Formalizing Euclid’s first axiom. Bulletin of Symbolic Logic. 20 (2014) 404–5. (Coauthor: Daniel Novotný) -/- Euclid [fl. 300 BCE] divides his basic principles into what came to be called ‘postulates’ and ‘axioms’—two words that are synonyms today but which are commonly used to translate Greek words meant by Euclid as contrasting terms. -/- Euclid’s postulates are specifically geometric: they concern geometric magnitudes, shapes, figures, etc.—nothing else. The first: “to draw a line from any point to any point”; the last: (...) the parallel postulate. -/- Euclid’s axioms are general principles of magnitude: they concern geometric magnitudes and magnitudes of other kinds as well even numbers. The first is often translated “Things that equal the same thing equal one another”. -/- There are other differences that are or might become important. -/- Aristotle [fl. 350 BCE] meticulously separated his basic principles [archai, singular archê] according to subject matter: geometrical, arithmetic, astronomical, etc. However, he made no distinction that can be assimilated to Euclid’s postulate/axiom distinction. -/- Today we divide basic principles into non-logical [topic-specific] and logical [topic-neutral] but this too is not the same as Euclid’s. In this regard it is important to be cognizant of the difference between equality and identity—a distinction often crudely ignored by modern logicians. Tarski is a rare exception. The four angles of a rectangle are equal to—not identical to—one another; the size of one angle of a rectangle is identical to the size of any other of its angles. No two angles are identical to each other. -/- The sentence ‘Things that equal the same thing equal one another’ contains no occurrence of the word ‘magnitude’. This paper considers the problem of formalizing the proposition Euclid intended as a principle of magnitudes while being faithful to the logical form and to its information content. (shrink)

Economic policy evaluations require social welfare functions for variable-size populations. Two important candidates are critical-level generalized utilitarianism (CLGU) and rank-discounted critical-level generalized utilitarianism, which was recently characterized by Asheim and Zuber (2014) (AZ). AZ introduce a novel axiom, existence of egalitarian equivalence (EEE). First, we show that, under some uncontroversial criteria for a plausible social welfare relation, EEE suffices to rule out the Repugnant Conclusion of population ethics (without AZ’s other novel axioms). Second, we provide a new characterization of (...) CLGU: AZ’s set of axioms is equivalent to CLGU when EEE is replaced by the axiom same-number independence. (shrink)

A homeomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That homeomorphism can be interpreted physically as the invariance to a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting at another way for proving it, more concise and meaningful physically. Furthermore, the conjecture can be generalized (...) and interpreted in relation to the pseudo-Riemannian space of general relativity therefore allowing for both mathematical and philosophical interpretations of the force of gravitation due to the mismatch of choice and ordering and resulting into the “curving of information” (e.g. entanglement). Mathematically, that homeomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. Philosophically, the same homeomorphism implies transcendentalism once the philosophical category of the totality is defined formally. The fundamental concepts of “choice”, “ordering” and “information” unify physics, mathematics, and philosophy and should be related to their shared foundations. (shrink)

A description of consciousness leads to a contradiction with the postulation from special relativity that there can be no connections between simultaneous event. This contradiction points to consciousness involving quantum level mechanisms. The Quantum level description of the universe is re- evaluated in the light of what is observed in consciousness namely 4 Dimensional objects. A new improved interpretation of Quantum level observations is introduced. From this vantage point the following axioms of consciousness is presented. Consciousness consists of two (...) distinct components, the observed U and the observer I. The observed U consist of all the events I is aware of. A vast majority of these occur simultaneously. Now if I were to be an entity within the space-time continuum, all of these events of U together with I would have to occur at one point in space-time. However, U is distributed over a definite region of space-time (region in brain). Thus, I is aware of a multitude of space-like separated events. It is seen that this awareness necessitates I to be an entity outside the space-time continuum. With I taken as such, a new concept called concept A is introduced. With the help of concept A a very important axiom of consciousness, namely Free Will is explained. Libet s Experiment which was originally seen to contradict Free will, in the light of Concept A is shown to support it. A variation to Libet s Experiment is suggested that will give conclusive proof for Concept A and Free Will. (shrink)

An isomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That isomorphism can be interpreted physically as the invariance between a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting another way for proving it, more concise and meaningful physically. Mathematically, the isomorphism means the invariance to (...) choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. (shrink)

It might be that the phrase ‘local holism’ covers a range of explanatory possibilities spreading to consistencies of theories generally, that we can take something from Peacocke’s caution about delimiting and differentiating modes of support for abstracts to sort something in the varieties of tensions at work in settling contents of theories self-determined to be consistent (facing a barrage of neo-consistencies). The subject-matter becomes then a holism in its entirety in self-consistent self-representation underpinned by that recognition operating over items formulated (...) in a vocabulary supporting consistent representations and operations. In A Study of Concepts scenarios—described by me as ‘containers’— associate a type of primitive to a grammar worked on transitions paralleling a proposed familiar egocentric type of orientation’s handling of objects in a vicinity. The value of scenarios though is distanced from origination, it lies in the depiction of support for and delineation and record of separable coordinate valuations and conceptual transcribing. (shrink)

The aim of this paper is to show that topology has a bearing on Leibniz’s Principle of the Identity of Indiscernibles (PII). According to (PII), if, for all properties F, an object a has property F iff object b has property F, then a and b are identical. If any property F whatsoever is permitted in PII, then Leibniz’s principle is trivial, as is shown by “identity properties”. The aim of this paper is to show that topology can make a (...) contribution to the problem of giving criteria of how to restrict the domain of properties to render (PII) non-trivial. In topology a wealth of different Leibnizian principles of identity can be defined - PII turns out to be just the weakest topological separation axiom T0 in disguise, stronger principles of can be defined with the aid of higher separation axioms Ti, i > 0. Topologically defined properties have a variety of nice features, in particular they are stable in a natural sense. Topologically defined properties do not have a monopoly on defining “good” properties. In the final section of the paper it is show that the topological approach is closely related to Gärdenfors’s approach of conceptual spaces based on the concept of convexity. (shrink)

The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski’s 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in first-order number theory where Peano’s second-order Induction Axiom is approximated by Herbrand’s Induction-Axiom (...) Schema [23]. Similarly, in first-order set theory, Zermelo’s second-order Separation Axiom is approximated by Fraenkel’s first-order Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is a template-text or scheme-template, a syntactic string composed of one or more “blanks” and also possibly significant words and/or symbols. In accordance with a side condition the template-text of a schema is used as a “template” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argument-texts, called instances of the schema. The side condition is a second component. The collection of instances may but need not be regarded as a third component. The instances are almost always considered to come from a previously identified language (whether formal or natural), which is often considered to be another component. This article reviews the often-conflicting uses of the expressions ‘schema’ and ‘scheme’ in the literature of logic. It discusses the different definitions presupposed by those uses. And it examines the ontological and epistemic presuppositions circumvented or mooted by the use of schemata, as well as the ontological and epistemic presuppositions engendered by their use. In short, this paper is an introduction to the history and philosophy of schemata. (shrink)

The aim of this paper is to show that (elementary) topology may be useful for dealing with problems of epistemology and metaphysics. More precisely, I want to show that the introduction of topological structures may elucidate the role of the spatial structures (in a broad sense) that underly logic and cognition. In some detail I’ll deal with “Cassirer’s problem” that may be characterized as an early forrunner of Goodman’s “grue-bleen” problem. On a larger scale, topology turns out to be useful (...) in elaborating the approach of conceptual spaces that in the last twenty years or so has found quite a few applications in cognitive science, psychology, and linguistics. In particular, topology may help distinguish “natural” from “not-so-natural” concepts. This classical problem that up to now has withstood all efforts to solve (or dissolve) it by purely logical methods. Finally, in order to show that a topological perspective may also offer a fresh look on classical metaphysical problems, it is shown that Leibniz’s famous principle of the identity of indiscernibles is closely related to some well-known topological separation axioms. More precisely, the topological perspective gives rise in a natural way to some novel variations of Leibniz’s principle. (shrink)

The aim of this paper is to show that topology has a bearing on<br><br>combinatorial theories of possibility. The approach developed in this article is “mapping account” considering combinatorial worlds as mappings from individuals to properties. Topological structures are used to define constraints on the mappings thereby characterizing the “really possible” combinations. The mapping approach avoids the well-known incompatibility problems. Moreover, it is compatible with atomistic as well as with non-atomistic ontologies.It helps to elucidate the positions of logical atomism and monism (...) with theaid of topological separation axioms. (shrink)

A description of consciousness leads to a contradiction with the postulation from special relativity that there can be no connections between simultaneous event. This contradiction points to consciousness involving quantum level mechanisms. The Quantum level description of the universe is re- evaluated in the light of what is observed in consciousness namely 4 Dimensional objects. A new improved interpretation of Quantum level observations is introduced. From this vantage point the following axioms of consciousness is presented. Consciousness consists of two (...) distinct components, the observed U and the observer I. The observed U consist of all the events I is aware of. A vast majority of these occur simultaneously. Now if I were to be an entity within the space-time continuum, all of these events of U together with I would have to occur at one point in space-time. However, U is distributed over a definite region of space-time (region in brain). Thus, I is aware of a multitude of space-like separated events. It is seen that this awareness necessitates I to be an entity outside the space-time continuum. With I taken as such, a new concept called concept A is introduced. With the help of concept A a very important axiom of consciousness, namely Free Will is explained. (shrink)

The link between the high-order metaphysics and abstractions, on the one hand, and choice in the foundation of set theory, on the other hand, can distinguish unambiguously the “good” principles of abstraction from the “bad” ones and thus resolve the “bad company problem” as to set theory. Thus it implies correspondingly a more precise definition of the relation between the axiom of choice and “all company” of axioms in set theory concerning directly or indirectly abstraction: the principle of abstraction, (...) axiom of comprehension, axiom scheme of specification, axiom scheme of separation, subset axiom scheme, axiom scheme of replacement, axiom of unrestricted comprehension, axiom of extensionality, etc. (shrink)

The idea of science being the best – or the only – way to reach the truth about our cosmos has been a major belief of modern civilization. Yet, science has grown tall on fragile legs of clay. Every scientific theory uses axioms and assumptions that by definition cannot be proved. This poses a serious limitation to the use of science as a tool to find the truth. The only way to search for the latter is to redefine the (...) former to its original glory. In the days well before Galileo and Newton, science and religion were not separated. They worked together to discover the truth and while the latter had God as its final destination, the former had God as its starting point. Science is based on the irrational (unproven) belief that the world is intelligible along many other assumptions. This poses a serious limitation to science that can only be overcome if we accept the irrationality of the cosmos. The motto “Credo quia absurdum” holds more truth than one can ever realize at first glance. There is nothing logical in logic, whereas there is deep wisdom in the irrational. For while the former tries to build castles on moving sand, the latter digs deep inside the depths of existence itself in order to build on the most concrete foundations that there can be: the cosmos itself. The only way forward is backwards. Backwards to a time when religion led the quest for knowledge by accepting what we cannot know, rather than trying to comprehend what we do not. Science was anyway based on that in the first place. (shrink)

The Pigou-Dalton (PD) principle recommends a non-leaky, non-rank-switching transfer of goods from someone with more goods to someone with less. This Article defends the PD principle as an aspect of distributive justice—enabling the comparison of two distributions, neither completely equal, as more or less just. It shows how the PD principle flows from a particular view, adumbrated by Thomas Nagel, about the grounding of distributive justice in individuals’ “claims.” And it criticizes two competing frameworks for thinking about justice that less (...) clearly support the principle: the veil-of-ignorance framework, and Larry Temkin’s proposal that fairer distributions are those concerning which individuals have fewer “complaints.” -/- The Article also clarifies the relation between the PD principle and prioritarianism. Prioritarians will surely endorse the PD principle (with the “good” individual well-being), but they are also committed to a distinct axiom of separability: the moral value of someone’s well-being change does not depend upon her position relative to others. The PD principle neither implies separability, nor is implied by it. Although prioritarianism is very plausible, the case for the PD principle is yet more compelling than for the combination of that principle with separability. In discussing prioritarianism, we should differentiate between these two, logically independent aspects of the view. -/- . (shrink)

For aggregative theories of moral value, it is a challenge to rank worlds that each contain infinitely many valuable events. And, although there are several existing proposals for doing so, few provide a cardinal measure of each world's value. This raises the even greater challenge of ranking lotteries over such worlds—without a cardinal value for each world, we cannot apply expected value theory. How then can we compare such lotteries? To date, we have just one method for doing so (proposed (...) separately by Arntzenius, Bostrom, and Meacham), which is to compare the prospects for value at each individual location, and to then represent and compare lotteries by their expected values at each of those locations. But, as I show here, this approach violates several key principles of decision theory and generates some implausible verdicts. I propose an alternative—one which delivers plausible rankings of lotteries, which is implied by a plausible collection of axioms, and which can be applied alongside almost any ranking of infinite worlds. (shrink)

The Sure-Thing Principle famously appears in Savage’s axiomatization of Subjective Expected Utility. Yet Savage introduces it only as an informal, overarching dominance condition motivating his separability postulate P2 and his state-independence postulate P3. Once these axioms are introduced, by and large, he does not discuss the principle any more. In this note, we pick up the analysis of the Sure-Thing Principle where Savage left it. In particular, we show that each of P2 and P3 is equivalent to a dominance (...) condition; that they strengthen in different directions a common, basic dominance axiom; and that they can be explicitly combined in a unified dominance condition that is a candidate formal statement for the Sure-Thing Principle. Based on elementary proofs, our results shed light on some of the most fundamental properties of rational choice under uncertainty. In particular they imply, as corollaries, potential simplifications for Savage’s and the Anscombe- Aumann axiomatizations of Subjective Expected Utility. Most surprisingly perhaps, they reveal that in Savage’s axiomatization, P3 can be weakened to a natural strengthening of so-called Obvious Dominance. (shrink)

We give two social aggregation theorems under conditions of risk, one for constant population cases, the other an extension to variable populations. Intra and interpersonal welfare comparisons are encoded in a single ‘individual preorder’. The theorems give axioms that uniquely determine a social preorder in terms of this individual preorder. The social preorders described by these theorems have features that may be considered characteristic of Harsanyi-style utilitarianism, such as indifference to ex ante and ex post equality. However, the theorems (...) are also consistent with the rejection of all of the expected utility axioms, completeness, continuity, and independence, at both the individual and social levels. In that sense, expected utility is inessential to Harsanyi-style utilitarianism. In fact, the variable population theorem imposes only a mild constraint on the individual preorder, while the constant population theorem imposes no constraint at all. We then derive further results under the assumption of our basic axioms. First, the individual preorder satisfies the main expected utility axiom of strong independence if and only if the social preorder has a vector-valued expected total utility representation, covering Harsanyi’s utilitarian theorem as a special case. Second, stronger utilitarian-friendly assumptions, like Pareto or strong separability, are essentially equivalent to strong independence. Third, if the individual preorder satisfies a ‘local expected utility’ condition popular in non-expected utility theory, then the social preorder has a ‘local expected total utility’ representation. Fourth, a wide range of non-expected utility theories nevertheless lead to social preorders of outcomes that have been seen as canonically egalitarian, such as rank-dependent social preorders. Although our aggregation theorems are stated under conditions of risk, they are valid in more general frameworks for representing uncertainty or ambiguity. (shrink)

The concept of quantum information is introduced as both normed superposition of two orthogonal sub-spaces of the separable complex Hilbert space and in-variance of Hamilton and Lagrange representation of any mechanical system. The base is the isomorphism of the standard introduction and the representation of a qubit to a 3D unit ball, in which two points are chosen. The separable complex Hilbert space is considered as the free variable of quantum information and any point in it (a wave function describing (...) a state of a quantum system) as its value as the bound variable. A qubit is equivalent to the generalization of ‘bit’ from the set of two equally probable alternatives to an infinite set of alternatives. Then, that Hilbert space is considered as a generalization of Peano arithmetic where any unit is substituted by a qubit and thus the set of natural number is mappable within any qubit as the complex internal structure of the unit or a different state of it. Thus, any mathematical structure being reducible to set theory is re-presentable as a set of wave functions and a subspace of the separable complex Hilbert space, and it can be identified as the category of all categories for any functor represents an operator transforming a set (or subspace) of the separable complex Hilbert space into another. Thus, category theory is isomorphic to the Hilbert-space representation of set theory & Peano arithmetic as above. Given any value of quantum information, i.e. a point in the separable complex Hilbert space, it always admits two equally acceptable interpretations: the one is physical, the other is mathematical. The former is a wave function as the exhausted description of a certain state of a certain quantum system. The latter chooses a certain mathematical structure among a certain category. Thus there is no way to be distinguished a mathematical structure from a physical state for both are described exhaustedly as a value of quantum information. This statement in turn can be utilized to be defined quantum information by the identity of any mathematical structure to a physical state, and also vice versa. Further, that definition is equivalent to both standard definition as the normed superposition and in-variance of Hamilton and Lagrange interpretation of mechanical motion introduced in the beginning of the paper. Then, the concept of information symmetry can be involved as the symmetry between three elements or two pairs of elements: Lagrange representation and each counterpart of the pair of Hamilton representation. The sense and meaning of information symmetry may be visualized by a single (quantum) bit and its interpretation as both (privileged) reference frame and the symmetries of the Standard model. (shrink)

Pettigrew offers new axiomatic constraints on legitimate measures of inaccuracy. His axiom called ‘Decomposition’ stipulates that legitimate measures of inaccuracy evaluate a credence function in part based on its level of calibration at a world. I argue that if calibration is valuable, as Pettigrew claims, then this fact is an explanandum for accuracy-rst epistemologists, not an explanans, for three reasons. First, the intuitive case for the importance of calibration isn’t as strong as Pettigrew believes. Second, calibration is a perniciously global (...) property that both contravenes Pettigrew’s own views about the nature of credence functions themselves and undercuts the achievements and ambitions of accuracy-rst epistemology. Finally, Decomposition introduces a new kind of value compatible with but separate from accuracy-proper in violation of Pettigrew’s alethic monism. introduction. (shrink)

The incompleteness of set theory ZFC leads one to look for natural extensions of ZFC in which one can prove statements independent of ZFC which appear to be "true". One approach has been to add large cardinal axioms. Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski- Grothendieck set theory TG [1]-[3] It is a non-conservative extension of ZFC and is obtaineed from other axiomatic set theories by the inclusion of Tarski's axiom which implies the (...) existence of inaccessible cardinals [1].Non-conservative extension of ZFC based on an generalized quantifiers considered in [4]. In this paper we look at a set theory NC_{∞^{#}}^{#},based on bivalent gyper infinitary logic with restricted Modus Ponens Rule [5]-[8]. In this paper we deal with set theory NC_{∞^{#}}^{#} based on gyper infinitary logic with Restricted Modus Ponens Rule. Set theory NC_{∞^{}}^{} contains Aczel's anti-foundation axiom [9]. We present a new approach to the invariant subspace problem for Hilbert spaces. Our main result will be that: if T is a bounded linear operator on an infinite-dimensional complex separable Hilbert space H,it follow that T has a non-trivial closed invariant subspace. Non-conservative extension based on set theory NC_{∞}^{#} of the model theoretical nonstandard analysis [10]-[12] also is considered. (shrink)

This is an attempt to axiomatise the natural laws. Note especially axiom 4, which is expressed in third order predicate logic, and which permits a solution to the problem of causation in nature without stating that “everything has a cause”. The undefined term “difference” constitutes the basic element and each difference is postulated to have an exact position and to have a discrete cause. The set of causes belonging to a natural set of dimensions is defined as a law. This (...) means that a natural law is determined by the discrete causes tied to a natural set of dimensions. A law is defined as “defined” in a point if a difference there has a cause. Given that there is a point for which the law is not defined it is shown that a difference is caused that connects two points in two separate sets of dimensions. (shrink)

The relationship between the One and the Multiple in mystic philosophy is a profound and central theme that explores the nature of existence, the cosmos, and the divine. This theme is present in various mystical traditions, including those of the East and West, and it addresses the paradoxical coexistence of the unity and multiplicity of all things. -/- In mystic philosophy, the **One** often represents the ultimate reality, the source from which all things emanate and to which all things return. (...) It is the absolute, the infinite, and the unchanging. The One is beyond all attributes and is often associated with the divine or the absolute truth. -/- The **Multiple**, on the other hand, represents the manifest world, the diversity of forms, and the realm of change and plurality. It is the world we experience through our senses, the domain of time and space, where differentiation and individuality are apparent. -/- The relationship between the One and the Multiple is not one of opposition but of emanation and unity. The Multiple is seen as a reflection, expression, or manifestation of the One. In this sense, the diversity of the world doesn’t contradict the unity of the One but rather demonstrates it in a myriad of forms. -/- Mystics seek to understand and experience this relationship through various practices and insights. They aim to transcend the illusion of separation and duality to experience the non-dual reality where the One and the Multiple are recognized as inseparable. -/- This concept can be illustrated by the metaphor of the ocean and its waves. The ocean represents the One—vast, deep, and all-encompassing—while the waves represent the Multiple—distinct, diverse, and ever-changing. Each wave is unique, yet it is not separate from the ocean. The wave’s existence is dependent on and made of the same substance as the ocean. In the same way, each individual entity in the Multiple is an expression of the One. -/- In summary, the relationship between the One and the Multiple in mystic philosophy is a dynamic interplay that challenges the conventional understanding of separation. It invites a deeper exploration of reality, where the apparent multiplicity of the world is a direct expression of the singular, underlying essence of all that is. -/- The relationship between the One and the Multiple in the context of mathematical philosophy is a profound topic that touches upon the very foundations of existence and knowledge. It’s a theme that has been explored by philosophers and mathematicians alike, often leading to the contemplation of unity and diversity within the structures of reality. -/- In mathematics, the concept of the One can be seen as the basis of unity from which all numbers derive. It’s the identity element in multiplication, the starting point in counting, and the foundation of dimension in geometry. The One is often associated with the concept of monism in philosophy, which posits that there is a single, underlying substance or principle that constitutes reality. -/- On the other hand, the Multiple represents the infinite variety and diversity of forms and numbers that arise from the One. It’s the embodiment of plurality and the complex interplay of different entities that mathematics seeks to understand and describe. This reflects the philosophical stance of pluralism, which acknowledges the existence of multiple realities or truths. -/- The interplay between the One and the Multiple can be seen in the mathematical concept of sets. A set can be thought of as a unity, a whole composed of distinct elements. Yet, each element within the set also retains its individuality, contributing to the diversity of the set’s composition. This duality is mirrored in the philosophical exploration of the universal and the particular, where the universal represents the One, and the particulars represent the Multiple. -/- In the realm of mathematical philosophy, this relationship often leads to questions about the nature of mathematical objects: Are they discovered as part of an objective reality (the One), or are they constructed by the human mind from a multitude of experiences (the Multiple)? This debate resonates with the philosophical inquiry into the nature of truth and reality. -/- Reflecting on the One and the Multiple can also lead to a deeper understanding of the self and the cosmos. Just as the number one is integral to the existence of all other numbers, the individual self can be seen as a unique expression of the universal whole. Similarly, the cosmos can be viewed as a grand unity composed of a multiplicity of forms and phenomena. -/- Gödel’s incompleteness theorems have a fascinating connection to the philosophical concepts of the One and the Multiple. These theorems, which are pivotal in mathematical logic and philosophy of mathematics, articulate the inherent limitations of formal axiomatic systems, particularly those sufficient to express the arithmetic of natural numbers. -/- Gödel’s incompleteness theorems have a fascinating connection to the philosophical concepts of the One and the Multiple. These theorems, which are pivotal in mathematical logic and philosophy of mathematics, articulate the inherent limitations of formal axiomatic systems, particularly those sufficient to express the arithmetic of natural numbers⁴. -/- The **first incompleteness theorem** reveals that within any such consistent system, there are propositions that are true but cannot be proven within the system itself. This reflects the idea of the Multiple in that there is an abundance of mathematical truths, a multiplicity that exceeds the unifying framework of any one system. It suggests that the realm of mathematical truth is more extensive than any single formal system can fully capture. -/- The **second incompleteness theorem** extends this by showing that a system cannot prove its own consistency. This relates to the concept of the One, as it implies that a system’s complete self-understanding, its unity and coherence, is unattainable from within. It must look beyond itself, to an external vantage point, to ascertain its consistency. -/- In the context of the One and the Multiple, Gödel’s theorems imply that the One (a consistent formal system) is inherently incomplete and cannot encompass the Multiple (the totality of mathematical truths). This resonates with philosophical discussions about the relationship between unity and plurality. Just as a single philosophical system cannot capture the entirety of truth, a single formal mathematical system cannot encapsulate all mathematical truths. -/- Moreover, Gödel’s work suggests that the pursuit of a single, unified theory of everything in mathematics—a One that encompasses all Multiples—is an inherently Sisyphean task. There will always be more truths (Multiples) than can be derived from any given set of axioms (the One). -/- In essence, Gödel’s incompleteness theorems provide a formal underpinning to the philosophical notion that the One cannot exist without the Multiple, and vice versa. They are interdependent, with the One giving rise to the Multiple, and the Multiple necessitating the One for its expression and comprehension. This interplay is a dance of limitations and possibilities, where the boundaries of logic, mathematics, and philosophy blur into one another. -/- The relationship between the One and the Multiple in the context of the mathematics of zero is a deeply philosophical inquiry that bridges the abstract world of numbers with the existential questions of being and non-being. -/- Zero, in mathematics, is a symbol of absence, a representation of nothingness, yet it holds a pivotal position as a number. It is the void from which all things emerge and to which they return. In the philosophy of mathematics, zero is the paradoxical junction where the One and the Multiple converge and diverge. -/- From the perspective of the One, zero can be seen as the origin—the singular point that precedes the existence of numbers. It is the empty set, the foundation upon which the edifice of mathematics is constructed. As the identity element in addition, zero maintains the integrity of numbers, for adding zero to any number leaves it unchanged, reflecting the immutable nature of the One. -/- Conversely, when we consider the Multiple, zero represents the infinite potentiality of creation. It is the canvas upon which the integers, both positive and negative, express their multitude. Zero is the balance point, the fulcrum around which the symphony of numbers dances. It embodies the plurality of possibilities, the beginning of the number line that stretches infinitely in both directions. -/- Philosophically, zero challenges our understanding of existence. It is both something and nothing—a number that quantifies the absence of quantity. This duality echoes the philosophical struggle to comprehend how the One gives rise to the Multiple. How does the unity of being manifest the diversity of the cosmos? Zero offers a mathematical metaphor for this mystery, as it encapsulates the transition from non-existence to existence, from the undifferentiated One to the differentiated Multiple. -/- In the realm of set theory, zero corresponds to the empty set—a set with no elements. This set is unique in that it is the only set that contains nothing, yet it is the foundation upon which all other sets are built. The empty set is the mathematical embodiment of the One, and all other sets, containing multiple elements, arise from it. -/- Reflecting on zero in the context of Gödel’s incompleteness theorems, we find a resonance with the idea that the One (a consistent formal system) cannot capture the entirety of the Multiple (the totality of mathematical truths). Zero, as the foundation of numbers, similarly suggests that from the nothingness of the One, the infinite complexity of the Multiple emerges—yet it can never be fully encompassed or expressed by any single system. -/- In conclusion, the mathematics of zero offers a profound reflection on the relationship between the One and the Multiple. It serves as a bridge between the abstract and the concrete, the known and the unknowable, challenging us to ponder the origins of existence and the nature of reality itself. -/- . (shrink)

This essay describes and evaluates the conception of mereological structure that underpins Deleuze’s account of ontogenesis in Difference and Repetition. A theory of mereology is a theory of composition: it asks what it is to be a part making a whole, what it is to be a whole collecting its parts; in short, in what the relation of making or composing consists. The locus classicus for modern mereology is the third of Husserl’s Logical Investigations (‘On the Theory of Wholes and (...) Parts’), which deduces the definitions, axioms and principles governing the relation of parthood and associated notions, such as wholeness, separation, simplicity and complexity. My question is this: what happens to these notions when they are no longer the terms of a ‘calculus of individuals’ and, instead, become the terms of a differential calculus of individuation? (shrink)

Non-commuting quantities and hidden parameters – Wave-corpuscular dualism and hidden parameters – Local or nonlocal hidden parameters – Phase space in quantum mechanics – Weyl, Wigner, and Moyal – Von Neumann’s theorem about the absence of hidden parameters in quantum mechanics and Hermann – Bell’s objection – Quantum-mechanical and mathematical incommeasurability – Kochen – Specker’s idea about their equivalence – The notion of partial algebra – Embeddability of a qubit into a bit – Quantum computer is not Turing machine – (...) Is continuality universal? – Diffeomorphism and velocity – Einstein’s general principle of relativity – „Mach’s principle“ – The Skolemian relativity of the discrete and the continuous – The counterexample in § 6 of their paper – About the classical tautology which is untrue being replaced by the statements about commeasurable quantum-mechanical quantities – Logical hidden parameters – The undecidability of the hypothesis about hidden parameters – Wigner’s work and и Weyl’s previous one – Lie groups, representations, and psi-function – From a qualitative to a quantitative expression of relativity − psi-function, or the discrete by the random – Bartlett’s approach − psi-function as the characteristic function of random quantity – Discrete and/ or continual description – Quantity and its “digitalized projection“ – The idea of „velocity−probability“ – The notion of probability and the light speed postulate – Generalized probability and its physical interpretation – A quantum description of macro-world – The period of the as-sociated de Broglie wave and the length of now – Causality equivalently replaced by chance – The philosophy of quantum information and religion – Einstein’s thesis about “the consubstantiality of inertia ant weight“ – Again about the interpretation of complex velocity – The speed of time – Newton’s law of inertia and Lagrange’s formulation of mechanics – Force and effect – The theory of tachyons and general relativity – Riesz’s representation theorem – The notion of covariant world line – Encoding a world line by psi-function – Spacetime and qubit − psi-function by qubits – About the physical interpretation of both the complex axes of a qubit – The interpretation of the self-adjoint operators components – The world line of an arbitrary quantity – The invariance of the physical laws towards quantum object and apparatus – Hilbert space and that of Minkowski – The relationship between the coefficients of -function and the qubits – World line = psi-function + self-adjoint operator – Reality and description – Does a „curved“ Hilbert space exist? – The axiom of choice, or when is possible a flattening of Hilbert space? – But why not to flatten also pseudo-Riemannian space? – The commutator of conjugate quantities – Relative mass – The strokes of self-movement and its philosophical interpretation – The self-perfection of the universe – The generalization of quantity in quantum physics – An analogy of the Feynman formalism – Feynman and many-world interpretation – The psi-function of various objects – Countable and uncountable basis – Generalized continuum and arithmetization – Field and entanglement – Function as coding – The idea of „curved“ Descartes product – The environment of a function – Another view to the notion of velocity-probability – Reality and description – Hilbert space as a model both of object and description – The notion of holistic logic – Physical quantity as the information about it – Cross-temporal correlations – The forecasting of future – Description in separable and inseparable Hilbert space – „Forces“ or „miracles“ – Velocity or time – The notion of non-finite set – Dasein or Dazeit – The trajectory of the whole – Ontological and onto-theological difference – An analogy of the Feynman and many-world interpretation − psi-function as physical quantity – Things in the world and instances in time – The generation of the physi-cal by mathematical – The generalized notion of observer – Subjective or objective probability – Energy as the change of probability per the unite of time – The generalized principle of least action from a new view-point – The exception of two dimensions and Fermat’s last theorem. (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)

Cyclic mechanic is intended as a suitable generalization both of quantum mechanics and general relativity apt to unify them. It is founded on a few principles, which can be enumerated approximately as follows: 1. Actual infinity or the universe can be considered as a physical and experimentally verifiable entity. It allows of mechanical motion to exist. 2. A new law of conservation has to be involved to generalize and comprise the separate laws of conservation of classical and relativistic mechanics, and (...) especially that of conservation of energy: This is the conservation of action or information. 3. Time is not a uniformly flowing time in general. It can have some speed, acceleration, more than one dimension, to be discrete. 4. The following principle of cyclicity: The universe returns in any point of it. The return can be only kinematic, i.e. per a unit of energy (or mass), and thermodynamic, i.e. considering the universe as a thermodynamic whole. 5. The kinematic return, which is per a unit of energy (or mass), is the counterpart of conservation of energy, which can be interpreted as the particular case of conservation of action “per a unit of time”. The kinematic return per a unit of energy (or mass) can be interpreted in turn as another particular law of conservation in the framework of conservation of action (or information), namely conservation of wave period (or time). These two counterpart laws of conservation correspond exactly to the particle “half” and to the wave “half” of wave-particle duality. 6. The principle of quantum invariance is introduced. It means that all physical laws have to be invariant to discrete and continuous (smooth) morphisms (motions) or mathematically, to the axiom of choice. The list is not intended to be exhausted or disjunctive, but only to give an introductory idea. (shrink)

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

Does individual desert matter for distributive justice? Is it relevant, for purposes of justice, that the pattern of distribution of justice’s “currency” (be it well-being, resources, preference-satisfaction, capabilities, or something else) is aligned in one or another way with the pattern of individual desert? -/- This paper examines the nexus between desert and distributive justice through the lens of individual claims. The concept of claims (specifically “claims across outcomes”) is a fruitful way to flesh out the content of distributive justice (...) so as to be grounded in the separateness of persons. A claim is a relation between a person and a pair of outcomes. If someone is better off in one outcome than a second, she has a claim in favor of the first. If she is equally well off in the two outcomes, she has a null claim between the two. In turn, whether one outcome is more just than a second depends upon the pattern of claims between them. -/- In prior work, I have elaborated the concept of claims across outcomes, and have used it to provide a unified defense of the Pareto and Pigou-Dalton axioms. Adding some further, plausible, axioms, we arrive at prioritarianism. -/- Here, I consider the possibility of desert-modulated claims—whereby the strength of an individual’s claim between two outcomes is determined not only by her well-being levels in the two outcomes, and her well-being difference between them, but also by her desert. This generalization of the notion of claims suggests a new axiom of justice: Priority for the More Deserving, requiring that, as between two individuals at the same well-being level, a given increment in well-being be allocated to the more deserving one. -/- If individual desert is intrapersonally fixed, this new axiom, together with a desert-modulated version of the Pigou-Dalton principle, and the Pareto axioms, yields a desert-modulated prioritarian account of distributive justice. Trouble arises, however, if an individual’s desert level can be different in different outcomes. In this case of intrapersonally variable desert, Priority for the More Deserving can conflict with the Pareto axioms (both Pareto indifference and strong Pareto). -/- This conflict, I believe, is sufficient reason to abandon the proposal to make claim strength a function of individual desert on top of well-being levels and differences. If distributive justice is truly sensitive to each individual’s separate perspective—if the justice ranking of outcomes is built up from the totality of individual rankings—we should embrace the Pareto axioms as axioms of justice and reject Priority for the More Deserving. In short: desert-modulated prioritarianism is a nonstarter. Rawls was right to sever distributive justice from desert. (shrink)

We investigate the conflict between the ex ante and ex post criteria of social welfare in a new framework of individual and social decisions, which distinguishes between two sources of uncertainty, here interpreted as an objective and a subjective source respectively. This framework makes it possible to endow the individuals and society not only with ex ante and ex post preferences, as is usually done, but also with interim preferences of two kinds, and correspondingly, to introduce interim forms of the (...) Pareto principle. After characterizing the ex ante and ex post criteria, we present a first solution to their conflict that extends the former as much possible in the direction of the latter. Then, we present a second solution, which goes in the opposite direction, and is also maximally assertive. Both solutions translate the assumed Pareto conditions into weighted additive utility representations, and both attribute to the individuals common probability values on the objective source of uncertainty, and different probability values on the subjective source. We discuss these solutions in terms of two conceptual arguments, i.e., the by now classic spurious unanimity argument and a novel informational argument labelled complementary ignorance. The paper complies with the standard economic methodology of basing probability and utility representations on preference axioms, but for the sake of completeness, also considers a construal of objective uncertainty based on the assumption of an exogeneously given probability measure. JEL classification: D70; D81. (shrink)

The orthodox theory of instrumental rationality, expected utility (EU) theory, severely restricts the way in which risk-considerations can figure into a rational individual's preferences. It is argued here that this is because EU theory neglects an important component of instrumental rationality. This paper presents a more general theory of decision-making, risk-weighted expected utility (REU) theory, of which expected utility maximization is a special case. According to REU theory, the weight that each outcome gets in decision-making is not the subjective probability (...) of that outcome; rather, the weight each outcome gets depends on both its subjective probability and its position in the gamble. Furthermore, the individual's utility function, her subjective probability function, and a function that measures her attitude towards risk can be separately derived from her preferences via a Representation Theorem. This theorem illuminates the role that each of these entities plays in preferences, and shows how REU theory explicates the components of instrumental rationality. (shrink)

We review our approach to quantum mechanics adding also some new interesting results. We start by giving proof of two important theorems on the existence of the A(Si) and i,±1 N Clifford algebras. This last algebra gives proof of the von Neumann basic postulates on the quantum measurement explaining thus in an algebraic manner the wave function collapse postulated in standard quantum theory. In this manner we reach the objective to expose a self-consistent version of quantum mechanics. In detail we (...) realize a bare bone skeleton of quantum mechanics recovering all the basic foundations of this theory on an algebraic framework. We give proof of the quantum like Heisenberg uncertainty relations using only the basic support of the Clifford algebra. In addition we demonstrate the well known phenomenon of quantum Mach Zender interference using the same algebraic framework, as well as we give algebraic proof of quantum collapse in some cases of physical interest by direct application of the theorem that we derive to elaborate the i,±1 N algebra. We also discuss the problem of time evolution of quantum systems as well as the changes in space location, in momentum and the linked invariance principles. We are also able to re-derive the basic wave function of standard quantum mechanics by using only the Clifford algebraic approach. In this manner we obtain a full exposition of standard quantum mechanics using only the basic axioms of Clifford algebra. We also discuss more advanced features of quantum mechanics. In detail, we give demonstration of the Kocken-Specher theorem, and also we give an algebraic formulation and explanation of the EPR paradox only using the Clifford algebra. By using the same approach we also derive Bell inequalities. Our formulation is strongly based on the use of idempotents that are contained in Clifford algebra. Their counterpart in quantum mechanics is represented by the projection operators that, as it is well known, are interpreted as logical statements, following the basic von Neumann results. Von Neumann realized a matrix logic on the basis of quantum mechanics. Using the Clifford algebra we are able to invert such result. According to the results previously obtained by Orlov in 1994, we are able to give proof that quantum mechanics derives from logic. We show that indeterminism and quantum interference have their origin in the logic. Therefore, it seems that we may conclude that quantum mechanics, as it appears when investigated by the Clifford algebra, is a two-faced theory in the sense that it looks from one side to “matter per se”, thus to objects but simultaneously also to conceptual entities. We advance the basic conclusion of the paper: There are stages of our reality in which we no more can separate the logic ( and thus cognition and thus conceptual entity) from the features of “matter per se”. In quantum mechanics the logic, and thus the cognition and thus the conceptual entity-cognitive performance, assume the same importance as the features of what is being described. We are at levels of reality in which the truths of logical statements about dynamic variables become dynamic variables themselves so that a profound link is established from its starting in this theory between physics and conceptual entities. Finally, in this approach there is not an absolute definition of logical truths. Transformations , and thus … “redefinitions”…. of truth values are permitted in such scheme as well as the well established invariance principles, clearly indicate . (shrink)

The discussion of the Principle of Non-Contradiction (PNC) in Aristotle’s Metaphysics Γ is usually taken to include three ‘versions’ of the principle: an ontological, psychological, and logical one. In this article I develop an interpretation of Metaphysics Γ3 and a parallel text, De interpretatione 14, in order to show that these texts are concerned with two related but different principles: a version of the Principle of Identity, and a corollary to this, which concerns the ability to accept two ‘opposite’ items (...) at the same time. I argue that these principles must be considered separately in order to properly understand Aristotle’s remarks about PNC in Metaphysics Γ, and in order to defend his approach in these remarks against certain objections raised against the discussion in Metaphysics Γ by Jan Łukasiewicz in his "Über den Satz des Widerspruchs bei Aristoteles". The main result of my interpretation is to distinguish in the discussion of PNC in Metaphysics Γ three principles: one concerning linguistic items (words and statements), one concerning thoughts (δόξαι), and one concerning objects or states of affairs. (shrink)

Axiom weakening is a technique that allows for a fine-grained repair of inconsistent ontologies. Its main advantage is that it repairs on- tologies by making axioms less restrictive rather than by deleting them, employing the use of refinement operators. In this paper, we build on pre- viously introduced axiom weakening for ALC, and make it much more irresistible by extending its definitions to deal with SROIQ, the expressive and decidable description logic underlying OWL 2 DL. We extend the definitions (...) of refinement operator to deal with SROIQ constructs, in particular with role hierarchies, cardinality constraints and nominals, and illustrate its application. Finally, we discuss the problem of termi- nation of an iterated weakening procedure. (shrink)

In this paper, I accomplish a conceptual task and a historical task. The conceptual task is to argue that (1) Einstein’s Principle of Separability (henceforth “separability”) is not a supervenience principle and that (2) separability and entanglement are compatible. I support (1) by showing that the conclusion of Einstein’s incompleteness argument would still follow even if one assumes that the state of a composite system does not supervene on the states of the subsystems, and by showing that what Einstein says (...) in “Quantum Mechanics and Reality” (1948) strongly suggests that separability is not a principle about how subsystem states relate to the state of composite systems. I support (2) by showing that if separability was incompatible with entanglement, then Einstein’s argument would be incoherent in a trivial way. Thus, by arguing for (1) and (2) I directly challenge what has been, and still is, a very common reading of separability. The historical task is to offer the first detailed review of the different ways in which separability has been defined by physicists and philosophers in the last 60 years. Among other things, such a review distinguishes three different definitions of the principle, and shows that since the 1990s and up until the present date, it became standard to take separability (as presented by Einstein) to be a supervenience principle. (shrink)

Separation of powers is one of the ideas with profound theoretical and practical significance, especially in the field of political science. The birth of the theory of separation of powers marked the transition from the barbaric use of power in authoritarian societies to the exercise of civilized power in democratic societies. Therefore, separation of powers is considered an objective necessity in democratic states, a condition to ensure the promotion of liberal values, and a criterion for assessing the (...) existence and development of the Rule of Law where people's sovereignty plays a paramount role. On the way to perfecting political thinking, the theory of separation of powers has developed into a complete system in the main philosophy of John Locke. Within the scope of the article, the author focuses on analyzing Locke's ideology of separation of powers in terms of the nature of separation of powers, the basis and some principles of power division, and the roles of the different parts of government power and the relationship between them. (shrink)

Spinoza's causal axiom is at the foundation of the Ethics. I motivate, develop and defend a new interpretation that I call the ‘causally restricted interpretation’. This interpretation solves several longstanding puzzles and helps us better understand Spinoza's arguments for some of his most famous doctrines, including his parallelism doctrine and his theory of sense perception. It also undermines a widespread view about the relationship between the three fundamental, undefined notions in Spinoza's metaphysics: causation, conception and inherence.

Although the concept of uncertainty is as old as Epicurus’s writings, and an excellent quantitative theory, with entropy as the measure of uncertainty having been developed in recent times, there has been little exploration of the qualitative theory. The purpose of the present paper is to give a qualitative axiomatization of uncertainty, in the spirit of the many studies of qualitative comparative probability. The qualitative axioms are fundamentally about the uncertainty of a partition of the probability space of events. (...) Of course, it is common to speak of the uncertainty, or randomness, of a random variable, but only the partition defined by the values of the random variable enter into the definition of uncertainty, not the actual values. It is straightforward to add axioms for decision making following the general line of Savage from the 1950s. Indeed, in the spirit of Epicurus, it is really our intuitive feeling about the uncertainty of the future that motivates much of our thinking about decisions. Here, the distinction between the concepts of probability and uncertainty can be made by citing many familiar examples. Without spelling out the technical details, the axiomatization of qualitative probability with uncertainty as the most important primitive concept, it is possible to raise a different kind of question about bounded rationality. This new question is whether or not one should bound the uncertainty in thinking and investigating any detailed framework of decision making. Discussion of this point is certainly different from the question of bounding rationality by not maximizing expected utility. In practice, we naturally bound uncertainty in our analysis of decision-making problems. As in the case of formulating an alternative for maximizing expected utility, so is the case of rational alternatives to maximizing uncertainty. There are several issues to consider. In the spirit of my other work in qualitative probability, I explore alternatives rather than attempt to give a definitive argument for one single solution. (shrink)

In a new study, Ben-Haim et al. use subliminal stimuli to separate conscious and unconscious perception in macaques. A programme of this type, using a range of cognitive tasks, is a promising way to look for conscious perception in more controversial cases.

Thick terms and concepts, such as honesty and cruelty, are at the heart of a variety of debates in philosophy of language and metaethics. Central to these debates is the question of how the descriptive and evaluative components of thick concepts are related and whether they can be separated from each other. So far, no empirical data on how thick terms are used in ordinary language has been collected to inform these debates. In this paper, we present the first empirical (...) study, designed to investigate whether the evaluative component of thick concepts is communicated as part of the semantic meaning or by means of conversational implicatures. While neither the semantic nor the pragmatic view can fully account for the use of thick terms in ordinary language, our results do favour the semanticist interpretation: the evaluation of a thick concept is only slightly easier to cancel than semantically entailed content. We further discovered a polarity effect, demonstrating that how easily an evaluation can be cancelled depends on whether the thick term is of positive or negative polarity. (shrink)

Legal positivism's ``separation thesis'' is usually taken in one of two ways: as an analytic claim about the nature of law – roughly, as some version of the Social Thesis; or as a substantive claim about the moral value of law – roughly, as some version of the Value Thesis. In this paper I argue that we should recognize a third kind of positivist separation thesis, one which complements, but is distinct from, positivism's analytic and moral claims. The (...) Neutrality Thesis says that the correct analytic claim about the nature of law does not by itself entail any substantive claims about the moral value of law. I give careful formulations of these three separation theses, explain the relationships between them, and sketch the role that each plays in the positivist approach to law. (shrink)

Peirce and Dewey were generally more concerned with the process of scientific activity than purely mathematical work. However, their accounts of knowledge production afford some insights into the epistemology of mathematical postulates, especially definition and axioms. Their rejection of rationalist metaphysics and their emphasis on continuity in inquiry provides the pretext for the pragmatic a priori – hypothetical and operational assumptions whose justification relies on their fruitfulness in the long run. This paper focuses on the application of this idea (...) to the epistemology of definitions and an account of progress in mathematics, although it has broader implications for the study of conceptual change and the function and basis of presuppositions in the sciences. (shrink)

In morally grounding a public justification requirement, public reason liberals frequently invoke the idea that persons should be construed as “free and equal.” But this tells us little with regard to what it is about us that makes us free or how a claim about our status as persons can ultimately ground a requirement of public justification. In light of this worry, I argue that a public justification requirement can be grounded in a Nozick-inspired argument from the separateness of persons (...) (one that is consistent with the idea that individuals are free and equal). As I claim, one particular feature of the fact of our separateness – the possession of a basic psychology consisting of beliefs, intentions, sentiments, and a variety of desire-like psychological states – does the most work in grounding both a principle of liberty (PL) and a requirement of public justification (RPJ). Together, PL and RPJ provide the basic framework for a theory of public reason liberalism. (shrink)

In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us, all directions are the same to us and all units of length we use to create geometric figures are the same to us. On the other hand, through the process of algebraic simplification, this system of axioms directly provides (...) the Weyl’s system of axioms for Euclidean geometry. The system of axioms, together with its a priori interpretation, offers new views to philosophy and pedagogy of mathematics: it supports the thesis that Euclidean geometry is a priori, it supports the thesis that in modern mathematics the Weyl’s system of axioms is dominant to the Euclid’s system because it reflects the a priori underlying symmetries, it gives a new and promising approach to learn geometry which, through the Weyl’s system of axioms, leads from the essential geometric symmetry principles of the mathematical nature directly to modern mathematics. (shrink)

Philosophy, it is said, has always been concerned with space and nature. Since Aristotle, even since the pre-Socratics, and up to Descartes and Leibniz at least, no one was worthy of the title of philosopher if he had not written something about meteors. And not only about meteors: observations on the movements of the Earth and changes in nature, changes of state or speed, transitions and contacts between strange elements, spatial geometrical properties, the order of coexistences - their limits - (...) and the correspondences between scales; force and matter, constructive typologies and geomorphology were an indissoluble part of the philosophical field. It is not that philosophy reflected on these external themes, but that these themes formed the fabric of the philosophical body. What happened, then, between the end of the seventeenth century and the first half of the eighteenth century? How did the themes related to space and nature become diluted and disappear from philosophical discourses? It is not a question of the split between man and nature, ancient and incommensurable, but of the separation of nature itself, modern and discontinuous, by means of which it was placed alongside the barren, the inert, as a category. -/- La filosofía, se dice, se ocupó siempre de los espacios y de la naturaleza. Desde Aristóteles, incluso desde los presocráticos, y hasta Descartes y Leibniz por lo menos, nadie era digno del título de filósofo si no había escrito algo sobre los meteoros. Y no sólo sobre los meteoros: las observaciones sobre los movimientos de la Tierra y la mudanza en la naturaleza, los cambios de estado o de velocidad, las transiciones y los contactos entre elementos extraños, las propiedades geométricas espaciales, el orden de las coexistencias —sus límites— y las correspondencias entre las escalas; la fuerza y la materia, las tipologías constructivas y la geomorfología formaban parte indisoluble del campo filosófico. No es que la filosofía reflexionase sobre esos temas externos, sino que esos temas formaban el tejido del cuerpo filosófico. ¿Qué sucedió, entonces, entre el final del siglo XVII y la primera mitad del siglo XVIII? ¿Cómo fue que los temas relacionados con el espacio y la naturaleza se diluyeron y desaparecieron de los discursos filosóficos? No se trata de la escisión entre hombre y naturaleza, antigua e inconmensurable, sino de la separación de sí de la naturaleza misma, moderna y discontinua, por medio de la cual fue puesta al lado de lo yerto, lo inerte, como una categoría. (shrink)

The difference between the unity of the individual and the separateness of persons requires that there be a shift in the moral weight that we accord to changes in utility when we move from making intrapersonal tradeoffs to making interpersonal tradeoffs. We examine which forms of egalitarianism can, and which cannot, account for this shift. We argue that a form of egalitarianism which is concerned only with the extent of outcome inequality cannot account for this shift. We also argue that (...) a view which is concerned with both outcome inequality and with the unfairness of inequality in individuals‘ expected utilities can account for this shift. Finally, we limn an alternative view, on.. (shrink)

Discussion of new axioms for set theory has often focused on conceptions of maximality, and how these might relate to the iterative conception of set. This paper provides critical appraisal of how certain maximality axioms behave on different conceptions of ontology concerning the iterative conception. In particular, we argue that forms of multiversism (the view that any universe of a certain kind can be extended) and actualism (the view that there are universes that cannot be extended in particular (...) ways) face complementary problems. The latter view is unable to use maximality axioms that make use of extensions, where the former has to contend with the existence of extensions violating maximality axioms. An analysis of two kinds of multiversism, a Zermelian form and Skolemite form, leads to the conclusion that the kind of maximality captured by an axiom differs substantially according to background ontology. (shrink)

We present an elementary system of axioms for the geometry of Minkowski spacetime. It strikes a balance between a simple and streamlined set of axioms and the attempt to give a direct formalization in first-order logic of the standard account of Minkowski spacetime in Maudlin and Malament. It is intended for future use in the formalization of physical theories in Minkowski spacetime. The choice of primitives is in the spirit of Tarski : a predicate of betwenness and a (...) four place predicate to compare the square of the relativistic intervals. Minkowski spacetime is described as a four dimensional ‘vector space’ that can be decomposed everywhere into a spacelike hyperplane—which obeys the Euclidean axioms in Tarski and Givant, 175–214 1999)—and an orthogonal timelike line. The length of other ‘vectors’ are calculated according to Pythagoras’ theorem. We conclude with a Representation Theorem relating models \ of our system \ that satisfy second order continuity to the mathematical structure \, called ‘Minkowski spacetime’ in physics textbooks. (shrink)

The Hyperuniverse Programme, introduced in Arrigoni and Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the `maximal iterative concept', and the programme identi fies higher-order statements motivated by the maximal iterative concept. The satisfaction (...) of these statements (H-axioms) in countable transitive models, the collection of which constitutes the `hyperuniverse' (H), has remarkable 1st-order consequences, some of which we review in section 5. (shrink)