Distinguishing the General from the Generic / Thinking as an Algebraic Mechanist

Illustration of Categoricity – preliminary notes on Herman Weyl’s categorial take on the circulus vitiosus of analysis

In this new category of posts called “Distinguishing the General from the Generic” I would like to publish posts that explore a twofold line (1) of why, for an understanding of computation as a literacy,  keeping the notions of generality and genericity distinct seems of crucial importance, and (2) how we might do so.

The approach which I would briefly like to delineate here is rooted in Herman Weyl’s Critical Examination of the Foundations of Analysis (1918). These roots do not mean that the suggested approach relies on sharing Weyl’s philosophical views on Intuition and Continuity; it merely imports Weyl’s categorial treatment of the foundation problem, which he shares with many of the algebraists among his contemporaries, like for example Richard Dedekind – whose procedure of how to define numbers (Dedekind-Cut) plays perhaps the crucial rôle in his book, but against whose philosophical views, best illustrated  in Dedekind’s often quoted statement from Was sind und was sollen Zahlen? (1888),  namely that “numbers are free creations of our mind”, Weyl dedicatedly distanced himself * (cf. the footnotes 19 and 32 on the first chapter in Critical Examination).

*(My own inclinations are with Dedekind on this, as I understand him to work towards a notion of  ‘intellectual’ ‘intuition’ which he seems to conceive in the sense of a literacy, as something that depends upon its (difficult) acquisition, development and mastership; while it is not entirely clear to me whether or how Weyl manages not to ‘naturalize‘ the natural numbers (for this would be severely at odds with his categorial take on analysis), Dedekind chooses the other option, namely to ‘intellectualize‘ the natural numbers. Yet – this may well turn out to be a false dichotomy, and naturalization and intellectualization actually express pretty much the same thing – once with a materialist inclination, and once with an idealist one. Yet, in a non-trivial sense of these philosophical legacies, is not the open dilemma that once generalization as a method is granted, every materialism turns out to be an idealism, just as every idealism turns out to be a materialism once induction as a method is granted? But this will be the topic of a separate post).

You can download a scanned pdf of Weyl’s text here: Hermann-Weyl-The-Continuum-A-Critical-Examination

The categorial treatment of the foundation problem in analysis

“Sense precedes existence” – this, it seems to me, is the aspect which is central to a categorial treatment of analysis. It is meant in a strictly formal way (namely that Frege’s notion of logical extension (sense) is not equivalent to a logical notion of existence (quantification)),  but  I have expressed it here with considerate reminiscence to the famous philosophical slogans with which Jean-Paul Sartre has mobilized his contemporaries to rise against the legacies of metaphysical doctrine and their jointly exercised oppression of societal modernization, namely that existence precedes essence and that of freedom, there is no essence. Surely in the same ethical spirit, yet also somewhat against these slogans (especially the latter concerning freedom), a categorial approach holds that every essence can only be articulated in categorial terms, and that the object of logical analysis is neither essence (metaphysical doctrines) nor existence, but sense. It conceives of sense in terms of Frege’s distinction between sense and meaning, as a statement’s extensionyet without sharing the equivalence between extension and existence as propagated by Frege.

Weyl’s suggestion is to conceive of propositions as constructed statements which express judgments, and judgements as affirmations of a state-of-affairs. Yet for him, the construction of proposition is not strictly logical, but also mathematical. The irreducibility of both aspects in the construction of propositions is Weyl’s crucial point. From a formalist point of view, the concepts of ‘existence’ and ‘universality’ can be applied to the natural numbers as well as to any sequence of natural numbers. This, Weyl criticizes as leading to a circulus vitiosus of analysis which necessitates the articulation of limits of analysis in order to exit it. These limits, Weyl suggests, can be articulated by restricting the application of the concepts of ‘existence’ and ‘universality’ strictly to the natural numbers, and not to any sequence of them. It is the sequentiality that involves the mathematical principles he suggests – those of substitution and iteration – and it is here where categoricity comes into play. In all brevity: every judgement (affirmation of a state-of-affairs) depends upon what he calls a ‘judgement scheme’; from his proposed 6 logical principles the judgement-schemes are constructible by his proposed 6 logical principles (governing the concepts of ‘negation’, ‘coincidence’, ‘and’, ‘or’, ‘closure’ (‘filling-in’ operation), and ‘saturation’ (Weyl does not actually call it the ‘saturation’ principle, he merely speaks of the concept of ‘being-there’; yet it seems helpful to call it by this name in order to stress that in Weyl’s table of principles, the ‘being-there’ concept is not equivalent to that ordinarily called ‘existence’). The combination of these principles can yield what he calls, algebraically, ‘identity schemes’; such identity schemes can be applied to what he calls ‘simple’ or ‘primitive’ judgement schemes (“those which correspond to the individual immediately given properties and relations”, p. 9). Like this, “an endless abundance of judgement schemes” (p. 11) can be obtained. While all of these judgement schemes produce sense (extensionality), not all of them are also meaningful. Hence, his starting point for a critical examination of the foundation of analysis is that concepts can be clearly and unambiguously defined, yet that this does not mean that they can be regarded as being intrinsically determined.

The sense of a clearly and unambiguously defined concept may indeed always assign the appropriate sphere of existence to the objects which share the essence expressed in the concept. But this certainly does not imply that this concept is extensionally determinate, i.e., that it is meaningful to speak of the existent objects falling under it as an ideally closed aggregate which is intrinsically determined and demarcated. (Appendix to Critical Examination, called “The circles vitiosus in the Current Foundation of Analysis”, p.109)

This is why Weyl conceives of the object of analysis strictly within what he calls ‘subject-ordered relations’ (instead of simply ‘relations’, as it is customary for formalist approaches). ‘Subject-ordered relations’ are governed by categories, and categories are ‘substantiated’ by the two mathematical principles that are complementing his six logical principles in the construction of propositions.

Essence, in an algebraist way of a categorial treatment differs from a metaphysical way of categorial treatment in that here, the categories precede essence. Categories are not extracted from substances, but the other way around: a meaningful application of categorial order needs to be substantiated before its deductions can claim to be  meaningful (and not only based on sense, i.e. extensive, clearly determinable). It’s promise is nothing less than that of a criticality – not one regarding the identification of positive knowledge (Kant), but one regarding the meaningfulness of abstractions.

The crucial problem then is how such ‘substantiation’ can be considered. For this, the distinction between generality and genericness plays a crucial role. For if we consider it from the standpoint of generality, we naturalize the categories. If we consider it from the standpoint of genericness, on the other hand, we absolutize the categories.

Herman Weyl’s Categoricity Illustrator

(He does not actually call the device he discusses here Categoricity Illustrator, but I think this would be a helpful name for it. Here is an excerpt from Weyl’s Appendix to Critical Examination, called “The circles vitiosus in the Current Foundation of Analysis”, p.114-117).

HERMAN WEYL THE CATEGORY ILLUSTRATOR

“Let us consider, say, the ternary relation ∈ (xy,Z) (“xy stand in the relation Z to one another”) in which the blanks xy are affiliated with the same basic category, while Z is affiliated with the category of binary relations between objects of this basic category. And, in accordance with note 4 of Chapter 1, let us represent the scheme of this relation by a wooden plate with one large and two small pegs corresponding to the blanks Z and xy respectively. The objects of the basic category are represented by balls, each equipped with a hole so that they can be stuck on the small pegs to represent filling the blanks xy. Suppose all this has been done. The blank Z t in ∈ must be filled by a binary relation R. But this in turn is represented by a plate with two pegs, a plate which, moreover, must be equipped with a hole as big as the large peg in ∈. If this plate is stuck on the large peg, then all three blanks in ∈ are filled. Nevertheless, no definite judgment emerges from ∈ in this way. For this purpose it is necessary that the two blanks x and y in ∈ or the objects which fill them be related in a definite way to the two blanks XY of the relation R ( x y) which fills the blank Z in ∈ or, as I prefer to say, the latter blanks must be “connected ” with the former. Accordingly, to the scheme of the relation e there belong two “connection wires” originating from the base of the pegs xy with whose help the connection of the “secondary” blanks to the “primary” ones xy, in the manner indicated by the figure, has to occur in the process of filling the blanks. Clearly, in the case of the given relation R, this connection can occur in two different ways. The fact that the wires are fed into the pegs XY from above is meant to indicate further that all blanks are “saturated” in the filling process. The existence of such “connection wires” in the scheme of a relation carries over naturally from ∈ to relations which are produced from ∈ and the primitive relations by means of the construction principles; the “secondary” blanks of the relations employed in the filling process must be connected by the wires in part to the primary blanks and in part to certain relational points. Some effort is required to understand how the scheme of a relation appears in general and what the process of filling the blanks, which produces a definite judgement scheme, amounts to. The model we have just employed will stand us in good stead here. But even if it is of great value to us in acquiring a comprehensive view of the syntax of relations, this rather involved schematism can be avoided by means of a simple and purely formal device, namely by introducing subject-ordered relations in place of relations. Altering somewhat the explanation given in my treatise, I here understand a subject-ordered relation to be one in whose scheme a definite order of succession is established within each group of blanks affiliated with one and the same category of object. If the blanks x and y in the above-mentioned relation ∈ are numbered in this way and if, further, a subject-ordered binary relation is always used to fill the blank Z, i.e., if the secondary blanks X and Y are likewise numbered, then the “connection” is superfluous, since it is self-evident that the secondary blank 1 is to be connected to the primary blank 1 and the secondary blank 2 to the primary blank 2. Because of this, ∈ and the relations to be derived from it are of the same sort, as I assumed from the very beginning of my treatise: in order to obtain a meaningful assertion from them, it is sufficient to fill each blank with an object of the relevant category (an object which may itself be a subject-ordered relation). I am even less hesitant to introduce subject-ordered relations, in spite of the formal and artificial character of this device, in view of the fact that, later on, if the transition from relations to sets is to be carried out, one must necessarily assume that relations are subject-ordered.

A quinary relation R (uv | xyz) (to pick a definite example) can also be regarded as a binary relation holding between u and v and depending on the three “arguments” xyz. It can then be used in a higher level relation to fill a blank Z affiliated with binary relations–in which case only the first two of the secondary blanks uv | xyz are saturated through connection, while xyz remain free blanks which wait to be filled by objects. This process is employed in the principle of substitution 7 (p. 35). If we are just interested in relations between objects of the basic categories, then clearly the introduction of ∈ and the principle of substitution by itself leads to no relations other than those which can be constructed with the help of the first six principles alone, without this expansion. ∈ and the principle of substitution become fruitful only when supplemented by the principle of iteration (8) of which they are the indispensable precursors. Iteration, however, is of the greatest significance for all mathematical concept-formation.

I have called those relations delimited which can be constructed, by the indicated means, from the intuitively exhibited, primitive relations of the underlying sphere of operation. In this way, we obtain the desired, extensionally determinate restriction of the concept “relation” which is required for founding a non-circular version of analysis. Only now, at the end, should the concept of set and Junction be introduced. And, of course, the sets and functions correspond to the (subject-ordered, delimited) relations in such a way that extension rather than sense determines their equality or distinctness; they form the mathematical superstructure over the intuitively grounded substructure of the basic categories.

As far as I can see, analysis provides no occasion for iterating this expanded mathematical process which includes the principles of construction 7 and 8.’ (Cf. p. 22 concerning the expression “mathematical process.”) And, furthermore, the schema we have secured proves to be comprehensive enough with regard to applications that on this basis a rational theory of the continuum can be constructed (as in Chapter 2).”

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