Recent years have seen an explosion of empirical data concerning arithmetical cognition. In this paper that data is taken to be philosophically important and an outline for an empirically feasible epistemological theory of arithmetic is presented. The epistemological theory is based on the empirically well-supported hypothesis that our arithmetical ability is built on a protoarithmetical ability to categorize observations in terms of quantities that we have already as infants and share with many nonhuman animals. It is argued here that (...) arithmetical knowledge developed in such a way cannot be totally conceptual in the sense relevant to the philosophy of arithmetic, but neither can arithmetic understood to be empirical. Rather, we need to develop a contextual a priori notion of arithmetical knowledge that preserves the special mathematical characteristics without ignoring the roots of arithmetical cognition. Such a contextual a priori theory is shown not to require any ontologically problematic assumptions, in addition to fitting well within a standard framework of general epistemology. (shrink)

In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational (...) models of human cognitive capacities. One interesting area of computational complexity theory is descriptive complexity, which connects the expressive strength of systems of logic with the computational complexity classes. In descriptive complexity theory, it is established that only first-order systems are connected to P, or one of its subclasses. Consequently, second-order systems of logic are considered to be computationally intractable, and may therefore seem to be unfit to model human cognitive capacities. This would be problematic when we think of the role of logic as the foundations of mathematics. In order to express many important mathematical concepts and systematically prove theorems involving them, we need to have a system of logic stronger than classical first-order logic. But if such a system is considered to be intractable, it means that the logical foundation of mathematics can be prohibitively complex for human cognition. In this paper I will argue, however, that this problem is the result of an unjustified direct use of computational complexity classes in cognitive modelling. Placing my account in the recent literature on the topic, I argue that the problem can be solved by considering computational complexity for humanly relevant problem solving algorithms and input sizes. (shrink)

Competence structure that pre-service teachers need to develop to become a future teacher has been defined since the 1930s. For pre-service mathematics teachers, their competence has its own characteristics. ICT integration in developing competence for pre-service mathematics teachers has been been proved to be effective in many previous studies. In Vietnam, the Ministry of Education and Training (MOET) has recommended the use of ICT to enhance teaching-learning activities in schools and universities, therefore, there have been many studies on (...) ICT intergration at different educational levels. However, there are only a few studies on the use ICT integration in developing competence for pre-service mathematics teachers. This paper presents results from a research on the feasibility of ICT integration in developing competence of Vietnamese pre-service mathematics teachers. The research was conducted by surveying on a randomly selected of 297 pre-service mathematics teachers and 40 mathematics lecturers in six universities which provide mathematics training programs. Results show that despite of the availability of technologies, ICT integration in training pre-service teachers is still limited. In addition, ICT integration in teaching activities of mathematics teachers is not really effective because the levels of their ICT skills are very different. Pre-service mathematics teachers have the ability to use ICT equipment, and good accessibility to learning systems, but their use of the learning management systems (LMS) is limited. This study can be a reference for future studies to test the effective of ICT integration in training pre-service teachers when designing activities to develop feasibly and effectively their competence. (shrink)

This thesis considers the following problem: What methods should the epistemology of science use to gain insight into the structure and behaviour of scientific knowledge and method in actual scientific practice? After arguing that the elucidation of epistemological and methodological phenomena in science requires a method that is rooted in formal methods, I consider two alternative methods for epistemology of science. One approach is the classical approaches of the syntactic and semantic views of theories. I show that typical approaches of (...) this sort are inadequate and inaccurate in their representation of scientific knowledge by showing how they fail to account for and misrepresent important epistemological structure and behaviour in science. The other method for epistemology of science I consider is modeled on the methods used to construct valid models of natural phenomena in applied mathematics. This new epistemological method is itself a modeling method that is developed through the detailed consideration of two main examples of theory application in science: double pendulum systems and the modeling of near-Earth objects to compute probability of future Earth impact. I show that not only does this new method accurately represent actual methods used to apply theories in applied mathematics, it also reveals interesting structural and behavioural patterns in the application process and gives insight into what underlies the stability of methods of application. I therefore conclude that for epistemology of science to develop fully as a scientific discipline it must use methods from applied mathematics, not only methods from pure mathematics and metamathematics as traditional formal epistemology of science has done. (shrink)

As a philosophy of mathematics, strict finitism has been traditionally concerned with the notion of feasibility, defended mostly by appealing to the physicality of mathematical practice. This has led the strict finitists to influence and be influenced by the field of computational complexity theory, under the widely held belief that this branch of mathematics is concerned with the study of what is “feasible in practice”. In this paper, I survey these ideas and contend that, contrary to popular (...) belief, complexity theory is not what the ultrafinitists think it is, and that it does not provide a theoretical framework in which to formalize their ideas —at least not while defending the material grounds for feasibility. I conclude that the subject matter of complexity theory is not proving physical resource bounds in computation, but rather proving the absence of exploitable properties in a search space. (shrink)

In Pantsar, an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain the (...) third one: why arithmetical knowledge appears to be necessary. A Kripkean analysis of necessity is used as an example to show that a proper analysis of the relevant possible worlds can explain arithmetical necessity in a sufficiently strong form. (shrink)

This paper presents proof that Buss's S22 can prove the consistency of a fragment of Cook and Urquhart's PV from which induction has been removed but substitution has been retained. This result improves Beckmann's result, which proves the consistency of such a system without substitution in bounded arithmetic S12. Our proof relies on the notion of "computation" of the terms of PV. In our work, we first prove that, in the system under consideration, if an equation is proved and either (...) its left- or right-hand side is computed, then there is a corresponding computation for its right- or left-hand side, respectively. By carefully computing the bound of the size of the computation, the proof of this theorem inside a bounded arithmetic is obtained, from which the consistency of the system is readily proven. This result apparently implies the separation of bounded arithmetic because Buss and Ignjatović stated that it is not possible to prove the consistency of a fragment of PV without induction but with substitution in Buss's S12. However, their proof actually shows that it is not possible to prove the consistency of the system, which is obtained by the addition of propositional logic and other axioms to a system such as ours. On the other hand, the system that we have considered is strictly equational, which is a property on which our proof relies. (shrink)

Department of Mathematics education curriculum implementation Based KKNI who have not provided the container development of character education for students. This can be seen from the learning process in college that still relies on aspects of increased knowledge. Achievement of learning on aspects of attitudes / values ​​still are administrative without being pushed on the feasibility of value investment education daily life. Applied learning models are still oriented to conventional learning eg discussions, lectures, discussion and assignment. It is necessary (...) to present innovations in the learning model for students. Model-oriented learning in the context of nationalism (educational values ​​Ki Hadjar Dewantara). so that by applying the learning model of value investment Ki Hajar Dewantara education can increase educational purposes at universities. The method used in this study is quasi-experimental design with a Two-Group Test post-only design. The population in this study that students of Department of Mathematics Education. The sample research nonprobability sampling using purposive sampling technique. Experiments in which the class is used as the Class A total of 39 students and the class is a class B control as many as 33 students. Results of research dianataranya that the student response to the application of learning models planting educational values ​​Ki Hadjar Dewantara is quite good. Students learning attitude was quite good with a big achievement average score of 64.41. Effect of application of learning models planting educational values ​​Ki Hadjar Dewantara attitude towards learning in mathematics education majors and significant effect on average there are differences in learning outcomes experimental class is higher than population control class learning outcomes. It states that the application of learning models planting educational values ​​Ki Hadjar Dewantara effectively implemented in the Department of Mathematics Education. (shrink)

In Pantsar (2014), an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain (...) the ... (shrink)

Our conscious minds exist in the Universe, therefore they should be identified with physical states that are subject to physical laws. In classical theories of mind, the mental states are identified with brain states that satisfy the deterministic laws of classical mechanics. This approach, however, leads to insurmountable paradoxes such as epiphenomenal minds and illusionary free will. Alternatively, one may identify mental states with quantum states realized within the brain and try to resolve the above paradoxes using the standard Hilbert (...) space formalism of quantum mechanics. In this essay, we first show that identification of mind states with quantum states within the brain is biologically feasible, and then elaborating on the mathematical proofs of two quantum mechanical no-go theorems, we explain why quantum theory might have profound implications for the scientific understanding of one's mental states, self identity, beliefs and free will. (shrink)

The current appeal of non-standard economic alternatives is backgrounded against the vulnerability of mainstream capitalism to meltdown and crisis as shown in recent times. There is an increasing number of governments, institutions, and civil societies (NGOs) that have been advocating economic systems, structures, or dynamics that would promote the good of the human person (dignity, personhood, values, and worth). People have started to realize that doing economics is not always within the realm of rationalized judgments and mathematized calculations (highly impersonal) (...) but must look after the human person, at all costs. This paper attempts to contribute to this agendum by employing an interdisciplinary dialogue between theology and economics; drawing a moral-cultural framework towards a compassion-based economics. Together with the positive traits of today’s economic alternatives and the salient features of Jesus’ praxis of compassion, this paper would offer fourteen (14) criteria as basis for what could be the most feasible base/locus for an economics of compassion. Eventually, what has been considered as the suitable base/locus is the Basic Ecclesial Communities (BECs). Hence, a BEC-based Economics of Compassion or BEC-EC. (shrink)

Hutto and Myin have proposed an account of radically enactive (or embodied) cognition (REC) as an explanation of cognitive phenomena, one that does not include mental representations or mental content in basic minds. Recently, Zahidi and Myin have presented an account of arithmetical cognition that is consistent with the REC view. In this paper, I first evaluate the feasibility of that account by focusing on the evolutionarily developed proto-arithmetical abilities and whether empirical data on them support the radical enactivist view. (...) I argue that although more research is needed, it is at least possible to develop the REC position consistently with the state-of-the-art empirical research on the development of arithmetical cognition. After this, I move the focus to the question whether the radical enactivist account can explain the objectivity of arithmetical knowledge. Against the realist view suggested by Hutto, I argue that objectivity is best explained through analyzing the way universal proto-arithmetical abilities determine the development of arithmetical cognition. (shrink)

The author has established a mathematical theory about the system of freedom in which components of freedom are ruled by the largest freedom principle, explaining how one invariant reality can be equated with the dynamical universe. Freedom as a whole is the reality, and components of freedom show variable phenomena and become a dynamic system. In freedom, component equality leads to sequence equality; therefore, various sequences coexist in the system. Because there are incompatible sequences for any sequence, the interior of (...) freedom cannot be a static sequence. In order for the system to be a whole, there must be some connecting sequences between any two sequences. Then, at every part of freedom, it is always possible to find a group of three independent sequences that, for most components, is located inside. For the sequence group, there is a sequence through which most components flow in and out. The most abundant three - sequence group and most abundant connecting sequence correspond to the space - time structure. Other incompatible sequences correspond to particles, and interactions between these sequences correspond to interactions between particles. The interactions have some symmetries similar to those in physics, such as SU (3) AND SU (2)×U (1), thus proving the feasibility of the hypothesis: the universe is equivalent with the system of freedom. (shrink)

Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein.

Lakatos's methodology, if analysed as belonging to the demarcationist-rationalist program launched by Popper gives some interesting conclusions concerning the feasibility of the project: (1) Rationalism cannot provide arguments against relativism. (2) A theory of scientific rationality cannot be defended without relying on scientific authorities. (3) A historical justification of scientific rationality does not show that the procedures that are rational according to the theory are truth-conducive.

Nền kinh tế nước ta đang chuyển biến mạnh mẽ từ nền kinh tế bao cấp sang kinh tế thị trường, nhất là từ khi nước ta gia nhập WTO. Đảng và chính phủ đã đề ra rất nhiều các chính sách để cải tiến các thể chế quản lý nền kinh tế và tài chính. Thị trường chứng khoán Việt Nam đã ra đời và đang đóng một vai trò quan trọng trong việc huy động vốn phục vụ cho (...) việc phát triển nền kinh tế quốc dân. Việc phân tích tác động của các chính sách đến nền kinh tế vĩ mô và nền tài chính, những đối sách của chính phủ ta và chính phủ các nước đối với cuộc khủng khoảng tài chính toàn cầu là các vấn đề rất thời sự đang thu hút sự chú ý của nhiều nhà kinh tế và toán học. (shrink)

I present and argue for a novel function-based account of feasibility - what I call the "Fitting Deliberation Account" - according to which whether an (individual or collective) action counts as feasible is a matter of whether it possesses those features that are required to make it a fitting object of practical reason or deliberation about what to do.

It is commonly taken for granted that questions of feasibility are highly relevant to our normative thinking – and perhaps especially our normative thinking about politics. But what exactly does this preoccupation with feasibility amount to, and in what forms if any is it warranted? This article aims to provide a critical introduction to, and clearer characterization of, the feasibility issue. I begin by discussing the question of how feasibility is to be understood. I then turn to the question of (...) feasibility’s role, suggesting that there are two quite different kinds of role questions that may be at play, though they are often run together: a question about feasibility’s normative significance; and a question about its proper use in informing our normative thinking. Finally, I consider how the feasibility issue differs from certain other related issues: the demandingness issue; the issue of whether “ought” implies “can;” and the “ideal versus non-ideal theory” issue. (shrink)

It is natural and relatively common to suppose that feasibility is a constraint on what we ought to do all-things-considered but not a constraint on what we ought to do as a matter of justice. I show that the combination of these claims entails an implausible picture of the relation between feasibility and desirability given an attractive understanding of the relation between what we ought to do as a matter of justice and what we ought to do all-things-considered.

It is common in political theory and practice to challenge normatively ambitious proposals by saying that their fulfillment is not feasible. But there has been insufficient conceptual exploration of what feasibility is, and very little substantive inquiry into why and how it matters for thinking about social justice. This paper provides one of the first systematic treatments of these issues, and proposes a dynamic approach to the relation between justice and feasibility that illuminates the importance of political imagination and (...) dynamic duties to expand agents’ power to fulfill ambitious principles of justice. (shrink)

There is a widespread belief that, conceptually, justice cannot require what we cannot achieve. This belief is sometimes used by defenders of so-called ‘non-ideal theories of justice’ to criticise so-called ‘ideal theories of justice’. I refer to this claim as ‘the feasibility constraint on the concept of justice’ and argue against it. I point to its various implausible implications and contend that a willingness to apply the label ‘unjust’ to some regrettable situations that we cannot fix is going to enhance (...) the action-guiding potential of a conception of justice, by providing an aspirational ideal. This is possible on the condition that, at all times, we cannot specify with certainty the limits of what is feasible for us collectively. The rejection of the feasibility constraint entails that there can be injustice without perpetrators; this is a theoretical price worth paying. (shrink)

The familiar complaint that some ambitious proposal is infeasible naturally invites the following response: Once upon a time, the abolition of slavery and the enfranchisement of women seemed infeasible, yet these things were actually achieved. Presumably, then, many of those things that seem infeasible in our own time may well be achieved too and, thus, turn out to have been perfectly feasible after all. The Appeal to History, as we call it, is a bad argument. It is not true (...) that if some desirable state of affairs was actually achieved, then it was feasible that it was achieved. “Actual” does not imply “feasible,” as we put it. Here is our objection. “Feasible” implies “not counterfactually fluky.” But “actual” does not imply “not counterfactually fluky.” So, “actual” does not imply “feasible.” While something like the Flukiness Objection is sometimes hinted at in the context of the related literature on abilities, it has not been developed in any detail, and both premises are inadequately motivated. We offer a novel articulation of the Flukiness Objection that is both more precise and better motivated. Our conclusions have important implications, not only for the admissible use of history in normative argument, but also by potentially circumscribing the normative claims that are applicable to us. (shrink)

Many of us feel internally conflicted in the face of certain normative claims that make infeasible demands: say, normative claims that demand that agents do what, given deeply entrenched objectionable character traits, they cannot bring themselves to do. On the one hand, such claims may seem false on account of demanding the infeasible, and insisting otherwise may seem to amount to objectionable unworldliness – to chasing “pies in the sky.” On the other hand, such claims may seem true in spite (...) of making infeasible demands, and insisting otherwise may seem to amount to treating the agents in question with undue lenience – to mistakenly letting them “off the hook.” What is going on? One possibility is that we are making a mistake. I explore the alternative hypothesis that our ambivalent reactions, far from involving any mistake, are entirely consistent and appropriate. Rather than some single privileged ought such that the idea that “ought” implies “feasible” is either true or false, there are simply different oughts that are supposed to be capable of operating in the service of, and playing distinct roles associated with, what I shall call different core normative practices. In particular, there is a) some salient core practice-serving ought for which it’s true that “ought” implies “feasible” and b) some other salient core practice-serving ought for which it’s false that “ought” implies “feasible.” I sketch a framework for understanding different core practice-serving oughts in general and then use this framework to consider which particular core practice-serving oughts might be capable of vindicating our ambivalence. I begin by considering and rejecting a prevalent and prima facie promising account according to which the relevant oughts are the prescriptive ought and the evaluative ought. I then propose a different account that holds that the oughts we need are instead the deliberative ought and the hypological ought. (shrink)

This study delves into the feasibility of introducing a market in Barangay Tanagan, Calatagan, Batangas, with a primary focus on assessing its viability and potential impact on local economic development. Utilizing survey questionnaires and employing statistical analysis techniques, the research gathered data from a carefully selected sample of 366 residents out of the total population of 4,224. The study evaluates both the potential benefits and challenges associated with establishing the proposed market. Findings, subjected to thorough analysis through methods such as (...) frequency counts and weighted mean calculations, provide valuable insights into discernible patterns, emerging trends, and relevant correlations. This analytical approach contributes to a pragmatic perspective that informs and guides future studies in the field. The study reveals an interest among respondents in the establishment of public markets in their vicinity, recognizing its potential to bolster the barangay and attract diverse opportunities. The anticipated positive impact extends to fostering economic growth and enhancing local commerce in Barangay Tanagan. Consequently, based on these compelling findings, the researchers advocate for a proposal outlining the establishment of a public market in Barangay Tanagan as a means of addressing the identified needs and aspirations of the community. (shrink)

Recent methodological debates regarding the place of feasibility considerations in normative political theory are hindered for want of a rigorous model of the feasibility frontier. To address this shortfall, I present an analysis of feasibility that generalizes the economic concept of a production possibility frontier and then develop a rigorous model of the feasibility frontier using the familiar possible worlds technology. I then show that this model has significant methodological implications for political philosophy. On the Target View, a political ideal (...) presents a long-term goal for morally progressive reform efforts and, thus, serves as an important reference point for our specification of normative political principles. I use the model to show that we can- not reasonably expect that adopting political ideals as long-term reform objectives will guide us toward the realization of morally optimal feasible states of affairs. I conclude by proposing that political philosophers turn their attention to the analysis of actual social failures rather than political ideals. (shrink)

Does ?ought? imply ?can? for collectives' obligations? In this paper I want to establish two things. The first, what a collective obligation means for members of the collective. The second, how collective ability can be ascertained. I argue that there are four general kinds of obligation, which devolve from collectives to members in different ways, and I give an account of the distribution of obligation from collectives to members for each of these kinds. One implication of understanding collective obligation and (...) ability in the proposed way is that a group can be blameworthy for failing to realize its obligation even when no member of the group is blameworthy for failing to realize her obligation. (shrink)

How feasible are conceptual engineering projects of social concepts that aim for the engineered concept to be widely adopted in ordinary everyday life? Predominant frameworks on the psychology of concepts that shape work on stereotyping, bias, and machine learning have grim implications for the prospects of conceptual engineers: conceptual engineering efforts are ineffective in promoting certain social-conceptual changes. Specifically, since conceptual components that give rise to problematic social stereotypes are sensitive to statistical structures of the environment, purely conceptual change (...) won’t be possible without corresponding world change. This tradition, however, tends to ignore that concepts don’t only encode statistical, but also causal information. Paying attention to this feature of concepts, I argue, shows that conceptual engineering is not only possible. There is an imperative to conceptually-engineer. (shrink)

A common objection to a proposal or theory in political philosophy is that it is not feasible to realise what it calls for. This is commonly taken to be sufficient to reject a proposal or theory: feasibility, on this common view, operates as a straightforward constraint on moral and political theory, whatever is not feasible is simply ruled out. This paper seeks to understand what we mean when we say that some proposal or outcome is or is not (...) feasible. It will argue that no single binary definition can be given. Rather, there is a whole range of possible specifications of the term ‘feasible’, each of which selects a range of facts of the world to hold fixed. No single one of these possible specifications, though, is obviously privileged as giving the appropriate understanding of ‘feasibility’ tout court. The upshot of my account of feasibility, then, will be that the common view of feasibility as a straightforward constraint cannot be maintained: in order to reject a moral theory, it will not be sufficient simply to say that it is not feasible. (shrink)

Many political philosophers hold the Feasible Alternatives Principle (FAP): justice demands that we implement some reform of international institutions P only if P is feasible and P improves upon the status quo from the standpoint of justice. The FAP implies that any argument for a moral requirement to implement P must incorporate claims whose content pertains to the causal processes that explain the current state of affairs. Yet, philosophers routinely neglect the need to attend to actual causal processes. (...) This undermines their arguments concerning moral requirements to reform international institutions. The upshot is that philosophers’ arguments must engage in causal analysis to a greater extent than is typical. -/- [Supplement: Handout available at http://db.tt/fyuVW3Xv]. (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)

This paper empirically raises and examines the question of ‘conceptual control’: To what extent are competent thinkers able to reason properly with new senses of words? This question is crucial for conceptual engineering. This prominently discussed philosophical project seeks to improve our representational devices to help us reason better. It frequently involves giving new senses to familiar words, through normative explanations. Such efforts enhance, rather than reduce, our ability to reason properly, only if competent language users are able to abide (...) by the relevant explanations, in language comprehension and verbal reasoning. This paper examines to what extent we have such ‘conceptual control’ in reasoning with new senses. The paper draws on psycholinguistic findings about polysemy processing to render this question empirically tractable and builds on recent findings from experimental philosophy to address it. The paper identifies a philosophically important gap in thinkers’ control over the key process of stereotypical enrichment and discusses how conceptual engineers can use empirical methods to work around this gap in conceptual control. The paper thus empirically demonstrates the urgency of the question of conceptual control and explains how experimental philosophy can empirically address the question, to render conceptual engineering feasible as an ameliorative enterprise. (shrink)

Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view (...) appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)

I argue that certain species of belief, such as mathematical, logical, and normative beliefs, are insulated from a form of Harman-style debunking argument whereas moral beliefs, the primary target of such arguments, are not. Harman-style arguments have been misunderstood as attempts to directly undermine our moral beliefs. They are rather best given as burden-shifting arguments, concluding that we need additional reasons to maintain our moral beliefs. If we understand them this way, then we can see why moral beliefs are vulnerable (...) to such arguments while mathematical, logical, and normative beliefs are not—the very construction of Harman-style skeptical arguments requires the truth of significant fragments of our mathematical, logical, and normative beliefs, but requires no such thing of our moral beliefs. Given this property, Harman-style skeptical arguments against logical, mathematical, and normative beliefs are self-effacing; doubting these beliefs on the basis of such arguments results in the loss of our reasons for doubt. But we can cleanly doubt the truth of morality. (shrink)

Disagreements that resist rational resolution, often termed “deep disagreements”, have been the focus of much work in epistemology and informal logic. In this paper, I argue that they also deserve the attention of philosophers of mathematics. I link the question of whether there can be deep disagreements in mathematics to a more familiar debate over whether there can be revolutions in mathematics. I propose an affirmative answer to both questions, using the controversy over Shinichi Mochizuki’s work on (...) the abc conjecture as a potential example of both phenomena. I conclude by investigating the prospects for the resolution of mathematical deep disagreements in virtue-theoretic approaches to informal logic and mathematical practice. (shrink)

Plausibly, mathematical claims are true, but the fundamental furniture of the world does not include mathematical objects. This can be made sense of by providing mathematical claims with paraphrases, which make clear how the truth of such claims does not require the fundamental existence of mathematical objects. This paper explores the consequences of this type of position for explanatory structure. There is an apparently straightforward relationship between this sort of structure, and the logical sort: i.e. logically complex claims are explained (...) by logically simpler ones. For example, disjunctions are explained by their (true) disjuncts, while generalizations are explained by their (true) instances. This would seem as plausible in the case of mathematics as elsewhere. Also, it would seem to be something that the anti-realist approaches at issue would want to preserve. It will be argued, however, that these approaches cannot do this: they lead not merely to violations of the familiar principles relating logical and explanatory structure, but even to reversals of these. That is, there are cases where generalizations explain their instances, or disjunctions their disjuncts. (shrink)

The purpose of this article is to examine the feasibility and working of the conciliatory means for settlement of intellectual property disputes in Jordan. Arbitration is the principal mechanism used.

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays (...) out and compares these options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)

There is a noticeable gap between results of cognitive neuroscientific research into basic mathematical abilities and philosophical and empirical investigations of mathematics as a distinct intellectual activity. The paper explores the relevance of a Wittgensteinian framework for dealing with this discrepancy.

Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of mathematical theorems can cover (...) at most one mathematical universe. Indispensability arguments may thus lose their central role in the debate about mathematical ontology. (shrink)

Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only used (...) to express mathematical concepts—they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition. (shrink)

The paper discusses Hilbert mathematics, a kind of Pythagorean mathematics, to which the physical world is a particular case. The parameter of the “distance between finiteness and infinity” is crucial. Any nonzero finite value of it features the particular case in the frameworks of Hilbert mathematics where the physical world appears “ex nihilo” by virtue of an only mathematical necessity or quantum information conservation physically. One does not need the mythical Big Bang which serves to concentrate all (...) the violations of energy conservation in a “safe”, maximally remote point in the alleged “beginning of the universe”. On the contrary, an omnipresent and omnitemporal medium obeying quantum information conservation rather than energy conservation permanently generates action and thus the physical world. The utilization of that creation “ex nihilo” is accessible to humankind, at least theoretically, as long as one observes the physical laws, which admit it in their new and wider generalization. One can oppose Hilbert mathematics to Gödel mathematics, which can be identified as all the standard mathematics until now featureable by the Gödel dichotomy of arithmetic to set theory: and then, “dialectic”, “intuitionistic”, and “Gödelian” mathematics within the former, according to a negative, positive, or zero value of the distance between finiteness and infinity. A mapping of Hilbert mathematics into pseudo-Riemannian space corresponds, therefore allowing for gravitation to be interpreted purely mathematically and ontologically in a Pythagorean sense. Information and quantum information can be involved in the foundations of mathematics and linked to the axiom of choice or alternatively, to the field of all rational numbers, from which the pair of both dual and anti-isometric Peano arithmetics featuring Hilbert arithmetic are immediately inferable. Noether’s theorems (1918) imply quantum information conservation as the maximally possible generalization of the pair of the conservation of a physical quantity and the corresponding Lie group of its conjugate. Hilbert mathematics can be interpreted from their viewpoint also after an algebraic generalization of them. Following the ideas of Noether’s theorem (1918), locality and nonlocality can be realized both physically and mathematically. The “light phase of the universe” can be linked to the gap of mathematics and physics in the Cartesian organization of cognition in Modernity and then opposed to its “dark phase”, in which physics and mathematics are merged. All physical quantities can be deduced from only mathematical premises by the mediation of the most fundamental physical constants such as the speed of light in a vacuum, the Planck and gravitational constants once they have been interpreted by the relation of locality and nonlocality. (shrink)

In this paper I make some distinctions, which I hope are of help for Laura Valentini and others. Are the recommendations of a theory of what the government should do possible and are they feasible? Is the project of the theorist possibility-sensitive and is the project feasibility-sensitive?

We demonstrate how real progress can be made in the debate surrounding the enhanced indispensability argument. Drawing on a counterfactual theory of explanation, well-motivated independently of the debate, we provide a novel analysis of ‘explanatory generality’ and how mathematics is involved in its procurement. On our analysis, mathematics’ sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies easier to grasp and reason with for creatures like us. This gives precise content (...) to key intuitions traded in the debate, regarding mathematics’ procurement of explanatory generality, and adjudicates unambiguously in favour of the nominalist, at least as far as explanatory generality is concerned. (shrink)

I discuss Gheaus's (2013) argument against the claim that the requirements of justice are not constrained by feasibility concerns. I show that the general strategy exemplified by this argument is not only dialectically puzzling, but also imposes a heavy cost on theories of justice -- puzzling because it simply sidesteps a presupposition of any plausible formulation of the so-called "feasibility requirement"; costly because it it deprives justice of its normative implications for action. I also show that Gheaus's attempt to recover (...) this normative force presupposes an epistemic dimension to the feasibility requirement that most proponents of that requirement would reject. (shrink)

Published in 1903, this book was the first comprehensive treatise on the logical foundations of mathematics written in English. It sets forth, as far as possible without mathematical and logical symbolism, the grounds in favour of the view that mathematics and logic are identical. It proposes simply that what is commonly called mathematics are merely later deductions from logical premises. It provided the thesis for which _Principia Mathematica_ provided the detailed proof, and introduced the work of Frege (...) to a wider audience. In addition to the new introduction by John Slater, this edition contains Russell's introduction to the 1937 edition in which he defends his position against his formalist and intuitionist critics. (shrink)

The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathematical and moral disagreement is not as straightforward as those arguments present it. In particular, I (...) argue that pluralist accounts of mathematics render fundamental mathematical disagreements compatible with mathematical realism in a way in which moral disagreements and moral realism are not. 11. (shrink)

Can agents be duty-bound towards ends that are infeasible? Some scholars have endorsed a “feasibility constraint” on justice and answered that we cannot be duty-bound to bring about the infeasible. In this paper, I question whether the feasibility constraint on justice should still be endorsed and whether we are duty-bound to pursue some aims regardless of this constraint. I ask: Can an ethical agent be duty-bound to work towards bringing about a state of affairs that is desirable but infeasible? I (...) consider the climate crisis: climate justice may require us to work towards ends that may be infeasible, such as maintaining global climate warming below 1.5°C or mitigating climate change in a way that treats all people affected fairly. I argue that we may be duty-bound or obliged to work towards some desirable goals when faced with the infeasible and conclude that we should reject feasibility constraints on climate justice. (shrink)

Callard (2007) argues that it is metaphysically possible that a mathematical object, although abstract, causally affects the brain. I raise the following objections. First, a successful defence of mathematical realism requires not merely the metaphysical possibility but rather the actuality that a mathematical object affects the brain. Second, mathematical realists need to confront a set of three pertinent issues: why a mathematical object does not affect other concrete objects and other mathematical objects, what counts as a mathematical object, and how (...) we can have knowledge about an unchanging object. (shrink)