We present the interval-valued intuitionistic fuzzy ordered weighted cosine similarity measure in this paper, which combines the interval-valued intuitionistic fuzzy cosine similarity measure with the generalized ordered weighted averaging operator. The main advantage of the IVIFOWCS measure provides a parameterized family of similarity measures, and the decision maker can use the IVIFOWCS measure to consider a lot of possibilities and select the aggregation operator in accordance with his interests. We have studied some of its main properties and particular (...) cases such as the interval-valued intuitionistic fuzzy ordered weighted arithmetic cosine similarity measure and the interval-valued intuitionistic fuzzy maximum cosine similarity measure. The IVIFOWCS measure not only is a generalization of some similarity measure, but also it can deal with the correlation of different decision matrices for interval-valued intuitionistic fuzzy values. Furthermore, we present an application of IVIFOWCS measure to the group decision-making problem. Finally the existing similarity measures are compared with the IVIFOWCS measure by an illustrative example. (shrink)

The Heisenberg-Gabor uncertainty principle defines the limits of information resolution in both time and frequency domains. The limit of resolution discloses unique properties of a time series by frequency decomposition. However, classical methods such as Fourier analysis are limited by spectral leakage, particularly in longitudinal data with shifting periodicity or unequal intervals. Wavelet transformation provides a workable compromise by decomposing the signal in both time and frequency through translation and scaling of a basis function followed by correlation or convolution with (...) the original signal. This study aimed to compare the accuracy of predictive models in mental arithmetic in time and frequency domains. Analysis of the author's response time at mental arithmetic using a soroban was modeled for two periods, an initial period, and a return period both separated by an interval of 370 days. The median response times in seconds was longer for all tasks during the TI compared to the TR period, for addition [CTAdd 62 vs 50 s] and summation [CTSum 68 vs 57 s]. Response times were longer for errors regardless of the study period or task. There was an increasing phase difference for the addition and summation tasks during the TI period toward the end of the series 49.65o compared to the TR period where the phase difference between the two tasks was only 2.05o, indicating that both tasks are likely demonstrating similar learning rates during the latter study period. A comparison between time and time/frequency domain forecasts for an additional 100 tasks demonstrated higher accuracy of the maximum overlap discrete wavelet transform model, where the mean absolute percentage error ranged between 5.48 and 8.19% and that for the time domain models [autoregressive integrated moving average, generalized autoregressive conditional heteroscedasticity ] was 6.16–10.80%. (shrink)

The objective of this study is to propose a new operation method based on the universal grey number to overcome the shortcomings of typical interval operation in solving system fault trees. First, the failure probability ranges of the bottom events are described according to the conversion rules between the interval number and universal grey number. A more accurate system reliability calculation is then obtained based on the logical relationship between the AND gates and OR gates of a fault (...) tree and universal grey number arithmetic. Then, considering an aircraft landing gear retraction system as an example, the failure probability range of the top event is obtained through universal grey operation. Next, the reliability of the aircraft landing gear retraction system is evaluated despite insufficient statistical information describing failures. The example demonstrates that the proposed method provides many advantages in resolving the system reliability problem despite poor information, yielding benefits for the function of the interval operation, and overcoming the drawback of solution interval enlargement under different orders of interval operation. (shrink)

We answer some problems set by Priest in [11] and [12], in particular refuting Priest's Conjecture that all LP-models of Th(N) essentially arise via congruence relations on classical models of Th(N). We also show that the analogue of Priest's Conjecture for I δ₀ + Exp implies the existence of truth definitions for intervals [0,a] ⊂ₑ M ⊨ I δ₀ + Exp in any cut [0,a] ⊂e K ⊆ M closed under successor and multiplication.

We present some general results concerning the topological space of cuts of a countable model of arithmetic given by a particular indicator Y.The notion of “indicator” is de.ned in a novel way, without initially specifying what property is indicated and is used to de.ne a topological space of cuts of the model. Various familiar properties of cuts are investigated in this sense, and several results are given stating whether or not the set of cuts having the property is comeagre.A (...) new notion of “generic cut” is introduced and investigated and it is shown in the case of countable arithmetically saturated models M ⊧ PA that generic cuts exist, indeed the set of generic cuts is comeagre in the sense of Baire, and furthermore that two generic cuts within the same “small interval” of the model are conjugate by an automorphism of the model.The paper concludes by outlining some applications to constructions of cuts satisfying properties incompatible with genericity, and discussing in model-theoretic terms those properties for which there is an indicator Y. (shrink)

We study the formalization within sybsystems of second-order arithmetic of theorems concerning periodic points in dynamical systems on the real line. We show that Sharkovsky's theorem is provable in WKL0. We show that, with an additional assumption, Sharkovsky's theorem is provable in RCA0. We show that the existence for all n of n-fold iterates of continuous mappings of the closed unit interval into itself is equivalent to the disjunction of Σ02 induction and weak König's lemma.

The authors present the main ideas of the computer-assisted proof of Mischaikow and Mrozek that chaos is really present in the Lorenz equations. Methodological consequences of this proof are examined. It is shown that numerical calculations can constitute an essential part of mathematical proof not only in the discrete mathematics but also in the mathematics of continua.

Gödel logics with truth sets being countable closed subsets of the unit real interval containing 0 and 1 are studied under their usual semantics and under the witnessed semantics, the latter admitting only models in which the truth value of each universally quantified formula is the minimum of truth values of its instances and dually for existential quantification and maximum. An infinite system of such truth sets is constructed such that under the usual semantics the corresponding logics have pairwise (...) different sets of tautologies, all these sets being non-arithmetical, whereas under the witnessed semantics all the logics have the same set of tautologies and it is Π2-complete. Further similar results are obtained. (shrink)

While every endpointed interval I in a linear order J is, considered as a linear order in its own right, trivially Muchnik‐reducible to J itself, this fails for Medvedev‐reductions. We construct an extreme example of this: a linear order in which no endpointed interval is Medvedev‐reducible to any other, even allowing parameters, except when the two intervals have finite difference. We also construct a scattered linear order which has many endpointed intervals Medvedev‐incomparable to itself; the only other known (...) construction of such a linear order yields an ordinal of extremely high complexity, whereas this construction produces a low‐level‐arithmetic example. Additionally, the constructions here are “coarse” in the sense that they lift to other uniform reducibility notions in place of Medvedev reducibility itself. (shrink)

The word ‘equality’ often requires disambiguation, which is provided by context or by an explicit modifier. For each sort of magnitude, there is at least one sense of ‘equals’ with its correlated senses of ‘is greater than’ and ‘is less than’. Given any two magnitudes of the same sort—two line segments, two plane figures, two solids, two time intervals, two temperature intervals, two amounts of money in a single currency, and the like—the one equals the other or the one is (...) greater than the other or the one is greater than the other [sc. in appropriate correlated senses of ‘equals’, ‘is greater than’ and ‘is less than’]. In case there are two or more appropriate senses of ‘equals’, the one intended is often indicated by an adverb. For example, one plane figure may be said to be equal in area to another and, in certain cases, one plane figure may be said to be equal in length to another. Each sense of ‘equality’ is tied to a specific domain and is therefore non-logical. Notice that in every cases ‘equality’ is definable in terms of ‘is greater than’ and also in terms of ‘is less than’ both of which are routinely considered domain specific, non-logical. The word ‘identity’ in the logical sense does not require disambiguation. Moreover, it is not correlated ‘is greater than’ and ‘is less than’. If it is not the case that a certain designated triangle is [sc. is identical to] an otherwise designated triangle, it is not necessary for the one to be greater than or less than the other. Moreover, if two magnitudes are equal then a unit of measure can be chosen and, no matter what unit is chosen, each magnitude is the same multiple of the unit that the other is. But identity does not require units. In this regard, congruence is like identity and unlike equality. In arithmetic, the logical concept of identity is coextensive with the arithmetic concept of equality. The logical concept of identity admits of an analytically adequate definition in terms of logical concepts: given any number x and any number y, x is y iff x has every property that y has. The arithmetical concept of equality admits of an analytically adequate definition in terms of arithmetical concepts: given any number x and any number y, x equals y iff x is neither less than nor greater than y. As Aristotle told us and as Frege retold us, just because one relation is coextensive with another is no reason to conclude that they are one. (shrink)

We study connections between classical asymptotic density, computability and computable enumerability. In an earlier paper, the second two authors proved that there is a computably enumerable set A of density 1 with no computable subset of density 1. In the current paper, we extend this result in three different ways: The degrees of such sets A are precisely the nonlow c.e. degrees. There is a c.e. set A of density 1 with no computable subset of nonzero density. There is a (...) c.e. set A of density 1 such that every subset of A of density 1 is of high degree. We also study the extent to which c.e. sets A can be approximated by their computable subsets B in the sense that A\B has small density. There is a very close connection between the computational complexity of a set and the arithmetical complexity of its density and we characterize the lower densities, upper densities and densities of both computable and computably enumerable sets. We also study the notion of "computable at density r" where r is a real in the unit interval. Finally, we study connections between density and classical smallness notions such as immunity, hyperimmunity, and cohesiveness. (shrink)

We study effectively inseparable prelattices $\wedge, \vee$ are binary computable operations; ${ \le _L}$ is a computably enumerable preordering relation, with $0{ \le _L}x{ \le _L}1$ for every x; the equivalence relation ${ \equiv _L}$ originated by ${ \le _L}$ is a congruence on L such that the corresponding quotient structure is a nontrivial bounded lattice; the ${ \equiv _L}$ -equivalence classes of 0 and 1 form an effectively inseparable pair of sets). Solving a problem in we show, that if (...) L is an e.i. prelattice then ${ \le _L}$ is universal with respect to all c.e. preordering relations, i.e., for every c.e. preordering relation R there exists a computable function f reducing R to ${ \le _L}$, i.e., $xRy$ if and only if $f\left{ \le _L}f\left$, for all $x,y$. In fact ${ \le _L}$ is locally universal, i.e., for every pair $a{ < _L}b$ and every c.e. preordering relation R one can find a reducing function f from R to ${ \le _L}$ such that the range of f is contained in the interval $\left\{ {x:a{ \le _L}x{ \le _L}b} \right\}$. Also ${ \le _L}$ is uniformly dense, i.e., there exists a computable function f such that for every $a,b$ if $a{ < _L}b$ then $a{ < _L}f\left{ < _L}b$, and if $a{ \equiv _L}a\prime$ and $b{ \equiv _L}b\prime$ then $f\left{ \equiv _L}f\left$. Some consequences and applications of these results are discussed: in particular for $n \ge 1$ the c.e. preordering relation on ${{\rm{\Sigma }}_n}$ sentences yielded by the relation of provable implication of any c.e. consistent extension of Robinson’s system R or Q is locally universal and uniformly dense; and the c.e. preordering relation yielded by provable implication of any c.e. consistent extension of Heyting Arithmetic is locally universal and uniformly dense. (shrink)

Cantor’s first set theory paper (1874) establishes the uncountability of ${\mathbb R}$. We study this most basic mathematical fact formulated in the language of higher-order arithmetic. In particular, we investigate the logical and computational properties of ${\mathsf {NIN}}$ (resp. ${\mathsf {NBI}}$ ), i.e., the third-order statement there is no injection resp. bijection from $[0,1]$ to ${\mathbb N}$. Working in Kohlenbach’s higher-order Reverse Mathematics, we show that ${\mathsf {NIN}}$ and ${\mathsf {NBI}}$ are hard to prove in terms of (conventional) comprehension (...) axioms, while many basic theorems, like Arzelà’s convergence theorem for the Riemann integral (1885), are shown to imply ${\mathsf {NIN}}$ and/or ${\mathsf {NBI}}$. Working in Kleene’s higher-order computability theory based on S1–S9, we show that the following fourth-order process based on ${\mathsf {NIN}}$ is similarly hard to compute: for a given $[0,1]\rightarrow {\mathbb N}$ -function, find reals in the unit interval that map to the same natural number. (shrink)

Kruskal proved that finite trees are well-quasi-ordered by hom(e)omorphic embeddability. Friedman observed that this statement is not provable in predicative analysis. Friedman also proposed (see in [Simpson]) some stronger variants of the Kruskal theorem dealing with finite labeled trees under home(e)omorphic embeddability with a certain gap-condition, where labels are arbitrary finite ordinals from a fixed initial segment of ω. The corresponding limit statement, expressing that for all initial segments of ω these labeled trees are well-quasi-ordered, is provable in Π 1 (...) 1 -CA, but not in the analogous theory Π 1 1 -CA 0 with induction restricted to sets. Schutte and Simpson proved that the one-dimensional case of Friedman's limit statement dealing with finite labeled intervals is not provable in Peano arithmetic. However, Friedman's gap-condition fails for finite trees labeled with transfinite ordinals. In [Gordeev 1] I proposed another gap-condition and proved the resulting one-dimensional modified statements for all (countable) transfinite ordinal-labels. The corresponding universal modified one-dimensional statement UM 1 is provable in (in fact, is equivalent to) the familiar theory ATR 0 whose proof-theoretic ordinal is Γ 0 . In [Gordeev 1] I also announced that, in the general case of arbitrarily-branching finite trees labeled with transfinite ordinals, in the proof-theoretic sense the hierarchy of the limit modified statements $\mathbf{M}_{ (which are denoted by LM λ in the present note) is as strong as the hierarchy of the familiar theories of iterated inductive definitions (more precisely, see [Gordeev 1, Concluding Remark 3]). In this note I present a "positive" proof of the full universal modified statement UM, together with a short proof of the crucial "reverse" results which is based on Okada's interpretation of the well-established ordinal notations of Buchholz corresponding to the theories of iterated inductive definitions. Formally the results are summarized in $\S5$ below. (shrink)

Between ca. 400 and 50 BCE, Babylonian astronomers used mathematical methods for predicting ecliptical positions, times and other phenomena of the moon and the planets. Until recently these methods were thought to be of a purely arithmetic nature. A new interpretation of four Babylonian astronomical procedure texts with geometric computations has challenged this view. On these tablets, Jupiter’s total distance travelled along the ecliptic during a certain interval of time is computed from the area of a trapezoidal figure (...) representing the planet’s changing daily displacement along the ecliptic. Moreover, the time when Jupiter reaches half the total distance is computed by bisecting the trapezoid into two smaller ones of equal area. In the present paper these procedures are traced back to precursors from Old Babylonian mathematics. Some implications of the use of geometric methods by Babylonian astronomers are also explored. (shrink)

This article attempts to broaden the phenomenologically motivated perspective of H. Weyl's Das Kontinuum in the hope of elucidating the differences between the intuitive and mathematical continuum and further providing a deeper phenomenological interpretation. It is known that Weyl sought to develop an arithmetically based theory of continuum with the reasoning that one should be based on the naturally accessible domain of natural numbers and on the classical first-order predicate calculus to found a theory of mathematical continuum free of impredicative (...) circularities only to stumble, to cite a key question, in the evident lack of intuitive support for the notion of points of the continuum. In this motivation, I set out to deal from a Husserlian viewpoint with the general notion of points as appearances reducible to individuals of pre-predicative experience in contrast with the notion of an interval of real numbers taken as an abstraction based on the intuition of time-flowing experience. I argue that the notions of points and of real intervals in the above sense are not by essence related to objective temporality and thus their incompatibility in mathematical terms is ultimately due to deeper constituting reasons independently of any causal and spatio-temporal constraints. (shrink)

Reverse mathematics is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, that is, non-set-theoretic, mathematics. As suggested by the title, this paper deals with the study of the topological notions of dimension and paracompactness, inside Kohlenbach’s higher-order RM. As to splittings, there are some examples in RM of theorems A, B, C such that A (...) ↔, that is, A can be split into two independent parts B and C, and the aforementioned topological notions give rise to a number of splittings involving highly natural A, B, C. Nonetheless, the higher-order picture is markedly different from the second-one: in terms of comprehension axioms, the proof in higher-order RM of, for example, the paracompactness of the unit interval requires full second-order arithmetic, while the second-order/countable version of paracompactness of the unit interval is provable in the base theory RCA 0. We obtain similarly “exceptional” results for the Urysohn identity, the Lindelöf lemma, and partitions of unity. We show that our results exhibit a certain robustness, in that they do not depend on the exact definition of cover, even in the absence of the axiom of choice. (shrink)

The origins of mathematics, a close connection and interpenetration of its parts, and uniform procedures of dealing with the mathematical matter - all of them speak in favour of the integrality of mathematics. It seems that a strong argument for such a view is a fundamental object of contemporary mathematics; namely a real line, which contains real numbers (so arithmetics as well) and constitutes a basis of geometry, mathematical analysis and all derivative branches. From the basic-structures perspective it is clear (...) that the real line is an exceptionally complex structure, for it contains the ordered structure (generated by the less-than relation), the algebraic structure (generated by addition and multiplication), the gemetrical structure (generated by translations and reflections) and the topological structure (generated by open intervals). This example explains, at least to some extent, the integrality and also the vivacity of mathematics. On the other hand, the integrality has not been confirmed by comprising the whole mathematics in one axiomatised, deductive theory. Moreover, the increasing „volume” of mathematics, unwillingness of mathematicians to cross the specialisation-barriers, emphasis on utility (models) and difficulties with axiomatisations of some parts of mathematics cast some doubts on the integrality in question. (shrink)

Since the discovery of incommensurability in ancient Greece, arithmeticism and geometricism constantly switched roles. After ninetieth century arithmeticism Frege eventually returned to the view that mathematics is really entirely geometry. Yet Poincaré, Brouwer, Weyl and Bernays are mathematicians opposed to the explication of the continuum purely in terms of the discrete. At the beginning of the twenty-first century ‘continuum theorists’ in France (Longo, Thom and others) believe that the continuum precedes the discrete. In addition the last 50 years witnessed the (...) revival of infinitesimals (Laugwitz and Robinson—non-standard analysis) and—based upon category theory—the rise of smooth infinitesimal analysis and differential geometry. The spatial whole-parts relation is irreducible (Russell) and correlated with the spatial order of simultaneity. The human imaginative capacities are connected to the characterization of points and lines (Euclid) and to the views of Aristotle (the irreducibility of the continuity of a line to its points), which remained in force until the ninetieth century. Although Bolzano once more launched an attempt to arithmetize continuity, it appears as if Weierstrass, Cantor and Dedekind finally succeeded in bringing this ideal to its completion. Their views are assessed by analyzing the contradiction present in Grünbaum’s attempt to explain the continuum as an aggregate of unextended elements (degenerate intervals). Alternatively a line-stretch is characterized as a one-dimensional spatial subject, given at once in its totality (as a whole) and delimited by two points—but it is neither a breadthless length nor the (shortest) distance between two points. The overall aim of this analysis is to account for the uniqueness of discreteness and continuity by highlighting their mutual interconnections exemplified in the nature of a line as a one-dimensional spatial subject, while acknowledging that points are merely spatial objects which are always dependent upon an extended spatial subject. Instead of attempting to reduce continuity to discreteness or discreteness to continuity, a third alternative is explored: accept the irreducibility of number and space and then proceed by analyzing their unbreakable coherence. The argument may be seen as exploring some implications of the view of John Bell, namely that the “continuous is an autonomous notion, not explicable in terms of the discrete.” Bell points out that initially Brouwer, in his dissertation of 1907, “regards continuity and discreteness as complementary notions, neither of which is reducible to each other.”. (shrink)

A famous theorem by van der Waerden states the following: Given any finite coloring of the integers, one color contains arbitrarily long arithmetic progressions. Equivalently, for every q,k, there is an N = N(q,k) such that for every q-coloring of an interval of length N one color contains a progression of length k. An obvious question is what is the growth rate of N = N(q,k). Some proofs, like van der Waerden's combinatorial argument, answer this question directly, while (...) the topological proof by Furstenberg and Weiss does not. We present an analysis of (Girard's variant of) Furstenberg and Weiss's proof based on monotone functional interpretation, both yielding bounds and providing a general illustration of proof mining in topological dynamics. The bounds do not improve previous results by Girard, but only—as is also revealed by the analysis—because the combinatorial proof and the topological dynamics proof in principle are identical. (shrink)

This study presents the evaluation of a computer-based learning program for children with developmental dyscalculia and focuses on factors affecting individual responsiveness. The adaptive training program Calcularis 2.0 has been developed according to current neuro-cognitive theory of numerical cognition. It aims to automatize number representations, supports the formation and access to the mental number line and trains arithmetic operations as well as arithmetic fact knowledge in expanding number ranges. Sixty-seven children with developmental dyscalculia from second to fifth grade (...) (mean age 8.96 years) were randomly assigned to one of two groups (Calcularis group, waiting control group). Training duration comprised a minimum of 42 training sessions à 20 minutes within a maximum period of 13 weeks. Compared to the waiting control group, children of the Calcularis group demonstrated a higher benefit in arithmetic operations and number line estimation. These improvements were shown to be stable after a 3-months post training interval. In addition, this study examines which predictors accounted for training improvements. Results indicate that this self-directed training was especially beneficial for children with low math anxiety scores and without an additional reading and/or spelling disorder. In conclusion, Calcularis 2.0 supports children with developmental dyscalculia to improve their arithmetical abilities and their mental number line representation. However, it is relevant to further adapt the setting to the individual circumstances. (shrink)

An exponential $\exp $ on an ordered field $$. The structure $$ is then called an ordered exponential field. A linearly ordered structure $$ is called o-minimal if every parametrically definable subset of M is a finite union of points and open intervals of M.The main subject of this thesis is the algebraic and model theoretic examination of o-minimal exponential fields $$ whose exponential satisfies the differential equation $\exp ' = \exp $ with initial condition $\exp = 1$. This study (...) is mainly motivated by the Transfer Conjecture, which states as follows:Any o-minimal exponential field $$ whose exponential satisfies the differential equation $\exp ' = \exp $ with initial condition $\exp =1$ is elementarily equivalent to $\mathbb {R}_{\exp }$.Here, $\mathbb {R}_{\exp }$ denotes the real exponential field $$, where $\exp $ denotes the standard exponential $x \mapsto \mathrm {e}^x$ on $\mathbb {R}$. Moreover, elementary equivalence means that any first-order sentence in the language $\mathcal {L}_{\exp } = \{+,-,\cdot,0,1, <,\exp \}$ holds for $$ if and only if it holds for $\mathbb {R}_{\exp }$.The Transfer Conjecture, and thus the study of o-minimal exponential fields, is of particular interest in the light of the decidability of $\mathbb {R}_{\exp }$. To the date, it is not known if $\mathbb {R}_{\exp }$ is decidable, i.e., whether there exists a procedure determining for a given first-order $\mathcal {L}_{\exp }$ -sentence whether it is true or false in $\mathbb {R}_{\exp }$. However, under the assumption of Schanuel’s Conjecture—a famous open conjecture from Transcendental Number Theory—a decision procedure for $\mathbb {R}_{\exp }$ exists. Also a positive answer to the Transfer Conjecture would result in the decidability of $\mathbb {R}_{\exp }$. Thus, we study o-minimal exponential fields with regard to the Transfer Conjecture, Schanuel’s Conjecture, and the decidability question of $\mathbb {R}_{\exp }$.Overall, we shed light on the valuation theoretic invariants of o-minimal exponential fields—the residue field and the value group—with additional induced structure. Moreover, we explore elementary substructures and extensions of o-minimal exponential fields to the maximal ends—the smallest elementary substructures being prime models and the maximal elementary extensions being contained in the surreal numbers. Further, we draw connections to models of Peano Arithmetic, integer parts, density in real closure, definable Henselian valuations, and strongly NIP ordered fields.Parts of this thesis were published in [2–5].Abstract prepared by Lothar Sebastian KrappE-mail: [email protected]: https://d-nb.info/1202012558/34. (shrink)

This Paper combines interval- valued neutrouphic sets and rough sets. It studies roughness in interval- valued neutrosophic sets and some of its properties. Finally we propose a Hamming distance between lower and upper approximations of interval valued neutrosophic sets.

In this paper, we investigate the expressiveness of the variety of propositional interval neighborhood logics , we establish their decidability on linearly ordered domains and some important subclasses, and we prove the undecidability of a number of extensions of PNL with additional modalities over interval relations. All together, we show that PNL form a quite expressive and nearly maximal decidable fragment of Halpern–Shoham’s interval logic HS.

This book develops arithmetic without the induction principle, working in theories that are interpretable in Raphael Robinson's theory Q. Certain inductive formulas, the bounded ones, are interpretable in Q. A mathematically strong, but logically very weak, predicative arithmetic is constructed. Originally published in 1986. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books (...) while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In (...) defense of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences. (shrink)

This paper begins with a response to Josh Gert’s challenge that ‘on a par with’ is not a sui generis fourth value relation beyond ‘better than’, ‘worse than’, and ‘equally good’. It then explores two further questions: can parity be modeled by an interval representation of value? And what should one rationally do when faced with items on a par? I argue that an interval representation of value is incompatible with the possibility that items are on a par (...) (a mathematical proof is given in the appendix). I also suggest that there are three senses of ‘rationally permissible’ which, once distinguished, show that parity does distinctive practical work that cannot be done by the usual trichotomy of relations or by incomparability. In this way, we have an additional argument for parity from the workings of practical reason. (shrink)

The oblivion of the interval -- Being in place -- The aporia between envelope and things -- Dualism in Bergson -- Interval, sexual difference -- Beyond man: rethinking life and matter -- Conclusion: interval as relation, interval as becoming.

SEMANTIC ARITHMETIC: A PREFACE John Corcoran Abstract Number theory, or pure arithmetic, concerns the natural numbers themselves, not the notation used, and in particular not the numerals. String theory, or pure syntax, concems the numerals as strings of «uninterpreted» characters without regard to the numbe~s they may be used to denote. Number theory is purely arithmetic; string theory is purely syntactical... in so far as the universe of discourse alone is considered. Semantic arithmetic is a broad (...) subject which begins when numerals are mentioned (not just used) and mentioned as names of numbers (not just as syntactic objects). Semantic arithmetic leads to many fascinating and surprising algorithms and decision procedures; it reveals in a vivid way the experiential import of mathematical propositions and the predictive power of mathematical knowledge; it provides an interesting perspective for philosophical, historical, and pedagogical studies of the growth of scientific knowledge and of the role metalinguistic discourse in scientific thought. (shrink)

Symbolic arithmetic is fundamental to science, technology and economics, but its acquisition by children typically requires years of effort, instruction and drill1,2. When adults perform mental arithmetic, they activate nonsymbolic, approximate number representations3,4, and their performance suffers if this nonsymbolic system is impaired5. Nonsymbolic number representations also allow adults, children, and even infants to add or subtract pairs of dot arrays and to compare the resulting sum or difference to a third array, provided that only approximate accuracy is (...) required6–10. Here we report that young children, who have mastered verbal counting and are on the threshold of arithmetic instruction, can build on their nonsymbolic number system to perform symbolic addition and subtraction11–15. Children across a broad socio-economic spectrum solved symbolic problems involving approximate addition or subtraction of large numbers, both in a laboratory test and in a school setting. Aspects of symbolic arithmetic therefore lie within the reach of children who have learned no algorithms for manipulating numerical symbols. Our findings help to delimit the sources of children’s difficulties learning symbolic arithmetic, and they suggest ways to enhance children’s engagement with formal mathematics. We presented children with approximate symbolic arithmetic problems in a format that parallels previous tests of non-symbolic arithmetic in preschool children8,9. In the first experiment, five- to six-year-old children were given problems such as ‘‘If you had twenty-four stickers and I gave you twenty-seven more, would you have more or less than thirty-five stickers?’’. Children performed well above chance (65.0%, t1952.77, P 5 0.012) without resorting to guessing or comparison strategies that could serve as alternatives to arithmetic. Children who have been taught no symbolic arithmetic therefore have some ability to perform symbolic addition problems. The children’s performance nevertheless fell short of performance on non-symbolic arithmetic tasks using equivalent addition problems with numbers presented as arrays of dots and with the addition operation conveyed by successive motions of the dots into a box (71.3% correct, F1,345 4.26, P 5 0.047)8.. (shrink)

We study the reverse mathematics of interval orders. We establish the logical strength of the implications among various definitions of the notion of interval order. We also consider the strength of different versions of the characterization theorem for interval orders: a partial order is an interval order if and only if it does not contain 2 \oplus 2. We also study proper interval orders and their characterization theorem: a partial order is a proper interval (...) order if and only if it contains neither 2 \oplus 2 nor 3 \oplus 1. (shrink)

All arithmetical identities involving 1, addition, multiplication and exponentiation will be true in a 2-element model of Tarski's system if a certain sequence of natural numbers is not bounded. That sequence can be bounded only if the set of Fermat's prime numbers is finite.

_The Foundations of Arithmetic_ is undoubtedly the best introduction to Frege's thought; it is here that Frege expounds the central notions of his philosophy, subjecting the views of his predecessors and contemporaries to devastating analysis. The book represents the first philosophically sound discussion of the concept of number in Western civilization. It profoundly influenced developments in the philosophy of mathematics and in general ontology.

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)

Carter (Noûs 55(1):171–198, 2021) argued that while most simple positive numerical sentences are literally false, they can communicate true contents because relevance has a weakening effect on their literal contents. This paper presents a challenge for his account by considering entailments between the imprecise contents of numerical sentences and the imprecise contents of comparatives. I argue that while Carter's weakening mechanism can generate the imprecise contents of plain comparatives such as `A is taller than B', it cannot generate the imprecise (...) contents of comparatives that quantify the difference between their arguments, such as `A is exactly n times as tall as B'. I then propose an alternative theory on which intervals serve as both thick degrees on a scale and denotations of numerical expressions. I argue that the alternative theory can account for the imprecise contents of both forms of comparatives as well as the data that motivate Carter's theory. (shrink)

This paper is part of a larger project about the relation between mathematics and transcendental philosophy that I think is the most interesting feature of Kant’s philosophy of mathematics. This general view is that in the course of arguing independently of mathematical considerations for conditions of experience, Kant also establishes conditions of the possibility of mathematics. My broad aim in this paper is to clarify the sense in which this is an accurate description of Kant’s view of the relation between (...) mathematics and transcendental philosophy. (shrink)

We propose a method to associate a coalitional interval game with each strategic game. The method is based on the lower and upper values of finite two-person zero-sum games. Associating with a strategic game a coalitional interval game we avoid having to take either a pessimistic or an optimistic approach to the problem. The paper makes two contributions to the literature: It provides a theoretical foundation for the study of coalitional interval games and it also provides, studies, (...) and characterizes a natural method of associating coalitional interval games with strategic games. (shrink)

A logic of intervals is proposed akin to the one published by Hamblin (Hamblin (1969) and (1971)). Like Hamblin's, the present system is also based on a single primitive. However, the work presented here differs from Hamblin's in a number of respects. Most importantly, the present system is explicitly based on mereological ideas in such a way that not only are the two notions of abutment and temporal order involved in Hamblin's primitive two-place relation "abuts at the earlier end" distinguished (...) ; the temporal ordering itself is built up from a symmetric mereological primitive. The present system is shown to be complete. (shrink)

We will introduce a partial ordering $\preceq_1$ on the class of ordinals which will serve as a foundation for an approach to ordinal notations for formal systems of set theory and second-order arithmetic. In this paper we use $\preceq_1$ to provide a new characterization of the ubiquitous ordinal $\epsilon _{0}$.

In this paper some of the history of the development of arithmetic in set theory is traced, particularly with reference to the problem of avoiding the assumption of an infinite set. Although the standard method of singling out a sequence of sets to be the natural numbers goes back to Zermelo, its development was more tortuous than is generally believed. We consider the development in the light of three desiderata for a solution and argue that they can probably not (...) all be satisfied simultaneously. (shrink)

General Arithmetic is the theory consisting of induction on a successor function. Normal arithmetic, say in the system called Peano Arithmetic, makes certain additional demands on the successor function. First, that it be total. Secondly, that it be one-to-one. And thirdly, that there be a first element which is not in its image. General Arithmetic abandons all of these further assumptions, yet is still able to prove many meaningful arithmetic truths, such as, most basically, Commutativity (...) and Associativity of Addition and Multiplication, but also Lagrange’s Four-Square Theorem. Adding one more axiom, the one-oneness of succession, one can prove many more theorems, such as Quadratic Reciprocity and Fermat’s Little Theorem. By looking at arithmetic in this general setting, one receives a deeper understanding of the underlying structures. (shrink)

In this paper we study the interpretations of a weak arithmetic, like Buss’s theory $\mathsf{S}^{1}_{2}$, in a given theory $U$. We call these interpretations the arithmetics of $U$. We develop the basics of the structure of the arithmetics of $U$. We study the provability logic of $U$ from the standpoint of the framework of the arithmetics of $U$. Finally, we provide a deeper study of the arithmetics of a finitely axiomatized sequential theory.

Orthodoxy holds that there is a determinate fact of the matter about every arithmetical claim. Little argument has been supplied in favour of orthodoxy, and work of Field, Warren and Waxman, and others suggests that the presumption in its favour is unjustified. This paper supports orthodoxy by establishing the determinacy of arithmetic in a well-motivated modal plural logic. Recasting this result in higher-order logic reveals that even the nominalist who thinks that there are only finitely many things should think (...) that there is some sense in which arithmetic is true and determinate. (shrink)

In arithmetic, if only because many of its methods and concepts originated in India, it has been the tradition to reason less strictly than in geometry, ...

In this paper, we first defined soft intervalvalued neutrosophic rough sets(SIVN- rough sets for short) which combines interval valued neutrosophic soft set and rough sets and studied some of its basic properties. This concept is an extension of soft interval valued intuitionistic fuzzy rough sets( SIVIF- rough sets). Finally an illustartive example is given to verfy the developped algorithm and to demonstrate its practicality and effectiveness.