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.score: 180.0
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  2. Charles Sayward (2002). A Conversation About Numbers. Philosophia 29 (1-4):191-209.score: 84.0
    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. A. A. Zenkin & A. Linear (2002). Goedel's Numbering of Multi-Modal Texts. Bulletin of Symbolic Logic 8 (1):180.score: 50.0
     
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  4. Frank Stephan & Jason Teutsch (2008). Immunity and Hyperimmunity for Sets of Minimal Indices. Notre Dame Journal of Formal Logic 49 (2):107-125.score: 26.0
    We extend Meyer's 1972 investigation of sets of minimal indices. Blum showed that minimal index sets are immune, and we show that they are also immune against high levels of the arithmetic hierarchy. We give optimal immunity results for sets of minimal indices with respect to the arithmetic hierarchy, and we illustrate with an intuitive example that immunity is not simply a refinement of arithmetic complexity. Of particular note here are the fact that there are three minimal index sets located (...)
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  5. Friederike Moltmann (2013). Reference to Numbers in Natural Language. Philosophical Studies 162 (3):499 - 536.score: 24.0
    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.score: 24.0
    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. Cian Dorr (2010). Of Numbers and Electrons. Proceedings of the Aristotelian Society 110 (2pt2):133-181.score: 24.0
    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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  8. 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.score: 24.0
    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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  9. Wojciech Krysztofiak (2012). Indexed Natural Numbers in Mind: A Formal Model of the Basic Mature Number Competence. [REVIEW] Axiomathes 22 (4):433-456.score: 24.0
    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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  10. Rafal Urbaniak (2012). Numbers and Propositions Versus Nominalists: Yellow Cards for Salmon & Soames. [REVIEW] Erkenntnis 77 (3):381-397.score: 24.0
    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. Brendan Balcerak Jackson (2013). Defusing Easy Arguments for Numbers. Linguistics and Philosophy 36 (6):447-461.score: 24.0
    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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  12. Charles Sayward (2002). A Conversation About Numbers and Knowledge. American Philosophical Quarterly 39 (3):275-287.score: 24.0
    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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  13. Steven M. Duncan, Platonism by the Numbers.score: 24.0
    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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  14. Fiona Woollard (2014). The New Problem of Numbers in Morality. Ethical Theory and Moral Practice 17 (4):631-641.score: 24.0
    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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  15. Yishai Cohen (2014). Don't Count on Taurek: Vindicating the Case for the Numbers Counting. Res Publica 20 (3):245-261.score: 24.0
    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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  16. 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.score: 24.0
    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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  17. Gabriel Wollner (2012). Egalitarianism, Numbers and the Dreaded Conclusion. Ethical Perspectives 19 (3):399-416.score: 24.0
    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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  18. Mohamed Chelli & Yves Gendron (2013). Sustainability Ratings and the Disciplinary Power of the Ideology of Numbers. Journal of Business Ethics 112 (2):187-203.score: 24.0
    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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  19. 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.score: 24.0
    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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  20. D. Pecher & I. Boot (2010). Numbers in Space: Differences Between Concrete and Abstract Situations. Frontiers in Psychology 2:121-121.score: 24.0
    Numbers might be understood by grounding in spatial orientation, where small numbers are represented as low or to the left and large numbers are represented as high or to the right. We presented numbers in concrete (seven shoes in a shoe shop) or abstract (29 – 7) contexts and asked participants to make relative magnitude judgments. Following the judgment a target letter was presented at the top or bottom (Experiments 1-3) or left or right (Experiment 4) (...)
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  21. Kristen Pilner Blair, Miriam Rosenberg-Lee, Jessica M. Tsang, Daniel L. Schwartz & Vinod Menon (2012). Beyond Natural Numbers: Negative Number Representation in Parietal Cortex. Frontiers in Human Neuroscience 6.score: 24.0
    Unlike natural numbers, negative numbers do not have natural physical referents. How does the brain represent such abstract mathematical concepts? Two competing hypotheses regarding representational systems for negative numbers are a rule-based model, in which symbolic rules are applied to negative numbers to translate them into positive numbers when assessing magnitudes, and an expanded magnitude model, in which negative numbers have a distinct magnitude representation. Using an event-related fMRI design, we examined brain responses in (...)
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  22. George Wolford Margaret M. Gullick (2013). Understanding Less Than Nothing: Children's Neural Response to Negative Numbers Shifts Across Age and Accuracy. Frontiers in Psychology 4.score: 24.0
    We examined the brain activity underlying the development of our understanding of negative numbers, which are amounts lacking direct physical counterparts. Children performed a paired comparison task with positive and negative numbers during an fMRI session. As previously shown in adults, both pre-instruction fifth graders and post-instruction seventh graders demonstrated typical behavioral and neural distance effects to negative numbers, where response times and parietal and frontal activity increased as comparison distance decreased. We then determined the factors impacting (...)
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  23. S. W. P. Steen (1972). Mathematical Logic with Special Reference to the Natural Numbers. Cambridge [Eng.]University Press.score: 24.0
    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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  24. John Corbett & Thomas Durt (2010). Spatial Localization in Quantum Theory Based on Qr-Numbers. Foundations of Physics 40 (6):607-628.score: 22.0
    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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  25. Øystein Linnebo (2009). The Individuation of the Natural Numbers. In Otavio Bueno & Øystein Linnebo (eds.), New Waves in Philosophy of Mathematics. Palgrave.score: 21.0
    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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  26. Andrei S. Morozov & Margarita V. Korovina (2008). On Σ‐Definability Without Equality Over the Real Numbers. Mathematical Logic Quarterly 54 (5):535-544.score: 21.0
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  27. Frank Restle (1970). Speed of Adding and Comparing Numbers. Journal of Experimental Psychology 83 (2p1):274.score: 21.0
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  28. Vasco Brattka (1997). Order‐Free Recursion on the Real Numbers. Mathematical Logic Quarterly 43 (2):216-234.score: 21.0
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  29. 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.score: 21.0
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  30. 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.score: 21.0
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  31. J. E. Rubin, K. Keremedis & Paul Howard (2001). Non-Constructive Properties of the Real Numbers. Mathematical Logic Quarterly 47 (3):423-431.score: 21.0
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  32. Frederick Bagemihl & F. Bagemihl (1992). Ordinal Numbers in Arithmetic Progression. Mathematical Logic Quarterly 38 (1):525-528.score: 21.0
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  33. Qingliang Chen, Kaile Su & Xizhong Zheng (2007). Primitive Recursive Real Numbers. Mathematical Logic Quarterly 53 (4‐5):365-380.score: 21.0
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  34. Iamblichus (1988). The Theology of Arithmetic: On the Mystical, Mathematical and Cosmological Symbolism of the First Ten Numbers. Phanes Press.score: 21.0
  35. Fred Richman (2008). Real Numbers and Other Completions. Mathematical Logic Quarterly 54 (1):98-108.score: 21.0
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  36. Cathy J. Price Roi Cohen Kadosh, Bahador Bahrami, Vincent Walsh, Brian Butterworth, Tudor Popescu (2011). Specialization in the Human Brain: The Case of Numbers. Frontiers in Human Neuroscience 5.score: 21.0
    How numerical representation is encoded in the adult human brain is important for a basic understanding of human brain organization, its typical and atypical development, its evolutionary precursors, cognitive architectures, education and rehabilitation. Previous studies have shown that numerical processing activates the same intraparietal regions irrespective of the presentation format (e.g. symbolic digits or non-symbolic dot arrays). This has led to claims that there is a single format independent, numerical representation. In the current study we used a functional magnetic resonance (...)
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  37. Stefan Slak (1970). Free Recall of Numbers with High- and Low-Rated Association Values. Journal of Experimental Psychology 83 (1p1):184.score: 21.0
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  38. Charles P. Thompson (1965). Effects of Interval Between Successive Numbers and Pattern in Verbal Learning. Journal of Experimental Psychology 70 (6):626.score: 21.0
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  39. Shunsuke Yatabe (2007). Distinguishing Non-Standard Natural Numbers in a Set Theory Within Łukasiewicz Logic. Archive for Mathematical Logic 46 (3-4):281-287.score: 21.0
    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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  40. John T. Sanders (1988). Why the Numbers Should Sometimes Count. Philosophy and Public Affairs 17 (1):3-14.score: 20.0
    John Taurek has argued that, where choices must be made between alternatives that affect different numbers of people, the numbers are not, by themselves, morally relevant. This is because we "must" take "losses-to" the persons into account (and these don't sum), but "must not" consider "losses-of" persons (because we must not treat persons like objects). I argue that the numbers are always ethically relevant, and that they may sometimes be the decisive consideration.
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  41. Pierre Pica & Alain Lecomte (2008). Theoretical Implications of the Study of Numbers and Numerals in Mundurucu. Philosophical Psychology 21 (4):507 – 522.score: 20.0
    Developing earlier studies of the system of numbers in Mundurucu, this paper argues that the Mundurucu numeral system is far more complex than usually assumed. The Mundurucu numeral system provides indirect but insightful arguments for a modular approach to numbers and numerals. It is argued that distinct components must be distinguished, such as a system of representation of numbers in the format of internal magnitudes, a system of representation for individuals and sets, and one-to-one correspondences between the (...)
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  42. Katharina Felka (2014). Number Words and Reference to Numbers. Philosophical Studies 168 (1):261-282.score: 20.0
    A realist view of numbers often rests on the following thesis: statements like ‘The number of moons of Jupiter is four’ are identity statements in which the copula is flanked by singular terms whose semantic function consists in referring to a number (henceforth: Identity). On the basis of Identity the realists argue that the assertive use of such statements commits us to numbers. Recently, some anti-realists have disputed this argument. According to them, Identity is false, and, thus, we (...)
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  43. David Landy, Noah Silbert & Aleah Goldin (2013). Estimating Large Numbers. Cognitive Science 37 (5):775-799.score: 20.0
    Despite their importance in public discourse, numbers in the range of 1 million to 1 trillion are notoriously difficult to understand. We examine magnitude estimation by adult Americans when placing large numbers on a number line and when qualitatively evaluating descriptions of imaginary geopolitical scenarios. Prior theoretical conceptions predict a log-to-linear shift: People will either place numbers linearly or will place numbers according to a compressive logarithmic or power-shaped function (Barth & Paladino, ; Siegler & Opfer, (...)
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  44. Martin H. Fischer, Marianna Riello, Bruno L. Giordano & Elena Rusconi (2013). Singing Numbers… in Cognitive Space — A Dual‐Task Study of the Link Between Pitch, Space, and Numbers. Topics in Cognitive Science 5 (2):354-366.score: 20.0
    We assessed the automaticity of spatial-numerical and spatial-musical associations by testing their intentionality and load sensitivity in a dual-task paradigm. In separate sessions, 16 healthy adults performed magnitude and pitch comparisons on sung numbers with variable pitch. Stimuli and response alternatives were identical, but the relevant stimulus attribute (pitch or number) differed between tasks. Concomitant tasks required retention of either color or location information. Results show that spatial associations of both magnitude and pitch are load sensitive and that the (...)
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  45. Suzana Herculano-Houzel (2009). The Human Brain in Numbers: A Linearly Scaled-Up Primate Brain. Frontiers in Human Neuroscience 3:31.score: 20.0
    The human brain has often been viewed as outstanding among mammalian brains: the most cognitively able, the largest-than-expected from body size, endowed with an overdeveloped cerebral cortex that represents over 80% of brain mass, and purportedly containing 100 billion neurons and 10x more glial cells. Such uniqueness was seemingly necessary to justify the superior cognitive abilities of humans over larger-brained mammals such as elephants and whales. However, our recent studies using a novel method to determine the cellular composition of the (...)
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  46. Claude Laflamme & Marion Scheepers (1999). Combinatorial Properties of Filters and Open Covers for Sets of Real Numbers. Journal of Symbolic Logic 64 (3):1243-1260.score: 20.0
    We analyze combinatorial properties of open covers of sets of real numbers by using filters on the natural numbers. In fact, the goal of this paper is to characterize known properties related to ω-covers of the space in terms of combinatorial properties of filters associated with these ω-covers. As an example, we show that all finite powers of a set R of real numbers have the covering property of Menger if, and only if, each filter on ω (...)
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  47. Andrea M. Loftus, Michael E. R. Nicholls, Jason B. Mattingley & John L. Bradshaw (2008). Left to Right: Representational Biases for Numbers and the Effect of Visuomotor Adaptation. Cognition 107 (3):1048-1058.score: 20.0
    Adaptation to right-shifting prisms improves left neglect for mental number line bisection. This study examined whether adaptation affects the mental number line in normal participants. Thirty-six participants completed a mental number line task before and after adaptation to either: left-shifting prisms, right-shifting prisms or control spectacles that did not shift the visual scene. Participants viewed number triplets (e.g. 16, 36, 55) and determined whether the numerical distance was greater on the left or right side of the inner number. Participants demonstrated (...)
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  48. Sven Lindberg Jan Lonnemann, Janosch Linkersdörfer, Telse Nagler, Marcus Hasselhorn (2013). Spatial Representations of Numbers and Letters in Children. Frontiers in Psychology 4.score: 20.0
    Different lines of evidence suggest that children’s mental representations of numbers are spatially organized in form of a mental number line. It is, however, still unclear whether a spatial organization is specific for the numerical domain or also applies to other ordinal sequences in children. In the present study, children (n = 129) aged 8-9 years were asked to indicate the midpoint of lines flanked by task-irrelevant digits or letters. We found that the localization of the midpoint was systematically (...)
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