Search results for 'Goedel numbers' (try it on Scholar)

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  1. Divisor Relative Numbers (1987). 3. The Monotone Series and Multiplier and Divisor Relative Numbers. International Logic Review: Rassegna Internazionale di Logica 15 (1):26.
     
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  2.  48
    Charles Sayward (2002). A Conversation About Numbers. Philosophia 29 (1-4):191-209.
    This is a dialogue in which five characters are involved. Various issues in the philosophy of mathematics are discussed. Among those issues are these: numbers as abstract objects, our knowledge of numbers as abstract objects, a proof as showing a mathematical statement to be true as opposed to the statement being true in virtue of having a proof.
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  3. Friederike Moltmann (2013). Reference to Numbers in Natural Language. Philosophical Studies 162 (3):499 - 536.
    A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are (...)
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  4.  6
    Mohamed Chelli & Yves Gendron (2013). Sustainability Ratings and the Disciplinary Power of the Ideology of Numbers. Journal of Business Ethics 112 (2):187-203.
    The main purpose of this paper is to better understand how sustainability rating agencies, through discourse, promote an “ideology of numbers” that ultimately aims to establish a regime of normalization governing social and environmental performance. Drawing on Thompson’s (Ideology and modern culture: Critical social theory in the era of mass communication, 1990 ) modes of operation of ideology, we examine the extent to which, and how, the ideology of numbers is reflected on websites and public documents published by (...)
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  5.  64
    Friederike Moltmann (2013). Reference to Numbers in Natural Language. Philosophical Studies 162 (3):499 - 536.
    A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are (...)
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  6. Rob Lawlor (2006). Taurek, Numbers and Probabilities. Ethical Theory and Moral Practice 9 (2):149 - 166.
    In his paper, “Should the Numbers Count?" John Taurek imagines that we are in a position such that we can either save a group of five people, or we can save one individual, David. We cannot save David and the five. This is because they each require a life-saving drug. However, David needs all of the drug if he is to survive, while the other five need only a fifth each.Typically, people have argued as if there was a choice (...)
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  7. Edward N. Zalta (1999). Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege"s Grundgesetze in Object Theory". [REVIEW] Journal of Philosophical Logic 28 (6):619-660.
    In this paper, the author derives the Dedekind-Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege's Grundgesetze. The proofs of the theorems reconstruct Frege's derivations, with the exception of the claim that every number has a successor, which is derived from a modal axiom that (...)
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  8. Cian Dorr (2010). Of Numbers and Electrons. Proceedings of the Aristotelian Society 110 (2pt2):133-181.
    According to a tradition stemming from Quine and Putnam, we have the same broadly inductive reason for believing in numbers as we have for believing in electrons: certain theories that entail that there are numbers are better, qua explanations of our evidence, than any theories that do not. This paper investigates how modal theories of the form ‘Possibly, the concrete world is just as it in fact is and T’ and ‘Necessarily, if standard mathematics is true and the (...)
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  9.  41
    Brendan Balcerak Jackson (2013). Defusing Easy Arguments for Numbers. Linguistics and Philosophy 36 (6):447-461.
    Pairs of sentences like the following pose a problem for ontology: (1) Jupiter has four moons. (2) The number of moons of Jupiter is four. (2) is intuitively a trivial paraphrase of (1). And yet while (1) seems ontologically innocent, (2) appears to imply the existence of numbers. Thomas Hofweber proposes that we can resolve the puzzle by recognizing that sentence (2) is syntactically derived from, and has the same meaning as, sentence (1). Despite appearances, the expressions ‘the number (...)
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  10.  24
    Rafal Urbaniak (2012). Numbers and Propositions Versus Nominalists: Yellow Cards for Salmon & Soames. [REVIEW] Erkenntnis 77 (3):381-397.
    Salmon and Soames argue against nominalism about numbers and sentence types. They employ (respectively) higher-order and first-order logic to model certain natural language inferences and claim that the natural language conclusions carry commitment to abstract objects, partially because their renderings in those formal systems seem to do that. I argue that this strategy fails because the nominalist can accept those natural language consequences, provide them with plausible and non-committing truth conditions and account for the inferences made without committing themselves (...)
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  11.  38
    Steven M. Duncan, Platonism by the Numbers.
    In this paper, I defend traditional Platonic mathematical realism from its contemporary detractors, arguing that numbers, understood as abstract, non-physical objects of rational intuition, are indispensable for the act of counting.
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  12. Cian Dorr (2010). Of Numbers and Electrons. Proceedings of the Aristotelian Society 110 (2pt2):133-181.
    According to a tradition stemming from Quine and Putnam, we have the same broadly inductive reason for believing in numbers as we have for believing in electrons: certain theories that entail that there are numbers are better, qua explanations of our evidence, than any theories that do not. This paper investigates how modal theories of the form ‘Possibly, the concrete world is just as it in fact is and T’ and ‘Necessarily, if standard mathematics is true and the (...)
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  13.  33
    Yishai Cohen (2014). Don’T Count on Taurek: Vindicating the Case for the Numbers Counting. Res Publica 20 (3):245-261.
    Suppose you can save only one of two groups of people from harm, with one person in one group, and five persons in the other group. Are you obligated to save the greater number? While common sense seems to say ‘yes’, the numbers skeptic says ‘no’. Numbers Skepticism has been partly motivated by the anti-consequentialist thought that the goods, harms and well-being of individual people do not aggregate in any morally significant way. However, even many non-consequentialists think that (...)
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  14.  13
    J. E. Rubin, K. Keremedis & Paul Howard (2001). Non-Constructive Properties of the Real Numbers. Mathematical Logic Quarterly 47 (3):423-431.
    We study the relationship between various properties of the real numbers and weak choice principles.
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  15.  25
    Fiona Woollard (2014). The New Problem of Numbers in Morality. Ethical Theory and Moral Practice 17 (4):631-641.
    Discussion of the “problem of numbers” in morality has focused almost exclusively on the moral significance of numbers in whom-to-rescue cases: when you can save either of two groups of people, but not both, does the number of people in each group matter morally? I suggest that insufficient attention has been paid to the moral significance of numbers in other types of case. According to common-sense morality, numbers make a difference in cases, like the famous Trolley (...)
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  16.  2
    Fred Richman (2008). Real Numbers and Other Completions. Mathematical Logic Quarterly 54 (1):98-108.
    A notion of completeness and completion suitable for use in the absence of countable choice is developed. This encompasses the construction of the real numbers as well as the completion of an arbitrary metric space. The real numbers are characterized as a complete Archimedean Heyting field, a terminal object in the category of Archimedean Heyting fields.
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  17.  16
    Catherine Rowett (2013). Philosophy's Numerical Turn: Why the Pythagoreans' Interest in Numbers is Truly Awesome. In Dirk Obbink & David Sider (eds.), Doctrine and Doxography: Studies on Heraclitus and Pythagoras. De Gruyter 3-32.
    Philosophers are generally somewhat wary of the hints of number mysticism in the reports about the beliefs and doctrines of the so-called Pythagoreans. It's not clear how much Pythagoras himself (as opposed to his later followers) indulged in speculation about numbers, or in more serious mathematics. But the Pythagoreans whom Aristotle discusses in the Metaphysics had some elaborate stories to tell about how the universe could be explained in terms of numbers—not just its physics but perhaps morality too. (...)
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  18.  32
    Wojciech Krysztofiak (2012). Indexed Natural Numbers in Mind: A Formal Model of the Basic Mature Number Competence. [REVIEW] Axiomathes 22 (4):433-456.
    The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in (...)
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  19.  3
    Marcelo E. Coniglio & Luís A. Sbardellini (2015). On the Ordered Dedekind Real Numbers in Toposes. In Edward H. Haeusler, Wagner Sanz & Bruno Lopes (eds.), Why is this a Proof? Festschrift for Luiz Carlos Pereira. College Publications 87-105.
    In 1996, W. Veldman and F. Waaldijk present a constructive (intuitionistic) proof for the homogeneity of the ordered structure of the Cauchy real numbers, and so this result holds in any topos with natural number object. However, it is well known that the real numbers objects obtained by the traditional constructions of Cauchy sequences and Dedekind cuts are not necessarily isomorphic in an arbitrary topos with natural numbers object. Consequently, Veldman and Waaldijk's result does not (...)
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  20.  9
    Gabriel Wollner (2012). Egalitarianism, Numbers and the Dreaded Conclusion. Ethical Perspectives 19 (3):399-416.
    Some contractualist egalitarians try to accommodate a concern for numbers by embracing a pluralist strategy. They incorporate the belief that the number of people affected matters for what distribution one ought to bring about by arguing that their primary contractualist concern for justifiability to each may be outweighed by aggregative considerations. The present contribution offers two arguments against such a pluralist strategy. First, I argue that advo- cates of the pluralist strategy are forced to abandon the rationale behind the (...)
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  21.  19
    Charles Sayward (2002). A Conversation About Numbers and Knowledge. American Philosophical Quarterly 39 (3):275-287.
    This is a dialogue in the philosophy of mathematics. The dialogue descends from the confident assertion that there are infinitely many numbers to an unresolved bewilderment about how we can know there are any numbers at all. At every turn the dialogue brings us only to realize more fully how little is clear to us in our thinking about mathematics.
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  22.  1
    Barbara W. Sarnecka (forthcoming). Learning to Represent Exact Numbers. Synthese:1-18.
    This article focuses on how young children acquire concepts for exact, cardinal numbers. I believe that exact numbers are a conceptual structure that was invented by people, and that most children acquire gradually, over a period of months or years during early childhood. This article reviews studies that explore children’s number knowledge at various points during this acquisition process. Most of these studies were done in my own lab, and assume the theoretical framework proposed by Carey. In this (...)
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  23.  2
    Qingliang Chen, Kaile Su & Xizhong Zheng (2007). Primitive Recursive Real Numbers. Mathematical Logic Quarterly 53 (4‐5):365-380.
    In mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure – Dedekind-complete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computable real numbers. Although the computable real numbers can be defined in various equivalent ways, if “computable” is replaced by “primitive recursive” , these definitions lead to a number of different (...)
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  24.  4
    Guram Bezhanishvili & Joel Lucero-Bryan (2012). More on D-Logics of Subspaces of the Rational Numbers. Notre Dame Journal of Formal Logic 53 (3):319-345.
    We prove that each countable rooted K4 -frame is a d-morphic image of a subspace of the space $\mathbb{Q}$ of rational numbers. From this we derive that each modal logic over K4 axiomatizable by variable-free formulas is the d-logic of a subspace of $\mathbb{Q}$ . It follows that subspaces of $\mathbb{Q}$ give rise to continuum many d-logics over K4 , continuum many of which are neither finitely axiomatizable nor decidable. In addition, we exhibit several families of modal logics finitely (...)
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  25.  1
    Robert Schwartzkopff (2011). Numbers as Ontologically Dependent Objects - Hume's Principle Revisited. Grazer Philosophische Studien 82:353-373.
    Adherents of Ockham’s fundamental razor contend that considerations of ontological parsimony pertain primarily to fundamental objects. Derivative objects, on the other hand, are thought to be quite unobjectionable. One way to understand the fundamental vs. derivative distinction is in terms of the Aristotelian distinction between ontologically independent and dependent objects. In this paper I will defend the thesis that every natural number greater than 0 is an ontologically dependent object thereby exempting the natural numbers from Ockham’s fundamental razor.
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  26.  1
    Frederick Bagemihl & F. Bagemihl (1992). Ordinal Numbers in Arithmetic Progression. Mathematical Logic Quarterly 38 (1):525-528.
    The class of all ordinal numbers can be partitioned into two subclasses in such a way that neither subclass contains an arithmetic progression of order type ω, where an arithmetic progression of order type τ means an increasing sequence of ordinal numbers γ r, δ ≠ 0.
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  27. Pierre‐Marie David (2016). Measurement, “Scriptural Economies,” and Social Justice: Governing HIV/AIDS Treatments by Numbers in a Fragile State, the Central African Republic. Developing World Bioethics 16 (1).
    Fragile states have been raising increasing concern among donors since the mid-2000s. The policies of the Global Fund to fight HIV/AIDS, Malaria, and Tuberculosis have not excluded fragile states, and this source has provided financing for these countries according to standardized procedures. They represent interesting cases for exploring the meaning and role of measurement in a globalized context. Measurement in the field of HIV/AIDS and its treatment has given rise to a private outsourcing of expertise and auditing, thereby creating a (...)
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  28. S. W. P. Steen (1972). Mathematical Logic with Special Reference to the Natural Numbers. Cambridge [Eng.]University Press.
    This book presents a comprehensive treatment of basic mathematical logic. The author's aim is to make exact the vague, intuitive notions of natural number, preciseness, and correctness, and to invent a method whereby these notions can be communicated to others and stored in the memory. He adopts a symbolic language in which ideas about natural numbers can be stated precisely and meaningfully, and then investigates the properties and limitations of this language. The treatment of mathematical concepts in the main (...)
     
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  29. Christian Helmut Wenzel (2010). Frege, the Complex Numbers, and the Identity of Indiscernibles. Logique Et Analyse 209:51-60.
    There are mathematical structures with elements that cannot be distinguished by the properties they have within that structure. For instance within the field of complex numbers the two square roots of −1, i and −i, have the same algebraic properties in that field. So how do we distinguish between them? Imbedding the complex numbers in a bigger structure, the quaternions, allows us to algebraically tell them apart. But a similar problem appears for this larger structure. There seems to (...)
     
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  30.  43
    Shunsuke Yatabe (2007). Distinguishing Non-Standard Natural Numbers in a Set Theory Within Łukasiewicz Logic. Archive for Mathematical Logic 46 (3-4):281-287.
    In ${\mathbf{H}}$ , a set theory with the comprehension principle within Łukasiewicz infinite-valued predicate logic, we prove that a statement which can be interpreted as “there is an infinite descending sequence of initial segments of ω” is truth value 1 in any model of ${\mathbf{H}}$ , and we prove an analogy of Hájek’s theorem with a very simple procedure.
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  31.  11
    Frank Restle (1970). Speed of Adding and Comparing Numbers. Journal of Experimental Psychology 83 (2p1):274.
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  32.  67
    John Corbett & Thomas Durt (2010). Spatial Localization in Quantum Theory Based on Qr-Numbers. Foundations of Physics 40 (6):607-628.
    We show how trajectories can be reintroduced in quantum mechanics provided that its spatial continuum is modelled by a variable real number (qr-number) continuum. Such a continuum can be constructed using only standard Hilbert space entities. In this approach, the geometry of atoms and subatomic objects differs from that of classical objects. The systems that are non-local when measured in the classical space-time continuum may be localized in the quantum continuum. We compare trajectories in this new description of space-time with (...)
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  33. Øystein Linnebo (2009). The Individuation of the Natural Numbers. In Otavio Bueno & Øystein Linnebo (eds.), New Waves in Philosophy of Mathematics. Palgrave
    It is sometimes suggested that criteria of identity should play a central role in an account of our most fundamental ways of referring to objects. The view is nicely illustrated by an example due to (Quine, 1950). Suppose you are standing at the bank of a river, watching the water that floats by. What is required for you to refer to the river, as opposed to a particular segment of it, or the totality of its water, or the current temporal (...)
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  34.  12
    Andrei S. Morozov & Margarita V. Korovina (2008). On Σ‐Definability Without Equality Over the Real Numbers. Mathematical Logic Quarterly 54 (5):535-544.
    In [5] it has been shown that for first-order definability over the reals there exists an effective procedure which by a finite formula with equality defining an open set produces a finite formula without equality that defines the same set. In this paper we prove that there exists no such procedure for Σ-definability over the reals. We also show that there exists even no uniform effective transformation of the definitions of Σ-definable sets into new definitions of Σ-definable sets in such (...)
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  35.  4
    Vasco Brattka (1997). Order‐Free Recursion on the Real Numbers. Mathematical Logic Quarterly 43 (2):216-234.
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  36.  2
    Allan L. Fingeret & W. J. Brogden (1973). Effect of Pattern in Display by Letters and Numerals Upon Acquisition of Serial Lists of Numbers. Journal of Experimental Psychology 98 (2):339.
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  37.  1
    E. C. Poulton & D. C. V. Simmonds (1963). Value of Standard and Very First Variable in Judgments of Reflectance of Grays with Various Ranges of Available Numbers. Journal of Experimental Psychology 65 (3):297.
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  38. Iamblichus (1988). The Theology of Arithmetic: On the Mystical, Mathematical and Cosmological Symbolism of the First Ten Numbers. Phanes Press.
  39. Stefan Slak (1970). Free Recall of Numbers with High- and Low-Rated Association Values. Journal of Experimental Psychology 83 (1p1):184.
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  40. Charles P. Thompson (1965). Effects of Interval Between Successive Numbers and Pattern in Verbal Learning. Journal of Experimental Psychology 70 (6):626.
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  41. John Bigelow (1988). The Reality of Numbers: A Physicalist's Philosophy of Mathematics. Oxford University Press.
    Challenging the myth that mathematical objects can be defined into existence, Bigelow here employs Armstrong's metaphysical materialism to cast new light on mathematics. He identifies natural, real, and imaginary numbers and sets with specified physical properties and relations and, by so doing, draws mathematics back from its sterile, abstract exile into the midst of the physical world.
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  42. Graham Oppy (1995). Inverse Operations with Transfinite Numbers and the Kalam Cosmological Argument. International Philosophical Quarterly 35 (2):219-221.
    William Lane Craig has argued that there cannot be actual infinities because inverse operations are not well-defined for infinities. I point out that, in fact, there are mathematical systems in which inverse operations for infinities are well-defined. In particular, the theory introduced in John Conway's *On Numbers and Games* yields a well-defined field that includes all of Cantor's transfinite numbers.
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  43.  82
    Jody Azzouni (2010). Talking About Nothing: Numbers, Hallucinations, and Fictions. Oxford University Press.
    Numbers -- Hallucinations -- Fictions -- Scientific languages, ontology, and truth -- Truth conditions and semantics.
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  44.  51
    Zach Weber (2010). Transfinite Numbers in Paraconsistent Set Theory. Review of Symbolic Logic 3 (1):71-92.
    This paper begins an axiomatic development of naive set theoryin a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will (...)
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  45.  30
    Philip Ehrlich (2012). The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small. Bulletin of Symbolic Logic 18 (1):1-45.
    In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including $-\omega, \,\omega/2, \,1/\omega, \sqrt{\omega}$ and $\omega-\pi$ to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG, it may be (...)
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  46.  14
    Karin Katz & Mikhail Katz (2012). Stevin Numbers and Reality. Foundations of Science 17 (2):109-123.
    We explore the potential of Simon Stevin’s numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis.
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  47. Paolo Mancosu (2009). Measuring the Size of Infinite Collections of Natural Numbers: Was Cantor's Theory of Infinite Number Inevitable? Review of Symbolic Logic 2 (4):612-646.
    Cantorsizesizesizewhole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the partdel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged (...)
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  48.  38
    Eric Margolis & Stephen Laurence (2008). How to Learn the Natural Numbers: Inductive Inference and the Acquisition of Number Concepts. Cognition 106 (2):924-939.
    Theories of number concepts often suppose that the natural numbers are acquired as children learn to count and as they draw an induction based on their interpretation of the first few count words. In a bold critique of this general approach, Rips, Asmuth, Bloomfield [Rips, L., Asmuth, J. & Bloomfield, A.. Giving the boot to the bootstrap: How not to learn the natural numbers. Cognition, 101, B51–B60.] argue that such an inductive inference is consistent with a representational system (...)
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  49. Saharon Shelah & Juris Steprāns (2001). The Covering Numbers of Mycielski Ideals Are All Equal. Journal of Symbolic Logic 66 (2):707-718.
    The Mycielski ideal M k is defined to consist of all sets $A \subseteq ^{\mathbb{N}}k$ such that $\{f \upharpoonright X: f \in A\} \neq ^Xk$ for all X ∈ [N] ℵ 0 . It will be shown that the covering numbers for these ideals are all equal. However, the covering numbers of the closely associated Roslanowski ideals will be shown to be consistently different.
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  50. Carlos Drews (2001). Wild Animals and Other Pets Kept in Costa Rican Households: Incidence, Species and Numbers. Society and Animals 9 (2):107-126.
    A nationwide survey that included personal interviews in 1,021 households studied the incidence, species, and numbers of nonhuman animals kept in Costa Rican households. A total of 71% of households keep animals.The proportion of households keeping dogs is 3.6 higher than the proportion of households keeping cats . In addition to the usual domestic or companion animals kept in 66% of the households, 24% of households keep wild species as pets. Although parrots are the bulk of wild species kept (...)
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