Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.

We give a new account of Björling’s contribution to uniform convergence in connection with Cauchy’s theorem on the continuity of an infinite series. Moreover, we give a complete translation from Swedish into English of Björling’s 1846 proof of the theorem. Our intention is also to discuss Björling’s convergence conditions in view of Grattan-Guinness’ distinction between history and heritage. In connection to Björling’s convergence theory we discuss the interpretation of Cauchy’s infinitesimals.

We formulate a restriction of Hindman’s Finite Sums Theorem in which monochromaticity is required only for sums corresponding to rooted finite paths in the full binary tree. We show that the resulting principle is equivalent to \-induction over \. The proof uses the equivalence of this Hindman-type theorem with the Pigeonhole Principle for trees \ with an extra condition on the solution tree.

We formulate a restriction of Hindman’s Finite Sums Theorem in which monochromaticity is required only for sums corresponding to rooted finite paths in the full binary tree. We show that the resulting principle is equivalent to Σ20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma ^0_2$$\end{document}-induction over RCA0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {RCA}_0$$\end{document}. The proof uses the equivalence of this Hindman-type theorem with the Pigeonhole Principle for trees TT1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...) \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {T}\,}{\mathsf {T}}^1$$\end{document} with an extra condition on the solution tree. (shrink)

Two results of elementary number theory, going back to Kürschák and Nagell, stating that the sums $\sum_{i=1}^k \frac{m_i}{n+i}$ (with $k\geq 1$, $(m_i, n+i)=1$, $m_i\lessthan n+i$) and $\sum_{i=0}^k \frac{1}{m+in}$ (with $n, m, k$ positive integers) are never integers, are shown to hold in $\mathrm{PA}^{-}$, a very weak arithmetic, whose axiom system has no induction axiom.

We study the role that the axiom of choice plays in Tychonoff's product theorem restricted to countable families of compact, as well as, Lindelöf metric spaces, and in disjoint topological unions of countably many such spaces.

Mathematical theorems are cultural artifacts and may be interpreted much as works of art, literature, and tool-and-craft are interpreted. The Fundamental Theorem of the Calculus, the Central Limit Theorem of Statistics, and the Statistical Continuum Limit of field theories, all show how the world may be put together through the arithmetic addition of suitably prescribed parts (velocities, variances, and renormalizations and scaled blocks, respectively). In the limit — of smoothness, statistical independence, and large N — higher-order parts, such (...) as accelerations, are, for the most, part irrelevant, affirming that, in the end, most of the world's particulars may be averaged over (a very un-Scriptural point of view). (We work out all of this in technical detail, including a nice geometric picture of stochastic integration, and a method of calculating the variance of the sum of dependent random variables using renormalization group ideas.) These fundamental theorems affirm a culture that is additive, ahistorical, Cartesian, and continuist, sharing in what might be called a species of modern culture. We understand mathematical results as useful because, like many other such artifacts, they have been adapted to fit the world, and the world has been adapted to fit their capacities. Such cultural interpretation is in effect motivation for the mathematics, and might well be offered to students as a way of helping them understand what is going on at the blackboard. Philosophy of mathematics might want to pay more attention to the history and detailed technical features of sophisticated mathematics, as a balance to the usual concerns that arise in formalist or even Platonist positions. (shrink)

Often, when one faces a choice between alternative actions, there are reasons both for and against each alternative. On one way of understanding these words, what one “ought to do all things considered (ATC)” is determined by the totality of these reasons. So, these reasons can somehow be “combined” or “aggregated” to yield an ATC verdict on these alternatives. First, various assumptions about this sort of aggregation of reasons are articulated. Then it is shown that these assumptions allow for the (...) proof of a “Reasons Aggregation Theorem” – parallel to John Harsanyi’s 1955 “Social Aggregation Theorem”. All reasons for action are grounded in reason-providing values; and, for every such reason-providing value, there is in principle a way of measuring how strongly this value counts against each alternative. The theorem tells us that the appropriate measure of how much reason ATC there is against each alternative is simply a weighted sum of the strengths with which these values count against that alternative; the agent ought not ATC to perform an action iff there is ATC more reason against it than against some available alternative. (shrink)

Dialetheism is the view that there exists a true contradiction. This paper ventures to suggest that Priest’s argument for Dialetheism from Gödel’s theorem is unconvincing as the lesson of Gödel’s proof (or Rosser’s proof) is that any sufficiently strong theories of arithmetic cannot be both complete and consistent. In addition, a contradiction is derivable in Priest’s inconsistent and complete arithmetic. An alternative argument for Dialetheism is given by applying Gödel sentence to the inconsistent and complete theory of arithmetic. We (...) argue, however, that the alternative argument raises a circularity problem. In sum, Gödel’s and its related theorem merely show the relation between a complete and a consistent theory. A contradiction derived by the application of Gödel sentence has the value of true sentences, i.e. the both-value, only under the inconsistent models for arithmetic. Without having the assumption of inconsistency or completeness, a true contradiction is not derivable from the application of Gödel sentence. Hence, Gödel’s and its related theorem never can be a ground for Dialetheism. (shrink)

I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of ±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of ±2 it sets out to prove. Once this oversight is ameliorated from Bell's argument (...) by identifying the impediment that leads to it and local realism is implemented correctly, the bounds on the Bell-CHSH sum of expectation values work out to be ±2√2 instead of ±2, thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by any of the Bell-test experiments is not local realism but the linear additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with. (shrink)

We prove the following theorems: (1) If X has strong measure zero and if Y has strong first category, then their algebraic sum has property s 0 . (2) If X has Hurewicz's covering property, then it has strong measure zero if, and only if, its algebraic sum with any first category set is a first category set. (3) If X has strong measure zero and Hurewicz's covering property then its algebraic sum with any set in APC ' is a (...) set in APC '. (APC ' is included in the class of sets always of first category, and includes the class of strong first category sets.) These results extend: Fremlin and Miller's theorem that strong measure zero sets having Hurewicz's property have Rothberger's property, Galvin and Miller's theorem that the algebraic sum of a set with the γ-property and of a first category set is a first category set, and Bartoszynski and Judah's characterization of SR M -sets. They also characterize the property (*) introduced by Gerlits and Nagy in terms of older concepts. (shrink)

1.1 In 1955, John Harsanyi proved a remarkable theorem:1 Suppose n agents satisfy the assumptions of von Neumann/Morgenstern (1947) expected utility theory, and so does the group as a whole (or an observer). Suppose that, if each member of the group prefers option a to b, then so does the group, or the observer (Pareto condition). Then the group’s utility function is a weighted sum of the individual utility functions. Despite Harsanyi’s insistence that what he calls the Utilitarian (...) class='Hi'>Theorem embeds utilitarianism into a theory of rationality, the theorem has fallen short of having the kind of impact on the discussion of utilitarianism for which Harsanyi hoped. Yet how could the theorem influence this discussion? Utilitarianism is as attractive to some as it is appalling to others. The prospects for this dispute to be affected by a theorem seem dim. Yet a closer look shows how the theorem could make a contribution. To fix ideas, I understand by utilitarianism the following claims: (1) Consequentialism: Actions are evaluated in terms of their consequences only. (2) Bayesianism: An agent's beliefs about possible outcomes are captured probabilistically. (3) Welfarism: The judgement of the relative goodness of states of affairs is based.. (shrink)

This chapter explains the interpersonal addition theorem. The theorem leads to two remarkable points. Firstly, it links the aggregation of good across the dimension of people with its aggregation across the dimension of states of nature. The result is that, in favourable circumstances, it links the value of equality in the distribution of good with the value of avoiding risk to good. The chapter also explains this link. The second point is even more remarkable. The theorem shows (...) that general utility is the sum of individual utilities. Since these utilities represent the general and individual betterness relations, this may seem, at first glance, to say that general good is the total of individual good. Joined with the addition theorem Bernoulli's hypothesis implies the utilitarian principle of distribution, which implies in turn that there is no value in avoiding inequality in the distribution of good between people. The chapter discusses the assessment of Bernoulli's hypothesis. (shrink)

A generalized Kochen-Specker theorem is proved. It is shown that there exist sets of n projection operators, representing n yes-no questions about a quantum system, such that none of the 2″ possible answers is compatible with sum rules imposed by quantum mechanics. Namely, if a subset of commuting projection operators sums up to a matrix having only even or only odd eigenvalues, the number of “yes” answers ought to he even or odd, respectively. This requirement may lead to contradictions. (...) An example is provided, involving nine projection operators in a 4-dimensional space. (shrink)

Bernoulli’s 1713 golden theorem is viewed retrospectively in the context of modern model-based frequentist inference that revolves around the concept of a prespecified statistical model Mθx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{M}}_{{{\varvec{\uptheta}}}} \left( {\mathbf{x}} \right)$$\end{document}, defining the inductive premises of inference. It is argued that several widely-accepted claims relating to the golden theorem and frequentist inference are either misleading or erroneous: (a) Bernoulli solved the problem of inference ‘from probability to frequency’, and thus (b) the (...) golden theorem cannot justify an approximate Confidence Interval (CI) for the unknown parameter θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document}, (c) Bernoulli identified the probability PA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P\left( A \right)$$\end{document} with the relative frequency 1n∑k=1nxk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{n}\sum\nolimits_{k = 1}^{n} {x_{k} }$$\end{document} of event A as a result of conflating f(x0|θ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\mathbf{x}}_{0} |\theta )$$\end{document} with f(θ|x0),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\theta |{\mathbf{x}}_{0} ),$$\end{document} where x0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{x}}_{0}$$\end{document} denotes the observed data, and (d) the same ‘swindle’ is currently perpetrated by the p value testers. In interrogating the claims (a)–(d), the paper raises several foundational issues that are particularly relevant for statistical induction as it relates to the current discussions on the replication crises and the trustworthiness of empirical evidence, arguing that: [i] The alleged Bernoulli swindle is grounded in the unwarranted claim θ^nx0≃θ∗,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\theta }_{n} \left( {{\mathbf{x}}_{0} } \right) \simeq \theta^{*},$$\end{document} for a large enough n, where θ^nX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\theta }_{n} \left( {\mathbf{X}} \right)$$\end{document} is an optimal estimator of the true value θ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta^{*}$$\end{document} of θ. [ii] Frequentist error probabilities are not conditional on hypotheses (H0 and H1) framed in terms of an unknown parameter θ since θ is neither a random variable nor an event. [iii] The direct versus inverse inference problem is a contrived and misplaced charge since neither conditional distribution f(x0|θ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\mathbf{x}}_{0} |\theta )$$\end{document} and f(θ|x0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\theta |{\mathbf{x}}_{0} )$$\end{document} exists (formally or logically) in model-based (Mθx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{M}}_{{{\varvec{\uptheta}}}} \left( {\mathbf{x}} \right)$$\end{document}) frequentist inference. (shrink)

Hirst investigated a natural restriction of Hindman’s Finite Sums Theorem—called Hilbert’s Theorem—and proved it equivalent over \ to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman’s Theorem provably much weaker than Hindman’s Theorem itself. We here introduce another natural restriction of Hindman’s Theorem—which we name the Adjacent Hindman’s Theorem with apartness—and prove it to be provable from Ramsey’s Theorem for pairs and strictly stronger (...) than Hirst’s Hilbert’s Theorem. The lower bound is obtained by a direct combinatorial implication from the Adjacent Hindman’s Theorem with apartness to the Increasing Polarized Ramsey’s Theorem for pairs introduced by Dzhafarov and Hirst. In the Adjacent Hindman’s Theorem homogeneity is required only for finite sums of adjacent elements. (shrink)

Among the many sorts of problems encountered in decision theory, allocation problems occupy a central position. Such problems call for the assignment of a nonnegative real number to each member of a finite set of entities, in such a way that the values so assigned sum to some fixed positive real number s. Familiar cases include the problem of specifying a probability mass function on a countable set of possible states of the world, and the distribution of a certain sum (...) of money, or other resource, among various enterprises. In determining an s-allocation it is common to solicit the opinions of more than one individual, which leads immediately to the question of how to aggregate their typically differing allocations into a single “consensual” allocation. Guided by the traditions of social choice theory decision theorists have taken an axiomatic approach to determining acceptable methods of allocation aggregation. In such approaches so-called “independence” conditions have been ubiquitous. Such conditions dictate that the consensual allocation assigned to each entity should depend only on the allocations assigned by individuals to that entity, taking no account of the allocations that they assign to any other entities. While there are reasons beyond mere simplicity for subjecting allocation aggregation to independence, this radically anti-holistic stricture has frequently proved to severely limit the set of acceptable aggregation methods. As we show in what follows, the limitations are particularly acute in the case of three or more entities which must be assigned nonnegative values summing to some fixed positive number s. For if the set V⊆[0,s] of values that may be assigned to these entities satisfies some simple closure conditions and Vis finite, then independence allows only for dictatorial or imposed aggregation. This theorem builds on and extends a theorem of Bradley and Wagner and, when V={0,1}, yields as a corollary an impossibility theorem of Dietrich on judgment aggregation. (shrink)

This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gábor Szego's classic 1915 theorem and its 1920 extension. Barry Simon emphasizes necessary and sufficient conditions, and provides mathematical background that until now has been available only in journals. Topics include background from the theory of meromorphic functions on hyperelliptic surfaces and the study of covering maps of the Riemann sphere with a finite number of slits removed. This allows (...) for the first book-length treatment of orthogonal polynomials for measures supported on a finite number of intervals on the real line. In addition to the Szego and Killip-Simon theorems for orthogonal polynomials on the unit circle and orthogonal polynomials on the real line, Simon covers Toda lattices, the moment problem, and Jacobi operators on the Bethe lattice. Recent work on applications of universality of the CD kernel to obtain detailed asymptotics on the fine structure of the zeros is also included. The book places special emphasis on OPRL, which makes it the essential companion volume to the author's earlier books on OPUC. (shrink)

In 1675, Leibniz elaborated his longest mathematical treatise he everwrote, the treatise ``On the arithmetical quadrature of the circle, theellipse, and the hyperbola. A corollary is a trigonometry withouttables''. It was unpublished until 1993, and represents a comprehensive discussion of infinitesimalgeometry. In this treatise, Leibniz laid the rigorous foundation of thetheory of infinitely small and infinite quantities or, in other words,of the theory of quantified indivisibles. In modern terms Leibnizintroduced `Riemannian sums' in order to demonstrate the integrabilityof continuous functions. The (...) article deals with this demonstration,with Leibniz's handling of infinitely small and infinite quantities,and with a general theorem regarding hyperboloids. (shrink)

Pierre de Fermat is known as the inventor of modern number theory. He invented–improved many methods useful in this discipline. Fermat often claimed to have proved his most difficult theorems thanks to a method of his own invention: the infinite descent. He wrote of numerous applications of this procedure. Unfortunately, he left only one almost complete demonstration and an outline of another demonstration. The outline concerns the theorem that every prime number of the form 4n + 1 is the (...) sum of two squares. In this paper, we analyse a recent proof of this theorem. It is interesting because: it follows all the elements of which Fermat wrote in his outline; it represents a good introduction to all logical nuances and mathematical variants concerning this method of which Fermat spoke. The assertions by Fermat will also be framed inside their historical context. Therefore, the aims of this paper are related to the history of mathematics and to the logic of proof-methods. (shrink)

In this note, we show that a partition of a cake is Pareto optimal if and only if it maximizes some convex combination of the measures used by those who receive the resulting pieces of cake. Also, given any sequence of positive real numbers that sum to one (which may be thought of as representing the players' relative entitlements), we show that there exists a partition in which each player receives either more than, less than, or exactly his or her (...) entitlement (according to his or her measure), in any desired combination, provided that the measures are not all equal. (shrink)

This paper is a commentary on the focal article by Grafen and on earlier papers of his on which many of the results of this focal paper depend. Thus it is in effect a commentary on the “formal Darwinian project”, the focus of this sequence of papers. Several problems with this sequence are raised and discussed. The first of these concerns fitness maximization. It is often claimed in these papers that natural selection leads to a maximization of fitness and that (...) this view is claimed in Fisher’s “fundamental theorem of natural selection”. These claims are refuted, and various incorrect statements about the meaning and interpretation of the fundamental theorem of natural selection, in this sequence and in other papers by other authors, are discussed. Next, much of the work in this sequence rests on the first Price equation. In the deterministic (infinite population) case this equation is no more than the standard classical equation relating to changes in gene frequencies. In the stochastic case the equation gives the change in gene frequencies as the sum of two terms (the second of which vanishes in the deterministic case). These two terms are of essentially equal importance in the situation considered in the focal article, yet one of Grafen’s results ignores the second term in the stochastic analysis. This is associated with a wavering between deterministic and stochastic analyses and the use of the Price fitness concept and the classical fitness concept. These comments cast doubts on Grafen’s optimization theory. (shrink)

Neutrices are convex additive subgroups of the nonstandard space , most of them are external sets. Because of the convexity and the invariance under some translations and multiplications, external neutrices are models for orders of magnitude. One dimensional neutrices have been applied to asymptotics, singular perturbations, and statistics. This paper shows that in , with standard k, every neutrix is the direct sum of k neutrices of . These components may be chosen to be orthogonal.

In this paper we study with proof-theoretic methods the function(al) s provably recursive relative to Ramsey's theorem for pairs and the cohesive principle (COH). Our main result on COH is that the type 2 functional provably recursive from $RCA_0 + COH + \Pi _1^0 - CP$ are primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact that $WKL_0 + \Pi _1^0 (...) - CP + COH$ is $\Pi _1^0 $ -conservative over PRA. Recent work of the first author showed that $\Pi _1^0 + CP + COH$ is equivalent to a weak variant of the Bolzano-Weierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs. For Ramsey's theorem for pairs and two colors $\left( {RT_2^2 } \right)$ we obtain the upper bounded that the type 2 functionals provable recursive relative to $RCA_0 + \sum\nolimits_2^0 {IA + RT_2^2 } $ are in T₁. This is the fragment of Gödel's system T containing only type 1 recursion—roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction of T₁-bounds from proofs that use RT2..... Moreover, this yields a new proof of the fact that $WKL_0 + \sum\nolimits_2^0 {IA} + RT_2^2 $ is $\Pi _1^0 $ -conservative over $RCA_0 + \sum\nolimits_2^0 { - IA} $ . The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel's functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using $\Pi _1^0 $ -comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard's ordinal analysis of bar recursion (for $RT_2^2 $. (shrink)

Let A be an admissible set. A sentence of the form ∀R̄φ is a ∀1(A) (∀s 1(A),∀1(Lω1ω)) sentence if φ ∈ A (φ is $\bigvee\Phi$ , where Φ is an A-r.e. set of sentences from A; φ ∈ Lω1ω). A sentence of the form ∃R̄φ is an ∃2(A) (∃s 2(A),∃2(Lω1ω)) sentence if φ is a ∀1(A) (∀s 1(A),∀1(Lω1ω)) sentence. A class of structures is, for example, a ∀1(A) class if it is the class of models of a ∀1(A) sentence. Thus (...) ∀1(A) is a class of classes of structures, and so forth. Let Mi be the structure $\langle i, 0$. Let Γ be a class of classes of structures. We say that a sequence $J_1,\ldots,J_i,\ldots, i < \omega$, of classes of structures is a Γ sequence if $J_i \in \Gamma, i < \omega$, and there is I ∈ Γ such that M ∈ Ji if and only if [M, Mi] ∈ I, where [,] is the disjoint sum. A class Γ of classes of structures has the easy uniformization property if for every Γ sequence $J_1,\ldots,J_i, \ldots, i < \omega$, there is a Γ sequence $J'_1,\ldots,J'_i,\ldots, i < \omega$, such that $J'_i \subseteq J_i, i < \omega, \bigcup J'_i = \bigcup J_i$, and the J'i are pairwise disjoint. The easy uniformization property is an effective version of Kuratowski's generalized reduction property that is closely related to Moschovakis's (topological) easy uniformization property. We show over countable structures that ∀1(A) and ∃2(A) have the easy uniformization property if A is a countable admissible set with an infinite member, that ∀s 1(Lα) and ∃s 2(Lα) have the easy uniformization property if α is countable, admissible, and not weakly stable, and that ∀1(Lω1ω) and ∃2(Lω1ω) have the easy uniformization property. The results proved are more general. The result for ∀s 1(Lα) answers a question of Vaught (1980). (shrink)

I begin, in section 1, with a presentation of the Interpersonal Addition Theorem. The theorem, due to John Broome (1991), is a re-formulation of the classical result by Harsanyi (1955). It implies that, given some seemingly mild assumptions, the overall utility of an uncertain prospect can be seen as the sum of its individual utilities. In sections 1 and 2, I discuss the theorem's connection with utilitarianism and in particular consider its implications for the Priority View, according (...) to which benefits to the worse off count for more, in terms of overall utility, than comparable benefits to the better off (cf. Parfit 1995 [1991]). Broome (1991) and Klint Jensen (1996) have argued that, in view of the Interpersonal Addition Theorem, the Priority View should be rejected for measurement-theoretical reasons. Therefore, it cannot be seen as a plausible competitor to utilitarianism (cf. section 1). I will suggest, however, that a proponent of the Priority View would be well-advised, on independent grounds, to reject one of the basic assumptions on which the Addition Theorem is based. I have in mind the so-called Principle of Personal Good for uncertain prospects (cf. sections 4 and 5). If the theorem is disarmed in this way, then, as a side benefit, the Priority View will avoid the afore-mentioned problems with measurement. (shrink)

This paper takes its departure from the Interpersonal Addition Theorem. The theorem, by John Broome, is a re-formulation of the classical result by Harsanyi. It implies that, given some seemingly mild assumptions, the overall utility of an uncertain prospect can be seen as the sum of its individual utilities. In sections 1 and 2, I discuss the theorem’s connection with utilitarianism and in particular the extent to which this theorem still leaves room for the Priority View. (...) According to the latter, the utilitarian approach needs to be modified: Benefits to the worse off should count for more, overall, than the comparable benefits to the better off.Broome and Jensen have argued that the Priority View cannot be seen as a plausible competitor to utilitarianism: Given the addition theorem, prioritarianism should be rejected for measurement-theoretical reasons. I suggest that this difficulty is spurious: The proponents of the Priority View would be well advised, on independent grounds, to reject one of the basic assumptions on which the addition theorem is based, the so-called Principle of Personal Good for uncertain prospects.According to the Principle of Personal Good, one prospect is better than another if it is better for everyone or at least better for some and worse for none. That the Priority View, as I read it, rejects this welfarist intuition may be surprising. Isn’t welfarism a common ground for prioritarians and utilitarians? Still, as I argue, this welfarist common ground is better captured by a restricted Principle of Personal Good that is valid for *outcomes*, but not necessarily for uncertain prospects. We obtain this surprising result if we take the priority weights imposed by prioritarians to be relevant only to *moral*, but not to *prudential*, evaluations of prospects. This makes it possible for a prospect to be morally better even though it is worse for everyone concerned. The proposed interpretation of the Priority View thus drives a sharp wedge between prudence and morality. (shrink)

We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several examples of colorings of the integers which do not have computable witnesses to Hindman's Theorem.

We say that a linear ordering is extendible if every partial ordering that does not embed can be extended to a linear ordering which does not embed either. Jullien’s theorem is a complete classification of the countable extendible linear orderings. Fraïssé’s conjecture, which is actually a theorem, is the statement that says that the class of countable linear ordering, quasiordered by the relation of embeddability, contains no infinite descending chain and no infinite antichain. In this paper we study (...) the strength of these two theorems from the viewpoint of Reverse Mathematics and Effective Mathematics. As a result of our analysis we get that they are equivalent over the basic system of .We also prove that Fraïssé’s conjecture is equivalent, over , to two other interesting statements. One that says that the class of well founded labeled trees, with labels from {+,−}, and with a very natural order relation, is well quasiordered. The other statement says that every linear ordering which does not contain a copy of the rationals is equimorphic to a finite sum of indecomposable linear orderings.While studying the proof theoretic strength of Jullien’s theorem, we prove the extendibility of many linear orderings, including ω2 and η, using just . Moreover, for all these linear orderings, , we prove that any partial ordering, , which does not embed has a linearization, hyperarithmetic in , which does not embed. (shrink)

We present finitary formulations of two well known results concerning infinite series, namely Abel's theorem, which establishes that if a series converges to some limit then its Abel sum converges to the same limit, and Tauber's theorem, which presents a simple condition under which the converse holds. Our approach is inspired by proof theory, and in particular Gödel's functional interpretation, which we use to establish quantitative versions of both of these results.

We give a polynomial time controlled version of a theorem of M. Hall: every real number can be written as the sum of two irrational numbers whose developments into a continued fraction contain only 1, 2, 3 or 4.

We examine the reverse mathematical strength of a variation of Hindman’s Theorem (HT) constructed by essentially combining HT with the Thin Set Theorem to obtain a principle that we call thin-HT. This principle states that every coloring c:N→N has an infinite set S⊆N whose finite sums are thin for c, meaning that there is an i with c(s)≠i for all nonempty sums s of finitely many distinct elements of S. We show that there is a computable instance of (...) thin-HT such that every solution computes ∅′, as is the case with HT, as shown by Blass, Hirst, and Simpson (1987). In analyzing this proof, we deduce that thin-HT implies ACA0 over RCA0+IΣ20. On the other hand, using Rumyantsev and Shen’s computable version of the Lovász Local Lemma, we show that there is a computable instance of the restriction of thin-HT to sums of exactly 2 elements such that any solution has diagonally noncomputable degree relative to ∅′. Hence there is a computable instance of this restriction of thin-HT with no Σ20 solution. (shrink)

The tridecompositional uniqueness theorem of Elby and Bub (1994) shows that a wavefunction in a triple tensor product Hilbert space has at most one decomposition into a sum of product wavefunctions with each set of component wavefunctions linearly independent. I demonstrate that, in many circumstances, the unique component wavefunctions and the coefficients in the expansion are both hopelessly unstable, both under small changes in global wavefunction and under small changes in global tensor product structure. In my opinion, this means (...) that the theorem cannot underlie lawlike solutions to the problems of the interpretation of quantum theory. I also provide examples of circumstances in which there are open sets of wavefunctions containing no states with various decompositions. (shrink)

The notions of permutable and weak-permutable convergence of a series|$\sum _{n=1}^{\infty }a_{n}$|of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann’s two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara’s principle BD-|$\mathbb {N}$|implies that every permutably convergent series is absolutely convergent. Since there are models of constructive mathematics in which the Riemann permutation theorem (...) for series holds but BD-|$\mathbb{N}$|does not, the best we can hope for as a partial converse to our first theorem is that the absolute convergence of series with a permutability property classically equivalent to that of Riemann implies BD-|$\mathbb {N}$|. We show that this is the case when the property is weak-permutable convergence. (shrink)

We describe an extension to our quantifier-free computational logic to provide the expressive power and convenience of bounded quantifiers and partial functions. By quantifier we mean a formal construct which introduces a bound or indicial variable whose scope is some subexpression of the quantifier expression. A familiar quantifier is the Σ operator which sums the values of an expression over some range of values on the bound variable. Our method is to represent expressions of the logic as objects in the (...) logic, to define an interpreter for such expressions as a function in the logic, and then define quantifiers as "mapping functions." The novelty of our approach lies in the formalization of the interpreter and its interaction with the underlying logic. Our method has several advantages over other formal systems that provide quantifiers and partial functions in a logical setting. The most important advantage is that proofs not involving quantification or partial recursive functions are not complicated by such notions as "capturing," "bottom," or "continuity." Naturally enough, our formalization of the partial functions is nonconstructive. The theorem prover for the logic has been modified to support these new features. We describe the modifications. The system has proved many theorems that could not previously be stated in our logic. Among them are. (shrink)

In this article, we give a summary of Leonhard Euler’s work on the pentagonal number theorem. First we discuss related work of earlier authors and Euler himself. We then review Euler’s correspondence, papers and notebook entries about the pentagonal number theorem and its applications to divisor sums and integer partitions. In particular, we work out the details of an unpublished proof of the pentagonal number theorem from Euler’s notebooks. As we follow Euler’s discovery and proofs of the (...) pentagonal number theorem, we pay attention to Euler’s ideas about when we can consider a mathematical statement to be true. Finally, we discuss related results in the theory of analytic functions. (shrink)

We prove the following theorems:1. IfX⊆ 2ωis aγ-set andY⊆2ωis a strongly meager set, thenX+Yis Ramsey null.2. IfX⊆2ωis aγ-set andYbelongs to the class ofsets, then the algebraic sumX+Yis anset as well.3. Under CH there exists a setX∈MGR* which is not Ramsey null.

A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right-angled triangle as the “Einstein sum” of the squares of the hyperbolic lengths of the other two sides, Fig. 1, thus completing the long path from Pythagoras to Einstein. Following the pioneering work of Varičak it is well known that relativistic (...) velocities are governed by hyperbolic geometry in the same way that prerelativistic velocities are governed by Euclidean geometry. Unlike prerelativistic velocity composition, given by the ordinary vector addition, the composition of relativistic velocities, given by the Einstein addition, is neither commutative nor associative due to the presence of Thomas precession. Following the discovery of the mathematical regularity that Thomas precession stores, it is now possible to extend Thomas precession by abstraction, (i) allowing hyperbolic geometry to be studied by means of analogies that it shares with Euclidean geometry; and, similarly (ii) allowing velocities and accelerations in relativistic mechanics to be studied by means of analogies that they share with velocities and accelerations in classical mechanics. The abstract Thomas precession, called the Thomas gyration, gives rise to gyrovector space theory in which the prefix gyro is used extensively in terms like gyrogroups and gyrovector spaces, gyroassociative and gyrocommutative laws, gyroautomorphisms, gyrotranslations, etc. We demonstrate the superiority of our gyrovector space formalism in capturing analogies by deriving the Hyperbolic Pythagorean Theorem in a form fully analogous to its Euclidean counterpart, thus contrasting it with the standard form in which the Hyperbolic Pythagorean Theorem is known in the literature. The hyperbolic metric, which supports the Hyperbolic Pythagorean Theorem, has a dual metric. We show that the dual metric does not support a Pythagorean theorem but, instead, it supports the π-Theorem according to which the sum of the three dual angles of a hyperbolic triangle is π. (shrink)

The home learning environment includes what parents do to stimulate children’s literacy and numeracy skills at home and their overall beliefs and attitudes about children’s learning. The home literacy and numeracy environments are two of the most widely discussed aspects of the home learning environment, and past studies have identified how socioeconomic status and parents’ own abilities and interest in these domains also play a part in shaping children’s learning experiences. However, these studies are mostly from the West, and there (...) has been little focus on the situation of homes in Asia, which captures a large geographical area and a wide diversity of social, ethnic, and linguistic groups. Therefore, this paper aims to review extant studies on the home literacy and numeracy environments that have been conducted in different parts of Asia, such as China, the Philippines, India, Iran, Turkey, and the United Arab Emirates. Specifically, we explore how parents in these places perceive their roles in children’s early literacy and numeracy development, the methods they regard as effective for promoting young children’s literacy and numeracy learning, and the frequency with which they engage their young children in different types of home literacy and numeracy activities. We also examine studies on the relationship of the home literacy and numeracy environment with young children’s developmental outcomes, and the effectiveness of parent training programs to improve the home literacy and numeracy environments in these contexts. By examining potential trends in findings obtained in different geographical areas, we can initially determine whether there are characteristics that are potentially unique to contexts in Asia. We propose future research directions that acknowledge the role of cultural values and social factors in shaping the home learning environment, and, by extension, in facilitating children’s early literacy and numeracy development. (shrink)

ABSTRACTWe examine Marx's critiques of language, politics, and capitalist political economy and show how these anticipated critical discourse and argumentation analysis and ‘cultural political economy’. Marx studied philology and rhetoric at university and applied their lessons critically. We illustrate this from three texts. The German Ideology critically explores language as practical consciousness, the division of manual and mental labor, the state, hegemony, intellectuals, and specific ideologies. The Eighteenth Brumaire studies the semantics and pragmatics of political language and how it represents (...) the class content of politics and contributes to social transformation. Capital deconstructs the categories of classical political economy and their constitutive role in capitalist social relations. This is one aspect of CPE. Capital also highlights the structural and agential aspects of these relations, their contradictory dynamic, and their crisis-prone character. We comment on this aspect too. This said, Marx held that social transformation is mediated through political imaginaries and highlighted the need for the proletariat to develop a ‘poetry’ of the future. We then consider the misleading ‘base-superstructure’ metaphor and note how, against the thrust of Marx's work, it tends to reify culture. The article concludes that Marx contributed to the critique of semiotic as well as political economy. (shrink)