The Quantum Strategy of Completeness: On the Self-Foundation of Mathematics

Gentzen’s approach by transfinite induction and that of intuitionist Heyting arithmetic to completeness and the self-foundation of mathematics are compared and opposed to the Gödel incompleteness results as to Peano arithmetic. Quantum mechanics involves infinity by Hilbert space, but it is finitist as any experimental science. The absence of hidden variables in it interpretable as its completeness should resurrect Hilbert’s finitism at the cost of relevant modification of the latter already hinted by intuitionism and Gentzen’s approaches for completeness. This paper investigates both conditions and philosophical background necessary for that modification. The main conclusion is that the concept of infinity as underlying contemporary mathematics cannot be reduced to a single Peano arithmetic, but to at least two ones independent of each other. Intuitionism, quantum mechanics, and Gentzen’s approaches to completeness an even Hilbert’s finitism can be unified from that viewpoint. Mathematics may found itself by a way of finitism complemented by choice. The concept of information as the quantity of choices underlies that viewpoint. Quantum mechanics interpretable in terms of information and quantum information is inseparable from mathematics and its foundation

A Mathematical Model of Quantum Computer by Both Arithmetic and Set Theory

A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. After the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent Peano arithmetics discussed in Section I. Many modifications of Turing machines cum quantum ones are researched in Section II for the Halting problem and completeness, and the model of two independent Turing machines seems to generalize them. Then, that pair can be postulated as the formal definition of reality therefore being complete unlike any of them standalone, remaining incomplete without its complementary counterpart. Representation is formal defined as a one-to-one mapping between the two Turing machines, and the set of all those mappings can be considered as “language” therefore including metaphors as mappings different than representation. Section III investigates that formal relation of “reality”, “representation”, and “language” modeled by (at least two) Turing machines. The independence of (two) Turing machines is interpreted by means of game theory and especially of the Nash equilibrium in Section IV. Choice and information as the quantity of choices are involved. That approach seems to be equivalent to that based on set theory and the concept of actual infinity in mathematics and allowing of practical implementations

Representation and Reality by Language: How to make a home quantum computer?

A set theory model of reality, representation and language based on the relation of completeness and incompleteness is explored. The problem of completeness of mathematics is linked to its counterpart in quantum mechanics. That model includes two Peano arithmetics or Turing machines independent of each other. The complex Hilbert space underlying quantum mechanics as the base of its mathematical formalism is interpreted as a generalization of Peano arithmetic: It is a doubled infinite set of doubled Peano arithmetics having a remarkable symmetry to the axiom of choice. The quantity of information is interpreted as the number of elementary choices (bits). Quantum information is seen as the generalization of information to infinite sets or series. The equivalence of that model to a quantum computer is demonstrated. The condition for the Turing machines to be independent of each other is reduced to the state of Nash equilibrium between them. Two relative models of language as game in the sense of game theory and as ontology of metaphors (all mappings, which are not one-to-one, i.e. not representations of reality in a formal sense) are deduced

A model of Peano's axioms in Euclidean geometry

The purpose of this paper is to show that arithmetic is consistent if Euclidean geometry is. Specifically, a model of Peano's axioms [2] is defined in the space of Euclidean geometry, where Hilberts axioms [3] are taken to be the axioms of Euclidean geometry

Selected Bibliography of Materials on Geometry Useful to High School Teachers and Students of Mathematics

Natural Scienc

Foundational Constructive Geometry

An ideal constructor produces geometry from scratch, modelled through the bottom-up assembly of a graph-like lattice within a space that is defined, bootstrap-wise, by that lattice. Construction becomes the problem of assembling a homogeneous lattice in three-dimensional space; that becomes the problem of resolving geometrical frustration in quasicrystalline structure; achieved by reconceiving the lattice as a dynamical system. The resulting construction is presented as the introductory model sufficient to motivate the formal argument that it is a fundamental structure; based on which, it is proposed that where mathematics’ numbers conventionally correspond to dimensionless points on the stateless number line, numbers more fundamentally correspond to an ordering of discrete objects constructed within the stateful number lattice. A second observation is that this fundamental lattice structure is helically configured with fractal character, which, as it relates to the geometry underlying spacetime, has relevance to questions in physics, particularly those involving wave-particle duality

Is Maddy's naturalism a defensible view of mathematics?

The thesis is an examination of Penelope Maddy's book Naturalism in Mathematics and of the defensibility of the arguments she presents in support of her version of naturalism, with emphasis on the philosophical significance of her work. In Part A I first give an overview of the aim of the book and the methods used to achieve this aim. I then set out and appraise the arguments Maddy advances showing how she supports these arguments with her appeals to historical and current scientific and mathematical practice, I discuss and appraise her comments on the work of the authors she cites and give examples of the way in which she presents her case. I examine the extent to which the work of these authors can be seen to give support to Maddy's arguments. I also examine the validity of her appeals to the analogy between naturalism in science and naturalism in mathematics with reference to her descriptions of scientific practice. In Part B I discuss the objections advanced against Maddy's version of naturalism by contemporary critics of her book, with reference to seven authors in particular, considering the similarities and differences of the approach taken by each to Maddy. I show how their objections fall under two specific heads and appraise the persuasiveness of their criticisms. Finaly, I assess the effect of the objections which can be raised against Maddy's naturalism and examine the question of how compelling they are and whether Maddy's naturalism is still a convincing approach to mathematics in the light of these criticisms

2022-2023 Course Catalog

An annual catalog of courses and course descriptions offered at the University of Montana.https://scholarworks.umt.edu/coursecatalogs_asc/1117/thumbnail.jp