Mathematical Proof and Genre Theory

The purpose of this study is to establish levels from an inductive proof to an algebraic demonstration in lower secondary school mathematics. I propose that we can establish six levels of proof in lower secondary school mathematics as steps from an inductive proof to an algebraic demonstration on the basis of three axes (contents of proof, representation of proof, and students' thinking). To reach this conclusion, I firstly examine the meaning of ‘demonstration in lower secondary school mathematics’ and ‘proof in lower secondary school mathematics’, and show the relationships between them. Secondly, I set out four basic levels of proof, as seen from two aspects (contents and representation of proof). Thirdly, I subdivide them into six levels from the third aspect of students' thinking. Finally, I illustrate my discussion with a 7th grader's activities.

Computability is one of the fundamental notions of mathematics and computer science, trying to capture the effective content of mathematics and its applications. Computability Theory explores the frontiers and limits of effectiveness and algorithmic methods. It has its origins in Godel's Incompleteness Theorems and the formalization of computability by Turing and others, which later led to the emergence of computer science as we know it today. Computability Theory is strongly connected to other areas of mathematics and theoretical computer science. The core of this theory is the analysis of relative computability and the induced degrees of unsolvability; its applications are mainly to Kolmogorov complexity and randomness as well as mathematical logic, analysis and algebra. Current research in computability theory stresses these applications and focuses on algorithmic randomness, computable analysis, computable model theory, and reverse mathematics (proof theory). Recent advances in these research directions have revealed some deep interactions not only among these areas but also with the core parts of computability theory. The goal of this Dagstuhl Seminar is to bring together researchers from all parts of computability theory and related areas in order to discuss advances in the individual areas and the interactions among those.

Several theorems about the equivalence of familiar theories of reverse mathematics with certain well-ordering principles have been proved by recursion-theoretic and combinatorial methods (Friedman, Marcone, Montalbán et al.) and with far-reaching results by proof-theoretic technology (Afshari, Freund, Girard, Rathjen, Thomson, Valencia Vizcaíno, Weiermann et al.), employing deduction search trees and cut elimination theorems in infinitary logics with ordinal bounds in the latter case. At type level 1, the well-ordering principles are of the form $$ \text { (*) } ``\text {if}\,X \text { is well-ordered then } \, f(X) \, \text { is well-ordered''} $$where f is a standard proof-theoretic function from ordinals to ordinals (such f’s are always dilators). One aim of the paper is to present a general methodology underlying these results that enables one to construct omega-models of particular theories from \((*)\) and even \(\beta \)-models from the type 2 version of \((*)\). As \((*)\) is of complexity \(\Pi ^1_2\) such a principle cannot characterize stronger comprehensions at the level of \(\Pi ^1_1\)-comprehension. This requires a higher order version of \((*)\) that employs ideas from ordinal representation systems with collapsing functions used in impredicative proof theory. The simplest one is the Bachmann construction. Relativizing the latter construction to any dilator f and asserting that this always yields a well-ordering turns out to be equivalent to \(\Pi ^1_1\)-comprehension. This result has been conjectured and a proof strategy adumbrated roughly 10 years ago, but a detailed proof has only been worked out in recent years.KeywordsInfinite proof theoryReverse mathematicsOrdinal analysisOrdinal representation systemsCut eliminationDilators\(\omega \)-models\(\beta \)-models

Uncomputability and undecidability in economic theory

Unknowability in Mathematics, Biology, and Physics Gregory J. Chaitin (bio) there are three fields in mathematics that deal with unknowability; in chronological order, they are probability theory, metamathematics, and algorithmic information theory. The theory of probabilities began in a series of letters between Pascal and Fermat. One of the first books in this field is A Philosophical Essay on Probabilities by Pierre-Simon Laplace, published in 1814, which is still very much worth reading. The theory of probability is about indeterminate events, but it is not about mathematical facts, which are always black or white, not grey, and which are necessary, not contingent, truths. However, the field of meta mathematics is actually about unknowability within mathematics itself—and a fascinating piece of intellectual history associated with the names David Hilbert, Kurt Gödel, and Alan Turing. These three gentlemen are mentioned elsewhere in this volume, so I will not dwell on them. The newest mathematical field dealing with unknowability is algorithmic information theory. The star of this theory is a real number, the halting probability Omega, which is defined as the probability that a self-contained computer program chosen at random will eventually halt. If all programs halted, the halting probability would be one. If no program halted, the halting probability would be zero. And since some programs eventually halt and others don't, Omega is actually between zero and one. [End Page 107] Omega has a rather straightforward mathematical definition, and it is a single, well-defined real number. Yet its numerical value is maximally unknowable. The base-two bits of the numerical value of the halting probability Omega provide a perfect simulation within pure mathematics of independent tosses of a fair coin. Whether an individual bit is a zero or a one is, to God or to an infinite mind, a single, well-determined individual mathematical truth, but to us down here, it appears maximally unknowable and very much like the result of independent tosses of a fair coin. Hilbert had hoped that there would be a theory of everything for pure mathematics, but Omega shows that pure math is more like biology, the domain of the irreducibly complex, than it is like fundamental physics, where there is still hope of finding a small set of simple equations that determine the universe. So that brings us to biology, a very messy, complicated field. I have a wife and a child, so I love messy biology, but from a mathematical point of view, biology looks rather intractable, rather opaque. Will we ever be able to prove mathematically that life must evolve? Does Darwin's theory provably account for the spectacular biological creativity that led to human beings, to you and me? Can one prove that random mutations and natural selection are enough to account for the richness and diversity of our biosphere? There is a highly developed field called systems biology that deals with computer simulations of complex biological systems, but there are very few mathematical proofs in biology. What can be done about this? Well, in a remarkable 1948 lecture published in 1951 and remarkably forgotten, The General and Logical Theory of Automata, John von Neumann identified the fundamental mathematical idea hidden in biology. That's the idea of software, which accounts both for the plasticity of computer technology and its overwhelming success, and for the plasticity of the biosphere with its spectacular diversity of forms. To put it bluntly, nature invented software millennia before humanity did. Biology deals with extremely ancient software. "Evodevo," or evolutionary developmental biology, is a kind of software archeology. [End Page 108] The new field I call "metabiology" recognizes the impossibility of a fundamental mathematical theory dealing with the effect of random mutations on natural software, on DNA, so instead it proposes studying the effect of random mutations on artificial software, on computer programs. That, in fact, turns out to be tractable mathematically, and it is probably as close as one can get to a theoretical biology as deeply impregnated with mathematics as theoretical physics already is. When I was a visiting professor at the Federal University of Rio de Janeiro, I sketched out these ideas, and they were published in a little...

The term proof theory, in German Beweistheorie, was introduced by Hilbert [1] in 1922 as the name of a subject which he had already described in a lecture [2] in 1917 as ‘an important, new field of research’ where ‘we must make the concept of specific mathematical proof itself object of investigation, just as also the astronomer pays attention to his place of observation, the physicist must care about the theory of his instrument, and the philosopher criticizes reason itself’.

In the 1990s Yankov published two papers containing important contributions to proof theory and based on the application of a dialogical interpretation of proofs. In both cases the method is used for providing constructive proofs of important metalogical results concerning classical logic and fundamental mathematical theories. In the first paper it is shown that impredicative extensions of intuitionistic versions of arithmetic, analysis and set theory, enriched with suitable bar induction schemata, are sufficiently strong for proving the consistency of their classical counterparts. In the second paper the same method is applied to provide a constructive proof of the completeness theorem for classical logic. In both cases a version of a one-sided sequent calculus in Schütte-style is used and cut elimination is established in the second case. Although the obtained results are important, and the applied method is original and interesting, they have not received the attention they deserve from the wider community of researchers in proof theory. In this paper we briefly recall the content of both papers. We focus on essential features of Yankov’s approach and provide comparisons with other results of similar character.KeywordsProof theoryDialogical interpretation of proofsConsistency proofsBar inductionArithmeticAnalysisSet theory

Using the example of the divide et impera program scheme we present a knowledge-assisted refinement process based on type theory that yields executable programs from given requirement specifications. Programming knowledge is described in terms of precise mathematical theories and proofs, and programs and knowledge about programs is expressed in the same language and are developed using the same techniques.

We discuss the halting probability Ω, whose bits are irreducible mathematical facts, that is, facts which cannot be derived from any principles simpler than they are. In other words, you need a mathematical theory with N bits of axioms in order to be able to determine N bits of Ω. This pathological property of Ω is difficult to reconcile with traditional philosophies of mathematics and with traditional views of the nature of mathematical proof and of mathematical knowledge. Instead Ω suggests a quasi-empirical view of math that emphasizes the similarities between mathematics and physics rather than the differences.

With the “mathematical watershed”, Logic had been transformed into a foundational theory for mathematics, a theory of truth and proofs—far away from its philosophical status of theory of the intellectual process of reasoning. With the recent substitution of the traditional proofs-as-discourses paradigm by the proofs-as-programs one, Logic is now becomming a foundational theory for computing. One could interpret this new watershed as being “yet another technological drift”, bringing Logic always closer to practical ingeneering, always further from the human intellectual process of reasoning. This article promotes the dual point of view: enlightened by the contemporary analysis of the dynamic of proofs, which bring us to a new understanding of the semantic counterpart of processes operationality (including the links between semantic dereliction due to inconsistency and computational exuberance), Logic has never appeared so close to being, finally, the theory of reasoning.

Destiny is the most complicated subject in our world and cannot be comprehended without taking a global view in both space and time. Therefore this paper takes a multi disciplinary approach using mathematics, physics, engineering, economics, and philosophy. The paper has two major related objectives (A) Provide a theory for the existence of destiny and (B) Explain away the counter logic. For part A we take the following approaches. (a1) The idea of destiny is well recognized by Newtonian physics. In our universe everything happens because of simultaneous interactions of everything. Thus destiny is not an individual concept. It is not my destiny or your destiny, it is our destiny. (a2) From this global concept of our destiny, we derive how the destiny plays out locally for an individual; how humans are really part of nature, work in cooperation with nature, and according to its dictates. In particular we show that our first reaction to every natural event is planned. (a3) Reincarnation theory of soul is an integral part of destiny theory. We prove giving real life examples, that reincarnation is a law of nature, and we are all reincarnated souls. Thus destiny is not only global in space; it is also global in time, covering all past lives. For part B we cover several important opposing views. (b1) The uncertainty principle of quantum mechanics says nothing can be certain, and therefore destiny cannot be true. We point out that the core idea behind Heisenberg's original mathematical proof is based on Fourier Transform theory. Fourier Transform uses infinity, which as we show is not meaningful in nature. Replacing infinity by any finite number changes the uncertainty to certainty, just like the Newtonian physics. (b2) Destiny is not in the mainstream, therefore it must be wrong. We show that it is the opposite. All Mathematical and physical theories have assumptions that are invalid in nature and therefore they cannot describe nature. These theories are not approximate theories, they are outright wrong. As a natural consequence, the real life engineering which uses these incorrect theories is full of kludges and patches. (b3) We must also understand money, money power, and the way it is controlled by the Central Bank. It is the same money that remote controls every activity and the mindset of every individual in our world. If anyone anywhere is allowed to find the truth on any subject, including destiny, then people will eventually discover the truth behind the central bank, which cannot be allowed. The central bank is a very secret private organization, cannot be controlled by any government, as is well known in economics. In the hierarchy of all activities, however, the soul theory and the laws of nature are at the top, above the central bank.

Mechanisms coupling the chemical reactions of oxidation and ATP synthesis in cellular metabolism by the fundamental biological process of oxidative phosphorylation (OX PHOS) in mitochondria provide > 90% of the energy requirements in living organisms. Mathematical graph theory methods have been extensively used to characterize various metabolic, regulatory, and disease networks in biology. However, networks of energy coupling mechanisms in OX PHOS have not been represented and analyzed previously by these approaches. Here, the problem of biological energy coupling is translated into a graph-theoretical framework, and all possible coupling schemes between oxidation and ATP synthesis are represented as graphs connecting these processes by various intermediates or states. The problem is shown to be transformed into the hard problem of finding a Hamiltonian tour in the networks of possible constituent mechanisms, given the constraints of a cyclical nature of operation of enzymes and biological molecular machines. Accessible mathematical proofs of three theorems that guarantee sufficient conditions for the existence of a Hamiltonian cycle in simple graphs are provided. The results of the general theorems are applied to the set of possible coupling mechanisms in OX PHOS and shown to (1) unequivocally differentiate between the major theories and mechanisms of energy coupling, (2) greatly reduce the possibilities for detailed consideration, and (3) deduce the biologically selected mechanism using additional constraints from the cumulative experimental record. Finally, an algorithm is constructed to implement the graph-theoretical procedure. In summary, the enormous power and generality of mathematical theorems and approaches in graph theory are shown to help solve a fundamental problem in biology.

An account of mathematical understanding should account for the differences between theorems whose proofs are “easy” to discover, and those whose proofs are difficult to discover. Though Hilbert seems to have created proof theory with the idea that it would address this kind of “discovermental complexity”, much more attention has been paid to the lengths of proofs, a measure of the difficulty of verifying of a given formal object that it is a proof of a given formula in a given formal system. In this paper we will shift attention back to discovermental complexity, by addressing a “topological” measure of proof complexity recently highlighted by Alessandra Carbone (2009). Though we will contend that Carbone’s measure fails as a measure of discovermental complexity, it forefronts numerous important formal and epistemological issues that we will discuss, including the structure of proofs and the question of whether impure proofs are systematically simpler than pure proofs.

Set theory is a rich and beautiful subject whose fundamental concepts permeate virtually every branch of mathematics. One could say that set theory is a unifying theory for mathematics, since nearly all mathematical concepts and results can be formalized within set theory. This textbook is meant for an upper undergraduate course in set theory. In this text, the fundamentals of abstract sets, including relations, functions, the natural numbers, order, cardinality, transfinite recursion, the axiom of choice, ordinal numbers, and cardinal numbers, are developed within the framework of axiomatic set theory. The reader will need to be comfortable reading and writing mathematical proofs. The proofs in this textbook are rigorous, clear, and complete, while remaining accessible to undergraduates who are new to upper-level mathematics. Exercises are included at the end of each section in a chapter, with useful suggestions for the more challenging exercises.

This book aims to bridge the gap between practising mathematicians and the practitioners of turbulence theory. It presents the mathematical theory of turbulence to engineers and physicists, and the physical theory of turbulence to mathematicians. The book is the result of many years of research by the authors to analyse turbulence using Sobolev spaces and functional analysis. In this way the authors have recovered parts of the conventional theory of turbulence, deriving rigorously from the Navier–Stokes equations what had been arrived at earlier by phenomenological arguments. The mathematical technicalities are kept to a minimum within the book, enabling the language to be at a level understood by a broad audience. Each chapter is accompanied by appendices giving full details of the mathematical proofs and subtleties. This unique presentation should ensure a volume of interest to mathematicians, engineers and physicists.