Paradoxes of set theory

There are passages in Wittgenstein where he compares his method to psychotherapy and one or two where he seems to suggest that the ‘patient’ has the last word on his ‘illness’ and ‘cure’.  This paper tries to take these seriously, especially in connexion with the idea that victims of philosophical confusion talk nonsense.

Philosophical accusations of talking nonsense made in a polemical spirit are bound to fail, since the only way one can effectively criticise another’s utterance is by attaching some meaning to it.  But if the other acknowledges the possibility that his puzzlement arises from his having been taken in by nonsense, then perhaps a co-operative investigation of this suggestion might be fruitful.  Indeed self-diagnosis ought in principle to be possible.  Ought it not to have happened at least once in the history of philosophy if philosophical nonsense has ever been exposed? 

But there are difficulties.  For example:
a) We must ask whether or not philosophical nonsense, when exposed as such, is supposed to lose all appearance of meaningfulness.  Either answer is problematic.
b) The acknowledged similarity of the methodology under investigation to that of psychoanalysis exposes it to many of the familiar objections to the methodology of psychoanalysis, especially those that draw attention to the misleading influence of suggestion by an authority-figure.
c) The proposal that relentless cross-questioning of a suspected nonsense-talker will eventually lead her to ‘dry up’ (if she really is talking nonsense) seems to involve a version of the so-called Socratic Fallacy.

Edward Witherspoon distinguishes Wittgenstein’s conception of nonsense from Rudolf Carnap’s.  The latter does not fully take into account the fact that, if something really is nonsense, it has no more meaning than ‘Ab sur ah’ and therefore cannot have any logical properties.  He finds the Carnapian conception even in professed Wittgensteinians such as Peter Hacker and Gordon Baker who think of themselves as rejecting it.  My question is where this leaves the notion of PHILOSOPHICAL nonsense.  Could anything that was utterly devoid of content play the crucial deceptive role that nonsensicalists, including Wittgensteinians, assign it?  Could it ever be detected if it did?

Metabiology is a mathematical theory mainly based on algorithmic information theory and allows us to study an open-ended evolution of programs, that is, to study how fast the organisms/programs become more complex or more creative without never stagnate at a maximum level of complexity or creativity. We take, as starting point, the first chaitinian metabiological models in which the organisms are computable systems, Nature (or environment) is a hyper-computable system, mutations are randomly generated computable systems (that is, random algorithmic mutations) and the fitnesses are measured by how big is the output of the organisms. In this context, this paper focuses on two questions: Assuming that organisms are uncomputable systems, should metabiology become unsuitable or unsound? Or, on the other hand, assuming that Nature itself is a computable system, should metabiology become unsuitable or unsound? Thus, the main purpose is to demonstrate that metabiology walks towards a general theory of open-ended evolutionary systems, whether these systems are sub-computable, computable or hyper-computable. As a secondary purpose we show that scientific and philosophical consequences of our mathematical results spread across fundamental questions about biology and artificial life, as well as, about physics and artificial intelligence. To satisfy these two objectives, we build two new metabiological models for two new different configurations of organisms and nature. First, we demonstrate it is possible to build a metabiological evolution in which nature is a computable system and, hence, it makes the organisms and mutations to be sub-computable systems. To show this we introduce the recursive relative uncomputability (a computable/recursive version of the relative uncomputability that occurs among different Turing degrees). We do with Turing, Radó and Chaitin the same Skolem did with Cantor by making a relativization of the Busy-Beaver´s uncomputability in relation to the Omega number, so that this new uncomputability
can exist between a computer and any of his sub-computers. As a consequence, we introduce the “paradox” of uncomputability which says that exists at least one function that is not computable if we “look from the inside”, but that is computable if we “look from the outside”. Further, we correlate this pseudo-paradox with problems of human creativity and artificial intelligence and with problems of consciousness, artificial life and the computability of the Universe. Second, we build a metabiological model for the evolution of hyper-programs in which the organisms are hyper-computable systems of any finite order and nature is a hyper-program of ordinal order omega. In this last model we demonstrate it is possible an evolution of organisms/hyper-programs, with just “blind” random algorithmic mutations (it means, that do not take into consideration previous organisms/hyper-programs), which progressively leads the organisms/hyper-programs to be able of solving any problem in the arithmetical hierarchy. Allied to this, we discuss about computability or uncomputability of “living beings” and upon the possibility of mathematically studying metabiological evolutionary systems that can surpass, as much as we want, the “frontier” of the computable.

I discuss Charles Pigden’s paper ‘Coercive theories of meaning, or why language shouldn’t matter (so much) to philosophy’ and ask whether theories of meaning whose primary purpose is to discredit rival philosophies as meaningless necessarily employ artificially narrow conceptions of meaningfulness.  In the course of my argument I ask whether the colloquial-rhetorical use of words like ‘nonsense’ and ‘meaningless’ has made it difficulty for philosophers to see just how novel and radical their own accusations of talking nonsense are.

A metabiologia é uma teoria matemática baseada, principalmente, na teoria da informação algorítmica, nos permitindo estudar a evolução open-ended de programas, isto é, estudar quão rápido os organismos/programas se tornam mais complexos ou mais criativos, sem nunca estagnar num nível máximo de complexidade ou criatividade. Tomamos como base os modelos metabiológicos originais chaitinianos, nos quais os organismos são sistemas computáveis (i.e., sistemas emuláveis por um computador), a natureza (ou meio-ambiente) é um sistema hipercomputável (i.e., um sistema emulável apenas por algum hipercomputador), as mutações são sistemas computáveis aleatoriamente gerados (i.e., mutações algorítmicas aleatórias) e a aptidão dos organismos é medida estritamente por quão grande é o output desses organismos. Nesse âmbito, este trabalho se foca em responder duas questões. A metabiologia seria inapropriada se os organismos não forem sistemas computáveis? Ou, na direção oposta, ela seria inapropriada se a própria Natureza for um sistema computável? Portanto, o objetivo principal é mostrar que a metabiologia caminha em direção a uma teoria geral para evolução open-ended de qualquer sistema, seja ele subcomputável, computável ou hipercomputável. Como objetivo secundário, mostramos que as consequências científicas e filosóficas dos nossos resultados matemáticos podem se alastrar pelas questões fundamentais da biologia e vida artificial, como também, da física e da inteligência artificial. Para satisfazer esses dois objetivos, construímos dois novos modelos metabiológicos para duas configurações, diferentes entre si, de organismos e natureza. Primeiro, mostramos que é possível a evolução metabiológica de subprogramas, dentro da qual tomamos a natureza como sendo computável, o que fez com que os organismos e as mutações passassem a sistemas subcomputáveis. Para essa demonstração, apresentamos e utilizamos o fenômeno da incomputabilidade relativa recursiva (uma versão computável/recursiva da incomputabilidade relativa que ocorre entre as diferentes ordens de hipercomputadores dentro da hierarquia dos graus de Turing). Fazemos com Turing, Radó e Chaitin o que Skolem fez com Cantor: relativizamos a incomputabilidade da função Busy-Beaver atrelada ao número Ω para ela existir, de modo análogo, entre um computador e qualquer subcomputador do mesmo. Em decorrência, apresentamos o “paradoxo” da computabilidade, o qual diz que existe, pelo menos, uma função que não é computável se “olhada de dentro”, mas é computável se “olhada de fora”. Então, correlacionamos esse pseudoparadoxo com problemas da criatividade humana e inteligência artificial, da consciência, da vida artificial e da computabilidade do Universo. Segundo, construímos um modelo metabiológico para a evolução de hiperprogramas, no qual os organismos são sistemas hipercomputáveis de qualquer ordem finita (dentro da hierarquia de Turing) e a natureza é um hiperprograma de ordem ordinal ω. Nesse último modelo, mostramos que é possível uma evolução de organismos/hiperprogramas por meio de mutações algorítmicas aleatórias “cegas” (i.e., que não levam em conta os organismos/hiperprogramas anteriores) que vai levando os organismos/hiperprogramas a serem capazes de resolver qualquer problema na hierarquia aritmética. Aliado a isso, discutimos sobre a computabilidade ou incomputabilidade dos seres vivos e sobre a possibilidade de estudar matematicamente a evolução metabiológica de sistemas que podem ultrapassar, o quanto se queira, a “barreira” do computável.

I discuss Charles Pigden’s paper ‘Coercive theories of meaning, or why language shouldn’t matter (so much) to philosophy’ and ask whether theories of meaning whose primary purpose is to discredit rival philosophies as meaningless necessarily employ artificially narrow conceptions of meaningfulness.  In the course of my argument I ask whether the colloquial-rhetorical use of words like ‘nonsense’ and ‘meaningless’ has made it difficulty for philosophers to see just how novel and radical their own accusations of talking nonsense are.