The meaning of quantifiers

Quantifiers very likely are equivocal. These differences in meaning are revealed when they occur within an intensional context. There are universal quantifiers which express the sum total of the instantiations (which must pertain to a finite or constructible domain) and those that do not (for instance mass nouns). There are quantifiers that are subordinate to relations of intension. Also in our previous post we argued that quantifiers express the ability to understand and follow rules, to do computation. All rules can be expressed via universal monadic second and first-order quantification (which is Aristotle's implicit metalogic in the Topics). There are also quantifiers that express generic properties.

It seems very plausible that all quantifiers are implicitly limited to some domain (even for 'everything'), that unbounded quantifiers are meaningless.  We should focus specifically on nested quantifiers $\forall (x:A) \exists (y : B). P(x,y)$ and $\exists (x: A)\forall (y :B). P(x,y)$. The first has meaning: every A has a P, and plausibly that P can be found through a computatable function.The second has meaning: there is an A which is the P of every B, which plausibly can be checked.

Perhaps Aristotle took pairs (combinations) of natural deduction rules for quantifiers as primitive rules.  Notice how $\exists E$ and $\exists I$ can combine, the latter being 'in the middle' of the former.

It is plausible that the Mitchell-Peirce system of quantifier logic (1885) is not only technically and philosophically superior to Frege's Begriffsschrift (1879) but was developed earlier (by Peirce's student O. Mitchell).  Modern dependent type theory uses Peirce's symbols $\Pi$ and $\Sigma$. Or even the Boole-De Morgan relation calculus approach can make this claim. For instance $\forall x \exists y R(x,y)$ can be formulated as $RU \cap \Delta = \Delta$ where $U$ is the universal relation and $\Delta$ is the diagonal relation. $\exists x\forall yR(x,y)$ can be expressed as $UR = \Delta$. $R(a,b)$ is expressed as $(a,b) \in R$. So we have an open theory expressing first-order logic for monadic and binary predicates. $R$ being transitive is simply expressed as $RR \subset R$. But what about when we leave relations as in $S(x,y,z) = R(x,y) \& R(y,z)$ ? Although obviously we cannot define such relations in our calculus it would appear (check this) that every first-order sentence or monadic or binary relation over binary predicates can be defined in the calculus (sentences via equations). Also how would we write $\forall x(G(x,x) \rightarrow \exists yH(y,x))$ which is equivalent to $\forall x\exists y(\neg G(x,x)\vee H(y,x))$ ? Maybe $((G\cap\Delta)^c \cup H^{-1})U \cap \Delta = \Delta$ ?

In Leibniz's Non inelegans specimen there are certainly definitions which employ $\exists$. The containment relation (let us write it $A\leq B$) is defined (Def. 3) as 

\[ A \leq B \equiv \exists N(B = A\oplus N)\]

But consider a general situation in which we have a defined predicate $P(x_1,...,x_n) \equiv \exists y_1...y_n\phi(x_1,...,x_n,y_1,...,y_n)$. Then there are situations in which  a $\psi$ involving occurrences of $P$ can be transformed into an equivalent (or rather stronger) sentence of the form

\[\forall z_1.....z_m\psi'(z_1,...,z_m)\]

in which $P$ does not occur, being replaced by $\phi(x_1,....x_n,y_1,....y_n)$ where the $y_i$ occur among the $z_i$.  For instance using Skolemization or constructivist techniques. Perhaps this fragment of Leibniz could be expressed in open logic. Transitivity of inclusions might be expressed as

\[\forall N,M((B = A \oplus N) \& (C = B \oplus M) \rightarrow C = A \oplus N \oplus M) \] Maybe Aristotle's ekthesis had to do with producing universal constructive sentences which are stronger than the sentence to be proven. Open formulas (unsaturated lekta) are problematic (and they not just universally quantified in disguise ?). An important task is to reformulate the axioms and rules of modern logic to deal only in closed formulas (sentences).

As for metalogic or metamathematics we must never forget that all results are themselves necessarily formalizable within certain recursive-axiomatic systems (including the very same system as the object system). Thus metalogic and metamathematics is just the study of formal reflections of systems into each other as well as self-reflection.  However knowledge and computation in recursive-axiomatic systems presupposes always a logical system at least as strong as monadic second-order logic. 

Hume's Treatise contains the following logical observations (when discussing causality): from the fact that every husband has a wife it does not follow that every man has a wife. Let $M(x,y)$ be the marriage (husband-wife) relation between a man $x$ and woman $y$. Then $Husband(a) \equiv \exists x. M(a,x)$ and $Wife(a) \equiv \exists y. M(y,a)$. Certainly $\forall x. Husband(x) \rightarrow \exists y. M(x,y)$ and $\forall x. Wife(x) \rightarrow \exists y. M(y,x)$. But it does not follow that $\forall x. \exists y. M(x,y)$.

Computability, Logic and Mind

Addendum:   Husserl's Philosophy of Arithmetic which is actually a treatise on some of the highest categories and operations of the understanding, in particular the act of combination or synthesis.

We begin with the psychological characterization of that abstraction which leads to the (authentic) concept of the multiplicity, and subsequently to the number concepts. We have already indicated the concreta on which the abstracting activity is based. They are totalities of determinate objects. We now add: "completely arbitrary" objects. For the formation of concrete totalities there actually are no restrictions at all with respect to the particular contents to be embraced. Any imaginable object, whether physical or psychical, abstract or concrete, whether given through sensation or phantasy, can be united with any and arbitrarily many others to form a totality, and accordingly can also be counted. For example, certain trees, the Sun, the Moon, Earth and Mars; or a feeling, an angel, the Moon, and Italy, etc. In these examples we can always speak of a totality, a multiplicity, and of a determinate number. The nature of the particular contents therefore makes no difference at all. (Husserl, PA, p. 17 Dillard tr.)

Thus we must distinguish between concrete multiplicities and abstract multiplicities.  Husserl explores the the aspects of combination and synthesis involved, including syntheses of syntheses and so forth.

A Kantian question about computability and the mind

 
On order for the human mind to act in a way equivalent to a universal Turing machine, what is the minimal logic that needs to be supposed ? The statement of this question lacks clarity and precision - to furnish it will be our task in the future. Another form of the question: what is the minimal logic required by the human mind to be able to follow rules of any complexity (discarding limitation of space and time) ?

It is interesting to consider monadic second-order logic (MSOL) (because of its well-known connection to automata theory and recursion theory), but specially when considered as an extension of monadic first-order logic only (and not full first-order logic as is more common), as in J. R. Büchi - Weak Second-Order Arithmetic and Finite Automata. Mathematical Logic Quarterly, 6(1-6):66–92, 1960.

https://arxiv.org/abs/2301.10519

Algorithms to search for and check  proofs in axiomatic-deductive systems can all be implemented in MSOL. So in a sense MSOL is transcendental logic and the other logics are rule-based games (a posteriori). Thus multiple generality and non-monadic relations need not be considered philosophically fundamental; thus many arguments for the superiority of modern logic over ancient logic fail. Also it is very evident that Aristotle's metalogic in the Organon is second-order or even higher.

Kant and Computability Theory

It is strange that few have noticed that it can be strongly argued that the abstract concept of computability and its allied notions are a candidate for being part of the pure a priori necessary concepts for all our cognition and experience (Husserl seems to have anticipated some recursion theory in his Philosophy of Arithmetic).

We have the intimately connected triad of formal logic, arithmetic-combinatorics and computability theory.  To write and check a formal proof we already are deploying computability concepts. But to investigate computability notions we need formal logic and arithmetic. Computation, proof  and the sequence of the natural numbers share the ordered directed time-like quality (linear with branching possibilities). Note: we are not suggesting that computability exhausts human cognition. Also by computability we include all classes in the arithmetical and analytic hierarchies, etc. In a future post we will critique the denigratory use of the term  'mechanistic'  showing it does not hold water when confronted by a serious mathematical and philosophical analysis of the use of differential equations in science.  Computability theory seems very close to Kant's notion of rule and of an architectonic of reason. Church's thesis is a transcendental condition for the possibility of knowledge.

Computability has to do with prescriptive normativity (method) rather than mere general normativity (rules).

We wonder if Kant's realm of pure synthetic a priori intuition of space does not actually correspond to graph theory and combinatorics - and whether category theory, and specially higher category theory  are not best viewed from this perspective (cf. simplicial sets and cubical sets). Category theoretic diagrams have a a kind of dynamic nature - at least in the way they are commonly used and visualized - which recall Kant's schematism.
 

The Church-Turing Thesis, Kripke and Kant

If we consider the abundance of hypothetical and counterfactual elements embedded irrevocably in our linguistic discourse, then a possible worlds semantician might be inclined to view the existence of Kripke's rigid designators as the transcendental conditions for the consistency and intelligibility of our discourse about the world.  But here we wish to discuss a Kantian turn in a different domain. What is it exactly that it means to follow a rule, a set of rules, to play a game,  learn how to use a language, carry out a logical debate, or in general to engage in the world ? For a subject  or mind to do this, it must be computationally competent, in other words, (at least) Turing-complete.  Secondly, it must be able to do this cross-platform, in an indefinite number of domains.  Thus the Church-Turing thesis, like Kripke rigid designators, appears as a transcendental condition for the possibility of our engaging in the world. It also suggests the a priori nature of a basic but fair portion of arithmetic, combinatorics and recursion theory.

Algorithms and Numbers

When we investigate the concept of computability we necessarily require arithmetic. When we investigate arithmetic from a logical point of view it is inevitable that we consider computability or are lead to it. Arithmetic and computability are inseparable notions and it is likely that the Turing-Church thesis is tied to the categoricity of the theory of natural numbers N. When one defines a natural number object in a topos the universal property gives us automatically the notion of a primitive recursive function. The fundamental concept in recursion theory is that of partial recursive function, which can be embodied or implemented in a variety of abstract machine models such as Turing machines. Partial recursive functions, which are partial functions (partial because the computation need not stop) from natural numbers to natural numbers can themselves be codified by natural numbers. This allows us to have a concept of constructive functional, extending the notion of recursivity to all finite types over the natural numbers. There are several possible constructions such as HRO and HEO. This entire process can be generalised to the algebraic setting of partial combinatoric algebras. Intuitionistic arithmetic in all finite types HA, the terms of which form Gödel's system T, is an alternative to set theory, category theory and type theory for doing mathematics. It has a model in ZFC given by interpreting terms of a given finite type by means set-theoretic maps and the natural numbers. This model is extremely complex (it can represent the real numbers) and very little can be said about it in general. What is amazing is that models of HA like HRO and HEO pack this entire structure into the microcosm of N ! This gives us a vision of the mathematical universe in which, as for the ancient Pythagoreans, all things are number.
Tichý has written a monumental work on the philosophy of logic and mathematics in which computationally inspired formal semantics is given allowing a consistent rectification, interpretation and extension of Frege's Begriffschrift and Russell's Ramified Theory of Types. In this system (which we call T) the structure of an expression is isomorphic to the computational process it signifies. Thus an expression in T is like a syntactically correct program in some (typed) programming language. Its meaning is the process of execution (evaluation) of such a program, or rather, a certain abstraction of this. For instance we can give a description of how a certain function (in the programming sense) is executed for an undefined value of its argument. But it is clear that a lambda-calculus type of formalism (or functional programming in general) is the most adequate for this approach and it is not difficult to see why Russell's and Church's systems should figure prominently. We could attempt to formalise in turn such descriptions of computational processes, but this will just evade the problem again. We must stop at an extra-formal foundation in terms of definite mental-cognitive processes and acts. Tichý in his anti-formalist argumentation underestimates just how much mathematics has to be used and assumed to treat combinations of symbols in a relevant and interesting way (for instance proving soundness, completeness and consistency). Hilbertian metamathematics is not consistent with relativism. 

What is Kant's logic ?

In this note we inquire into the nature of Kant's logic in the CPR. More specifically, we investigate the formal syntax and expressive capacity of Kant's conception of 'general logic' and 'transcendental logic'. Clues are furnished by concrete examples given by Kant, by the expressions occurring in the numerous proofs throughout the CPR and above all by the famous and perplexing table of the forms of judgment A70-76 (B95-101). We also ask if there be an overall consistency in the structure of the logic. Maybe the logic introduced via the table is weaker than the logic implicitly deployed in other places of the CPR, that is, the table-logic cannot cope with the expression of multiple generality found in the formulation, for instance, of the analogies of experience.
The original version of the second analogy in A was: for everything that happens there is something which succeeds it, according to a rule. Alles, was geschiet (anhebt zu sein) setzt etwas voraus, worauf es nach einer Regel folgt.  This expression has a logical form  which exceed the capacities of the classical theory of judgment if this last be identified, as is usual, with monadic first-order logic.

The basic unit of Kant's logic is the judgment. Our inquiry is aimed at  the general forms of judgments. Attending to form means abstracting from content. But is can also mean taking heed of the function of the understanding involved - this is what delimits the transcendental logic from the general logic. It is an appeal to the understanding which allows Kant to justify the separate modes of singular and infinite judgment.

Was Kant conservative or innovative  in his logic ? There appears to be conflicting evidence in the CPR.
In a footnote Kant considers categorical  judgments and the four syllogistic figures as a mere fragment of logic, a fragment that does not take into account  hypothetic and disjunctive judgments (on p.78 (6) Hanna seems to forget that not all judgments are directly about relations of concepts; hypothetical judgments are first of all relations of judgments).

A crucial question is how  we are to interpret the table of the forms of judgment.

The table has four 'Titeln' (which we translate as 'rubrics') each having three modes. Thus there are 12 entries in the table.  Did Kant intend that any judgment must fall within exactly one of the 12 entries ? Or possibly more than one ? Does the table have some (perhaps) limited similarity to the inductive definitions of the syntax of formal languages, for instance, first-order logic ? These are the questions we shall investigate.

It is clear that judgments can fall into modes of more than one rubric and but that they cannot fall into modes of all rubrics at once. Also that evidently they can fall into only one mode of each rubric (the third mode of each rubric is already conceived as a kind of synthesis between the first two). For instance 'necessarily all men are mortal' would seem to fall into definite modes  of the rubric of quantity and of the rubric of modality. We can wonder about how Kant conceived the negation mode of the rubric of quality. For instance "all men are not mortal"  can mean that all men are in fact not moral or that some men are not mortal. In other words, is Kant considering judgment negation and term negation ? Do we have in this example a combination of two modes of the rubrics of quantity and quality respectively ?


Take the hypothetical mode of the rubric of relation.  Kant's example is 'If perfect justice exists then the obstinately wicked will be punished'.  Such judgments would seem to be of the form 'if $J_1$ then $J_2$' where $J_1$ and $J_2$ are themselves judgments. There does not seem to be further evidence in the CPR about whether $J_1$ and $J_2$ can in fact be any judgments (and thus we would have an anticipation of modern inductive definitions) or if there is some kind of constraint.  Anyhow someone may want to express the judgment that 'it is not the case that if perfect justice exists then the obstinately wicked will be punished'. It is difficult to see how Kant would have rejected this judgment which fits both in the rubric of relation and that of quality under the negation mode.  There is also the important question of whether the disjunctive mode of relation corresponds to our modern propositional disjunctive connective.  It seems that the table presents an attempt (perhaps not entirely successful) at an amalgam of the Aristotelic syllogistic (in its full modal version) and the full propositional calculus (which already received a sophisticated development in the Stoics).

We have shown that judgments can fall into modes of more than one rubric.  But it also seems certain that  no judgments that fall under certain combinations of modes of all four rubrics. For instance: what mode of quantity should we assign to hypothetical judgments ?  But Kant's explanation in B100 seems to suggest that the modality modes apply only to assertoric judgments (he speaks of an affirmation or negation).

There is no doubt that Kant distinguished between judgment and proposition Satz. Judgments involve intentional attitudes to propositions. Let us see how B100/A75 can  refine our understanding of the hypothetical judgment. Kant writes that both the antecedent and consequent are problematic but the consequence relation itself is assertoric. Similar considerations seem to apply for the disjunctive judgment. This suggests that hypothetical judgments cannot themselves fall within the problematic mode and thus it is difficult to see how Kant would have interpreted Aristotle's modal syllogistic.

Another question: how does logical equivalence enter into the classification of the table ?  For instance is 'John is not not-handsome' affirmative or negative in quantity ?

Kant uses 'extension' of a term in its traditional sense. And at least as far back as Porphyry's Eisagogê we a correlative to extension called 'comprehension', the accumulation of differences applied to a given genus defining a species. The greater the comprehension the smaller the extension and vice-versa. For Kant individual terms have no extension, they are Begriffe ohne Ausnahme.

Now comprehension determines extension, but this is a many-to-one relation. Thus a synthetic judgment, as explained in B11,  must be in a certain sense 'extensional', that is, equivalent to a relation between extensions such as: the extension of term A is contained in the extension of term B, even though this is not deducible from the comprehension of A and B as in 'a rational animal is an animal'.  Compare how property and inseparable accident differed from definition in Porphyry and Aristotle.


Consider a judgment such as 'if the world is infinite and the world is eternal then the world is eternal and the world is infinite'.

This is an 'analytic' judgment which does seem directly related to the comprehension of terms or conceptual 'containment' of terms.

In conclusion we find that the determination of the exact formal nature of Kant's logic in the CPR - specifically with regards to  expressive power and formal nature of the syntax - is a difficult open problem.

We propose that various interpretations be weighed by the aptness to express key judgments in the CPR (the work \cite{lam} would appear to be relevant to this project). Of course this is being  charitable to Kant as the insufficiency of his logic could be taken at face value as a simply a flaw, specially if we consider how the table of categories derived from the judgment table permeates and guides the entire architectonic of the CPR.


Consider the 'analogies of experience'.  The original version of the second analogy in A was: for everything that happens there is something which succeeds it, according to a rule - Alles, was geschiet (anhebt zu sein) setzt etwas voraus, worauf es nach einer Regel folgt,

\[\forall x (Hap(x) \rightarrow \exists y RegSuc(x,y))\]

At first sight this would seem to transcend the expressive capacities of the logic defined by the table of the forms of judgments.

But we note that according to Bobzien and Shogry Stoic logic could handle this. In their view the Stoics would have expressed it in a regimented expression employing anaphora thus

if something happens then something follows it according to a rule.

The section on Subjective Notion in Hegel's 'Encyclopedia Logic' has a structure which appears very 'conservative' in that is does not seem to differ much from the core conceptual architecture of Kant's logic (although of course Hegel's interpretation and agenda is vastly different). Perhaps Hegel's treatment could shed light on some details of Kant's logic in the CPR.

What would the table of the forms of judgment be in modern terms, if we considered modal monadic first-order logic ?

We could say that any sentence of the logic is in exactly one of the following forms:

(1)$\exists x \phi$, (2)$\forall x \phi$, (3)$A(c)$,  (4)$\neg \phi$, (5)$\phi_1\rightarrow \phi_2$, (6)$\phi_1 \vee \phi_2$,  (7)$\square \phi$, (8)$\lozenge \phi$.

where the $\phi$ and $\phi_i$ can in turn be any of the 8 types. Here $A$ is an atomic predicate and $c$ a constant. Evidently we can organize these in a sequence of pairs corresponding to Kant's 4 rubrics. In classical logic these can be reduced to 5 forms. If each of $\forall$ and $\exists$ can be defined in terms of the other using $\neg$ then which one is to be considered a 'primitive' concept of the understanding ? William Lawvere might have answered: none of them, but rather the concept of adjunction of which both are a case of. We retort: the concept of adjunction depends on both quantifiers for its intelligibility. Furthermore: this problem can be seen simply as a strong argument for the primordiality of intuitionistic logic (in which such a problem does not arise) over classical logic. A topos is naturally intuitionistic, the classical version is a special case. If Kant's antinomies raise many difficulties for a rigorous formal interpretation, perhaps the profound metalogical thought behind them was simply that $A \vee \neg A$ is not a law of reason in general. Thus there need not be a proof either than the world infinite or that the world is not infinite.

Find evidence for second-order monadic logic in Kant.

Notes on Transcendental Subjective Idealism

There is mind-stuff and there is non-mind-stuff, for instance some kind of Fregean objectivities or natural world objects. The problem is clarifying exactly what we mean by 'mind-stuff' (consider for instance Frege's view of the 'subjective' component or thought).  There is a lot of different mind stuff and this stuff is not easy no sort out or define - but it is most certainly distinct from object-stuff though it is object-like in its cognitive-linguistic referentiability. But besides all that there is stuff which is neither mind-stuff nor the objective kind of non-mind-stuff referenced above which we call object-stuff. We call this the self-stuff which is the great object and greater mystery of philosophy. Up to now only a negative definition, we know almost nothing about it.  Let us say that is pertains to the impression of the 'I' , 'me', 'mine' or 'self' . The self-stuff is certainly in relation to mind-stuff and mind-stuff is said to be intentionally directed to object-stuff. All these 'categories'  of things are tightly connected.   If we can say that the objective world manifests itself in or through mind-stuff. But the self-stuff also manifests itself through mind-stuff.  Kant can be read as saying: the objective world could not manifest without the help of the self-stuff. We can think of self-stuff not as an object but as a tendency, a power, a force (mine-making, I-making) that constitutes itself in another in the likeness of an object. Mind-stuff is self-stuff imbued. Self-stuff is usually mind-stuff imbued. This self-stuff pseudo-object is not an inter-subjective object in the way of object-stuff and never can be such.  It is an object only to itself usually through the medium of mind-stuff.  The self-stuff object subsists not through the act of knowing but through ignorance. The usual self-stuff self-presentation object is the archetypal illusion and delusion but not itself self-stuff  which can be seen as tendency, power or force - which is itself not mind-stuff but 'contaminated with' and 'identified with' and 'fueled' by  mind-stuff.  These considerations are all of the uttermost subtlety and difficulty and this is only the tip of the iceberg.  Indeed we have not even scratched the surface of what could be said about self-stuff about the limit-category beyond object-stuff and mind-stuff. The immediate use  of this system of categories is as a means of classifying and clarifying philosophical concepts (such as 'consciousness' or 'subject') and systems  such as various forms of idealism and realism. Let us just say for now that saying that object-stuff is only real and mind-stuff is an illusion or can be reduced to it is sheer and utter nonsense. Likewise saying that all object-stuff is not real and can be reduced to mind-stuff is highly dubious (this supposing we are keeping silence about self-stuff and its sphere of influence). For instance we must avoid solipsism and acknowledge that there is a universally binding moral law. Higher philosophy begins when begin to make an inquiry into and criticism of our personal first-person impression, experience and concept of self or ego or identity. In particular trying to phenomenologically grasp it: what exactly is this, what do I mean by 'I' ? What is present in my consciousness ?  This domain seems also to furnish to foundations for moral philosophy.

A note on 'neuro-idealism'. It is claimed that we perceive the world and act on the world exclusively through the efferent and afferent nerves (spinal cord + cranial nerves + ...). That is our perception and acts are through coded signals in multiple parallel channels. Now:

1) the concept of signal and parallel channel assumes the concepts of space, time and causality.

2) what is the decoder, what is its origin and how does it work ? How is it even possible ? 

3) some kind of minimal decoder would be needed at the beginning to bootstrap the whole process....we can conceive the instructions to build a more complex decoder as transmitted from the world...

4) it dos not seem that space, time and causality were learnt.

5) mind-stuff is like an UI to a self-stuff. A high level symbolic abstraction of the object-world. But all this is itself symbol. The scientific worldview is itself symbolic and provisional.

6) meaning understood objectively...how can 'this chair' in the mind be connected to a definite space-time region of energy and matter ? The latter is not what is 'meant' although it is a reference in some way...reference perhaps is something entirely objective and not determined by the mental-stuff process of 'meaning'. Reference has to be contemplated from without.

6.1. 'The chair...'  and 'The ripple in the lake' : think about the vastly different meanings and meaning-structure. Who ever gave a ripple to someone or put a ripple in their pocket ? But scientific worldview 'objects' are like ripples (timelessly conceived).

7) and yet everything in 6) is mental-stuff and meaning ! Also we do not know much about 'energy' and 'matter' (just enough for technology and weapons) since unfortunately our physics took a wrong turn and/or stagnated.

8) There is syntax and semantics. The syntax cannot in itself reach out to the world of semantics. But it can represent and talk about semantics and a relationship to semantics. Just so the scientific worldview is coded and represented within the common-sense life-world.

9) Learn to question, question, question and learn the humility of knowing your not knowing.

New Update (10/9/2024) : we have had radical new insights about the matters discussed below. The discussions below are not quite right,  not precise enough or incomplete if not wrong.  The new insights need to be expounded with great caution and rigour.

Update (10/9/2024): there are serious dangers, ambiguities and confusions surrounding subjective idealism which are impossible to avoid.  Both the individual psychological self and the phenomenal world must be considered illusory and relative at the same time, at once. There is no question here of a psychological phenomenism. An enlightened sort of cosmic realism, in the Stoic sense, can be equally a good way to overcome the illusions of the individual psychological self (Man cannot bear too much reality).   There is confusion here with the term 'subjective'.  'Realism' is also a very vague term. For it is not a denial of realism to posit that reality has infinitely more dimensions and modes of interrelation than given immediately to the individual psychological self and its ordinary categories of experience. Hegel was right to see that consciousness and its world must evolve together - and this does not involve denying the reality of a world even if ultimately becoming a higher form of consciousness itself.

Or better: consider the following interpretation couched in terms of the allegory of the cave:

The prisoner's spectacle of the shadows is the 'phenomenal world'. But not 'phenomenal' in the sense
of a mere chimera, a meaningless, arbitrary delusion. The dance of shadows is an 'illusion' but a special kind of 'illusion' which implies a intelligible real causal link to something else beyond the shadows that is indeed real - the objects of which the shadows are shadows of. The objects are real and the process of projection, of casting a shadow, is real and part of the rational order of the world of objects and the light source. To say 'the shadows are the objects' or 'the objects are the shadows' is false. But rather the shadows point to the objects and so as thus they are 'manifesting reality'. Kant's transcendental aesthetic describes the screen, the wall of the cave on which the shadows are projected. The categories reflect the condition and position of the prisoners.

Perhaps 'subjective idealism' (whatever one's position regarding it) is a dangerously ambiguous term because it could be interpreted as : the shadows are projections of the prisoners themselves !

My (broadly Hegelian) position is: there is an objective world more real, more intelligible, different yet connected to the one we ordinarily perceive and there is a form of consciousness and intellect more accurate and truthful than the one we usually employ - which is correlated to our usual manifest world. But this trans-objective objectivity and trans-subjective subjectivity does not contradict the relative validity (within their own limits) of the initial subject and its object. 

The great object of philosophy is the self, the criticism and analysis and inquiry into the self and the operations, projections, constitution, identifications, dynamics, appropriations of the self. From thence must follow the moral law.

(end of update)

How are we to understand the analysis of experience ? Is the analysis of experience the same as the analysis of consciousness ? What is the ultimate goal of such an analysis, is it purely theoretical (like for any other science) or does the goal contain something else in addition, for instance an integral transformation of human consciousness and life-experience itself ? And how precisely are we to understand such an integral transformation which would seem to have roots in classical philosophy ? Arguably some traces of this goal are to be found in Kant and Husserl, perhaps more in the ethics-centered Kant than in Husserl where a transformation of consciousness occurs in a more methodological context rather than as a goal itself.
For us the analysis of consciousness and its goal will be centered on the concepts of 'illusion', 'delusion' and 'suffering'. These concepts permeate and play a far more fundamental role in western philosophy than is generally realized or acknowledged.
For us the goal of the analysis of consciousness will be knowledge which dispels the natural illusions and delusions of consciousness as well as the bringing of an end to suffering. These two goals are connected. One way to look at the illusions and delusions of consciousness is through the phenomenological concept of 'claim-of-being' or Hume's theory of habit and association. Consciousness somehow presents to itself objects which embody claims and beliefs not immediately justified by what is immediately given in consciousness itself. Rather these objects are saturated with the structuring and coloring and association of mental material which contribute to objectification in the realm of being.
Another central aspect of our analysis of consciousness will be importance of the human body. This is a very subtle and nuanced concept with which we must be careful to steer away from simplistic generalizations or physicalist dogma. Our approach to the human body involves the first-person experience of the body, that is, the body qua fundamental component of natural consciousness and experience. Any analysis of consciousness must involve an analysis of body-experience and body-consciousness. The overcoming of delusion and suffering must involve the analysis of body-consciousness and very importantly a special shift of focus and attitude towards this body-consciousness. The subtlety lies in that on one hand we are deluded in our ordinary experience by forgetting the essential conditionality and foundational role of body-consciousness for the whole architecture and dynamics of our consciousness, and on the other hand body-consciousness is ultimately itself delusion and mere claim-of-being alongside other objects of consciousness. It would be interested to compare this to Schopenhauer's parallel considerations on this paradox.

What are the most general divisions or analyses that can be made of consciousness as a whole ? The division in the Pali suttas into the five khandas can been compared to analysis of cognition and experience in Kant, specially bearing in mind their close dependency. Viñanna has the characteristics of Kant's transcendental unity of apperception. And yet the suttas speak of an infinite, absolute, all-illuminating consciousness as well. Kant makes an analysis of the different concepts of 'subject' and 'soul' (the empirical ego) which are in his view confused in the paralogisms of reason. But he also introduces the 'intelligible character' unknowable and yet the foundation of morality and freedom.
We might say that satipathâna (the setting up or establishment of mindfulness) is the central methodology of the process of analysis and integral transformation of consciousness in the Pali suttas, playing a role parallel to Husserl's transcendental reduction or epokhê. And yet, unlike for Husserl, we are left with the question: what is the subject of cognition of satipathâna ? Just as for Husserl, it is patently not a naturalistic ego or the ordinary cognitive subject. Instead of trying to answer this question directly let us consider Hegel's theory of the beginning and method of philosophy. Hegel does not take the ordinary empirical ego as a foundation for knowledge; the empirical ego is considered somewhat 'illusory' and needing to be overcome or led first to the state of absolute consciousness which is no longer the standpoint of the said individual empirical ego. The idea of a pure logic, a pure consciousness, a pure cognition, etc. which many associate with Husserl are all found in Hegel. If Hanna argues that Kant should be considered the father of modern anti-psychologism then in Hegel it certainly had arguably one of its most remarkable developments. Note also the almost Zen-like utterance at the beginning of the Greater Logic: the purpose of this introduction is to convey the point that Knowledge needs no introduction or beginning. But let is return to Kant. The suttas point out a path for overcoming the dualism and limitation of concepts and conceptual proliferation (papanca) yet express important insights and truths in an evidently conceptual and logical form. All the khandas are mutually conditioned and dependent. In the same way Kant reduces and relativizes the understanding to phenomenality and yet admits a transcendental critical use of reason which can attain definite knowledge of the structure, conditions and dependencies of phenomenal cognition and consciousness. Also though the noumenon is unknowable by the understanding it is thinkable. In the suttas there is the concept of truth and knowledge regarding an unconditioned which transcends the khandas. There is also the analogue of critical knowledge (vipassana, pañña), an attainable understanding of how the khandas originate and are mutually co-dependent (paticcasamupada).

But what is satipathâna, or more generally, what is the method for the analysis of consciousness and experience ? We will not attempt to give a complete answer. Rather we can safely say than an important aspects involves the transformation of consciousness's own consciousness of it itself, its own self-relation. While all consciousness arguably involves an essential element of self-reflection or self-directedness, this transformation involves the relationship of consciousness as whole to itself. Consciousness must become aware of itself as it really is (self-transparency), wherein it is presented naked, immanent, here and now, in its dynamics and structure without assent or belief in its ontological projections. Consciousness becomes a consciousness of consciousness itself qua immanent stream of 'thoughts', anchored in the here and now of internal experience, but ontologically projecting - in claims-of-being- indefinitely into past and future and possibilities. The awareness of the total sphere of thought from the perspective of the here and now and contentment, is the true transcendental viewpoint in which the primordial role and nature of temporality (neglected in ordinary naturalistic consciousness) is clearly discerned. We will return further ahead to the structure of this stream of 'thoughts'.

The goal of overcoming illusion, delusion and suffering depends on both the analysis and coming face-to-face with this primordial thought-flow it its pristine originality and giveness, and on the possibility of conscious suspension of this flow itself which implies a radical in-folding, self-folding and inversion of the deep structure of consciousness. The idea of such a possibility is found in classical and Hellenistic philosophy as well as, in a certain partial form, in Hegel and Schopenhauer.
In this process we obtain direct intuitive knowledge of the deep structure of consciousness, how components apparently distinct and independent in ordinary consciousness are unified in a common essence and origin. For instance 'ego', 'thought' and 'will'.
Ordinary thought is like a whirlwind or stormy sea, opaque and extremely difficult to see and analyze and much less to control and ultimately suspend and calm. This is where the beginner's stages of satipathâna come in. The body, the breath, concentration of the present moment, the here-and-now, contentment, letting-go and so forth, are so many powerful anchors leading to calm and self-transparency of consciousness. But all these anchors are far from sufficient, they are no substitute for transcendental philosophical insight. That is, seeing consciousness as it really is. Seeing the world as immanent in consciousness, and consciousness as temporally conditioned, as subjective, as filled with being-claims but no actual being-verification. Seeing that thought is all about make-believe projections into past and future and what is far away but is really just here and now and subjective. What is really here, what is really is here now: the stream of mental phenomena and their projections or seemings. This was Hume's great discovery. The world turns out to be an immanent world, the stream of consciousness and its percepts (i.e. elements of sensory-data compounded and woven in consciousness).

We find the common root and unity between the will, reason and the ego-self-positing - and then the unity between will, reason, ego and the world-consciousness. The illusion of world and self are dissolved by insight-analysis. By the focus on the here and now. What are bliss and freedom but the abrogation of the ego-self and with its will, reason and its world ? Bliss and freedom are characterized by mastery over the illusion of space and time, by the power of ubiquity and timelessness.


Consciousness is woven on the senses and specially inner sense, imagination, recollection, anticipation, etc. The beginner's cultivation of satipathâna is sharpened on the body, then the breath, then outward perception and finally inner perception thereby leading to inner silence (corresponding to hearing) and inner emptyness (corresponding to sight). The process must begin with the body and breath and with awareness of general processes of nature as pertaining to the body as manifest in transience, compositeness, etc.

We now consider the important question of the role of psychology in the process of analysis and integral transformation of consciousness. Psychology can be seen as non-philosophical or at least less-philosophical analysis of certain more concrete and well-known aspect of consciousness. But psychology is not only theoretical, it also aims to be practical, therapeutical, to effect beneficial permanent change on consciousness itself. Just as the body-consciousness has been wrongly neglected in much of the western philosophical tradition, so too has the vital integration of psychology into philosophy: that is, a certain psychology is a vital methodological component of an integral analysis and transformation of consciousness, as long we avoid 'psychologism' or confusing consciousness with the ordinary empirical ego, as we saw above. Phenomenological consciousness must first be an ordinary psychological consciousness focused on feelings as feelings, mental states as such and the 5 hindrances accompanied by some insight analysis and their active uprooting, examining consciousness in function of obsessions, attachments, ill-will, etc.Consciousness is so complex and vast that it is important to know where to begin, where to look at first, what to do first. Only then can genuine philosophical phenomenology and awareness of thought as such take place. This was an error of many past philosophers, neglecting psychological preparation. We now consider a philosophical-psychological analysis of consciousness relating to the overcoming of attachment and obsession. Consider an object which the mind is deeply attached to or obsessed with. This object dwells in the mind and is constantly intentionally directed at and becomes the self-reinforcing center of an entire growing web of associations. But what is this object really, immanently and purely in consciousness as such ? Nothing more than a complex web and circuit of recollection, anticipation, etc. all woven from percepts, units of recollected or imagined sensory input. But there was a point in time, a moment, in which a different kind of percept occurred, the full act of perception here and now, the spark which occasioned the explosion of a continuous stream of reverberation and self-amplification in consciousness. Yet the object itself is habitual self-reinforcing web of recollection and anticipation, imagination, variation, recombination, based on 'faint copies' of a sparse moments of full perception. Thus the 'object' of attachment is both an illusion and a delusion. In the suttas when the 'mind' (manas) is considered as a 'sense' then it is to be understood as pertaining to the inner reverberations and conceptual articulations of raw sense-perception data. What about a world outside of consciousness, a thing-in-itself, a hypothetical cause for the full act of perception in the here and now ? We will not go into this deep question here. We merely remark that this hypothetical 'real world' cannot be changing or in time, it cannot have a 'now' and 'afterwards' - for these concepts only make sense as immanent in consciousness.
Our philosophical claims are that there are beyond any shadow of a doubt multiple different consciousnesses many of which are caught in the same web of illusion, delusion and suffering as our own consciousness. We reject solipsism. The moral categorical imperative is to realize that you and other beings are in such a web, to strive to liberate both yourself and others from it. The first steps are following the universal duty of compassion (see the work of S. Shapshay on the compatibility of Kantian and Schopenhauerian ethics) and teaching and arguing for transcendental subjective idealism.

Category Theory and Philosophy

Prolegomena to a future logico-mathematical metaphysics

The pure categories (captured by higher order categorical logic, etc.) must be unfolded and specified via schematism and regional ontologies. Such can only be geometric and geometro-modal-dynamic although in a suitable categorical framework.  On the other hand a formal axiomatic philosophy (a theory of 'manifolds') can be obtained via abstraction from certain geometro-modal-dynamic frameworks found in applications. By modal we mean everything pertaining to phase spaces, configuration spaces, moduli spaces, the only right way to treat modality.  These geometro-modal-dynamic frameworks are not only found in mathematical physics or mathematical biology but in the study of concepts, in semantics, in mathematical linguistics. The question is, can we follow Husserl and Gödel this way to study all possible worlds to all possible minds ?

Lawvere's theory of smooth toposes and their use for formulating differential geometry and differential equations can be seen as an unveiling of fundamental synthetic a priori categories and principles. The adequacy and conformity to scientific experience (in the spirit of critical idealism) is a vital dimension in the philosophical deployment of category theory.

The sequence seems to be : topos $->$ ringed topos $->$ lined topos $->$  smooth topos.  Our task is to elucidate the phenomenological and categorical (in the philosophical sense) meaning of the concept of topos.  Also to understand why the internal version of the concepts of commutative ring, commutative algebra, linear map (cf. Weil algebras and the Kock-Lawvere axiom) and an infinitesimal version of simplicial objects (used for defining the dg-algebra of differential forms) appear to be of so fundamental and vast a scope as a priori conditions for mathematical physics and other branches of natural science. And how are we to understand Kant's concept of space in the light of synthetic differential geometry, it being  point-free ? Seeing a vector field on $X$ as an infinitesimal deformation of the identity map on $X$ seems very intuitive.

Kant's theory of schematism could be interpreted in particular as implying that any line-shape $R \rightarrow X$ must be seen as a solution of a differential equation; we give the Weil algebra a generative dynamic interpretation.

Let us look at the first two antinomies of pure reason in the transcendental dialectic, involving time and composition. The point-free nature of space is presupposes for the argument to work. Can causality involving a previous moment of time be captured in terms of the infinitesimal path $D$ via a prolongation principle ?  The infinity which cannot be object of a synthesis appears to be best captured by a set dense set for $<$.   Otherwise why cannot for instance the points $\frac{1}{n}$ be objects of a synthesis ?

Kant appears to be saying that every state of the universe must have a temporally previous state but at the same time there cannot be infinitely many previous states to a given state.  

We must compare the antinomies to various intuitionistic principles and classical principles not intuitionistically valid.

It could be tempting to see a sheaf as a derived concept relative to an abstract bundle (which is a very Kantian notion). Sheaves are sections of some bundle. But we must check if this carries over for Grothendieck topologies too, that there is an equivalence of categories between sheaves over a site and étale bundles defined not in terms of ordinary topology but in terms of Grothendieck topologies (as in étale cohomology).  However there are other entities in geometry which are more naturally seen as sheaves than as a bundles: for instance the sheaf of continuous or holomorphic functions. A sheaf is essentially a phase space, a space of phenomenal possibilities which expresses how these possibilities flow locally and cohere (se our post https://chryssipus.blogspot.com/2024/06/cognition-and-states-of consciousness.html). But in actual situations the number of different sheaves (over a given space) is very definite and determinate. For instance, sheaves of smooth functions on a manifold and smooth $k$-forms and other functions relevant to analysis. Many of these actual sheaves form natural complexes, so much so, that the complex itself can be seen as adequate kind of generalized sheaf (cf. the derived category).  Since it is the cohomology of the sequence that mostly of interest, complexes are identified if they have the same cohomology : this is the basis of the derived category constructions which turns morphisms of complexes module homotopy into 'fractions' where  in the denominator quasi-isomorphisms are inverted.

But how are actual sheaves, sheaves of concrete interest, 'generated' , beyond the basic ones discussed above ? By functors generated by continuous maps $f: X \rightarrow Y$.  Of great interest is the study of neighbourhoods of fibers $f^{-1}(x)$ as $x$ varies.  $Y$ is often seen as parametrization space or base space. Hence the presheaf  on $Y$ given bt $V \rightarrow \Gamma(f^{-1}(V), F)$ for $F$ a sheaf on $X$. This carries over to a derived functor $Rf_{\star}$ taking complexes of sheaves on $X$ to complexes of sheaves on $Y$ by which is studied the cohomological variation of the fibers along $Y$. This is a main source of the generation of interesting sheaves, the sheaves used in practice.  Another source of is the functor $f^{-1}$ by which is studied the cohomology in an infinitesimal neighbourhood of the images of open sets via $f$. These two functors are abstract versions of integration and differentiation. Another important operation is the "restriction" composition $R_\star j j^{-1}$ for an inclusion $j : V \rightarrow X$. This takes a sheaf $F$ and yields a new sheaf which, roughly speaking, takes into account only the nature of $F$ on $V$ or in infinitesimal neighbourhoods of $V$.

Thus a category of sheaves becomes interesting and intelligible by its relation to other categories of sheaves.  So considering at once the 2-category of sheaf-categories, or more generally, toposes, is very natural and imposes itself in the nature of things. The adjunction between toposes and locally presentable categories is discussed in Marta Bunge, Aurelio Carboni, The symmetric topos, Journal of Pure and Applied Algebra 105:233-249, (1995).

Having a whole category of sheaves leaves a vast amount of elbow-room. A category of sheaves represents a spectrum of different spaces of possible manifestation (cf. how locally constant sheaves can be identified with covering spaces). The category of sheaves over a given topological space represents every mode of phenomenal possibility space of that space - thus in a way the category can be identified with the space itself.

A remarkable property of sheaves is their homogeneity for scaling. Given an open set $U \subset X$ we get automatically from a category of sheaves on $X$ a new category of sheaves on $O$, $\Gamma_U : Shv(X) \rightarrow Shv(U)$.

Categories do not apparently have the vertical hierarchical structure of the classical genus-species classification. For instance: group is a species of monoid and abelian group is a species of group. There are corresponding categories Mon, Grp and Ab which form a chain of subcategories. Our construction from our paper can be interpreted in terms of successively taking equivalence classes of equivalence classes of equivalence classes.  In category theoretic terms this translates as a sequence of categories $C_0, C_1,C_2,...,C_n$ and a sequence  $F_1, F_2,...,F_n$ of sets of morphisms in $C_{i-1}$ such that $C_i$ is the quotient category of $C_{i-1}$ via $F_i$.  Consider how a given infima species might be described in terms of a protoype $P$, membership  of an object $X$ being ensured by the existence of a deformation $f : P \rightarrow X$. We can also think of an interpretation in terms of $\infty$-groupoids but it is more subtle; it is a top-down approach using connected components and $n$-contractibility.

But sections of a sheaf are like individuals of a given infima species. The category of sheaves is like the genus of the infima species. Then the 2-category of (small) sheaf categories is like the yet higher genus of this genus. Functor categories are like the category of relations.

So: section $->$ sheaf $->$ sheaf category $->$ 2-category of sheaf categories. Aquinas (in De Ente et Essentia) views genus as a space of possibilities rather than as a minimal matter to which difference is added as a form. Difference determines or 'picks out' a latent possibility of the genus. This agrees with the concept of sheaf and category: we choose a section of a sheaf or a definite sheaf in the category of sheaves.

Formal axiomatic metaphysics in the sense of Zalta all depends on a multi-valued logic having truth-values $\Omega^W$ were $W$ represents possible worlds.  Or rather, consider that $W$ must be endowed with a topology or Grothendieck topology so we can consider $\Omega$ for $Shv(W)$.

Thus we have a fundamental axiom of metaphysics: the set of possible worlds must be endowed with a topology. The idea of alternative situation always depends on a more or less strong continuous deformation of the actual world. If a sentence $P$ holds at a world $w$ then it must hold in some neighbourhood $U(w)$ of $w$. Thus it is natural to capture modal logic by the topos of sheaves over $W$, $Shv(W)$.

To do: in the section on understanding in the phenomenology of spirit there are many interesting considerations about 'force' and exteriorization and the unity of motion determined by a law.  The germ of a sheaf is the unity of between interiority and exterior manifestation (for there is not definition for the point itself independently of its neighbourhood).   Let $W$ be a category $\mathbb{W}$ with a Grothendieck topology.  Then take modal logic to be a functor $ m: \mathbb{W} \rightarrow \mathbf{Cat}$.

Topos theory is (very roughly) the study of how logic coheres and varies with space and time and  possibility. All predicates $X\rightarrow \Omega$ have a domain which is a 'type' $X$.  But in general $X$ is a sheaf. Types as spaces can be expressed as 'types as sheaves'.  For instance a type parametrized by possible worlds. Thus predicates are in this case coherently parametrized according to possible worlds. But notice how in general the objects $X(U)$ can vary for $U$ in the base category. If $X$ were an atomic type of sets of individuals then the actual individuals could very according to the possible world. An equivalent way to see this is as the subobject fibration in which we view propositions on $X$ in correspondence with Sub(X).

Zalta's encoding could be descriped as a morphism $enc_X : PX \rightarrow PX$.

Jean Hyppolite lays emphasis on Hegel's positing of  'the identity of identity and difference'. But it is difficult not to think here of the univalence axiom: 'the equivalence of equivalence and identity'. Hegel's logic, despite this going against the surface intention of Hegel himself, may well be capable of a formal axiomatic treatment. This will depend on a proper account of identity and equality.

In Jacobs' Categorical Logic and Type Theory there is the idea of giving a categorical semantics for untyped lambda calculus ($\lambda$-categories on p.155-156) related to Scott's reflexive objects. We take an object $\Omega$ in a Cartesian closed category for which $\Omega = \Omega \rightarrow \Omega$. This expresses that $\Omega$ has a mediation within itself, is self-mediating. Is in and for itself.

Cognition and States of Consciousness

Husserl wanted us to develop a state of consciousness which also, of course, has a cognitive aspect - indeed the cognitive aspect might be seen as its raison d'être. But it is more than this. A state of consciousness implies a permanent habit, a transformation of character. Both Husserl and the oldest Buddhist texts dwell on (analytic) insight, disidentification, suspension and distancing (abgeschiedenheit).

If conscious experience is normally present in unreflected 'globs' , the goal of analytic insight is to unmask and be continuously aware of the ternary structure present in consciousness $\bullet \rightarrow \bullet$ and its subsequent higher-order unfoldings.

We mentioned before the archetypal structures and processes of consciousness. Here is an incomplete tentative list (with an implicitly Kantian basis):

Synthesis - gluing, covers, the sheaf-condition = extensibility on $j$-dense objects for a topos with a Joyal-Tierney topology.

- different orders of wholes (higher groupoids)

Self-reflection - a system which can represent (partially at least) higher order aspects of itself within itself.  This is the original synthetic unity of apperception = I know that I am knowing. This is found in recursive definitions, inductive types, the successive powers of the $\lambda$-cube wherein external aspects of the system become internalized and internally represented, also the subobject classifier, truth-value object $\Omega$ in a topos. See also our post on the meaning of the logical connectives.

Return-to-self, that is, Kant's trinary structure in the CPR.  This is related to the negation of the negation, double negation as the third (synthesis).  In topos theory this relates to the dense topology and in particular to forcing.  The idea is simple. In rough terms it is as follows: consider $U\Vdash\phi$ as signifying that the sentence $\phi$ holds in region $U$. We define $U \Vdash \neg \phi$ if $\phi$ does not hold on any subregion $V\subset U$.  Then $U\Vdash \neg\neg \phi$ means that for any subregion $V\subset U$ we choose we must have that there exists a $W\subset V$ such that $W\Vdash \phi$.

Double-negation can also be connected to temporality: something must pass to reveal itself, ti to einai, quod quid erat.

But this is assuming a static consciousness, a fixed state of consciousness with its corresponding archetypal structures and processes. But what about the transformation into other states (such as found in Schopenhauer and Hegel) ? Do the archetypes change ? Or must we find further higher-level archetypes that govern and characterize this transformation ? To self-reflection we should add self-negation and self-transcendence whereby the correlative self of consciousness is abrogated and transposed to more universal and wide-encompassing modalities and states.

Kant also had a Leibnizean dream, a complete axiomatic-deductive system of the pure a priori concepts and principles of the understanding. What is not clear is how he envisioned deduction and the interplay of the analytic and the synthetic.  Could the synthetic be exhausted in a finite set of axioms and all the rest be entirely analytic, Frege-style ? How could Kant explain that in mathematics there is often a convergence between intuition and formal deduction ?

The history of transcendental idealism is yet to be written, specially as regards to the time between Kant and Husserl. Schopenhauer, Von Hartmann and Spir are far more important than Fichte or even Schelling. Tolstoy wrote of Spir: "reading Thought and Reality has been a great joy for me. I do not know a philosopher so profound and at the same time so precise, I mean scientific, accepting only what is strictly necessary and clear for everybody. I am sure that his doctrine will be understood and appreciated as it deserves and that the destiny of his work will be similar to that of Schopenhauer, who became known and admired only after his death".

We can view Husserl' transcendental subjective idealism and Fregean-Leibnizean objective platonism as not mutually opposing by complementary or at least compatible. Also these two need not be considered exhaustive of reality,  as an important place should be given to ethics, to philosophy of art and to naturphilosophie and above all the practice and psychological basis of meditation (higher ethics).

Theory of theories

Given a theory, a systematic theory, we can analyze i) its intrinsic logical-conceptual structure, ii) the process by which a person comes to learn and understand the theory, and iii) the historical or personal biographical genesis of the theory (which of course can involve i) and ii) at previous times).

Regarding approach i) we can ask to what extent is the organization of the theory drawn by necessity and each logical step or 'development' (in an asynchronic sense) guided by implicitness or inner necessity ? (These considerations seem to have played an important role for Fichte and Schelling).

Speculation: can ii) and iii) somehow shed light on this question regarding i) ?  What is the relationship of this to Aristotle's distinction between things clear to us and things clear in themselves and his methodology of starting with the former ?

Speculation: can the study of biological organization or general systems theoretic concepts help with i) ?  What are the most important metatheoretic concepts we need to consider (for instance the idea of something external and ad hoc becoming internalized, the discussions in our post about reflection-into-self, etc.) . Category theory and categorical logic offer a very important paradigm and key. The bare concept of category (and higher category) functions like a supreme genus. As more properties are added these are mirrored in the nature of the internal logic. They way successive relevant properties emerge is certainly not arbitrary but seems to conform to basic meta-theoretic archetypes, if we are careful to unfold them in a gradual and ordered way.

But let us look at the processes and archetypes of consciousness (such as unification, return-to-self, negation, intentionality, etc.). How are they reflected in or determine theories ? Does the logical-conceptual structure of theories reveal the structure and processes of the mind and vice-versa ? The formula for Aufhebung $A \rightarrow \neg\neg A$. However this process stops after the first iteration. Subpresheaves (subfunctors) of a presheaf over a category $C$ form a Heyting algebra. It is interesting to look at $\neg\neg A$. This is related to density (the dense or double negation Grothendieck topology).  Given a subset $A$ of a set $X$ we can look for the smallest set $B$ for which $A$ is dense in $B$, that is $B$ is the closure or completion of $A$.

Sheaf theory recalls Kantian schematism: it is the synthetic realization (in particular topological) of an abstract category. The sheaf axioms express Kantian synthesis.

Of particular importance are theories of wholes, of different kinds of wholes, in particular non-distributive (mass-noun-like) and constructive/computational wholes.  All quantifiers (in dependent type theory) are intensions related to wholes and it is important to know what kind of whole is under consideration.

On a Formalization of Kant

The paper by Van Lambalgen and Pinosio 'The logic and topology of Kant's temporal continuum' (which is just one of a series of papers by Van Lambalgen on Kant)  opens with a nice discussion and careful justification of the general idea of the formalization of philosophical systems. The coined expression 'virtuous circle'  is particularly fortunate. In this post, which will be continuously updated, we will critically explore the above paper and make some connections with our own work on Aristotle's theory of the continuum.

The primitives are called 'events', self-affectations of the mind, which must be brought into order by fixed rules.  The authors work over finite sets of events which is justified by textual evidence from the CPR (we will return to this later).  Their task is to formalize relations between events - and to thus develop a point-free theory of the linear temporal continuum.

We find that that their notation could be improved and the axioms better justified. Instead of the confusingly asymmetric (all for the sake of the substitution principle, I suppose, or for the transitivity axiom) $aR_- b$ and $cR_+ d$  let us write $a{}_\bullet \leq b$ and $d\leq_\bullet c$. Instead of $a\oplus b$ we write $a\leftarrow b$ and insead of $a\ominus b$ we write $a\rightarrow b$.

The basic idea is that : $x{}_\bullet\leq y$ does not need to imply that $x\leq_\bullet y$ or vice-versa.

Kant's concept of causality implies that in order for a part $x$ of $a$ to influence $b$ we must have $x{}_\bullet\leq b$.  Thus the following axiom is expected

\[  a\ominus b{}_\bullet\leq b\]

But let us look at axiom 4 for event structures (in our notation):

\[ cOb\,\&\, a\leq_\bullet c \,\&\, b{}_\bullet \leq a \Rightarrow aOb \]

Our task is to make sense of this by offering a more satisfactory account of the primitive relations. Let us consider the set of connected (hence simply connected) subsets of the real line $\mathbb{R}$ and the interpretations:

\[ a{}_\bullet\leq b \equiv \forall x \in a. \exists y\in b. x\leq y  \]

\[ a \leq_\bullet b \equiv \forall x \in b. \exists x\in a. x\leq y  \]

But this does not work for  $a{}_\bullet\leq b \Rightarrow a\leq_\bullet b$. But let us take our events to be bounded open intervals $(a,b)$ and consider

\[ (a,b){}_\bullet\leq (c,d) \equiv  b < d  \]

\[ (a,b) \leq_\bullet (c,d) \equiv a < c \]

\[(a_1,a_2)O(b_1,b_2) \equiv a_2 > b_1\,\&\, a_1 < b_2\]

Then if we consider $(0,1)$ and $(0,2)$ we have that $(0,1){}_\bullet\leq (0,2)$ but not $(0,1)\leq_\bullet (0,2)$. The inequalities must be strict for allowing  $(a,b){}_\bullet\leq (a,b)$ is absurd, for then we could not associate any clear or definite Kantian philosophical concept with the relation.

Now let us look at axiom 4:

\[ (c_1,c_2)O(b_1,b_2)\,\&\, (a_1,a_2)\leq_\bullet (c_1,c_2) \,\&\, (b_1,b_2){}_\bullet \leq (a_1,a_2) \Rightarrow (a_1,a_2)O(b_1,b_2) \] which becomes

\[ c_2 > b_1\,\&\,  c_1 < b_2   \,\&\,a_1< c_1\,\&\, b_2 < a_2 \Rightarrow a_2 > b_1\,\&\, a_1 < b_2\]

But this follows immediately, using in addition the fact that $b_2 > b_1$. The condition $c_2 > b_1$ appears not to be needed.

We could try defining $(a_1,a_2)\rightarrow (b_1,b_2) := (a_1,b_2)$ when $a_1 < b_2$ and $(a_1,a_2)\leftarrow (b_1,b_2) :=  (b_1,a_2)$ when $b_1 < a_2$.

This models should be introduced right at the start of the paper to motivate the the definition of event structure. Notice that the set of events is here identified with the (infinite) subset $E \subset \mathbb{R}\times\mathbb{R} = \{(x,y): x < y\}$ but we could take only a finite subset.

We must check the axioms for event-structures for our model and also give a geometrical interpretation of the relations and operations above in terms of the identification of $E$ as a subset of the plane above.

Are analogies adjunctions ?

Consider an analogy : day is to night as life is to death.  Surely in any analogy there is implicit a correspondence, a set of functions which takes the pairs to each other: F: {day,night} $\rightarrow$ {life,death} and G: {life,death} $\rightarrow$ {day,night}.  Consider the statement: death is the night of life. If this is so then surely:  night is the death of day and vice-versa. We take $of\, life$ and $of\, day$ to be the functors $F$ and $G$ respectively - here we venture that 'life' and 'day' designate the whole genus, the set of both elements of the respective pairs. Thus it would make sense to day : day is the life of day  and life is the day of life. And we read this as $death \rightarrow F\,night \Leftrightarrow G\, death \rightarrow night$ which (together with the case for day and life) corresponds to the definition of an adjunction (we can assume we are in a groupoid in which 'is' is an isomorphism). We can also write symbolically:

$\frac{day}{night} = \frac{life}{death} \Leftrightarrow day\times death = life \times night$

Update: work-in-progress on new papers

New papers:

1. Transcendental Subjective Idealism

2. Category Theory and Philosophy

3. Computability, Logic and Mind

(We have removed many important posts as their material in now being incorporated in the above papers)

4. Inquiry into Kant's logic (this is related to 3)

5. Formal metaphysics. As far as the project of a formal axiomatic-deductive philosophy we have been investigating the role category theory might play in this endeavor which at the same time is fully aligned with Kant's critical idealism and Husserl's transcendental phenomenology. But there are other possible approaches and perspectives which preserve this alignment. One that takes dependent type theory to be more fundamental, universal and fine-grained than category theory. Another is based on the philosophy of computability and second-order monadic logic. One problem is that we still do not a have a completely clear understanding of quantifiers or connectives or intensionality.

Topos theory (as is well known) is the natural environment for higher-order logic. My idea is that if we consider the type of possible worlds endowed with a kind of 'topology' (this could be an abstract Grothendieck topology defined by a Joyal-Tierney topology) then it seems natural to consider the environment for object logic to be not a general elementary topos but a Grothendieck topos, a topos of sheaves over the 'space' of possible worlds. The objects of this topos are sheaves, that is, they are types parametrized over possible worlds. The great challenge is to seem how core notions of Zalta's object logic, specially the encoding relation, can be expressed in this context and whether they have natural categorical descriptions...

6. Expand S. Shapshay's work on the compatibility of Kantian and Schopenhauerian ethics. In particular by the drastic constrast with Hegel's implicit or explicit historico-evolutionary relativism and pragmatism. Also address the pertinent questions raised by Shapshay regarding Schopenhauer's theory of the 'denial of the will' and show how the internal tensions in Schopenhauer's thought can be resolved through understanding the process of personal development in original Buddhism and its counterparts in ancient philosophy.

 The first project involves understanding the famous enmity between Schopenhauer and Hegel. My thesis is that this antagonism was not rooted merely in personal animosity but rather in irreconcilable ethical positions. So it is important to bring to light the ethics implicit in Hegel's Phenomenology of Spirit and show how they constitute the greatest contrast both with Schopenhauer's and Kant's ethics. My tentative position is that, very roughly, Hegel is basically a kind of pragmatist, evolutionist and historical relativist while for Schopenhauer and Kant ethics must be universal, valid for all people at all times. For example, for Hegel slavery was 'good' for its time and indeed a 'necessary' phase of the historical-development of human consciousness while for Kant and Schopenhauer it is intrinsically and universally 'bad' - a violation of basic human rights.
My second project involves addressing the internal tensions and contradictions you discussed involving the theory of the 'negation of the will' and world-rejecting asceticism which you contrast with the more hopeful aspect of Schopenhauer anchored in compassionate reason-guided morality and aesthetic contemplation. My thesis is that this internal tension is due primarily to Schopenhauer's idea of 'negation of the will' and 'asceticism' being almost entirely derived from the Christian mystics, his understanding of oriental traditions and practices being colored and distorted by this preconceived idea. I wish to put forward that original Buddhism (for example) offers an entirely distinct concept and practice of 'bhavana' (which can be translated as personal development or asceticism) which is radically incompatible with Christian mysticism and asceticism. This radical incompatibility is centered on divergent attitudes towards and role of suffering, specially with regards to the points you so brilliantly analysed in your book. The conclusion is that if we take this concept of bhavana then the internal contradictions and inner tensions largely disappear.

7. Deployment of a theory of knowledge, philosophy of language and philosophy of logic within the framework of 1-3. In particular continuing the work of Ortiz Hill, Rosado Haddock and several others...showing the superiority of Kant and Husserl over Russell, Carnap, Quine and Rorty.  The Frege-Husserl relationship and divergence can be greatly illuminated by a true understanding of Kant. That is, Frege's anti-psychologism (quite distinct from Husserl's)  which became the original impetus for so much of subsequent analytic philosophy is seen to be questionable and based on serious confusions. This is one of our fundamental problems: reconciling epistemic absolutism and objectivism (with regards to fundamental philosophical and scientific domains, including ethics) with full transcendental subjective idealism. Kant and Husserl offered solutions (and we should consider Brouwer as well). Frege just could not see any.  The other problem involves the historical consciousness of logic, the nature of transcendental logic,  multiple generality, intensionality and a rigorous theory of symbolism and analogy. We can reframe this historically as the problem of reconciling neoplatonic philosophy (understood as a synthesis of Plato, Aristotle and the Stoics) and Kantian and post-Kantian subjective idealism.  In the East such a reconciliation seems to have been as the basis of both the Vedânta and certain schools of Mahâyâna Buddhism.  Also in the west there are certain esoteric writers.

We are planning to write a series of notes and commentaries on Word and Object and Rethinking Identity (perhaps as blog posts). The  discussion of the Frege-Husserl relationship is very inspiring and thought provoking.  There is a perfect consistency between the early 'objective' Husserl and the later 'subjective idealist' Husserl - and it is Kant (in particular as read in the light of subsequent philosophy) that furnishes important clues. This all hinges on the different varieties and nuances of psychologism and anti psychologism...The book by Robert Hanna, Kant and the origins of Analytic Philosophy, is an interesting read, specially as it argues that there was already a strong anti-psychologism in Kant. We continue to investigate some central problems in the philosophy and history of logic, but right now focused on multiple generality and quantifier logic. With regards to intensionality we have come up with a new idea about the set of possible worlds. We think that just considering a set of possible worlds is wrong; rather there must also be a concept of nearness between worlds, based on the possibility (are we being circular here ?) of continuous deformation and variation. This can be justifies both on Husserlian grounds and by ordinary linguistic usage: our counterfactuals are usually based on more or less small variations (deformations) of the actual world or state-of-affairs.

 

8. Critical study of the Neoplatonic school both from a philosophical perspective and from an historical and archaeological perspective, that is, studying the views of the Neoplatonists themselves regarding the nature of ancient cultures and traditions: Chaldean, Egyptian, Greek, Syrian, Phoenician, etc.

 

https://antiqualux.blogspot.com/

 

9. The meaning of the Abhidhamma literature which, independently of the particular views held, is a logical, epistemological and psychological monument of enormous importance. By 'atomism' we can understand a theory of a given domain of reality which is logically based on primitive conceptual units and their relations, in particular the causal relation for temporal processes. It represents the dawn of authentic relational, non-monadic logic and systematic evidence-based rationality. Study also the relationship to Hume.

The collected works of F.W. Lawvere

https://github.com/mattearnshaw/lawvere