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 concrete world (...) is just as it in fact is, then T’ bear on this claim. It concludes that, while analogies with theories that attempt to eliminate unobservable concrete entities provide good reason to regard theories of the former sort as explanatorily bad, this reason does not apply to theories of the latter sort. (shrink)
This paper criticizes the view that number words in argument position retain the meaning they have on an adjectival or determiner use, as argued by Hofweber :179–225, 2005) and Moltmann :499–534, 2013a, 2013b). In particular the paper re-evaluates syntactic evidence from German given in Moltmann to that effect.
The question whether numbers are objects is a central question in the philosophy of mathematics. Frege made use of a syntactic criterion for objethood: numbers are objects because there are singular terms that stand for them, and not just singular terms in some formal language, but in natural language in particular. In particular, Frege (1884) thought that both noun phrases like the number of planets and simple numerals like eight as in (1) are singular terms referring to numbers as (...) abstract objects. (shrink)
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 not primarily treated (...) abstract objects, but rather 'aspects' of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted. (shrink)
Do scientific theories limit human knowledge? In other words, are there physical variables hidden by essence forever? We argue for negative answers and illustrate our point on chaotic classical dynamical systems. We emphasize parallels with quantum theory and conclude that the common real numbers are, de facto, the hidden variables of classical physics. Consequently, real numbers should not be considered as ``physically real" and classical mechanics, like quantum physics, is indeterministic.
I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's. I give a detailed mathematical demonstration that 0 is {} and for every natural number n, n is the set of all natural numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals.
Is the way we use propositions to individuate beliefs and other intentional states analogous to the way we use numbers to measure weights and other physical magnitudes? In an earlier paper [2], I argued that there is an important disanalogy. One and the same weight can be 'related to' different numbers under different units of measurement. Moreover, the choice of a unit of measurement is arbitrary,in the sense that which way we choose doesn't affect the weight attributed to the object. (...) But it makes little sense to say that one and the same belief can be related to different propositions: different proposition means different belief. So there is no analogous arbitrary choice. (shrink)
You ought to save a larger group of people rather than a distinct smaller group of people, all else equal. A consequentialist may say that you ought to do so because this produces the most good. If a non-consequentialist rejects this explanation, what alternative can he or she give? This essay defends the following explanation, as a solution to the so-called numbers problem. Its two parts can be roughly summarised as follows. First, you are morally required to want the survival (...) of each stranger for its own sake. Secondly, you are rationally required to achieve as many of these ends as possible, if you have these ends. (shrink)
The low representation (< 30%) of women in philosophy in English-speaking countries has generated much discussion, both in academic circles and the public sphere. It is sometimes suggested (Haslanger 2009) that unconscious biases, acting at every level in the field, may be grounded in gendered schemas of philosophers and in the discipline more widely, and that actions to make philosophy a more welcoming place for women should address such schemas. However, existing data are too limited to fully warrant such an (...) explanation, which therefore will not satisfy those in favor of the status quo or those who argue against the need to address gender imbalance. In this paper, we propose measures to improve the profession that ought to be implemented without referring explicitly to this underrepresentation or to the climate for women and other underrepresented groups. Such recommendations are based on empirical research already carried out in other disciplines and do not rest on whether it is possible to identify the cause of this low representation. We argue that we need not wait for new or better data to ensure that fairer practices are enacted for women, other underrepresented groups, and everybody else, if only out of precaution. (shrink)
David Hilbert's finitistic standpoint is a conception of elementary number theory designed to answer the intuitionist doubts regarding the security and certainty of mathematics. Hilbert was unfortunately not exact in delineating what that viewpoint was, and Hilbert himself changed his usage of the term through the 1920s and 30s. The purpose of this paper is to outline what the main problems are in understanding Hilbert and Bernays on this issue, based on some publications by them which have so far (...) received little attention, and on a number of philosophical reconstructions of the viewpoint (in particular, by Hand, Kitcher, and Tait). (shrink)
I argue that the human mind includes an innate domain-specific system for representing precise small numerical quantities. This theory contrasts with object-tracking theories and with domain-general theories that only make use of mental models. I argue that there is a good amount of evidence for innate representations of small numerical quantities and that such a domain-specific system has explanatory advantages when infants’ poor working memory is taken into account. I also show that the mental models approach requires previously unnoticed domain-specific (...) structure and consequently that there is no domain-general alternative to an innate domain-specific small number system. (shrink)
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.
Numbers are concepts whose content, structure, and organization are influenced by the material forms used to represent and manipulate them. Indeed, as argued here, it is the inclusion of multiple forms (distributed objects, fingers, single- and two-dimensional forms like pebbles and abaci, and written notations) that is the mechanism of numerical elaboration. Further, variety in employed forms explains at least part of the synchronic and diachronic variability that exists between and within cultural number systems. Material forms also impart characteristics (...) like linearity that may persist in the form of knowledge and behaviors, ultimately yielding numerical concepts that are irreducible to and functionally independent of any particular form. Material devices used to represent and manipulate numbers also interact with language in ways that reinforce or contrast different aspects of numerical cognition. Not only does this interaction potentially explain some of the unique aspects of numerical language, it suggests that the two are complementary but ultimately distinct means of accessing numerical intuitions and insights. The potential inclusion of materiality in contemporary research in numerical cognition is advocated, both for its explanatory power, as well as its influence on psychological, behavioral, and linguistic aspects of numerical cognition. (shrink)
In this paper, I suggest that infinite numbers are large finite numbers, and that infinite numbers, properly understood, are 1) of the structure omega + (omega* + omega)Ө + omega*, and 2) the part is smaller than the whole. I present an explanation of these claims in terms of epistemic limitations. I then consider the importance, part of which is demonstrating the contradiction that lies at the heart of Cantorian set theory: the natural numbers are too large to be counted (...) by any finite number, but too small to be counted by any infinite number – there is no number of natural numbers. (shrink)
Is there some large number of very mild hangnail pains, each experienced by a separate person, which would be worse than two years of excruciating torture, experienced by a single person? Many people have the intuition that the answer to this question is No. However, a host of philosophers have argued that, because we have no intuitive grasp of very large numbers, we should not trust such intuitions. I argue that there is decent intuitive support for the No answer, (...) which does not depend on our intuitively grasping or imagining very large numbers. (shrink)
The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and (...) logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education. (shrink)
Philosophers of science are increasingly arguing for the importance of doing scientifically- and socially-engaged work, suggesting that we need to reduce barriers to extra-disciplinary engagement and broaden our impact. Yet, we currently lack empirical data to inform these discussions, leaving a number of important questions unanswered. How common is it for philosophers of science to engage other communities, and in what ways are they engaging? What barriers are most prevalent when it comes to broadly disseminating one’s work or collaborating (...) with others? To what extent do philosophers of science actually value an engaged approach? Our project addresses this gap in our collective knowledge by providing empirical data regarding the state of philosophy of science today. We report the results of a survey of 299 philosophers of science about their attitudes towards and experiences with engaging those outside the discipline. Our data suggest that a significant majority of philosophers of science think it is important for non-philosophers to read and make use of their work; most are engaging with communities outside the discipline; and many think philosophy of science, as a discipline, has an obligation to ensure it has a broader impact. Interestingly, however, many of these same philosophers believe engaged work is generally undervalued in the discipline. We think these findings call for cautious optimism on the part of those who value engaged work—while there seems to be more interest in engaging other communities than many assume, significant barriers still remain. (shrink)
Pythagoras’s number doctrine had a great effect on the development of science. Number – the key to the highest reality, and such approach allowed Pythagoras to transform mathematics from craft into science, which continues implementation of its project of “digitization of being”. Pythagoras's project underwent considerable transformation, but it only means that the plan in knowledge is often far from result.
A Commonplace of recent philosophy of mind is that intentional states are relations between thinkers and propositions. This thesis-call it the 'Relational Thesis'-does not depend on any specific theory of propositions. One can hold it whether one believes that propositions are Fregean Thoughts, ordered n-tuples of objects and properties or sets of possible worlds. An assumption that all these theories of propositions share is that propositions are abstract objects, without location in space or time...
A god is a cosmic designer-creator. Atheism says the number of gods is 0. But it is hard to defeat the minimal thesis that some possible universe is actualized by some possible god. Monotheists say the number of gods is 1. Yet no degree of perfection can be coherently assigned to any unique god. Lewis says the number of gods is at least the second beth number. Yet polytheists cannot defend an arbitrary plural number of (...) gods. An alternative is that, for every ordinal, there is a god whose perfection is proportional to it. The n -th god actualizes the best universe(s) in the n -th level of an axiological hierarchy of possible universes. Despite its unorthodoxy, ordinal polytheism has many metaphysically attractive features and merits more serious study. (shrink)
A multiverse is comprised of many universes, which quickly leads to the question: How many universes? There are either finitely many or infinitely many universes. The purpose of this paper is to discuss two conceptions of infinite number and their relationship to multiverses. The first conception is the standard Cantorian view. But recent work has suggested a second conception of infinite number, on which infinite numbers behave very much like finite numbers. I will argue that that this second (...) conception of infinite number is the correct one, and analyze what this means for multiverses. (shrink)
All humans share a universal, evolutionarily ancient approximate number system (ANS) that estimates and combines the numbers of objects in sets with ratio-limited precision. Interindividual variability in the acuity of the ANS correlates with mathematical achievement, but the causes of this correlation have never been established. We acquired psychophysical measures of ANS acuity in child and adult members of an indigene group in the Amazon, the Mundurucú, who have a very restricted numerical lexicon and highly variable access to mathematics (...) education. By comparing Mundurucú subjects with and without access to schooling, we found that education significantly enhances the acuity with which sets of concrete objects are estimated. These results indicate that culture and education have an important effect on basic number perception. We hypothesize that symbolic and nonsymbolic numerical thinking mutually enhance one another over the course of mathematics instruction. (shrink)
It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is (...) empirically equivalent to classical mechanics, but uses only finite-information numbers. This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory. Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic. Most physicists straightforwardly supplement classical theory with real numbers to which they attribute physical existence, while most physicists reject Bohmian mechanics as supplemented quantum theory, arguing that Bohmian positions have no physical reality. (shrink)
This paper is about how to aggregate outside opinion. If two experts are on one side of an issue, while three experts are on the other side, what should a non-expert believe? Certainly, the non-expert should take into account more than just the numbers. But which other factors are relevant, and why? According to the view developed here, one important factor is whether the experts should have been expected, in advance, to reach the same conclusion. When the agreement of two (...) (or of twenty) thinkers can be predicted with certainty in advance, their shared belief is worth only as much as one of their beliefs would be worth alone. This expectational model of belief dependence can be applied whether we think in terms of credences or in terms of all-or-nothing beliefs. (shrink)
In this paper I introduce an objection to normative evidentialism about reasons for belief. The objection arises from difficulties that evidentialism has with explaining our reasons for belief in unstable belief contexts with a single fixed point. I consider what other kinds of reasons for belief are relevant in such cases.
Quantity is the first category that Aristotle lists after substance. It has extraordinary epistemological clarity: "2+2=4" is the model of a self-evident and universally known truth. Continuous quantities such as the ratio of circumference to diameter of a circle are as clearly known as discrete ones. The theory that mathematics was "the science of quantity" was once the leading philosophy of mathematics. The article looks at puzzles in the classification and epistemology of quantity.
Context: The infinite has long been an area of philosophical and mathematical investigation. There are many puzzles and paradoxes that involve the infinite. Problem: The goal of this paper is to answer the question: Which objects are the infinite numbers (when order is taken into account)? Though not currently considered a problem, I believe that it is of primary importance to identify properly the infinite numbers. Method: The main method that I employ is conceptual analysis. In particular, I argue that (...) the infinite numbers should be as much like the finite numbers as possible. Results: Using finite numbers as our guide to the infinite numbers, it follows that infinite numbers are of the structure w + (w* + w) a + w*. This same structure also arises when a large finite number is under investigation. Implications: A first implication of the paper is that infinite numbers may be large finite numbers that have not been investigated fully. A second implication is that there is no number of finite numbers. Third, a number of paradoxes of the infinite are resolved. One change that should occur as a result of these findings is that “infinitely many” should refer to structures of the form w + (w* + w) a + w*; in contrast, there are “indefinitely many” natural numbers. Constructivist content: The constructivist perspective of the paper is a form of strict finitism. (shrink)
It's currently fashionable to take Putnamian model theoretic worries seriously for mathematics, but not for discussions of ordinary physical objects and the sciences. But I will argue that (under certain mild assumptions) merely securing determinate reference to physical possibility suffices to rule out nonstandard models of our talk of numbers. So anyone who accepts realist reference to physical possibility should not reject reference to the standard model of the natural numbers on Putnamian model theoretic grounds.
Faced with the choice between saving one person and saving two others, what should we do? It seems intuitively plausible that we ought to save the two, and many forms of consequentialists offer a straightforward rationale for the intuition by appealing to interpersonal aggregation. But still many other philosophers attempt to provide a justification for the duty to save the greater number without combining utilities or claims of separate individuals. I argue against one such attempt proposed by Iwao Hirose. (...) Despite being consequentialist, his argument is aggregation-free since it relies on a non-aggregative value judgement method, instead of interpersonal aggregation, to establish that (other things being equal) a state of affairs is better when more people survive therein. I do not take issue with its consequentialist element; rather, I claim that there is no good reason to adopt the method in question, and thus no good reason to be moved by his argument overall. What we are in search of is not merely a logically possible method that can produce the conclusion that we already want, but one that we have good reason to adopt. Hirose's argument elegantly demonstrates how it could possibly be true that it is right to save the greater number; but it fails to show that we have reason to believe so - even when we do not combine the interests of different individuals. (shrink)
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 Numbers (...) Skepticism goes too far in rejecting the claim that you ought to save the greater number. Besides the prima facie implausibility of Numbers Skepticism, Michael Otsuka has developed an intriguing argument against this position. Otsuka argues that Numbers Skepticism, in conjunction with an independently plausible moral principle, leads to inconsistent choices regarding what ought to be done in certain circumstances. This inconsistency in turn provides us with a good reason to reject Numbers Skepticism. Kirsten Meyer offers a notable challenge to Otsuka’s argument. I argue that Meyer’s challenge can be met, and then offer my own reasons for rejecting Otsuka’s argument. In light of these criticisms, I then develop an improved, yet structurally similar argument to Otsuka’s argument. I argue for the slightly different conclusion that the view proposed by John Taurek that ‘the numbers don’t count’ leads to inconsistent choices, which in turn provides us with a good reason to reject Taurek’s position. (shrink)
Hartry Field has argued that mathematical realism is epistemologically problematic, because the realist is unable to explain the supposed reliability of our mathematical beliefs. In some of his discussions of this point, Field backs up his argument by saying that our purely mathematical beliefs do not ‘counterfactually depend on the facts’. I argue that counterfactual dependence is irrelevant in this context; it does nothing to bolster Field's argument.
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.
I begin with a personal confession. Philosophical discussions of existence have always bored me. When they occur, my eyes glaze over and my attention falters. Basically ontological questions often seem best decided by banging on the table--rocks exist, fairies do not. Argument can appear long-winded and miss the point. Sometimes a quick distinction resolves any apparent difficulty. Does a falling tree in an earless forest make noise, ie does the noise exist? Well, if noise means that an ear must be (...) there to hear it, then the answer to the question is evidently "no." But if noise means that, if there were (counterfactually) someone there, then he would hear it, then just as obviously, the answer becomes "yes.". (shrink)
On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique—the arguments from congruency, confounds, and imprecision—and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes (...) for number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind(s) of number being represented. In response, we propose that the ANS represents not only natural numbers (e.g. 7), but also non-natural rational numbers (e.g. 3.5). It does not represent irrational numbers (e.g. 2), however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research. (shrink)
This article accomplishes two goals. First, the paper clarifies Edmund Husserl’s investigation of the historical inception of the number system from his early works, Philosophy of Arithmetic and, “On the Logic of Signs (Semiotic)”. The article explores Husserl’s analysis of five historical developmental stages, which culminated in our ancestor’s ability to employ and enumerate with number signs. Second, the article reveals how Husserl’s conclusions about the history of the number system from his early works opens up a (...) fusion point with his investigations from his mature texts, The Crisis of the European Sciences and “The Origin of Geometry”. On the one hand, the essay shows that Husserl’s methodology was similar, as he sought in both his early and late writings to uncover the essence of the history of the formal sciences and was not executing mere intellectual history. On the other hand, the article discloses that Husserl’s insights from both time periods are strikingly analogous. Already in his early texts, Husserl saw that the sciences emerged from pre-theoretical experiences of the world and that the sciences are the result of a historical process, which involves the psychic activities of past individuals and the maintaining of discoveries over time by intersubjective communities. I conclude by showing how, in light of the analysis of this paper, we can rethink the evolution of Husserl’s philosophy. (shrink)
A moderately risk averse person may turn down a 50/50 gamble that either results in her winning $200 or losing $100. Such behaviour seems rational if, for instance, the pain of losing $100 is felt more strongly than the joy of winning $200. The aim of this paper is to examine an influential argument that some have interpreted as showing that such moderate risk aversion is irrational. After presenting an axiomatic argument that I take to be the strongest case for (...) the claim that moderate risk aversion is irrational, I show that it essentially depends on an assumption that those who think that risk aversion can be rational should be skeptical of. Hence, I conclude that risk aversion need not be irrational. (shrink)
The characterization of early token-based accounting using a concrete concept of number, later numerical notations an abstract one, has become well entrenched in the literature. After reviewing its history and assumptions, this article challenges the abstract–concrete distinction, presenting an alternative view of change in Ancient Near Eastern number concepts, wherein numbers are abstract from their inception and materially bound when most elaborated. The alternative draws on the chronological sequence of material counting technologies used in the Ancient Near East—fingers, (...) tallies, tokens, and numerical notations—as reconstructed through archaeological and textual evidence and as interpreted through Material Engagement Theory, an extended-mind framework in which materiality plays an active role (Malafouris 2013). (shrink)
Numbers without Science opposes the Quine-Putnam indispensability argument, seeking to undermine the argument and reduce its profound influence. Philosophers rely on indispensability to justify mathematical knowledge using only empiricist epistemology. I argue that we need an independent account of our knowledge of mathematics. The indispensability argument, in broad form, consists of two premises. The major premise alleges that we are committed to mathematical objects if science requires them. The minor premise alleges that science in fact requires mathematical objects. The most (...) common rejection of the argument denies its minor premise by introducing scientific theories which do not refer to mathematical objects. Hartry Field has shown how we can reformulate some physical theories without mathematical commitments. I argue that Field’s preference for intrinsic explanation, which underlies his reformulation, is ill-motivated, and that his resultant fictionalism suffers unacceptable consequences. I attack the major premise instead. I argue that Quine provides a mistaken criterion for ontic commitment. Our uses of mathematics in scientific theory are instrumental and do not commit us to mathematical objects. Furthermore, even if we accept Quine’s criterion for ontic commitment, the indispensability argument justifies only an anemic version of mathematics, and does not yield traditional mathematical objects. The first two chapters of the dissertation develop these results for Quine’s indispensability argument. In the third chapter, I apply my findings to other contemporary indispensabilists, specifically the structuralists Michael Resnik and Stewart Shapiro. In the fourth chapter, I show that indispensability arguments which do not rely on Quine’s holism, like that of Putnam, are even less successful. Also in Chapter 4, I show how Putnam’s work in the philosophy of mathematics is unified around the indispensability argument. In the last chapter of the dissertation, I conclude that we need an account of mathematical knowledge which does not appeal to empirical science and which does not succumb to mysticism and speculation. Briefly, my strategy is to argue that any defensible solution to the demarcation problem of separating good scientific theories from bad ones will find mathematics to be good, if not empirical, science. (shrink)
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 numerosity expressed by (...) the number and its metrics. It is shown that while many-number systems involve a compositionality of units, sets and sets composed of units, few-number languages, such as Mundurucu, do not have access to sets composed of units in the usual way. The nonconfigurational character of the Mundurucu language, which is related to a property for which we coin the term 'low compositionality power', accounts for this and explains the curious fact that Mundurucus make use of marked one-to-one correspondence strategies in order to overcome the limitations of the core system at the perceptual/motor interface of the language faculty. We develop an analysis of a particular construction, parallel numbers, which has not been studied before, elucidating the whole system. This analysis, we argue, sheds new light on classical philosophical, psychological and linguistic debates about numbers and numerals and their relation to language, and more particularly, sheds light on few-number languages. (shrink)
This paper demonstrates that Edmund Husserl’s frequently overlooked 1890 manuscript, “On the Logic of Signs,” when closely investigated, reveals itself to be the hermeneutical touchstone for his seminal 1891 Philosophy of Arithmetic. As the former comprises Husserl’s earliest attempt to account for all of the different kinds of signitive experience, his conclusions there can be directly applied to the latter, which is focused on one particular type of sign; namely, number signs. Husserl’s 1890 descriptions of motivating and replacing signs (...) will be respectively employed to clarify his 1891 understanding of the authentic and inauthentic presentations of numbers via number signs. Moreover, his schematic classification of replacement-signs in Semiotic will illuminate the reasons why he believed the number system to be necessary for the operation of replacing number signs. (shrink)
Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers can be generated by (...) a successor function and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Munduruc (an Amazonian language), and young Western children (3-4 years old) understand these fundamental properties of numbers. (shrink)
What are numbers, and where do they come from? A novel answer to these timeless questions is proposed by cognitive archaeologist Karenleigh A. Overmann, based on her groundbreaking study of material devices used for counting in the Ancient Near East—fingers, tallies, tokens, and numerical notations—as interpreted through the latest neuropsychological insights into human numeracy and literacy. The result, a unique synthesis of interdisciplinary data, outlines how number concepts would have been realized in a pristine original condition to develop into (...) one of the ancient world’s greatest mathematical traditions, a foundation for mathematical thinking today. In this view, numbers are abstract from their inception and materially bound at their most elaborated. The research updates historical work on Neolithic tokens and interpretations of Mesopotamian numbers, challenging several longstanding assumptions about numbers in the process. The insights generated are also applied to the role of materiality in human cognition more generally, including how concepts become distributed across and independent of the material forms used for their representation and manipulation; how societies comprised of average individuals use material structures to create elaborated systems of numeracy and literacy; and the differences between thinking through and thinking about materiality. (shrink)
Number is a major object in mathematics. Mathematics is a discipline which studies the properties of a number. The object is expressible by mathematical language, which has been devised more rigorously than natural language. However, the language is not thoroughly free from natural language. Countability of natural number is also originated from natural language. It is necessary to understand how language leads a number into mathematics, its’ main playground.
Since independence, at least 28 African countries have legalized some form of gambling. Yet a range of informal gambling activities have also flourished, often provoking widespread public concern about the negative social and economic impact of unregulated gambling on poor communities. This article addresses an illegal South African numbers game called fahfee. Drawing on interviews with players, operators, and regulatory officials, this article explores two aspects of this game. First, it explores the lives of both players and runners, as well (...) as the clandestine world of the Chinese operators who control the game. Second, the article examines the subjective motivations and aspirations of players, and asks why they continue to play, despite the fact that their aggregate losses easily outstrip their aggregate gains. In contrast with those who reduce its appeal simply to the pursuit of wealth, I conclude that, for the (mostly) black, elderly, working class women who play fahfee several times a week, the associated trade-off—regular, small losses, versus the social enjoyment of playing and the prospect of occasional but realistic windfalls—takes on a whole new meaning, and preferences for relatively lumpy rather than steady consumption streams help explain the continued attraction of fahfee. This reinforces the need to understand players’ own accounts of gambling utility rather than simply to moralistically condemn gambling or to dismiss gamblers behaviour as irrational. (shrink)
This paper discusses the "numbers problem," the problem of explaining why you should save more people rather than fewer when forced to choose. Existing non-consequentialist approaches to the problem appeal to fairness to explain why. I argue that this is a mistake and that we can give a more satisfying answer by appealing to requirements of charity or beneficence.
The present paper deals with the ontological status of numbers and considers Frege ́s proposal in Grundlagen upon the background of the Post-Kantian semantic turn in analytical philosophy. Through a more systematic study of his philosophical premises, it comes to unearth a first level paradox that would unset earlier still than it was exposed by Russell. It then studies an alternative path, that departin1g from Frege’s initial premises, drives to a conception of numbers as synthetic a priori in a more (...) Kantian sense. On this basis, it tentatively explores a possible derivation of basic logical rules on their behalf, suggesting a more rudimentary basis to inferential thinking, which supports reconsidering the difference between logical thinking and AI. Finally, it reflects upon the contributions of this approach to the problem of the a priori. (shrink)
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