω-sequence would do.

Benacerraf’s second worry is perhaps even more pressing for assertory views. The possibility of a gap between our mathematical theories and their intended subject matter raises the question, “How do we know that our mathematical theories have things right about their subject matter?”. To answer this, we need to consider the nature of the purported objects about which our theories are supposed to assert truths. It seems that our best characterization of mathematical objects is negative: to account for the extent of our mathematical theories, and the timelessness of mathematical truths, it seems reasonable to suppose that mathematical objects are non-physical, non- spatiotemporal (and, it is sometimes added, mind- and language-independent) objects – in short, mathematical objects are abstract. But this negative characterization makes it difficult to say anything positive about how we could know anything about how things are with these objects. Assertory, Platonist views of mathematics are thus challenged to explain just how we are meant to evaluate our mathematical assertions – just how do the kinds of evidence these Platonists present in support of their theories succeed in ensuring that these theories track the truth?

Intuition

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During his attempt to axiomatize the category of all categories, Lawvere says

Our intuition tells us that whenever two categories exist in our world, then so does the corresponding category of all natural transformations between the functors from the first category to the second (The Category of Categories as a Foundation).

However, if one tries to reduce categorial constructions to set theory, one faces some serious problems in the case of a category of functors. Lawvere (who, according to his aim of axiomatization, is not concerned by such a reduction) relies here on “intuition” to stress that those working with categorial concepts despite these problems have the feeling that the envisaged construction is clear, meaningful and legitimate. Not the reducibility to set theory, but an “intuition” to be specified answers for clarity, meaningfulness and legitimacy of a construction emerging in a mathematical working situation. In particular, Lawvere relies on a collective intuition, a common sense – for he explicitly says “our intuition”. Further, one obviously has to deal here with common sense on a technical level, for the “we” can only extend to a community used to the work with the concepts concerned.

In the tradition of philosophy, “intuition” means immediate, i.e., not conceptually mediated cognition. The use of the term in the context of validity (immediate insight in the truth of a proposition) is to be thoroughly distinguished from its use in the sensual context (the German Anschauung). Now, language is a manner of representation, too, but contrary to language, in the context of images the concept of validity is meaningless.

Obviously, the aspect of cognition guiding is touched on here. Especially the sensual intuition can take the guiding (or heuristic) function. There have been many working situations in history of mathematics in which making the objects of investigation accessible to a sensual intuition (by providing a Veranschaulichung) yielded considerable progress in the development of the knowledge concerning these objects. As an example, take the following account by Emil Artin of Emmy Noether’s contribution to the theory of algebras:

Emmy Noether introduced the concept of representation space – a vector space upon which the elements of the algebra operate as linear transformations, the composition of the linear transformation reflecting the multiplication in the algebra. By doing so she enables us to use our geometric intuition.

Similarly, Fréchet thinks to have really “powered” research in the theory of functions and functionals by the introduction of a “geometrical” terminology:

One can [ …] consider the numbers of the sequence [of coefficients of a Taylor series] as coordinates of a point in a space [ …] of infinitely many dimensions. There are several advantages to proceeding thus, for instance the advantage which is always present when geometrical language is employed, since this language is so appropriate to intuition due to the analogies it gives birth to.

Mathematical terminology often stems from a current language usage whose (intuitive, sensual) connotation is welcomed and serves to give the user an “intuition” of what is intended. While Category Theory is often classified as a highly abstract matter quite remote from intuition, in reality it yields, together with its applications, a multitude of examples for the role of current language in mathematical conceptualization.

This notwithstanding, there is naturally also a tendency in contemporary mathematics to eliminate as much as possible commitments to (sensual) intuition in the erection of a theory. It seems that algebraic geometry fulfills only in the language of schemes that essential requirement of all contemporary mathematics: to state its definitions and theorems in their natural abstract and formal setting in which they can be considered independent of geometric intuition (Mumford D., Fogarty J. Geometric Invariant Theory).

In the pragmatist approach, intuition is seen as a relation. This means: one uses a piece of language in an intuitive manner (or not); intuitive use depends on the situation of utterance, and it can be learned and transformed. The reason for this relational point of view, consists in the pragmatist conviction that each cognition of an object depends on the means of cognition employed – this means that for pragmatism there is no intuitive (in the sense of “immediate”) cognition; the term “intuitive” has to be given a new meaning.

What does it mean to use something intuitively? Heinzmann makes the following proposal: one uses language intuitively if one does not even have the idea to question validity. Hence, the term intuition in the Heinzmannian reading of pragmatism takes a different meaning, no longer signifies an immediate grasp. However, it is yet to be explained what it means for objects in general (and not only for propositions) to “question the validity of a use”. One uses an object intuitively, if one is not concerned with how the rules of constitution of the object have been arrived at, if one does not focus the materialization of these rules but only the benefits of an application of the object in the present context. “In principle”, the cognition of an object is determined by another cognition, and this determination finds its expression in the “rules of constitution”; one uses it intuitively (one does not bother about the being determined of its cognition), if one does not question the rules of constitution (does not focus the cognition which determines it). This is precisely what one does when using an object as a tool – because in doing so, one does not (yet) ask which cognition determines the object. When something is used as a tool, this constitutes an intuitive use, whereas the use of something as an object does not (this defines tool and object). Here, each concept in principle can play both roles; among two concepts, one may happen to be used intuitively before and the other after the progress of insight. Note that with respect to a given cognition, Peirce when saying “the cognition which determines it” always thinks of a previous cognition because he thinks of a determination of a cognition in our thought by previous thoughts. In conceptual history of mathematics, however, one most often introduced an object first as a tool and only after having done so did it come to one’s mind to ask for “the cognition which determines the cognition of this object” (that means, to ask how the use of this object can be legitimized).

The idea that it could depend on the situation whether validity is questioned or not has formerly been overlooked, perhaps because one always looked for a reductionist epistemology where the capacity called intuition is used exclusively at the last level of regression; in a pragmatist epistemology, to the contrary, intuition is used at every level in form of the not thematized tools. In classical systems, intuition was not simply conceived as a capacity; it was actually conceived as a capacity common to all human beings. “But the power of intuitively distinguishing intuitions from other cognitions has not prevented men from disputing very warmly as to which cognitions are intuitive”. Moreover, Peirce criticises strongly cartesian individualism (which has it that the individual has the capacity to find the truth). We could sum up this philosophy thus: we cannot reach definite truth, only provisional; significant progress is not made individually but only collectively; one cannot pretend that the history of thought did not take place and start from scratch, but every cognition is determined by a previous cognition (maybe by other individuals); one cannot uncover the ultimate foundation of our cognitions; rather, the fact that we sometimes reach a new level of insight, “deeper” than those thought of as fundamental before, merely indicates that there is no “deepest” level. The feeling that something is “intuitive” indicates a prejudice which can be philosophically criticised (even if this does not occur to us at the beginning).

In our approach, intuitive use is collectively determined: it depends on the particular usage of the community of users whether validity criteria are or are not questioned in a given situation of language use. However, it is acknowledged that for example scientific communities develop usages making them communities of language users on their own. Hence, situations of language use are not only partitioned into those where it comes to the users’ mind to question validity criteria and those where it does not, but moreover this partition is specific to a particular community (actually, the community of language users is established partly through a peculiar partition; this is a definition of the term “community of language users”). The existence of different communities with different common senses can lead to the following situation: something is used intuitively by one group, not intuitively by another. In this case, discussions inside the discipline occur; one has to cope with competing common senses (which are therefore not really “common”). This constitutes a task for the historian.

|, ||, |||, ||||| . 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.).

Of Magnitudes, Metrization and Materiality of Abstracto-Concrete Objects.

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The possibility of introducing magnitudes in a certain domain of concrete material objects is by no means immediate, granted or elementary. First of all, it is necessary to find a property of such objects that permits to compare them, so that a quasi-serial ordering be introduced in their set, that is a total linear ordering not excluding that more than one object may occupy the same position in the series. Such an ordering must then undergo a metrization, which depends on finding a fundamental measuring procedure permitting the determination of a standard sample to which the unit of measure can be bound. This also depends on the existence of an operation of physical composition, which behaves additively with respect to the quantity which we intend to measure. Only if all these conditions are satisfied will it be possible to introduce a magnitude in a proper sense, that is a function which assigns to each object of the material domain a real number. This real number represents the measure of the object with respect to the intended magnitude. This condition, by introducing an homomorphism between the domain of the material objects and that of the positive real numbers, transforms the language of analysis (that is of the concrete theory of real numbers) into a language capable of speaking faithfully and truly about those physical objects to which it is said that such a magnitude belongs.

Does the success of applying mathematics in the study of the physical world mean that this world has a mathematical structure in an ontological sense, or does it simply mean that we find in mathematics nothing but a convenient practical tool for putting order in our representations of the world? Neither of the answers to this question is right, and this is because the question itself is not correctly raised. Indeed it tacitly presupposes that the endeavour of our scientific investigations consists in facing the reality of “things” as it is, so to speak, in itself. But we know that any science is uniquely concerned with a limited “cut” operated in reality by adopting a particular point of view, that is concretely manifested by adopting a restricted number of predicates in the discourse on reality. Several skilful operational manipulations are needed in order to bring about a homomorphism with the structure of the positive real numbers. It is therefore clear that the objects that are studied by an empirical theory are by no means the rough things of everyday experience, but bundles of “attributes” (that is of properties, relations and functions), introduced through suitable operational procedures having often the explicit and declared goal of determining a concrete structure as isomorphic, or at least homomorphic, to the structure of real numbers or to some other mathematical structure. But now, if the objects of an empirical theory are entities of this kind, we are fully entitled to maintain that they are actually endowed with a mathematical structure: this is simply that structure which we have introduced through our operational procedures. However, this structure is objective and real and, with respect to it, the mathematized discourse is far from having a purely conventional and pragmatic function, with the goal of keeping our ideas in order: it is a faithful description of this structure. Of course, we could never pretend that such a discourse determines the structure of reality in a full and exhaustive way, and this for two distinct reasons: In the first place, reality (both in the sense of the totality of existing things, and of the ”whole” of any single thing), is much richer than the particular “slide” that it is possible to cut out by means of our operational manipulations. In the second place, we must be aware that a scientific object, defined as a structured set of attributes, is an abstract object, is a conceptual construction that is perfectly defined just because it is totally determined by a finite list of predicates. But concrete objects are by no means so: they are endowed with a great deal of attributes of an indefinite variety, so that they can at best exemplify with an acceptable approximation certain abstract objects that are totally encoding a given set of attributes through their corresponding predicates. The reason why such an exemplification can only be partial is that the different attributes that are simultaneously present in a concrete object are, in a way, mutually limiting themselves, so that this object does never fully exemplify anyone of them. This explains the correct sense of such common and obvious remarks as: “a rigid body, a perfect gas, an adiabatic transformation, a perfect elastic recoil, etc, do not exist in reality (or in Nature)”. Sometimes this remark is intended to vehiculate the thesis that these are nothing but intellectual fictions devoid of any correspondence with reality, but instrumentally used by scientists in order to organize their ideas. This interpretation is totally wrong, and is simply due to a confusion between encoding and exemplifying: no concrete thing encodes any finite and explicit number of characteristics that, on the contrary, can be appropriately encoded in a concept. Things can exemplify several concepts, while concepts (or abstract objects) do not exemplify the attributes they encode. Going back to the distinction between sense on the one hand, and reference or denotation on the other hand, we could also say that abstract objects belong to the level of sense, while their exemplifications belong to the level of reference, and constitute what is denoted by them. It is obvious that in the case of empirical sciences we try to construct conceptual structures (abstract objects) having empirical denotations (exemplified by concrete objects). If one has well understood this elementary but important distinction, one is in the position of correctly seeing how mathematics can concern physical objects. These objects are abstract objects, are structured sets of predicates, and there is absolutely nothing surprising in the fact that they could receive a mathematical structure (for example, a structure isomorphic to that of the positive real numbers, or to that of a given group, or of an abstract mathematical space, etc.). If it happens that these abstract objects are exemplified by concrete objects within a certain degree of approximation, we are entitled to say that the corresponding mathematical structure also holds true (with the same degree of approximation) for this domain of concrete objects. Now, in the case of physics, the abstract objects are constructed by isolating certain ontological attributes of things by means of concrete operations, so that they actually refer to things, and are exemplified by the concrete objects singled out by means of such operations up to a given degree of approximation or accuracy. In conclusion, one can maintain that mathematics constitutes at the same time the most exact language for speaking of the objects of the domain under consideration, and faithfully mirrors the concrete structure (in an ontological sense) of this domain of objects. Of course, it is very reasonable to recognize that other aspects of these things (or other attributes of them) might not be treatable by means of the particular mathematical language adopted, and this may imply either that these attributes could perhaps be handled through a different available mathematical language, or even that no mathematical language found as yet could be used for handling them.

Concepts – Intensional and Extensional.

Omega-exp-omega-labeled

Let us start in this fashion: objects to which concepts apply (or not). The first step in arriving at a theory for this situation is, to assume that the objects in question are completely arbitrary (as urelements in set theory). This assumption is evidently wrong in empirical experience as also in mathematics itself, e.g., in function theory. So to admit this assumption forces us to build our own theory of sets to take care of the case of complex objects later on.

Concepts are normally given to us by linguistic expressions, disregarding by abstraction the origin of languages or signals or what have you. Now we can develop a theory of concepts as follows. We idealize our language by fixing a vocabulary together with logical operators and formulate expressions for classes, functions, and relations in the way of the λ-calculus. Here we have actually a theory of concepts, understood intensionally. Note that the extensional point of view is by no means lost, since we read for e.g., λx,yR(x,y) as the relation R over a domain of urelements; but either R is in the vocabulary or given by a composed expression in our logical language; equality does not refer to equal extensions but to logical equivalence and reduction processes. By the way, there is no hindrance to apply λ-expressions again to λ-expressions so that hierarchies of concepts can be included.

Another approach to the question of obtaining a theory of concepts is the algebraic one. Here introducing variables for extensions over a domain of urelements, and calling them classes helps develop the axiomatic class calculus. Adding (two-place) relations again with axioms, and we can obtain the relation calculus. One could go a step further to polyadic algebra. These theories do not have a prominent role nowadays, if one compares them with the λ-calculus or set theory. This is probably due to the circumstance that it seems difficult, not to say actually against the proper idea behind these theories, to allow iteration in the sense of classes of classes, etc.

For the mathematical purposes and for the use of logics, the appropriate way is to restrict a theory of concepts to a theory of their extensions. This has a good reason, since in an abstract theory we are interested in being as neutral as possible with respect to a description or factual theory given beforehand. There is a philosophical principle behind this, namely that logical (and in this case set theoretical) assumptions should be as far as possible distinguishable from any factual or descriptive assumption.

Category Theory of a Sketch. Thought of the Day 50.0

Untitled

If a sketch can be thought of as an abstract concept, a model of a sketch is not so much an interpretation of a sketch, but a concrete or particular instantiation or realization of it. It is tempting to adopt a Kantian terminology here and say that a sketch is an abstract concept, a functor between a sketch and a category C a schema and the models of a sketch the constructions in the “intuition” of the concept.

The schema is not unique since a sketch can be realized in many different categories by many different functors. What varies from one category to the other is not the basic structure of the realizations, but the types of morphisms of the underlying category, e.g., arbitrary functions, continuous maps, etc. Thus, even though a sketch captures essential structural ingredients, others are given by the “environment” in which this structure will be realized, which can be thought of as being itself another structure. Hence, the “meaning” of some concepts cannot be uniquely given by a sketch, which is not to say that it cannot be given in a structuralist fashion.

We now distinguish the group as a structure, given by the sketch for the theory of groups, from the structure of groups, given by a category of groups, that is the category of models of the sketch for groups in a given category, be it Set or another category, e.g., the category of topological spaces with continuous maps. In the latter case, the structure is given by the exactness properties of the category, e.g., Cartesian closed, etc. This is an important improvement over the traditional framework in which one was unable to say whether we should talk about the structure common to all groups, usually taken to be given by the group axioms, or the structure generated by “all” groups. Indeed, one can now ask in a precise manner whether a category C of structures, e.g., the category of (small) groups, is sketchable, that is, whether there exists a sketch S such that Mod(S, Set) is equivalent as a category to C.

There is another category associated to a sketch, namely the theory of that sketch. The theory of a sketch S, denoted by Th(S), is in a sense “freely” constructed from S : the arrows of the underlying graph are freely composed and the diagrams are imposed as equations, and so are the cones and the cocones. Th(S) is in fact a model of S in the previous sense with the following universal property: for any other model M of S in a category C there is a unique functor F: Th(S) → C such that FU = M, where U: S → Th(S). Thus, for instance, the theory of groups is a category with a group object, the generic group, “freely” constructed from the sketch for groups. It is in a way the “universal” group in the sense that any other group in any category can be constructed from it. This is possible since it contains all possible arrows, i.e., all definable operations, obtained in a purely internal or abstract manner. It is debatable whether this category should be called the theory of the sketch. But that may be more a matter of terminology than anything else, since it is clear that the “free” category called the theory is there to stay in one way or another.

Noneism. Part 1.

Meinong

Noneism was created by Richard Routley. Its point of departure is the rejection of what Routley calls “The Ontological Assumption”. This assumption consists in the explicit or, more frequently, implicit belief that denoting always refers to existing objects. If the object, or objects, on which a proposition is about, do not exist, then these objects can only be one: the null entity. It is incredible that Frege believed that denoting descriptions without a real (empirical, theoretical, or ideal) referent denoted only the null set. And it is also difficult to believe that Russell sustained the thesis that non-existing objects cannot have properties and that propositions about these objects are false.

This means that we can have a very clear apprehension of imaginary objects, and quite clear intellection of abstract objects that are not real. This is possible because to determine an object we only need to describe it through its distinctive traits. This description is possible because an object is always chacterized through some definite notes. The amount of traits necessary to identify an object greatly varies. In some cases we need only a few, for instance, the golden mountain, or the blue bird; in other cases we need more, for instance, the goddess Venus or the centaur Chiron. In other instances the traits can be very numerous, even infinite. For instance the chiliedron, and the decimal number 0,0000…009, in which 9 comes after the first million zeros, have many traits. And the ordinal omega or any Hilbert space have infinite traits (although these traits can be reckoned through finite definitions). These examples show, in a convincing manner, that the Ontological Assumption is untenable. We must reject it and replace it with what Routley dubbs the Characterization Postulate. The Characterization Postulate says that, to be an object means to be characterized by determined traits. The set of the characterizing traits of an object can be called its “characteristic”. When the characteristic of an object is set up, the object is perfectly recognizable.

Once this postulate is adopted, its consequences are far reaching. Since we can characterize objects through any traits whatsoever, an object can not only be inexistent, it can even be absurd or inconsistent. For instance, the “squond” (the circle that is square and round). And we can make perfectly valid logical inferences from the premiss: x is the sqound:

(1) if x is the squond, then x is square
(2) if x is the squond, then x is round

So, the theory of objects has the widest realm of application. It is clear that the Ontological Assumption imposes unacceptable limits to logic. As a matter of fact, the existential quantifier of classical logic could not have been conceived without the Ontological Assumption. The expression “(∃x)Fx” means that there exists at least an object that has the property F (or, in extensional language, that there exists an x that is a member of the extension of F). For this reason, “∃x” is unappliable to non existing objects. Of course, in classical logic we can deny the existence of an Object, but we cannot say anything about Objects that have never existed and shall never exist (we are strictly speaking about classical logic). We cannot quantify individual variables of a first order predicate that do not refer to a real, actual, past or future entity. For instance, we cannot say “(∃x) (x is the eye of Polyphemus)”. This would be false, of course, because Polyphemus does not exist. But if the Ontological Assumption is set aside, it is true, within a mythological frame, that Polyphemus has a single eye and many other properties. And now we can understand why noneism leads to logical material-dependence.

As we have anticipated, there must be some limitations concerning the selection of the contradictory properties; otherwise the whole theory becomes inconsistent and is trivialized. To avoid trivialization neutral (noneist) logic distinguishes between two sorts of negation: the classical propositional negation: “8 is not P”, and the narrower negation: “8 is non-P”. In this way, and by applying some other technicalities (for instance, in case an universe is inconsistent, some kind of paraconsistent logic must be used) trivialization is avoided. With the former provisions, the Characterization Postulate can be applied to create inconsistent universes in which classical logic is not valid. For instance, a world in which there is a mysterious personage, that within determined but very subtle circumstances, is and is not at the same time in two different places. In this case the logic to be applied is, obviously, some kind of paraconsistent logic (the type to be selected depends on the characteristic of the personage). And in another universe there could be a jewel which has two false properties: it is false that it is transparent and it is false that it is opaque. In this kind of world we must use, clearly, some kind of paracomplete logic. To develop naive set theory (in Halmos sense), we must use some type of paraconsistent logic to cope with the paradoxes, that are produced through a natural way of mathematical reasoning; this logic can be of several orders, just like the classical. In other cases, we can use some kind of relevant and, a fortiori, paraconsistent logic; and so on, ad infinitum.

But if logic is content-dependent, and this dependence is a consequence of the Ontological Assumption’s rejection, what about ontology? Because the universes determined through the application of the Characterization Postulate may have no being (in fact, most of them do not), we cannot say that the objects that populate such universes are entities, because entities exist in the empirical world, or in the real world that underpins the phenomena, or (in a somewhat different way), in an ideal Platonic world. Instead of speaking about ontology, we should speak about objectology. In essence objectology is the discipline founded by Meinong (Theory of Objects), but enriched and made more precise by Routley and other noneist logicians. Its main division would be Ontology (the study of real physical and Platonic objects) and Medenology (the study of objects that have no existence).

Rhizomatic Topology and Global Politics. A Flirtatious Relationship.

 

rhizome

Deleuze and Guattari see concepts as rhizomes, biological entities endowed with unique properties. They see concepts as spatially representable, where the representation contains principles of connection and heterogeneity: any point of a rhizome must be connected to any other. Deleuze and Guattari list the possible benefits of spatial representation of concepts, including the ability to represent complex multiplicity, the potential to free a concept from foundationalism, and the ability to show both breadth and depth. In this view, geometric interpretations move away from the insidious understanding of the world in terms of dualisms, dichotomies, and lines, to understand conceptual relations in terms of space and shapes. The ontology of concepts is thus, in their view, appropriately geometric, a multiplicity defined not by its elements, nor by a center of unification and comprehension and instead measured by its dimensionality and its heterogeneity. The conceptual multiplicity, is already composed of heterogeneous terms in symbiosis, and is continually transforming itself such that it is possible to follow, and map, not only the relationships between ideas but how they change over time. In fact, the authors claim that there are further benefits to geometric interpretations of understanding concepts which are unavailable in other frames of reference. They outline the unique contribution of geometric models to the understanding of contingent structure:

Principle of cartography and decalcomania: a rhizome is not amenable to any structural or generative model. It is a stranger to any idea of genetic axis or deep structure. A genetic axis is like an objective pivotal unity upon which successive stages are organized; deep structure is more like a base sequence that can be broken down into immediate constituents, while the unity of the product passes into another, transformational and subjective, dimension. (Deleuze and Guattari)

The word that Deleuze and Guattari use for ‘multiplicities’ can also be translated to the topological term ‘manifold.’ If we thought about their multiplicities as manifolds, there are a virtually unlimited number of things one could come to know, in geometric terms, about (and with) our object of study, abstractly speaking. Among those unlimited things we could learn are properties of groups (homological, cohomological, and homeomorphic), complex directionality (maps, morphisms, isomorphisms, and orientability), dimensionality (codimensionality, structure, embeddedness), partiality (differentiation, commutativity, simultaneity), and shifting representation (factorization, ideal classes, reciprocity). Each of these functions allows for a different, creative, and potentially critical representation of global political concepts, events, groupings, and relationships. This is how concepts are to be looked at: as manifolds. With such a dimensional understanding of concept-formation, it is possible to deal with complex interactions of like entities, and interactions of unlike entities. Critical theorists have emphasized the importance of such complexity in representation a number of times, speaking about it in terms compatible with mathematical methods if not mathematically. For example, Foucault’s declaration that: practicing criticism is a matter of making facile gestures difficult both reflects and is reflected in many critical theorists projects of revealing the complexity in (apparently simple) concepts deployed both in global politics.  This leads to a shift in the concept of danger as well, where danger is not an objective condition but “an effect of interpretation”. Critical thinking about how-possible questions reveals a complexity to the concept of the state which is often overlooked in traditional analyses, sending a wave of added complexity through other concepts as well. This work seeking complexity serves one of the major underlying functions of critical theorizing: finding invisible injustices in (modernist, linear, structuralist) givens in the operation and analysis of global politics.

In a geometric sense, this complexity could be thought about as multidimensional mapping. In theoretical geometry, the process of mapping conceptual spaces is not primarily empirical, but for the purpose of representing and reading the relationships between information, including identification, similarity, differentiation, and distance. The reason for defining topological spaces in math, the essence of the definition, is that there is no absolute scale for describing the distance or relation between certain points, yet it makes sense to say that an (infinite) sequence of points approaches some other (but again, no way to describe how quickly or from what direction one might be approaching). This seemingly weak relationship, which is defined purely ‘locally’, i.e., in a small locale around each point, is often surprisingly powerful: using only the relationship of approaching parts, one can distinguish between, say, a balloon, a sheet of paper, a circle, and a dot.

To each delineated concept, one should distinguish and associate a topological space, in a (necessarily) non-explicit yet definite manner. Whenever one has a relationship between concepts (here we think of the primary relationship as being that of constitution, but not restrictively, we ‘specify’ a function (or inclusion, or relation) between the topological spaces associated to the concepts). In these terms, a conceptual space is in essence a multidimensional space in which the dimensions represent qualities or features of that which is being represented. Such an approach can be leveraged for thinking about conceptual components, dimensionality, and structure. In these terms, dimensions can be thought of as properties or qualities, each with their own (often-multidimensional) properties or qualities. A key goal of the modeling of conceptual space being representation means that a key (mathematical and theoretical) goal of concept space mapping is

associationism, where associations between different kinds of information elements carry the main burden of representation. (Conceptual_Spaces_as_a_Framework_for_Knowledge_Representation)

To this end,

objects in conceptual space are represented by points, in each domain, that characterize their dimensional values. A concept geometry for conceptual spaces

These dimensional values can be arranged in relation to each other, as Gardenfors explains that

distances represent degrees of similarity between objects represented in space and therefore conceptual spaces are “suitable for representing different kinds of similarity relation. Concept

These similarity relationships can be explored across ideas of a concept and across contexts, but also over time, since “with the aid of a topological structure, we can speak about continuity, e.g., a continuous change” a possibility which can be found only in treating concepts as topological structures and not in linguistic descriptions or set theoretic representations.