is a fragment of first-order aritlimetic so weak that it cannot prove the totality of an iterated exponential fimction. Surprisingly, however, the theory is remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context.

Lighten up about mathematics! Have fun. If you read this book, you will have to endure bad math puns and jokes and out-of-date pop culture references. You'll learn some really cool mathematics to boot. In the process, you will immerse yourself in living, thinking, and breathing logical reasoning. We like to call this proofs, which to some is a bogey word, but to us it is a boogie word. You will learn how to solve problems, real and imagined. After all, (...) math is a game where, although the rules are pretty much set, we are left to our imaginations to create. Think of this book as blueprints, but you are the architect of what structures you want to build. Make sure you lay a good foundation, for otherwise your buildings might fall down. To help you through this, we guide you to think and plan carefully. Our playground consists of basic math, with a loving emphasis on number theory. We will encounter the known and the unknown. Ancient and modern inquirers left us with elementary-sounding mathematical puzzles that are unsolved to this day. You will learn induction, logic, set theory, arithmetic, and algebra, and you may one day solve one of these puzzles. (shrink)

The present article discusses the computational tools (both conceptual and material) used in various attempts to deal with individual cases of FLT, as well as the changing historical contexts in which these tools were developed and used, and affected research. It also explores the changing conceptions about the role of computations within the overall disciplinary picture of number theory, how they influenced research on the theorem, and the kinds of general insights thus achieved. After an overview of Kummer’s (...) contributions and its immediate influence, I present work that favored intensive computations of particular cases of FLT as a legitimate, fruitful, and worth-pursuing number-theoretical endeavor, and that were part of a coherent and active, but essentially low-profile tradition within nineteenth century number theory. This work was related to table making activity that was encouraged by institutions and individuals whose motivations came mainly from applied mathematics, astronomy, and engineering, and seldom from number theory proper. A main section of the article is devoted to the fruitful collaboration between Harry S. Vandiver and Emma and Dick Lehmer. I show how their early work led to the hesitant introduction of electronic computers for research related with FLT. Their joint work became a milestone for computer-assisted activity in number theory at large. (shrink)

1.1 Some examples of rule induction on permutations . . . . . . . 6 1.2 Ways of making new permutations . . . . . . . . . . . . . . . 7 1.3 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Removing elements . . . . . . . . . . (...) . . . . . . . . . . . . . . 8.. (shrink)

We present a new method for expressing Chaitin’s random real, Ω, through Diophantine equations. Where Chaitin’s method causes a particular quantity to express the bits of Ω by fluctuating between finite and infinite values, in our method this quantity is always finite and the bits of Ω are expressed in its fluctuations between odd and even values, allowing for some interesting developments. We then use exponential Diophantine equations to simplify this result and finally show how both methods can also be (...) used to create polynomials which express the bits of Ω in the number of positive values they assume. (shrink)

We give an overview of some remarkable connections between symmetric informationally complete measurements and algebraic number theory, in particular, a connection with Hilbert’s 12th problem. The paper is meant to be intelligible to a physicist who has no prior knowledge of either Galois theory or algebraic number theory.

Belief in the divine origin of the universe began to wane most markedly in the nineteenth century, when scientific accounts of creation by natural law arose to challenge traditional religious doctrines. Most of the credit - or blame - for the victory of naturalism has generally gone to Charles Darwin and the biologists who formulated theories of organic evolution. Darwinism undoubtedly played the major role, but the supporting parts played by naturalistic cosmogonies should also be acknowledged. Chief among these was (...) the nebular hypothesis proposed by Pierre Simon Laplace in 1796, which explained the origin of the solar system as a natural development over extended periods of time. Ronald Numbers focuses on Laplace's theory as it affected American scientific thought. he first traces the history of Laplace's cosmogony chronologically, from its European inception to its demise about 1900. the last three chapters explore some of the theological and scientific consequences resulting from the acceptance of this cosmogony. Most significant was the change in the status of supernatural doctrine. When the nebular hypothesis lost credence at the end of the nineteenth century, those who had before tried to accommodate natural theory with supernatural doctrine no longer felt compelled to do so when faced with succeeding theories. The nebular hypothesis, it seems, had established natural law in the heavens. (shrink)

Real-world information is imperfect and is usually described in natural language (NL). Moreover, this information is often partially reliable and a degree of reliability is also expressed in NL. In view of this, the concept of a Z-number is a more adequate concept for the description of real-world information. The main critical problem that naturally arises in processing Z-numbers-based information is the computation with Z-numbers. Nowadays, there is no arithmetic of Z-numbers suggested in existing literature. This book is the (...) first to present a comprehensive and self-contained theory of Z-arithmetic and its applications. Many of the concepts and techniques described in the book, with carefully worked-out examples, are original and appear in the literature for the first time. The book will be helpful for professionals, academics, managers and graduate students in fuzzy logic, decision sciences, artificial intelligence, mathematical economics, and computational economics. (shrink)

ABSTRACT It is currently fashionable to take Putnamian model-theoretic worries seriously for mathematics, but not for discussions of ordinary physical objects and the sciences. However, I will argue that merely securing determinate reference to physical possibility suffices to rule out the kind of nonstandard interpretations of our number talk Putnam invokes. So, anyone who accepts determinate reference to physical possibility should not reject determinate reference to the natural numbers on Putnamian model-theoretic grounds.

Despite the efforts of many individuals, the disciplines of modern physics and number theory have remained largely divorced, in the sense that the experimentally verified theories of quantum physics and gravity are written in the language of linear algebra and advanced calculus, without reference to several established branches of pure mathematics. This absence raises questions as to whether or not pure mathematics has undiscovered application to physical modeling that could have far reaching implications for human scientific understanding. In (...) this paper, we review physical interpretations of number theoretic concepts developed in the twentieth century, in an attempt to help bridge the divide between pure mathematical truth and testable physical theory. Specifically, the relevance of L-functions and modular forms to the physics of quantum and classical physical systems is addressed, with the objective of motivating general interest among physicists in these mathematical concepts. (shrink)

We prove that the theory IΔ0, extended by a weak version of the Δ0-Pigeonhole Principle, proves that every integer is the sum of four squares (Lagrange's theorem). Since the required weak version is derivable from the theory IΔ0 + ∀x (xlog(x) exists), our results give a positive answer to a question of Macintyre (1986). In the rest of the paper we consider the number-theoretical consequences of a new combinatorial principle, the ‘Δ0-Equipartition Principle’ (Δ0EQ). In particular we give (...) a new proof, which can be formalized in IΔ0 + Δ0EQ, of the fact that every prime of the form 4n + 1 is the sum of two squares. (shrink)

This paper reports recent progress in applying nonstandard analysis to additive number theory, especially to problems involving upper Banach density.

Given classical (2 valued) structures and and a homomorphism h of onto , it is shown how to construct a (non-degenerate) 3-valued counterpart of . Classical sentences that are true in are non-false in . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (...) (the proof for number theory was obtained earlier by R. K. Meyer and suggested the present abstract development). (shrink)

ArgumentThis article gives the background to a public lecture delivered in Hebrew by Edmund Landau at the opening ceremony of the Hebrew University in Jerusalem in 1925. On the surface, the lecture appears to be a slightly awkward attempt by a distinguished German-Jewish mathematician to popularize a few number-theoretical tidbits. However, quite unexpectedly, what emerges here is Landau's personal blend of Zionism, German nationalism, and the proud ethos of pure, rigorous mathematics – against the backdrop of the situation of (...) Germany after World War I. Landau's Jerusalem lecture thus shows how the Zionist cause was inextricably linked to, and determined by political agendas that were taking place in Europe at that time. The lecture stands in various historical contexts - Landau's biography, the history of Jewish scientists in the German Zionist movement, the founding of the Hebrew University in Jerusalem, and the creation of a modern Hebrew mathematical language. This article provides a broad historical introduction to the English translation, with commentary, of the original Hebrew text. (shrink)

Philosophers have explored objective interpretations of probability mainly by considering empirical probability statements. Because of this focus, it is widely believed that the logical interpretation and the actual-frequency interpretation are unsatisfactory and the hypothetical-frequency interpretation is not much better. Probabilistic assertions in pure mathematics present a new challenge. Mathematicians prove theorems in number theory that assign probabilities. The most natural interpretation of these probabilities is that they describe actual frequencies in finite sets and limits of actual frequencies in (...) infinite sets. This interpretation vindicates part of what the logical interpretation of probability aimed to establish. (shrink)

Originally published in 1995, The Selected Works of George McCready Price is the seventh volume in the series, Creationism in Twentieth Century America, reissued in 2019. The volume brings together the original writings and pamphlets of George McCready Price, a leading creationist of the early antievolution crusade of the 1920s. McCready Price labelled himself the 'principal scientific authority of the Fundamentalists' and as a self-taught scientist he enjoyed more scientific repute amongst fundamentalists of the time. This interesting and unique collection (...) of original source material includes five of his writings between 1906 and 1924, challenging the new Darwinian theory of evolution and natural selection through his writings on the natural sciences. His literature covers the topics of evolution and biology and critiques biological arguments for evolution. He also wrote widely on geology offering his own alternative argument of 'flood geography' in opposition to the Darwinian theory concerning palaeontology and geology. This volume will be of interest to historians of natural history and the creationism movement, as well as scholars of religion and American history. (shrink)

As the crusade to outlaw the teaching of evolution changed to a battle for equal time for creationism, the ideological defenses of that doctrine also shifted from primarily biblical to more scientific grounds. This essay describes the historical development of “scientific creationism” from a variety of late–nineteenth– and early–twentieth–century creationist reactions to Charles Darwin's theory of evolution, through the Scopes trial and the 1960s revival of creationism, to the current spread of strict creationism around the world.