A note on formal philosophy

The following preliminary text needs to be corrected and the final considerations clarified and expanded in light of Platonic dialectics.

Is a philosophy a subject, an activity of authentic value, capable of genuine progress, worthy to stand alongside mathematics, the sciences and engineering ? This has been much discussed. One very few have proposed is that maybe mainly only ancient philosophy (both of the west and the east) is of value – is authentic philosophy – and that authentic philosophy in the western post-classical era has remained a very illusive, hidden tradition. The iconoclastic position is not so difficult to reconcile with our exposition of phenomenological metaphilosophy. But here we take a radically distinct point of view – a view which does not however discard the psycho-therapeutic and ethical value of phenomenology.

So then what is ‘good’ or ‘authentic’ philosophy (we refer to this metaphilosophical ideal as logical formalism LF) ? . Here are its essential characteristics:

1. it keeps mental habits, ill-defined concepts and prejudices from insinuating themselves into philosophy, in particular in a cloaked or transposed form.

2. it is deeply concerned with the question: What is an argument ? (in particular: What is a valid argument ?)

3. it is deeply concerned with the question: How does language work ?

4. it holds up pure mathematics as the canon of knowledge and it follows that philosophical concepts, theories and arguments (proofs) in order to be valid must be able to be presented, expounded and checked in exactly the same way as mathematics.

Furthermore we can divide 4 into

4a. acknowledging 4 as the canon and goal of philosophy

4b. actually realizing this goal in partial or full detail

A corollary of 4 is:

Authentic philosophy is not possible without an adequate formalization of a sufficiently rich fragment of natural language.

We see also that ‘linguistics’ (in the post-Saussurean and contemporary sense) is a major part of philosophy.

Here we wish to present the antiquity thesis:

By and large we find a larger presence of LF in pre-modern philosophy than in modern philosophy (with several very important exceptions).

To show this we can study the 4 characteristics in the Peripatetic, Stoic, P latonic/Academic/Neoplatonic and Pyrrhonian schools. To show this is the case for Aristotle is the ultimate motivation for our paper ‘Aristotle’s Second-Order Logic’.

But there were those heroes of early modernity that had this metaphilosophical ideal, philosophers, logicians, linguists/lexicographers (like Wilkins)– however lacking they were in the actual realization of this philosophy (if not falsely presenting their work as being more geometrico when it is not even close). It is unnecessary to go through the luminaries of the much maligned “rationalist” tradition of early modern Europe. We wish however to make the following points:

1. The alleged failure of characteristic 1. The religious influence in rationalism is in fact far less (and less specifically Christian rather than Hellenic) than in all the powerful concealed or transposed forms which it took in subsequent philosophy.

2. The genial insight and far-ranging influence of Descartes is not appreciated enough (and the same goes for medieval philosopher Jean Buridan).

3. The rival “empiricist” tradition is also surprisingly aligned to the ideal and rigor of LF.

Paradoxically there is far more religious influence in the specifically 19th and 20th century evolutionary kinds of naturalism (as well as Heidegger) than in 17th century rationalism.

It is trendy to blame Descartes for introducing so-called “mechanism”, “mathematization of nature” and much of what is bad in western civilization: we reply with the challenge to define what exactly they mean by “mechanism” and refer the reader to the discussion on determinism, computability and differential models of nature. Descartes’ low point is his abhorrent view on animals (found also in Malebranche) which would seem to proceed not from logical argument but from inherited scholastic dogma. In fact Descartes’ (comparative) physiology might be easily interpreted as furnishing powerful arguments for animal rights (cf. the improved views of Leibniz) which already found a 17th century voice in Shakespeare.

We can question whether German idealism be not actually very far from this metaphilosophical ideal and if we do not find also a frequent conceptual and naturalistic transposition of Christianity into this philosophy order to make it more palatable and apparently compatible with the perceived progress of science and social changes (as well as the tastes of Romantic art). The conceptual and argumentative aspects of its texts do not seem, at first glance, very close to the ideal of mathematics: sometimes this is explicitly acknowledged, taken as a virtue (as in several passages in Hegel). We find alleged ‘deductions’ (in Kant and Fichte) which are difficult to see as proofs in the logical or mathematical sense.

Jules Vuillemin wrote a book about Kant’s intuitionism. While it is certainly reasonable to allow for a relation between primitive concepts (and axioms) and intuition, Kant’s use of intuition in the form of the synthetic a priori is very different. Schopenhauer has pointed out the inconsistent definitions of many key terms in the Critique of Pure Reason.

It would be impossible to discuss Bolzano, Cantor, Frege, Peirce, Peano and other important figures in the second half of the 19th century as well as 20th century philosophy without going into detail about points 2,3 and 4, something which would go far beyond the scope of this short note. If Frege represents a revival of Leibniz’s characteristica project (another aspect was developed in Roget’s Thesaurus, an underrated work with strong philosophical roots) he also represents (according to Bobzien) a conscious re-emergence of some of the core elements of Stoic philosophy. We argue in “Aristotle’s Second-Order Logic” that Frege’s second-order logic is simply the logic and metalogic of Aristotle’s Organon (although we need an alternative way of presenting natural deduction closer to natural language reasoning).

We must make the important observation that so-called formal and symbolic logic became part of the education and interest of certain philosophical schools, but as a rule in a very deceptive and misleading way if we are looking for the kind of metaphilosophical ideal in question (Wittgenstein does not seem to have much in common with it). It is very important to study certain non-mainstream philosophical currents in French philosophy of 19th century and the first half of the 20th century (among both the “spiritualist” and ontologist schools but also among such noted thinkers as Brunschvicg, Rougier, Vuillemin, Cavaillès, etc.). Neokantianism however fails because of its defective logic inherited from Kant, its confused account of intuition and its typical Kantian dogmatic assumptions about the limits of reason.

After so-called ‘early analytic philosophy’ (Frege, the early Russell, Carnap but also lesser known contributions by Hilbert, Mally, the Polish school of mereology, etc.) anything approximating LF was lost sight of in the analytic philosophy mainstream and has to be careful looked for and investigated. The project of formalizing natural language has been carried out in ways less interested in logic and in the definition of philosophically relevant concepts. LF -relevant work is found outside official academic philosophy among linguists and researchers in artificial intelligence and knowledge representation (like John Sowa) – and most specially in mathematical general systems theory – a mathematical model theory encompassing consciousness, living systems, social organization and every kind of scientific and engineering domain.

It is worthwhile to examine in detail the re-emergence of metaphysics in analytic philosophy since the 1980s (specially the work of Timothy Williamson and Edward Zalta).

We must find a reconciliation between LF and universal phenomenology (UP). Notice how Descartes is a key figure for both and how both share the same high regard for ancient philosophy. They both esteem Hume. They both are opposed to inferentialism and meaning-as-use theories. If Husserl has an obvious connection to UP, the work of Claire Ortiz Hill has shown that LF-related concerns run deep as well in Husserl with a close connection to Hilbert. A similar situation is found Gödel (see J. Kennedy and Mark van Atten: Gödel’s Philosophical Development). Gödel was not only enthusiastic about the phenomenological method but considered also the quest for the primitive terms and their axioms to be a viable alternative. Even Kant never ceased to dream of a kind of Leibnizean project.

To effect this synthesis or reconciliation we can take inspiration from how there is a mutually helping and corrective feedback loop between insight and formal deduction in actual mathematical practice. Descartes called deduction the intuition of the relation between intuitions.

It would seem however that LF cannot itself furnish the higher or ultimate foundations for logic or mathematics itself, specifically with regards to combinatorics, number theory and recursion theory – thus it would seem that LF already assumes that a large portion of logical and linguistic issues have been settled and thus it serves more as a tool for second philosophy. It would appear thus that LF cannot in itself completely solve the problems in the philosophy of logic, philosophy of language, theory of knowledge and metaphysics.

We are led to the difficult problems of the self-reflection of formal systems and the self-foundation of LF. The idea of self-foundation and self-positing. Category theory seems to be relevant here as a conceptually rich and multidimensional formal system which yet differs in its structure and use from classical logico-deductive systems. In Category theory concepts can be co-implicit in each other; there is a facility of passing to the meta-level inside the system, proofs are more analytic in the sense of involving generally an unpacking of concepts employing only minimal logic. Category theory’s ascent into abstraction bears a similar relation to ordinary mathematics as Descartes’ analytic geometry did to Euclidean geometry.

Maybe we need an entirely new self-reflective concept of formal systems and the role of formal systems. Maybe the activity itself of doing LF can manifest or show something higher though this can never be expressed or deduced in a formal system of LF. This again is an instance of the feedback loop aforementioned, which echoes the famous letter of Plato.