Afrikan Spir (1837-1890) - Denken und Wirklichkeit (1873)

https://archive.org/details/penseetralit00spiruoft

This work is of considerable interest to the concerns of this blog. Spir belongs to that fascinating category of  'forgotten thinkers' who are only mentioned in the context of biographies of famous literary figures.

Spir (the name seems to be of Greek origin) was born in present Ukraine, but wrote in German. His father was a protestant military surgeon who was knighted.

Spir seems to have partially realized the goal of expressing the core philosophy of early buddhism within the tradition  of western philosophy with a particular emphasis on Hume (whom he calls 'the wisest of men') and the spirit (rather than the letter and specific content) of Kantian criticism.  Hume and epistemic relativism are placed on a consistent, coherent basis. However remarkably there is no specific mention of buddhism in the work above (though moral, humanitarian and spiritual concerns are central to his thought), but we note the equivalence between 'the norm' and dhamma/dharma.  Spir is certainly one of the most interesting 'Humeans' alongside Husserl and Meinong. Indeed his treatment of 'sensations' seems quite analogous to Brentano's and later Husserl's theory of intentionality.

A defect of much of phenomenological and empirical philosophy and psychology was not primarily considering 'impressions' and 'ideas'  as 'objects' of desire, attachment, obsession. And not considering the remarkable phenomenology of the process of overcoming such desires which is at once the most difficult of task and yet relatively easy if guided by the right insight - an insight that should be our key philosophical guide.

Poetry (such as Novalis and Hölderlin) should offer an alternative complementary mode and path to philosophical knowledge - giving equal importance to the scientific and to the literary/artistic.

Hegel's logic might be interpreted as extracting (in a kind of depth psychology) the schemata of thought, of consciousness. Hegel's logic in turn can be expressed, according to some, in category theory.  Hegel's absolute knowing is awareness of the multiplicity of structures of consciousness as well as their relativity and passing away into another - just as for Sextus spiritual peace involves awareness of the multiplicity of hypotheses and their equipollence without contradicting this awareness itself - this is itself similar to the Spirian-Humean process of enlightenment which finds the norm imitated by yet fundamentally incompatible with phenomena.

An interesting link to Hume is provided by the work of Sowa on conceptual structures (the percepts) - and more specifically the Humean-style work in cognitive psychology discussed in his book (and it is possible that some French philosophers and semioticians (Greimas) of the second half of the 20th century might be of interest as well) - and the connection to category theory might be provided by Goguen (initial algebras, institutions):

The lattice of theories of Sowa and the formal concept analysis of Wille each address certain formal aspects of concepts, though for different purposes and with different technical apparatus. Each is successful in part because it abstracts away from many difficulties of living human concepts. Among these difficulties are vagueness, ambiguity, flexibility, context dependence, and evolution. The purpose of this paper is first, to explore the nature of these difficulties, by drawing on ideas from contemporary cognitive science, sociology, computer science, and logic. Secondly, the paper suggests approaches for dealing with these difficulties, again drawing on diverse literatures, particularly ideas of Peirce and Latour. The main technical contribution is a unification of several formal theories of concepts, including the geometrical conceptual spaces of Gärdenfors, the symbolic conceptual spaces of Fauconnier, the information flow of Barwise and Seligman, the formal concept analysis of Wille, the lattice of theories of Sowa, and the conceptual integration of Fauconnier and Turner; this unification works over any formal logic at all, or even multiple logics. A number of examples are given illustrating the main new ideas. A final section draws implications for future research. One motivation is that better ways for computers to integrate and process concepts under various forms of heterogeneity, would help with many important applications, including database systems, search engines, ontologies, and making the web more semantic.

 https://link.springer.com/chapter/10.1007/11524564_4

Thus we wish for a theory of language and theory of mind based on concepts on a phenomenist-phenomenological (introspective-intuitive)  basis with roots in Hume, Kant, Mill, Brentano, Meinong and philosophical psychology (and cognitive psychology) possibly guided by  idealism, structuralism, computer science (in particular the relationship between assembly and high-level languages) and category theory (and categorical logic) - ultimately to result in a western elaboration of the philosophy of original buddhism.  We are radically opposed to  naturalism,  behaviourism and in general philosophies that reject the primordial foundational role of meaning and consciousness.  This philosophy is distinct from metaphysics or natural science.

And yet we do not even begin to have a clear solid idea of what self- introspection is or how it can be achieved in order to effect a systematic phenomenism and  philosophy of mind. That is, how does or can consciousness observe and know its own immanent process ? We can start by inquiring into belief. What beliefs do we hold ? And if so what does it mean for us to hold such a belief or rather what are the consciousness-contents which accompany or indeed constitute the state or act of believing something ?

Important figures: Brentano, Wundt (and other structuralists), Meinong, Lotze, Stumpf, Köhler,  Is the psychology of early Buddhism gestalt ? It is this work the constitutes the true 'phenomenology' or what we call 'philosophical introspective psychology'.

Philosophy embraces the whole

Philosophy concerns the entirety of human experience and human existence.  Artificial, shallow, abstracted segregated collections of concepts and their 'puzzles'  (cut off from ethics, anthropology, or the theory of the Platonic dialectic) cannot be authentic philosophy or have very much value. This false abstraction is not the abstraction of  mathematics, logic or theoretical science which on the contrary is living - has an intrinsic open-horizon - and permeates, even if implicitly, all of experience.

The Platonic dialectic is closely connected to ethics.  But few have explored the deeper significance of the ethical thought found in the dialogues of Plato and other key thinkers of antiquity (both eastern and western).  Popper's view is well known. Simone Weil (sister of mathematician André Weil) sensed that there was something very significant in the ethical thought of Plato and tried to argue for its proximity to Christianity.  How could Popper and Weil come to such drastically incompatible and divergent views regarding Plato's ethics ? Few have noticed the irony, circumspection as well as blunt iconoclasm in the dialogues - or indeed how radically progressive and enlightened some parts were relative to the social-political conditions of the time. It difficult to judge how 'shocking' certain views expounded by Socrates were perceived to be (such as, in the Gorgias, that it is better to suffer injustice than to commit it).  Although Socrates praised the frankness of those that expounded Thrasymachus-type views, it is still difficult to gauge just how mainstream such views might have been contemporary Athenian society.

Platonic insights into ethics are perhaps what is missing in our project of reconciliation of Kant and Schopenhauer (as well as the tantalizing question of Hegel's ethics).  Our ethics is one of the timeless universality and absoluteness of human and animal rights (cf. Plutarch, Porphyry and the account of Pythagoras given by Ovid)  as well as the duty to uphold and defend them.

Platonism offers us key insights into the investigation of the concept of 'intelligence'.

Our conclusion (which is ancient and is expressed for instance in the Theaetetus) is that intelligence is before all else and essentially the possession of the knowledge of what is right and wrong, the knowledge of what should and should not be done, the knowledge of what ought to be done or not done. And this includes not only how we should treat other human beings and animals but also the knowledge of the duty of self-cultivation, the knowledge that it is a duty to develop certain mental habits and exercises which are a preparation for Platonic dialectic (cf. the passage in the Republic beginning with: . But when a man's pulse is healthy and temperate, and when before going to sleep he has awakened his rational powers, and fed them on noble thoughts and enquiries, collecting himself in meditation (...)). If we make an analogy of these to diet, then we have to distinguish between intellectual 'health food' (such as problems in pure mathematics, formal logic, theoretical science, poetry, classical music, the fine arts, games such as chess and go, etc. - these are anagogic, they build, clarify and refine higher concepts)  and pseudo-intellectual 'junk food' (word games and puzzles based on the arbitrariness, vagueness, homophony and ambiguity of a specific natural language, magic tricks,  riddles and puns, games of gambling and chance, puzzles based on perceptual illusions, legal equivocation,  games of psychological manipulation, etc.). These last can be seen as the exercises and the bag of tricks of the sophist (although at a basic level there can be general strategies). No question of relevance to intelligence can depend on being formulated in a specific language (i.e. its ambiguity or fluid contingent semantic/phonetic associations).

But, O my friend, you cannot easily convince mankind that they should pursue virtue or avoid vice, not merely in order that a man may seem to be good, which is the reason given by the world, and in my judgment is only a repetition of an old wives' fable. Whereas, the truth is that God is never in any way unrighteous—he is perfect righteousness; and he of us who is the most righteous is most like him. Herein is seen the true cleverness of a man, and also his nothingness and want of manhood. For to know this is true wisdom and virtue, and ignorance of this is manifest folly and vice. All other kinds of wisdom or cleverness, which seem only, such as the wisdom of politicians, or the wisdom of the arts, are coarse and vulgar. The unrighteous man, or the sayer and doer of unholy things, had far better not be encouraged in the illusion that his roguery is clever; for men glory in their shame(...) - Theaetetus.

Indeed, is there anything more monstrous and ignoble than dominating and harming others (or wanting to do so)  or the appropriation and accumulation of resources (beyond one's basic needs) ? Or calling 'intelligence' the ability or practice of doing so ? Or having a 'culture' based on valuing this ? 

If every just man that now pines with want
Had but a moderate and beseeming share
Of that which lewdly-pampered Luxury
Now heaps upon some few with vast excess,
Nature's full blessings would be well dispensed
In unsuperfluous even proportion, (Milton, Comus 768-773)  

We plan to analyze carefully Popper's criticisms of Plato (which are rather obvious points) and the role of Sparta in Plato's thought. Also how contemporary ideology and junk psychology can hinder the appreciation and practice of platonic dialectic (for instance by denying that 'analytic' and 'intuitive' thought are inseparable).

In the previous post we wrote 'overcoming the illusion of the ordinary self and consciousness'.   Hegel's Phenomenology of Spirit can be seem as the path leading from the illusion of ordinary consciousness and self to that of absolute consciousness and its self-knowledge as spirit. Plotinus on the other hand makes a strict connection between the Platonic dialectic and the anagogic process whereby the embodied and mixed soul is brought back to its essential unity with the nous and the One. 

The exercises, the exercise of dialectic itself  transfigures, unveils and unmasks ordinary consciousness and the ordinary self (the Pali term vipassanâ) - this is what 19th-century philosophical psychology (cf. Brentano) struggled with, the inability to objectively observe the phenomena of consciousness (and Brentano is very frank about this). Understood in this light,  Sextus Empiricus and Hume can be given a consistent interpretation. There is some important literature on the relationship between Pyrrhonism and Buddhism  (Beckwith) as well as between Hume and Buddhism (sp. the Abdhidhamma). And of course there is the difficult matter of the evolution of the Platonic Academy and its convergence to something (apparently) similar to Pyrrhonism. 

The exercises discussed above obviously seem to involve 'concentration' -  but what exactly is this (how does the concept of concentration in Pali buddhism and Plotinus relate to intellectual concentration of study and problem-solving) and how does it relate to Platonic dialectic ? 

And do we need a critique of mathematics and a definition of what 'good' mathematics is (as opposed to mere addition to the repertoire of proofs and results and adding new definitions). Good mathematics depends on the ideal of axiomatic, formal, logical clarity and precision - but also of intuitive clarity and relevance to philosophy and science -  and on the ideal of elegance and simplicity of proof -  and on the possibility of an adequate explanation to others. There is nothing wrong with calling for a radical reformation of mathematics, for example via homotopy type theory or so-called 'formal mathematics' projects based on various proof assistants, or the reverse mathematics project, or a computable, constructive or finitary mathematics, etc.

A genuine mathematician must be half a philosopher, a genuine philosopher half a mathematician: examples Frege, Hilbert, Brouwer, Gödel.

Nothing is worse than aiming at proving certain results by whatever means, no matter how tortuous, artificial, obscure, convoluted, technical and lengthy.  This is not the mathematics of  interest to platonism. This is sledge-hammer mathematics, a huge rugged artificial contraption,  not the path of the philosopher.  We have also written between the difference between good (natural, logical, intuitive) and bad abstraction.

Just as for Plato we hope to show the profound connection between good (philosophical) mathematics and ethics.

The universality of computability, algorithms, combinatorics, graphs, core number theory and their inseparable 'logic' - this is what becomes manifest in mathematics seen in particular as a foundation for Platonic dialectic. 

We need a scientific philosophical linguistics which studies natural language seriously from a formal and mathematical perspective and at the same time is acutely conscious of and focused on the discrepancies and illusions of posing simplistic correspondences between the mathematical (i.e. grammatical) structure of a natural language and the actual conscious semantic content and intentions present in linguist utterance or internal discourse.  What does it even mean to use formal logic to analyze or express natural language ? Is it a translation in the same way we would translate into another natural language ? Or are we merely making more explicit certain aspects of the logical structure of an expression ? But what guarantee is there that even this logical structure of the natural language expression is reflected faithfully in the current systems of formal logic ? 

Maybe the psychologism opposed by Frege and Husserl was a strawman naturalized psychologism distinct from a pure psychologism which ironically has some similarity with what Husserl later espoused. Pyrrhonism and psychologism can be given a pure consistent foundation precisely if we abandon unproven naturalist assumptions or take positivism in its true sense as did Jayatilleke. And the Platonic dialectic is perhaps a kind of fluid general intelligence, the application of a universal method for solving any kind of problem and specially for psychological introspection: vipassana.

In original buddhism and its later development we find different perspectives: the abhidhamma, madhyamaka and yogacara - all of which have very close connections to their western counterparts, both ancient and modern: pyrrhonism (including the academy and sextus), stoicism and platonism (including middle and neoplatonism) and for the moderns specially: Hume, Kant (as read by Dennis Schulting), Hegel, Schopenhauer and Brentano. All this can be clarified and brought together into a consistent whole and nature and significance of the platonic dialectic be understood.

Fundamental problem in the philosophy of logic

The fundamental problem in the philosophy of logic is understanding the nature and meaning of formal logic, that is,  so-called mathematical or symbolic logic.

The key notion involved is that of self-representation and self-reflection.

We have informal but rigorous proofs concerning abstract axiomatic systems. Then we have abstract axiomatic systems representing reasoning and proof concerned with abstract axiomatic systems. But then we must prove that a given structure is a proof of a proposition in the same way we prove a proposition in the object axiomatic system. And we require an abstract axiomatic system to reason about proofs in the deductive system - or to prove soundness and consistency.  But how do we prove that what we informally can prove we can also formally prove ?

In order to carry out deductions we must have the concepts of rule and what it means to apply a rule correctly. Likewise we must have the concepts of game and goal. The concept of rule is tied to logic and computability. 

The concept of game includes counting, computing and reasoning.

Kant's question: how is pure mathematics possible ? should not have gone the way of synthetic a priori intuitions but rather to the question: how is formal mathematical proof possible ? That is, how would Leibniz's characteristica be possible ?

Hilbert's treatment of geometry vs. Kant.

Another problem involves the countability of linguistic expressions vs. the possible uncountability of objects.  It follows that there are uncountably many indefinable objects which hence cannot be uniquely identified. Any property they have they must share with other such objects.

We find  the term 'sociologism' very apt to describe the 'linguistic turn'  (meaning-as-use, inferentialism) of Wittgenstein, Ryle, Austin and it continuation in Sellars, Brandom, etc. There is a strict parallelism with the earlier psychologism. It is likewise untenable. It is part of the physicalist assault against the mind, consciousness, individually accessible knowledge and truth (for example a priori moral, logical and mathematical truth) and moral conscience and freedom. It is a pseudo-scepticism and pseudo-relativism/conventionalism  and is ultimately nonsensical. It is reductionism (grabbed from neuroreductionism and functionalism) and is circular.  While sociology is a legitimate scientific discipline, sociologism is not based on science and is bad philosophy.

The idea that meaning of the term 'and' can be given by exhibiting a rule does not appear to be very cogent.

A: What does 'and' mean ?
B. That's simple. IF you postulate a sentence A as being true *AND* a sentence B as being true THEN you can postulate that the sentence "A and B" is true (and vice-versa).
A: I asked for you to define 'and' and you gave me an explanation that uses 'and', 'if...then', 'being true' and the concept of judgment. Sorry, that just won't do ! 

 It is also obvious that A may be possible to infer from B but that a person that accepts A is not sociologically obliged in anyway to state or defend B, for example, Fermat's last theorem before its proof by Wiles.  Any adequate language for fully describing the full range of sociological behavior, norms and practices is at least Turing complete.  So appeals to sociology cannot be used to furnish foundations for either logic or language.

Sociologism stands Frege on his head. It is a transposition to the social plane of the false dogma of functionalism and behaviourism.

Given a sentence S we can consider the recursively enumerable (but not recursive) set I(S) of all sentence which can be inferred from S in a system T.  Clearly I(S) cannot count as the meaning of S. Elementary number theory abounds in statements involving only elementary concepts the truth and inferentiability of which is not known.

Recommended reading: C. W. Mundle - A Critique of Linguistic Philosophy (Oxford, 1970).

Another strand of linguistic philosophy which seeks to undermine the certainty, clarity, objectivity and a priority of knowledge has roots in the later Wittgenstein's theories of polymorphism and his assault on definitions and meanings (but see the discussion in the Theatetus). In its current form it revolves around what we call 'the cult of vagueness'.

The cult of vagueness attempts to undermine the clarity, precision and non-ambiguity of language, and most importantly the language of philosophy, ethics, psychology - not to mention logic, mathematics and science.  Two of its sources are the  'paradoxes' and obvious peculiarities of certain natural language elements, specially the more homely and down-to-earth terms like 'bald' and 'cup' - there is nothing strange about certain adjectives having a trifold decomposition.  Of course to do this it has to assume a certain doctrine about language and its relation to the mind and the world.

The meaning of a property can be crystal clear and yet the application of the property can be difficult and uncertain. And it is only uncertain because the meaning is clear.

The cult of vagueness has its own peculiar rhetorical style which involves never stating one's assumptions clearly but only insinuating them.  

Erroneous theory of 'semantic relations' including 'speech acts' like 'whispering'.   What do they mean by act (and old Aristotelean metaphysical concept)  ? And whispering is a quality of speech not a semantic relation. For instance 'Mary whispered the nonsense spell she read in the book' has no semantic component. 

Anna Wierzbicka's distinction between folk and scientific concept demolishes the cult of vagueness.  Our low level concepts do not have definitions in the technical sense, they have stories. They are also dynamic and socio-specific.  Thus it is a category mistake to concoct arguments which ignore this distinction.

Linguistics depends on psychology and the philosophy of mind but these last depend on language.

Most adjectives and many nouns are not analogous to mathematical properties such as 'prime number'.  Negation functions differently. Often the adjective property has a tripartite structure, for instance 'tall', 'short' and 'medium height'.  Thus is somebody is not tall is does not mean they are short.  These folk concepts (having the possibility of a fair range of adjectival and adverbial degree modifiers) can give place to scientific ones which generally will involve scale, a measure.  Temperature is measured by different instruments. There is a limit of precision and variations across measurements by different instruments or the same instrument at different times.  But this does not make the concept of temperature vague or ambiguous. In fact statistical concepts are not vague even if as properties they cannot describe the state of a system in a unique way.

We can transpose Gödel's arguments to Zalta's Object Logic.  Instead of numerical coding of formulas we use the encoding relation for properties and objects.  We can thus define predicates for an object codifying only a certain property, only a certain sentence, and only a proof of a certain sentence $Proof(p,a)$ where $p$ is to be seen as codifying a sequence of sentences.  Then we can define $Diag(a,b)$ iff $a$ encodes the proposition $Bb$ where $b$ encodes only property $B$.  Then we can construct the Gödel sentence by taking the formula $G$ (property) $\lambda z.\neg \exists x,y Proof(x,y) \& Diag(z,y)$ which is encoded by $g$ to construct the Gödel sentence $Gg$.

Consider a reference relation between expressions and objects. Suppose that there were uncountably infinitely many objects.  Then:

i) either there are objects which cannot be referred to by any definite description

ii) or there are objects which share all their properties with infinitely other objects (indiscernability)

Or infinitely many objects with one binary relation. There are uncountably infinitely many possible states of affairs which cannot thus be referred to in a unique way. The same argument applies.  And of course arguments involving categoricity.

"Speech acts", the vagueness of ordinary terms...this is already found in Husserl's Logical Investigation (see for instance vol II, Book I). And previously in Benno Erdmann. 

Meaning and psychology: the great question.  Consciousness is so much more than the lower sphere of (mainly audio-visual) fantasy and imagination processes.  When we think of the concept of prime number or the concept of 'meaningless sentence'...and of course there is the Fregean view.

Multiplicity of psychological experience in the meaning phenomenon. But we can abstract a type, a species of what is invariable. Husserl is lead from here to ideal objects à la Frege, the space of pure meanings. But in the first Logical Investigations when Husserl discusses the psychological content of abstract expressions, how these are very poor, fluctuating and even totally non-existent and hence cannot be identified with meanings. But Husserl mentions the hypothesis of a rich subconscious psychological content involved. What is going on really when we think of "prime number" ? Do we have a subconscious web of experience reaching back to when we first learnt the concept ? And could not all this ultimately correspond to a kind of formal rule such as : if a divides p then a is 1 or p,  or if a is not 1 or p then a does not divide p ? There is nothing social here or only in the most vague and general way. An extended and rectified Hilbertian view can be seen as depth phenomenology perhaps, specially in light of modern formal mathematics projects.

A priority, certainty, as well as intersubjective agreement - all this depends on recursion theory and arithmetic or its 'deep logic'. Logos is a web of relations which is not relative. 

Meinong's Hume Studies: Part I: Meinong's Nominalism

Meinong's Hume Studies: Part II. Meinong's Analysis of Relations

The deep meaning of Gödel's incompleteness theorem is the mutual inclusion of the triad: logic, arithmetic and recursion theory. 

Gödel's rotating universe.  Individual subjective time that parametrizes a path needs to have any simple correspondence with cosmic time which implies a global foliation by hypersurfaces.

Miscellany of philosophical observations

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

2. 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. 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.

5. 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 ? An analogy of experience: how the very notion of measurable function supposes the standard topology and Borel structure on the real line $\mathbb{R}$.

6.  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. 

Distributions are clearly in themselves meant to be idealizations and abstractions of actual functions with their ultimate aim being approximation results. What is a dirac function ? This will depend on the scale. Dirac functions in nature are only approximate.

7. Study differential geometry as type theory; dispel all difficulties in a general understanding of mathematics as a language. It is of utmost importance to give physics, specially quantum theory, great formal logical and mathematical and philosophical rigour. Outstanding example: Peter Bongaarts' book.

8. 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.

9.  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.

10. How do mathematicians think, actually prove theorems and have insight and intuition - all of which is very different from a low-level proof-search for some formal axiomatic-deductive system ? In particular how can formal logic and intuition agree ?  If logic is the science of valid thought, then it just cannot ignore this question.  We certainly think immediately using admissible rules.

Consider a formal logic $L$ in which we have the concept of atomic predicate, equivalence and equality. Let $T$ and $P$ be a countably infinite set of symbols no occurring in the language of $L$. By a prelogic we mean a finite set $(t,p)$ consisting of a finite sets $t,p$ of formulas in $L(T,P)$ of the form $q(x_1,...,x_n) \equiv ...$ and of the form $t(x_1,...,x_n) = ...$. We write $(t_1,p_1) \leq (t_2,p_2)$ iff the symbols in the left sides of $t_2,p_2$ all occur in $t_1$ and $p_1$ and furthermore if $t_1 \subset t_2$ and $p_1 \subset p_2$.  For each prelogic $(t,s)$ we further associate a set of intuitively valid sentences $ISen \subset Sen(t,s)$ and intuitively valid inferences  $IDed$ which are subsets of $Sen(t,s) \times Sen(t,s)$, where $Sen(t,s)$ denotes the set of sentence whose symbols occur all in $t,s$.

11. The problem of the denotation of the selection of one of two orientations of vector space or one of the square roots of $-1$. 

12. Some important authors to study: Albert Lautman and Jean Petitot. A synthesis of Kant and Husserl within the framework of an enlightened mathematical structuralism.

13. Determinism may be only local. Determinism (think analytic continuation) is like a covering space. Only one continuation and lifting of a path for a chosen point in the fiber. But we can have instead of a locally constant sheaf a constructible sheaf. There is a stratification in which non-deterministic switches or choices take place (although they can be perfectly continuous).

14. What is completeness for a logical-deductive system ? And relative to a class of models ? Take intuitionistic propositional logic.  The classical logical-deductive notion of completeness does not apply anymore. Only a model theoretic one.  And the model theoretic one needs to change to become multi-valued, i.e. as in topos theory or at least the Heyting algebra of truth-values. This was the insight behind Kant's transcendental dialectic: that $A \vee \neg A$ is not a universal law of reason.

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.