Three metaphilosophies

We have proposed three metaphilosophies.  Phenomenological metaphilosophy involves understanding the timeless and universal principles of the phenomenological method and program which are found across a great variety of different philosophical systems, times and places.  Formal metaphilosophy takes a highly skeptical view of common philosophical practice with a focus on the logical and linguistic aspects and proposes a methodology based on axiomatic-deductive systems and rigorous definition of all concepts involved in all philosophical arguments and debates. Critical metaphilosophy (inspired by Frege, the early Husserl, Gödel, Gellner (backed by Russell), Mundle, Preston, Findlay, A. Wierzbicka, J. Fodor, C. Ortiz Hill, Rosado Haddock and Unger, John W. Cook and also some considerations of Marcuse) questions the value of much of 20th and 21st century philosophy from a predominantly logical and linguistic point of view as well paying great attention to presupposed or insinuated materialist hypotheses found therein (Preston and Unger should have engaged in exhibiting substantial textual evidence for  'scientiphicalism').  Every accusation is a confession and if linguistic philosophy/ordinary language philosophy is patently bad philosophy, the kind of condemnations it engaged in towards previous philosophy prophetically turned out to apply remarkable well to itself. And indeed this linguistic philosophy never ceased to be an underlying powerful force until today despite its various disguises and apparently sanitized versions, including analytic metaphysics. A true scientific linguistics deployed in a critical metaphilosophical way  is what is called for - the study of the psychology, sociology and linguistics of professional philosophy/sophistry. Even in non-orthodox philosophers in this tradition (Robert Hanna, George Bealer)  we find a strong presence of many of its assumptions and rhetorical-argumentative patterns.

We do not loose sight of the hard problems and limitations involved both in phenomenological and formal metaphilosophy.  

Why use the term phenomenology rather than psychology or introspective psychology.  For the greatest of problems involves what is most primordially given. And how can truth be found or based on anything but this ? The goal of philosophy is to see fully, to know fully, it is self-transparency and liberation.  Locke, Berkeley and Hume dealt with the deepest and most fundamental, most fertile of all questions. They looked in the right direction and had the right perspective. The great question: what is a concept ? Without concepts there is no logic, no language, no reason, no knowledge.  Can we admit knowledge without any conceptuality ? Or mind without conceptuality ? Certainly ordinary knowledge involves concepts. Even asking about knowledge and truth, are we not asking about concepts ? Is not truth a concept ? Is not knowledge a concept ? And do we have a concept of a concept even it is an unclear, vague, definition-lacking concept ?

And what about ethics, specially an ethics based on compassion ? Schopenhauer, as we mentioned before, offers us a purely phenomenological ethics based on compassion which thus would appear to have a non-conceptual anchor.

If philosophy is foremost a quest for individual clarity and knowledge regarding one's own consciousness, how can we express the truth we find to others ?  How can we argue, how can we persuade ? What are the rules which must govern or direct this argument or persuasion ? There are no arguments without concepts. If we do not know what a concept is, we do not know what an argument is. We have a concept of concept yet this concept is not an adequate concept. We can know things and yet not know how to define them. Sentences express concepts (they can be nominalized) just as adjectives, adverbs, nouns, verbs, pronouns, etc.  And concepts are not vague. It is difficult to find two different words (from hundreds of thousands) which have exactly the same meaning. If meaning boundaries were fluid we would not expect this to happen.

Without concepts there is no language. It is erroneous and foolish to go about theories of language and profess to talk about the mind without first venturing into the vast realm of the philosophy of concepts. 

Our stream of consciousness is not a stream of sensations or recollected images of simple sensations but includes a stream of concepts (in-consciousness concepts, not Fregean concepts obviously).

As a temporary remedy for this state of affairs we propose formal philosophy, carrying out philosophical arguments in an entirely mathematical fashion.

Husserl's Logical Investigations is a great textbook in philosophy, a kind of summa of the best psychological introspective, logical, linguistic and ontological work of the 19th and 18th centuries.  Likewise Frege is a model of clarity and elegance - regardless of one's views.

The danger of philosophical introspective (and transformative) psychology is turning into mere psychotherapy or psychoanalysis or becoming uncritically influenced by occultism and religion. Equally harmful are naturalism, neuro-reductionism, behaviorism and the dogmas of 'linguistic philosophy' or 'ordinary language philosophy' (now called inferentialism and theories of vagueness). Speech acts and language games are still necessarily abstracted, isolated, analyzed and understood conceptually.

The dilemma here seems to be between staying safely at the periphery or venturing to where lurks the great danger of religion, occultism and cults.  Philosophy is indeed a psychotherapy which aims heroically to overcome the conditioning of religion and materialism alike (cf. Gödel's statement: religion for the masses, materialism for the intellectuals). There are no royal roads or shortcuts in philosophy.  See this essay by Tragasser and van Atten on Gödel, Brouwer and the Common Core thesis. Gödel's theory, as recounted by the authors, is of utmost significance. Gödel was promoting the restoration of the authentic meaning of Plato's dialectics and the role of mathematics expounded in the Republic and other texts.  Perhaps Gödel has pointed out the best path (at once philosophical and self-developmental) (for so-called "Western man" ) which avoids the double pitfall of materialism and religion, psychotherapy and occultism. In the 21st century (inheriting from the 20th century) we are inundated by the cult of the irrational, by anti-rationalism in every subtle and insidious form. The "rational" is only allowed to thrive in a limited and watered-down form, harnessed generally to  materialistic/technological/economical/military goals.  And the technological and economical goals here often do not even aim at the common good and equal and fair distribution of the earth's resources.

And here is what is remarkable about the Platonic-Gödelian method: the confluence between pure mathematical thought and introspective transformative philosophical psychology.  But this project can be discerned in Husserl's Logical Investigations and Claire Ortiz Hill has written extensively about the objective, formal and logicãl aspect of this work, in particular the important connection to Hilbert's lesser known philosophical thought.  However the psychological and phenomenological aspect is just as important, just not in the way of the later Husserl, rather in the Platonic-Gödelian and transformative philosophical psychological way.

The epokhê as Husserl outlined is not possible (and even less is the Heidegger alternative valid), rather such a clarity and 'transcendental experience'  is possible through the Platonic-Gödelian method. 

For a good summary of the role of mathematics in Plato see Sir Thomas Heath, A History of Greek Mathematics, Vol.1, Ch. IX.

George Bealer (1944-2025)

 https://dailynous.com/2025/01/22/george-bealer-1944-2025/

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) $\lambda z.\mathrm{\neg }\mathrm{\exists }x,yProof(x,y)\mathrm{\&}Diag(z,y)$ 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 need to have any simple correspondence with cosmic time which implies a global foliation by hypersurfaces.

Computability, determinism and analyticity

An overlooked but nevertheless very important problem concerns the role of the differentiable and smooth categories in mathematical physics, that is, the category of maps having continuous up to a certain order or all order. Our question is: why use such a class of maps rather than (real) analytic ones (or semi-analytic) ?  The equations of physics have analytic coeficients.  Known solutions are analytic (for simplicity we do not distinguish analytic from meromorphic).  In fact known solution are analytic having power series representations with computable coeficients (for a standard notion of computability for sequences of real numbers).  And in fact all numerical methods for mathematical physics depend on working in the domain of computable analytic functions.

The class of computable analytic functions is related to the problem of integration of elementary functions. It is also very elegant and simple in itself as it reduces the problems of the foundation of analysis (infinitesimals, non-standard analysis) to the algebra of  (convergent) power series.

Deep results in the theory of smooth maps depend on the theory of several complex variables.

The existence and domain of analytic solutions to analytic equations is an interesting and difficult area of mathematics.  Several results rule out associating analytic equations with any kind of global determinism (in the terminology of Poincaré, solutions in power series diverge). That is if we wish to equate determinism with computability and thus with computable analytic functions in physics. Thus that computable determinism is essentially local is not philosophy but a hard result in mathematics.

An interesting mathematical questions: are there analytic equations which (locally) admit smooth but analytic solutions ? 

Another vexing question: why are there not abundantly more applications of the theory of functions of several complex variables (and complex analytic geometry) to mathematical physics ?

There are objections against the analytic class.  For instance it rules out the test function used in distribution theory or more generally functions with compact support.  Thus we cannot represent a completely localized field or soliton wave (but notice how Newton's law of gravitation posits that a single mass will influence the totality of space).  And yet the most general functions constructed  (like the test function) are often simply the result of gluing together analytic functions along a certain boundary. Most concrete examples of  smooth function but not analytic functions are precisely of this sort. We could call these piecewise analytic maps. Thus additional arguments are required to justify why we have to go beyond piecewise analytic maps.  An obvious objection would be: distributions and weak solutions. But here again we invoke the theory of hyperfunctions.  It seems plausible that there could be a piecewise analytic version of distribution theory (using sheaf cohomology) - even a computable piecewise analytic version.

In another note we investigate other incarnations of computability in mathematical physics (and their possible role in interpreting quantum theory).  Can we consider measurable but not continuous maps which are yet computable ? Together with obvious examples of locally or almost-everyhwere (except on a computable analytic set) computable analytic maps we can seek for examples of nowhere continuous measurable maps which are yet computable (in some adequate sense). The philosophy behind this is the computable determinism may go beyond the differential and analytic category, the equations of physics in this case however only expressing (in a non-exhaustively determining way) measure-theoretic properties of the solutions. 

We end with a discussion of what constitutes exactly a computable real analytic function and how we can define the most interesting and natural classes of such functions. Obvious examples are so-called ’elementary functions’ which have very simple coeficient series in their Taylor expansions. Also it is clearly interesting to study real analytic functions whose coeficient series  are computable in terms of n. And can we decide mechanically when one of these functions is elementary ?
Consider the class of real elementary functions defined on a real interval I. These are real analytic functions. How can we characterise their power series ? That is, what can we say about the series of their coeficients ? For instance there are coeficients an given by rational functions in n , or given by combinations of rational functions and factorials functions, primitive recursive coeficients, coeficients given by recurrence relations, etc. It is easy to give an example of a real analytic function which is not elementary. Just solve the equation x′′ − tx = 0 using power series. This equation is known not to have any non-trivial elementary solution, in fact it has no Liouville solution (indefinite integrals of elementary functions).
Let ELEM be the problem: given a convergent Taylor series, does it represent an elementary function ? Let INT be the problem: given an elementary function does it have a primitive/indefinite integral which is an elementary function ? An observation that can be made is that if ELEM is decidable then so too is INT. Given an elementary function write down its Taylor series and integrate each term. Then apply the decision procedure for ELEM (of course we must be more precise here, this is just the general idea). Thus to show that ELEM is undecidable it suffices to show that INT is.
In the literature there is defined the class holomonic functions which can be characterised
either as:

1) Being solutions of a homogenous linear differential equation with polynomial coeficients.

2) Having Taylor series coeficients given by polynomial recurrence relations.

There is an algorithm to pass between these two presentations. The holonomic class includes elementary functions, the hypergeometric functions, the Bessel function, etc. The question naturally arises: given a sequence of real numbers, is it decidable if they obey a polynomial recurrence relation ?

Do all polynomial equations have computable analytic solutions ? (to do: check Wronski's work).

Whitehead's A Treatise on Universal Algebra

 https://archive.org/details/atreatiseonuniv00goog

The Extension theory of Hermann Grassmann

Burnside - Theory of Groups of Finite Order

Additions to 'Hegel and modern topology'

The something and the other. In the Science of Logic Hegel thinks of the idea of two things bearing the same relation to each other and indistinguishable in any other way. Now a good illustration of this situation in found in the two possible orientations of a vector space. Each bears the same relation to the other and there is absolutely no way to uniquely identify one of them in distinction to the other.  In the same place Hegel talks about the meaning of proper names and states that they have none !

Hegel's strategy. All of Hegel is based on the dynamics and structure of consciousness as it reveals itself to itself. But equally important it is all based on positing this structure of consciousness  to be essentially objective and not subjective.  Thus 'being a subject' is seen as a stage in the development of the object while mere  'subjectivity' is seen as a failure and partiality of objective consciousness in not living up to its full objectivity or actuality (or knowledge and realization thereof).

Thus ideality and infinity are the key structure of an object which has developed to a point of being a subject. 

Finitude and limit:  a closed set.  Yet the boundary also defines what the set is not (cf. discussion on Heyting algebra of open sets, etc.). Thus the boundary contains implicitly its own negation.

Limitation : an open set (also analytic continuation). Limit outside itself.

The ought... manifestation of the germ of a space through a given open set representation of the equivalence class. Any particular one is insufficient and can be replaced by another which is also insufficient.

False infinity: taking simply the set (or diagram) of such  open set representatives.  True infinity taking the limit (equivalence class) or limit (in the categorical sense)..

Degrees of interpenetration and transparency between the individual and the universal (and between self-consciousness and essence) in the Phenomenology of Spirit.  From the rudimentary form in the ethical life to: the universal is in the individual, the individual in the universal and the universal is in the relation between individuals and the relation between individuals is in the universal (mutual confession and forgiveness).

A central problem of philosophy

To us a central problem of philosophy is to elucidate the relationship between the following three domains of (apparent) reality /experience:

1. logic and language

2. mind and consciousness

3. an objective or external world 

While the the relationship between 2 and 3 is a classical topic which has produced an immense literature,  the deeper problem seems to be the relationship between 1 and 2 and 1 and 3.

My question is: how can my anti-inferentialism and anti-anti-representationalism and anti-functionalism be expressed in terms of such relationships ? 

A preliminary and useful questions: what was logic and language for Leibniz, Kant, Fichte, Hegel and Schopenhauer ? What was logic and language for Frege, Brentano and Husserl ? 

Why was not anti-psychologism accompanied by a corresponding anti-physicalism ? 

Another question: how does one's view of the relationship between 2 and 3 condition one's view of the relations 1-2 and 1-3 ? For instance, is the physicalist or idealist somehow conditioned in (or by) their views on logic and language ?

How are we to understand the theory that logic and language are precisely aspects of the self-interaction (self-reflection)  of a universal (super-individual) consciousness when to understand any process we must presuppose logic and language ?

Die Logik dagegen kann keine dieser Formen der Reflexion oder Regeln und Gesetze des Denkens voraussetzen, denn sie machen einen Theil ihres Inhalts selbst aus und haben erst innerhalb ihrer begründet zu werden. Nicht nur aber die Angabe der wissenschaftlichen Methode, sondern auch der Begriff selbst der Wissenschaft überhaupt gehört zu ihrem Inhalte, und zwar macht er ihr letztes Resultat aus; was sie ist, kann sie daher nicht voraussagen, sondern ihre ganze Abhandlung bringt dieß Wissen von ihr selbst erst als ihr Letztes und als ihre Vollendung hervor. (Hegel)

How can we apply logic and language to determine the relationship between logic and language themselves and something which is beyond logic or language ?

Can we develop the theory that a certain super-logical super-linguistic, non-logical and non-logical consciousness and cognition is necessary and useful ?  We have sketched a theory of analyticity based on computability and from this perspective the super-logical can been seen as unfolded in the hierarchy of degrees of hyper-computability. A logical pluralism which yet has nothing conventional or arbitrary about it.

Consciousness, experience, cognition, life...these are (at least potentially) infinitely more vast than abstract conceptual 'thought' in the ordinary sense of the word. We need to see thought as a multilayered structure and process part of a larger enveloping and grounding structure and process...

Also: I do not see any weighty argument against my own contention that the central problem of the philosophy of logic is simply: what is an argument, in particular what is a so-called 'valid' or 'persuasive' argument ? What is a sophistical argument (or a sophistical worldview) ?