Principia Logico-Mathematica

https://mally.stanford.edu/principia.pdf

Zalta uses second-order logic extended with what might be called 'third-order' predicates which express the encoding relation between $n$ first-order objects and $n$-ary predicates. These he calls the 'encoding' relation (inspired by Meinong and Mally). To us the great interest of the work of Zalta and his collaborators is that it offers a solid example of 'formal philosophy', philosophy carried out in entirely in a formal language standing in for natural language. But the real significance of Zalta's work is not so much a setting up a particular metaphysical-philosophical system or giving plausible charitable (re)interpretations of Plato, Leibniz and Frege but rather furnishing a framework for purely formal dialectics, argumentation and debate. The challenge is to formalize and express in this context the process of debate (we could take for instance the formal game outlined in Aristotle's Topics). This will involve a certain theory $T$ which both players must accept (which will include the core axiomatic-deductive system), a sequence of choices of assumptions which one player or both players must be forced either to accept or reject.  Perhaps the resulting logic is like temporal logic? We have given an example of the formalization of philosophical dialectic (debate) in our preprint "A formalization of a fragment of Plato's Lysis".

The heart of madhyamaka - kicking away the ladder

Philosophical discourse may be logically, linguistically, psychologically, sociologically and ideologically analyzed. One can question whether philosophical discourse has any claim to veracity or value. One can be amazed by the hubris and confidence of philosophers and question the epistemic or intrinsic public value of the discourse they produce.  One can inquire whether philosophers fail to question the assumptions they tacitly assume, or whether some positions be never stately clearly but only insinuated (cf. Ernst Gellner's Words and Things). One can question whether self-proclaimed "rigorous" philosophy is actually as "rigorous" as it claims to be and question whether such a "rigor" be not entirely illusory and only based on stylistic mimicry and convoluted jargon masking the lack of any possibility of formalization in an axiomatic-deductive system. One can question whether philosophical discourse be not a mere language game (like the crossword) based on vagueness, ambiguity and unconscious association, tolerated due to the loftiness of the object of discourse, the stylistically "scientific" semblance of the discourse itself, specially when focused on technical topics in logic and language. Surely we must reject a game based on the dynamics and structure of human consciousness which yet denies outright the underlying role of human consciousness in the spirit of: pay no attention to the man behind the curtain! Philosophy can well seem scientific yet be not so and produce arguments that on the surface seem to be like a logical mathematical proof but fall utterly short of it. And, most importantly, be completely lacking criteria for veracity and value. And what are we to make of philosophers who downplay  or trivialize introspective psychology, mathematical linguistics and mathematical linguistic analysis (the only way to go beyond circularity in linguistics) or the universal ethical principles of human and animal rights? All human activities with value have  public, objective criteria. Philosophy does not have such criteria. Contrary to most academic critiques of academic philosophy the central flaw is not ignoring fundamental domains of human experience and the human condition (which is a valid enough point) but the claim to be analytical and rigorous while being not so and lacking any objective public criteria of veracity or value. A mathematical proof can be formalized and checked in a proof assistant such as Agda. Introspective psychological descriptions can be confirmed, verified or corrected by comparison and discussion among researchers.  A theory in natural science can be confronted with experimental data and evidence. A medication or medical procedure has the objective criterion of the data of clinical trials.  A work of art is received by the public.. But philosophy has no criteria beyond "deemed acceptable to be published in journal X by the editor and reviewers A,B,C,...". A tacit assumption and error is that the mathematical formalization of (a relevant fragment of) natural language is neither possible nor desirable for philosophy. Another one is (often tied to physicalist dogmatism): introspective psychology (as in Hume, William James and Brentano)  is of little value and not a source of philosophical insight and progress. And if  to this we add to this the rejection of the marvelous metaphoric, poetic and analogical capabilities of natural language, we come to the conclusion that such philosophy is mere verbiage, a linguistic production without veracity or value which merits careful linguistic, psychological and cultural diagnosis and analysis.

A note on the formalization of natural language. When two people sit down to play a game they have reached an agreement. The pieces on the board and the rules will represent the game but are not themselves the game.  The agreement is that the board, pieces and rules will stand for the game. A formalization of a fragment of natural language will take the form of a formal system incorporating a series of formal definitions which a group of people will agree represents a certain fragment of natural language. This group will agree (based on a perceived contextual adequacy) to abide by the rules of the game of such a formal system for the purposes of certain arguments or debates - they agree that the formal system and its moves stand  for the linguistic activity of the debate. The formalization of natural language is in a way a return to the original essence of natural language. Indeed as mentioned in the conclusion of our Analyticity, Computability and the A Priori, it is through the reliving and showing forth of the process of our psycho-cognitive development that we may hope to ultimately see the light regarding the problems of philosophy - without naively confusing the developmental path of a cognitive structure with its intrinsic constitution(*). Just as Kant distinguished between a legitimate and illegitimate use of reason (the transcendental illusion once the basic foothold of logic and introspective intuition are abandoned) so too we must distinguish between a perfectly legitimate use and application of language (in which the adequacy of language is unquestionable, independently of any formalization) and an illegitimate use and domain as exemplified by the naive linguistic productions of philosophy.  The formalization of natural language is a necessary critique, a judgment, a challenge and a sifting.

(*) A complementary method will involve what we described as a "linguistic phenomenology". We are given an object and a restricted vocabulary (which may be or not be a set of proposed semantic primes) and the task is to define, describe or tell a story about that object using only such a vocabulary. 

The three times in human aging

In human aging we can distinguish three 'times'. Physical time, biological time and intrinsic, lived time (ontochronology).  The biological time can be written as a smooth function $b(t)$ of physical time. Ontochronology - which is in a sense the real age for a human being - is measured by the length of the graph $b(t)$ starting from the moment of birth $(t_0, b(t_0))$. Thus

$o(t) = \int_{t_0}^t\sqrt{1 + b'(t)^2} dt$

If we consider the total chronological time of a person's life as fixed, then corresponding to this constraint there can be great variability in the corresponding smooth graph for $b(t)$ (subject to further biological constraints). For instance $b(t)$ may for a significant portion of physical time lie beneath the diagonal $\Delta(t) = t$ and thus that person will have in total a longer youth determined by periods in which their biological age is less than their chronological age. There is a certain similarity between these considerations and the measurement of time in special relativity.

The method of meditation and the philosophical method

There is something to be said about books presenting a "basic method of meditation" such Ajahn Brahm's "Mindfulness, Bliss and Beyond". Such books are part of marketing ploy to sell retreats organized by religious institutions (which serve ultimately as recruiting grounds for new monks and nuns). As such the methods described - which may have unquestionable value - are only suited to the specific conditions of the retreat. Such conditions are inaccessible to the vast majority of us. This includes those who have the luck of having access to the seclusion and peace of nature for some limited amount of time, but certainly not for weeks - the time needed for such methods to be effective. These methods count on philosophical insight about consciousness manifesting on its own during the long periods of seclusion and silence, or they count on the guidance of a teacher. These methods in themselves fall under the category of  "samatha" rather than "vipassana", even if there are also "vipassana" -themed retreats. 

We note that Ajahn Brahm himself probably became aware of the Herculean task of detaching oneself from the world (abandoning past and future to abide in the eternal present) and so introduced an alternative method which amounts to calming the body and mind through body-scanning, a sweeping focus built on psycho-somatic feedback (this method can also be traced to the earliest suttas). Rather than trying to tear the stubborn agitated child from his toys or tearing the toys from the child one engages in calming and lulling the child to sleep.  What is involved here is a doctrine concerning the fundamental importance of the first-person experience of the body and concerning the fundamental unity and cyclic dependency between this first-person experience of the body and all aspects of consciousness. The first-person experience of the body involves localized 'feeling'  and the cultivation of consciousness of such feeling can open up vast domains of very subtle and significant experience. The doctrine is that such domains of subtle feeling in first-person body experience are the key to total mastery of all domains of consciousness (cf. how the flux of consciousness is woven with sense impressions and sense impressions are intimately psycho-somatically related to sense organs, receptors and their neural pathways). The doctrine appears to be present in a more-or-less veiled way for instance in the sutta MN 119. We can think of the psycho-somatic feedback as creating a kind of field, container (cf. the simile of the soap and bath in MN 119) in which the rest of consciousness is then immersed to become pacified and purified (cf. again the similes in MN 119). One can say that the initial process of both local and global psycho-somatic feedback prepares a thoroughly purified and pacified 'container',  'field' or  'body' (there are many meanings of khaya in the suttas).  Once this is thoroughly achieved then piti and sukha will 'dawn' and 'well up'.  We then interpret the similes in MN 119 as involving the further purification, 'drenching',  'irrigation' and total 'covering' of this 'container-field-body' with piti and sukha. There are interesting connections that could be made with Plotinus as well.

Hence the necessity of a "method of meditation"  which is still powerful and effective in the context of ordinary life in the world.  Such a method must be based - alongside the considerations above -  on introspective-philosophical psychology founded on phenomenological insight. A method of philosophical psychological analysis, introspection and insight which aims at liberation from both the "world" and the "self" whose starting point is the positivist neutral consideration of consciousness, the facts of consciousness, as they are in themselves, as manifest in their manifestation, without the screen of metaphysical reification or the habitual oblivion of their fundamental co-conditioning factors such as temporality or the underlying mental directedness in recollection and anticipation. A powerful tool for this method is hearing or reading the original suttas of the Pali Canon or foundational philosophical texts of the Mahâyâna tradition. It is in this spirit that much of what we have written should be read.

Another point concerns the difference between Western and Eastern art. It does seem to us indeed that in some schools and works of Eastern Art (specially traditional Japanese art inspired by Zen) have a very strong connection to the pure neutral phenomenological-philosophical insight discussed above. In some sense, there are schools of Eastern Art which manifest anatta and the holistic flux of a neutral phenomenological consciousness (for instance the traditional Shakuhachi)  which is rarely found in classical Western art (but we do find such for instance in more contemporary compositions such as Brian Eno or the Polish ambient band How to Disappear Completely). 

Should Type Theory Replace Set Theory as the Foundation of Mathematics?

Mathematicians often consider Zermelo-Fraenkel Set Theory with Choice (ZFC) as the only foundation of Mathematics, and frequently don’t actually want to think much about foundations. We argue here that modern Type Theory, i.e. Homotopy Type Theory (HoTT), is a preferable and should be considered as an alternative (Thorsten Altenkirch).

https://www.researchgate.net publication/368474630_Should_Type_Theory_Replace_Set_Theory_as_the_Foundation_of_Mathematics 

We refer the reader also to Bealer's interesting critique of sets in his book Quality and Concept. Our own finitist and computationalist position concerning foundations, as we have discussed before, is very critical of ZFC and highly favors dependent type theory (but not necessarily homotopy type theory) but also attaches value to Girard's system F (or its extensions) and second-order logic (cf. Simpson's reverse mathematics project) and to systems such as $HA^\omega$ (Heyting Arithmetic in all finite types) built over Gödel's system T.  We consider the concept of list (finite sequence) to be more fundamental than that of finite set - this is reflected in combinatorics. It would also be interesting to investigate a 'dialogical' foundations of mathematics (inspired by linear logic and functional interpretations as well as automatic theorem provers) in which proofs are replaced by dialogues in the style of Plato's Meno.

Category theory is not a foundations of mathematics, but it is close. Some suitable extension of dependent type theory can serve as a good foundations which we can liken to machine code (or assembly language) together with the system calls of an operating system kernel.  Category theory is then like the C programming language and its compiler.  

Is a mathematical model of consciousness possible?

No, a mathematical model of consciousness is not possible. First we must distinguish between the natural consciousness of Dasein studied according to Heidegger's analytic (in which there are indeed certain proto-topological notions present) and the awakened insight-imbued ego-absolving consciousness. In both cases the idea of a mathematical model of consciousness is a serious mistake because all mathematics assumes that the fundamental problems regarding circularity in the a priori and analytic we discussed have been solved (i.e. by modelling consciousness via mathematics we are already making assumptions about consciousness necessary to justify the use of mathematics). This is analogous to the error of trying to give a philosophical account of language via meaning-as-use theories. As we wrote in the conclusion of our paper on Analyticity, Computability and the A Priori:

How can we shed greater light on the questions and problems discussed in this note? What methods should be employed to go beyond the seemingly unbreakable circularity between logic, computation, arithmetic and combinatorial intuition? A possible answer lies in adopting a rigorous phenomenological approach or the combination of such an approach with something along the lines of Piaget's genetic epistemology. We need to investigate thoroughly what it means for us to learn something (like counting, calculating), to know how to do something, to understand rules, to know how to play a game and to understand a game. All of this is at present unclear and difficult. It is naive and dogmatic to take these topics that demand extensive investigation and elucidation as some kind of philosophical explanation as done for instance in meaning-as-use theories. The elucidation in question needs to proceed not by formal abstract reification (which leads to the same circularity) but by investigation of the matter itself in its concrete lived experience, both personal and interpersonal.

 

We of course must be careful about what we mean by 'model' or in what sense we are discussing the 'modelling' of something. The specific sense we had in mind here is that of theoretical physics: the model, a theoretical system consisting predominantly of mathematical structures, is posited as representing or reflecting the actual fundamental structure and process of the physical world, so much so that physicists in possession of complex mathematical theories (which do not involve any trace of ordinary physical intuition) usually speak of attaining a deeper 'understanding of nature' . This is, of course, very different from the 'models' used in engineering simulations or in applied mathematics.  All talk about consciousness, from the theories used in behaviorist or Gestalt psychology, theories of personality, Heidegger's analytic of Dasein, Piaget's genetic epistemology, the Buddhist abhidhamma - are in a sense 'models' of consciousness and as such perfectly legitimate expressions of the human desire to understand. There is nothing wrong with some of the most technical or sophisticated of these models employing mathematics as an extension of their language, mode of description and theoretical articulation (*). However this all very different from the idea of a mathematical model of consciousness conceived in a fundamental, philosophical, all-embracing sense analogous to the way mathematical models of theoretical physics are posited as models of nature. This is, as we saw above, a fundamental error and category mistake (we cannot relegate consciousness to a self-contained isolated regional ontology as done in the domains of the sciences) which ignores the fundamental question of the foundations of mathematics itself (in the circle of the analytic a priori). Mathematics is a fundamental problem of consciousness, not itself a key to solving the philosophical problems of consciousness (in Kantian language: the question of how a priori synthetic judgments are possible is essentially a problem of consciousness). There can be consciousness, conscious experience, without mathematics or mathematical activity, but there can be no mathematics or mathematical activity without consciousness. Thus consciousness, not mathematics, is the ontological primitive. Again, consciousness is Turing complete and has inherent meta-postulates regarding meta-interpretation, reflection and hypercomputation: in particular it represents a term-rewriting system which can produce a $n$-length derivation of any term-rewriting system for a given initial word.  But any mathematical model represents only a particular term-rewriting system (recursive axiomatic-deductive system) which cannot express the universal computational capacity of consciousness or the general inherent meta-postulates of consciousness. Thus no mathematical model can be an adequate model of consciousness. Or, in more concrete terms, consciousness is the power of playing all games, while a mathematical model can represent only a particular game. Thus a mathematical model cannot adequately capture consciousness.

 

Positions which would deny that mathematics is a fundamental problem of consciousness are among the following. Positions like those of Carnap, Quine and meaning-as-use we have critiqued in our Analyticity, Computability and the A Priori. Our critique also applies to versions of Fregean logicism. But the most obvious examples are  forms of physicalism: eliminationism, theories of emergence, functionalism and its (computationalist) neuro-reductionist variants. In this context one needs but to be in possession of a sophisticated enough physical-mathematical-computational model (which may be stochastic or quantum) of the brain in order to have an adequate mathematical model of consciousness, for consciousness is considered to be an emergent phenomenon and its fundamental structures and processes are considered to be theoretically deducible from such a model. Some theories introduce a twist in which the ontological primitives are considered to be 'mind-like' (for instance so-called panpsychism) but which in their articulation and theories of actual consciousness are indistinguishable from physicalism.

 

(*)  In the domains of consciousness involved in psycho-somatic feedback the theory of partial differential equations (the study of oscillatory, radiative, diffusive, equilibrium, deformation and transport phenomena) may play an important role in the study of certain aspects of consciousness - thus lending a scientific basis to many of the linguistic metaphors employed for describing the mind (and the process of meditation). The design of certain complex software such as the Linux Kernel or a RPG game may reflect the structure of consciousness once we abstract away the underlying architecture of the hardware and consider other forms of hardware (truly concurrent hardware, neural nets or celullar automata - something like contemporary GPUs). This being the case, it is very plausible that formal systems accounting for dynamic concurrent interacting processes are important for studying consciousness. While some aspects of LLMs may be interesting for the study of cognitive-semantic architecture (and these aspects very likely could be much improved) there is also much which clearly is meant to have only a practical design and significance. 

The path to peace

On second thought the considerations put forward previously could be quite mistaken both on an existential and historical-anthropological level. The golden chain in the West must indeed proceed from the East. But not through neoplatonism and certainly not through Origen, Eckhart and Heidegger. Rather it is in such figures as Sextus and Hume and Kant. There are true gems in the West but part of Western hubris is the great difficulty in coming to terms that its mainstream traditions are flawed and rotten to the core. They cannot be redeemed or purified or idealized or abstracted. The western religious spirit is so rotten than even in its own physicalist negation it propagates the same essence. Dasein is sakkayaditthi (cf. Merleau-Ponty). A detailed phenomenological analysis of an illusory limiting experience is no less illusory for that. But more brilliant is the Buddhist philosophical insight on how to overcome Dasein. Thus much of our work can be described as attempting to present some aspects of such a super-phenomenology. Anatta and Atheism in its genuine sense (the absence of introducing false constructions, separations and selections in the purity of thusness), are at the heart of Buddhism and is found in many priceless original sources in the West, and are humanity's fundamental step to individual and collective authenticity, progress and liberation.  The direct liberating knowledge and insight of anatta is inconceivable in the context of Heidegger's analytic of Dasein. Anatta is the joyous dissolving of Dasein.

The currencies with which we pay taxes to the world: attention and erotic interest. Do not pay. The task of Hercules is the first stage described in Ajahn Brahm's book Mindfulness, Bliss and Beyond. The rest is easy. Overcoming the world, not engaging in the world, is authenticity - which renders the ego transparent and flimsy and ready to be transcended. Care is the fundamental ontological-existential power. Do not care beyond what is necessary for your daily bread or helping others according to your means. The first ontological-existential step, detachment from the world, relinquishing the world, abandoning the world, returning unto the authentic being-there which is a being-beyond, a being-far-away, a being which is reconciled with itself. An authentic being which does not pay the tax and tribute to the world (which is attention, care, concern, as well as erotic interest). The authentic being which does not sully its care and concern with the world but has returned into the freedom and purity of its own self. The inauthentic being-in-the-world is being thrown in time, drowning in the ocean of time, dragged by time and its storm of past and future. In authentic being the world-time and time-in-the-world are dissolved into the present moment illuminated by the pure manifestation of the universal flux of consciousness.  In this state of transcendent withdrawal, the fabric of temporality becomes transparent and the eternal present shines through.This authentic being-there is the pure manifestation of the world as the universal flux of consciousness, manifest as such. The inauthentic being is thrown into the world and in world-time and also throws itself and loses itself in this same world and its stormy ocean of time. In authentic being, the transcendent being-there, in the universal flux of consciousness shining in the eternal present, the self itself becomes flimsy and tenuous and ripe to be overcome. There is the beacon of freedom. Although a given religion or spiritual practice can be the occasional cause for the launch into authentic being-there can be no doubt that freedom and authentic being in itself are radically incompatible and antagonistic to any religion. Religions are essentially part of the world and dependent on worldly care and intention and their talk is only intelligible for human inauthentic being-in-the-world.  

Philosophical updates

Heidegger's Being and Time must be considered a brilliant and profound work and if not a vast improvement over Husserl then at least a realization of what in Husserl was just reified meta-talk and illusive promise. Being and Time actually shows in a clear, concrete and beautiful way how phenomenology is possible and indeed phenomenology itself is rectified in several essential ways (by recovering the indisputable central importance of being and "existence") and set on solid and clear foundations which in fact are deeply rooted in Hellenic thought.

Being and Time also offers a section which might be described as the proto-topology of the Dasein's Umwelt which certainly echoes some of our previous work.

This work retains great value even if we consider the Dasein considered to be deluded and unenlightened Dasein (the essentially worldly Dasein). Also we can argue that Heidegger misread Hegel and that the Hegel-Heidegger connection might be a rich field of investigation.

Heidegger's approach to language, logic, thought and intellect is deep and illuminating and suggests (besides offering overwhelmingly powerful tools for the strangely non-embarked upon task of  destroying analytic philosophy once and for all)  some radical developments for our paper on Analyticity, Computability and the A Priori. What does it mean for us to learn something, to learn to do a thing, for instance, to learn to add, to learn to calculate, to learn the rules of game vs. to learn how to play the game? This is the principal question of our paper and Heidegger's method (combined perhaps with the work of Piaget) seems well worth pursuing. When we learn how to do something then when we do a thing, the thing itself becomes far from us having been close when we struggled to learn it. We have at present no clear understanding what it means for us to calculate or to play a game.  

Heidegger also offers some interesting insight to our spiritual and anthropological investigations: a shift of focus from early eastern traditions (like Pali Buddhism) to the overwhelming deep luminous source of Hellenic traditions - in which genuine questioning and freedom is found without the shadow of despotism and projected absolutism as in the typology of a 'lord of the world' (which was overcome somewhat in the purely philosophical Mahayana schools and more so in the sublime transformation in Chinese Buddhism grounded in Daoism: if you meet a Buddha on the road, kill him - could not have been easily uttered by an Indian Sage, only a Chinese or a Greek one).  A study of the mysteries, Orphism, Greek Drama, the pre-Socratics, the Platonic texts reveals more and more and shameful plagiarism and perversion which Hellenic thought and traditions (and their sources and influences) were subject to by certain later currents of Alexandrian and Mediterranean Hellenistic cultures (specially the so-called "gnostics").

We point out that Heidegger also has some very disappointing and serious anthropological blunders wherein he reveals himself to be a child of his own time blinded by typical German pseudoscientific racism and nationalism. In the Black Notebooks we find anti-universalist  anti-humanist   Blut und Boden type utterances which are unworthy of a philosopher. But at the same time the correction of these errors is of immense philosophical importance - and it seems it is up to us to engage in this.

There can be no authentic inquiry into being and into man unless the selfsame dignity of every single human being (and indeed animal) is acknowledged without exception (to reject the Dasein of another is to reject one's own Dasein).  All human beings are equal before Being though evidently not all human beings and cultures at a given time have the same purity and openness to Being. 

There can be no valid onto-theology without the fundamental doctrine of Apokatastasis. The eventual universal salvation of all beings can be taken as the fundamental ground of the very essence of divinity without which we have nothing but monstruosities and human calculation and construction.

Heidegger's reading of Eckhart suggests that Eckhart is deeply Hellenic and - if we allow some key clarification and rectification so as to leave no doubt about the unconditional adoption of Origen's doctrine of Apokatastasis - then Eckhart can indeed by considered an important spiritual guide providing a concrete practical complement to Plotinus. Happiness consists in dwelling peacefully in the clearing of Being. We must show in detail how Eckhart's thought is admirably compatible and suggestive of Origen's doctrine of Apokatastasis. Indeed in Wagner's Parsifal the redemption of all of nature is celebrated, even of the redeemer Himself. Spiritual love means liberation from the limitations of place, time and finite fixed embodiment and yields a measure of intemporality, omnipresence and multidimensional integration, reflection, flow and radiation.

And of course with regards to the philosophical understanding of technology it almost seems that Heidegger was a philosopher that was prophetically born not so much for his own time but for our present time. Heidegger is vastly relevant and important to understand the horrors of the contemporary technocratic dystopia and what is designated by the misnomer artificial intelligence. In fact generative AI, large language models, those electrically animated Frankenstein corpses of adulterated human language and instrumentality are the most perfect embodiment of everything Heidegger wrote about technology.

Another interesting anthropological-spiritual source concerns the Persian and Hermetic traditions understood through Henry Corbin's interesting Heideggerian hermeneutics.