This chapter presents arguments for two slightly different versions of the thesis that the value of persons is incomparable. Both arguments allege an incompatibility between the demands of a certain kind of practical reasoning and the presuppositions of value comparisons. The significance of these claims is assessed in the context of the “Numbers problem”—the question of whether one morally ought to benefit one group of potential aid recipients rather than another simply because they are greater in number. It (...) is argued that many of the popular approaches to this problem—even ones that avoid the aggregation of personal value—are imperiled by the incomparability theses. -/- . (shrink)

Suppose we can save either a larger group of persons or a distinct, smaller group from some harm. Many people think that, all else equal, we ought to save the greater number. This article defends this view (with qualifications). But unlike earlier theories, it does not rely on the idea that several people's interests or claims receive greater aggregate weight. The argument starts from the idea that due to their stakes, the affected people have claims to have a say in (...) the rescue decision. As rescuers, our primary duty is to respect these procedural claims, which we must do by doing what these people would decide, in a process where each is given an equal vote on the matter. So in cases where each votes in their own self-interest, respect for their equal right to decide, or their autonomy, will lead us to save the greater number. The argument is explained in detail, with special attention to the questions of how, exactly, it avoids aggregation, and of why majority rule is superior to lottery procedures. The view has further advantages. Especially, it explains the “partial” relevance of numbers in cases involving unequal harms, and it does so in a way that dissolves the appearance of paradox that besets theories of “partial aggregation.”. (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)

This paper aims to show that Selim Berker’s widely discussed prime number case is merely an instance of the well-known generality problem for process reliabilism and thus arguably not as interesting a case as one might have thought. Initially, Berker’s case is introduced and interpreted. Then the most recent response to the case from the literature is presented. Eventually, it is argued that Berker’s case is nothing but a straightforward consequence of the generality problem, i.e., the problematic aspect (...) of the case for process reliabilism (if any) is already captured by the generality problem. (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.

This chapter examines the pioneering work of Gottfried Wilhelm Leibniz (1646-1716) on various number systems, in particular binary, which he independently invented in the mid-to-late 1670s, and hexadecimal, which he invented in 1679. The chapter begins with the oft-debated question of who may have influenced Leibniz’s invention of binary, though as none of the proposed candidates is plausible I suggest a different hypothesis, that Leibniz initially developed binary notation as a tool to assist his investigations in mathematical problems that were (...) exercising him at the time, namely those concerning the divisibility of composite numbers, primality, and perfect numbers. The chapter then explores Leibniz’s development of binary, his little-known work on binary fractions and expansions, his use of binary as a symbol for creation in his philosophical-theology, and his response to the suggestion that there was a correlation between binary numeration and the hexagrams of the ancient Chinese divinatory text, the Yijing. The chapter then focuses on Leibniz’s work on other number systems, in particular his invention and exploration of hexadecimal as well as his work on duodecimal. The chapter concludes by revealing a hitherto unknown practical application of binary that Leibniz devised in the last year of his life. (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)

Many philosophers still countenance senses or meanings in the broadly Fregean vein. However, it is difficult to posit the existence of senses without positing quite a lot of them, including at least one presenting every entity in existence. I discuss a number of Cantorian paradoxes that seem to result from an overly large metaphysics of senses, and various possible solutions. Certain more deflationary and nontraditional understanding of senses, and to what extent they fare better in solving the problems, are also (...) discussed. In the end, it is concluded that one must divide senses into various ramified-orders in order to avoid antinomy, but that the philosophical justification of such orders is, as yet, still somewhat problematic. (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 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)

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)

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)

Hofweber (Ontology and the ambitions of metaphysics, Oxford University Press, 2016) argues for a thesis he calls “internalism” with respect to natural number discourse: no expressions purporting to refer to natural numbers in fact refer, and no apparent quantification over natural numbers actually involves quantification over natural numbers as objects. He argues that while internalism leaves open the question of whether other kinds of abstracta exist, it precludes the existence of natural numbers, thus establishing what he (...) calls “restricted nominalism” about natural numbers. We argue that Hofweber’s internalism fails to establish restricted nominalism. Not only is his primary argument for restricted nominalism invalid, the analysis of quantification proposed threatens to collapse internalism into either a traditional form of error theory or realism. (shrink)

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)

I respond to P. McLaughlin and O. Schlaudt’s critique of my analysis of the cross-cultural origins of numbers, noting that my work draws extensively upon number systems as ethnographically attested around the globe, and thus is based only in part on the important Mesopotamian case study. I place the work of Peter Damerow in its historical context, noting its genesis in Piaget’s genetic epistemology and the problems associated with applying Piaget’s developmental theory to societies. While Piaget assumed numeracy involves (...) invariant mental transformations, ongoing research in numerical cognition has been largely unsuccessful in identifying specific brain-bound mechanisms for numerical structure. Accordingly, I suggest the extended mind paradigm from the philosophy of mind may be a more fruitful approach, and detail such an approach using Material Engagement Theory. (shrink)

What is a number? Answering this will answer questions about its philosophical foundations - rational numbers, the complex numbers, imaginary numbers. If we are to write or talk about something, it is helpful to know whether it exists, how it exists, and why it exists, just from a common-sense point of view [Quine, 1948, p. 6]. Generally, there does not seem to be any disagreement among mathematicians, scientists, and logicians about numbers existing in some way, but (...) currently, in the mainstream arena only definitions, descriptions of properties, and effects are presented as evidence. Enough historical description of numbers in history provides an empirical basis of number, although a case can be made that numbers do not exist by themselves empirically. Correspondingly, numbers exist as abstractions. All the while, though, these "descriptions" beg the question of what numbers are ontologically. Advocates for numbers being the ultimate reality have the problem of wrestling with the nature of reality. I start on the road to discovering the ontology of number by looking at where people have talked about numbers as already existing: history. Of course, we need to know not only what ontology is but the problems of identifying one, leading to the selection between metaphysics and provisional approaches. While we seem to be dimensionally limited, at least we can identify a more suitable bootstrapping ontology than mere definitions, leading us to the unity of opposites. The rest of the paper details how this is done and modifies Peano's Postulates. (shrink)

Reductionist realist accounts of certain entities, such as the natural numbers and propositions, have been taken to be fatally undermined by what we may call the problem of arbitrary identification. The problem is that there are multiple and equally adequate reductions of the natural numbers to sets (see Benacerraf, 1965), as well as of propositions to unstructured or structured entities (see, e.g., Bealer, 1998; King, Soames, & Speaks, 2014; Melia, 1992). This paper sets out to solve (...) the problem by canvassing what we may call the arbitrary reference strategy. The main claims of such strategy are 2. First, we do not know which objects are the referents of proposition and numerical terms since their reference is fixed arbitrarily. Second, our ignorance of which object is picked out as the referent does not entail that no object is referred to by the relevant expression. Different articulations of the strategy are assessed, and a new one is defended. (shrink)

There are mathematical structures with elements that cannot be distinguished by the properties they have within that structure. For instance within the field of complex numbers the two square roots of −1, i and −i, have the same algebraic properties in that field. So how do we distinguish between them? Imbedding the complex numbers in a bigger structure, the quaternions, allows us to algebraically tell them apart. But a similar problem appears for this larger structure. There seems (...) to be always a background and a context that we rely upon. Thus mathematicians naturally make use of Kantian intuition and references fixed by names and denotations. I argue that such features cannot be avoided. (shrink)

In this paper, we explore the properties and optimization techniques related to polyhedral cones and energy numbers with a focus on the cone of positive semidefinite matrices and efficient computation strategies for kernels. In Part (a), we examine the polyhedral nature of the cone of positive semidefinite matrices, , establishing that it does not form a polyhedral cone for due to its infinite dimensional characteristics. In Part (b), we present an algorithm for efficiently computing the kernel function on-the-fly, leveraging (...) a polyhedral description of the convex hull generated by the feature mappings and . By restructuring the problem and using gradient-based optimization techniques, our approach minimizes memory usage and computational overhead, thus enabling scalable computation. Through examples and visualizations, we demonstrate the practical applications and efficiency of the proposed algorithm in optimizing these kernel computations. (shrink)

This article offers an interpretation of a controversial aspect of Frege’s The Foundations of Arithmetic, the so-called Julius Caesar problem. Frege raises the Caesar problem against proposed purely logical definitions for ‘0’, ‘successor’, and ‘number’, and also against a proposed definition for ‘direction’ as applied to lines in geometry. Dummett and other interpreters have seen in Frege’s criticism a demanding requirement on such definitions, often put by saying that such definitions must provide a criterion of identity of a (...) certain kind. These interpretations are criticized and an alternative interpretation is defended. The Caesar problem is that the proposed definitions fail to well-define ‘number’ and ‘direction’. That is, the proposed definitions, even when taken together with the extra-definitional facts, fail to fix unique semantic extensions for ‘number’ and ‘direction’, and thereby fail to fix unique truth-values for sentences like ‘Caesar is a number’ and ‘England is a direction’. A minor modification of the criticized definitions well-defines ‘0’, ‘successor’ and ‘number’, thereby avoiding the Caesar problem as Frege understands it, but without providing any criterion of number identity in the usual sense. (shrink)

The boundless nature of the natural numbers imposes paradoxically a high formal bound to the use of standard artificial computer programs for solving conceptually challenged problems in number theory. In the context of the new cognitive foundations for mathematics' and physics' program immersed in the setting of artificial mathematical intelligence, we proposed a refined numerical system, called the physical numbers, preserving most of the essential intuitions of the natural numbers. Even more, this new numerical structure additionally possesses (...) the property of being a bounded object allowing us to work with quite similar axioms like the classic Peano axioms, but in a finite environment and with an enriched physical dimension. Finally, we present several enlightening examples and we conclude that the physical numbers provide a natural formal setting for approaching classic problems in number theory from a more hybrid perspective, i.e. with a potential participation in the generation of solutions, not only of working mathematicians but also of more sophisticated artificial interactive (mathematical) intelligent agents (within the context of cognitive-computational metamathematics or artificial mathematical intelligence) and more controlled by physically-inspired principles. Finally, this paper addresses a highly new, bottom-up and paradigm-shifting approach to the foundations of physics: One should refine and improve the conceptual formal setting of the language that we use for understanding physical phenomena (e.g. the mathematical grounding concepts and theories) for being able to understand better more subtle and complex physical phenomena. (shrink)

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)

We recall, "a priori," numeric energy expression: -/- Energy Numbers -/- $\begin{gathered}\mathcal{V}=\left\{f \mid \exists\left\{e_1, e_2, \ldots, e_n\right\} \in E \cup R\right\} \\ \mathcal{V}=\left\{f \mid \exists\left\{e_1, e_2, \ldots, e_n\right\} \in E, \text { and }: E \mapsto r \in R\right\} \\ \mathcal{V}=\left\{E \mid \exists\left\{a_1, \ldots, a_n\right\} \in E, E \not \neg r \in R\right\}\end{gathered}$ -/- We now introduce the set of optimized energy numbers: -/- ($H_a \in \mathcal{H}$ or $P^n = NP$ or $(P,\mathcal{L},F) = NP$). -/- Based on our (...) formulation of the bi-objective optimization task, we can make the following mathematical inferences: -/- 1. If the optimized energy numbers set $\mathcal{N}_H$ is equal to the original energy numbers set $\mathcal{E}$, i.e. $\mathcal{N}_H = \mathcal{E}$, then the maximum optimization score is achieved, i.e. the bi-objective optimization task is solved. This implies that there exists at least one solution to the optimization problem and $H_a \in \mathcal{H}$, where $H_a$ is the hypothesis that states the existence of an efficient algorithm to solve the problem. -/- 2. If the optimized energy numbers set $\mathcal{N}_H$ is a subset of the original energy numbers set $\mathcal{E}$, i.e. $\mathcal{N}_H \subset \mathcal{E}$, then the optimization score is less than the maximum score. This indicates that there may exist more efficient algorithms to solve the problem, and the hypothesis $H_a$ is still possible. -/- 3. If the optimized energy numbers set $\mathcal{N}_H$ is a superset of the original energy numbers set $\mathcal{E}$, i.e. $\mathcal{N}_H \supset \mathcal{E}$, then the optimization score is higher than the maximum score. This implies that the optimization problem may be easier than initially thought, and $P^n = NP$, or at least some form of $NP$-completeness. -/- 4. If the optimized energy numbers set $\mathcal{N}_H$ is a strict subset of the original energy numbers set $\mathcal{E}$, i.e. $\mathcal{N}_H \subsetneq \mathcal{E}$, and $P^n \neq NP$, then it can be concluded that the optimization problem is complex but there may exist algorithms that can efficiently approximate the solution. -/- 5. If the optimized energy numbers set $\mathcal{N}_H$ is empty, i.e. $\mathcal{N}_H = \emptyset$, then it can be inferred that the optimization problem is infeasible, i.e. no efficient algorithm exists to solve it, and $H_a$ is false. -/- 6. Comparing the two objectives in the bi-objective optimization task, we can make the following statements: -/- - The first objective, $\frac{\delta_{v(f)}(\mathbf{v}, \mathbf{w_{max}})} {\langle \mathbf{v_f}, \mathbf{1_f}\rangle}$, measures the efficiency of the algorithm and its ability to find low energy numbers. - The second objective, $\rho(\mathcal{N}_H)$, measures the accuracy of the algorithm in terms of loss and perplexity on the HyperLanguageModel. - Therefore, by optimizing both objectives simultaneously, we aim to find an efficient algorithm that also minimizes the loss and perplexities on the HyperLanguageModel. - If the optimization task is successfully solved, then the algorithm achieves both high efficiency and high accuracy. This would imply that the algorithm is able to find low energy numbers effectively and also generalize well on the HyperLanguageModel. -/- The optimized energy numbers aim to find a set of numbers $\mathbf{v}$ that maximize the bi-objective optimization task, while also minimizing the loss and perplexity of the HyperLanguageModel. This is achieved by finding the set of numbers that have the highest delta value and the lowest perplexity, resulting in a more optimized and efficient set of energy numbers. -/- By comparing the optimized energy numbers to the original set, we can see that the optimized set may have a higher delta value and a lower perplexity, indicating that it is a better set of numbers for the given task. This shows that the optimized energy numbers have successfully achieved their goal of maximizing efficiency while minimizing loss and perplexity. (shrink)

The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two di erent ways. We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural (...) class='Hi'>numbers and relation of inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and mathematician, in 1932. The axioms W are those of ordered sets without largest element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent. There follows a set of intuitive axioms PI of integers arithmetic, modelled on P and proposed by B. Iwanuś, as well as a set of axioms WI of this arithmetic, modelled on the W axioms, PI and WI being also equivalent, categorical and consistent. We also discuss the problem of independence of sets of axioms, which were dealt with earlier. (shrink)

This short essay outlines the problem Taurek responds to and the argument he uses in Should the Numbers Count. His argument posits that in a situation where you can either prevent harm to one stranger or five strangers but you cannot prevent harm to all six, the best thing to do is is give each person an equal chance of survival by flipping a coin. Although this paper is largely an explication, I do provide a short critique of (...) Taurek's argument. I claim flipping a coin overlooks the worth of the individual people at risk, and other solutions may be better suited to solve this ethical dilemma. (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.

This study focuses on implementing electric vehicles (EVs) in developing countries where energy production is mainly based on fossil fuels. Although for these countries the environmental short-run benefits of the EVs cannot offset the short-run costs, it may still be the best option to implement the EVs as soon as possible. Hence, it is necessary to evaluate the alternatives to introducing EVs to the market due to the environmental concerns that created an opportunity for some developing countries to catch up (...) with the international competition. Therefore, we develop a case scenario to explore the decision-making process in implementing the EVs with three alter natives and twelve criteria. We solve the decision-making problem by using Type-2 neutrosophic numbers (T2NNs) based on the RAFSI (Ranking of Alternatives through Functional mapping of criterion sub-intervals into a Single Interval) method. The proposed model combines the advantages of the RAFSI technique, and it applies T2NNs to address the uncertainties. The results show that the alternatives that may suspend the implementation of the EVs are inferior. Direct implementation of EVs is prioritized. The policy implications of the results are discussed in the study. (shrink)

An enduring puzzle in philosophy and developmental psychology is how young children acquire number concepts, in particular the concept of natural number. Most solutions to this problem conceptualize young learners as lone mathematicians who individually reconstruct the successor function and other sophisticated mathematical ideas. In this chapter, I argue for a crucial role of testimony in children’s acquisition of number concepts, both in the transfer of propositional knowledge (e.g., the cardinality concept), and in knowledge-how (e.g., the counting routine).

The paper considers a new type of objects – blinking fractals – that are not covered by traditional theories studying dynamics of self-similarity processes. It is shown that the new approach allows one to give various quantitative characteristics of the newly introduced and traditional fractals using infinite and infinitesimal numbers proposed recently. In this connection, the problem of the mathematical modelling of continuity is discussed in detail. A strong advantage of the introduced computational paradigm consists of its well-marked (...) numerical character and its own instrument – Infinity Computer – able to execute operations with infinite and infinitesimal numbers. (shrink)

We give a brief overview of the evolution of mathematics, starting from antiquity, through Renaissance, to the 19th century, and the culmination of the train of thought of history’s greatest thinkers that lead to the grand unification of geometry and algebra. The goal of this paper is not a complete formal description of any particular theoretical framework, but to show how extremisation of mathematical rigor in requiring everything be drivable directly from first principles without any arbitrary assumptions actually leads to (...) relaxing the computational difficulty along with maximal conceptual clarity. With this, we consider a revision of the foundations of elementary geometry and algebra based on the work of Grassmann and Clifford and apply it to conceptual and practical problems of past and present modern mathematics and mathematical physics. (shrink)

In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism.

I argue that a crucial point has been overlooked in the debate over the “numbers problem.” The initial arrangement of parties in the problem can be thought of as chancy, and whatever considerations of fairness recommend the reliance on something like a coin toss in approaching this problem equally recommend treating the initial distribution as a kind of lottery. This fact, I suggest, undermines one of the principal arguments against saving the greater number.

Building on theoretical insights and rich experimental data of our preprints, we present here new theoretical and experimental results in three interrelated approaches to the Collatz problem and its generalizations: algorithmic decidability, random behavior, and Diophantine representation of related discrete dynamical systems, and their cyclic and divergent properties.

A number of Christian theologians and philosophers have been critical of overly moralizing approaches to the doctrine of sin, but nearly all Christian thinkers maintain that moral fault is necessary or sufficient for sin to obtain. Call this the “Moral Consensus.” I begin by clarifying the relevance of impurities to the biblical cataloguing of sins. I then present four extensional problems for the Moral Consensus on sin, based on the biblical catalogue of sins: (1) moral over-demandingness, (2) agential unfairness, (3) (...) moral repugnance, and (4) moral atrocity. Next, I survey several partial solutions to these problems, suggested by the recent philosophical literature. Then I evaluate two largely unexplored solutions: (a) genuine sin dilemmas and (b) defeasible sinfulness. I argue that (a) creates more problems than it solves and that, while (b) is well-motivated and solves or eases each of the above problems, (b) leaves many biblical ordinances about sin morally misleading, creating (5) a pedagogical problem of evil. I conclude by arguing that (5) places hefty explanatory burdens on those who would appeal to (b) to resolve the four extensional problems discussed in this paper. So Christian thinkers may need to consider a more radical separation of sin and moral fault. (shrink)

The Problem of Too Many Thinkers is the result, implied by several “permissive” ontologies, that we spatiotemporally overlap with a number of intrinsically person-like entities. The problem, as usually formulated, leaves open a much-neglected question: do we literally share our mental lives, i.e. each of our mental states, with these person-like entities, or do we instead enjoy mental lives that are qualitatively indistinguishable but numerically distinct from theirs? The latter option raises the worry that there is an additional (...) Problem of Too Many Mental Tokens. This paper argues that there is indeed such a problem, at least in fission cases. In the course of articulating this problem, we will make a number of surprising discoveries about the relationship between personal persistence and the metaphysics of mental entities. We will also see that the Problem of Too Many Mental Tokens has significant epistemological and ethical implications, which will haunt us even once we have addressed the Problem of Too Many Thinkers. (shrink)

Many believe that we ought to save a large number from being permanently bedridden rather than save one from death. Many also believe that we ought to save one from death rather than a multitude from a very minor harm, no matter how large this multitude. I argue that a principle I call “Aggregate Relevant Claims” satisfactorily explains these judgments. I offer a rationale for this principle and defend it against objections.

The GOOGLE and XPRIZE $5,000,000 for the practical and socially useful utilization of the quantum computer is the starting point for ontomathematical reflections for what it can really serve. Its “output by measurement” is opposed to the conjecture for a coherent ray able alternatively to deliver the ultimate result of any quantum calculation immediately as a Dirac -function therefore accomplishing the transition of the sequence of increasingly narrow probability density distributions to their limit. The GOOGLE and XPRIZE problem’s solution (...) needs the initial understanding that the result of any quantum calculation is a wave function, or respectively, a probability (density or not) distribution unlike the Turing machine one, and only a “calculating ray” is able to transform the former into the later without any “curving” disturbances. Thus, the unique capability of the quantum computer due to its inherent quantum parallelism can be conserved for all Gödel unresolvable problems, only on the fast “NP” track of which the quantum computer “Achilles” is able practically to overrun the Turing machine “Tortoise” since the latter can “sprint” only by any “P” calculating speed even on it and unlike “Achilles” himself able for “NP” velocities as well. The way for any material body to “calculate” the certain trajectory of least action also resolves the “traveling salesman problem” in fact, therefore illustrating furthermore what the “NP” output of a quantum computer by a “calculating ray” should mean. Another example is the problem of the number of all prime numbers less than “N”, which is suggested to visualize once again the quantum computer capability to resolve problems by a “NP” speed and thus qualitatively to overrun any competing Turing machine. The technically implemented idea of laser is now reinterpreted to “radiate a wave function” in order to explain the “calculating ray” option for a quantum computer “NP” output. The generalization of an entanglement “trigger ray” described by any non-Hermitian operator (unlike a laser) is suggested as well as those “sci-fi” horizons which it reveals. One of them is meant for elucidating the practical meaning of the “Yang-Mills existence and mass gap problem”, one more of the CMI “Millennium Problems” after that of “P vs NP”. -/- . (shrink)

The ability to solve problems is a prerequisite in preparing mathematics preservice teachers. This study assessed preservice teachers’ problem-solving difficulties and performance, particularly in worded problems on number sense, measurement, geometry, algebra, and probability. Also, academic profile differences in the preservice teacher’s problem-solving performance and common errors were determined. A descriptive-comparative research design was employed with 158 random respondents. Data were gathered face-to-face during the first semester of the school year 2022-2023, and data were analyzed with the aid (...) of jamovi software, ensuring ethical measures. Overall findings revealed that the preservice teachers experienced average difficulty in solving problems. The low performance of the preservice teachers on the given problems was also demonstrated. Further analysis revealed a significant difference between the preservice teachers’ problem-solving performance based on their subject preference and program. Moreover, the error analysis revealed that the preservice teachers incurred comprehension errors in misrepresentation, misinterpretation, and miscalculation. These results will serve as a measure for policymakers and curriculum developers of the teacher education institution concerned to make relevant enhancements to the math courses offered in the elementary and secondary education programs. (shrink)

The study determined the problem-solving performance and skills of prospective elementary teachers (PETs) in the Northern Philippines. Specifically, it defined the PETs’ level of problem-solving performance in number sense, measurement, geometry, algebra, and probability; significant predictors of their problem-solving performance in terms of sex, socio-economic status, parents’ educational attainment, high school graduated from and subject preference; and their problem-solving skills. The PETs’ problem-solving performance was determined by a problem set consisting of word problems with (...) number sense, measurement, geometry, algebra, and probability. A mixed-method research design was employed. Senior PETs purposively served as a sample where they mostly preferred to teach other subjects than mathematics. PETs who preferred math performed satisfactorily, while prospective teachers who opted for other subjects performed unsatisfactorily. The PETs’ unsatisfactory output indicates the need for remediation to advance the mathematical material skills and enrich the problem-solving abilities of these primary schools' potential teachers. Besides, results showed that subject preference strongly affected and predicted the problem-solving success of the PETs. PETs who preferred to teach mathematics performed significantly better than their counterparts; hence, mathematics as a field of specialization in the Bachelor of Elementary Education program may be considered by teacher education institutions. Further, most PETs displayed lack of problem-solving skills; thus, a Problem-Solving course is recommended for them. (shrink)

This problem has already been discussed by a number of authors.[i] Typically, however, authors take one of two extreme positions: they hold that all welfare should be taken at face value, or they hold that "suspect" welfare should be completely ignored. My contribution here is the following: First, I introduce the notion of unauthorized (suspect) welfare, of which welfare from meddlesome preferences, offensive tastes, expensive tastes, etc. are special cases. Second, I formulate four conditions of adequacy, applicable to any (...) welfare-based theory, for dealing with unauthorized welfare. These conditions require that unauthorized welfare be "discounted" (play a restricted role) but not be completely ignored. Thus, I shall be exploring a position intermediate between taking "unauthorized" welfare at face value and simply ignoring it. Moreover, the four conditions jointly determine exactly how existing welfare-based theories need to be revised so as to be appropriately sensitive to unauthorized welfare. (shrink)

In this paper, we introduce the integer programming in neutrosophic environment, by considering coffecients of problem as a triangulare neutrosophic numbers. The degrees of acceptance, indeterminacy and rejection of objectives are simultaneously considered. The Neutrosophic Integer Programming Problem (NIP) is transformed into a crisp programming model, using truth membership (T), indeterminacy membership (I), and falsity membership (F) functions as well as single valued triangular neutrosophic numbers. To measure the efficiency of the model, we solved several numerical (...) examples. (shrink)

The main goal of this paper is to investigate what explanatory resources Robert Brandom’s distinction between acknowledged and consequential commitments affords in relation to the problem of logical omniscience. With this distinction the importance of the doxastic perspective under consideration for the relationship between logic and norms of reasoning is emphasized, and it becomes possible to handle a number of problematic cases discussed in the literature without thereby incurring a commitment to revisionism about logic. One such case in particular (...) is the preface paradox, which will receive an extensive treatment. As we shall see, the problem of logical omniscience not only arises within theories based on deductive logic; but also within the recent paradigm shift in psychology of reasoning. So dealing with this problem is important not only for philosophical purposes but also from a psychological perspective. (shrink)

The New Evil Demon Problem presents a serious challenge to externalist theories of epistemic justification. In recent years, externalists have developed a number of strategies for responding to the problem. A popular line of response involves distinguishing between a belief’s being epistemically justified and a subject’s being epistemically blameless for holding it. The apparently problematic intuitions the New Evil Demon Problem elicits, proponents of this response claim, track the fact that the deceived subject is epistemically blameless for (...) believing as she does, not that she is justified for so believing. This general strategy—which I call the “unjustified-but-blameless maneuver”—is motivated, in part, by the assumption that the distinction between epistemic justification and blamelessness is merely an extension of the familiar distinction between moral justification and blamelessness. In this paper, I consider three ways of drawing the distinction between justification and blamelessness familiar from the moral domain: the first in terms of a connection with reactive attitudes, the second in terms of the distinction between wrongness and wronging, and the third in terms of reasons-responsiveness. All three ways of drawing the distinction, I argue, make it difficult to see how an analogous distinction in the epistemic domain could help externalists explain away the intuitions which underwrite the New Evil Demon Problem. Motivating the unjustified-but-blameless maneuver, I conclude, is a much less straightforward task than its proponents tend to assume. (shrink)

A number of theorists have argued that Scanlon's contractualist theory both "gets around" and "solves" the non-identity problem. They argue that it gets around the problem because hypothetical deliberation on general moral principles excludes the considerations that lead to the problem. They argue that it solves the problem because violating a contractualist moral principle in one's treatment of another wrongs that particular other, grounding a person-affecting moral claim. In this paper, I agree with the first claim (...) but note that all it shows is that the act is impersonally wrong. I then dispute the second claim. On Scanlon's contractualist view, one wrongs a particular other if one treats the other in a way that is unjustifiable to that other on reasons she could not reasonably reject. We should think of person-affecting wronging in terms of the reasons had by the actual agent and the actual person affected by the agent's action. In non-identity cases, interpersonal justifiability is therefore shaped both by the reason to reject the treatment provided by the bad suffered and the reason to affirm the treatment provided by the goods had as a result of existing. I argue it would be reasonable for the actual person to find the treatment justifiable, and so I conclude that Scanlon's contractualist metaethics does not provide a narrow person-affecting solution to the non-identity problem on its own terms. I conclude that the two claims represent a tension within Scanlon's contractualist theory itself. (shrink)

The threshold problem is the task of adequately answering the question: “Where does the threshold lie between knowledge and lack thereof?” I start this paper by articulating two conditions for solving it. The first is that the threshold be neither too high nor too low; the second is that the threshold accommodate the significance of knowledge. In addition to explaining these conditions, I also argue that it is plausible that they can be met. Next, I argue that many popular (...) accounts of knowledge cannot meet them. In particular, I lay out a number of problems that standard accounts of knowledge face in trying to meet these conditions. Finally, near the end of this paper, I argue that there is one sort of account that seems to evade these problems. This sort of account, which is called a cluster account of knowledge, says that knowledge is to be accounted for in terms of truth, belief and a cluster of epistemic properties and also that knowledge doesn’t require having all members of the cluster, but merely some subset. (shrink)

To speak of being religious lucky certainly sounds odd. But then, so does “My faith holds value in God’s plan, while yours does not.” This book argues that these two concerns — with the concept of religious luck and with asymmetric or sharply differential ascriptions of religious value — are inextricably connected. It argues that religious luck attributions can profitably be studied from a number of directions, not just theological, but also social scientific and philosophical. There is a strong tendency (...) among adherents of different faith traditions to invoke asymmetric explanations of the religious value or salvific status of the home religion vis-à-vis all others. Attributions of good/bad religious luck and exclusivist dismissal of the significance of religious disagreement are the central phenomena that the book studies. Part I lays out a taxonomy of kinds of religious luck, a taxonomy that draws upon but extends work on moral and epistemic luck. It asks: What is going on when persons, theologies, or purported revelations ascribe various kinds of religiously-relevant traits to insiders and outsiders of a faith tradition in sharply asymmetric fashion? “I am saved but you are lost”; “My religion is holy but yours is idolatrous”; “My faith tradition is true, and valued by God, but yours is false and valueless.” Part II further develops the theory introduced in Part I, pushing forward both the descriptive/explanatory and normative sides of what the author terms his inductive risk account. Firstly, the concept of inductive risk is shown to contribute to the needed field of comparative fundamentalism by suggesting new psychological markers of fundamentalist orientation. The second side of what is termed an inductive risk account is concerned with the epistemology of religious belief, but more especially with an account of the limits of reasonable religious disagreement. Problems of inductively risky modes of belief-formation problematize claims to religion-specific knowledge. But the inductive risk account does not aim to set religion apart, or to challenge the reasonableness of religious belief tout court. Rather the burden of the argument is to challenge the reasonableness of attitudes of religious exclusivism, and to demotivate the “polemical apologetics” that exclusivists practice and hope to normalize. Lexington Books Pages: 290 978-1-4985-5017-8 • Hardback • December 2018 • $95.00 • (£65.00) 978-1-4985-5018-5 • eBook • December 2018 • $90.00 • (£60.00) ISBN 978-1-4985-5018-5 (pbk: alk. paper) (coming 2020) [Download the 30% personal use Discount Order Form I uploaded for hardcover or e-book, and please ask your library to purchase a copy for their collection.]. (shrink)

In its original form, Nozick’s experience machine serves as a potent counterexample to a simplistic form of hedonism. The pleasurable life offered by the experience machine, its seems safe to say, lacks the requisite depth that many of us find necessary to lead a genuinely worthwhile life. Among other things, the experience machine offers no opportunities to establish meaningful relationships, or to engage in long-term artistic, intellectual, or political projects that survive one’s death. This intuitive objection finds some support in (...) recent research regarding the psychological effects of phenomena such as video games or social media use. After a brief discussion of these problems, I will consider a variation of the experience machine in which many of these deficits are remedied. In particular, I’ll explore the consequences of a creating a virtual world populated with strongly intelligent AIs with whom users could interact, and that could be engineered to survive the user’s death. The presence of these agents would allow for the cultivation of morally significant relationships, and the world’s long-term persistence would help ground possibilities for a meaningful, purposeful life in a way that Nozick’s original experience machine could not. While the creation of such a world is obviously beyond the scope of current technology, it represents a natural extension of the existing virtual worlds provided by current video games, and it provides a plausible “ideal case” toward which future virtual worlds will move. While this improved experience machine would seem to represent progress over Nozick’s original, I will argue that it raises a number of new problems stemming from the fact that that the world was created to provide a maximally satisfying and meaningful life for the intended user. This, in turn, raises problems analogous in some ways to the problem(s) of evil faced by theists. In particular, I will suggest that it is precisely those features that would make a world most attractive to potential users—the fact that the AIs are genuinely moral agents whose well-being the user can significantly impact—that render its creation morally problematic, since they require that the AIs inhabiting the world be subject to unnecessary suffering. I will survey the main lines of response to the traditional problem of evil, and will argue that they are irrelevant to this modified case. I will close by considering by consider what constraints on the future creation of virtual worlds, if any, might serve to allay the concerns identified in the previous discussion. I will argue that, insofar as the creation of such worlds would allow us to meet morally valuable purposes that could not be easily met otherwise, we would be unwise to prohibit it altogether. However, if our processes of creation are to be justified, they must take account of the interests of the moral agents that would come to exist as the result of our world creation. (shrink)

Despite the growth in research in philosophy of religion over the past several decades, recent years have seen a number of critical studies of this subfield in an effort to redirect the methods and topics of inquiry. This article argues that in addition to problems of religious parochialism described by critics such as Wesley Wildman, the subfield is facing a problem of relevance. In responding to this problem, it suggests that philosophers of religion should do three things: first, (...) be critically self-aware about their aims of inquiry; second, investigate concepts used by other philosophers, scientists, and religious studies scholars to identify and dispel confusion about religions; and third, following the model of applied ethics, work to clarify concepts and advance arguments of contemporary practical urgency. (shrink)

The Caesar problem arises for abstractionist views, which seek to secure reference for terms such as ‘the number of Xs’ or #X by stipulating the content of ‘unmixed’ identity contexts like ‘#X = #Y’. Frege objects that this stipulation says nothing about ‘mixed’ contexts such as ‘# X = Julius Caesar’. This article defends a neglected response to the Caesar problem: the content of mixed contexts is just as open to stipulation as that of unmixed contexts.

The new evil demon problem is often considered to be a serious obstacle for externalist theories of epistemic justification. In this paper, I aim to show that the new evil demon problem also afflicts the two most prominent forms of internalism: moderate internalism and historical internalism. Since virtually all internalists accept at least one of these two forms, it follows that virtually all internalists face the NEDP. My secondary thesis is that many epistemologists – including both internalists and (...) externalists – face a dilemma. The only form of internalism that is immune to the NEDP, strong internalism, is a very radical and revisionary view – a large number of epistemologists would have to significantly revise their views about justification in order to accept it. Hence, either epistemologists must accept a theory that is susceptible to the NEDP or accept a very radical and revisionary view. (shrink)