This book continues the series of volumes containing reprints of the papers in the original Cabal Seminar volumes of the Springer Lecture Notes in Mathematics series [Cabal i, Cabal ii, Cabal iii, Cabal iv], unpublished material, and new papers. The first volume, [Cabal I], contained papers on games, scales and Suslin cardinals. The second volume, [Cabal II], contained papers on Wadge degrees and pointclasses and projective ordinals. In this volume, we continue with Parts V and VI of the project: Ordinal definability in models of determinacy and Recursion theory. As in our first two volumes, each of the parts contains an introductory survey (written by John Steel for Part V and by Leo Harrington and Ted Slaman for Part VI) putting the papers into a present-day context.

In addition to the reprinted papers, this volume contains a paper by Kechris and Martin (On the theory of Π13sets of reals, II) that dates back to the period of the original Cabal publications but was not included in the old volumes. Neeman contributed a new paper, An inner models proof of the Kechris–Martin theorem, related to this paper. Steel and Woodin contributed two new papers (A theorem of Woodin on mouse sets, authored by Steel, and HODas a core model, jointly) with recent results that fit well with the topics of Part V. There is also a new paper by Marks, Slaman and Steel (Martin's conjecture, arithmetic equivalence, and countable Borel equivalence relations) that contains earlier, unpublished, as well as new results related to the theme of Part VI. Table 1 gives an overview of the papers in this volume with their original references.

As emphasized in our first two volumes, our project is not to be understood as a historical edition of old papers. In the retyping process, we uniformized and modernized notation and numbering of sections and theorems. As a consequence, references to papers in the old Cabal volumes will not always agree with references to their reprinted versions. In this volume, references to papers that already appeared in reprinted form will use the new numbering. In order to help the reader to easily cross-reference old and new numberings, we provide a list of changes after the preface.

Let us presume as a working hypothesis that the views expressed in Spinoza's Ethics are in broad strokes metaphysically and ontologically correct. We thus presume that what really exist are relational structures of infinite variety and that these structures everywhere enter in turn into higher-order relations with one another and so on without limit. We and all the things around us are both immersed within relations of these kinds and thoroughly saturated by them. At any particular scale of investigation, then, we may take relatively stable linkages of more or less determinate local relations to constitute ‘things’ or ‘objects’ in a quite general sense, and we may expect to find a variety of relations linking such scare-quoted objects to one another depending upon which we decide to select. Furthermore such relations frequently come packaged together in internally differentiated systems of some common type: chemical, linguistic, military-strategic, stellar-galactic and so forth. Our aim is to develop a workable method for plunging philosophically into this immanent relational sea.

We proceed accordingly at two distinct levels. At an initial level, we will develop an informal diagrammatic notation for representing and analysing arbitrary systems of relations, with objects represented by dots and relations of the relevant type by arrows between dots. At a secondary, reflective level, we will treat the same notation in a more regimented way in order to introduce category theoretical mathematics. At both levels, we will be using systems of partly determined and partly undetermined relations (a variety of dot-and-arrow diagrams) to represent and investigate systems of partly determined and partly undetermined relations (worldly phenomena and mathematical categories). To some extent these representations and patterns of investigation will be reversible, that is, the represented will by virtue of the very form of representation at work serve as a potential representing medium in its own right. It will be a diagram too. The fact that the same notation serves as object, method and mathematical formalism provides the primary link here to Spinoza's philosophy of immanence.

A distinctive feature of Spinoza's thought is that it rejects any explanatory mechanisms grounded in mystery or de jure unknowability, in particular any explanatory criteria of experience and knowledge that would rely on ‘objects’ external to the mind. While Spinoza concedes that ‘a true idea must agree with its object’, he understands philosophical explanation to be grounded properly not in truth but in adequacy: ‘By adequate idea I understand an idea which, insofar as it is considered in itself, without relation to an object, has all the properties, or intrinsic denominations of a true idea.’ From a Spinozist point of view, then, empiricism in anything like the Lockean style is a philosophical non-starter. For Spinoza, it will never be sufficient to rest any explanation of experience, knowledge or power on the sheer fact that it is given. No doubt one always begins with what is given, but on Spinoza's terms philosophy fails to think adequately if the given functions for it as an answer and not solely as a relative starting-point. Thought itself is a transitive activity and a continuous process, and so beginnings are as such exterior to thought. Because for a philosophy of immanence nothing can be absolutely exterior to thought, such thinking cannot countenance absolute beginnings. Among other reasons this is why the undivided term ‘God, or Nature’ in Spinoza must be understood not as a foundation or ultimate principle but rather as an incontrovertible milieu: real immanence.

If certainly no Lockean empiricist, Spinoza does seem to be grouped readily among the early modern ‘rationalists’. And sure enough, Spinoza shares with Descartes and Leibniz a resolute willingness to blur if not entirely efface the distinction of logic – and mathematics – from metaphysics. For reason as such, conceptual and formal relations, not the experiential contents of the senses, are eminently knowable. Hence metaphysics as the rational science of reality's ultimate structure is feasible. Yet unlike Descartes and Leibniz, Spinoza makes no important contributions to mathematics and indeed demonstrates no exceptional aptitude in that arena.

The problems of representation and reference under conditions of radical immanence are particularly striking, as both of these appear to require at least a minimal transcendence in order to function at all. The category-theoretical framework continued in this chapter addresses these abstract yet still strictly philosophical problems from a purely formal standpoint in which the ‘interiority’ and ‘exteriority’ of categorical determinations serve as a basis for reconceiving the internal and external relations of entities understood as diagrammatically structured. The key insight is a continuation and extension of that begun in Chapter 2, namely that within the framework of category theory categories themselves may appear as objects with welldefined systems of relations to one another via functors. Such systems of objects and relations may themselves constitute categories in their own right. This formal ‘immanence’ – the collapse of meta-systems and meta-relations into the same type of diagrammatically tractable systems and relations – may itself be lifted to the level of functors themselves: whereas functors in category theory are essentially maps between categories, natural transformations are structure-preserving maps between functors (maps between maps of systems of maps). Thus the relation between functors and natural transformations is a useful general model of the distinction between systems of relations and systems of meta-relations that yet treats these in a structurally identical manner. Examples in this chapter are developed in terms of demonstrating the ways in which the diagrammatic and structural methods of category theory thus model immanence, building up to a formal model of diagrammatic signification based in presheaves, a type of functor, and the functor categories – presheaf categories – that are constructed from them. This model will cast the Peircean characterisation of diagrammatic semiotics from the previous chapter in a categorical context.

Consider an abstract ‘slice’ of the cocktail party encountered in Chapter 2, selecting from among its innumerable constituent things, properties and relations two couples in the midst of two conversations. With dots representing people and arrows standing for the relation ‘is speaking to’, we have a diagram like that on the left in Figure 4.1.

We have followed two largely independent lines of thought, one philosophical and one mathematical. The overall proposal of diagrammatic immanence is meant to chart their possible long-term convergence. Immanent metaphysics has been characterised as intrinsically relational (Spinoza), semiotic (Peirce) and differential (Deleuze). Because diagrams – understood in a sufficiently broad sense – investigate relations by way of expressing relations and experimenting semiotically with differences, diagrams suggest themselves as a very general philosophical method conforming to the requirements of immanence. In elaborating the basic concepts of category theory we have uncovered a highly developed and extremely general field of mathematics that both uses its own type of diagrams as a formal notation and is also particularly adept at tracking how diagrams in general work. The schema of diagrammatic signification expressed in terms of the triad selection-experimentation-evaluation has served as a common locus, an overlap, of the philosophical and mathematical territories explored. As a schema expressible both in terms of Peirce's triadic theory of signs and as a category theoretical construction based in presheaves, it strongly suggests that further coordination along similar lines is possible. No doubt the formal constructions of adjunctions and topoi will be useful in testing this hypothesis. In any case, the pragmatic criterion remains crucial. As an entwined philosophical and mathematical approach to thinking in diagrams, what can diagrammatic immanence potentially do?

First of all, by shifting philosophical terrain both substantively and methodologically from language and textuality to diagrammatic relations and practices, an opening is made for enabling new constructive relations within philosophy itself and between philosophy and a variety of other fields. Obviously, the present proposal sets up multiple ways for metaphysics, formal logic and contemporary mathematics to communicate. In this respect, the present proposal may be broadly aligned with recent work by Badiou, Williamson, Zalamea and others, opening such work to productive dialogue with the Spinozist, Peircean and Deleuzian traditions in particular. But category theory is of course not just a tool for philosophers. It applies in its own distinctive manner to virtually any system of relations.

In his 1868 article for the Journal of Speculative Philosophy, ‘Some Consequences of Four Incapacities’, Charles Peirce lays out a systematic rejection of the principles of Cartesianism. His rationale consists of the assertion and defence of the ‘four incapacities’ of his title, four powers that the Cartesian tradition has in one way or another presumed to exist and which Peirce himself denies the philosophical inquirer (and by extension, the community of inquirers) to possess. Peirce summarises his anti-Cartesian quartet as follows:

1. 1. We have no power of Introspection, but all knowledge of the internal world is derived by hypothetical reasoning from our knowledge of external facts.

2. 2. We have no power of Intuition, but every cognition is determined logically by previous cognitions.

3. 3. We have no power of thinking without signs.

4. 4. We have no conception of the absolutely incognizable.

It is worth citing Peirce's defence of the fourth and final claim at some length since in it he raises a number of issues that will be crucial in what follows. Essentially, in justifying his rejection of any concept of the ‘absolutely incognizable’, Peirce moves from an initial conception of the mind-independent thing-in-itself that serves as a sort of ideal original limit of cognitive experience to a terminal understanding of the mind-independent real as a futural projection made on the basis of thought's presumptive tendency to exclude progressively the idiosyncratic aspects of individual cognitions. In other words, he orders the conceptual dynamics of thought itself in terms of a continuum or open interval stretched from the minimally to the maximally general, yet comprising neither endpoint within it.