Miscellany of philosophical observations

1. Quantum theory gave us the idea of introducing negative probabilities, i.e. signed measures. 

2. Category theory is intensional (non-extensionalist) mathematics based on minimal logic, thus hyper-constructive.  We ask about a natural number object (the concept of an 'element'  is not taken as a primitive; rather we have only generalized elements $1\to A$) in a given category, that is, about its universal property;  we construct concrete generalized element 'numbers' through composition of primitive morphisms $z:1\to N$ and $s:N\to N$. Recall how the concept of primitive recursive function emerges naturally from this definition...

4. There have always been different notions of 'quantification' (and the corresponding determiners) which were conflated by extensionalist logicians.  This is clear in the distinction between intensional, conceptual universal quantification and extensional quantification. Also such distinctions are brought to light by the behaviour of quantifiers in propositional attitudes.  Constructivism tried to bridge the gap between extension and intension via a kind schematism (see previous post). We must bring all the different kinds of quantification to light again. 'Some' seem to be even richer in nuances than 'for all'. The distinction between the classical and intuitionistic/constructivist 'some' is deeply rooted in and reflected in cognition and natural language semantics. For instance, the intuitionistic interpretation fails for existential formulas in the scope of propositional attitudes. I may believe that the money in a book in the library without there being a specific book in which I believe the money is in.

Are set-theoretic extensions are atomistic structureless heaps, like the extreme abstract atomic alienated negativity in certain stages of Hegel's phenomenology of spirit ? This is not really so, they can have a very definite tree-like structure. Groupoids have more organic unity. We must investigate what it means to quantify over groupoids.

5. Some people are scared of homotopy type theory, higher category theory or of Coq and Agda. I respect that.  I feel the same about fractal calculus. But perhaps fractal calculus has something to do with the following important question. Numerical, discrete, computational methods are routinely used to find approximate solutions of differential (and integral-differential) equations. But we also need in a turn a theory of how differential and smooth systems can be seen as approximations of non-differential and non-smooth systems. Is this not what we do when we apply the Navier-Stokes equations to model real fluids ? Recall how continuous functions with compact support are dense in the $L^p$ spaces of integrable measurable functions (but see also Lusin' theorem).  Can all this be given a Kantian interpretation ? An analogy of experience: how the very notion of measurable function supposes the standard topology and Borel structure on the real line $\mathbb{R}$.

6.  What are distributions ? They allow a mathematical treatment of the vague notion of particle. Indeed particles are just euphemisms for certain kinds of stable self-similar field-phenomena. The great geniuses in physics were those who helped build geometric physics (which is what is most developed and sophisticated in modern physics):  Leibniz, Lagrange, Euler, Hamilton, Gauss, Riemann, Poincaré, Minkowsky and many others.  But it is no use playing around with highly sophisticated geometric physics (which looses all connection to experiment)  if you haven't solved the problem of quantum theory first. 

Distributions are clearly in themselves meant to be idealizations and abstractions of actual functions with their ultimate aim being approximation results. What is a dirac function ? This will depend on the scale. Dirac functions in nature are only approximate.

7. Study differential geometry as type theory; dispel all difficulties in a general understanding of mathematics as a language. It is of utmost importance to give physics, specially quantum theory, great formal logical and mathematical and philosophical rigour. Outstanding example: Peter Bongaarts' book.

8. Many of our concepts have a tripartite nature $(A,A^{\circ },\overline{A})$ expressing certain $A$, certain not $A$ and the grey neutral area $\overline{A}$. For instance: bald, not bald, sort of bald but not really bold. Each one in turn will depend on an individual and a possible situation of affairs. But this is not enough. In order to do any kind of 'logic' here we need some kind of quantified probability measure, for instance the ability to measure quantities of individuals and states of affairs. Then the sorites is resolved by presenting a tripartite distribution.  Thus it is interesting to have a logic which can express probability distributions.

9.  The goal is to pass from language-based philosophy to pure logic based philosophy. But this needs a mediator. The mediator can only be advanced, sophisticated, mathematical models, qualitative, essential, extending to all domains of reality (deformations, moduli are the right way to study possible worlds). All aspects of Kant and Husserl can be given their mathematical interpretation and from thence their logical-axiomatic interpretation. The same goes for naturphilosophie via René Thom and Stephen Smale. Theoretical platonism and idealism is not enough. We need this realized applied platonism. Mathematics furnishes a rigorous way of dealing with analogy and integrating analogy into philosophy. Also mathematics furnishes the deeper meaning and interpretation of Kant's theory of categories and schematism. Mathematics furnishes us with a way of studying concepts which is not divorced from the conceiving mind but at the same time is not psychologistic.

10. How do mathematicians think, actually prove theorems and have insight and intuition - all of which is very different from a low-level proof-search for some formal axiomatic-deductive system ? In particular how can formal logic and intuition agree ?  If logic is the science of valid thought, then it just cannot ignore this question.  We certainly think immediately using admissible rules.

Consider a formal logic $L$ in which we have the concept of atomic predicate, equivalence and equality. Let $T$ and $P$ be a countably infinite set of symbols no occurring in the language of $L$. By a prelogic we mean a finite set $(t,p)$ consisting of a finite sets $t,p$ of formulas in $L(T,P)$ of the form $q(x_1,...,x_n)\equiv ...$ and of the form $t(x_1,...,x_n)=...$. We write $(t_1,p_1)\le (t_2,p_2)$ iff the symbols in the left sides of $t_2,p_2$ all occur in $t_1$ and $p_1$ and furthermore if $t_1\subset t_2$ and $p_1\subset p_2$.  For each prelogic $(t,s)$ we further associate a set of intuitively valid sentences $ISen\subset Sen(t,s)$ and intuitively valid inferences  $IDed$ which are subsets of $Sen(t,s)\times Sen(t,s)$, where $Sen(t,s)$ denotes the set of sentence whose symbols occur all in $t,s$.

11. The problem of the denotation of the selection of one of two orientations of vector space or one of the square roots of $-1$. 

12. Some important authors to study: Albert Lautman and Jean Petitot. A synthesis of Kant and Husserl within the framework of an enlightened mathematical structuralism.

13. Determinism may be only local. Determinism (think analytic continuation) is like a covering space. Only one continuation and lifting of a path for a chosen point in the fiber. But we can have instead of a locally constant sheaf a constructible sheaf. There is a stratification in which non-deterministic switches or choices take place (although they can be perfectly continuous).

14. What is completeness for a logical-deductive system ? And relative to a class of models ? Take intuitionistic propositional logic.  The classical logical-deductive notion of completeness does not apply anymore. Only a model theoretic one.  And the model theoretic one needs to change to become multi-valued, i.e. as in topos theory or at least the Heyting algebra of truth-values. This was the insight behind Kant's transcendental dialectic: that $A\vee \mathrm{\neg }A$ is not a universal law of reason.