In the paper “Math Anxiety,” Aden Evens explores the manner by means of which concepts are implicated in the problematic Idea according to the philosophy of Gilles Deleuze. The example that Evens draws from Difference and Repetition in order to demonstrate this relation is a mathematics problem, the elements of which are the differentials of the differential calculus. What I would like to offer in the present paper is an historical account of the mathematical problematic that Deleuze deploys in (...) his philosophy, and an introduction to the role that this problematic plays in the develop- ment of his philosophy of difference. One of the points of departure that I will take from the Evens paper is the theme of “power series.”2 This will involve a detailed elaboration of the mechanism by means of which power series operate in the differential calculus deployed by Deleuze in Difference and Repetition. Deleuze actually constructs an alternative history of mathematics that establishes an historical conti- nuity between the differential point of view of the infinitesimal calculus and modern theories of the differential calculus. It is in relation to the differential point of view of the infinitesimal calculus that Deleuze determines a differential logic which he deploys, in the form of a logic of different/ciation, in the development of his proj- ect of constructing a philosophy of difference. (shrink)
How to accommodate the possibility of lucky true beliefs in necessary (or armchair) truths within contemporary modal epistemology? According to safety accounts luck consists in the modal proximity of a false belief, but a belief in a true mathematical proposition could not easily be false because a proposition believed could never be false. According to Miščević modal stability of a true belief under small changes in the world is not enough, stability under small changes in the cognizer should also (and (...) primarily) be considered. I argue for a more traditional modal reliabilism based on the critical question: how easy is it for a belief to be false, given the way it was formed? A belief (a priori or a posteriori) is then agent-lucky when based on a specific method which might easily lead to a false belief in the target proposition. Miščević suggests a unifying approach in terms of virtue epistemology. It seems to me that this approach, if successful, will undermine the project that he started with: formulate an anti-luck condition in the frame of a modal theory of luck. (shrink)
We attribute three major insights to Hegel: first, an understanding of the real numbers as the paradigmatic kind of number ; second, a recognition that a quantitative relation has three elements, which is embedded in his conception of measure; and third, a recognition of the phenomenon of divergence of measures such as in second-order or continuous phase transitions in which correlation length diverges. For ease of exposition, we will refer to these three insights as the R First Theory, Tripartite Relations, (...) and Divergence of Measures. Given the constraints of space, we emphasize the first and the third in this paper. (shrink)
This article proposes a brief argument for why mathematics is transcendental in so far as the concept of infinity emerges from it; this ultimately relies on the understanding that math gives a metaphysical justification for non-existence or "0".
This paper sought to investigate the fundamental differences in mathematics education through a comparison of curriculum of 2 countries—Singapore and Canada (as represented by Ontario)—in order to discover what the Ontario education system may learn from Singapore in terms of mathematics education. Mathematics curriculum were collected for Grades 1 to 8 for Ontario, and the equivalent in Singapore. The 2 curriculums were textually analyzed based on both the original and the revised Bloom’s taxonomy to expose their foci. The difference in (...) focus was then compared and discussed to find the best ways to improve the Ontario mathematics curriculum. With one of the best education systems in North America, the Ontario mathematics curriculum would only need to refocus its attention towards a more balanced approach, with greater focus on understanding through practices. Ontario would benefit greatly from a deeper research into the Singaporean math curriculum. (shrink)
In this paper I offer a reconstruction of the account of meaning and language the Cyrenaics appear to have defended on the basis of a famous passage of Sextus, as well as showing the philosophical parentage of that account.
A work on the philosophy of mathematics (2017) -/- ‘Number’, such a simple idea, and yet it fascinated and absorbed the greatest proportion of human geniuses over centuries, not to mention the likes of Pythagoras, Euclid, Newton, Leibniz, Descartes and countless maths giants like Euler, Gauss and Hilbert, etc.. Einstein thought of pure maths as the poetry of logical ideas, the exactitude of which, although independent of experience, strangely seems to benefit the study of the objects of reality. And, interestingly (...) as well as surprisingly we are nowhere near any clear understandings of numbers despite discoveries of many productive usages of numbers. This is - rightly or wrongly - a humble attempt to approach the subject from an angle hitherto unthought-of. (shrink)
This article had its start with another article, concerned with measuring the speed of gravitational waves - "The Measurement of the Light Deflection from Jupiter: Experimental Results" by Ed Fomalont and Sergei Kopeikin (2003) - The Astrophysical Journal 598 (1): 704–711. This starting-point led to many other topics that required explanation or naturally seemed to follow on – Unification of gravity with electromagnetism and the 2 nuclear forces, Speed of electromagnetic waves, Energy of cosmic rays and UHECRs, Digital string theory, (...) Dark energy+gravity+binary digits, Cosmic strings and wormholes from Figure-8 Klein bottles, Massless and massive photons and gravitons, Inverse square+quantum entanglement = God+evolution, Binary digits projected to make Prof. Greene’s cosmic holographic movie, Renormalization of infinity, Colliding subuniverses, Unifying cosmic inflation, TOE (emphasizing “EVERYthing”) = Bose-Einstein renormalized. The text also addresses (in a nonmathematical way) the wavelength of electromagnetic waves, the frequency of gravitational waves, gravitational and electromagnetic waves having identical speed, the gamma-ray burst designated GRB 090510, the smoothness of space, and includes these words – “Gravity produces electromagnetism. Retrocausally (by means of humans travelling into the past of this subuniverse with their electronics); this “Cosmic EM Background” produces base-2 mathematics, which produces gravity. EM interacts with gravity to produce particles, mass – gravity/EM could be termed “the Higgs field” - and the nuclear forces associated with those particles. It makes gravity using BITS that copy the principle of magnetism attracting and repelling, before pasting it into what we call the strong force and dark energy.” . (shrink)
This paper discusses the phenomenon of Kāpil Maṭh (Madhupur, India), a Sāṃkhyayoga āśrama founded in the early twentieth century by the charismatic Bengali scholar-monk Swāmi Hariharānanda Ᾱraṇya (1869–1947). While referring to Hariharānanda’s writings I will consider the idea of the re-establishment of an extinct spiritual lineage. I shall specify the criteria for identity of this revived Sāṃkhyayoga tradition by explaining why and on what assumptions the modern reinterpretation of this school can be perceived as continuation of the thought of Patañjali (...) and Īśvarakṛṣṇa. The starting point is, however, the question whether it is possible at all to re-establish a philosophical tradition which had once broken down and disappeared for centuries. In this context, one ought to ponder if it is likely to revitalise the same line of thinking, viewing, philosophy-making and practice in accordance with the theoretical exposition of the right insight achieved by an accomplished teacher, a master, the founder of a “new”revived tradition declared to maintain a particular school identity. Moreover, I refer to a monograph of Knut A. Jacobsen (2018) devoted to the tradition of Kāpil Maṭh interpreted as a typical product of the nineteenth-century Bengali renaissance. (shrink)
In his précis of a recent book, Richard Joyce writes, “My contention…is that…any epistemological benefit-of-the-doubt that might have been extended to moral beliefs…will be neutralized by the availability of an empirically confirmed moral genealogy that nowhere…presupposes their truth.” Such reasoning – falling under the heading “Genealogical Debunking Arguments” – is now commonplace. But how might “the availability of an empirically confirmed moral genealogy that nowhere… presupposes” the truth of our moral beliefs “neutralize” whatever “epistemological benefit-of-the-doubt that might have been extended (...) to” them? In this article, I argue that there appears to be no satisfactory answer to this question. The problem is quite general, applying to all arguments with the structure of Genealogical Debunking Arguments aimed at realism about a domain meeting two conditions. The Benacerraf-Field Challenge for mathematical realism affords an important special case. (shrink)
This paper gives a framework for understanding causal counterpossibles, counterfactuals imbued with causal content whose antecedents appeal to metaphysically impossible worlds. Such statements are generated by omissive causal claims that appeal to metaphysically impossible events, such as “If the mathematician had not failed to prove that 2+2=5, the math textbooks would not have remained intact.” After providing an account of impossible omissions, the paper argues for three claims: (i) impossible omissions play a causal role in the actual world, (ii) (...) causal counterpossibles have broad applications in philosophy, and (iii) the truth of causal counterpossibles provides evidence for the nonvacuity of counterpossibles more generally. (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)
Can we do science without numbers? How much contingency is there? These seemingly unrelated questions--one in the philosophy of math and science and the other in metaphysics--share an unexpectedly close connection. For as it turns out, a radical answer to the second leads to a breakthrough on the first. The radical answer is new view about modality called compossible immutabilism. The breakthrough is a new strategy for doing science without numbers. One of the chief benefits of the new strategy (...) is that, unlike the existing substantialism approach from Field (1980), the new strategy naturally generalizes to theories formulated in terms of state space. (shrink)
Is math in harmony with existence? Is it possible to calculate any property of existence over math? Is exact proof of something possible without pre-acceptance of some physical properties? This work is realized to analysis these arguments somehow as simple as possible over short cuts, and it came up with some compatible results finally. It seems that both free space and moving bodies in this space are dependent on the same rule as there is no alternative, and the (...) rule is determined by mathematics. (shrink)
Nietzsche has a surprisingly significant and strikingly positive assessment of mathematics. I discuss Nietzsche's theory of the origin of mathematical practice in the division of the continuum of force, his theory of numbers, his conception of the finite and the infinite, and the relations between Nietzschean mathematics and formalism and intuitionism. I talk about the relations between math, illusion, life, and the will to truth. I distinguish life and world affirming mathematical practice from its ascetic perversion. For Nietzsche, (...) class='Hi'>math is an artistic and moral activity that has an essential role to play in the joyful wisdom. (shrink)
In this paper a Knowledge-Based System (KBS) for determining the appropriate students major according to his/her preferences for sophomore student enrolled in the Faculty of Engineering and Information Technology in Al-Azhar University of Gaza was developed and tested. A set of predefined criterions that is taken into consideration before a sophomore student can select a major is outlined. Such criterion as high school score, score of subject such as Math I, Math II, Electrical Circuit I, and Electronics I (...) taken during the student freshman year, number of credits passed, student cumulative grade point average of freshman year, among others, were then used as input data to KBS. KBS was designed and developed using Simpler Level Five (SL5) Object expert system language. KBS was tested on three generation of sophomore students from the Faculty of Engineering and Information Technology of the Al-Azhar University, Gaza. The results of the evaluation show that the KBS is able to correctly determine the appropriate students major without errors. (shrink)
Swami Vivekananda is considered as one of the most influential spiritual educationist and thinker of India. He was disciple of Ramakrishna Paramahamsa and the founder of Ramakrishna Math and Ramakrishna Mission. He is considered by many as an icon for his fearless courage, his positive exhortations to the youth, his broad outlook to social problems, and countless lectures and discourses on Vedanta philosophy. For him, “Education is not the amount of information that is put into your brain and runs (...) riots there, undigested all your life. We must have life-building, man-making, character-making, assimilation of ideas.” It is rightly said that, “The Swami’s mission was both national and international. A lover of mankind, he strove to promote peace and human brotherhood on the spiritual foundation of the Vedantic Oneness of existence. (shrink)
This is the first book in a two-volume series. The present volume introduces the basics of the conceptual foundations of quantum physics. It appeared first as a series of video lectures on the online learning platform Udemy.]There is probably no science that is as confusing as quantum theory. There's so much misleading information on the subject that for most people it is very difficult to separate science facts from pseudoscience. The goal of this book is to make you able to (...) separate facts from fiction with a comprehensive introduction to the scientific principles of a complex topic in which meaning and interpretation never cease to puzzle and surprise. An A-Z guide which is neither too advanced nor oversimplified to the weirdness and paradoxes of quantum physics explained from the first principles to modern state-of-the-art experiments and which is complete with figures and graphs that illustrate the deeper meaning of the concepts you are unlikely to find elsewhere. A guide for the autodidact or philosopher of science who is looking for general knowledge about quantum physics at intermediate level furnishing the most rigorous account that an exposition can provide and which only occasionally, in few special chapters, resorts to a mathematical level that goes no further than that of high school. It will save you a ton of time that you would have spent searching elsewhere, trying to piece together a variety of information. The author tried to span an 'arch of knowledge' without giving in to the temptation of taking an excessively one-sided account of the subject. What is this strange thing called quantum physics? What is its impact on our understanding of the world? What is ‘reality’ according to quantum physics? This book addresses these and many other questions through a step-by-step journey. The central mystery of the double-slit experiment and the wave-particle duality, the fuzzy world of Heisenberg's uncertainty principle, the weird Schrödinger's cat paradox, the 'spooky action at a distance' of quantum entanglement, the EPR paradox and much more are explained, without neglecting such main contributors as Planck, Einstein, Bohr, Feynman and others who struggled themselves to come up with the mysterious quantum realm. We also take a look at the experiments conducted in recent decades, such as the surprising "which-way" and "quantum-erasure" experiments. Some considerations on why and how quantum physics suggests a worldview based on philosophical idealism conclude this first volume. This treatise goes, at times, into technical details that demand some effort and therefore requires some basics of high school math (calculus, algebra, trigonometry, elementary statistics). However, the final payoff will be invaluable: Your knowledge of, and grasp on, the subject of the conceptual foundations of quantum physics will be deep, wide, and outstanding. Additionally, because schools, colleges, and universities teach quantum physics using a dry, mostly technical approach which furnishes only superficial insight into its foundations, this manual is recommended for all those students, physicists or philosophers of science who would like to look beyond the mere formal aspect and delve deeper into the meaning and essence of quantum mechanics. The manual is a primer that the public deserves. (shrink)
A solution to the question "Why is there something rather than nothing?" is proposed that also entails a proposed solution to the question "Why does a thing exist?". In brief, I propose that a thing exists if it is a grouping. A grouping ties stuff together into a unit whole and, in so doing, defines what is contained within that new unit whole. For outside-the-mind groupings, like a book, the grouping is physically present and visually seen as an edge, boundary, (...) or enclosing surface that defines this unit whole/existent entity. For inside-the-mind groupings, like the concept of a car, the grouping may be better thought of as the top-level label the mind gives to the mental construct that groups together other constructs into a new unit whole (i.e., the mental construct labeled “car” groups together the constructs of engine, car chassis, tires, use for transportation, etc.). The grouping, or enclosing surface/label, gives "substance" and existence to the thing as a new unit whole that's a different existent entity than any components contained within considered individually. Next, in regard to the question "Why is there something rather than nothing?", when we get rid of all known existent entities including matter, energy, space/volume, time, abstract concepts, laws or constructs of physics/math/logic, possible worlds/possibilities, counteracting forces, properties, consciousness, and minds, including the mind of the person trying to imagine this, we think what is left is the lack of all existent entities, or "nothing" (here, “nothing” does not mean the mind's conception of "nothing", but "nothing" itself, in which all minds would be gone). But once everything is gone, and the mind of the person thinking about this is gone, this "nothing" would, by its very nature, be the complete definition of the situation. That is, the very lack of all would itself be the entirety. Is there anything else besides that "nothing"? No. It is "nothing", and it is “the all”. Completely-defines-the-situation/entirety/”the all” is a grouping, which means, by the definition given here, that the situation we previously considered to be "nothing" is itself an existent entity. The surface of this entity isn't some separate structure; instead, it is the complete-definition-of-the-situation/entirety/"the all" grouping itself that is the surface. Said another way, by its very nature, "nothing" defines itself and is therefore the beginning point in the chain of being able to define existent entities in terms of other existent entities. One objection might be that a grouping is a property so how can it be there in "nothing"? The answer is that it is only once all known existent entities, including all properties and the mind of the person visualizing this “nothing”, are gone does this “nothing” completely define the situation and become the “the all” grouping and, therefore, an existent entity, or a “something”. The above argument can be restated by making the analogy between the question "Why is there something rather than nothing?" and the idea that you that you start with a 0 (e.g., "nothing") and end up with a 1 (e.g., "something"). Because you can't change a 0 into a 1, the only way you can do this is if that 0 really wasn't a 0 but was actually a 1 in disguise, even though it looks like a 0 on the surface. In conclusion, it is argued that "something" is necessary because even what we previously considered to be "nothing" is a "something". This isn't a new idea, but, to the best of the author's knowledge, providing a mechanism for why "nothing" is a "something" is. (shrink)
How do axioms, or first principles, in ethics compare to those in mathematics? In this companion piece to G.C. Field's 1931 "On the Role of Definition in Ethics", I argue that there are similarities between the cases. However, these are premised on an assumption which can be questioned, and which highlights the peculiarity of normative inquiry.
Bill D'Alessandro talks to Kenny Easwaran about fractal music, Zoom conferences, being a good referee, teaching in math and philosophy, the rationalist community and its relationship to academia, decision-theoretic pluralism, and the city of Manhattan, Kansas.
According to Cantor (Mathematische Annalen 21:545–586, 1883 ; Cantor’s letter to Dedekind, 1899 ) a set is any multitude which can be thought of as one (“jedes Viele, welches sich als Eines denken läßt”) without contradiction—a consistent multitude. Other multitudes are inconsistent or paradoxical. Set theoretical paradoxes have common root—lack of understanding why some multitudes are not sets. Why some multitudes of objects of thought cannot themselves be objects of thought? Moreover, it is a logical truth that such multitudes do (...) exist. However we do not understand this logical truth so well as we understand, for example, the logical truth $${\forall x \, x = x}$$ . In this paper we formulate a logical truth which we call the productivity principle. Rusell (Proc Lond Math Soc 4(2):29–53, 1906 ) was the first one to formulate this principle, but in a restricted form and with a different purpose. The principle explicates a logical mechanism that lies behind paradoxical multitudes, and is understandable as well as any simple logical truth. However, it does not explain the concept of set. It only sets logical bounds of the concept within the framework of the classical two valued $${\in}$$ -language. The principle behaves as a logical regulator of any theory we formulate to explain and describe sets. It provides tools to identify paradoxical classes inside the theory. We show how the known paradoxical classes follow from the productivity principle and how the principle gives us a uniform way to generate new paradoxical classes. In the case of ZFC set theory the productivity principle shows that the limitation of size principles are of a restrictive nature and that they do not explain which classes are sets. The productivity principle, as a logical regulator, can have a definite heuristic role in the development of a consistent set theory. We sketch such a theory—the cumulative cardinal theory of sets. The theory is based on the idea of cardinality of collecting objects into sets. Its development is guided by means of the productivity principle in such a way that its consistency seems plausible. Moreover, the theory inherits good properties from cardinal conception and from cumulative conception of sets. Because of the cardinality principle it can easily justify the replacement axiom, and because of the cumulative property it can easily justify the power set axiom and the union axiom. It would be possible to prove that the cumulative cardinal theory of sets is equivalent to the Morse–Kelley set theory. In this way we provide a natural and plausibly consistent axiomatization for the Morse–Kelley set theory. (shrink)
Many theorists have focused on Wittgenstein’s use of examples, but I argue that examples form only half of his method. Rather than continuing the disjointed style of his Cambridge lectures, Wittgenstein returns to the techniques he employed while teaching elementary school. Philosophical Investigations trains the reader as a math class trains a student—‘by means of examples and by exercises’ (§208). Its numbered passages, carefully arranged, provide a series of demonstrations and practice problems. I guide the reader through one such (...) series, demonstrating how the exercises build upon one another and give us ample opportunity to hone our problem-solving skills. Through careful practice, we learn to pass the test Wittgenstein poses when he claims that something is ‘easy to imagine’ (§19). Whereas other critics have viewed the Investigations as merely a diagnosis of our philosophical delusions, I claim that Wittgenstein also writes a prescription for our disease: Do your exercises. (shrink)
Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics (QM) is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets (or partitions) which are category-theoretically dual (...) to one another and which are developed in two mathematical logics, the usual Boolean logic of subsets and the more recent logic of partitions. Our sense-making strategy is "follow the math" by showing how the logic and mathematics of set partitions can be transported in a natural way to Hilbert spaces where it yields the mathematical machinery of QM--which shows that the mathematical framework of QM is a type of logical system over ℂ. And then we show how the machinery of QM can be transported the other way down to the set-like vector spaces over ℤ₂ showing how the classical logical finite probability calculus (in a "non-commutative" version) is a type of "quantum mechanics" over ℤ₂, i.e., over sets. In this way, we try to make sense out of objective indefiniteness and thus to interpret quantum mechanics. (shrink)
Since Benacerraf’s ‘What Numbers Could Not Be, ’ there has been a growing interest in mathematical structuralism. An influential form of mathematical structuralism, modal structuralism, uses logical possibility and second order logic to provide paraphrases of mathematical statements which don’t quantify over mathematical objects. These modal structuralist paraphrases are a useful tool for nominalists and realists alike. But their use of second order logic and quantification into the logical possibility operator raises concerns. In this paper, I show that the work (...) of both these elements can be done by a single natural generalization of the logical possibility operator. (shrink)
This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ①. The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of these (...) notions and proposes a more physical treatment of mathematical objects separating the objects from tools used to study them. It both offers a possibility to create new numerical methods using infinities and infinitesimals in floating-point computations and allows one to study certain mathematical objects dealing with infinity more accurately than it is done traditionally. In these notes, we explain that even though both methodologies deal with infinities and infinitesimals, they are independent and represent two different philosophies of Mathematics that are not in a conflict. It is proved that texts :539–555, 2017; Gutman and Kutateladze in Sib Math J 49:835–841, 2008; Kutateladze in J Appl Ind Math 5:73–75, 2011) asserting that the ①-based methodology is a part of non-standard analysis unfortunately contain several logical fallacies. Their attempt to show that the ①-based methodology can be formalized within non-standard analysis is similar to trying to show that constructivism can be reduced to the traditional Mathematics. (shrink)
Alfred Tarski (1901--1983) is widely regarded as one of the two giants of twentieth-century logic and also as one of the four greatest logicians of all time (Aristotle, Frege and Gödel being the other three). Of the four, Tarski was the most prolific as a logician. The four volumes of his collected papers, which exclude most of his 19 monographs, span over 2500 pages. Aristotle's writings are comparable in volume, but most of the Aristotelian corpus is not about logic, whereas (...) virtually everything written by Tarski concerns logic more or less directly. There is no doubt that Tarski wrote more on logic than any other author; he started publishing on logic in 1921 at the age of 20 and continued until his death at the age of 82. Two of his works appeared posthumously [Hist. Philos. Logic 7 (1986), no. 2, 143--154; MR0868748 (88b:03010); Tarski and Givant, A formalization of set theory without variables, Amer. Math. Soc., Providence, RI, 1987; MR0920815 (89g:03012)]. Tarski's voluminous writings were widely scattered in numerous journals, some quite rare. It has been extremely difficult to study the development of Tarski's thought and to trace the interconnections and interdependence of his various papers. Thanks to the present collection all this has changed, and it is likely that the increased accessibility of Tarski's papers will have the effect of increasing Tarski's already enormous influence. (shrink)
This book is a new translation of Jivanmukti Viveka by Vidyaranya by Swami Harshananda, Ramakrishna Math, Bangalore. This translation is lucid and helps one to understand clearly the various subtle nuances of the original Sanskrit text. The original translation was into Kannada, which has been translated into English by H Ramachandra Swamy.
I came late to philosophy and even later to normative ethics. When I started my undergraduate studies at the University of Toronto in 1970, I was interested in mathematics and languages. I soon discovered, however, that my mathematical talents were rather meager compared to the truly talented. I therefore decided to study actuarial science (the applied mathematics of risk assessment for insurance and pension plans) rather than abstract math. After two years, however, I dropped out of university, went to (...) work for a life insurance company, and started studying on my own for the ten professional actuarial exams. When not studying, I would often go to the public library and I was drawn to the philosophy section—although I had no idea of what philosophy was about. I there saw Logical Positivism, edited by A.J. Ayer. I was interested in logical thinking and I also favored an optimistic attitude towards life (!) and so I thought that the book might be interesting. I checked it out and was absolutely enthralled with the writings of Bertrand Russell, Rudolf Carnap, Carl Hempel and others (if I’m remembering correctly). Of course, I didn’t really understand much of what they were doing, but I did see that they were addressing important problems in a systematic and rigorous manner. I liked it! (shrink)
Mathematical thinking skills are very important in mathematics, both to learn math or as learning goals. Thinking skills can be seen from the description given answers in solving mathematical problems faced. Mathematical thinking skills can be seen from the types, levels, and process. Proportionally questions given to students at universities in Indonesia (semester I, III, V, and VII). These questions are a matter of description that belong to the higher-level thinking. Students choose 5 of 8 given problem. Qualitatively, the (...) answers were analyzed by descriptive to see the tendency to think mathematically used in completing the test. The results show that students tend to choose the issues relating to the calculation. They are more use cases, examples and not an example, to evaluate the conjecture and prove to belong to the numeric argumentation. Used mathematical thinking students are very personal (intelligence, interest, and experience), and the situation (problems encountered). Thus, the level of half of the students are not guaranteed and shows the level of mathematical thinking. (shrink)
Years ago, when I was an undergraduate math major at the University of Wyoming, I came across an interesting book in our library. It was a book of counterexamples t o propositions in real analysis (the mathematics of the real numbers). Mathematicians work more or less like the rest of us. They consider propositions. If one seems to them to be plausibly true, then they set about to prove it, to establish the proposition as a theorem. Instead o f (...) setting out to prove propositions, the psychologists, neuroscientists, and other empirical types among us, set out to show that a proposition is supported by the data, and that it is the best such proposition so supported. The philosophers among us, when they are not causing trouble by arguing that AI is a dead end or that cognitive science can get along without representations, work pretty much like the mathematicians: we set out to prove certain propositions true on the basis of logic, first principles, plausible assumptions, and others' data. But, back to the book of real analysis counterexamples. If some mathematician happened t o think that some proposition about continuity, say, was plausibly true, he or she would then set out to prove it. If the proposition was in fact not a theorem, then a lot of precious time would be wasted trying to prove it. Wouldn't it be great to have a book that listed plausibly true propositions that were in fact not true, and listed with each such proposition a counterexample to it? Of course it would. (shrink)
Games of chance are developed in their physical consumer-ready form on the basis of mathematical models, which stand as the premises of their existence and represent their physical processes. There is a prevalence of statistical and probabilistic models in the interest of all parties involved in the study of gambling – researchers, game producers and operators, and players – while functional models are of interest more to math-inclined players than problem-gambling researchers. In this paper I present a structural analysis (...) of the knowledge attached to mathematical models of games of chance and the act of modeling, arguing that such knowledge holds potential in the prevention and cognitive treatment of excessive gambling, and I propose further research in this direction. (shrink)
I discuss the structure of genealogical debunking arguments. I argue that they undermine our mathematical beliefs if they undermine our moral beliefs. The contrary appearance stems from a confusion of arithmetic truths with (first-order) logical truths, or from a confusion of reliability with justification. I conclude with a discussion of the cogency of debunking arguments, in light of the above. Their cogency depends on whether information can undermine all of our beliefs of a kind, F, without giving us direct reason (...) to doubt that our F-beliefs are modally secure. (shrink)
Alternate Universes: Religion assumes the other world after death: paradise, hell, nirvana, karma.. Our world is incomplete, because there is truer universe, replicating Plato: behind something is something.. till the true idea - last judgment, karma.. R. Descartes's "I think, therefore I am", is independent of Plato. I'm thinking, regardless of there is truer idea or not. As I'm thinking, I can realize my first idea was false (eg. solving a math problem), and then the Plato's truer idea reappears. (...) Plato and Descartes precede each other as chicken and egg.. Paradox of Religion: Religions' rules (e.g. don't lie) increase chances of future paradise. Hume's: "A" preceding "B" doesn't need to cause "B" turns to: Expectation of "B" can cause "A" NOW Believers expect paradise to forever delay it. Paradise can paradoxically occur only by deviation from the religion... Exhibited in Holland Park, W8 6LU, The Ice House between 18. Oct - 3. Nov. 2013. (shrink)
Reid, Constance. Hilbert (a Biography). Reviewed by Corcoran in Philosophy of Science 39 (1972), 106–08. -/- Constance Reid was an insider of the Berkeley-Stanford logic circle. Her San Francisco home was in Ashbury Heights near the homes of logicians such as Dana Scott and John Corcoran. Her sister Julia Robinson was one of the top mathematical logicians of her generation, as was Julia’s husband Raphael Robinson for whom Robinson Arithmetic was named. Julia was a Tarski PhD and, in recognition of (...) a distinguished career, was elected President of the American Mathematics Society. https://en.wikipedia.org/wiki/Julia_Robinson http://www.awm-math.org/noetherbrochure/Robinson82.html. (shrink)
In the framework of materialism, the major attention is to find general organizational laws stimulated by physical sciences, ignoring the uniqueness of Life. The main goal of materialism is to reduce consciousness to natural processes, which in turn can be translated into the language of math, physics and chemistry. Following this approach, scientists have made several attempts to deny the living organism of its veracity as an immortal soul, in favor of genes, molecules, atoms and so on. However, advancement (...) in various fields of biology has repeatedly given rise to questions against such a denial and has supplied more and more evidence against the completely misleading ideological imposition that living entities are particular states of matter. In the recent past, however, the realization has arisen that cognitive nature of life at all levels has begun presenting significant challenges to the views of materialism in biology and has created a more receptive environment for the soul hypothesis. Therefore, instead of adjudicating different aprioristic claims, the development of an authentic theory of biology needs both proper scientific knowledge and the appropriate tools of philosophical analysis of life. In a recently published paper the first author of present essay made an attempt to highlight a few relevant developments supporting a sentient view of life in scientific research, which has caused a paradigm shift in our understanding of life and its origin [1]. The present essay highlights the uniqueness of biological systems that offers a considerable challenge to the mainstream materialism in biology and proposes the Vedāntic philosophical view as a viable alternative for development of a biological theory worthy of life. (shrink)
William Oliver Martin published "The Order and Integration of Knowledge" in 1957 to address the problem of the nature and the order of various kinds of knowledge; in particular, the theoretical problem of how one kind of knowledge is related to another kind. Martin characterizes kinds of knowledge as being either autonomous or synthetic. The latter are reducible to two or more of the autonomous (or irreducible) kinds of knowledge, viz., history (H), metaphysics (Meta), theology (T), formal logic (FL), mathematics (...) (Math), and generalizations of experimental science (G). Metaphysics and theology constitute the two domains of the ontological context while history and experimental science are the two domains of the phenomenological context. The relation of one kind of knowledge to another may be instrumental, constitutive, and/or regulative. For instance, historical propositions are constitutive of G, metaphysical propositions are regulative of G, and propositions in formal logic and mathematics are instrumental to G. Theological propositions are not related to G and so there is no conflict between science and theology. Martin's work sheds light on the possible areas of incompatibility between science and religion. (shrink)
The movement called Experimental Philosophy (‘x-Phi’) has now passed its tenth anniversary. Its central insight is compelling: When an argument hinges on accepting certain ‘facts’ about human perception, knowledge, or judging, the evoking of relevant intuitions by thought experiments is intended to make those facts seem obvious. But these intuitions may not be shared universally. Experimentalists propose testing claims that traditionally were intuition-based using real experiments, with real samples. Demanding that empirical claims be empirically supported is certainly reasonable; though experiments (...) are not necessarily the only means available. When experiments are conducted, adequately interpreting their results requires understanding the study’s design (and possibly flaws) that produced them. Experiment-based reports should document the design clearly. If Plato, writing his Meno, replaced accounts of Socrates demonstrating geometry to a slave-boy, with a survey of 100 real boys—some grasping his demonstrations, others not—what conclusions could be reached? Before answering, the reader needs details on key design questions, including (among others): (a) What population are these samples intended to represent? (E.g., ‘all slave boys’?; ‘math-ignorant people’?; ‘everyone’?) (b) What statistical tests were conducted, on what assumptions? (c) How was ‘significance’ of results determined? (d) Was the test instrument’s validity established? For readers wishing to explore x-Phi’s potentials, as contributors or as interpreters of their findings, this paper offers some cautionary considerations. Throughout their literature, examples can be found casting doubt on some experimentalists’ findings, due to design-related issues. Increased awareness of methodological questions would tighten the x-phi literature, going forward . (shrink)
I start with some famous comments by the philosopher (psychologist) Ludwig Wittgenstein because Pinker shares with most people (due to the default settings of our evolved innate psychology) certain prejudices about the functioning of the mind and because Wittgenstein offers unique and profound insights into the workings of language, thought and reality (which he viewed as more or less coextensive) not found anywhere else. The last quote is the only reference Pinker makes to Wittgenstein in this volume, which is most (...) unfortunate considering that he was one of the most brilliant and original analysts of language. -/- In the last chapter, using the famous metaphor of Plato’s cave, he beautifully summarizes the book with an overview of how the mind (language, thought, intentional psychology) –a product of blind selfishness, moderated only slightly by automated altruism for close relatives carrying copies of our genes--works automatically, but tries to end on an upbeat note by giving us hope that we can nevertheless employ its vast capabilities to cooperate and make the world a decent place to live. -/- Pinker is certainly aware of but says little about the fact that far more about our psychology is left out than included. Among windows into human nature that are left out or given minimal attention are math and geometry, music and sounds, images, events and causality, ontology (classes of things), dispositions (believing, thinking, judging, intending etc) and the rest of intentional psychology of action, neurotransmitters and entheogens, spiritual states (e.g, satori and enlightenment, brain stimulation and recording, brain damage and behavioral deficits and disorders, games and sports, decision theory (incl. game theory and behavioral economics), animal behavior (very little language but a billion years of shared genetics). Many books have been written about each of these areas of intentional psychology. The data in this book are descriptions, not explanations that show why our brains do it this way or how it is done. How do we know to use the sentences in their various way (i.e., know all their meanings)? This is evolutionary psychology that operates at a more basic level –the level where Wittgenstein is most active. And there is scant attention to context. -/- Nevertheless this is a classic work and with these cautions is still well worth reading. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Human Behavior (2019), and Suicidal Utopian Delusions in the 21st Century 4th ed (2019). (shrink)
In a recent article in this journal [Phil. Math., II, v.4 (1989), n.2, pp.?- ?] J. Fang argues that we must not be fooled by A.J. Ayer (God rest his soul!) and his cohorts into believing that mathematical knowledge has an analytic a priori status. Even computers, he reminds us, take some amount of time to perform their calculations. The simplicity of Kant's infamous example of a mathematical proposition (7+5=12) is "partly to blame" for "mislead[ing] scholars in the direction (...) of neglecting the temporal element"; yet a brief instant of time is required to grasp even this simple truth. If Kant were alive today, "and if he had had a little more mathematical savvy", Fang explains, he could have used the latest example of the largest prime number (391,581 x 2 216,193 - 1) as a better example of the "synthetic a priori" character of mathematics. The reason Fang is so intent upon emphasizing the temporal character of mathematics is that he wishes to avoid "the uncritical mixing of ... a theology and a philosophy of mathematics." For "in the light of the Computer Age today: finitism is king!" Although Kant's aim was explicitly "to study the 'human' ... faculty", Fang claims that even he did not adequatley emphasize "the clearly and concretely distinguishable line of demarcation between the human and divine faculties.". (shrink)
Girolamo Saccheri (1667--1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. He earned a permanent place in the history of mathematics by discovering and rigorously deducing an elaborate chain of consequences of an axiom-set for what is now known as hyperbolic (or Lobachevskian) plane geometry. Reviewer's remarks: (1) On two pages of this book Saccheri refers to his previous and equally original book Logica demonstrativa (Turin, 1697) to which 14 of the 16 pages of the editor's "Introduction" are devoted. (...) At the time of the first edition, 1920, the editor was apparently not acquainted with the secondary literature on Logica demonstrativa which continued to grow in the period preceding the second edition \ref[see D. J. Struik, in Dictionary of scientific biography, Vol. 12, 55--57, Scribner's, New York, 1975]. Of special interest in this connection is a series of three articles by A. F. Emch [Scripta Math. 3 (1935), 51--60; Zbl 10, 386; ibid. 3 (1935), 143--152; Zbl 11, 193; ibid. 3 (1935), 221--333; Zbl 12, 98]. (2) It seems curious that modern writers believe that demonstration of the "nondeducibility" of the parallel postulate vindicates Euclid whereas at first Saccheri seems to have thought that demonstration of its "deducibility" is what would vindicate Euclid. Saccheri is perfectly clear in his commitment to the ancient (and now discredited) view that it is wrong to take as an "axiom" a proposition which is not a "primal verity", which is not "known through itself". So it would seem that Saccheri should think that he was convicting Euclid of error by deducing the parallel postulate. The resolution of this confusion is that Saccheri thought that he had proved, not merely that the parallel postulate was true, but that it was a "primal verity" and, thus, that Euclid was correct in taking it as an "axiom". As implausible as this claim about Saccheri may seem, the passage on p. 237, lines 3--15, seems to admit of no other interpretation. Indeed, Emch takes it this way. (3) As has been noted by many others, Saccheri was fascinated, if not obsessed, by what may be called "reflexive indirect deductions", indirect deductions which show that a conclusion follows from given premises by a chain of reasoning beginning with the given premises augmented by the denial of the desired conclusion and ending with the conclusion itself. It is obvious, of course, that this is simply a species of ordinary indirect deduction; a conclusion follows from given premises if a contradiction is deducible from those given premises augmented by the denial of the conclusion---and it is immaterial whether the contradiction involves one of the premises, the denial of the conclusion, or even, as often happens, intermediate propositions distinct from the given premises and the denial of the conclusion. Saccheri seemed to think that a proposition proved in this way was deduced from its own denial and, thus, that its denial was self-contradictory (p. 207). Inference from this mistake to the idea that propositions proved in this way are "primal verities" would involve yet another confusion. The reviewer gratefully acknowledges extensive communication with his former doctoral students J. Gasser and M. Scanlan. ADDED 14 March 14, 2015: (1) Wikipedia reports that many of Saccheri's ideas have a precedent in the 11th Century Persian polymath Omar Khayyám's Discussion of Difficulties in Euclid, a fact ignored in most Western sources until recently. It is unclear whether Saccheri had access to this work in translation, or developed his ideas independently. (2) This book is another exemplification of the huge difference between indirect deduction and indirect reduction. Indirect deduction requires making an assumption that is inconsistent with the premises previously adopted. This means that the reasoner must perform a certain mental act of assuming a certain proposition. It case the premises are all known truths, indirect deduction—which would then be indirect proof—requires the reasoner to assume a falsehood. This fact has been noted by several prominent mathematicians including Hardy, Hilbert, and Tarski. Indirect reduction requires no new assumption. Indirect reduction is simply a transformation of an argument in one form into another argument in a different form. In an indirect reduction one proposition in the old premise set is replaced by the contradictory opposite of the old conclusion and the new conclusion becomes the contradictory opposite of the replaced premise. Roughly and schematically, P,Q/R becomes P,~R/~Q or ~R, Q/~P. Saccheri’s work involved indirect deduction not indirect reduction. (3) The distinction between indirect deduction and indirect reduction has largely slipped through the cracks, the cracks between medieval-oriented logic and modern-oriented logic. The medievalists have a heavy investment in reduction and, though they have heard of deduction, they think that deduction is a form of reduction, or vice versa, or in some cases they think that the word ‘deduction’ is the modern way of referring to reduction. The modernists have no interest in reduction, i.e. in the process of transforming one argument into another having exactly the same number of premises. Modern logicians, like Aristotle, are concerned with deducing a single proposition from a set of propositions. Some focus on deducing a single proposition from the null set—something difficult to relate to reduction. (shrink)
Why microscopic objects exhibit wave properties (are delocalized), but macroscopic do not (are localized)? Traditional quantum mechanics attributes wave properties to all objects. When complemented with a deterministic collapse model (Quantum Stud.: Math. Found. 3, 279 (2016)) quantum mechanics can dissolve the discrepancy. Collapse in this model means contraction and occurs when the object gets in touch with other objects and satisfies a certain criterion. One single collapse usually does not suffice for localization. But the object rapidly gets in (...) touch with other objects in a short time, leading to rapid localization. Decoherence is not involved. (shrink)
Consider the following. The first is a one-premise argument; the second has two premises. The question sign marks the conclusions as such. -/- Matthew, Mark, Luke, and John wrote Greek. ? Every evangelist wrote Greek. -/- Matthew, Mark, Luke, and John wrote Greek. Every evangelist is Matthew, Mark, Luke, or John. ? Every evangelist wrote Greek. -/- The above pair of premise-conclusion arguments is of a sort familiar to logicians and philosophers of science. In each case the first premise is (...) logically equivalent to the set of four atomic propositions: “Matthew wrote Greek”, “Mark wrote Greek”, “Luke wrote Greek”, and “John wrote Greek”. The universe of discourse is the set of evangelists. We presuppose standard first-order logic. -/- As many logic texts teach, the first of these two premise-conclusion arguments—sometimes called a complete enumerative induction— is invalid in the sense that its conclusion does not follow from its premises. To get a counterargument, replace ‘Matthew’, ‘Mark’, ‘Luke’, and ‘John’ by ‘two’,’four’, ‘six’ and ‘eight’; replace ‘wrote Greek’ by ‘are even’; and replace ‘evangelist’ by ‘number’. This replacement converts the first argument into one having true premises and false conclusion. -/- But the same replacement performed on the second argument does no such thing: it converts the second premise into the falsehood “Every number is two, four, six, or eight”. As many logic texts teach, there is no replacement that converts the second argument into one with all true premises and false conclusion. The second is valid; its conclusion is deducible from its two premises using an instructive natural deduction. -/- This paper “does the math” behind the above examples. The theorem could be stated informally: the above examples are typical. (shrink)
Continuing his series of books on the mathematics of gambling, the author shows how a simple-rule game such as roulette is suited to a complex mathematical model whose applications generate improved betting systems that take into account a player's personal playing criteria. The book is both practical and theoretical, but is mainly devoted to the application of theory. About two-thirds of the content is lists of categories and sub-categories of improved betting systems, along with all the parameters that might stand (...) as the main objective criteria in a personal strategy - odds, profits and losses. The work contains new and original material not published before. The mathematical chapter describes complex bets, the profit function, the equivalence between bets and all their properties. All theoretical results are accompanied by suggestive concrete examples and can be followed by anyone with a minimal mathematical background because they involve only basic algebraic skills and set theory basics. The reader may also choose to skip the math and go directly to the sections containing applications, where he or she can pick desired numerical results from tables. The book offers no new so-called winning strategies, although it discusses them from a mathematical point of view. It does, however, offer improved betting systems and helps to organize a player's choices in roulette betting, according to mathematical facts and personal strategies. It is a must-have roulette handbook to be studied before placing your bets on the turn of either a European or American roulette wheel. (shrink)
This book presents not only the mathematical concept of probability, but also its philosophical aspects, the relativity of probability and its applications and even the psychology of probability. All explanations are made in a comprehensible manner and are supported with suggestive examples from nature and daily life, and even with challenging math paradoxes.
Much problem solving and learning research in math and science has focused on formal representations. Recently researchers have documented the use of unschooled strategies for solving daily problems -- informal strategies which can be as effective, and sometimes as sophisticated, as school-taught formalisms. Our research focuses on how formal and informal strategies interact in the process of doing and learning mathematics. We found that combining informal and formal strategies is more effective than single strategies. We provide a theoretical account (...) of this multiple strategy effect and have begun to formulate this theory in an ACT-R computer model. We show why students may reach common impasses in the use of written algebra, and how subsequent or concurrent use of informal strategies leads to better problem-solving performance. Formal strategies facilitate computation because of their abstract and syntactic nature; however, abstraction can lead to nonsensical interpretations and conceptual errors. Reapplying the formal strategy will not repair such errors; switching to an informal one may. We explain the multiple strategy effect as a complementary relationship between the computational efficiency of formal strategies and the sense-making function of informal strategies. (shrink)
The history of the house in Shyampukur, Kolkata, India, where Sri Ramakrishna lived for sometime when he was ailing. And the history of the place till the present-day, when it is a branch centre of the Ramakrishna Math.
I give a detailed review of 'The Outer Limits of Reason' by Noson Yanofsky 403(2013) from a unified perspective of Wittgenstein and evolutionary psychology. I indicate that the difficulty with such issues as paradox in language and math, incompleteness, undecidability, computability, the brain and the universe as computers etc., all arise from the failure to look carefully at our use of language in the appropriate context and hence the failure to separate issues of scientific fact from issues of how (...) language works. I discuss Wittgenstein's views on incompleteness, paraconsistency and undecidability and the work of Wolpert on the limits to computation. -/- Those wishing a comprehensive up to date account of Wittgenstein, Searle and their analysis of behavior from the modern two systems view may consult my article The Logical Structure of Philosophy, Psychology, Mind and Language as Revealed in Wittgenstein and Searle (2016). Those interested in all my writings in their most recent versions may download from this site my e-book ‘Philosophy, Human Nature and the Collapse of Civilization Michael Starks (2016)- Articles and Reviews 2006-2016’ by Michael Starks First Ed. 662p (2016). -/- All of my papers and books have now been published in revised versions both in ebooks and in printed books. -/- Talking Monkeys: Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet - Articles and Reviews 2006-2017 (2017) https://www.amazon.com/dp/B071HVC7YP. -/- The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle--Articles and Reviews 2006-2016 (2017) https://www.amazon.com/dp/B071P1RP1B. -/- Suicidal Utopian Delusions in the 21st century: Philosophy, Human Nature and the Collapse of Civilization - Articles and Reviews 2006-2017 (2017) https://www.amazon.com/dp/B0711R5LGX . (shrink)
Purpose – In the last half-century, individual sensory neurons have been bestowed with characteristics of the whole human being, such as behavior and its oft-presumed precursor, consciousness. This anthropomorphization is pervasive in the literature. It is also absurd, given what we know about neurons, and it needs to be abolished. This study aims to first understand how it happened, and hence why it persists. Design/methodology/approach – The peer-reviewed sensory-neurophysiology literature extends to hundreds (perhaps thousands) of papers. Here, more than 90 (...) mainstream papers were scrutinized. Findings – Anthropomorphization arose because single neurons were cast as “observers” who “identify”, “categorize”, “recognize”, “distinguish” or “discriminate” the stimuli, using math-based algorithms that reduce (“decode”) the stimulus-evoked spike trains to the particular stimuli inferred to elicit them. Without “decoding”, there is supposedly no perception. However, “decoding” is both unnecessary and unconfirmed. The neuronal “observer” in fact consists of the laboratory staff and the greater society that supports them. In anthropomorphization, the neuron becomes the collective. Research limitations/implications – Anthropomorphization underlies the widespread application to neurons Information Theory and Signal Detection Theory, making both approaches incorrect. Practical implications – A great deal of time, money and effort has been wasted on anthropomorphic Reductionist approaches to understanding perception and consciousness. Those resources should be diverted into more-fruitful approaches. Originality/value – A long-overdue scrutiny of sensory-neuroscience literature reveals that anthropomorphization, a form of Reductionism that involves the presumption of single-neuron consciousness, has run amok in neuroscience. Consciousness is more likely to be an emergent property of the brain. (shrink)
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