In formal logic and theoretical computer science not enough attention has been payed to the explicit study of resource limitations and their implications (cf. the study of the length of proofs). We should reform these disciplines by explicitly postulating finite bounds in every aspect of the object of study and attach importance to the quantitative study of the interdependence of these bounds. For example: study the length of proofs in function of the formula to be proved in system based on a finite language. Study resource limitation in the grammar of natural language. Study the computational capacities of automata seen as finite approximations of Turing machines. This may be a way a developing abstract theories with a higher degree of fundamenta in re. Study the bounds and limitations of encodings of information and attempt to understand the question: can there be provable mathematical statements which have proofs which are too big to be written down or processed by a computer given our physical limitations ? Are there numbers to big to be represented by any means in the physical universe ? Finitism means not only finitely many basic or 'atomic' elements but finitely many higher-order relations between them. This all is related to the mysterious problem of qualitative changes in scale (and also the relativity of scale as illustrated in Gulliver's travels). Remaining in the finite domain by increase the quantity of a certain parameter (size, speed, etc.) suddenly beyond a certain limit the behavior of the system can change radically. This is usually studied using the infinitary models of mathematical analysis and geometry, but surely this occurs in a finitary context at a basic level. The question of finitism is related to some of the most fundamental questions both of phenomenology and the very concept of 'analysis' and 'scientific consciousness'. An authentic phenomenology must give a central place to the concepts of 'illusion' and 'delusion' as well as the two-pronged nature of the 'social' and the 'cultural'.
To understand the synthesis of phenomenology, logic and analytic scientific consciousness (in a playful symbolic way we could say: the synthesis between Hume, Sextus and Democritus - and we would add Epictetus for moral theory) it is good to look at the structure and dynamics of the theories of biology and biochemistry and their surprising correspondence with certain aspects of contemporary mathematics. The analogy between a tissue and its cells and the definition of algebraic variety (or better sub-analytic objects or general stratified objects) which has both a 'smooth' and 'extensive' aspect and a local, discrete, algebraic 'intensive' aspect. The algebraic locality is much like the discrete biochemistry of a cell and homology theory a kind of genome or centrifuge of the algebraic structure (so too are graded ring constructions). What is lacking is of course the dynamic, transformative aspect. The connection between biochemistry and logic (or between processes and inferences) is profound and Girard explicitly acknowledges this in his foundational papers on linear logic. We propose that the basic local-global (sheaf theoretic) concepts of geometry be applied to logic as well. We have an organism made up from local logics considered as individual cells. For each cell hypothesis are what are given from without and what is deduced (asserted) is what exits the cell. The difference from standard sheaf (and topos) theory is that we have a dynamic global aspect related to (rapid) transport and distant interactions (cf. the $\pi$-calculus). This TeX package is interesting because it bridges the gap between logical (and linguistic) syntax and chemical syntax by developing a linear system to represent two and three dimensional chemical diagrams (just as logical expressions codify syntactic trees).
If reality is represented by $U$ then a 'perspective' or 'approximation' or 'abstraction' or 'construction' is a system of relational logical atomism $A \rightarrow U$. These perspectives can be organized by a partial orders $A < A'$ signifying that $A'$ extends $A$ or that $A$ abstracts from details of $A'$. A very interesting aspect of this abstraction are the convenient fictions of mathematical analysis, the 'passage to the infinite', seen also in statistical mechanics and the kinetic theory of gases. Some comparisons might be made with Jain logic. The theory of scale is fundamental here.
All formal systems involve the characteristic of proliferation, generativity (cf. our previous discussion on semi-Thue systems). This is also omnipresent in algebra and geometry. The philosophical and scientific ideal of a 'system'. This reproduces fundamental characteristics of mental processes (note how Chatterjee's book on the Yogacara makes a connection to the broadly Hegelian subjective idealism of Gentile). But this generativity is limited by resources and is also error prone. Pyrrho and Sextus have a radically distinct approach and goal.
Open systems of logics (of whatever order) - essentially the relational syllogism in Galen which we argue to be present much earlier - seem good candidates for pure a priori logic and the for expressing the cognitive nature of human Turing completeness. This is (combined with Stoic propositional logic) is the natural logic to express semi-Thue, term rewriting systems and indeed an adequate concept of law and rules of a game (cf. as Panini's grammar). If existential quantifiers are interpreted constructively then we can interpret their presence in a theorem as an abbreviation for a more complex expression which is part of a purely open formula (let us say, second-order).
We must not look for linear progress in the history of logic, but rather cycles, periods of decadence and progress. There appears of have been a huge step backwards in many epochs in the history of logic. For instance the Kathâvatthu is much more sophisticated (possibly) than the later standard Nyâya theory of inference before it subsequently becomes very sophisticated and interesting in the Nava-Nyâya school - which Ganeri argues to represent a kind of graph-theoretic ontology and dialethic logic which also represents polyadic predicates (numbers are interpreted as such).
The syllogism $Fa$ and $\forall xFx\rightarrow Gx$ hence $Ga$ is epistemically faulty with a substitutional reading of the universal quantifier, because then it includes as a particular case $Fa\rightarrow Ga$ with means that we already know that $Ga$ since $Fa$. Also $a$ may be the only $x$ such that $Fx$ - hence the requirement that $F$ and $G$ be non-empty with a non-$a$ instance.
Addendum: Boethius and the medieval tradition support out interpretation of a topic as a 'maximal proposition', i.e. as an axiom. There is an interesting notion of definition as an 'unfolding'. We should mention Boethius' 'hypothetical syllogisms' in our note on Kant's logic. Can we find where Boethius is more explicit about a universally quantified conditionals ? Note that his indefinite propositions is our indefinite article selector and that singular propositions are introduced. Disjunction seems to have been studied from a profoundly mereological perspective (see also Kant and Hegel) which echoes our treatment.
Addendum: to our post on Hume we should add that there is much epistemically humility and agnosticism in Hume, direct reference to a mysterious, magical, unknown and unknowable 'human nature' which recalls very much Kant's similar remarks on the schematism of the pure concepts of the understanding.
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