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.

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)

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)

On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system, 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 of number being represented. In response, we propose that the ANS represents not only natural numbers, but also non-natural rational numbers. It does not represent irrational numbers, 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)

One of the most important abilities we have as humans is the ability to think about number. In this chapter, we examine the question of whether there is an essential connection between language and number. We provide a careful examination of two prominent theories according to which concepts of the positive integers are dependent on language. The first of these claims that language creates the positive integers on the basis of an innate capacity to represent real numbers. The (...) second claims that language’s function is to integrate contents from modules that humans share with other animals. We argue that neither model is successful. (shrink)

Formal epistemologists often claim that our credences should be representable by a probability function. Complete probabilistic coherence, however, is only possible for ideal agents, raising the question of how this requirement relates to our everyday judgments concerning rationality. One possible answer is that being rational is a contextual matter, that the standards for rationality change along with the situation. Just like who counts as tall changes depending on whether we are considering toddlers or basketball players, perhaps what counts as rational (...) shifts according to whether we are considering ideal agents or creatures more like ourselves. Even though a number of formal epistemologists have endorsed this type of solution, I will argue that there is no way to spell out this contextual account that can make sense of our everyday judgments about rationality. Those who defend probabilistic coherence requirements will need an alternative account of the relationship between real and ideal rationality. (shrink)

Once we have distinguished between beauty and aesthetic value, we are faced with the question of whether beauty is a thing of value in itself. A number of theorists have suggested that the answer might be no. They have thought that the pursuit of beauty is just the indulgence of one particular taste: a taste that has, for contingent historical reasons, been privileged. This paper attempts to resist a line of thought that leads to that conclusion. It does so by (...) arguing that there really are objective facts about beauty. To do this, the paper draws distinctions between objectivity and subjectivity, and between realism and anti-realism. It argues that, regarding attributions of beauty, we should be realists and objectivists. This is shown to be compatible with taking the semantic content of such attributions to vary between contexts. This form of context sensitivity is able to account of those features of beauty-attributions that have been taken as evidence for its subjectivity. (shrink)

Arguments against the Russellian theory of definite descriptions based on cases that involve failures of uniqueness are a recurrent theme in the relevant literature. In this paper, I discuss a number of such arguments, from Strawson (1950), Ramachandran (1993) and Szabo (2005). I argue that the Russellian has resources to account for these data by deploying a variety of mechanisms of quantifier domain restrictions. Finally, I present a case that is more problematic for the Russellian. While the previous cases all (...) involve referential uses of descriptions (or some variations of such uses), the most effective objection to the uniqueness condition draws on genuine attributive uses. (shrink)

We provide an intuitive motivation for the hyperreal numbers via electoral axioms. We do so in the form of a Socratic dialogue, in which Protagoras suggests replacing big-oh complexity classes by real numbers, and Socrates asks some troubling questions about what would happen if one tried to do that. The dialogue is followed by an appendix containing additional commentary and a more formal proof.

Reading Kripke's "Wittgenstein on Rules and Private Language", at first one can easily get confused about his claim that the problem discovered was a sort of ontological skepticism. Contrary to the opinion of a great number of contemporary philosophers who hold that rule-following brings up merely epistemological problems I will argue that the scepticism presented by Kripke really is ontological because it is concerned with the exclusion of certain facts. The first section in this paper is dedicated to a presentation (...) of Kripke's paradox with a clarification of the position of "plus/quus-talk" in the argument. Section two is engaged in one of his classical direct solutions: the dispositional theory which will serve as a preparation for the last section. Section three is concerned with Kripke's solution to the skeptical problem, ending with the question of whether he is giving a real solution. In section four, I will try to give an answer to those questions, distinguishing between two different versions of the problem given by the paradox; a wrong one and a correct one. Readers who are really fed up with the sceptical problem and its sceptical solution can skip section one, two and three, concentrating instead on my own argument for the real nature of the problem. In section five, I will pick up on some ideas from Norwich, who tries to give a "straight solution" to the paradox reanimating some weaker version of a dispositional theory of meaning. I will argue that Horwich's solution is misleading because he aims at the wrong version of the paradox. (shrink)

When consumers choose to abstain from purchasing meat, they face some uncertainty about whether their decisions will have an impact on the number of animals raised and killed. Consequentialists have argued that this uncertainty should not dissuade consumers from a vegetarian diet because the “expected” impact, or average impact, will be predictable. Recently, however, critics have argued that the expected marginal impact of a consumer change is likely to be much smaller or more radically unpredictable than previously thought. This objection (...) to the consequentialist case for vegetarianism is known as the “causal inefficacy” (or “causal impotence”) objection. In this paper, we argue that the inefficacy objection fails. First, we summarize the contours of the objection and the standard “expected impact” response to it. Second, we examine and rebut two contemporary attempts (by Mark Budolfson and Ted Warfield) to defeat the expected impact reply through alleged demonstrations of the inefficacy of abstaining from meat consumption. Third, we argue that there are good reasons to believe that single individual consumers—not just individual consumers taken as an aggregate—really do make a positive difference when they choose to abstain from meat consumption. Our case rests on three economic observations: (i) animal producers operate in a highly competitive environment, (ii) complex supply chains efficiently communicate some information about product demand, and (iii) consumers of plant-based meat alternatives have positive consumption spillover effects on other consumers. (shrink)

This paper develops and explores a new framework for theorizing about the measurement and aggregation of well-being. It is a qualitative variation on the framework of social welfare functionals developed by Amartya Sen. In Sen’s framework, a social or overall betterness ordering is assigned to each profile of real-valued utility functions. In the qualitative framework developed here, numerical utilities are replaced by the properties they are supposed to represent. This makes it possible to characterize the measurability and interpersonal comparability (...) of well-being directly, without the use of invariance conditions, and to distinguish between real changes in well-being and merely representational changes in the unit of measurement. The qualitative framework is shown to have important implications for a range of issues in axiology and social choice theory, including the characterization of welfarism, axiomatic derivations of utilitarianism, the meaningfulness of prioritarianism, the informational requirements of variable-population ethics, the impossibility theorems of Arrow and others, and the metaphysics of value. (shrink)

According to Karl Popper, science cannot verify its theories empirically, but it can falsify them, and that suffices to account for scientific progress. For Popper, a law or theory remains a pure conjecture, probability equal to zero, however massively corroborated empirically it may be. But it does just seem to be the case that science does verify empirically laws and theories. We trust our lives to such verifications when we fly in aeroplanes, cross bridges and take modern medicines. We can (...) do some justice to this apparent capacity of science to verify if we make a number of improvements to Popper’s philosophy of science. The key step is to recognize that physics, in accepting unified theories only, thereby makes a big metaphysical assumption about the nature of the universe. The outcome is a conception of scientific method which facilitates the criticism and improvement of metaphysical assumptions of physics. This view provides, not verification, but a perfect simulacrum of verification indistinguishable from the real thing. (shrink)

While I was working about some basic physical phenomena, I discovered some geometric relations that also interest mathematics. In this work, I applied the rules I have been proven to P=NP? problem over impossibility of perpendicularity in the universe. It also brings out extremely interesting results out like imaginary numbers which are known as real numbers currently. Also it seems that Euclidean Geometry is impossible. The actual geometry is Riemann Geometry and complex numbers are real.

The nature of time is perceived by intellectuals variedly. An attempt is made in this paper to reconcile such varied views in the light of the Upanishads and related Indian spiritual and philosophical texts. The complex analysis of modern mathematics is used to represent the nature and presentation physical and psychological times so differentiated. Also the relation between time and energy is probed using uncertainty relations, forms of energy and phases of matter. Implications to time-dependent Schrodinger wave equation and uncertainty (...) principle are hinted. (shrink)

In ‘The Train Paradox’, I argued that sequential random selections from the natural numbers would grow through time. I used this claim to present a paradox. In response to this proposed paradox, Jon Pérez Laraudogoitia has argued that random selections from the natural numbers do not grow through time. In this paper, I defend and expand on the argument that random selections from the natural numbers grow through time. I also situate this growth of random selections in (...) the context of my overall work on infinite number, which involves two main claims: 1) infinite numbers, properly understood, are the infinite natural numbers in a nonstandard model of the reals, and 2) ω is potentially infinite. (shrink)

The philosophy of mathematics has largely abandoned foundational studies, but is still fixated on theorem proving, logic and number theory, and on whether mathematical knowledge is certain. That is not what mathematics looks like to, say, a knot theorist or an industrial mathematical modeller. The "computer revolution" shows that mathematics is a much more direct study of the world, especially its structural aspects.

When mathematicians think of the philosophy of mathematics, they probably think of endless debates about what numbers are and whether they exist. Since plenty of mathematical progress continues to be made without taking a stance on either of these questions, mathematicians feel confident they can work without much regard for philosophical reflections. In his sharp–toned, sprawling book, David Corfield acknowledges the irrelevance of much contemporary philosophy of mathematics to current mathematical practice, and proposes reforming the subject accordingly.

This paper is on Aristotle's conception of the continuum. It is argued that although Aristotle did not have the modern conception of real numbers, his account of the continuum does mirror the topology of the real number continuum in modern mathematics especially as seen in the work of Georg Cantor. Some differences are noted, particularly as regards Aristotle's conception of number and the modern conception of real numbers. The issue of whether Aristotle had the notion (...) of open versus closed intervals is discussed. Finally, it is suggested that one reason there is a common structure between Aristotle's account of the continuum and that found in Cantor's definition of the real number continuum is that our intuitions about the continuum have their source in the experience of the real spatiotemporal world. A plea is made to consider Aristotle's abstractionist philosophy of mathematics anew. (shrink)

In a recent article, Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably infinite set. He rejects the conclusion of Cantor’s diagonal argument for the higher, non-denumerable infinity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only ‘well-defined’ real numbers as proper objects of (...) study. In practice, this means excluding as inadmissible all those real numbers whose decimal expansions cannot be calculated in as much detail as one would like by some rule. We argue against Ormell that the classical realist account of the continuum has explanatory power in mathematics and should be accepted, much in the same way that "dark matter" is posited by physicists to explain observations in cosmology. In effect, the indefinable real numbers are like the "dark matter" of real analysis. (shrink)

It is argued that colour name strategy, object name strategy, and chunking strategy in memory are all aspects of the same general phenomena, called stereotyping, and this in turn is an example of a know-how representation. Such representations are argued to have their origin in a principle called the minimum duplication of resources. For most the subsequent discussions existence of colour name strategy suffices. It is pointed out that the BerlinA- KayA universal partial ordering of colours and the frequency of (...) traffic accidents classified by colour are surprisingly similar; a detailed analysis is not carried out as the specific colours recorded are not identical. Some consequences of the existence of a name strategy for the philosophy of language and mathematics are discussed: specifically it is argued that in accounts of truth and meaning it is necessary throughout to use real numbers as opposed to bi-valent quantities; and also that the concomitant label associated with sentences should not be of unconditional truth, but rather several real-valued quantities associated with visual communication. The implication of real-valued truth quantities is that the Continuum Hypothesis of pure mathematics is side-stepped, because real valued quantities occur ab initio. The existence of name strategy shows that thought/sememes and talk/phonemes can be separate, and this vindicates the assumption of thought occurring before talk used in psycho-linguistic speech production models. (shrink)

The concept of ‘ideas’ plays central role in philosophy. The genesis of the idea of continuity and its essential role in intellectual history have been analyzed in this research. The main question of this research is how the idea of continuity came to the human cognitive system. In this context, we analyzed the epistemological function of this idea. In intellectual history, the idea of continuity was first introduced by Leibniz. After him, this idea, as a paradigm, formed the base of (...) several fundamental scientific conceptions. This idea also allowed mathematicians to justify the nature of real numbers, which was one of central questions and intellectual discussions in the history of mathematics. For this reason, we analyzed how Dedekind’s continuity idea was used to this justification. As a result, it can be said that several fundamental conceptions in intellectual history, philosophy and mathematics cannot arise without existence of the idea of continuity. However, this idea is neither a purely philosophical nor a mathematical idea. This is an interdisciplinary concept. For this reason, we call and classify it as mathematical and philosophical invariance. (shrink)

This paper suggests that time could have a much richer mathematical structure than that of the real numbers. Clark & Read (1984) argue that a hypertask (uncountably many tasks done in a finite length of time) cannot be performed. Assuming that time takes values in the real numbers, we give a trivial proof of this. If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is (...) an ultratask (a sequence which includes one task done for each ordinal number—thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible. (shrink)

We define a notion of the intelligence level of an idealized mechanical knowing agent. This is motivated by efforts within artificial intelligence research to define real-number intelligence levels of compli- cated intelligent systems. Our agents are more idealized, which allows us to define a much simpler measure of intelligence level for them. In short, we define the intelligence level of a mechanical knowing agent to be the supremum of the computable ordinals that have codes the agent knows to be (...) codes of computable ordinals. We prove that if one agent knows certain things about another agent, then the former necessarily has a higher intelligence level than the latter. This allows our intelligence no- tion to serve as a stepping stone to obtain results which, by themselves, are not stated in terms of our intelligence notion (results of potential in- terest even to readers totally skeptical that our notion correctly captures intelligence). As an application, we argue that these results comprise evidence against the possibility of intelligence explosion (that is, the no- tion that sufficiently intelligent machines will eventually be capable of designing even more intelligent machines, which can then design even more intelligent machines, and so on). (shrink)

In this ninth book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, – in email messages to research colleagues, or replies, and personal notes, some handwritten on the planes to, and from international conferences, about topics on Neutrosophy and its applications, such as: Neutrosophic Bipolar Set, Linguistic Neutrosophic Set, Neutrosophic Resonance Frequency, n-ary HyperAlgebra, n-ary NeutroHyperAlgebra, n-ary AntiHyperAlgebra, Plithogenic Crisp Graph, Plithogenic Fuzzy Graph, Plithogenic Intuitionistic Fuzzy Graph, Plithogenic Neutrosophic Graph, Plithogenic (...) Real Number Graph, Plithogenic Complex Number Graph, Plithogenic Neutrosophic Number Graph, and many more. -/- Exchanging ideas with: Tareq Al-Shami, Riad Khidr Al-hamido, Mohammad Akram, B. De Baets, Robert Neil Boyd, Said Broumi, Terman Frometa-Castillo, Yilmaz Ceven, D. Dubois, Harish Garg, L. Godo, Erik Gonzalez, Yanhui Guo, Mohammad Hamidi, E. Hüllermeier, W. B. Vasantha Kandasamy, Mary Jansi, Nivetha Martin, Mani Parimala, Akbar Rezaei, Bouzina Salah, Christy Vincent, Jun Ye.‬‬‬. (shrink)

After generalizing the Archimedean property of real numbers in such a way as to make it adaptable to non-numeric structures, we demonstrate that the real numbers cannot be used to accurately measure non-Archimedean structures. We argue that, since an agent with Artificial General Intelligence (AGI) should have no problem engaging in tasks that inherently involve non-Archimedean rewards, and since traditional reinforcement learning rewards are real numbers, therefore traditional reinforcement learning probably will not lead to (...) AGI. We indicate two possible ways traditional reinforcement learning could be altered to remove this roadblock. (shrink)

Several areas of welfare economics seek to evaluate states of affairs as a function of interpersonally comparable individual utilities. The aim is to map each state of affairs onto a vector of individual utilities, and then to produce an ordering of these vectors that can be represented by a mathematical function assigning a real number to each. When this approach is used in intertemporal contexts, a central theoretical question concerns the evaluative weight to be applied to utility coming at (...) different times. This question concerns the rate of pure time preference, which is one key determinant of the social discount rate. This article argues that the standard philosophical account of pure time preference is mistaken, because it ascribes to economists a methodological commitment they need not, and often do not, accept. This in turn undercuts the most common philosophical objection to pure time preference, which traces at least to Rawls’s A Theory of Justice. The article then evaluates three further objections to pure time preference, concluding that it might still be defensible under certain circumstances. The article closes by articulating a final argument that is suggested by the “Social, Economic and Ethical Concepts and Methods” chapter of the most recent Intergovernmental Panel on Climate Change report. If this further argument is sound, it would constitute a decisive objection to pure time preference as it currently figures in much intertemporal welfare economics. (shrink)

We prove a representation theorem for preference relations over countably infinite lotteries that satisfy a generalized form of the Independence axiom, without assuming Continuity. The representing space consists of lexicographically ordered transfinite sequences of bounded real numbers. This result is generalized to preference orders on abstract superconvex spaces.

What the world needs now is another theory of vagueness. Not because the old theories are useless. Quite the contrary, the old theories provide many of the materials we need to construct the truest theory of vagueness ever seen. The theory shall be similar in motivation to supervaluationism, but more akin to many-valued theories in conceptualisation. What I take from the many-valued theories is the idea that some sentences can be truer than others. But I say very different things to (...) the ordering over sentences this relation generates. I say it is not a linear ordering, so it cannot be represented by the real numbers. I also argue that since there is higher-order vagueness, any mapping between sentences and mathematical objects is bound to be inappropriate. This is no cause for regret; we can say all we want to say by using the comparative truer than without mapping it onto some mathematical objects. From supervaluationism I take the idea that we can keep classical logic without keeping the familiar bivalent semantics for classical logic. But my preservation of classical logic is more comprehensive than is normally permitted by supervaluationism, for I preserve classical inference rules as well as classical sequents. And I do this without relying on the concept of acceptable precisifications as an unexplained explainer. The world does not need another guide to varieties of theories of vagueness, especially since Timothy Williamson (1994) and Rosanna Keefe (2000) have already provided quite good guides. I assume throughout familiarity with popular theories of vagueness. (shrink)

In formal epistemology, we use mathematical methods to explore the questions of epistemology and rational choice. What can we know? What should we believe and how strongly? How should we act based on our beliefs and values? We begin by modelling phenomena like knowledge, belief, and desire using mathematical machinery, just as a biologist might model the fluctuations of a pair of competing populations, or a physicist might model the turbulence of a fluid passing through a small aperture. Then, we (...) explore, discover, and justify the laws governing those phenomena, using the precision that mathematical machinery affords. For example, we might represent a person by the strengths of their beliefs, and we might measure these using real numbers, which we call credences. Having done this, we might ask what the norms are that govern that person when we represent them in that way. How should those credences hang together? How should the credences change in response to evidence? And how should those credences guide the person’s actions? This is the approach of the first six chapters of this handbook. In the second half, we consider different representations—the set of propositions a person believes; their ranking of propositions by their plausibility. And in each case we ask again what the norms are that govern a person so represented. Or, we might represent them as having both credences and full beliefs, and then ask how those two representations should interact with one another. This handbook is incomplete, as such ventures often are. Formal epistemology is a much wider topic than we present here. One omission, for instance, is social epistemology, where we consider not only individual believers but also the epistemic aspects of their place in a social world. Michael Caie’s entry on doxastic logic touches on one part of this topic, but there is much more. Relatedly, there is no entry on epistemic logic, nor any on knowledge more generally. There are still more gaps. These omissions should not be taken as ideological choices. This material is missing, not because it is any less valuable or interesting, but because we v failed to secure it in time. Rather than delay publication further, we chose to go ahead with what is already a substantial collection. We anticipate a further volume in the future that will cover more ground. Why an open access handbook on this topic? A number of reasons. The topics covered here are large and complex and need the space allowed by the sort of 50 page treatment that many of the authors give. We also wanted to show that, using free and open software, one can overcome a major hurdle facing open access publishing, even on topics with complex typesetting needs. With the right software, one can produce attractive, clear publications at reasonably low cost. Indeed this handbook was created on a budget of exactly £0 (≈ $0). Our thanks to PhilPapers for serving as publisher, and to the authors: we are enormously grateful for the effort they put into their entries. (shrink)

Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are our internally imagined objects, some of which, at least approximately, we can realize or represent; (ii) mathematical truths (...) are not truths about the external world but specifications (formulations) of mathematical conceptions; (iii) mathematics is first and foremost our imagined tool by which, with certain assumptions about its applicability, we explore nature and synthesize our rational cognition of it. (shrink)

There is no uniquely standard concept of an effectively decidable set of real numbers or real n-tuples. Here we consider three notions: decidability up to measure zero [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382], which we abbreviate d.m.z.; recursive approximability [or r.a.; K.-I. Ko, Complexity Theory of Real Functions, Birkhäuser, Boston, 1991]; and decidability ignoring boundaries [d.i.b.; W.C. Myrvold, The decision problem (...) for entanglement, in: R.S. Cohen et al. (Eds.), Potentiality, Entanglement, and Passion-at-a-Distance: Quantum Mechanical Studies fo Abner Shimony, Vol. 2, Kluwer Academic Publishers, Great Britain, 1997, pp. 177–190]. Unlike some others in the literature, these notions apply not only to certain nice sets, but to general sets in Rn and other appropriate spaces. We consider some motivations for these concepts and the logical relations between them. It has been argued that d.m.z. is especially appropriate for physical applications, and on Rn with the standard measure, it is strictly stronger than r.a. [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382]. Here we show that this is the only implication that holds among our three decidabilities in that setting. Under arbitrary measures, even this implication fails. Yet for intervals of non-zero length, and more generally, convex sets of non-zero measure, the three concepts are equivalent. (shrink)

According to Quine’s indispensability argument, we ought to believe in just those mathematical entities that we quantify over in our best scientific theories. Quine’s criterion of ontological commitment is part of the standard indispensability argument. However, we suggest that a new indispensability argument can be run using Armstrong’s criterion of ontological commitment rather than Quine’s. According to Armstrong’s criterion, ‘to be is to be a truthmaker (or part of one)’. We supplement this criterion with our own brand of metaphysics, 'Aristotelian (...) (...) realism', in order to identify the truthmakers of mathematics. We consider in particular as a case study the indispensability to physics of real analysis (the theory of the real numbers). We conclude that it is possible to run an indispensability argument without Quinean baggage. (shrink)

In this paper a solution of Whitehead’s problem is presented: Starting with a purely mereological system of regions a topological space is constructed such that the class of regions is isomorphic to the Boolean lattice of regular open sets of that space. This construction may be considered as a generalized completion in analogy to the well-known Dedekind completion of the rational numbers yielding the real numbers . The argument of the paper relies on the theories of continuous (...) lattices and “pointless” topology.

. (shrink)

This paper examines the mechanisms involved in the interpretation of utterances that are both metaphorical and ironical. For example, when uttering 'He's a real number-cruncher' about a total illiterate in maths, the speaker uses a metaphor with an ironic intent. I argue that in such cases both logically and psychologically, the metaphor is prior to irony. I hold that the phenomenon is then one of ironic metaphor, which puts a metaphorical meaning to ironic use, rather than an irony used (...) metaphorically (§1). This result is then used to argue for the claim that in metaphor, it is metaphorical, not literal, meaning that determines the utterance‘s truth conditions. Gricean accounts, which exclude metaphorical meaning from truth conditional content and rely entirely on conversational implicature, are seen to be unsatisfactory. Five contextualist arguments are briefly discussed to the conclusion that metaphorical content is part of truth-conditional content, rather than implicated (§2). (shrink)

In this paper, I present a puzzle involving special relativity and the random selection of real numbers. In a manner to be specified, darts thrown later hit reals further into a fixed well-ordering than darts thrown earlier. Special relativity is then invoked to create a puzzle. I consider four ways of responding to this puzzle which, I suggest, fail. I then propose a resolution to the puzzle, which relies on the distinction between the potential infinite and the actual (...) infinite. I suggest that certain structures, such as a well-ordering of the reals, or the natural numbers, are examples of the potential infinite, whereas infinite integers in a nonstandard model of arithmetic are examples of the actual infinite. (shrink)

Using Riemann’s Rearrangement Theorem, Øystein Linnebo (2020) argues that, if it were possible to apply an infinite positive weight and an infinite negative weight to a working scale, the resulting net weight could end up being any real number, depending on the procedure by which these weights are applied. Appealing to the First Postulate of Archimedes’ treatise on balance, I argue instead that the scale would always read 0 kg. Along the way, we stop to consider an infinitely jittery (...) flea, an infinitely protracted border conflict, and an infinitely electric glass rod. (shrink)

I explore some ways in which one might base an account of the fundamental metaphysics of geometry on the mathematical theory of Linear Structures recently developed by Tim Maudlin (2010). Having considered some of the challenges facing this approach, Idevelop an alternative approach, according to which the fundamental ontology includes concrete entities structurally isomorphic to functions from space-time points to real numbers.

In a calculation involving imaginary numbers, we begin with real numbers that represent concrete measures and we end up with numbers that are equally real, but in the course of the operation we find ourselves walking “as if on a bridge that stands on no piles”. How is that possible? How does that work? And what is involved in the as-if stance that this metaphor introduces so beautifully? These are questions that bother Törless deeply. And (...) that Törless is bothered by such questions is a central question for any reader of Törless. Here I offer my interpretation, along with a reconstruction of the philosophical intuition that lies behind it all. (shrink)

Years ago, when I was an undergraduate math major at the University of Wyoming, I came across an interesting book in our library. It was a book of counterexamples t o propositions in real analysis (the mathematics of the real numbers). Mathematicians work more or less like the rest of us. They consider propositions. If one seems to them to be plausibly true, then they set about to prove it, to establish the proposition as a theorem. Instead (...) o f setting out to prove propositions, the psychologists, neuroscientists, and other empirical types among us, set out to show that a proposition is supported by the data, and that it is the best such proposition so supported. The philosophers among us, when they are not causing trouble by arguing that AI is a dead end or that cognitive science can get along without representations, work pretty much like the mathematicians: we set out to prove certain propositions true on the basis of logic, first principles, plausible assumptions, and others' data. But, back to the book of real analysis counterexamples. If some mathematician happened t o think that some proposition about continuity, say, was plausibly true, he or she would then set out to prove it. If the proposition was in fact not a theorem, then a lot of precious time would be wasted trying to prove it. Wouldn't it be great to have a book that listed plausibly true propositions that were in fact not true, and listed with each such proposition a counterexample to it? Of course it would. (shrink)

We attribute three major insights to Hegel: first, an understanding of the real numbers as the paradigmatic kind of number ; second, a recognition that a quantitative relation has three elements, which is embedded in his conception of measure; and third, a recognition of the phenomenon of divergence of measures such as in second-order or continuous phase transitions in which correlation length diverges. For ease of exposition, we will refer to these three insights as the R First Theory, (...) Tripartite Relations, and Divergence of Measures. Given the constraints of space, we emphasize the first and the third in this paper. (shrink)

The objective of this document is to present three introductory notes on set theory: The first note presents an overview of this discipline from its origins to the present, in the second note some considerations are made about the evaluation of reasoning applying the first-order Logic and Löwenheim's theorems, Church Indecidibility, Completeness and Incompleteness of Gödel, it is known that the axiomatic theories of most commonly used sets are written in a specific first-order language, that is, they are developed within (...) the framework of first-order logic, so this note is relevant; and the third note refers to the presence of mathematical platonism in the axioms of ZFC and in the axioms of "complete ordered field", it is known that the last axioms mentioned characterize (except isomorphism) the real number system and are currently used to develop the real Analysis in the context of set theory. It is hoped that this article will be of pedagogical utility for students interested in set theory and in the philosophy of mathematics (that are beginning in the subject). (shrink)

If the total state of the universe is encodable by a real number, Hardin and Taylor have proved that there is a solution to one version of the problem of induction, or at least a solution to a closely related epistemological problem. Is this philosophical application of the Hardin-Taylor result modest enough? The paper advances grounds for doubt. [A longer and more detailed sequel to this paper, 'Proving Induction', was published in the Australasian Journal of Logic in 2011.].

A computational methodology called Grossone Infinity Computing introduced with the intention to allow one to work with infinities and infinitesimals numerically has been applied recently to a number of problems in numerical mathematics (optimization, numerical differentiation, numerical algorithms for solving ODEs, etc.). The possibility to use a specially developed computational device called the Infinity Computer (patented in USA and EU) for working with infinite and infinitesimal numbers numerically gives an additional advantage to this approach in comparison with traditional methodologies (...) studying infinities and infinitesimals only symbolically. The grossone methodology uses the Euclid’s Common Notion no. 5 ‘The whole is greater than the part’ and applies it to finite, infinite, and infinitesimal quantities and to finite and infinite sets and processes. It does not contradict Cantor’s and non-standard analysis views on infinity and can be considered as an applied development of their ideas. In this paper we consider infinite series and a particular attention is dedicated to divergent series with alternate signs. The Riemann series theorem states that conditionally convergent series can be rearranged in such a way that they either diverge or converge to an arbitrary real number. It is shown here that Riemann’s result is a consequence of the fact that symbol ∞ used traditionally does not allow us to express quantitatively the number of addends in the series, in other words, it just shows that the number of summands is infinite and does not allows us to count them. The usage of the grossone methodology allows us to see that (as it happens in the case where the number of addends is finite) rearrangements do not change the result for any sum with a fixed infinite number of summands. There are considered some traditional summation techniques such as Ramanujan summation producing results where to divergent series containing infinitely many positive integers negative results are assigned. It is shown that the careful counting of the number of addends in infinite series allows us to avoid this kind of results if grossone-based numerals are used. (shrink)

Since the “last day” of 2019, a new virus emerged in Asia, which in Feb./2020 was called by the World Health Organization (WHO, 2020) as Coronavirus disease (Covid-19). Due to its fast transmission, after eight months since the first global official case, at 23:59 (GMT) on August 31, 2020, the world has accounted for about 25,620,737 new confirmed cases with 854,222 deaths and 17,921,063 recovered cases (WORLDOMETERS, 2020). The pandemic is the newest challenge for all nations, most of them eager (...) to learn from countries that are successful against the virus. However, until now, no methodology was developed to identify them by taking into account a holistic approach with international rankings concerned to health, innovation, sustainability, image, and competitiveness, as well as the estimated real number of fatal cases by one million population during the first 180 days of facing the pandemic. Thus, the main objective is to develop a holistic methodology to identify twenty benchmark countries that are saving people's lives against Covid-19. The research is applied, as its results and recommendations are useful for academy, government policymakers and authorities. It is descriptive, with a qualitative and quantitative approach, based on bibliographic and documentary research, involving the study of official sites, articles, reports, manuals, and other technical documents related to 13 international rankings. As a result, the fifteen phases of the methodology, far from perfect, shows that among 108 well-evaluated countries, the top six benchmark countries are from Asia (1) Vietnam; 2) Taiwan; 3) Thailand; 4) China; 5) Malaysia; 6) Singapore), which suffered from fatal cases from first SARS-CoV in 2002/2003, followed by 7) South Korea; 8) New Zealand; 9) Australia; 10) Japan; 11) Hong Kong; 12) Cyprus; 13) Greece; 14) Latvia; 15) Iceland; 16) the United Arab Emirates; 17) Czech; 18) Lithuania; 19) Norway, and 20) Estonia. (shrink)

A brief but rigorous description of the logical structure of mathematical truth.

We develop a simple framework called ‘natural topology’, which can serve as a theoretical and applicable basis for dealing with real-world phenomena.Natural topology is tailored to make pointwise and pointfree notions go together naturally. As a constructive theory in BISH, it gives a classical mathematician a faithful idea of important concepts and results in intuitionism. -/- Natural topology is well-suited for practical and computational purposes. We give several examples relevant for applied mathematics, such as the decision-support system Hawk-Eye, and (...) various real-number representations. -/- We compare classical mathematics (CLASS), intuitionistic mathematics (INT), recursive mathematics (RUSS), Bishop-style mathematics (BISH) and formal topology, aiming to reduce the mutual differences to their essence. To do so, our mathematical foundation must be precise and simple. There are links with physics, regarding the topological character of our physical universe. -/- Any natural space is isomorphic to a quotient space of Baire space, which therefore is universal. We develop an elegant and concise ‘genetic induction’ scheme, and prove its equivalence on natural spaces to a formal-topological induction style. The inductive Heine-Borel property holds for ‘compact’ or ‘fanlike’ natural subspaces, including the real interval [g, h]. Inductive morphisms respect this Heine-Borel property, inversely. This partly solves the continuous-function problem for BISH, yet pointwise problems persist in the absence of Brouwer’s Thesis. -/- By inductivizing the definitions, a direct correspondence with INT is obtained which allows for a translation of many intuitionistic results into BISH. We thus prove a constructive star-finitary metrization theorem which parallels the classical metrization theorem for strongly paracompact spaces. We also obtain non-metrizable Silva spaces, in infinite-dimensional topology. Natural topology gives a solid basis, we think, for further constructive study of topological lattice theory, algebraic topology and infinite-dimensional topology. The final section reconsiders the question of which mathematics to choose for physics. Compactness issues also play a role here, since the question ‘can Nature produce a non-recursive sequence?’ finds a negative answer in CTphys . CTphys , if true, would seem at first glance to point to RUSS as the mathematics of choice for physics. To discuss this issue, we wax more philosophical. We also present a simple model of INT in RUSS, in the two-players game LIfE. (shrink)

Classical interpretations of Goedels formal reasoning, and of his conclusions, implicitly imply that mathematical languages are essentially incomplete, in the sense that the truth of some arithmetical propositions of any formal mathematical language, under any interpretation, is, both, non-algorithmic, and essentially unverifiable. However, a language of general, scientific, discourse, which intends to mathematically express, and unambiguously communicate, intuitive concepts that correspond to scientific investigations, cannot allow its mathematical propositions to be interpreted ambiguously. Such a language must, therefore, define mathematical truth (...) verifiably. We consider a constructive interpretation of classical, Tarskian, truth, and of Goedel's reasoning, under which any formal system of Peano Arithmetic---classically accepted as the foundation of all our mathematical Languages---is verifiably complete in the above sense. We show how some paradoxical concepts of Quantum mechanics can, then, be expressed, and interpreted, naturally under a constructive definition of mathematical truth. (shrink)

The iroha song of human concepts (2021) The iroha is a Japanese poem of a perfect pangram and isogram, containing each character of the Japanese syllabary exactly once. It also mimics an ultimate conceptual engineering, in that there is more and more restricted scope for meaningful expressions, given more and more condensed means of description. This culminates in crystallizations of human values by auto-condensations of meaningful concepts. Instead of distilling Japanese values of 11th century, I try for those of human (...) concepts, given our merging mind, language and culture. (shrink)

I summarize John Hick’s pluralistic theory of the world’s great religions, largely in his own voice. I then focus on the core posit of his theory, what he calls “the Real,” but which I less tendentiously call “Godhick”. Godhick is supposed to be the ultimate religious reality. As such, it must be both possible and capable of explanatory and religious significance. Unfortunately, Godhick is, by definition, transcategorial, i.e. necessarily, for any creaturely conceivable substantial property F, it is neither an (...) F nor a non-F. As a result, Godhick is impossible, as shown by the Self-Identity Problem, the Number Problem, and the Pairing Problem. Moreover, even if Godhick is possible, it faces the Insignificance Problem. The upshot is that, so far as I can see, John Hick’s God is unworthy of any further interest. (shrink)