Genetic logic and epistemology of theories

  By 'theory' we mean a body of knowledge capable of a formal or semi-formal presentation. But here we focus on the kind of theory which can be formalized in the standard way using  for instance first or second order logic or dependent type theory. A theory is not merely the set of sentences which can be deduced from a given axiomatic-deductive system. Rather some sentences are marked off as being more meaningful and relevant than others (theorems, lemmas and corollaries and also examples and counterexamples). We call these 'relevant' sentences. Mere tautologies are a priori excluded.  And most importantly a theory is built up from an dependently ordered (non-circular) hierarchy of definitions. Relevant sentences and the definition system can be organized in a dependency tree which is usually flattened or made linear when the theory is presented (for simplicity we do not dwell here on the case in which a definition depends on a previous relevant sentences, for instance a uniqueness result). This intrinsically linear or tree-like structure is analogous to the process of organic (ontogenetic) or cultural development. We call it the genetic logic of the theory. This leads to the question of justifying a genetic logic. Is it somehow implicit in the axiomatic-deductive system, built into its 'essence' ? Or does the genetic logic and the axiomatic-deductive system both derive from some third principle, something that well-known incompleteness results as well as the fact that the 'same' theory can be presented in terms of different axiomatic-deductive systems would naturally suggest ? In what sense are the successive stage or branchings of the genetic logic tree dependent on each other or the branching elements contained implicitly in or are an unfolding of the previous elements ? We call this the 'unfolding impulse'. There is the remarkable fact that many previous definitions turn out to be particular cases of a single more universal definition. For instance the case of Kan extensions in category theory.  

 Light is shed on this question by considering the genetic epistemology of a theory. That is, the (optimal) process by which human beings come to learn and understand a theory. This can be either on a individual level or on an historical-cultural level. On an individual level at present the genetic logic largely coincides with the genetic epistemology of a theory. But there are important nuances involved in the distinction between research papers, introductions, fundamental treatises and reference works. The very term 'relevant' already suggests a subjective human dimension. It seems interesting to investigate to what extent the mirroring of the genetic dimension in the epistemological dimension can explain the 'unfolding impulse' mentioned above (we do bear in mind the non-linearity of the human learning process and the necessity of 'cyclic return' in the form of revision and refinement). How does our knowledge of a concept or theorem (or a particular kind of philosophical reflection on this knowledge)  already in itself lead to the thrust or impulse to find the subsequent concepts or theorems ? For instance reflecting on the concept of 'initial object' (or reflecting on our own understanding of such a concept) we could easily be lead to propose the dual concept of 'terminal object' (and vice-versa). Aristotle however distinguished between a method starting from things more clear and fundamental 'in themselves' and a method starting from things more clear and fundamental 'to us'. Thus maybe the concept of Kan extension is more fundamental 'in itself' (relative to category theory) but for cognitive-epistemic reasons 'for us' it is better to first go through a series of concrete cases of this concept. We must investigate the relationship between historical genetic epistemology and  individual genetic epistemology (both for adults and for the process of child development). For instance last century children were taught Euclid's Elements at school, thus manifesting a kind of cultural law of ontogenetic recapitulation. We do not think it an abuse of terminology (pace Piaget) to refer to the process by which an adult comes to learn a given theory 'genetic epistemology'. Also we must answer the objection that genetic logic itself can be historically and culturally dependent. For instance we can perhaps show that in outline the genetic logic of the calculus has remained fairly constant since the 17th-century.  And we must study the relationship, both synchronic and diachronic, between theories and the justification of taking a body of knowledge as a single theory (i.e. study subtheories, branches of theories on one hand and on the other interdisciplinarity and theories of theories).  This includes the study of how the 'same' concept can appear in entirely different theories. This is what gave birth to category theory and the notion of functor in the early developments of algebraic topology.

  We propose the definition of 'general logic' as the study of the genetic logic and epistemology of theories.  And we define 'pure logic' as the study of the genetic logic and epistemology of the system of universal concepts present in all theories.  If we restrict ourselves to mathematical theories then in what sense does category theory contribute to this goal and surpass model theory ? 

 Consider the category theoretic concepts of 'product' and 'co-product'.  The co-product of two objects $A$ and $B$ an object $A + B$ (actually an isomorphism class) and two morphisms $i_1 : A \rightarrow A + B$, $i_2 : B \rightarrow A + B$ which corresponds to the simplest way of collecting $A$ and $B$ together into a whole much as in addition and counting. $A$ and $B$ are treated as unity and there is no relation involved. In natural language this corresponds to some uses of 'and'. For instance  the meaning of 'Alice and Bob' is the meaning of both proper names gathered side by side, collected into a whole in the simplest, freest way.  Type theoretically and in algebraic logic the coproduct corresponds to intuitionistic disjunction.  The coproduct (and pushout) is of great importance in homotopy theory . Also topos theory for defining the generalized concept of 'connectivity'.

 Now the product $A\times B$ and its associated morphisms $\pi_\ : A\times B \rightarrow A$ and $\pi_2 : A\times B \rightarrow B$ is a subtly different concept.  In fact we believe that it is only dependent type theory that can give us the clearest expression of its meaning as a degenerate case of the more general type $\Pi (x : A) B(x)$. The product expresses the idea of the simplest type of mixture and distributive combination of the objects $A$ and $B$, a template for all the relations an $A$ can have with a $B$. We can also consider the notions of 'interaction' and 'entanglement'.

Or there is a mereological analysis:  a part of $A+ B$ may be a part only of  $A$ or  only  of $B$ but a part of $A\times B$ must always involve both $A$ and $B$. Thus the mereological analysis overlaps with the standard Curry-Howard interpretation.

But these considerations can be refined by considering linear logic and the further duality present in each of the basic logic connectives.

  Older note: In formalizing concepts one must ask: scientific (including logical and mathematical) concepts or the semantics of ordinary human life-world concepts ? How do these two relate ? What is their dependency and priority ?  We can investigate  'Semantic Primes and Universals' to use the terminology of A. Wierzbicka. Each folk-concept is given a 'definition', a 'story', employing a fixed set of 'semantic primes'. Piaget's genetic epistemology is also worth exploring from this perspective.