The universal numbers. From Biology to Physics

Abstract

I will explain how the mathematicians have discovered the universal numbers, or abstract computer, and I will explain some abstract biology, mainly self-reproduction and embryogenesis. Then I will explain how and why, and in which sense, some of those numbers can dream and why their dreams can glue together and must, when we assume computationalism in cognitive science, generate a phenomenological physics, as part of a larger phenomenological theology (in the sense of the greek theologians). The title should have been "From Biology to Physics, through the Phenomenological Theology of the Universal Numbers", if that was not too long for a title. The theology will consist mainly, like in some (neo)platonist greek-indian-chinese tradition, in the truth about numbers' relative relations, with each others, and with themselves. The main difference between Aristotle and Plato is that Aristotle (especially in its common and modern christian interpretation) makes reality WYSIWYG (What you see is what you get: reality is what we observe, measure, i.e. the natural material physical science) where for Plato and the (rational) mystics, what we see might be only the shadow or the border of something else, which might be non physical (mathematical, arithmetical, theological, …). Since Gödel, we know that Truth, even just the Arithmetical Truth, is vastly bigger than what the machine can rationally justify. Yet, with Church's thesis, and the mechanizability of the diagonalizations involved, machines can apprehend this and can justify their limitations, and get some sense of what might be true beyond what they can prove or justify rationally. Indeed, the incompleteness phenomenon introduces a gap between what is provable by some machine and what is true about that machine, and, as Gödel saw already in 1931, the existence of that gap is accessible to the machine itself, once it is has enough provability abilities. Incompleteness separates truth and provable, and machines can justify this in some way. More importantly incompleteness entails the distinction between many intensional variants of provability. For example, the absence of reflexion (beweisbar(⌜A⌝) → A with beweisbar being Gödel's provability predicate) makes it impossible for the machine's provability to obey the axioms usually taken for a theory of knowledge. The most important consequence of this in the machine's possible phenomenology is that it provides sense, indeed arithmetical sense, to intensional variants of provability, like the logics of provability-and-truth, which at the propositional level can be mirrored by the logic of provable-and-true statements (beweisbar(⌜A⌝) ∧ A). It is incompleteness which makes this logic different from the logic of provability. Other variants, like provable-and-consistent, or provable-and-consistent-and-true, appears in the same way, and inherits the incompleteness splitting, unlike beweisbar(⌜A⌝) ∧ A. I will recall thought experience which motivates the use of those intensional variants to associate a knower and an observer in some canonical way to the machines or the numbers. We will in this way get an abstract and phenomenological theology of a machine M through the true logics of their true self-referential abilities (even if not provable, or knowable, by the machine itself), in those different intensional senses. Cognitive science and theoretical physics motivate the study of those logics with the arithmetical interpretation of the atomic sentences restricted to the "verifiable" (Σ1) sentences, which is the way to study the theology of the computationalist machine. This provides a logic of the observable, as expected by the Universal Dovetailer Argument, which will be recalled briefly, and which can lead to a comparison of the machine's logic of physics with the empirical logic of the physicists (like quantum logic). This leads also to a series of open problems.