Domain extension in mathematics occurs whenever a given mathematical domain is augmented so as to include new elements. Manders argues that the advantages of important cases of domain extension are captured by the model-theoretic notions of existential closure and model completion. In the specific case of domain extension via ideal elements, I argue, Manders’s proposed explanation does not suffice. I then develop and formalize a different approach to domain extension based on Dedekind’s Habilitationsrede, to which Manders’s (...) account is compared. I conclude with an examination of three possible stances towards extensions via ideal elements. (shrink)

(...) However, whether we chose a weak or strong approximation, the set would not make any sense at all, if (once more) this choice would not be justified in either temporal or spatial sense or given the context of possible applicability of the set in different circumstances. This would obviously represent a dualism in itself as we would (for instance) posit and apply a full identity-equality-equivalence of x and y when applying Newtonian physics to certain observations we make (it would (...) be the case of neural correlates), and we would posit and apply a non - identity-equality-equivalence of x and y when applying Quantum mechanics to other observations. Following this dualism in and of theories, the same sate would need to be slightly modified: U2:(x~y)∪(x≈y); U3:(x~y)∩(x≈y); U4a:(x~y)⊆(x≈y) vs. U4b:(x~y)⊇(x≈y). (shrink)

The paper discusses some changes in Bolzano's definition of mathematics attested in several quotations from the Beyträge, Wissenschaftslehre and Grössenlehre: is mathematics a theory of forms or a theory of quantities? Several issues that are maintained throughout Bolzano's works are distinguished from others that were accepted in the Beyträge and abandoned in the Grössenlehre. Changes are interpreted as a consequence of the new logical theory of truth introduced in the Wissenschaftslehre, but also as a consequence of the overcome (...) of Kant's terminology, and of the radicalization of Bolzano's anti‐Kantianism. Bolzano's evolution is understood as a coherent move, once the criticism expressed in the Beyträge on the notion of quantity is compared with a different and larger notion of quantity that Bolzano developed already in 1816. This discussion is enriched by the discovery that two unknown texts mentioned by Bolzano in the Beyträge can be identified with works by von Spaun and Vieth respectively. Bolzano's evolution is interpreted as a radicalization of the criticism of the Kantian definition of mathematics and as an effect of Bolzano's unaltered interest in the Leibnizian notion of mathesis universalis. As a conclusion, the author claims that Bolzano never abandoned his original idea of considering mathematics as a scientia universalis, i.e. as the science of quantities in general, and suggests that the question of ideal elements in mathematics, apart from being a main reason for the development of a new logical theory, can also be considered as a main reason for developing a different definition of quantity. (shrink)

Any philosophy of science ought to have something to say about the nature of mathematics, especially an account like constructive empiricism in which mathematical concepts like model and isomorphism play a central role. This thesis is a contribution to the larger project of formulating a constructive empiricist account of mathematics. The philosophy of mathematics developed is fictionalist, with an anti-realist metaphysics. In the thesis, van Fraassen's constructive empiricism is defended and various accounts of mathematics are considered (...) and rejected. Constructive empiricism cannot be realist about abstract objects; it must reject even the realism advocated by otherwise ontologically restrained and epistemologically empiricist indispensability theorists. Indispensability arguments rely on the kind of inference to the best explanation the rejection of which is definitive of constructive empiricism. On the other hand, formalist and logicist anti-realist positions are also shown to be untenable. It is argued that a constructive empiricist philosophy of mathematics must be fictionalist. Borrowing and developing elements from both Philip Kitcher's constructive naturalism and Kendall Walton's theory of fiction, the account of mathematics advanced treats mathematics as a collection of stories told about an ideal agent and mathematical objects as fictions. The account explains what true portions of mathematics are about and why mathematics is useful, even while it is a story about an ideal agent operating in an ideal world; it connects theory and practice in mathematics with human experience of the phenomenal world. At the same time, the make-believe and game-playing aspects of the theory show how we can make sense of mathematics as fiction, as stories, without either undermining that explanation or being forced to accept abstract mathematical objects into our ontology. All of this occurs within the framework that constructive empiricism itself provides the epistemological limitations it mandates, the semantic view of theories, and an emphasis on the pragmatic dimension of our theories, our explanations, and of our relation to the language we use. (shrink)

This collection of articles was written over the last 10 years and the most important and longest within the last year. Also I have edited them to bring them up to date (2016). The copyright page has the date of this first edition and new editions will be noted there as I edit old articles or add new ones. All the articles are about human behavior (as are all articles by anyone about anything), and so about the limitations of having (...) a recent monkey ancestry (8 million years or much less depending on viewpoint) and manifest words and deeds within the framework of our innate psychology as presented in the table of intentionality. As famous evolutionist Richard Leakey says, it is critical to keep in mind not that we evolved from apes, but that in every important way, we are apes. If everyone was given a real understanding of this (i.e., human ecology and psychology) in school, maybe civilization would have a chance. -/- In my view these articles and reviews have many novel and highly useful elements, in that they use my own version of the recently (ca. 1980’s) developed dual systems view of our brain and behavior to lay out a logical system of rationality (personality, psychology, mind, language, behavior, thought, reasoning, reality etc.) that is sorely lacking in the behavioral sciences (psychology, philosophy, literature, politics, anthropology, history, economics, sociology etc.). The philosophy centers around the two writers I have found the most important, Ludwig Wittgenstein and John Searle, whose ideas I combine and extend within the dual system (two systems of thought) framework that has proven so useful in recent thinking and reasoning research. As I note, there is in my view essentially complete overlap between philosophy, in the strict sense of the enduring questions that concern the academic discipline, and the descriptive psychology of higher order thought (behavior). Once one has grasped Wittgenstein’s insight that there is only the issue of how the language game is to be played, one determines the Conditions of Satisfaction (what makes a statement true or satisfied etc.) and that is the end of the discussion. -/- Now that I think I understand how the games work I have mostly lost interest in philosophy, which of course is how Wittgenstein said it should be. But since they are the result of our innate psychology, or as Wittgenstein put, it due to the lack of perspicuity of language, the problems run throughout all human discourse, so there is endless need for philosophical analysis, not only in the ‘human sciences’ of philosophy, sociology, anthropology, political science, psychology, history, literature, religion, etc., but in the ‘hard sciences’ of physics, mathematics, and biology. It is universal to mix the language game questions with the real scientific ones as to what the empirical facts are. Scientism is ever present and the master has laid it before us long ago, i.e., Wittgenstein (hereafter W) beginning principally with the Blue and Brown Books in the early 1930’s. -/- "Philosophers constantly see the method of science before their eyes and are irresistibly tempted to ask and answer questions in the way science does. This tendency is the real source of metaphysics and leads the philosopher into complete darkness." (BBB p18) -/- Nevertheless, a real understanding of Wittgenstein’s work, and hence of how our psychology functions, is only beginning to spread in the second decade of the 21st century, due especially to P.M.S. Hacker (hereafter H) and Daniele Moyal-Sharrock (hereafter DMS),but also to many others, some of the more prominent of whom I mention in the articles. -/- When I read ‘On Certainty’ a few years ago I characterized it in an Amazon review as the Foundation Stone of Philosophy and Psychology and the most basic document for understanding behavior, and about the same time DMS was writing articles noting that it had solved the millennia old epistemological problem of how we can know anything for certain. I realized that W was the first one to grasp what is now characterized as the two systems or dual systems of thought, and I generated a dual systems (S1 and S2) terminology which I found to be very powerful in describing behavior. I took the small table that John Searle (hereafter S) had been using, expanded it greatly, and found later that it integrated perfectly with the framework being used by various current workers in thinking and reasoning research. -/- Since they were published individually, I have tried to make the book reviews and articles stand by themselves, insofar as possible, and this accounts for the repetition of various sections, notably the table and its explanation. I start with a short article that presents the table of intentionality and briefly describes its terminology and background. Next, is by far the longest article, which attempts a survey of the work of W and S as it relates to the table and so to an understanding or description (not explanation as W insisted) of behavior. -/- The key to everything about us is biology, and it is obliviousness to it that leads millions of smart educated people like Obama, Chomsky, Clinton and the Pope to espouse suicidal utopian ideals that inexorably lead straight to Hell On Earth. As W noted, it is what is always before our eyes that is the hardest to see. We live in the world of conscious deliberative linguistic System 2, but it is unconscious, automatic reflexive System 1 that rules. This is Searle’s The Phenomenological Illusion (TPI), Pinker’s Blank Slate and Tooby and Cosmides Standard Social Science Model. Democracy and equality are wonderful ideals, but without strict controls, selfishness and stupidity gain the upper hand and soon destroy any nation and any world that adopts them. The monkey mind steeply discounts the future, and so we sell our children’s heritage for temporary comforts. -/- The astute may wonder why we cannot see System 1 at work, but it is clearly counterproductive for an animal to be thinking about or second guessing every action, and in any case there is no time for the slow, massively integrated System 2 to be involved in the constant stream of split second ‘decisions’ we must make. As W noted, our ‘thoughts’ (T1 or the thoughts of System 1) must lead directly to actions. It is my contention that the table of intentionality (rationality, mind, thought, language, personality etc.) that features prominently here describes more or less accurately, or at least serves an heuristic for how we think and behave, and so it encompasses not merely philosophy and psychology but everything else (history, literature, mathematics, politics etc.). Note especially that intentionality and rationality as I (along with Searle, Wittgenstein and others) view it,includes both conscious deliberative System 2 and unconscious automated System 1 actions or reflexes. -/- Thus all the articles, like all behavior, are intimately connected if one knows how to look at them. As I note, The Phenomenological Illusion (oblivion to our automated System 1) is universal and extends not merely throughout philosophy but throughout life. I am sure that Chomsky, Obama, Zuckerberg and the Pope would be incredulous if told that they suffer from the same problem as Hegel, Husserl and Heidegger, but it’s clearly true. While the phenomenologists only wasted a lot of people’s time, they are wasting the earth. -/- My writings are available in updated versions as paperbacks and Kindles on Amazon. -/- Talking Monkeys: Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet - Articles and Reviews 2006-2017 (2017) ASIN B071HVC7YP. -/- The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle--Articles and Reviews 2006-2016 (2017) ASIN B071P1RP1B. -/- Suicidal Utopian Delusions in the 21st century: Philosophy, Human Nature and the Collapse of Civilization - Articles and Reviews 2006-2017 (2017) 2nd printing with corrections (Feb 2018) ASIN B0711R5LGX -/- Suicide by Democracy: an Obituary for America and the World (2018) ASIN B07CQVWV9C . (shrink)

In this dissertation, I present a novel account of the components that have a peculiar epistemic role in our scientific inquiries, since they contribute to establishing a form of coordination. The issue of coordination is a classic epistemic problem concerning how we justify our use of abstract conceptual tools to represent concrete phenomena. For instance, how could we get to represent universal gravitation as a mathematical formula or temperature by means of a numerical scale? This problem is particularly pressing when (...) justification for using these abstract tools comes, in part or entirely, from knowledge which is not independent from them, thus leading to threats of circularity. Achieving coordination between some abstract conceptual tools and the concrete phenomena that they are supposed to represent is usually a complex process, which involves several epistemic components. Some of these components eventually provide stable conditions for applying those abstract representations to concrete phenomena. It is in this sense of providing certain conditions of applicability that different philosophical traditions, as well as some contemporary reappraisals, view these components as constitutive or a priori. In this work, I present a new gradualist, contextualist, and relational approach to understand these constitutive components of scientific inquiry. It is gradualist inasmuch as the degree to which some component is constitutive depends on three quantifiable features: quasi-axiomaticity, generative potential, and empirical shielding. Since the quantification of these three features impinges on the history and practice of using these components in a scientific context, my approach is a contextualist one. Finally, my approach is relational in a double sense: first, it identifies ordinal relationships among epistemic components with respect to their constitutive character; second, these relationships are relative to a scientific framework of inquiry. After introducing my account and a classic example of constitutively a priori principles, i.e., Friedman’s (2001) analysis of Newtonian mechanics, I turn to my own case studies to demonstrate the advantages of my approach. Firstly, I discuss Okasha’s (2018) view of endogenization as a pervasive theoretical strategy in evolutionary biology and suggest that the constitutive character of the core Darwinian principles progressively increases with endogenization. Secondly, I apply a conceptual distinction between two varieties or scopes of coordination – general coordination and coordination in measurement – to Ohm’s work on electrical conductivity. This distinction allows me to pinpoint to what extent components along different dimensions (e.g., instrumentation, measurement, theorising, etc.) were constitutive of the forms of coordination which Ohm relied on. Thirdly, I discuss the epistemic function of the Hardy-Weinberg principle in the history and practice of population genetics. I assess this principle in terms of my account and identify approximation and stability as two components that are highly constitutive, in that they contribute to justifying its use in population genetics. Finally, applying my account to these case studies enables me to identify at least three qualitatively different types of constitutive components: domain-specific theoretical principles, material components, and domain-independent assumptions underlying reasoning abilities. In the light of my results, I draw some general conclusions on epistemic justification and scientific knowledge. (shrink)

Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late nineteenth century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis (in the sense of the infinitesimal calculus) received much attention in the nineteenth century. They helped instigate what Hans Hahn called a “crisis of intuition”, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this “crisis” (...) as follows: Mathematicians had for a long time made use of supposedly geometric evidence as a means of proof in much too naive and much too uncritical a way, till the unclarities and mistakes that arose as a result forced a turnabout. Geometrical intuition was now declared to be inadmissible as a means of proof... (p. 67) Avoiding geometrical evidence, Hahn continued, mathematicians aware of this crisis pursued what he called “logicization”, “when the discipline requires nothing but purely logical fundamental concepts and propositions for its development.” On this view, an epistemically ideal mathematics would minimize, or avoid altogether, appeals to visual representations. This would be a radical reformation of past practice, necessary, according to its advocates, for avoiding “unclarities and mistakes” like the one exposed by Peano. (shrink)

Techne and Truth. The problem of techne in the dispute between Gorgias and Plato -/- The source of the problem matter of the book is the Plato’s dialogue „Gorgias”. One of the main subjects of the discussion carried out in this multi-aspect work is the issue of the art of rhetoric. In the dialogue the contemporary form of the art of rhetoric, represented by Gorgias, Polos and Callicles, is confronted with Plato’s proposal of rhetoric and concept of art (techne). The (...) ingenuous and dramatic structure of the dialogue composed of three acts, in each of which Socrates’ consecutive interlocutors emerge, is difficult to interpret. It seems, however, that even though the first conversation between Socrates and Gorgias constitutes a small part of the dialogue, it forms the foundations for the discussion. This part, dealing with such important issues as the subject of art, knowledge, power and relation between art and good, is the starting point for the conversations which follow. What do the differences between the Gorgias’ and Plato’s concepts of art consist in? Any attempt to answer the question requires presentation of the background for the discussion taking place in Plato’s work as well as the reconstruction of Gorgias’ vision of the art of rhetoric and art in general, defining the relation between art and knowledge. The latter may be undertaken on the basis of two paraphrases of the treatise “On Non-existence”, two epideictic speeches entitled “Helen” and “Palamedes” and some smaller fragments which have been preserved from Gorgias’ creative output. The treatise “On Non–existence”, very paradoxical in its philosophical meaning, testifies to new, sophistic thinking represented by Gorgias of Leontinoi. The treatise is the polemic and attack on Eleatism based on certain principles and methods elaborated by the Eleatics themselves. It also shows that Gorgias – using traditional Eleatic schemes – is opposed to this tradition. Even though the treatise does not discuss the problem of art directly, it anticipates the considerations from epideictic speeches. “Helen” and “Palamedes” constitute further stages in the development of Gorgias’ thought and concept of art combining typically epideictic elements with some philosophical convictions. Gorgias refers to the Eleatic theme considering the issues of truth, knowledge and belief, yet defining them in the empirical spirit. Solutions in the cognitive sphere determine the horizon of Gorgias’ rhetoric. The word is separated from the existence. Human cognition is limited and therefore the word and visual images affect the belief. This promotes the great importance of rhetoric, painting and sculpture, i.e. the areas which influence in a specific way – creating the “delusion” (ἀπάτη). Through idealization art provides pleasure, which is its objective. Therefore the task of art is to create delusion. The rhetor of Leontinoi discusses the issue of “how?” to use the word ‘rhetoric’. Gorgias is aware that rhetoric may be used to talk about what is unknown and that technical correctness of a rhetoric statement is not identical with the truthfulness of the proclaimed message. Mentioning it, Gorgias shows his respect to the truth as a value which should be respected in speech on the one hand, and on the other emphasizes that the truth in itself has no persuasive power. For this reason it is not enough to proclaim the truth, but, to convince the listeners, the orator should possess the skill of speaking. What is then the relation between truth and art emerging from epideictic speeches? Gorgias undoubtedly emphasizes the formal aspect of speech, as rhetoric is art, thus an ability based on certain principles. According to Gorgias, the truth is the value which should be respected in speeches. Rhetoric is able to create a fictive world, as a great range of reality remains outside direct human experience. Rhetoric does not oppose the truth but discussing not only what is directly given concerns the area outside the sphere of direct experience. This concept of rhetoric is the basis of the discussion about art carried out in the dialogue “Gorgias”. Even though the first of the discussions in the dialogue – the conversation between Socrates and Gorgias – is directly concerned with rhetoric itself, numerous remarks testify to its much wider character. This is best exemplified by the way of carrying out the conversation by Socrates, who, asking questions about rhetoric practised by Gorgias, throughout the whole conversation compares it with other skills. Additionally, the manner and the structure of the conversation prove that Plato considers the issue of rhetoric from a wider perspective determined by the issue of art. The specific feature of the conversation is ordering it around main issues – such notions as πρᾶγμα, ἐπιστήμη, δύναμις, each of which has its own meaning in Plato. In the first part of Gorgias Plato presents the principles previously occurring in other dialogues in a certain order, thus constructing the structure of the conversation with Gorgias. The notions are not novum in Plato but the fact that he clearly wants to systematise them and present a certain concept of art is a new and significant aspect. This tendency is of paramount importance for the discussion because Gorgias’ answers are evaluated by Plato from the perspective of this clearly set concept consisting of not only established notions but also certain principles. The discussion between Socrates and Gorgias introduces a theme which is crucial for the dialogue: the relation between art and good, i.e. the technical and ethical spheres. The issue is introduced by Gorgias, who mentions that rhetoric may be used for evil purposes. On this basis Socrates proves that if rhetoricians can use rhetoric for evil purposes, they have no knowledge concerning the subject of their art – justice, because one who knows what is just, always acts justly. Thus the relation between the technical and ethical spheres transpires to be the main point of contention. The question is whether a technician following the principles of the art he practices may act against good. This paradoxical statement based on the well-known Socrates’ principle "nemo sua sponte peccat" acquires philosophical foundations in the dialogue. In its remaining two parts Plato presents a justification, placing the principle within the framework of a certain concept of art, as a result of which not only rhetoric but also many other skills are criticised. The notions on which the conversation with Gorgias focuses are more precisely defined in the subsequent parts of the dialogue. Additionally, Plato presents twice the conditions which art should fulfill. They oblige the “technicians” to know the nature and purpose of the art and to know the causes of undertaken activities and to be able to justify them. These conditions determine the relation between the two spheres – technical and ethical. According to Plato, art may not act against good because it should be the objective of its activities. Certain concepts of good and knowledge outlined in the dialogue are the guarantees of “technicality”. Good, which should be respected by every art, is understood objectively and is contrasted with subjective pleasure. Knowledge, which should be the basis of art, opposes experience, which uses observation and recollection. Additionally, the definition of knowledge, significance of order in the world assuming the form of “geometrical equality”, significance of mathematics becoming the ideal of exactness indicate that Plato’s concept of knowledge is formed in Gorgias. These conclusions render it impossible that the results of any activity deserving the name of art should oppose good. And this is Plato’s main reproach of Gorgias’ concept, who wrote about false speeches possessing the power of conviction because they were written according to the principles proof of art or who defined tragedy as “a deceit, where he who cheats is more just than he who does not and the cheated is wiser than he who was not cheated”. Such determination of the relation between the technical and ethical sphere resulted from Gorgias’ concept of the world and fundamental philosophical principles. The whole concept of art described by Plato in the dialogue opposes the view presented by Gorgias. According to Plato, every technical activity has a defined object (ἔργον), is based on knowledge (ἐπιστήμη) and is always aimed at the good of human soul or body. The danger revealed and presented in the dialogue by Plato exists in the very concept of art presented by Gorgias, i.e. in its empirical foundations and the resulting understanding of the relation between art and good. Gorgias’ concept of art based on empirical cognition, which concerns the world of phenomena and rejects the possibility of perfect knowledge, leads, according to Plato, to a dangerous falsification of the relation between art and good. Plato proves that this weakness of Gorgias’ concept is revealed when confronted with these representatives of “the magnificent art” who are not protected against radicalism by the adherence to the moral tradition. The comparison of two, completely contrasting concepts of art based on different philosophical understanding of knowledge and good is decisive for the criticism of rhetoric in Gorgias. The concept presented by Plato criticises contemporary rhetoric and sophistry but determines the direction of research of the real art of rhetoric. (shrink)

This article deals with how moral freedom relates to historicity and contingency by comparing Kierkegaard's theory of the anthropological synthesis to Kant's concept of moral character. The comparison indicates that there are more Kantian elements in Kierkegaard's anthropology than shown by earlier scholarship. More specifically, both Kant and Kierkegaard see a true change in the way one lives as involving not only a revolution in the way one thinks, but also that one takes over—and tries to reform—both oneself and (...) human society. Also, Kierkegaard relies on the ideality of ethics and the doctrines of moral rigorism and radical evil. However, Kierkegaard can be seen as trying to find amore systematic role for historicity and contingency than Kant by developing the concept of facticity and by analyzing the so-called “despair of possibility.”. (shrink)

The Upanishads reveal that in the beginning, nothing existed: “This was but non-existence in the beginning. That became existence. That became ready to be manifest”. (Chandogya Upanishad 3.15.1) The creation began from this state of non-existence or nonduality, a state comparable to (0). One can add any number of zeros to (0), but there will be nothing except a big (0) because (0) is a neutral number. If we take (0) as Nirguna Brahman (God without any form and attributes), then (...) from where and how did the universe come into existence? -/- The neutral power (0) cannot produce anything without having an element of duality in it. Although Nirguna Brahman is neutral, it has a positive, negative, and neutral pole, constituting its Prakriti or nature. Prakriti has three latent Gunas (modes or qualities): Satva, Rajas, and Tamas. They are related to Gyana Shakti (the power of knowledge) Sankalpa Shakti (the power of ideation), and Kriya Shakti (the power of action). Science says that Atom is the basic element from which the universe evolved. The Atom has three nuclei- electron, pluton, and neutron. The Satva, Rajas, and Tamas in Indian spirituality are nothing but the mystical names for the nuclei of an atom. -/- According to Bhagavata Purana, Prakriti is also constituted of the elements of Time, Karma (action/destiny) and Swabhava (innate nature). Time disturbs the equilibrium of Gunas. From the aspect of Karma is produced an entity called Mahat, in the form of intelligence. Mahat is dominated by Satva and Rajas. From Mahat manifests the next evolute dominated by Tamas with three predominant qualities – Dravya (substance), Kriya (action), as well as intelligence. It forms to become the Ego principle (Ahamkara). -/- Ahamkara has three modes – Satvic, Rajasic and Tamasic. Satvic is Jnana oriented, Rajasic action-oriented and Tamasic Dravya (substance) oriented. From the Tamasic Ahankara, five gross elements are produced (ether, air, fire, water, and earth - Akash, Vayu, Agni, Jal and Prithvi). From the Satvic Ahamkara, guardian deities (the sun and moon, deities of sense organs, and organs of action) and from Rajasic Ahamkara, ten senses (Indriyas), five senses of perception, five organs of action, the faculties of intellect and Prana (life breath) are produced. -/- Bhagavata says that since these energies, elements, and faculties remained disassociated, they were combined to form a Cosmic Egg. The egg floats in the primal waters for a thousand years. Then God enters this Cosmic Egg and manifests himself as the Cosmic Purusha. He is the first nucleus, the God particle equivalent to the number (1) which is the embodiment of everything in the universe. The concept of creation and dissolution in Hinduism can be compared to the waves in an ocean that appear and disappear incessantly. The Manvantaras are such successive episodes of creation emerging from the Cosmic Person, the Manu who is embodied God-Consciousness, God himself. These episodes of creation are measured in Hinduism in terms of Manvantaras, the epochs of Manus. -/- Manu is the ‘First-Born’ (1) of God, the Cosmic Purusha from whom the world has originated. This Cosmic Person (Purusha) has been described as having fourteen biospheres (heavens) that are inhabited by various life forms in the order of evolution of consciousness. If we begin from the human biosphere (Bhuloka), there are ten astral biospheres that a man must transcend to attain liberation or Mukti from the cycle of births and deaths. -/- The number (1) produces the primary numbers up to (9) through the transmutation of the creational energies and qualities. The process can be related to the descent and cyclical evolution of (1) through ten spiritual stages, finally merging with the nondual (0). The number (10) marks the merger of (1) with (0). The Hindu worldview is that all life forms in an episode of creation must evolve to become one with the non-dual state (0) to attain liberation from the cycle of births and deaths in a cyclical process. -/- . (shrink)

In this paper a parallel is drawn between the problem of epistemic access to abstract objects in mathematics and the problem of epistemic access to idealized systems in the physical sciences. On this basis it is argued that some recent and more traditional approaches to solving these problems are problematic.

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

The quest to understand the natural and the mathematical as well as philosophical principles of dynamics of life forms are ancient in the human history of science. In ancient times, Pythagoras and Plato, and later, Copernicus and Galileo, correctly observed that the grand book of nature is written in the language of mathematics. Platonism, Aristotelian logism, neo-realism, monadism of Leibniz, Hegelian idealism and others have made efforts to understand reasons of existence of life forms in nature and the underlying (...) principles through the lenses of philosophy and mathematics. In this paper, an approach is made to treat the similar question about nature and existential life forms in view of mathematical philosophy. The approach follows constructivism to formulate an abstract model to understand existential life forms in nature and its dynamics by selectively combining the elements of various schools of thoughts. The formalisms of predicate logic, probabilistic inference and homotopy theory of algebraic topology are employed to construct a structure in local time-scale horizon and in cosmological time-scale horizon. It aims to resolve the relative and apparent conflicts present in various thoughts in the process, and it has made an effort to establish a logically coherent interpretation. (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)

This article explains Kant’s claim that sciences must take, at least as their ideal, the form of a ‘system’. I argue that Kant’s notion of systematicity can be understood against the background of de Jong & Betti’s Classical Model of Science (2010) and the writings of Georg Friedrich Meier and Johann Heinrich Lambert. According to my interpretation, Meier, Lambert, and Kant accepted an axiomatic idea of science, articulated by the Classical Model, which elucidates their conceptions of systematicity. I show (...) that Kant’s critique of the mathematical method is compatible with his adherence to this axiomatic conception of science. I further show that systematicity furthers traditionally accepted logical ideals of scientific knowledge, which explains why Meier and Kant think that sciences must be ‘systematic’. (shrink)

Curriculum integration is frequently promoted as a means of enabling deeper and more authentic learning, with Mathematics often considered a suitable subject for doing so. This review investigates which elements contribute to the effectiveness of Mathematics integration in secondary education. Teacher factors include attitudes towards integrative practices and knowledge of both disciplinary content and curriculum integration theory. Pedagogy factors concern utilising activities that best synthesise concepts from multiple subjects to enhance learning, especially projects. Institutional factors relate to (...) curriculum and assessment requirements as well as school-level variables such as access to planning time and professional development. Limitations in the research literature include a narrow focus on connecting Mathematics with scientific disciplines, a reliance on qualitative methods and small sample sizes, and unknown generalisability to education systems other than those in which studies were conducted. Even so, attempting to navigate the factors involved in effective curriculum integration with Mathematics appears worthwhile. (shrink)

For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What could (...) mathematics be knowledge about? (2) How do we distinguish significant from insignificant mathematics? This is a fundamental philosophical problem concerning the nature of mathematics. But it is also a practical problem concerning mathematics itself. In the absence of the solution to the problem, there is the danger that genuinely significant mathematics will be lost among the unchecked growth of a mass of insignificant mathematics. This second problem cannot, it would seem, be solved granted knowledge-inquiry. For, in order to solve the problem, mathematics needs to be related to values, but this is, it seems, prohibited by knowledge-inquiry because it could only lead to the subversion of mathematical rigour. Both problems are solved, however, when mathematics is viewed from the perspective of wisdom-inquiry. (1) Mathematics is not a branch of knowledge. It is a body of systematized, unified and inter-connected problem-solving methods, a body of problematic possibilities. (2) A piece of mathematics is significant if (a) it links up to the interconnected body of existing mathematics, ideally in such a way that some problems difficult to solve in other branches become much easier to solve when translated into the piece of mathematics in question; (b) it has fruitful applications for (other) worthwhile human endeavours. If ever the revolution from knowledge to wisdom occurs, I would hope wisdom mathematics would flourish, the nature of mathematics would become much more transparent, more pupils and students would come to appreciate the fascination of mathematics, and it would be easier to discern what is genuinely significant in mathematics (something that baffled even Einstein). As a result of clarifying what should count as significant, the pursuit of wisdom mathematics might even lead to the development of significant new mathematics. (shrink)

Faik Kurtulmuş’s exploration of the epistemic basic structure (EBS) invites us to think about the generation, dissemination, and absorption of knowledge in a society, emphasizing the role of institutions in determining epistemic outcomes. Moreover, Kurtulmuş—in joint work with Gürol Irzık—offers a normative take on the EBS from the viewpoint of the theory of justice and does not shy away from drawing specific policy recommendations. Thus, a powerful, innovative concept is used to extend an influential theory and draw out its practical (...) implications. What more is there to wish for? A project of such ambition inevitably faces serious challenges. I address the issue of the alignment between the ideal and non-ideal elements of Kurtulmuş’s theory with an eye toward the practicalities of social reform. My central claim is that the philosophers striving to promote distributive epistemic justice need to think carefully about whom they intend to address with their proposals and how to persuade such addressees. I also maintain that emphasis on a holistic approach to social reform represents a major hindrance to its pragmatic success. (shrink)

Many students consider mathematics as the most dreaded subject in their curriculum, so much so that the term “math phobia” or “math anxiety” is practically a part of clinical psychological literature. This symptom is widespread and students suffer mental disturbances when facing mathematical activity because understanding mathematics is a great task for them. This paper described the students’ cognitive skills performance in Basic Mathematics based on the following logical operations: Classification, Seriation, Logical Multiplication, Compensation, Ratio and Proportional (...) Thinking, Probability Thinking and Correlation Thinking as it serves a critical element of teaching and learning, that is determining the current position of the learners’ mathematical capacity. Its implications for mathematics education were also revealed. A descriptive quantitative design was used in this study. The study included 1011 first-year college students from six state universities in the Philippines who were enrolled in the Bachelor of Science of Secondary Education Program during the first semester of the Academic Year 2019-2020. This study made use of the teacher-made test called the Test on Logical Operations. Findings revealed that the cognitive skills achievement of the first year BSE students from different state universities in the Philippines falls under the category of late concrete operational stage. As a result, students are unable to perform the logical operational skills expected of their age. At their age level, they are expected to be under the formal operational stage based on Piaget’s stages of cognitive development. These findings revealed a weak mathematical education foundation among students which requires immediate attention. This reality should be recognized by educational planners and implementers when making curricular and other instructional decisions. (shrink)

The reconstruction of Leibniz’s metaphysics that Deleuze undertakes in The Fold provides a systematic account of the structure of Leibniz’s metaphysics in terms of its mathematical foundations. However, in doing so, Deleuze draws not only upon the mathematics developed by Leibniz—including the law of continuity as reflected in the calculus of infinite series and the infinitesimal calculus—but also upon developments in mathematics made by a number of Leibniz’s contemporaries—including Newton’s method of fluxions. He also draws upon a number (...) of subsequent developments in mathematics, the rudiments of which can be more or less located in Leibniz’s own work—including the theory of functions and singularities, the Weierstrassian theory of analytic continuity, and Poincaré’s theory of automorphic functions. Deleuze then retrospectively maps these developments back onto the structure of Leibniz’s metaphysics. While the Weierstrassian theory of analytic continuity serves to clarify Leibniz’s work, Poincaré’s theory of automorphic functions offers a solution to overcome and extend the limits that Deleuze identifies in Leibniz’s metaphysics. Deleuze brings this elaborate conjunction of material together in order to set up a mathematical idealization of the system that he considers to be implicit in Leibniz’s work. The result is a thoroughly mathematical explication of the structure of Leibniz’s metaphysics. This essay is an exposition of the very mathematical underpinnings of this Deleuzian account of the structure of Leibniz’s metaphysics, which, I maintain, subtends the entire text of The Fold. (shrink)

There is a tension in Emilie Du Châtelet’s thought on mathematics. The objects of mathematics are ideal or fictional entities; nevertheless, mathematics is presented as indispensable for an account of the physical world. After outlining Du Châtelet’s position, and showing how she departs from Christian Wolff’s pessimism about Newtonian mathematical physics, I show that the tension in her position is only apparent. Du Châtelet has a worked-out defense of the explanatory and epistemic need for mathematical objects, (...) consistent with their metaphysical nonfundamentality. I conclude by sketching how Du Châtelet’s conception of mathematical indispensability differs interestingly from many contemporary approaches. (shrink)

In this paper I study the development of arithmetical cognition with the focus on metaphorical thinking. In an approach developing on Lakoff and Núñez, I propose one particular conceptual metaphor, the Process → Object Metaphor, as a key element in understanding the development of mathematical thinking.

In the last couple of years, a few seemingly independent debates on scientific explanation have emerged, with several key questions that take different forms in different areas. For example, the questions what makes an explanation distinctly mathematical and are there any non-causal explanations in sciences (i.e., explanations that don’t cite causes in the explanans) sometimes take a form of the question of what makes mathematical models explanatory, especially whether highly idealized models in science can be explanatory and in virtue of (...) what they are explanatory. These questions raise further issues about counterfactuals, modality, and explanatory asymmetries: i.e., do mathematical and non-causal explanations support counterfactuals, and how ought we to understand explanatory asymmetries in non-causal explanations? Even though these are very common issues in the philosophy of physics and mathematics, they can be found in different guises in the philosophy of biology where there is the statistical interpretation of the Modern Synthesis theory of evolution, according to which the post-Darwinian theory of natural selection explains evolutionary change by citing statistical properties of populations and not the causes of changes. These questions also arise in philosophy of ecology or neuroscience in regard to the nature of topological explanations. The question here is can the mathematical (or more precisely topological) properties in network models in biology, ecology, neuroscience, and computer science be explanatory of physical phenomena, or are they just different ways to represent causal structures. The aim of this special issue is to unify all these debates around several overlapping questions. These questions are: are there genuinely or distinctively mathematical and non-causal explanations?; are all distinctively mathematical explanations also non-causal; in virtue of what they are explanatory; does the instantiation, implementation, or in general, applicability of mathematical structures to a variety of phenomena and systems play any explanatory role? This special issue provides a platform for unifying the debates around several key issues and thus opens up avenues for better understanding of mathematical and non-causal explanations in general, but also, it will enable even better understanding of key issues within each of the debates. (shrink)

Anyone who has read Plato’s Republic knows it has a lot to say about mathematics. But why? I shall not be satisfied with the answer that the future rulers of the ideal city are to be educated in mathematics, so Plato is bound to give some space to the subject. I want to know why the rulers are to be educated in mathematics. More pointedly, why are they required to study so much mathematics, for so (...) long? (shrink)

Willem Jacob ’s Gravesande is widely remembered as a leading advocate of Isaac Newton’s work. In the first half of the eighteenth century, ’s Gravesande was arguably Europe’s most important proponent of what would become known as Newtonian physics. ’s Gravesande himself minimally described this discipline, which he called «physica», as studying empirical regularities mathematically while avoiding hypotheses. Commentators have as yet not progressed much beyond this view of ’s Gravesande’s physics. Therefore, much of its precise nature, its methodology, and (...) its relation to Newton’s actual work remains unclear. This article discusses one particular methodological element that ’s Gravesande himself often stressed in detail, namely the use of mathematics in philosophy and physics. In doing so, it takes exception to the claim that mathematics played only a minor role in ’s Gravesande’s work, a view put forward in recent historiography. Besides that, this article casts new light on the interpretation of ’s Gravesande’s philosophical notion of «mathematical reasoning», a notion that has remained somewhat obscure thus far. (shrink)

(English abstract further below.) -/- Wer denkt, Polyamorie erfordere ein geringeres Maß an Verpflichtung als Zweierbeziehungen, der liegt gründlich daneben. Wie aber gestaltet sich polyamoröse wechselseitige Verpflichtung idealerweise? In diesem Beitrag untersuche ich, ob sich ein bestimmtes, auf Iris Murdochs Konzeption von Liebe als gerechter Aufmerksamkeit beruhendes Ideal wechselseitiger Verpflichtung in romantischen Partnerschaften fruchtbar auf polyamoröse Beziehungsgeflechte anwenden lässt. Ich beginne damit, Murdochs im deutschsprachigen Raum kaum rezipierte Liebeskonzeption ausführlich darzustellen und diese dabei von Simone Weils Position abzugrenzen, der (...) Murdoch wesentliche Elemente entnimmt. In einem zweiten Schritt skizziere ich das von Murdochs Position inspirierte Ideal wechselseitiger Verpflichtung. In Auseinandersetzung mit Überlegungen, die John Enman-Beech und Julienne Obadia mit Blick auf die in polyamorösen Beziehungsgeflechten verbreitete Praxis angestellt haben, Beziehungsvereinbarungen einzugehen, werbe ich drittens für die skizzierte Idealkonzeption, indem ich zeige, dass sich mit ihr den von Enman-Beech und Obadia aufgeworfenen Herausforderungen, die sich im Zuge intrapolykularer Beziehungsvereinbarungen stellen, in zwei Hinsichten mit Erfolg begegnen lässt. Erstens hebe ich hervor, dass eine am skizzierten Ideal orientierte Praxis bereits die Art von prozeduralen Normen implementiert, auf deren Bedeutung Enman-Beech zu Recht hinweist. Zweitens argumentiere ich dafür, dass das skizzierte Ideal nicht den Schwierigkeiten ausgesetzt ist, die sich nach Obadia mit denjenigen Elementen intrapolykularer Beziehungsvereinbarungen verbinden, die sie als Vertragskomplex bezeichnet, und ein weniger pessimistisches Bild davon nahelegt, wie vermittelst solcher Beziehungsvereinbarungen konstituierte Individuen aufzufassen sind. -/- ENGLISH ABSTRACT: In this paper, I focus on a certain ideal conception of how lovers may commit to each other in polyamorous relationships. Drawing on an analysis of Iris Murdoch’s adaptation of Simone Weil’s ideas, I sketch what I have elsewhere called the ideal lovers’ pledge—a conception of a commitment between lovers that is spelled out in terms of Weil’s and Murdoch’s shared notion of love as just attention. I then argue that this conception is well-suited to meet certain challenges that arise in the context of thinking about relationship agreements in polyamorous relationships, challenges that have been brought out in recent work by John Enman-Beech and Julienne Obadia. First, I show that practices of forging relationship agreements in polyamorous relationships that are oriented toward the ideal I propose will naturally implement the kind of procedural norms that Enman-Beech rightly highlights. Second, I argue that the ideal I propose does not generate the kinds of problems that according to Obadia arise due to a constellation of assumptions that she thinks underlie the practice of forging relationship agreements in polyamorous relationships – a constellation she dubs the contract complex. The proposed ideal thus suggests an alternative under-standing of such agreements and implies a less pessimistic account of what kind of individuals are constituted by way of forging such agreements. (shrink)

It is modernly debated whether application of the free will has potential to cause harm to nature. Power possessed to the discourse, sensory/perceptual, physical influences on life experience by the slow moving machinery of change is a viral element in the problems of civilization; failed resolution of historical paradox involving mind and matter is a recurring source of problems. Reference is taken from the writing of Euclid in which a oneness of nature as an indivisible point of thought is made (...) prerequisite in criteria of interpretation to demonstrate that contemporary scientific methodologies alternately ensue from the point of empirically centered induction. A qualification for conceptualizations is proposed that involves a physically describable form bound to energy in addition to contemporary notions of energy bound to form and a visually based mathematical-physical form is elaborated and discussed with respect to biological and natural processes. (shrink)

Siyaves Azeri (2020) quite well shows that arithmetical thinking emerges on the basis of specific social practices and material engagement (clay tokens for economic exchange practices beget number concepts, e.g.). But his discussion here is relegated mostly to Neolithic and Bronze Age practices. While surely such practices produced revolutions in the cognitive abilities of many humans, much of the cognitive architecture that allows normative conceptual thought was already in place long before this time. This response, then, is an attempt to (...) sketch the deep prehistory of human cognition in order to show the inter-social bases of normative thought in general. To do this, I will look first to the work of Vygotsky and Leontiev, two often neglected psychologists whose combined efforts culminate in a developmental account of human cognitive origins. Then, I will review some key insights from the contemporary comparative psychologist Michael Tomasello—whose project is admittedly a Vygotskian one—in order to further shed light on the social-practical basis of abstract thought, of which mathematical cognition is surely a part. (shrink)

In this paper I argue that constructivism in mathematics faces a dilemma. In particular, I maintain that constructivism is unable to explain (i) the application of mathematics to nature and (ii) the intersubjectivity of mathematics unless (iii) it is conjoined with two theses that reduce it to a form of mathematical Platonism. The paper is divided into five sections. In the first section of the paper, I explain the difference between mathematical constructivism and mathematical Platonism and I (...) outline my argument. In the second, I argue that the best explanation of how mathematics applies to nature for a constructivist is a thesis I call Copernicanism. In the third, I argue that the best explanation of how mathematics can be intersubjective for a constructivist is a thesis I call Ideality. In the fourth, I argue that once constructivism is conjoined with these two theses, it collapses into a form of mathematical Platonism. In the fifth, I confront some objections. (shrink)

A reasoned argument or tarka is essential for a wholesome vāda that aims at establishing the truth. A strong tarka constitutes of a number of elements including an anumāna based on a valid hetu. Several scholars, such as Dharmakīrti, Vasubandhu and Dignāga, have worked on theories for the establishment of a valid hetu to distinguish it from an invalid one. This paper aims to interpret Dignāga’s hetu-cakra, called the wheel of grounds, from a modern philosophical perspective by deconstructing it (...) into a simple probabilistic mathematical model. The objective is to understand how and why a vāda based on a probabilistically weaker hetu can degrade into a Jalpa or vitaṇḍā. To do so, the paper maps the concept of ‘Bounded Rationality’ onto the hetu-cakra. Bounded Rationality, an idea coined by the management thinker Herbert Simon, is often employed in understanding decision-making processes of rational agents. In the context of this paper, the concept would state that the prativādin and ālocaka (debater) may not hold unbounded information to back their pratijñā (proposition). The paper argues that within the probabilistically deconstructed hetu-cakra model, most people argue in the ‘Zone of Bounded Rationality’, and thus, the probability of a debate degrading into Jalpa or vitaṇḍā is high. -/- . (shrink)

The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity (...) in set theory. Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering “theorem”, transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for “Hilbert mathematics” accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödel’s papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part. (shrink)

The aim of the paper is to investigate the relations between the basic moral categories, namely those of norms, ideals and supererogation. The subject of discussion is, firstly, the ways that these categories are understood; secondly, the possible approaches towards moral acting that appear due to their use; and thirdly, their relationship within the moral system. However, what is of a special importance here is the relationship between the categories of norms and ideals (or in a wider aspect — laudable (...) acts) and a thesis about their complementary character. For it seems that the omission of one of the elements that are analysed within a moral system must result in an essential limitation of both the possibilities of finding an adequate approach to a given moral problem (on a descriptive level), as well as defining the full set of normative guidelines. Therefore, as I shall argue, resigning from a two-level method of moral analysis may lead to the atrophy of the very idea of ethics itself. (shrink)

In 1804, when asked by the aspiring writer Clemens Brentano why she had chosen to publish her work, Karoline von Günderrode wrote that she longed “mein Leben in einer bleibenden Form auszusprechen, in einer Gestalt, die würdig sei, zu den Vortreflichsten hinzutreten, sie zu grüssen und Gemeinschaft mit ihnen zu haben.” In light of this kind of statement, it is perhaps not surprising if, despite some exceptions, much of the still relatively scant literature on Günderrode reads her works largely in (...) terms of how they articulate and manifest Günderrode’s desires, frustrations, and character, for the most part ignoring their imaginary, creative, and intellectual aspects. This interpretation of the author’s works as biography is, in Günderrode’s case, often accompanied by an interpretation of her biography, particularly her suicide, as literary work. This paper is not the first to question the conflation of Günderrode’s life, death, and writing, but it is one of only a handful that aim to address the autopoietic element of Günderrode’s work in a way that does not reduce her writings to biographical and psychological expressions, or Günderrode herself to an image – or a legend – encapsulated by her writings and her relationship to them. This paper argues that Günderrode’s own position on what the self is has been largely neglected as a result of this conflation, and that taking this position into account changes how we understand Günderrode’s articulations of self in her writings. Thus this paper has two goals: to address difficulties in articulating and even constituting oneself sincerely when one’s efforts are unrecognized, belittled, censored, and forced to conform to the conventions of a society in which one is marginalized; and to unearth a neglected and potentially rich account of the modern self. (shrink)

In this essay, we consider the formal and ontological implications of one specific and intensely contested dialectical context from which Deleuze’s thinking about structural ideal genesis visibly arises. This is the formal/ontological dualism between the principles, ἀρχαί, of the One (ἕν) and the Indefinite/Unlimited Dyad (ἀόριστος δυάς), which is arguably the culminating achievement of the later Plato’s development of a mathematical dialectic.3 Following commentators including Lautman, Oskar Becker, and Kenneth M. Sayre, we argue that the duality of the One (...) and the Indefinite Dyad provides, in the later Plato, a unitary theoretical formalism accounting, by means of an iterated mixing without synthesis, for the structural origin and genesis of both supersensible Ideas and the sensible particulars which participate in them. As these commentators also argue, this duality furthermore provides a maximally general answer to the problem of temporal becoming that runs through Plato’s corpus: that of the relationship of the flux of sensory experiences to the fixity and order of what is thinkable in itself. Additionally, it provides a basis for understanding some of the famously puzzling claims about forms, numbers, and the principled genesis of both attributed to Plato by Aristotle in the Metaphysics, and plausibly underlies the late Plato’s deep considerations of the structural paradoxes of temporal change and becoming in the Parmenides, the Sophist, and the Philebus. After extracting this structure of duality and developing some of its formal, ontological, and metalogical features, we consider some of its specific implications for a thinking of time and ideality that follows Deleuze in a formally unitary genetic understanding of structural difference. These implications of Plato’s duality include not only those of the constitution of specific theoretical domains and problematics, but also implicate the reflexive problematic of the ideal determinants of the form of a unitary theory as such. We argue that the consequences of the underlying duality on the level of content are ultimately such as to raise, on the level of form, the broader reflexive problem of the basis for its own formal or meta-theoretical employment. We conclude by arguing for the decisive and substantive presence of a proper “Platonism” of the Idea in Deleuze, and weighing the potential for a substantive recuperation of Plato’s duality in the context of a dialectical affirmation of what Deleuze recognizes as the “only” ontological proposition that has ever been uttered. This is the proposition of the univocity of Being, whereby “being is said in the same sense, everywhere and always,” but is said (both problematically and decisively) of difference itself. (shrink)

Indigenous communities have a rich cultural heritage encompassing diverse ways of knowing, learning, and understanding the world around them. This mixed methods study utilized the explanatory sequential design to determine the level and relationship of the IP mathematics teachers' beliefs and teaching practices and gain a deeper insight into these beliefs and attitudes. There are 115 respondents for the quantitative phase, while 10 participants in the qualitative phase. Data were collected through survey and key informant interviews and were analyzed (...) through mean, standard deviation, Pearson product moment correlation, and thematic analysis. Results showed that the IP mathematics teachers possess high levels for both beliefs and teaching practices, and a significant moderate positive relationship exists between the two variables. Qualitative data analysis revealed that teachers' beliefs include the ideal qualities of an effective mathematics teacher, respect for students' culture and background, and a constructivist approach to teaching. Teaching practices include discussing student performance and intervention during meetings, implementing culturally relevant approaches, technology, varied activities, and peer mentoring and coaching. Further, it is concluded that teachers’ belief, which influences teaching practices, is shaped by different environmental factors. It is recommended that IP mathematics teachers continue to participate in personal and professional development activities and strengthen collaboration and reflection. (shrink)

At the beginning of the present century, a series of paradoxes were discovered within mathematics which suggested a fundamental unclarity in traditional mathemati­cal methods. These methods rested on the assumption of a realm of mathematical idealities existing independently of our thinking activity, and in order to arrive at a firmly grounded mathematics different attempts were made to formulate a conception of mathematical objects as purely human constructions. It was, however, realised that such formulations necessarily result in a (...) class='Hi'>mathematics which lacks the richness and power of the old ‘platonistic’ methods, and the latter are still defended, in various modified forms, as embodying truths about self-existent mathematical entities. Thus there is an idealism-realism dispute in the philosophy of mathematics in some respects parallel to the controversy over the existence of the experiential world to the settle­ment of which lngarden devoted his life. The present paper is an attempt to apply Ingarden’s methods to the sphere of mathematical existence. This exercise will reveal new modes of being applicable to non-real objects, and we shall put forward arguments to suggest that these modes of being have an importance outside mathematics, especially in the areas of value theory and the ontology of art. (shrink)

In this article I examine two mathematical definitions of observational equivalence, one proposed by Charlotte Werndl and based on manifest isomorphism, and the other based on Ornstein and Weiss’s ε-congruence. I argue, for two related reasons, that neither can function as a purely mathematical definition of observational equivalence. First, each definition permits of counterexamples; second, overcoming these counterexamples will introduce non-mathematical premises about the systems in question. Accordingly, the prospects for a broadly applicable and purely mathematical definition of observational equivalence (...) are unpromising. Despite this critique, I suggest that Werndl’s proposals are valuable because they clarify the distinction between provable and unprovable elements in arguments for observational equivalence. (shrink)

Errett Bishop's work in constructive mathematics is overwhelmingly regarded as a turning point for mathematics based on intuitionistic logic. It brought new life to this form of mathematics and prompted the development of new areas of research that witness today's depth and breadth of constructive mathematics. Surprisingly, notwithstanding the extensive mathematical progress since the publication in 1967 of Errett Bishop's Foundations of Constructive Analysis, there has been no corresponding advances in the philosophy of constructive mathematics (...) Bishop style. The aim of this paper is to foster the philosophical debate about this form of mathematics. I begin by considering key elements of philosophical remarks by Bishop, especially focusing on Bishop's assessment of Brouwer. I then compare these remarks with ``traditional'' philosophical arguments for intuitionistic logic and argue that the latter are in tension with Bishop's views. ``Traditional'' arguments for intuitionistic logic turn out to be also in conflict with significant recent developments in constructive mathematics. This rises pressing questions for the philosopher of mathematics, especially with regard to the possibility of offering alternative philosophical arguments for constructive mathematics. I conclude with the suggestion to look anew at Bishop's own remarks for inspiration. (shrink)

Abstract: In this paper, we study a new notion of fuzzy subimplicative ideal of a BH-algebra, namely fuzzy subimplicative ideal with respect to an element in BH-algebra is introduced and some related properties are investigated.

The denial of the intrinsic value of acts apart from both motives and consequences lies at the heart of Ross’s deontology and his opposition to ideal utilitarianism. Moreover, the claim that acts can have intrinsic value is a staple element of early and contemporary attempts to “consequentialise” all of morality. I first show why Ross’s denial is relevant both for his philosophy and for current debates. Then I consider and reject as inconclusive some of Ross’s explicit and implicit motivations (...) for his claim, stemming from his philosophy of action, his axiology, and his concept of intrinsic value, or a combination of these. I also criticize Ross’s later view that all right acts somehow produce some good, but that the value of some of these goods is explained by the prior rightness of the act. In the course of the discussion, the idea that acts can have intrinsic value apart from motives and consequences gains credibility both from the weaknesses in Ross’s arguments and from some putative examples. So, finally, I distinguish two attitudes in the history of ideal utilitarianism towards the necessity or not to give a detailed account of the intrinsic value of acts, and suggest that a Why Bother attitude is more promising than a Constructive one. (shrink)

in this paper we return to Marshall Clagett’s view about the existence of an ancient Greek geometry of motion. It can be read in two ways. As a basic presentation of ancient Greek geometry of motion, followed by some aspects of its further development in landmark works by Galileo and Newton. Conversely, it can be read as a basic presentation of aspects of Galileo’s and Newton’s mathematics that can be considered as developments of a geometry of motion that was (...) first conceived by ancient Greek mathematicians. (shrink)

This paper addresses the issue of finite versus countable additivity in Bayesian probability and decision theory -- in particular, Savage's theory of subjective expected utility and personal probability. I show that Savage's reason for not requiring countable additivity in his theory is inconclusive. The assessment leads to an analysis of various highly idealised assumptions commonly adopted in Bayesian theory, where I argue that a healthy dose of, what I call, conceptual realism is often helpful in understanding the interpretational value of (...) sophisticated mathematical structures employed in applied sciences like decision theory. In the last part, I introduce countable additivity into Savage's theory and explore some technical properties in relation to other axioms of the system. (shrink)

In his book On What Matters, Derek Parfit defends a version of moral non-naturalism, a view according to which there are objective normative truths, some of which are moral truths, and we have a reliable way of discovering them. These moral truths do not exist, however, as parts of the natural universe nor in Plato’s heaven. While explaining in what way these truths exist and how we discover them, Parfit makes analogies between morality on the one hand, and mathematics (...) and logic on the other. Moral truths “exist” in a way that numbers exist, and we discover these truths in a similar way as we discover truths about numbers. By the end of the second volume, Parfit also responds to a powerful objection against his view, an objection based on the phenomenon of moral disagreement. If people widely and deeply disagree about what is the moral truth, it is doubtful whether we have a reliable way of discovering it. In his reply, he claims that in ideal conditions for thinking about moral questions, we would all have sufficiently similar moral beliefs. However, we often find ourselves in less-than-ideal conditions due to various factors that distort our ability to agree. Therefore, differences in moral opinion can be expected. In this paper, I draw a connection between these parts of Parfit’s theory and comment on them. Firstly, I argue that Parfit’s analogy with mathematics and logic and his answer to the disagreement objection are in tension because there are important epistemic differences between morality and these fields. If one would try to account for the differences, one would have to sacrifice some measure of similarity between morality and them. Secondly, I comment on Parfit’s reply to the disagreement objection itself. I believe that, although his description of ideal conditions has some potential for reaching moral agreement, it may be difficult to tell if ideal conditions prevail. This obscurity spells further trouble for Parfit’s overall theory. (shrink)

Are mathematical objects affected by their historicity? Do they simply lose their identity and their validity in the course of history? If not, how can they always be accessible in their ideality regardless of their transmission in the course of time? Husserl and Foucault have raised this question and offered accounts, both of which, albeit different in their originality, are equally provocative. Both acknowledge that a scientific object like a geometrical theorem or a chemical equation has a history because it (...) is only constituted in and transmitted through history. But they see that history as a part of its ideality, so that, although historical, a scientific object retains its identity as one and the same object. (shrink)

: The debate between proponents of ideal and non-ideal approaches to political philosophy has thus far been framed as a meta-level debate about normative theory. The argument of this essay will be that the ideal/non-ideal debate can be helpfully reframed as a ground-level debate within normative theory. Specifically, it can be understood as a debate within the applied normative field of professional ethics, with the profession being examined that of political philosophy itself. If the community of (...) academic political theorists and philosophers cannot help us navigate the problems we face in actual political life, they have not lived up to the moral demands of their vocation. A moderate form of what David Estlund decries as “utopophobia” is therefore an integral element of a proper professional ethic for political philosophers. The moderate utopophobe maintains that while devoting scarce time and resources to constructing utopias may sometimes be justifiable, it is never self-justifying. Utopianism is defensible only insofar as it can reasonably be expected to help inform or improve nonutopian political thinking. (shrink)

Deliberative Democracy theory is an ever-expanding field in political theory. In the present article, I aim to present the significance of Gandhian thought for the theory of deliberative democracy. Gandhi never used the term deliberation or articulated a theory of deliberative democracy specifically while expressing his notion of ideal democracy. For him, discussion, exchange of thoughts, reasoning, etc. was instinctive for democracy and not something that required to be defended within the boundaries of scholarship. I trace the central (...) class='Hi'>elements of democracy in Gandhian thought and examine them through the lens of deliberative democracy theory. I also examine the implication of Gandhi’s formulations in India. In doing so, I would develop a richer understanding of democracy by bringing clarity to the contribution of Gandhi to the cause of deliberative democracy. (shrink)

Over the past two decades, gamblers have begun taking mathematics into account more seriously than ever before. While probability theory is the only rigorous theory modeling the uncertainty, even though in idealized conditions, numerical probabilities are viewed not only as mere mathematical information, but also as a decision-making criterion, especially in gambling. This book presents the mathematics underlying the major games of chance and provides a precise account of the odds associated with all gaming events. It begins by (...) explaining in simple terms the meaning of the concept of probability for the layman and goes on to become an enlightening journey through the mathematics of chance, randomness and risk. It then continues with the basics of discrete probability, combinatorics and counting arguments for those interested in the supporting mathematics. These mathematic sections may be skipped by readers who do not have a minimal background in mathematics; these readers can skip directly to the Guide to Numerical Results to pick the odds and recommendations they need for the desired gaming situation. Doing so is possible due to the organization of that chapter, in which the results are listed at the end of each section, mostly in the form of tables. The chapter titled The Mathematics of Games of Chance presents these games not only as a good application field for probability theory, but also in terms of human actions where probability-based strategies can be tried to achieve favorable results. Through suggestive examples, the reader can see what are the experiments, events and probability fields in games of chance and how probability calculus works there. The main portion of this work is a collection of probability results for each type of game. Each game s section is packed with formulas and tables. Each section also contains a description of the game, a classification of the gaming events and the applicable probability calculations. The primary goal of this work is to allow the reader to quickly find the odds for a specific gaming situation, in order to improve his or her betting/gaming decisions. Every type of gaming event is tabulated in a logical, consistent and comprehensive manner. The complete methodology and complete or partial calculations are shown to teach players how to calculate probability for any situation, for every stage of the game for any game. Here, readers can find the real odds, returned by precise mathematical formulas and not by partial simulations that most software uses. Collections of odds are presented, as well as strategic recommendations based on those odds, where necessary, for each type of gaming situation. The book contains much new and original material that has not been published previously and provides great coverage of probabilities for the following games of chance: Dice, Slots, Roulette, Baccarat, Blackjack, Texas Hold em Poker, Lottery and Sport Bets. Most of games of chance are predisposed to probability-based decisions. This is why the approach is not an exclusively statistical one, but analytical: every gaming event is taken as an individual applied probability problem to solve. A special chapter defines the probability-based strategy and mathematically shows why such strategy is theoretically optimal.". (shrink)

Hilbert's axiomatic approach was an optimistic take over on the side of the logical foundations. It was also a response to various restrictive views of mathematics supposedly bounded by the reaches of epistemic elements in mathematics. A complete axiomatization should be able to exclude epistemic or ontic elements from mathematical theorizing, according to Hilbert. This exclusion is not necessarily a logicism in similar form to Frege's or Dedekind's projects. That is, intuition can still have a role (...) in mathematical reasoning. Nevertheless, this role is to be given a structural orientation with the help of explications of the underlying logic of axiomatization. (shrink)

Can an AGI create a more intelligent AGI? Under idealized assumptions, for a certain theoretical type of intelligence, our answer is: “Not without outside help”. This is a paper on the mathematical structure of AGI populations when parent AGIs create child AGIs. We argue that such populations satisfy a certain biological law. Motivated by observations of sexual reproduction in seemingly-asexual species, the Knight-Darwin Law states that it is impossible for one organism to asexually produce another, which asexually produces another, and (...) so on forever: that any sequence of organisms (each one a child of the previous) must contain occasional multi-parent organisms, or must terminate. By proving that a certain measure (arguably an intelligence measure) decreases when an idealized parent AGI single-handedly creates a child AGI, we argue that a similar Law holds for AGIs. (shrink)