3,910 research outputs found
On a Perceived Expressive Inadequacy of Principia Mathematica
This paper deploys a Cantor-style diagonal argument which indicates that there is more possible mathematical content than there are propositional functions in Russell and Whitehead's Principia Mathematica and similar formal systems. This technical result raises a historical question: "How did Russell, who was himself an expert in diagonal arguments, not see this coming?" It turns out that answering this question requires an appreciation of Russell's understanding of what logic is, and how he construed the relationship between logic and Principia Mathematica
Predicativity and parametric polymorphism of Brouwerian implication
A common objection to the definition of intuitionistic implication in the
Proof Interpretation is that it is impredicative. I discuss the history of that
objection, argue that in Brouwer's writings predicativity of implication is
ensured through parametric polymorphism of functions on species, and compare
this construal with the alternative approaches to predicative implication of
Goodman, Dummett, Prawitz, and Martin-L\"of.Comment: Added further references (Pistone, Poincar\'e, Tabatabai, Van Atten
Wittgensteinâs ânotorious paragraphâ about the Gödel Theorem
In §8 of Remarks on the Foundations of Mathematics
(RFM), Appendix 3 Wittgenstein imagines what
conclusions would have to be drawn if the Gödel formula P
or ÂŹP would be derivable in PM. In this case, he says, one
has to conclude that the interpretation of P as âP is
unprovableâ must be given up. This ânotorious paragraphâ
has heated up a debate on whether the point Wittgenstein
has to make is one of âgreat philosophical interestâ
revealing âremarkable insightâ in Gödelâs proof, as Floyd
and Putnam suggest (Floyd (2000), Floyd (2001)), or
whether this remark reveals Wittgensteinâs
misunderstanding of Gödelâs proof as Rodych and Steiner
argued for recently (Rodych (1999, 2002, 2003), Steiner
(2001)). In the following the arguments of both
interpretations will be sketched and some deficiencies will
be identified. Afterwards a detailed reconstruction of
Wittgensteinâs argument will be offered. It will be seen that
Wittgensteinâs argumentation is meant to be a rejection of
Gödelâs proof but that it cannot satisfy this pretension
Mathematical Foundations of Consciousness
We employ the Zermelo-Fraenkel Axioms that characterize sets as mathematical
primitives. The Anti-foundation Axiom plays a significant role in our
development, since among other of its features, its replacement for the Axiom
of Foundation in the Zermelo-Fraenkel Axioms motivates Platonic
interpretations. These interpretations also depend on such allied notions for
sets as pictures, graphs, decorations, labelings and various mappings that we
use. A syntax and semantics of operators acting on sets is developed. Such
features enable construction of a theory of non-well-founded sets that we use
to frame mathematical foundations of consciousness. To do this we introduce a
supplementary axiomatic system that characterizes experience and consciousness
as primitives. The new axioms proceed through characterization of so- called
consciousness operators. The Russell operator plays a central role and is shown
to be one example of a consciousness operator. Neural networks supply striking
examples of non-well-founded graphs the decorations of which generate
associated sets, each with a Platonic aspect. Employing our foundations, we
show how the supervening of consciousness on its neural correlates in the brain
enables the framing of a theory of consciousness by applying appropriate
consciousness operators to the generated sets in question
The 1900 Turn in Bertrand Russellâs Logic, the Emergence of his Paradox, and the Way Out
Russellâs initial project in philosophy (1898) was to make mathematics rigorous reducing it to logic. Before August 1900, however, Russellâs logic was nothing but mereology. First, his acquaintance with Peanoâs ideas in August 1900 led him to discard the part-whole logic and accept a kind of intensional predicate logic instead. Among other things, the predicate logic helped Russell embrace a technique of treating the paradox of infinite numbers with the help of a singular concept, which he called âdenoting phraseâ. Unfortunately, a new paradox emerged soon: that of classes. The main contention of this paper is that Russellâs new conception only transferred the paradox of infinity from the realm of infinite numbers to that of class-inclusion.
Russellâs long-elaborated solution to his paradox developed between 1905 and 1908 was nothing but to set aside of some of the ideas he adopted with his turn of August 1900: (i) With the Theory of Descriptions, he reintroduced the complexes we are acquainted with in logic. In this way, he partly restored the pre-August 1900 mereology of complexes and simples. (ii) The elimination of classes, with the help of the âsubstitutional theoryâ, and of propositions, by means of the Multiple Relation Theory of Judgment, completed this process
Consequences of a Goedel's misjudgment
The fundamental aim of the paper is to correct an harmful way to interpret a
Goedel's erroneous remark at the Congress of Koenigsberg in 1930. Despite the
Goedel's fault is rather venial, its misreading has produced and continues to
produce dangerous fruits, as to apply the incompleteness Theorems to the full
second-order Arithmetic and to deduce the semantic incompleteness of its
language by these same Theorems. The first three paragraphs are introductory
and serve to define the languages inherently semantic and its properties, to
discuss the consequences of the expression order used in a language and some
question about the semantic completeness: in particular is highlighted the fact
that a non-formal theory may be semantically complete despite using a language
semantically incomplete. Finally, an alternative interpretation of the Goedel's
unfortunate comment is proposed. KEYWORDS: semantic completeness, syntactic
incompleteness, categoricity, arithmetic, second-order languages, paradoxesComment: English version, 19 pages. Fixed and improved terminolog
Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?
Classical interpretations of Goedel's formal reasoning imply that the truth
of some arithmetical propositions of any formal mathematical language, under
any interpretation, is essentially unverifiable. However, a language of
general, scientific, discourse cannot allow its mathematical propositions to be
interpreted ambiguously. Such a language must, therefore, define mathematical
truth verifiably. We consider a constructive interpretation of classical,
Tarskian, truth, and of Goedel's reasoning, under which any formal system of
Peano Arithmetic is verifiably complete. We show how some paradoxical concepts
of Quantum mechanics can be expressed, and interpreted, naturally under a
constructive definition of mathematical truth.Comment: 73 pages; this is an updated version of the NQ essay; an HTML version
is available at http://alixcomsi.com/Do_Goedel_incompleteness_theorems.ht
The semantics of jitter in anticipating time itself within nano-technology
The development of nano-technology calls for a careful examination of anticipatory systems at this small scale. For the characteristics of time at the boundary between classical and quantum domains are quite critical for the advancement of the new technology. It has long been well recognised that time is not absolute even in classical subjects like navigation and dynamics where idealised concepts like mean solar time, International Atomic Time and Newtonâs dynamical time have had to be used. Time is the data of the Universe and belongs in the semantics of its extensional form. At the boundary between classical and quantum behaviour the uncertainty of time data becomes a significant effect and this is why it is of great importance in nanotechnology, in areas such as the interoperability of different time domains in hardware, where noise in the form of jitter causes a system to behave in an unpredictable fashion, a severe and expensive problem for anticipating how time is to be handled. A fundamental difficulty is that jitter is represented using numbers, giving rise to undecidability and incompleteness according to Gödelâs theorems. To escape the clutches of Gödel undecidability it is necessary to advance to cartesian closed categories beyond the category of sets to represent the relationship between different times as adjoint endofunctors in monad and comonad constructions
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