Quodlibet

 1. René Thom called quantum mechanics 'the greatest intellectual scandal of the 20th century'. Maybe this was too harsh, but quantum theory was meant originally just as to be crude very, provisional proto-theory to give place to something to better (which has not due to political, military, economic and industrial reasons ?). Consider the double-slit experiment. The 20th century was also the century of dynamical systems and chaos theory. It is clear to us that the random aspect of the double-slit experiment must be explained in light of chaos theory, thus of an underlying deterministic system. In a classical setting there will also be a pseudo-random aspect for particles traversing the two slits (but without the interference pattern). Nobody would think of interpreting this as a probabilistic collapse of a wave-function. In the non-classical situation it would occur to almost anyone to see the wave-function as real physical field associated to particles (a "pilot-wave"). If we rule out local hidden variables (but do we really need to ?), then we are lead no non-local yet deterministic non-linear systems which generate the pseudo-random phenomena of quantum theory in a standard way.

2. Quantum theory gave us the idea of introducing negative probabilities, i.e. signed measures. 

3. Category theory is intensional (non-extensionalist) mathematics based on minimal logic, thus hyper-constructive.  We ask about a natural number object (the concept of an 'element'  is not taken as a primitive; rather we have only generalized elements $1 \rightarrow A$) in a given category, that is, about its universal property;  we construct concrete generalized element 'numbers' through composition of primitive morphisms $z : 1 \rightarrow N$ and $s : N\rightarrow N$. Recall how the concept of primitive recursive function emerges naturally from this definition...

4.  Sights, sounds, etc. These are always sight-thoughts, sound-thoughts, formatted by the mind into objects integrated into a continuous semantic-temporal context. Sense-data are abstract results of conscious reflection.  Rupa is essentially a thought-constituted object modulo a sense-modality; only thus can it be an object of attachment and associated with feeling and consciousness. A raw abstracted sense-datum could never be the object of attachment or desire, except in rare pathological situations (the theory of the rupa-khanda is an ontology and general theory of categories for representations). Thus did the yogacarins refute the abhidhamma. We must investigate this carefully in the ancient texts. This must have been the original view which after becoming distorted  was then recovered in later sutras such as the Shurangama-sutra where a sky-flower object is given as an example of khanda-rupa.  Plotinus prefigured Kant.

What we know through sense perception is a simulacrum of the thing and perception does
not receive the thing itself. The later (the thing itself ) remains outside.  Enn. V, 5, 1.

De Anima 425a27: We already have (before actual perception) a common perception of the ”commons” (the categories of the sensus communis). Aristotle also wrote that it is impossible for time to exist without the mind (Physics 223a26).

Bhâvanâ cannot start with low-level abstractions. Husserl taught us to see the true categories at work in experience.

5. There have always been different notions of 'quantification' (and the corresponding determiners) which were conflated by extensionalist logicians.  This is clear in the distinction between intensional, conceptual universal quantification and extensional quantification. Also such distinctions are brought to light by the behaviour of quantifiers in propositional attitudes.  Constructivism tried to bridge the gap between extension and intension via a kind schematism (see previous post). We must bring all the different kinds of quantification to light again. 'Some' seem to be even richer in nuances than 'for all'. The distinction between the classical and intuitionistic/constructivist 'some' is deeply rooted in and reflected in cognition and natural language semantics. For instance, the intuitionistic interpretation fails for existential formulas in the scope of propositional attitudes. I may believe that the money in a book in the library without there being a specific book in which I believe the money is in.

Are set-theoretic extensions are atomistic structureless heaps, like the extreme abstract atomic alienated negativity in certain stages of Hegel's phenomenology of spirit ? This is not really so, they can have a very definite tree-like structure. Groupoids have more organic unity. We must investigate what it means to quantify over groupoids.

6. Some people are scared of homotopy type theory, higher category theory or of Coq and Agda. I respect that.  I feel the same about fractal calculus. But perhaps fractal calculus has something to do with the following important question. Numerical, discrete, computational methods are routinely used to find approximate solutions of differential (and integral-differential) equations. But we also need in a turn a theory of how differential and smooth systems can be seen as approximations of non-differential and non-smooth systems. Is this not what we do when we apply the Navier-Stokes equations to model real fluids ? Recall how continuous functions with compact support are dense in the $L^p$ spaces of integrable measurable functions (but see also Lusin' theorem).  Can all this be given a Kantian interpretation ?

7.  What are distributions ? They allow a mathematical treatment of the vague notion of particle. Indeed particles are just euphemisms for certain kinds of stable self-similar field-phenomena. The great geniuses in physics were those who helped build geometric physics (which is what is most developed and sophisticated in modern physics):  Leibniz, Lagrange, Euler, Hamilton, Gauss, Riemann, Poincaré, Minkowsky and many others.  But it is no use playing around with highly sophisticated geometric physics (which looses all connection to experiment)  if you haven't solved the problem of quantum theory first.

8. Study differential geometry as type theory; dispel all difficulties in a general understanding of mathematics as a language.

9. Many of our concepts have a tripartite nature $(A, A^\circ, \bar{A})$ expressing certain $A$, certain not $A$ and the grey neutral area $\bar{A}$. For instance: bald, not bald, sort of bald but not really bold. Each one in turn will depend on an individual and a possible situation of affairs. But this is not enough. In order to do any kind of 'logic' here we need some kind of quantified probability measure, for instance the ability to measure quantities of individuals and states of affairs. Then the sorites is resolved by presenting a tripartite distribution.  Thus it is interesting to have a logic which can express probability distributions.

10.  The goal is to pass from language-based philosophy to pure logic based philosophy. But this needs a mediator. The mediator can only be advanced, sophisticated, mathematical models, qualitative, essential, extending to all domains of reality (deformations, moduli are the right way to study possible worlds). All aspects of Kant and Husserl can be given their mathematical interpretation and from thence their logical-axiomatic interpretation. The same goes for naturphilosophie via René Thom and Stephen Smale. Theoretical platonism and idealism is not enough. We need this realized applied platonism. Mathematics furnishes a rigorous way of dealing with analogy and integrating analogy into philosophy. Also mathematics furnishes the deeper meaning and interpretation of Kant's theory of categories and schematism. Mathematics furnishes us with a way of studying concepts which is not divorced from the conceiving mind but at the same time is not psychologistic.

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.

See also:

https://chryssipus.blogspot.com/2023/11/the-church-turing-thesis-kripke-and-kant.html

https://chryssipus.blogspot.com/2024/01/algorithms-and-numbers.html

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).

Schopenhauer on the Content of Compassion

https://phil.washington.edu/research/publications/schopenhauer-content-compassion

Instead of simply saying that the compassionate person perceives no distinction between herself and the object of her compassion, we should say that she perceives them to not be distinct spatiotemporal individuals. That is, she perceives them to be distict only in the way that Platonic ideas are distinct. The latter distinctness is not sufficient for individuality in the normal sense, for Schopenhauer, since he calls space and time alone the principle of individuation (OBM 4:267, p. 250). The key difference is that, at the level of ideas, things metaphysically overlap with each other in ways that they do not at the spatiotemporal level. (p.7)

So we have here a holology which also suggests comparison with Plotinus' theory of how ideas are unified in the nous.

Hegel's phenomenology of spirit is the antithesis of both Kantian and Schopenhauerian ethics. It is essentially anti-transcendent, pragmatist, relativist, collectivist, deterministic and culminates in a totalitarian-statist mysticism (a fascism based on an esoteric Christianity which subsumes and buries the the possibility of the resurgence of the enlightenment).  This is seen in the treatment of the phase of practical reason and its transition into "spirit". Hegel turns Kant's noumenon into his secular pragmatic collectivist fatalistic thing itself Sache Selbst. From thenceforth it is no longer about the individual but only the drama of the collective. The individuality of the individual is allowed to manifest in its "negativity" only for the sake of, and in function of, the development of the power and self-transparency of the collective.  What corresponds to this 'spirit' is the third section of what is inappropriately called 'Begriff' in the Science of Logic, the weakest and most ad hoc part,  which appropriates the far deeper insights into the structure of consciousness found in Kant and Fichte. 

At the basis of ethics must be a consciousness which does not make a distinction between self and others or between today, yesterday and tomorrow. Ethics concerns the timeless individually directly cognizable universal ought which is completely independent from any hypothetical necessary developmental law or process just as it has nothing to do relativistically with arbitrary convention. At a social-cultural level there can indeed be ethical progress, but this should never be seen as the working of some kind of natural law or the result of necessity. Confusing the ideal of human ethical progress with speculation about evolution in natural science has been a very serious error.  Human ethical progress is a normative ideal never a law or natural necessity. A normative ideal that remains invariant throughout recorded historical humanity, even if tragically it seems to be less and less realized in the world today. There are non-human sentient beings which cannot be subject to the normativity of the moral law. But we could explore how there is an implicit, albeit imperfect, morality already at work in nature. What we must reject are arbitrary speculations attempting to link non-human and human beings whereby such a link serves as a foundation or explanation of morality.

The correct theory of ethics is much like Frege's  philosophy of logic. Or, to paraphrase Husserl:

Whatever is a moral duty, is absolutely, intrinsically a moral duty: the moral law is one and the same for men or non-men, angels or gods. Moral laws speak of the ought in this ideal unity, set over against the real multiplicity of races, individuals and experiences, and it is of this ideal unity that we all speak when we are not confused by relativism.

The moral law implies that we should strengthen our historical organizations dedicated to the universal unconditional upholding of human rights and international law.

Theory of theories revisited

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.

Analytic metaphilosophy

By analytic metaphilosophy we mean a methodology which aims to apply mathematical and formal logical rigor and the full force of linguistic analysis to philosophical texts in order to assess their argumentative and epistemic value. Analytic metaphilosophy has a strong affinity and connection to early classical analytic philosophy (though it has many brilliant precursors before that time) but no connection to subsequent post-classical analytic philosophy - indeed it can be conceived as the ultimate tool to thoroughly refute and show the nonsense and vacuity of its various currents as well as some of those that claim to break with it.  A novel aspect is that it attaches an enormous importance to the style of philosophical texts and aims to be far more wary and careful about the pitfalls, delusions and psychological foibles of the whole process of literary creation. It never forgets that the philosophical writer is never far from the precipice of literary fantasy often exhibiting egocentric and sycophantic qualities geared to socio-economic advantages rather than epistemic and ethical goals.  Analytic metaphilosophy studies in particular the sociology and psychology of sophistry and literary delusion. More Platonico we shamelessly proclaim that it is impossible to engage in analytic metaphilosophy without either a solid undergraduate background in mathematics and mathematical logic or at least a couple of years of experience working on a formal mathematics project employing a proof assistant.  In a nutshell: analytic metaphilosophy applies mathematical standards of logic and rigour to philosophical texts and refuses to be impressed by the mirages and artifices of language (though an ideal philosophical text can have also beauty, elegance and clarity of style as in the writings of Frege and Claire Ortiz Hill). Jargon-laden and convoluted texts rarely betray deep, complex, rigorous or valid thoughts. Such pseudo-difficulty is of an entirely different nature from the 'difficulty' of mathematical texts. Analytic metaphilosophy can also be seen as a kind of prolegomena and justification for the formal philosophy project we described in previous posts.

In that most rigorous, clear and certain of the sciences, mathematics,  mistakes and confusions still arise, there are gaps in proofs, there are unjustified assumptions, careless generalizations, confusions in terminology, silly oversights, circular reasoning, etc. As the length of the proof increases so does the probability of error, even for the best mathematicians and Fields Medalists like Vladimir Voevodsky. Careful checking by several experts is absolutely necessary. In some areas the length of the proofs become almost too long for this to be feasible, so proofs are formalized and run through specialized proof checking software.

Now consider philosophy, the least rigorous, clear and certain of human epistemic enterprises. Once a philosophical 'argument' becomes long, convoluted and (on the surface level) complex one can be almost certain that it is wrong or inconclusive. The same goes for texts with an elevated number of technical terms or  'jargon density', so to speak. 

Mathematicians have an artist's liberty to use and invent symbols for their primitive and defined notions and variables.  The philosopher, shackled by natural language and lack of mathematical training, is in a very dire situation, terms are pathologically and enormously semantically overloaded and the resulting terminology is opaque, ambiguous and stylistically repugnant. Perhaps this can be partially overcome by the construction of an artificial language for philosophical terms.

The majority of philosophical texts have nothing to do with the logical standards and rigour of mathematics or even the exact sciences. What reason is there to attribute to them any meaning or epistemic value at all ? Or even social value ?  And many of their authors are aware of this, don't care, or consider it a virtue.  They have their communities with their founding narratives and (non-reflectively) received doctrines and they happily engage in their 'language-games'  and strictly controlled boxed (bottled ?) 'innovations'. They have their own 'logic' and 'rules' and 'criteria' for parsing and deciding the value or legitimacy of textual-output - and this output is a torrent, an endless deluge and logorrhoea that seemingly might be generated by large language models. Ex falso quodlibet. 

Analytic metaphilosophy (which favours Martin-Löf type theory as an adequate intensional and modal logic) is entirely immune from objections culled from mathematical logic itself such as all-too-common misunderstandings and misappropriations of Gödel's incompleteness theorems.  Although mathematics can be conceived as a subset  of logic (by assuming special axioms such as univalence), logic can also be conceived as an application of mathematics. There is no concept of computability without involving arithmetic and no concept of arithmetic which does not involve some notion of computability.

Recall that we hold that logic is embodied in a closely unified (organic) family of formal systems which are related to each other by gradation or (mutual) embeddability and reflection. There is no trace of convention or arbitrariness.

And in nowise does our metaphilosophy claim to be philosophy itself or a substitute for it. However knowledge of applied mathematics at an advanced theoretical level has huge conceptual advantages for thinking about possible worlds, possibility, causality, identity and states of being which far surpass the crude models used in analytic philosophy.

Analytic metaphilosophy can be integrated seamlessly into the Gödelian and Husserlian frameworks as complementing and helping the metaphilosophy and methodology of ultimate evidences and intuitions as well as ethical metaphilosophy*. It works alongside it and provides powerful aid by refuting anti-idealist arguments.

Schopenhauer's detailed criticism of Kant in the WWR and T. H. Green's long introduction to Hume's Treatise are  good examples of pre-Fregean analytic metaphilosophy. The investigation of the expression of multiple generality and its associated reasoning in ancient philosophy is clearly a cornerstone to analytic metaphilosophy's  approach to ancient and early modern philosophy.

*ethical metaphilosophy focuses on the explicit and implicit content relating to human and animal rights, in particular the status, dignity and consciousness of animals,  in historical philosophy - and thereby comes to general conclusion about the value and merit of different philosophical systems.  Leaving aside ancient eastern philosophy, the cases of Porphyry, Schopenhauer and Husserl are enough to de-fang ethics-based anti-idealist arguments whilst arch-anti-idealist Nietzsche's raving praise of Descartes' view of animals delivers a fatal counter-blow.

Medieval philosophy has been criticized for being the handmaiden of theology and merely a tool for the apologetics of the dogma of organized religion.  If this is justified then philosophy also cannot be allowed to be the handmaiden for para-scientific ideology and dogma either, which is what we find predominantly in the so-called 'philosophy of mind'. Is there anything more silly than  a priori arguments for speculations  based on incomplete or faulty experimental science ? Image the money it saves on equipment and resources for the materialist neuroscientist.

Nevertheless Kant's language is often indistinct, indefinite, inadequate, and sometimes obscure. Its obscurity, certainly, is partly excusable on account of the difficulty of the subject and the depth of the thought; but he who is himself clear to the bottom, and knows with perfect distinctness what he thinks and wishes, will never write indistinctly, will never set up wavering and indefinite conceptions, compose most difficult and complicated expressions from foreign languages to denote them, and use these expressions constantly afterwards, as Kant took words and formulas from earlier philosophy, especially Scholasticism, which he combined with each other to suit his purposes; as, for example, "transcendental synthetic unity of apperception," and in general "unity of synthesis" (_Einheit der Synthesis_), always used where "union" (_Vereinigung_) would be quite sufficient by itself. Moreover, a man who is himself quite clear will not be always explaining anew what has once been explained, as Kant does, for example, in the case of the understanding, the categories, experience, and other leading conceptions. In general, such a man will not incessantly repeat himself, and yet in every new exposition of the thought already expressed a hundred times leave it in just the same obscure condition, but he will express his meaning once distinctly, thoroughly, and exhaustively, and then let it alone. "_Quo enim melius rem aliquam concipimus eo magis determinati sumus ad eam unico modo exprimendam_," says Descartes in his fifth letter. But the most injurious result of Kant's occasionally obscure language is, that it acted as _exemplar vitiis imitabile_; indeed, it was misconstrued as a pernicious authorisation. The public was compelled to see that what is obscure is not always without significance; consequently, what was without significance took refuge behind obscure language. - Schopenhauer, WWR (II).

Updated Project

We authored papers on ancient logic and on Aristotle's theory of the continuum.

1. Investigate Ortiz Hill's theories on identity, equality, intensionality and extensionality in the light of dependent type theory and homotopy type theory.

a) intensions and modalities are irreducible parts of logic and the structure of reality

b) we must never confuse identity with equality which is better to call equivalence, there being several notions of equivalence

c) extensions (and completed infinities) can be problematic and do not have logical or ontological priority 

d) type theory was a step in the right direction  which became perfected by Martin-Löf

2. Investigate embeddings of formalizations of philosophical systems (specially modal type theories) into dependent type theory.

3. Defend Husserl's philosophy of logic and theory of knowledge against its opponents.

3a. Continue Gödel's philosophy based on a unique interpretation and reconciliation of Leibniz and Husserl.

4. Continue the project of a virtuous hermeneutic circle between higher category theory and Hegel's logic.

5. Give a detailed philosophical intepretation of Pali Buddhism in terms of ancient and modern western philosophy (Skeptics, Stoics, Plato, Aristotle, Plotinus, Kant, Schopenhauer and Husserl) and show how popular accounts of original Buddhism are mistaken.  This involves in particular

5a. Develop a synthesis between Kantian and Schopenhauerian ethics.

On Van Lambalgen et al.'s 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.

A filosofia de Ludwig Wittgenstein à luz do diagnóstico de autismo

 https://philarchive.org/rec/SILAFD-6

Philosophy of Mind

Consciousness is.  But we must reflect and become aware first of all that there is consciousness. Make consciousness itself an object for consciousness.  But in this awareness of consciousness itself we must make sure that we see consciousness in its purity and totality. That all experience, past, present, future, in whatever mode, the world itself, is immanent in consciousness.  Thus does consciousness stand outside itself and perceive itself in itself by itself - the primordial stream which has sometimes been called the stream of thought, although to be accurate we must include all modalities of consciousness experience under the heading of cogitata.  Temporality is unveiled in its primordial role. This shift of awareness* can have a sobering and even liberating effect, like an exit from Plato's cavern.  We are left with an agent-self standing both inside and outside consciousness, the universal sphere of consciousness as an object, and the relation between the agent-self and the universal sphere.

But here we take a practical turn which will hopefully lead us to truth and allow us to avoid many errors. 

There is the primordial transcendental knowledge that there is something seriously wrong with our consciousness that needs to be corrected, there is something that needs to be overcome.  The agent-self must somehow act on consciousness and hence ultimately act on itself to remedy this situation.  Part of such primordial wrong is the powerlessness and passivity of the self-agent, specially in its naive unreflected self in which it inhabits naturalistically and in oblivion  a world unknowingly constituted by consciousness. And no, we are not confusing merely 'psychological' problems with philosophical ones. The divorce of psychology and philosophy is itself philosophically questionable.

The transcendental ego in the Husserlian sense is merely passive, shallow and theoretical (in the sense of being talk about knowledge rather than knowledge, the menu rather the the actual dish).  Because it lacks power and it cannot act, it cannot attain to knowledge, indeed despite some rupture it does not leave the western conceit that genuine knowledge can be divorced from a vital transformation of the subject.

How can the transcendental ego gain a grasp or foot-holding on consciousness itself so that it does not remain a vain chimera ? The answer involves the theory of embodiment, but in a way radically different from any naturalism or existentialism. To us the theory of embodiment is the theory of the body from a first-person perspective, the phenomenology of the body as experienced exclusively from the inside (the external theory being developed from and based on the internal theory). We claim that the phenomenology of  inner-body consciousness is necessary and of immense importance to any correct philosophy of consciousness.  The transcendental ego must first of all focus, concentrate and investigate inner-body consciousness.

Focus on the inner-body consciousness is the solution, the vital foot-holding, for  the transcendental ego becoming active and gaining power over the total immanent sphere of consciousness and thus to at last access the long sought-for clarity in knowledge.  Inner-body consciousness is a kind of organic crystallization of consciousness which reveals itself to be very deep with roots in manifold aspects of ordinary human psychological  experience - even neurological ones fall within the scope of a strictly internal point of view. It is the necessary path to pass through in order to be able to  grasp and bring order to other spheres of more subtle conscious modalities. That is: we must bring to the light of consciousness what we are in fact deeply identified with without knowing in order to achieve dis-identification from it (we will not go into here the philosophy of self and its modalities of unification, synthesis, negativity and subsumption).

The body is thus the initial and essential tool and means by which the transcendental ego attains the correct relationship to its own total immanent sphere of consciousness. It is also, as we shall see in a future essay, the key to a Kantian-Schopenhauerian ethics of universal compassion, which involves the overcoming of solipsism. It is also the basis for  a new philosophical top-down organicist architectonic of the sciences which is not without strong connections to Goethe's,  Schopenhauer's and von Uexküll's theories of natural science.

Our use of  'body' is perhaps misleading as it is used in a very technical sense: that (Kant's noumenal X) which enters into a particular kind of relationship to consciousness which characterizes inner, first-person body-experience and body-consciousness. Consider the section dedicated to the so-called 'refutation of idealism' in Kant's CPR. Our goal is to show in an analogous (or transposed) way that the certainty of our own consciousness and its reflected embodied experience - in particular the reflected embodied experience of pain - necessarily presupposes the existence of other similarly embodied consciousnesses external to us, when we consider ourselves as particular embodied consciousnesses. The moral law we wish to establish and develop (which has both Kantian and Schopenhauerian foundations) has two inseparable and necessary components: compassion for self and compassion for others, the duty to alleviate one's own suffering and to alleviate the sufferings of others, or, in the words of the Gospels, to 'love thy neighbor as thyself', supposing, of course, that by 'neighbor' is meant every human (or in general sentient) being.

*The path to  transcendental consciousness involves the detachment-inducing awareness of temporality (including past and futurity), of the transience, change, frailty and compositeness of all domains of our embodiment and experience (psycho-physicality) as well as scrutiny of the now, the actual, what in fact is really there in the midst and in light of this transience. This should become ever more subtle, anchoring in the realm of pure thought.

We can also say that transcendental consciousness is attained by a decision to examine what is really present before us, what is positively pristinely there rather than a projection, anticipation, interpretation and extension conditioned by volition, etc. This leads to a conversion inward, a unification which is also an inversion and shift of the entire domain of consciousness. 

Transcendental consciousness involves the process by which we come face to face - but in a detached way - with the current of our own thoughts (possibly entwined with empirical-sensual content). This is what Husserl describes in the first chapters of the Cartesian Meditations.

Or rather transcendental consciousness is a freely flowing unified state beyond the stopping (or self-limitation) of consciousness in intentional acts.

See also:  Scarfe, A. (2006). Hegelian “Absolute Idealism” with Yogācāra Buddhism on Consciousness, Concept (Begriff), and Co-dependent Origination (Pratītyasamutpāda). Contemporary Buddhism, 7(1), 47–73. doi:10.1080/14639940600877994

Serious philosophy might be said to be simply studying Kant and finding  ways or reconciling Fichte and Schelling, Schopenhauer, Frege and Husserl...except that i) modern mathematics has opened the way to the development of the genuine philosophical logic which few bother to learn and understand: categorical logic and type theory and ii) we now have access to immense Sanskrit libraries of Buddhist philosophy which few bother to learn and understand.

Meaning of the logical connectives

The meaning of the implication/conditional operator $A\rightarrow B$ is simply that of a relation of truth values (as Kant described the hypothetical judgment in the CPR). It has nothing to do with causality, inference or relevance.  If we take $0$ as false and $1$ as true then $A\rightarrow B$ is simply the proposition which states that the truth value of $A\leq $  truth value of $B$. What is paradoxical about the fact that for any proposition $A$ we have that 0 $\leq$ truth value of $A$ ? What is paradoxical about the fact that given any two propositions $A$ and $B$ we have that either truth value of $A\leq$ truth value of $B$ or truth value of $B\leq $ truth value of $A$ ? Relevancy is irrelevant in the face of propositions regarding the relationship of the truth-values of propositions - which are purely  mathematical. Logical connectives are in a way a reflection-into-self of logic, they are propositions - having truth values - about the truth values of propositions. This is clear even in the semantics of linear logic, interpreted as a many-valued logic.  And the many-valued truth value of $A\& \sim A$ can be seen for instance the the result of a voting process. There can be a draw between $A$ and $\sim A$ and this itself be a value.

In general implication means that there is some computable function that takes terms inhabiting in $A$ into terms inhabiting $B$.  That is, we can compute $B$s in terms of $A$s.  Connectives are semantically truth-value based or in general witness based. Their legitimacy and value is untouched.  We can however think of an additional, alternative theory of intensional connectives, relevance, inference and causality. Notice that if an effect is unique to its cause then classical logical connectives cannot capture causality.

Category theory has since decades developed a useful tool for dealing with contextualism and pragmatics: fibered category theory.

Even Girard's linear logic can be understood in terms of phase semantics; as an algebraic many-valued logic.  $\multimap$ is interpreted much like in realizability or dependent type theory.

In case you didn't know

Subjective idealism or the idealism in general found in Kant (correctly interpreted), Schopenhauer and Husserl has nothing to do with relativism or psychologism and is immune to all anti-psychologist arguments (including Moore's arguments against Berkeley-style 'idealism'). Nor can objectivity be founded on any simplistic and confused empiricism; nor a logic based on nominalist, conventionalist  of socio-pragmatic premises - which ignores the irreducible reality of intensional entities - even deserve that name. Sophisticated formal languages such as dependent type theory are meaningful and epistemically adequate in their own right, existing alongside natural language.  There is nothing cognitively or ontologically normative about natural language, let alone the English language.  One of the most unjustifiable and harmful  errors is stating that mathematics is justified solely by its applications in natural science or its indispensability in the language of natural science. This thesis is laughable to anybody with any serious knowledge of the history of mathematics and theoretical physics.

As in Kant, philosophy may form a tightly-knit organic whole. There is no good reason why ethics and the theory of knowledge might not exhibit vital connections to each other. Departments do not have to mean water-tight compartments. Divisions need a justification from a higher perspective, just as the species of a genus share both community and difference. Ethics rests on theory. The activity of theory can itself have a deeply ethical significance: sapere aude !

Historical progress will always be only an ought rooted in individual freedom and endeavor, never an automatic necessity.

Conceptual engineering is just the sociological version of the old psychologism. As sociology it is interesting and has its merits. But it does not contribute anything of philosophical value per se, although it can suggest problems such as the critical analysis of sociologically accepted fallacies and contradictions implicit in popularly used jargon and terminology.

Socrates and Husserl suffered similar fates: they were 'killed' by their times. The message and spirit of Husserl's philosophy (after Husserl himself was banned by the Nazis) was killed and then appropriated by existentialism, naturalism and Catholic neoscholasticism.

Philosophy has its stand-up comedians such as those that argue for logical nihilism.

The stronger one's scientism the greater the probability of having a very little knowledge either of scientific theories or the scientific method(s). Philosophy, logic, mathematics and ethics are epistemically (cognitively), semantically, ontologically (topically) robustly independent, preeminent (i.e. a priori) valid disciplines in their own right and do not depend on nor are subservient in any way on natural science - rather they furnish vital elements necessary for the progress of natural science (and this includes ethics, of course).

There is one name for the synthesis of many of the worst philosophical errors of the past: (neo)pragmatism, linguistic pragmatism, meaning-as-use - Pittsburgh School conceptualism and inferentialism.

To the most advanced among the exponents of the New Age logic even this is not enough. Why, they ask, cling dogmatically to consistency ? Why not jettison the law of non-contradiction (...) Men of action (the Lenins and Hitlers of this world) have long been familiar with the advantages of embracing contradictions. They know that it not only neatly solves all problems in logic proper, but provides an intellectual key to 'final solutions' in other fields of human endeavour. (Pawel Tichý)

Just because Rorty was 'canceled' by the reigning philosophy does not make him ipso facto into some kind of hero of truth valiantly defending a radical alternative; on the contrary it can well be that his program was actually worse that the status quo and just represented in a more thorough way its ultimate consequences or original motivations. Rorty was the Trotsky of analytic philosophy.

https://chryssipus.blogspot.com/2023/10/pyrrhonian-strategy-in-rortys-mirror-of.html

Regarding Rorty let us quote from J.N. Mohanty's The possibility of transcendental philosophy (1985) p.59 :

Impressive as he is in his scholarship, he has given very few arguments of his own. He uses Sellars' arguments against the given and Quine's against meaning, as though they cannot be answered, but he has done little to show they cannot be. He plays one philosopher against the other, and would have one or both dismissed, according as it suits his predelineated moral. These are rhetorically effective but argumentatively poor techniques. What does it matter if Sellars rejects the concept of the given - one may equally rhetorically ask - if there are other good philosophers who accept the viability of that concept? There is also an implied historicist, argument that has little cutting edge. If the Cartesian concept of the mental had a historical genesis (who in fact ever wanted to say that any philosophical concept or philosophy itself did not have one?) whatever and however that origin may be, that fact is taken to imply  that there is something wrong about the concept.

The Young Carnap's Unknown Master

https://www.routledge.com/The-Young-Carnaps-Unknown-Master-Husserls-Influence-on-Der-Raum-and-Der-logische-Aufbau-der-Welt/Haddock/p/book/9780754661580

Examining the scholarly interest of the last two decades in the origins of logical empiricism, and especially the roots of Rudolf Carnap’s Der logische Aufbau der Welt (The Logical Structure of the World), Rosado Haddock challenges the received view, according to which that book should be inserted in the empiricist tradition. In The Young Carnap's Unknown Master Rosado Haddock, builds on the interpretations of Aufbau propounded by Verena Mayer and of Carnap's earlier thesis Der Raum propounded by Sahotra Sarkar and offers instead the most detailed and complete argument on behalf of an Husserlian interpretation of both of these early works of Carnap, as well as offering a refutation of the rival Machian, Kantian, Neo-Kantian, and other more eclectic interpretations of the influences on the work of the young Carnap. The book concludes with an assessment of Quine's critique of Carnap's 'analytic-synthetic' distinction and a criticism of the direction that analytic philosophy has taken in following in the footsteps of Quine's views.

Stephen Hicks in Explaining postmodernism

Showing that a movement leads to nihilism is an important part of understanding it, as is showing how a failing and nihilistic movement can still be dangerous. Tracing postmodernism’s roots (...) explains how all of its elements came to be woven together. Yet identifying postmodernism’s roots and connecting them to contemporary bad consequences does not refute postmodernism.

What is still needed is a refutation of those historical premises, and an identification and defense of the alternatives to them. The Enlightenment was based on premises opposite to those of postmodernism, but while the Enlightenment was able to create a magnificent world on the basis of those premises, it articulated and defended them only incompletely. That weakness is the sole source of postmodernism’s power against it. Completing the articulation and defense of those premises is therefore essential to maintaining the forward progress of the Enlightenment vision and shielding it against postmodern strategies.

The names of the postmodern vanguard are now familiar: Michel Foucault, Jacques Derrida, Jean-François Lyotard, and Richard Rorty. They are its leading strategists.

Members of this elite group set the direction and tone for the postmodern intellectual world.

Michel Foucault has identified the major targets: “All my analyses are against the idea of universal necessities in human existence.” Such necessities must be swept aside as baggage from the past: “It is meaningless to speak in the name of—or against—Reason, Truth, or Knowledge.”

Richard Rorty has elaborated on that theme, explaining that that is not to say that postmodernism is true or that it offers knowledge. Such assertions would be self-contradictory, so postmodernists must use language “ironically.”

Against this Kantian ethics postulates:

1. Moral dignitarianism, the anti-egoistic, anti-utilitarian, and anti-relativistic universalist ethical idea that every rational human animal possesses dignity, i.e., an absolute, non-denumerably infinite, intrinsic, objective value or worth, beyond every merely hedonistic, self-interested, instrumental, economic, or utilitarian value, which entails that we always and everywhere ought to treat everyone as persons and never as mere means or mere things, and therefore always and everywhere with sufficient respect for their dignity, no matter what merely prudential reasons there are to do otherwise.

2.  Political dignitarianism, the anti-despotic, anti-totalitarian, and anti-Hobbesian- liberal yet also liberationist, radically enlightened idea that all social institutions based on coercion and authoritarianism, whether democratic or not-so- democratic, are rationally unjustified and immoral, and that in resisting, devolving, and/or transforming all such social institutions, we ought to create and sustain a worldwide or cosmopolitan ethical community beyond all borders and nation-States, consisting of people who who think, care, and act for themselves and also mutually sufficiently respect the dignity of others and themselves, no matter what their race, sex, ethnicity, language, age, economic status, or abilities.

Husserl:

 Whatever is true, is absolutely, intrinsically true: truth is one and the same whether men or non-men, angels or gods apprehend and judge it. Logical laws speak of truth in this ideal unity, set over against the real multiplicity of races, individuals and experiences, and it is of this ideal unity that we all speak when we are not confused by relativism.  

P. Tichý (Foundations of Frege's Logic):

Fate has not been kind to Gottlob Frege and his work. His logical achievement, which dwarfed anything done by logicians over the preceding two thousand years, remained all but ignored by his contemporaries. He liberated logic from the straight-jacket of psychologism only to see others claim credit for it. He expounded his theory in a monumental two-volume work, only to find an insidious error in the very foundations of the system. He successfully challenged the rise of Hilbert-style formalism in logic only to see everybody follow in the footsteps of those who had lost the argument. Ideas can live with lack of recognition. Even ignored and rejected, they are still there ready to engage the minds of those who find their own way to them. They are in danger of obliteration, however, if they are enlisted to serve conceptions and purposes incompatible with them. This is what has been happening to Frege's theoretical bequest in recent decades. Frege has become, belatedly, something of a philosophical hero. But those who have elevated him to this status are the intellectual heirs of Frege's Hilbertian adversaries, hostile to all the main principles underlying Frege's philosophy. They are hostile to Frege's platonism, the view that over and above material objects, there are also functions, concepts, truth-values, and thoughts. They are hostile to Frege's realism, the idea that thoughts are independent of their expression in any language and that each of them is true or false in its own right. They are hostile to the view that logic, just like arithmetic and geometry, treats of a specific range of extra-linguistic entities given prior to any axiomatization, and that of two alternative logics—as of two alternative geometries—only one can be correct. And they are no less hostile to Frege's view that the purpose of inference is to enhance our knowledge and that it therefore makes little sense to infer conclusions from premises which are not known to be true. We thus see Frege lionized by exponents of a directly opposing theoretical outlook.

Schopenhauer vs. Schopenhauer

The questions Shapshay asks in her book and her theory of an internal contradiction or tension in Schopenhauer regarding compassion vs. renunciation are very incisive and relevant to our thesis, which is as follows.

1. Schopenhauer had an imperfect grasp of ancient Indian philosophical and spiritual traditions.

2. Schopenhauer's theory of renunciation was largely colored by Christian mysticism (from the middle ages to the 17th-century) and in particular by Eckhart and Luther.

3. This lead to a miscomprehension  and misreprentation of ancient Indian traditions due to a falsely postulating their essential unity with Christian mysticism in so far as being brought together under the heading of the common phenomenon of renunciation and negation of the will.

4. Christian mysticism and many important ancient Indian traditions (in particular original Pali Buddhism, Samkhya and the Yoga of Patanjali) are mutually antagonistic and irreconcilable. For the practice promoted by such traditions (called in Pali bhâvanâ) can be seen as the consequence to one the two fundamental sides of the positive ethics of compassion : compassion for others and compassion for self. For instance, the corner-stone of original Buddhism is the rejection of causing suffering to others and practices involving self-torment or causing suffering to self. We have the duty both to cultivate the alleviation of the suffering of others and the suffering of our own self (to do: investigate Kantian aspect).  This is radically opposite to Christian mysticism. For suffering (of the agent) is an instrumental, circumstantial, empirical cause for practicing compassion and self-development but never an essential or sufficient cause; there is no ethical or social value in suffering per se be it voluntary or involuntary.  This completely rules out the legitimacy of the concepts of vindictive (as opposed to preventive and corrective) justice as well as vicarious atonement and of course all misguided forms of asceticism based on mental or physical self-harm.

5. Such Indian traditions completely evade the important objections raised by Shapshay and are fully compatible with the ethics of compassion and hope.  It is the theory of art in WWR3 (rather than the theory of renunciation in the fourth book) that offers a far more accurate philosophical interpretation of the effects and ultimate aim of self-development.

Project

1. Natural deduction and quantifier logic in ancient philosophy.

2. What was Kant's logic in the CPR ? Was it adequate even to express the transcendental analytic ?

3. Claire Ortiz-Hill's analysis of equality and identity in Frege and Husserl in the light of dependent type theory and in particular homotopy type theory. How Gentzen and Martin-Löf show us the most promising path in philosophy.

https://chryssipus.blogspot.com/2023/11/equality-and-sameness-from-frege-to.html

Our considerations on 'holology' and higher category theory - are in fact extremely relevant to the philosophy of concepts, objects, extensions, abstractions and intensions all concerning which ancient philosophy has many important things to say. Why should the 'object' that is an 'extension' of a 'concept' be a set rather than an $n$-groupoid ? How do we account for 'some' in mass-nouns and propositional attitudes ?

4. (Book) Kant, Schopenhauer, Husserl (both of the earlier and later phase) and certain ancient eastern traditions: logic, epistemology and ethics with reference to the interpretations of Hanna, Schulting and Shapshay.

Kant and Husserl in their 'cognitive semantics' agree remarkably well  with the basic architecture of the mind (or consciousness) layed out in the Pali suttas. But in some fundamental points, in which he differs or corrects Kant, Husserl is closer and in other points (logical, dialectical) Kant is. As for ethics, we might consider a synthesis of Kant and Schopenhauer.

4a. Original Buddhism was neither empiricist (in modern terms) nor relativist. And neither were Pyrrho and Sextus. 

5. All philosophers at the table: what is axiomatic philosophy, why it matters and how it is possible. 

5a. Computer assisted axiomatic philosophy using dependent type theory.

6. Theory of theory, theory of proof and genealogy of the theory of definition.

7. The Hegelian Kant, Husserl and category theory as universal ontology.

8. In defense of analyticity and refutation of inferentialism/proof-theoretic semantics and of pragmatic, social, relativist and coherentist accounts of truth, meaning and inference.

9. Formalize Porphyry's Eisagoge (done) and pin-point difficult questions and uncertain aspects.

(...)

Leibniz's dream is more than a dream

Leibniz's mathesis universalis, characteristica universalis and calculus ratiocinator are more than dreams or utopias. Nor is talk of formal philosophy mere metaphilosophical speculation and wishful thinking.

Zalta's Object Logic in its three degrees of unfolding (each subsuming the previous one) offers a non-trivial axiomatization and formal proofs of some interesting aspects of three great systems of ontology: Plato, Leibniz and Frege. Both the series of Object Logics and the series of these three ontologies can be given a Hegelian interpretation.

Furthermore this axiomatic metaphysics can be embedded and expressed in dependent-type theory. Here are some examples in Coq.

Plato: 

https://github.com/owl77/CoqFormalisations/blob/main/zalta2.v

Leibniz:

https://github.com/owl77/CoqFormalisations/blob/main/zalta3.v

Frege (for now just an embedding of modal typed object logic)

https://github.com/owl77/CoqFormalisations/blob/main/zalta4.v 

Kant, Husserl and beyond

We recommend Corijn van Mazik's paper Husserl’s covert critique of Kant in the sixth book of Logical
Investigations (2018).  Also Robert Hanna's  Kant and the Foundations of Analytic Philosophy (2001).  

Both Kant and Husserl (of the 5th and 6th Logical Investigations) lay out a theory of the structure of the human mind, of consciousness, cognition and experience which does not forget ontology nor (bodily) sensation.  Both Kant and Husserl offer a logic and a theory of objectivity, perhaps both platonic and constructivist (using dependent type theory shown to express (a substantial) part of universally valid laws of reason for rational agents).  There is a Leibnizean mathesis (universal ontology) project looming in both Kant and Husserl - and Hegel's Logic combined with modern category theory (and model theory) seems a promising way of realizing it.

We find that both architectures, while insightful and brilliant, are yet radically insufficient, both from below and from above.  

From below because they neglect, unlike classical philosophers like Aristotle (see our paper on De Anima), to take into account many crucial psychological and physiological elements in the structure of consciousness, such as a the key role of feeling, desire and habit in cognition and mental experience, as well as the psycho-physiological act of perception.

From above: this radical insufficiency was overcome by Husserl, following the supremely important critique of Kant implicit in the Logical Investigations, in his famous so-called transcendental turn. But in Husserl's discovery there still lurked the danger of not understanding how transcendental subjectivism is at the same time objective absolutism. Also there are some dangerous equivocations surrounding the term ego in 'transcendental ego'.  It would have been better and safer to adopt a purely negative approach and to speak of a transcendental consciousness not conditioned by an ego. Also missing in Husserl is the crucial connection between the awareness of universal temporality/temporalization and the transcendentality of transcendental consciousness. Missing is the description of how the worldly ego is constituted.

However there are also many important aspects in Kant's transcendental dialectic which were more or less overlooked or not given prominence by Husserl: the notions of inconsistency and incompleteness (undecidability) of conceptual and logical systems viewed formally. Kant's transcendental dialectic is in fact a more developed and consequential version of certain elements of Pyrrhonism. How can Kant's critical (transcendental) knowledge escape the bounds set by his theory of knowledge ?

Hanna talks a lot of 'embodiment' and throws the charge of solipsism at Husserl.  But the correct unfoldment of transcendental subjectivism involves apodeictic realism regarding other consciousnesses, human and otherwise,  together with the reality of the first-person (human or otherwise) experience of the body, of embodiment. Can we reconcile Kantian ethics and Schopenhauer's ethics of universal compassion ?

This is how Robert Hanna articulates Kantian morality and its political implications:

1. Moral dignitarianism, the anti-egoistic, anti-utilitarian, and anti-relativistic universalist ethical idea that every rational human animal possesses dignity, i.e., an absolute, non-denumerably infinite, intrinsic, objective value or worth, beyond every merely hedonistic, self-interested, instrumental, economic, or utilitarian value, which entails that we always and everywhere ought to treat everyone as persons and never as mere means or mere things, and therefore always and everywhere with sufficient respect for their dignity, no matter what merely prudential reasons there are to do otherwise.

2.  Political dignitarianism, the anti-despotic, anti-totalitarian, and anti-Hobbesian- liberal yet also liberationist, radically enlightened idea that all social institutions based on coercion and authoritarianism, whether democratic or not-so- democratic, are rationally unjustified and immoral, and that in resisting, devolving, and/or transforming all such social institutions, we ought to create and sustain a worldwide or cosmopolitan ethical community beyond all borders and nation-States, consisting of people who who think, care, and act for themselves and also mutually sufficiently respect the dignity of others and themselves, no matter what their race, sex, ethnicity, language, age, economic status, or abilities.

Finally we must give an account of aesthetics (including platonic and neoplatonic theories) and how it positively harmonizes with philosophy and ethics. We must value beauty greatly in itself but be realistic about its context and the way it manifests in the process of human life.

Theory of mind

The fate of modern western philosophy involves, it seems, the conflict (and attempt at harmonization) between psychology, logic and the perceived 'laws of nature'.  It has always had strong tendency to dualisms, meta-isms (confusing going beyond something with overcoming it),  reductionisms (in the materialist or idealist directions) or ad hoc amalgmations, constructions, neologisms (intensionality, schematism, etc). There has always been a deep confusion and misguided mixture between the mental and the logical which seems hardwired into the terms themselves like 'concept' and 'idea'.  Schopenhauer accuses Kant of a heilosen Vermischung der intuitiven und abstrakten Erkenntnis. Frege and Husserl both struggled with this. Earlier analytic philosophy likewise attempted to evade the psychological in ways less cogent. It seems that by trying to analyze the psychological and the logical the mind inevitably puts both into each one of them and also into their very relationship.

Let us drop  logicism, psychologism and meaning-as-use for the moment, and drop all reductionist dogma. Let us engage in the philosophy of mind, starting from basic, down-to-earth, common-sense considerations.  From the most basic and obvious and neglected it is possible to ascend to the most subtle, all-encompassing and unexpected.  An organic system-theoretic thinking about the larger role of sensation and perception and its multilateral connection without any artificial assumptions. Bringing in the importance of the body and physiology to the mind without trying to restrict or reduce the mind.  For the mental has a continuous gradation (analogous to the gradations in energy and frequency in physics) of yet qualitatively distinct levels all bearing an organic inter-dependence.

While we distinguish between formal and conceptual-philosophical clarity we hold that in philosophy both are even more indispensable and fundamental than in mathematics. Clarity and transparency are the highest virtues in philosophy. And, without implying completeness or achievement,  formal clarity in philosophy would be, in perspective, a huge step forward. We do not see modern philosophy as excelling ancient philosophy in clarity or rigour nor do we see "analytic" philosophy (excluding some notable exceptions) as exhibiting greater clarity (formal or conceptual) than "continental" philosophy. Lack of clarity, formal, conceptual, not to mention deductive, has been the original sin of modern philosophy.  The clearest philosophical "concept" is still murkier and vaguer than the most complex scientific or mathematical one.  There is hardly a single example of a definite philosophical proof. The idea of focusing on the analysis of language as a surrogate for philosophy is circular  (the metalanguage is not less complex than the object language) and leads to even greater confusion and ambiguity. This is in ironic and tragic contrast to the huge development of mathematics in the direction of formal and deductive clarity. 

We need a new language for philosophy. And we need a new methodology and a new form of intuition and insight.

If it is one of the tasks of philosophy to break the domination of the word over the human spirit by laying bare the misconceptions that through the use of language often almost unavoidably arise concerning the relations between concepts and by freeing thought from that with which only the means of expression of ordinary language, constituted as they are, saddle it, then my ideography, further developed for these purposes, can become a useful tool for the philosopher. -  Frege, Begriffsschrift, Preface.

There have been two really significant discoveries and projects in philosophy. The first, objective, corresponds to Leibniz's idea of a characteristica universalis, calculus ratiocinator and mathesis universalis. This objective philosophy was, although this is little known, given substantial and valuable development in the earlier work of Edmund Husserl (though mixed and alongside other concerns and investigations: the unfortunate mixture and confusion we discussed above). It is thanks to the pioneering work of Claire Ortiz-Hill that the purely objective Husserl has been unearthed and brought to light. This aspect of Husserl is not incompatible with formalization and with Leibniz's project, rather it amounts to a substantial contribution to it. It offers devastating arguments against Carnap and Quine within an illuminating historical perspective on Fregean extensionalism.  We equate this objective Husserl with the Leibnizean project of a formal philosophy and it is an urgent task to gather together all subsequent significant work which can be seen as contributing to this project (for instance, Zalta's project).

This is no vague utopian project.  A modal type theory with special distinguished metaphysical predicates (for platonic participation, abstract objects, representation, relativization, etc.) can already accomplish something. There is nothing wrong with initial modesty, with employing weaker or partial systems first to elaborate solid results in axiomatic philosophy. Type theory is a kind of paradise of thought and yields substantial results - this without claiming any type theory to be complete or absolute.

We posted previously about a Hegelian interpretation of the various interrelated systems of type theory. Zalta's book Axiomatic Metaphysics book can be given a Hegelian interpretation. Indeed three systems are presented in order of successive strength, each subsuming the previous one. These three systems are deployed to formalize three successively more complex ontologies which also reflect a historical progression: the Platonic theory of forms, Leibniz's theory of possible worlds and monads and Frege's theory of objects, concepts and senses. To do: embed all these systems in dependent type theory or more specifically in Coq.

The second really significant progress in philosophy was Husserl's discovery of transcendental consciousness, which we might conveniently called the 'subjective' Husserl.  It is our task to unearth to true meaning of Husserl's discovery and to guard it against subsequent misunderstandings and appropriations (existentialist, naturalist, etc.).  The objective and subjective Husserl do not contradict each other, they are complementary, though the subjective Husserl remains the most profound, all-encompassing and important perspective. The key to re-establishing the 'subjective' Husserl against modern philosophical aberrations lies in its connection to ancient philosophy, specially ancient Eastern philosophy.

Pippin is right to criticize a reading of Kant’s Deduction that takes there to be a nonconceptual content that (1) is in and of itself objectively valid and (2) is built upon by means of subsequent acts of judgement (or understanding), such as for example Robert Hanna (2008) believes. Such readings do not make sense in the Kantian context, where it is precisely the goal of the Deduction to demonstrate how it is possible that we can determine, a priori , how thought content and sensory content hook up inwardly , which justifies our conceptual claims about empirical objects. If the contribution by sensible content, more precisely, ‘transcendental content’, were really supplied ‘from the outside’, one would be none the wiser from any argument in the Deduction that supposedly showed how we are justified in making claims about objects, how pure concepts are justifiably (necessarily) employed in any judgement that says that some a is F. If it were true that such content is supplied from the outside, Kant could not have shown the fundamental intimacy between the pure concepts and empirical knowledge of objects, precisely the goal of the Deduction. Apperception and Self-Consciousness (2021).

Also we need to take heed of Bolzano's and Brentano's criticism of Kant's notion of analyticity. True logics form a tightly-knit organic whole or family (there is nothing remotely arbitrary about when a formal system constitutes or not a genuine logic).  Thus analyticity = a certain type of logical consequence. There is minimal analyticity, intuitionistic analyticity, classical analyticity, linear analyticity, relevant analyticity, etc.  Kant's notion is mistaken in the literal sense (which seems to betray strictly monadic thinking). However 'containment' can be given an interpretation in terms of 'unpacking' definitions and constructive type theory. This will be elaborated in our theory of definition and proof. We will examine Robert Hanna's slightly pedantic and convoluted treatment of analyticity in 'Kant and the Foundations of Analytic Philosophy'.  Hanna is spot on about 'microstructure'.  However the framing of analyticity in terms of necessary extensional equivalence is at least questionable as is several features of what he says about formal logic in general.

We have finally the correct purification and separation between the objective and the subjective. But both belong intrinsically to our 'rationality'. But wait...what happened to 'formalism is not clarity' ?  This clarity cannot be sought for directly in the objective beyond the axiomatic-deductive method. The subjective Husserl gives the sought for clarity, that is, shows it, rather than expresses it.

Gödelianism (see M. van Atten and J. Kennedy, The Philosophical Development of Kurt Gödel) postulates that there can be more than one valid method for attaining philosophical truth, more specifically,  both Husserl's transcendental subjectivism and Leibnizean objectivism can be equally valid and complementary philosophical paths. For Gödel the Kantian project extends to all possible minds, not just to the conditions of the specifically human mind.

Category theoretic view of definition

 View the objects of a category as Aristotelic species/genera.  Why not compare the process of definition with canonical algebraic constructions ? That is, given $A$ seen as a genus, one of its species $B$ results from a particularization of $A$ through some condition. Can we view this as being expressed in the standard generators and relations presentation (or quotient construction) ?

\[ R \hookrightarrow A \mapsto B \]

As in our upcoming paper we propose that modern mathematical definitions (specially as practiced in proof assistants) go beyond orthodox Aristotelic ones in that they can consider products of genera -   new species are generated also from inducing conditions of products of genera and taking the quotient:

 \[ R \hookrightarrow \Pi_I A_i \mapsto B \]

Think of the different notions of 'generators' of a category.  Of course this works better for strong logical restrictions of the kinds of objects we are considering (consider the HSP theorem in Universal Algebra). This approach could also have strong roots in ancient philosophy, for instance seeing the generators as basic kinds of logoi in a Stoic (see post on Stoic categories) and neoplatonic sense. The above considerations are not meant to be rigorous but only a kind of logico-mathematical metaphor.  But we could be more precise in a topos-theoretic context.

Plato's Sophist and Type Theory (older post)

Suppose we had a type $A \rightarrow B$. Then application $a : A, f : A\rightarrow B \vdash f a : B$ can be 'internalized' as a type \[ A\rightarrow (A\rightarrow B) \rightarrow B \] But application of this type can likewise be internalised as\[ A \rightarrow (A \rightarrow B) \rightarrow (A\rightarrow (A\rightarrow B) \rightarrow B ) \rightarrow B \] and so forth. This is similar to the 'third man' argument of the Parmenides. Take $B$ to be the 'truth-value' type $\Omega$. The 'canopy' argument in fact seems to herald the idea that a type (i.e. a 'form' or 'unsaturated' propositional function) should be seen not only a 'set' but as a 'space' as in homotopy type theory. In a passage of the Sophist 237 there is a remarkable discussion on 'non-being'. You cannot talk about non-being because by doing so you already attribute it implicitly the mark of a something, an ought - both unity and being. Now this passage is in many ways an anticipation of the rule of contradiction (for $\bot$) in natural deduction as well as its logical deployment in axiomatic set theory, specially dealing with $\emptyset$ in formal proofs. But most interesting is the connection to Martin-Löf type theory: the zero type or empty type $\mathbb{O}$. This type is not inhabited by anything. But yet to use this type, to reason with it, you must assume that it is inhabited $ a : \mathbb{O}$.  Martin-Löf type theory allows us to flesh out Plato's intuition about the connection between falsehood, nothingness and absurdity. The empty set is a paradigm of the initial object of a category. There is a unique set theoretic function $f : \emptyset \rightarrow X$ for a given set $X$. Correspondingly the type $\mathbb{O}\rightarrow A$ is inhabited where $A$ is for example some non-empty inductive type. Thus we can speak meaningfully about nothing. Consider the very difficult passage Sophist 243-246 that clearly picks up on and rectifies the Parmenides. But for Plato what are 'Being', 'Unity' and the 'Whole' ?  We can consider the unit type $\mathbb{I}$ as corresponding to 'Unity' and think of it as either a set-theoretic singleton $\{u\}$ or as a contractible type where equality is interpreted as a homotopy path. In this sense it is a homogenous space much like the 'sphere' in Parmenides' poem. Plato does indeed distinguish between pure unity and a whole participating of unity. The singleton set is a paradigm of a terminal object in a category.
The 'Whole' is clearly a 'universe' type $U$ or $Type$.  It is a difficult problem to relate the 'Whole' to 'Unity'. What does it mean even for the 'Whole' to participate of 'Unity' ? That there is one supreme universe $Unity$ with only one inhabitant $Type : Unity$ ? But then we could not have accumulativity : $ a : Type$ implying that $ a : Unity$. Or, categorically, how do we interpret that all objects $A$ admit a unique morphism $A \rightarrow 1$ ? Category theoretically a ($\infty-\enspace$) groupoid is a candidate for a category 'participating in unity'. There is a tension between unital being $\mathbb{I}$ and the being-whole, the being spread out and shared by all beings $Type$. The logic in this passage is apparently 'type-free' and impredicative. 'Being' $: \Pi ( X: Type), \Omega$ and 'Unity': $\Pi (X : Type), \Omega$ seem to be able to be applied meaningfully to anything (indeed $Type : Type$). The passages on 'names' and 'reference' is quite striking. Specially when Plato conjures up 'names that refer to names' and a name referring only to itself (imagine a type only inhabited by itself). Part of Plato's argument is that no matter what in reality 'is' the fact is that there is a plurality of \names. In the later section on the five suprema genera we may ask: why is there not also a form for 'participation' itself ?What are some fundamental kinds of types ? The empty type $\mathbb{O}$, the unit type $\mathbb{I}$, the universe(s)  $\mathcal{U} $ (or $Type_i$), equality $ \Pi ( X\enspace Y: Type), Prop $, number $Nat : Type$ and the interval $\mathbf{I}$ for path spaces or cubical sets in homotopy type theory. Paths represent change, the interval represents temporality. Plato speaks of equality being or not being equal to something, just as Voevodsky speaks of equality being equivalent to equivalence.

Note on the Stoic Categories (older post)

We plan to study the formalization of Stoic logic and the Stoic categories.  I had made a close study of Bealer's  first-order intensional  logic such as presented in his book Quality and Concept. Unlike most approaches  Bealer takes propositions, unary predicates, relations, etc. to be primitive entities, units of meaning which are woven together by complex logico-combinatoric relations including the attribution of truth-value depending on states-of-affairs.  Afterwards I read Susanne Bobzien's papers on Stoic logic and specially  Did Frege plagiarize the Stoics which offered insights into both the Stoics and Frege. I then realized that the Stoic theory of lekta was an intensional version of Frege's logic and that not only did it agree with Bealer but my experience working with  dependent type theory in the Coq proof assistant suggested to me an alternative intensional type-theoretic  version of the semantics of Bealer's logic which is closely aligned to Stoic logic. There is a primitive type of 'saturared' lekta $\Lambda$  corresponding to Bealer's set of propositions $\mathcal{D}_0$,   a primitive type of truth-values $\Omega$ and a primitive type $W$ which can be thought of as indexing possible states-of-affairs or 'possible worlds'.  Our logic is essentially a logic of meaning and all operations are defined primitively on $\Lambda$ or types involving $\Lambda$ rather than on $\Omega$ (like in topos theory). The logic of truth and extensions is mediated by what I call the alethic term $\alpha : W \times \Lambda \rightarrow \Omega$ which specifies which assertibles hold in a given situation or state-of-affairs. As Bobzien writes in the Cambridge Companion (2006): The most far-reaching one is that truth and falsehood are temporal properties of assertibles. They can belong to an assertible at one time but not at another.  Note that in the Calculus of Constructions upon which Coq is based we could consider taking the type Prop as $\Lambda$. I found that  section II  of Logic and General Theory of Science (lectures from 1917/1918)  Husserl carries out an analysis of natural language which corresponds closely to the type-theoretic intensional logic I had in mind. I noted that in paragraph of section II Husserl writes: A grammatical distinction passed down from Scholasticism, and otherwise going back to the Stoics, can serve as our point of departure. This is the distinction between independent and dependent expressions.

There is also the enigma of the Stoic categories. An old paper by Margaret Reesor (1957) presents some interesting information about the category of 'relation'  pros ti  and specially pros ti pôs ekhôn translated as 'relative disposition'.   It seems that the older Stoics considered such  terms as 'intelligence' and 'virtue' both in themselves simply and qualified and related to something else.  For instance for Zeno courage is wisdom in things to be endured and justice is wisdom in things to be distributed.  It does not seem straightforward to present this in terms of the modern predicate calculus, for instance a binary relation $\exists y.R(x,y)$ instantiated differently as $ R_1(x):=R(x,a),  R_2(x) := R(x,b)$, etc. Rather this situation recalls polymorphism in modern type theory or its more general form involving dependent type theory with universes.  This can be understood by an example from programming languages where there is a general class for 'list'  taking a type as a parameter which then becomes a concrete object according to the specified type (we get lists of integers, lists of boolean values, lists of strings, etc.).  Thus dependent types such as $\Pi_{T: Type} Type$ would seem to capture the Stoic category of relation which includes 'virtue' and 'intelligence'. These are both  types in themselves and  types which can be further determined through specification of an external type $S$ which we obtain through application $(\Pi_{T: Type} Type) S : Type$.  Thus we can speak of 'list' simply or 'list of $X$s', 'list of $Y$s', etc. Likewise we have both 'wisdom' and 'wisdom in $X$', 'wisdom in $Y$' for types X and Y.

What of the concepts of  dependent and independent meaning ?  The type $\Lambda$ is certainly independent. The corresponding dependent types are those that can be written equivalently (i.e. via currying) in the form $X \rightarrow \Lambda$. Thus for example 'or' would have type $\Lambda\rightarrow(\Lambda \rightarrow \Lambda)$ which is equivalent to $\Lambda\times\Lambda \rightarrow \Lambda$.  But there are other types of independent and dependent meanings.  A preliminary proposal might be that independent meanings are atomic types and dependent meanings non-atomic types.

A dependent meaning can be transformed into a independent one, for instance when we talk about 'the connective or'.  This can be handled by a term $ n_X : (X\rightarrow \Lambda) \rightarrow N$ which transforms a dependent meaning type into for instance a noun term of the atomic type $N$. This is of course also Bealer's  intensional abstraction /nominalization operator.

Formalism is not clarity

Ich setze also voraus, daß man sich nicht damit begnügen will, die reine Logik in der bloßen Art unserer mathematischen Disziplinen als ein in naiv-sachlicher Geltung erwachsendes Sätze system auszubilden, sondern daß man in eins damit philosophische Klarheit in betreff dieser Sätze anstrebt (...) - Husserl I Log. Unt.

On a surface level Aristotle's Organon and Physics are formally impressive and from a contemporary mathematical point of view quite suggestive.  Yet, if we analyze things very carefully we find that at a deeper level we are in the presence of a big step backwards from Plato which also cannot really be compared to the sophistication and brilliance of the Stoics. For in Aristotle the key fundamental terms and concepts ("term", "concept", "predication", "essential predication",  "proposition",  "huparkhein", "quality", etc.) are never defined, elucidated, clarified  and perhaps not even used consistently. There is also a serious lack of grammatical and linguistic analysis . To study Aristotle it is not enough either to engage in traditional "classicist"  or commentary-based methods of exhaustive textual analysis and nitpicking or to think that somehow modern mathematical or symbolic logic in itself is sufficient tool to clarify all problems.  Rather we must deploy what is scientific and sophisticated in modern philosophy to bring to light what lies beneath the surface of the Aristotelic texts. Fortunately we do have a kind of philosophical Principia Mathematica, and this is Husserl's Logical Investigations and other subsequently published and equally important texts complementing and developing this work.

Recall Husserl's distinction between judgments of existence and judgment of essence.  Can this help us understand the universal quantifier ? Consider:

1.All ducks can swim.

2.All people in this room are under 30.

3. All prime numbers greater than 2 are odd.

 What does 1) mean, what do we mean by 1).  That swimming is part of the definition of duck, that being able to swim is a logical consequence of the definition of duck - and here we are assuming an artificial consensus - or that everything belonging to the extension of duck (for instance, we could just take a heap of things and label it "duck") happens to have the property of being able to swim (this is unlikely or at most genetic, plausible) ?  For 2) we cannot state that being under 30 somehow is a logical consequence of the concept of being in this room.  2) is definitely a Husserlian 'judgment of existence'.  3) can be given an extensional reading but it also could be given a logical reading in the sense that being odd follows from the definition of being prime and the condition of being greater than 2.  Thus 3) differs from 1) and 2) by allowing both interpretations. 3) can also mean: there is an algorithm which takes as input a prime number and a proof that this number is greater than 2 and yields as output a proof that it is odd.

But consider a model-theoretic approach.  For a model $M$, representing the current world, or current global state-of-affairs, we may well have that $M \Vdash \forall x. \phi(x) \rightarrow \psi(x)$ without it being the case that for our theory $T$ we have  that $T \vdash \forall x. \phi(x) \rightarrow \psi(x)$. But the statement $M \Vdash \forall x. \phi(x) \rightarrow \psi(x)$ must itself be proven in some metatheory $T'$ and is thus again purely logical.

The extensional interpretation of 1) can be: i) that things in the extension of "duck" have the property of being able to swim. ii) that the extension of "duck" is contained as a set in the extension of "being able to swim". 

More profound is the dependent-type theoretic interpretation $\vdash p: \Pi_{ x : D} S(x)$ which reads: there is a function $p$ which takes as input a duck and yields a proof that that particular duck can swim. Compare this to Bobzien and Shogry's interpretation of Stoic quantification:

If something is a duck then that duck can swim. 

How far we are from understanding quantifiers, concepts, extensions and predication in general !

Radical mathematical logicism is the position that logic (or pure rationality) only exists fully in mathematics (and mathematical models in science).  Natural language can only attain an approximate rationality via a mathematical pragmatics (as in computer science).

There is, at first sight at least,  a huge chasm between our mathematics and the complex organic self-directed concreteness of living systems and consciousness. But this chasm can be bridged if we study mathematical theories qua theories, their diachronic and synchronic systemic articulation and organicity seen as an abstract version of consciousness and life. 

If in mathematics both formal and conceptual clarity are of great importance, in philosophy they are even more so.  While agreeing with the quote of Husserl we do not undermine the greatness of formal clarity and the huge progress in philosophy that, in the scale of things, would be achieved by a formal philosophy even if this not mean the ultimate clarity and the highest development of the philosophical project.