This paper deploys a Cantor-style diagonal argument which indicates that there is more possible mathematical content than there are propositional functions in Russell and Whitehead's Principia Mathematica and similar formal systems. This technical result raises a historical question: "How did Russell, who was himself an expert in diagonal arguments, not see this coming?" It turns out that answering this question requires an appreciation of Russell's understanding of what logic is, and how he construed the relationship between logic and (...) Principia Mathematica. (shrink)

Mi objetivo es discutir las principales dificultades que Frank P. Ramsey encontró en Principia Mathematica y la solución que, vía el Tractatus Logico-Philosophicus, propuso al respecto. Sostengo que las principales dificultades que Ramsey encontró en Principia Mathematica están, todas, relacionadas con que Russell y Whitehead desatendieron la forma lógica de las proposiciones matemáticas, las cuales, según Ramsey, deben ser tautológicas.

In the Foundations of Mathematics, Ramsey attempted to amend Principia Mathematica’s logicism to meet serious objections raised against it. While Ramsey’s paper is well known, some questions concerning Ramsey’s motivations to write it and its reception still remain. This paper considers these questions afresh. First, an account is provided for why Ramsey decided to work on his paper instead of simply accepting Wittgenstein’s account of mathematics as presented in the Tractatus. Secondly, evidence is given supporting that Wittgenstein was (...) not moved by Ramsey’s objection against the Tractarian account of arithmetic, and a suggestion is made to explain why Wittgenstein reconsidered Ramsey’s account in the early thirties on several occasions. Finally, a reading is formulated to understand the basis on which Wittgenstein argues against Ramsey’s definition of identity in his 1927 letter to Ramsey. (shrink)

Conceptual engineers have made hay over the differences of their metaphilosophy from those of conceptual analysts. In this article, I argue that the differences are not as great as conceptual engineers have, perhaps rhetorically, made them seem. That is, conceptual analysts asking ‘What is X?’ questions can do much the same work that conceptual engineers can do with ‘What is X for?’ questions, at least if conceptual analysts self-understand their activity as a revisionary enterprise. I show this with a study (...) of Russell's metaphilosophy, which was just such a revisionary conception of conceptual analysis. (shrink)

According to Russell, the notation in Principia Mathematica has been designed to avoid the vagueness endemic to our natural language. But what does Russell think vagueness is? My argument is an attempt to show that his views on vagueness evolved and that the final conception he adopts is not coherent. Three phases of his conception of vagueness are identified, the most significant being the view that he articulates on vagueness in his 1923 address to the Jowett Society. My (...) central thesis is that the 1912 conception of vagueness -- what I characterize as "semantic egalitarianism" -- seriously conflicts with the later paradoxical account of vagueness. (shrink)

Russell’s role in the controversy about the paradoxes of material implication is usually presented as a tale of how even the greatest minds can fall prey to basic conceptual confusions. Quine accused him of making a silly mistake in Principia Mathematica. He interpreted “if- then” as a version of “implies” and called it material implication. Quine’s accusation is that this decision involved a use-mention fallacy because the antecedent and consequent of “if-then” are used instead of being mentioned as (...) the premise and the conclusion of an implication relation. It was his opinion that the criticisms and alternatives to the material implication presented by C. I. Lewis and others would never be made in the first place if Russell simply called the Philonian construction “material conditional” instead of “material implication”. Quine’s interpretation of the topic became hugely influential if not universally accepted. This paper will present the following criticisms against this interpretation: (1) the notion of material implication does not involve a use-mention fallacy, since the components of “if-then” are mentioned and not used; (2) Quine’s belief that the components of “if-then” are used was motivated by a conditional-assertion view of conditionals that is widely controversial and faces numerous difficulties; (3) if anything, it was Quine who could be accused of fallacious reasoning: he ignored that in the assertion of a conditional is the whole proposition that is asserted and not its constituents; (4) the Philonian construction remains counter-intuitive even if it is called “material conditional”; (5) the Philonian construction is more plausible when it is interpreted as a material implication. (shrink)

Higher-order logic, with its type-theoretic apparatus known as the simple theory of types (STT), has increasingly come to be employed in theorizing about properties, relations, and states of affairs—or ‘intensional entities’ for short. This paper argues against this employment of STT and offers an alternative: ordinal type theory (OTT). Very roughly, STT and OTT can be regarded as complementary simplifications of the ‘ramified theory of types’ outlined in the Introduction to Principia Mathematica (on a realist reading). While STT, (...) understood as a theory of intensional entities, retains the Fregean division of properties and relations into a multiplicity of categories according to their adicities and ‘input types’ and discards the division of intensional entities into different ‘orders’, OTT takes the opposite approach: it retains the hierarchy of orders (though with some modifications) and discards the categorisation of properties and relations according to their adicities and input types. In contrast to STT, this latter approach avoids intensional counterparts of the Epimenides and related paradoxes. Fundamental intensional entities lie at the base of the proposed hierarchy and are also given a prominent part to play in the individuation of non-fundamental intensional entities. (shrink)

When Newton articulated the concept of absolute time in his treatise, Philosophae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), along with its correlate, absolute space, he did not present it as anything controversial. Whereas his references to attraction are accompanied by the self- protective caveats that typically signal an expectation of censure, the Scholium following Principia’s definitions is free of such remarks, instead elaborating his ideas as clarifications of concepts that, in some manner, we already possess. (...) This is not surprising. The germ of the concept emerged naturally from astronomers’ findings, and variants of it had already been formulated by other seventeenth century thinkers. Thus the novelty of Newton’s absolute time lay mainly in the use to which he put it. (shrink)

Newton frequently characterized his methodology as distinctive and capable of achieving greater evidential support than that of his contemporaries, due to its mathematical character. Newton's pronouncements reflect a striking position regarding the role of mathematics in natural philosophy. We can give an initial characterization of his position by considering two questions central to seventeenth century debates about the applicability of mathematics. First, how are we to understand the distinctive universality and necessity of mathematical reasoning? One common way to preserve the (...) demonstrative character of mathematics was to restrict its domain, as far as possible, to pure abstractions. The subject matter of mathematics is then taken to be abstracted from the changeable natural world, consisting of quantity and magnitude themselves rather than the objects bearing quantifiable properties. Yet restricting the domain in this way leaves it difficult to see how mathematics relates to natural phenomena. How could the book of nature be written in a language of pure abstractions? Second, what is the proper role of mathematical reasoning in natural philosophy? Many followed Aristotle in consigning mathematics to a subordinate role. Mathematics could not fulfill the main aim of natural philosophy, namely to provide demonstrations reflecting the essences of things and nature's causal order. A related concern was also pressing for the mechanical philosophers: a merely mathematical demonstration fails to provide an intelligible mechanical explanation. The very title of Newton's masterpiece, Philosophiae Naturalis Principia Mathematica, would have been extremely perplexing: how could natural philosophy be based on mathematical principles? My aim is to articulate Newton's position regarding the mathematization of nature in response to these two concerns. (shrink)

C. I. Lewis (I883-I964) was the first major figure in history and philosophy of logic—-a field that has come to be recognized as a separate specialty after years of work by Ivor Grattan-Guinness and others (Dawson 2003, 257).Lewis was among the earliest to accept the challenges offered by this field; he was the first who had the philosophical and mathematical talent, the philosophical, logical, and historical background, and the patience and dedication to objectivity needed to excel. He was blessed with (...) many fortunate circumstances, not least of which was entering the field when mathematical logic, after only six decades of toil, had just reaped one of its most important harvests with publication of the monumental Principia Mathematica. It was a time of joyful optimism which demanded an historical account and a sober philosophical critique. Lewis was one of the first to apply to mathematical logic the Aristotelian dictum that we do not understand a living institution until we see it growing from its birth. (shrink)

In recent decades, plural logic has established itself as a well-respected member of the extensions of first-order classical logic. In the present paper, I draw attention to the fact that among the examples that are commonly given in order to motivate the need for this new logical system, there are some in which the elements of the plurality in question are internally singularized (e.g. ‘Whitehead and Russell wrote Principia Mathematica’), while in others they are not (e.g. ‘Some philosophers (...) wrote Principia Mathematica’). Then, building on previous work, I point to a subsystem of plural logic in which inferences concerning examples of the first type can be adequately dealt with. I notice that such a subsystem (here called ‘discrete plural logic’) is in reality a mere variant of first-order logic as standardly formulated, and highlight the fact that it is axiomatizable while full plural logic is not. Finally, I urge that greater attention be paid to discrete plural logic and that discrete plurals are not used in order to motivate the introduction of full-fledged plural logic—or, at least, not without remarking that they can also be adequately dealt with in a considerably simpler system. (shrink)

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework (...) recognizes multiple universes of discourse serving as different ranges of the individual variables in different interpretations?as in post-WWII model theory. In the early 1960s, many logicians?mistakenly, as we show?held the ?contrary alternative? that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege?Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes. (shrink)

In he Problems of Philosophy and other works of the same period, Russell claims that every proposition must contain at least one universal. Even fully general propositions of logic are claimed to contain “abstract logical universals”, and our knowledge of logical truths claimed to be a species of a priori knowledge of universals. However, these views are in considerable tension with Russell’s own philosophy of logic and mathematics as presented in Principia Mathematica. Universals generally are qualities and relations, (...) but if, for example, PM’s disjunction (∨) is a relation, what is it a relation between? There is no obvious answer to this given Russell’s other philosophical commitments at this time, although hints are left in some of the pre-PM manuscripts. In this paper, I explore this tension in Russell's philosophy and relate it to developments both before and after Problems. (shrink)

É bem conhecida a divergência entre as posições de Gottlob Frege e Bertrand Russell com relação ao tratamento semântico dado a sentenças contendo termos singulares indefinidos, ou seja, termos singulares sem referência ou com referência ambígua, tais como ‘Papai Noel’ ou ‘o atual rei da França’ ou ‘1/0 ’ ou ‘√4’ ou ‘o autor de Principia Mathematica’. Para Frege, as sentenças da linguagem natural que contêm termos indefinidos não formam declarações e portanto não são nem verdadeiras nem falsas. (...) Já para as sentenças da matemática, Frege defende que elas precisam ser corrigidas através da convenção forçada de uma referência não ambígua. Russell, por outro lado, aceita os termos indefinidos e propõe, através de sua teoria das descrições definidas, uma maneira de avaliar as sentenças em que eles ocorrem; e Quine amplia a teoria de Russell para abranger também os nomes com problemas de referência. Na prática da matemática são comuns os termos singulares indefinidos, sem referência, tais como ‘1/0 ’, ou com referência ambígua, tais como ‘√4’. Apesar de não haver uma sistematização rigorosa desta situação entre os matemáticos, há, no entanto, um conjunto de regras convencionais que tradicionalmente costumam ser aplicadas no tratamento matemático dos termos indefinidos. Nossa proposta é tomar a convenção matemática como inspiração e modelo para apresentar uma interpretação semântica formal para as descrições definidas e os nomes e utilizá-la como um argumento que favorece a abordagem de Russell relativamente à de Frege. (shrink)

Bertrand Russell’s conception of philosophy evolved dramatically in 1911 — the year he fell in love with Lady Ottoline Morrell. For many years Russell had been an ardent advocate of the view that philosophers ought to look for truths that are certain. The co-author of Principia Mathematica altered his commitment to certainty in philosophy in 1911. An analysis of his published views and correspondence from this time strongly suggests that the radical transformation was induced by an epiphany brought (...) about by his emotional entanglement with his lover. (shrink)

It is widely agreed by philosophers that the so-called “Frege-Russell definition of natural number” is actually an assertion concerning the nature of the numbers and that it cannot be regarded as a definition in the ordinary mathematical sense. On the basis of the reasoning in this paper it is clear that the Frege-Russell definition contradicts the following three principles (taken together): (1) each number is the same entity in each possible world, (2) each number exists in each possible world, (3) (...) some entities existing in the actual world do not exist in every possible world. Since these principles seem to be true, the paper is a refutation of the Frege-Russell definition. The paper does more. It shows that the contradictory of the Frege-Russell definition follows even when principles 2 and 3 are replaced by one considerably weaker principle. The ideas contained in the paper are related to two earlier objections to the definition. The first, sometimes attributed to the mathematician, C. S. Keyser, is that existence of the numbers as defined implies the existence of infinitely many particulars in each possible world. The second is, in effect, an idea which is said to have led Whitehead to reject the definition of number to which he had subscribed in Principia Mathematica. Whitehead is supposed to have said that he could not believe that the number two changes every “time twins are born”. The mathematician H. Jeffreys expressed similar ideas [Philos. of Sci. 5 (1938), 434–451]. One of the merits of the author’s work is that it refutes the Frege-Russell definition without the need to take sides on controversial points presupposed by the Keyser and Whitehead objections. The objections made by the author are therefore not to be identified with the Keyser and Whitehead objections. Even if the author’s work is to be regarded as a refinement and integration of previous ideas, it is nevertheless a contribution—not only because the basic points are well worth repeating but also because the refinements are logically significant improvements and because the author has stated them clearly and concisely in the idiom of contemporary philosophy. (shrink)

Bertrand Russell was one of the best-known proponents of logicism: the theory that mathematics reduces to, or is an extension of, logic. Russell argued for this thesis in his 1903 The Principles of Mathematics and attempted to demonstrate it formally in Principia Mathematica (PM 1910–1913; with A. N. Whitehead). Russell later described his work as a further “regressive” step in understanding the foundations of mathematics made possible by the late 19th century “arithmetization” of mathematics and Frege’s logical definitions (...) of arithmetical concepts. The logical system of PM sought to improve on earlier attempts by solving the contradictions found in, e.g., Frege’s system, by employing a theory of types. In this article, I also consider and critically evaluate the most common objections to Russell’s logicism, including the claim that it is undermined by Gödel’s incompleteness results, and Putnam's charge of “if-thenism”. I suggest that if we are willing to accept a slightly revisionist account of what counts as a mathematical truth, these criticisms do not obviously refute Russell’s claim to have established that mathematical truths generally are a species of logical truth. (shrink)

It is well known that the circumflex notation used by Russell and Whitehead to form complex function names in Principia Mathematica played a role in inspiring Alonzo Church's “lambda calculus” for functional logic developed in the 1920s and 1930s. Interestingly, earlier unpublished manuscripts written by Russell between 1903–1905—surely unknown to Church—contain a more extensive anticipation of the essential details of the lambda calculus. Russell also anticipated Schönfinkel's combinatory logic approach of treating multiargument functions as functions having other functions (...) as value. Russell’s work in this regard seems to have been largely inspired by Frege’s theory of functions and “value-ranges”. This system was discarded by Russell due to his abandonment of propositional functions as genuine entities as part of a new tack for solving Russell’s paradox. In this article, I explore the genesis and demise of Russell’s early anticipation of the lambda calculus. (shrink)

In his 1927 Analysis of Matter and elsewhere, Russell argued that we can successfully infer the structure of the external world from that of our explanatory schemes. While nothing guarantees that the intrinsic qualities of experiences are shared by their objects, he held that the relations tying together those relata perforce mirror relations that actually obtain (these being expressible in the formal idiom of the Principia Mathematica). This claim was subsequently criticized by the Cambridge mathematician Max Newman as (...) true but trivial, insofar as from a closed body of observations (or “Ramsey sentence”) one can always generate other equally-satisfactory networks of relations, provided they respect the original set’s cardinality. Since any model thus generated will be empirically adequate, “[t]he defence is therefore driven back from the fairly safe fictitious-real classification to the much less tenable ‘trivial’ and ‘important’” (Newman 1928). Given the definitional rigour afforded by the initial appeal to isomorphism (via one-to-one correspondences in extension), the received assessment, shared by Russell himself, is that retreating to a pragmatic adjudication would betoken a fatal blow. However, I suggest that reliance on “importance” can be avoided if we incorporate an impersonal criterion of diachronic precedence. When collecting observations, an ordinality emerges alongside the cardinality which gives that underived structure an irrevocable epistemological privilege. Hence, I argue that, all other things being equal, any construct parasitic on an antecedent theory ought to be regarded as inferior/dispensable, since it was generated by an algorithm lacking the world-involving pedigree of its host structure. (shrink)

A deconstruction of the implicit notion of Absolute space that dominates modern physics. The deconstruction is enacted by juxtaposing the common notion of Absolute space abstracted from Newton’s Philosophiae Naturalis Principia Mathematica with Levinas’ particular present treatment of space in Otherwise than Being: Or Beyond Essence.

Bis vor kurzem wurden die Namen Alfred North Whitehead und Bertrand Russell zumeist in einem Atemzug genannt. Im Anschluss an die gemeinsam verfassten Principia Mathematica gingen beide jedoch dezidiert eigene philosophische Wege. Wahrend Russell maageblich zur Entstehung der analytischen Philosophie beitrug, markiert Whiteheads spate Philosophie den Beginn der bis heute virulenten prozessmetaphysischen Tradition. Stand Whitehead dabei lange im Schatten seines langjahrigen Freundes und Kollegen, zeichnet sich spatestens seit Beginn des 21. Jahrhunderts ein bemerkenswerter Umschwung ab. Der vorliegende Band (...) nimmt die Entwicklung zum Anlass, um Gemeinsamkeiten und Differenzen der beiden Klassiker der angelsachsischen Philosophie neu zu sondieren. (shrink)

Contents: Anna Lemańska, The Autonomous Philosophy of Nature ; Anna Latawiec, The Progress in the Philosophy of Nature ; Grzegorz Bugajak, Philosophy of Nature, Realism, and the Postulated Ontology of Scientific Theories ; Maria-Magdalena Weker, Multi-dimensional Image of the World ; Jarosław Kukowski, The Application of Principia Mathematica and Principia Metaphisica for Resolving the Incommensurate Paradigms. The Comparative Study in Philosophy of Physics and Philosophy of Nature ; Danuta Ługowska, The Dilemmas Concerning the Essence of Life ; (...) Adam Świeżyński, The Beginning of the Universe in the Concept of Creation and in Contemporary Cosmology. The Philosopher’s of Nature Considerations . (shrink)

Fizik disiplininde astronomiye yönelik sorular Johannes Kepler (1571-1630) ve Isaac Newton (1643-1727) gibi fizik alanında devrim yaratan düşüncelerin ortaya çıkmasına yol açmıştır. Kepler ‘Yeni Astronomi’ (Astronomia Nova), ‘Dünya'nın Uyumu’ (Harmonice Mundi) ve ‘Kopernik Astronomisinin Özeti’ (Epitome Astronomiae Copernicanae) adlı çalışmalarından hareketle ileri sürdüğü gezegensel hareket yasaları, Newton ise 1687 yılında yayımlanan ‘Doğa Felsefesinin Matematiksel İlkeleri’ (Philosophiae Naturalis Principia Mathematica) isimli çalışmasında yer verdiği hareket yasaları ile fizik disiplininde Aristoteles fiziğinin terk edilerek yeni bir fizik anlayışının ortaya çıkmasına ve (...) böylece fizik alanında bir devrimin meydana gelmesine sebep olmuşlardır. Fizik disiplininde gerçekleşen bu devrimin bir benzeri ise canlıların çalışılması ya da biyoloji disiplini söz konusu olduğunda 1800’lerden biraz daha geç bir zamana rastlamaktadır. Biyolojide gerçekleşen devrimsel düşünce doğal seçilim mekanizması ile canlılarda zaman içerisinde gerçekleşen değişim ve dönüşüm sürecini açıklayan Charles Darwin’in evrim kuramına yönelik görüşlerini ortaya koymasına tekabül etmektedir. Bu kapsamda çalışmada öncelikli olarak canlılarda meydana gelen değişim süreçlerine yönelik tespitleri ile Darwin’in görüşlerinin olgunlaşmasına katkıda bulunan Fransız Transformistler (Dönüşümcüler) olarak da bilinen Georges-Louis Leclerc (Comte de Buffon), Jean-Baptiste Lamarck ve George Cuvier’in görüşlerine yer verilmektedir. Ardından bu bilim insanlarının canlılardaki dönüşümsel ya da evrimsel değişime ilişkin argümanlarının Darwin’in evrim kuramını nasıl etkilediği ve bu etkilerin sonucunda biyoloji biliminde nasıl bir devrimin ortaya çıkmasına yardımcı oldukları aktarılmakta ve bu çerçevede Darwin’in on dokuzuncu yüzyılda biyoloji biliminde devrime yol açan tespitlerine ve evrim kuramının ayrıntılarına yer verilmektedir. (shrink)

Based on Galileo's experiments, Newton develops the theory of gravity in his first book Philosophiæ Naturalis Principia Mathematica ("Principia") of 1686. Immediately after, Robert Hooke accused Newton of plagiarism, claiming that he unduly assumed his "notion" of "the rule of the decrease of Gravity, being reciprocally as the squares of the distances from the Center". But, according to Edmond Halley, Hooke agreed that "the demonstration of the curves generated by it" belongs entirely to Newton.

The System of the World, Isaac Newton's first draft for the third volume of his classic work Philosophiae Naturalis Principia Mathematica (1687; Mathematical Principles of Natural Philosophy), is an important document in the history of science, using the principles of mechanics to construct the first complete scientific system on the workings of the universe in human history. -/- This book of 78 treatises briefly describes the principles established in the first two volumes of the Mathematical Principles of Natural (...) Philosophy, then applies these principles to the derivation of the actual orbits of the solar system and comets, laying out in layman's terms the universality of the law of gravitation and, from this, studying the shape of the earth, explaining the planetary precession and the tides of the oceans, probing the motion of the moon, and determining the orbits of comets. In this book, Newton does not construct abstract mathematical models, but analyses them in relation to astronomical phenomena, using only a small amount of mathematical language to explain these intriguing elements, making its content suitable for a general audience. -/- The appendices of this Chinese translation by Haixuan Pan(潘海璇) include Life of Sir Issac Newton by N. W. Chittenden, Newtonian Studies by Alexandre Koyré, which introduce Newton's life and main achievements, as well as the debates and evaluations of the main academic views of Newton by the leading philosophers and scientists of the time, and are important supplementary and reference materials when reading The System of the World. (shrink)

Standard interpretations of Descartes' work, especially those inspired by analytic philosophy, assume the syllogism and Principia Mathematica's artificial language are appropriate tools for interpreting Cartesianism, particularly the ontological argument. The dissertation shows technical problems arise when those logics are used for that purpose. An intentianal interpretation along Meinongian lines is proposed, as an alternative to deductive exegeses. This interpretation is developed after arguing that Russell did not identify what I call "the epistemic use" of Meinong's theory of objects. (...) The proposed interpretation is nonlogical in the sense that intentionality alone, not deductive arguments, explains the cogito and res cogitans' belief in god's existence. God is the only one of res cogitan's intentionalia in which existence is analytically included; thus, there is no need for an ontological argument. Such an interpretation is consistent with Descartes' rejection of syllogistics as a truth-discovering tool and provides a framework to evaluate Cartesianism and Kant's criticism of it. Moreover, some problems afflicting logico-deductive readings cannot arise in this interpretation. (shrink)

Dark Matter was not matter at all. It was a theoretical brainteaser that finally philosophy had to unscramble. Scientists of today do not like this idea but philosophy is capable to deal with theoretical conundrums like dark matter. First chapter which is like a combat between mathematical counterintuitive physics and human commonsense, explains that human commonsense equipped with proper philosophical approach is capable to deal with the problem of dark matter. -/- After making a case for philosophical method, this book (...) then challenges the fundamental convictions of the established Cosmology and explains that even many visible galaxies are located at (light travel) distance of many hundred billion light years. There is no dark matter in any of the so-called 'proofs' of the existence of dark matter and MOND is also an engineered and artificial solution. -/- This book has solved Galactic Rotation problem using Newton's theory and have shown that available theory was capable to explain the flat rotation curves of galaxies without necessitating the existence of dark matter. Thus theory itself is not challenged, blamed or modified. However understanding of scientists of their so-called counterintuitive theories is blamed. For example, to deal with the Galactic Rotation problem, the relevant part of Newton's Principia Mathematica was Proposition LXXIII, Theorem XXXIII. Whereas to deal with this problem, scientists had wrongfully applied Proposition LXXI, Theorem XXXI. Obviously, inaccurate application of available theory resulted in a fake problem and dark matter only served as a ghost solution to that bogus problem. -/- Not only the Galactic Rotation, other so-called indicators of Dark Matter like Cluster Dynamics, Gravitational Lensing, Bullet Cluster, Dark Matter Ring, Fluctuations in CMB Temperature and Structures Formation etc. also have been explained without requiring the need for Dark Matter. -/- Overall this book has presented a strong case of the failure of counterintuitive regime of established Cosmology and Physics. (shrink)

The formal ontology here presented is what we might call a typed combinatorial Meinongian mereology. Its author seeks to formulate the laws, here called ‘canons’, regulating how entities can combine together in wholes of different sorts. The method, as in Bergmann’s earlier works, involves the construction of an ideal language of such a sort that the analysis of complex wholes can be achieved by transforming our natural-language representations of reality into what we might think of as artificial characteristic maps or (...) diagrams which allow the relevant ontological structures to be read off immediately from the symbolic representations which results. In former works Bergmann had held that the symbolic language of Principia Mathematica could serve as the appropriate diagrammatic device for the standard first-order functional calculus and develops instead a new sort diagrammatic language. (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 paper critically scrutinizes the widespread idea that Russell subscribes to a "Universalist Conception of Logic." Various glosses on this somewhat under-explained slogan are considered, and their fit with Russell's texts and logical practice examined. The results of this investigation are, for the most part, unfavorable to the Universalist interpretation.

One of the questions that any future history of British moral philosophy in the twentieth century should investigate and document is how it came about that Moore's Principia Ethica was appropriated by what we can call the Humean tradition of moral philosophy. I shall not trace that development now but only argue that there was no excuse or justification for it.

This 4-page review-essay—which is entirely reportorial and philosophically neutral as are my other contributions to MATHEMATICAL REVIEWS—starts with a short introduction to the philosophy known as mathematical structuralism. The history of structuralism traces back to George Boole (1815–1864). By reference to a recent article various feature of structuralism are discussed with special attention to ambiguity and other terminological issues. The review-essay includes a description of the recent article. The article’s 4-sentence summary is quoted in full and then analyzed. The point (...) of the quotation is to make clear how murky, incompetent, and badly written the paper is. There is no way to determine from the article whether the editor or referees suggests improvements. (shrink)

Newton veut simplement réaffirmer la vérité sur l'omniprésence de Dieu sans l'impliquer directement dans la physique du système du monde. Newton veut se distancer d'un concept cartésien de Dieu et convaincre les athées que Dieu est une présence réelle dans le monde. Dieu doit exister dans l'espace pour exister l'espace, mais Dieu n'agit pas seulement par contact. Newton a toujours supposé que Dieu agît par le biais de causes secondaires. Dans l'édition de 1687 des Principes mathématiques de la philosophie naturelle, (...) Newton déclare clairement qu'il n'attribue pas de cause particulière à l'attraction gravitationnelle. DOI: 10.13140/RG.2.2.27735.42402 . (shrink)

The Oxford Handbook of the History of Physics brings together cutting-edge writing by more than twenty leading authorities on the history of physics from the seventeenth century to the present day. By presenting a wide diversity of studies in a single volume, it provides authoritative introductions to scholarly contributions that have tended to be dispersed in journals and books not easily accessible to the general reader. While the core thread remains the theories and experimental practices of physics, the Handbook contains (...) chapters on other dimensions that have their place in any rounded history. These include the role of lecturing and textbooks in the communication of knowledge, the contribution of instrument-makers and instrument-making companies in providing for the needs of both research and lecture demonstrations, and the growing importance of the many interfaces between academic physics, industry, and the military. (shrink)

A New Italian translation of Principia Ethica coming almost 60 years after the 1964 translation by Gianni Vattimo with Nicola Abbagnano’s preface. The new edition makes room for the English text alongside the Italian translation; it includes the 1922 Preface, a bibliography of Moore’s ethical writings with critical literature, a chronology of Moore’s life and works, and an Index. The Introduction by Sergio Cremaschi reconstructs the background of ideas, concerns and intentions from which Moore’s Principia Ethica originated. It (...) stresses the role of religious concerns, as well as that of the Idealist legacy. I argue that PE is more a patchwork of incompatible lines of argument than a consistent work. It compares PE with Rashdall’s almost contemporary ethical work, suggesting that the latter defends the same general claims in a different and more plausible way, which manages to answer decisive objections. It ends with the suggestion that the emergence of Analytic Ethics was a more multi-faceted phenomenon than the received view would make us believe and the analytic tradition's wheat, that is, what logic can do for philosophy, has to be separated from the chaff, that is, the confused, and besides mutually incompatible, legacies of Utilitarianism and Idealism. (shrink)

In this paper, I discuss how Newton’s inductive argument of the Principia can be defended against criticisms levelled against it by Duhem, Popper and myself. I argue that Duhem’s and Popper’s criticisms can be countered, but mine cannot. It requires that we reconsider, not just Newton’s inductive argument in the Principia, but also the nature of science more generally. The methods of science, whether conceived along inductivist or hypothetico-deductivist lines, make implicit metaphysical presuppositions which rigour requires we make (...) explicit within science so that they can be critically assessed, alternatives being developed and assessed, in the hope that they can be improved. Despite claiming to derive his law of gravitation by induction from phenomena without resource to hypotheses, Newton does nevertheless acknowledge in the Principia that his rules of reasoning make metaphysical presuppositions. To this extent, Newton has a more enlightened view of scientific method than most 20th and 21st century scientists and historians and philosophers of science. (shrink)

A core feature of Kant’s Critical account of moral motivation is that pure reason can be practical by itself. I argue that Kant developed this view in the 1770s concerning the principium diiudicationis and principium executionis. These principles indicate the normative and performative aspects of moral motivation. I demonstrate that cognition of the normative principle effects the moral incentive. So, the hallmark of Kant’s Critical account of motivation was contained in his pre-Critical view. This interpretation resolves a controversy about Kant’s (...) apparent eudaimonism in the first Critique and shows that he developed his account of moral autonomy in the 1770s. (shrink)

Spinoza’s Principles of Cartesian Philosophy is often presented simply as an interpretation of Descartes’ Principia that does not reveal anything significant about Spinoza’s philosophy and its development. This paper, however, shows that Spinoza altered Descartes’ text in a way congruent with what he would later write in his Theological Political Treatise and the Ethics. More precisely, this paper concentrates not on what Spinoza added to Descartes’ texts but on how he presented them. The paper furthermore examines questions that were (...) obviously important for Descartes but absent in Spinoza’s interpretation. Finally, this paper examines two concrete examples to show that Spinoza’s adaptations function as an unmasking of Descartes’ physics. (shrink)

As John Henry states, Newton simply wants to reaffirm the truth of God's omnipresence without directly involving him in the physics of the world system. Newton simply wants to distance himself from a Cartesian concept of God and convince the atheists that God is a real presence extended in the world. God must exist in space for the space to exist, but God does not only act through contact. Henry believes that Andrew Janiak and Hylarie Kochiras give us a wrong (...) picture of a Newton who believes in opportunism. Newton, Henry asserts, has always assumed that God acted through secondary causes: DOI: 10.13140/RG.2.2.35495.39846. (shrink)

Bei den korpuskulartheoretischen Erklärungen von Naturphänomenen, wie sie Descartes in den Principia philosophiae vornimmt und im Discours de la methode anspricht, lassen sich zwei verschiedene und nur teilweise miteinander vereinbare Bedeutungsgruppen des Hypothesenbegriffs nachweisen. Sie verbinden sich mit unterschiedlichen Bewertungen des Status von Hypothesen im wissenschaftlichen Erkenntnisprozeß. Einerseits findet man eine Verwandtschaft zum heute wissenschaftstheoretisch verbreiteten Verständnis von Hypothesen als positivem und integralem Bestandteil der Naturerkenntnis. Typischer für Descartes' Naturphilosophie ist jedoch die andererseits von ihm vertretene Vorstellung, daß der (...) Umfang des bloß hypothetischen Wissens im Fortgang der Naturerkenntnis zu minimieren sei. Mit dieser Forderung erweist sich Descartes als Vertreter einer neuzeitlichen Wissenschaftsauffassung, die deutlich vom modernen Verzicht auf ein ausschließlich geltendes System der Naturerkenntnis abgehoben ist. (shrink)

I argue that The Metaethical Moral Relativism has been being used as a form of support in order to commit desire–belief actions against Human Biology Principia. The lacking of Moral Epistemology allows me illustrate and explain practical downsides in the Ontological part of Moral Knowledge. I recommend a departure from The Biological Basis of Morality in favor to avoid contemporary misuses in the Justification of The Metaethical Moral Relativism.

Moore's moral programme is increasingly unpopular. Judith Jarvis Thomson's attack has been especially influential; she says the Moorean project fails because ‘there is no such thing as goodness’. I argue that her objection does not succeed: while Thomson is correct that the kind of generic goodness she targets is incoherent, it is not, I believe, the kind of goodness central to the Principia. Still, Moore's critics will resist. Some reply that we cannot understand Moorean goodness without generic goodness. Others (...) claim that even if Moore does not need Thomson's concept, he still requires the objectionable notion of absolute goodness. I undermine both these replies. I first show that we may dispense with generic goodness without losing Moorean intrinsic goodness. Then, I argue that though intrinsic goodness is indeed a kind of absolute goodness, the objections marshalled against the concept are unsound. (shrink)

Newton rested his theory of mechanics on distinct metaphysical and epistemological foundations. After Leibniz's death in 1716, the Principia ran into sharp philosophical opposition from Christian Wolff and his disciples, who sought to subvert Newton's foundations or replace them with Leibnizian ideas. In what follows, I chronicle some of the Wolffians' reactions to Newton's notion of absolute space, his dynamical laws of motion, and his general theory of gravitation. I also touch on arguments advanced by Newton's Continental followers, such (...) as Leonhard Euler, who made novel attempts to defend his mechanical foundations against the pro-Leibnizian attack. This examination grants us deeper insight into the fate of Newton's mechanics on the Continent during the early eighteenth century and, more specifically, sheds needed light on the conflicts and tensions that characterized the reception of Newton's philosophy of mechanics among the Leibnizians. (shrink)

Dialectica: Mathematica and Physica, Truth and Justice, Trick and Life. Mathematica as the Constructive Metaphysica and Ontology. Mathematica as the constructive existential method. Сonsciousness and Mathematica: Dialectica of "eidos" and "logos". Mathematica is the Total Dialectica. The basic maternal Structure - "La Structure mère". Mathematica and Physica: loss of existential certainty. Is effectiveness of Mathematica "unreasonable"? The ontological structure of space. Axiomatization of the ontological basis of knowledge: one axiom, one principle and one (...) mathematical object. The main ideas and concepts of the ontological construction/ "Point with a vector germ" and "heavenly triangle". "Ordo geometricus" and "Ordo onto-topological". Architecture of the onto-topological basis of knowledge: general framework structure, carcass and foundation. The absolute space and the absolute field. The absolute (natural) system of coordinates of Universum. Eidos of "idea of ideas", the symbol and the "formula of Justice". (shrink)

DEFINING OUR TERMS A “paradox" is an argumentation that appears to deduce a conclusion believed to be false from premises believed to be true. An “inconsistency proof for a theory" is an argumentation that actually deduces a negation of a theorem of the theory from premises that are all theorems of the theory. An “indirect proof of the negation of a hypothesis" is an argumentation that actually deduces a conclusion known to be false from the hypothesis alone or, more commonly, (...) from the hypothesis augmented by a set of premises known to be true. A “direct proof of a hypothesis" is an argumentation that actually deduces the hypothesis itself from premises known to be true. Since `appears', `believes' and `knows' all make elliptical reference to a participant, it is clear that `paradox', `indirect proof' and `direct proof' are all participant-relative. PARTICIPANT RELATIVITY In normal mathematical writing the participant is presumed to be “the community of mathematicians" or some more or less well-defined subcommunity and, therefore, omission of explicit reference to the participant is often warranted. However, in historical, critical, or philosophical writing focused on emerging branches of mathematics such omission often invites confusion. One and the same argumentation has been a paradox for one mathematician, an inconsistency proof for another, and an indirect proof to a third. One and the same argumentation-text can appear to one mathematician to express an indirect proof while appearing to another mathematician to express a direct proof. WHAT IS A PARADOX’S SOLUTION? Of the above four sorts of argumentation only the paradox invites “solution" or “resolution", and ordinarily this is to be accomplished either by discovering a logical fallacy in the “reasoning" of the argumentation or by discovering that the conclusion is not really false or by discovering that one of the premises is not really true. Resolution of a paradox by a participant amounts to reclassifying a formerly paradoxical argumentation either as a “fallacy", as a direct proof of its conclusion, as an indirect proof of the negation of one of its premises, as an inconsistency proof, or as something else depending on the participant's state of knowledge or belief. This illustrates why an argumentation which is a paradox to a given mathematician at a given time may well not be a paradox to the same mathematician at a later time. -/- The present article considers several set-theoretic argumentations that appeared in the period 1903-1908. The year 1903 saw the publication of B. Russell's Principles of mathematics, [Cambridge Univ. Press, Cambridge, 1903; Jbuch 34, 62]. The year 1908 saw the publication of Russell's article on type theory as well as Ernst Zermelo's two watershed articles on the axiom of choice and the foundations of set theory. The argumentations discussed concern “the largest cardinal", “the largest ordinal", the well-ordering principle, “the well-ordering of the continuum", denumerability of ordinals and denumerability of reals. The article shows that these argumentations were variously classified by various mathematicians and that the surrounding atmosphere was one of confusion and misunderstanding, partly as a result of failure to make or to heed distinctions similar to those made above. The article implies that historians have made the situation worse by not observing or not analysing the nature of the confusion. -/- RECOMMENDATION This well-written and well-documented article exemplifies the fact that clarification of history can be achieved through articulation of distinctions that had not been articulated (or were not being heeded) at the time. The article presupposes extensive knowledge of the history of mathematics, of mathematics itself (especially set theory) and of philosophy. It is therefore not to be recommended for casual reading. AFTERWORD: This review was written at the same time Corcoran was writing his signature “Argumentations and logic”[249] that covers much of the same ground in much more detail. https://www.academia.edu/14089432/Argumentations_and_Logic . (shrink)

Teaching Newtonian physics involves the replacement of students’ ideas about physical situations with precise concepts appropriate for mathematical applications. This paper focuses on the concepts of ‘matter’ and ‘mass’. We suggest that students, like some pre-Newtonian scientists we examine, use these terms in a way that conflicts with their Newtonian meaning. Specifically, ‘matter’ and ‘mass’ indicate to them the sorts of things that are tangible, bulky, and take up space. In Newtonian mechanics, however, the terms are defined by Newton’s Second (...) Law: ‘mass’ is simply a measure of the acceleration generated by an impressed force. We examine the relationship between these conceptions as it was discussed by Newton and his editor, Roger Cotes, when analyzing a series of pendulum experiments. We suggest that these experiments, as well as more sophisticated computer simulations, can be used in the classroom to sufficiently differentiate the colloquial and precise meaning of these terms. (shrink)