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)

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)

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 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)

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)

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)

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)

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 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)

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)

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)

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)

An important theme in recent experimental philosophy is that certain judgements (e.g. our judgements involving intentional action and causation) exhibit a kind of normative penetration whereby, in spite of a not-obviously-normative subject matter, they turn out to be sensitive to, and co-vary with, our normative attitudes in interesting and surprising ways. We present the results of several new experimental studies that suggest that our judgements about feasibility also appear to exhibit this kind of normative penetration in at least some cases; (...) that the best explanation of it involves attributing to subjects a certain kind of extensional error that results from conflating the question at hand with another question that is more salient given their normative attitudes; and that these conclusions have significant, though not straightforward, implications for our understanding of both normative penetration and feasibility, respectively. (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 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.

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)

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)

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)

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)

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.

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)

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)

There is a major debate as to whether there are non-causal mathematical explanations of physical facts that show how the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws. We focus on Marc Lange’s account of distinctively mathematical explanations to argue that purported mathematical explanations are essentially causal explanations in disguise and are no different from ordinary applications of mathematics. This is because these explanations work not by appealing to what (...) the world must be like as a matter of mathematical necessity but by appealing to various contingent causal facts. (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)

K denotes both the knowledge predicate satisfied by every currently known theorem and the finite set of all currently known theorems. The set K is time-dependent, publicly available, and contains theorems both from formal and constructive mathematics. Any theorem of any mathematician from past or present forever belongs to K. Mathematical statements with known constructive proofs exist in K separately and form the set K_c⊆K. We assume that mathematical sets are atemporal entities. They exist formally in ZFC theory although (...) their properties can be time-dependent (when they depend on K) or informal. The true statement "K is non-empty" is outside K forever because K is not a formal set. Paul Cohen proved in 1963 that the equality 2^{\aleph_0}=\aleph_1 is independent of ZFC. Before 1963, the statement "There is a known constructively defined integer n⩾1 such that 2^{\aleph_0}=\aleph_n" was outside K. Since 1963, this statement is outside K forever. The true statement "There exists a set X⊆{1,...,49} such that card(X)=6 and X never occurred as the winning six numbers in the Polish Lotto lottery" refers to the current non-mathematical knowledge and is outside K forever. In Martin-Löf's terminology, every currently known theorem is actually true whereas every theorem (known or unknown) is potentially true. Actual truth is knowledge dependent and tensed. Potential truth is knowledge independent and tenseless. Algorithms always terminate. We explain the distinction between algorithms whose existence is provable in ZFC and constructively defined algorithms which are currently known. By using this distinction, we obtain non-trivially true statements on decidable sets X⊆N that belong to constructive and informal mathematics and refer to the current mathematical knowledge on X. For every such statement, its truth concerning the current mathematical knowledge is implied by a true statement of the form: (the conjunction of i conditions of the form α∈K)∧(the conjunction of j conditions of the form β∉K)∧(the conjunction of k conditions of the form γ∈K_c)∧(the conjunction of l conditions of the form δ∉K_c), where i,j,k,l∈N and α,β,γ,δ are mathematical statements. In constructive mathematics and the traditional Brouwerian intuitionism, the truth of a mathematical statement means that we know a constructive proof. Therefore, the truth of a mathematical statement depends on time, where the statement is formally stated in the classical mathematics without the predicate K. In this article, mathematical statements on decidable sets X⊆N refer to time because they refer to the current mathematical knowledge on X. They cannot be formally stated in the classical mathematics without the predicate K and their logical values may change in time. For a set X⊆N whose infiniteness is false or unproven, we define which elements of X are classified as known. No known set X⊆N satisfies Conditions (1)-(4) and is widely known in number theory or naturally defined, where this term has only informal meaning. (1) A known algorithm with no input returns an integer n satisfying card(X)<ω ⇒ X⊆(-∞,n]. (2) A known algorithm for every k∈N decides whether or not k∈X. (3) For every known algorithm A with no input, the statement "A returns the logical value of the statement card(X)=ω" is outside K. (4) There are many elements of X and it is conjectured, though so far unproven, that X is infinite. (5) X is naturally defined. The infiniteness of X is false or unproven. X has the simplest definition among known sets Y⊆N with the same set of known elements. No set X⊆N will satisfy Conditions (1)-(4) forever, if for every algorithm with no input, at some future day, a computer will be able to execute this algorithm in 1 second or less. The physical limits of computation disprove the assumption of the above statement. We prove that the set X={n∈N: the interval [-1,n] contains more than 29.5+(11!/(3n+1))∙sin(n) primes of the form k!+1} satisfies Conditions (1)-(5) except the requirement that X is naturally defined. We present a table that shows satisfiable conjunctions of the form #(Condition1)∧(Condition 2)∧#(Condition 3)∧(Condition 4)∧#(Condition 5), where # denotes the negation ¬ or the absence of any symbol. We prove that there exists a naturally defined set C⊆N which satisfies the following conditions (6)-(11). (6) A known and simple algorithm for every k∈N decides whether or not k∈C. (7) For every known algorithm A with no input, the statement "A returns the logical value of the statement card(C)=ω" is outside K. (8) For every known algorithm A with no input, the statement "A returns the logical value of the statement card(N\C)=ω" is outside K. (9) It is conjectured, though so far unproven, that C is infinite. (10) For every known algorithm A with no input, the statement "A returns an integer n satisfying card(C)<ω ⇒ C⊆(-∞,n]" is outside K. (11) For every known algorithm A with no input, the statement "A returns an integer m satisfying card(N\C)<ω ⇒ N\C⊆(-∞,m]" is outside K. The set C={k∈N: 2^{2^k}+1 is composite} is naturally defined and satisfies conditions (6)-(11). (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.

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.

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)

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)

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)

In these days, there is an increasing technological development in intelligent tutoring systems. This field has become interesting to many researchers. In this paper, we present an intelligent tutoring system for teaching mathematics that help students understand the basics of math and that helps a lot of students of all ages to understand the topic because it's important for students of adding and subtracting. Through which the student will be able to study the course and solve related problems. An (...) evaluation of the intelligent tutoring systems was carried out and the results were encouraging. (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)

I introduce an argument for Platonism based on intra-mathematical explanation: the explanation of one mathematical fact by another. The argument is important for two reasons. First, if the argument succeeds then it provides a basis for Platonism that does not proceed via standard indispensability considerations. Second, if the argument fails it can only do so for one of three reasons: either because there are no intra-mathematical explanations, or because not all explanations are backed by dependence relations, or because some form (...) of noneism-the view according to which non-existent entities possess properties and stand in relations-is true. The argument thus forces a choice between nominalism without noneism, intra-mathematical explanation and a backing conception of explanation. You can have any two, but not all three. (shrink)

This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal (...) science, mathematical ontology: what does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies. (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)

Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and computer science. This introduction provides some background to their work.

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)

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)

William L. Rowe’s argument against divine freedom has drawn considerable attention from theist philosophers. One reply to Rowe’s argument that has emerged in the recent literature appeals to modified accounts of libertarian freedom which have the result that God may be free even if he necessarily actualizes the best possible world. Though in many ways attractive, this approach appears to lead to the damning consequence of modal collapse i.e., that the actual world is the only possible world. But appearances can (...) be deceiving, and in this paper I argue that the threat of modal collapse dissolves when we consider Alvin Plantinga’s critique of the purportedly Leibnizian notion that God can actualize any possible world, and incorporate the implications of this critique into the divine freedom debate. Developing a suggestion by Edward R. Wierenga, I argue first that the modal collapse objection fails within a Molinist context, and then I extend the discussion beyond that context to show that the objection also fails on the assumption that Molinism is false. (shrink)

_ Source: _Volume 6, Issue 2-3, pp 120 - 142 This paper aims to contribute to the debate over epistemic versus non-epistemic readings of the ‘hinges’ in Wittgenstein’s _On Certainty_. I follow Marie McGinn’s and Daniele Moyal-Sharrock’s lead in developing an analogy between mathematical sentences and certainties, and using the former as a model for the latter. However, I disagree with McGinn’s and Moyal-Sharrock’s interpretations concerning Wittgenstein’s views of both relata. I argue that mathematical sentences as well as certainties are (...) true and are propositions; that some of them can be epistemically justified; that in some senses they are not prior to empirical knowledge; that they are not ineffable; and that their primary function is epistemic as much as it is semantic. (shrink)

An advanced degree in criminal justice can open doors far outside traditional criminal justice practice, making it a highly in-demand course. This current study aimed to assess the viability of J.H. Cerilles State College to offer a Master of Science in Criminal Justice with Specialization in Criminology (MSCJ) in 2021. A descriptive survey type of research was employed as the methodology for this study. The 215 respondents from students, graduates, and professionals in the field from private and government institutions were (...) chosen via random sampling to answer the online survey questionnaire. Management viability, market viability, operation viability, location viability, and industry viability were all evaluated. There was evidence to suggest that offering MSCJ in the college is doable, provided that the curriculum will be tailored to account for community needs and industry demands. In meeting future demand for this field of study, constant improvements in school facilities, instruction, research, and extension should be made. The methodology utilized in this study can be replicated in a future feasibility assessment, which may also include other areas that were not covered in this study. z. (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)

Moral obligation and permissibility are usually thought to be interdefinable. Following the pattern of the duality definitions of necessity and possibility, we have that something’s being permissible could be defined as its not being obligatory to not do it. And that something’s being obligatory could be defined as its not being permissible to not do it. In this paper, I argue that neither direction of this alleged interdefinability works. Roughly, the problem is that a claim that some act is obligatory (...) or permissible entails that there is a moral law, whereas a negative claim that some act is not obligatory or not permissible does not. Nevertheless, one direction of the interdefinability can potentially be salvaged. I argue that, if we do not require the conceptual possibility of moral dilemmas, then there is a way to plausibly define obligation in terms of permissibility. I conclude that permissibility is the only feasible deontic primitive. (shrink)

We present an approach in which ancient Greek mathematical proofs by Hippocrates of Chios and Euclid are addressed as a form of (guided) intentional reasoning. Schematically, in a proof, we start with a sentence that works as a premise; this sentence is followed by another, the conclusion of what we might take to be an inferential step. That goes on until the last conclusion is reached. Guided by the text, we go through small inferential steps; in each one, we go (...) through an autonomous reasoning process linking the premise to the conclusion. The reasoning process is accompanied by a metareasoning process. Metareasoning gives rise to a feeling-knowing of correctness. In each step/cycle of the proof, we have a feeling-knowing of correctness. Overall, we reach a feeling of correctness for the whole proof. We suggest that this approach allows us to address the issues of how a proof functions, for us, as an enabler to ascertain the correctness of its argument and how we ascertain this correctness. (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)