An influential position in the philosophy of biology claims that there are no biological laws, since any apparently biological generalization is either too accidental, fact-like or contingent to be named a law, or is simply reducible to physical laws that regulate electrical and chemical interactions taking place between merely physical systems. In the following I will stress a neglected aspect of the debate that emerges directly from the growing importance of mathematical models of biological phenomena. My main aim (...) is to defend, as well as reinforce, the view that there are indeed laws also in biology, and that their difference in stability, contingency or resilience with respect to physical laws is one of degrees, and not of kind . (shrink)

Time is a complex category not only in philosophy but also in mathematics and physics. In one thought about time, the author accidentally discovered a new way to explain and solve problems related to time dilation, such as solving the problem of Muon particle when moving from a height of 10 km to the earth’s surface, while the Muon’s lifespan is only 2.2 microseconds, or explaining Michelson-Morley experiment using the new method. In addition, the author also prove that the (...) speed of light in vacuum is the maximum speed in the universe, and discovered the red shift effect while there is no increase in distance between objects. To do this, the author has built two axioms based on the discontinuity in the motion of the object and draw two consequences along with the law of conservation of time. (shrink)

George Boole emerged from the British tradition of the “New Analytic”, known for the view that the laws of logic are laws of thought. Logicians in the New Analytic tradition were influenced by the work of Immanuel Kant, and by the German logicians Wilhelm Traugott Krug and Wilhelm Esser, among others. In his 1854 work An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, Boole argues that the (...) class='Hi'>laws of thought acquire normative force when constrained to mathematical reasoning. Boole’s motivation is, first, to address issues in the foundations of mathematics, including the relationship between arithmetic and algebra, and the study and application of differential equations (Durand-Richard, van Evra, Panteki). Second, Boole intended to derive the laws of logic from the laws of the operation of the human mind, and to show that these laws were valid of algebra and of logic both, when applied to a restricted domain. Boole’s thorough and flexible work in these areas influenced the development of model theory (see Hodges, forthcoming), and has much in common with contemporary inferentialist approaches to logic (found in, e.g., Peregrin and Resnik). (shrink)

In this paper I review three different positions on the wave function, namely: nomological realism, dispositionalism, and configuration space realism by regarding as essential their capacity to account for the world of our experience. I conclude that the first two positions are committed to regard the wave function as an abstract entity. The third position will be shown to be a merely speculative attempt to derive a primitive ontology from a reified mathematical space. Without entering any discussion about nominalism, I (...) conclude that an elimination of abstract entities from one’s ontology commits one to instrumentalism about the wave function, a position that therefore is not as unmotivated as it has seemed to be to many philosophers. (shrink)

This paper commences from the critical observation that the Turing Test (TT) might not be best read as providing a definition or a genuine test of intelligence by proxy of a simulation of conversational behaviour. Firstly, the idea of a machine producing likenesses of this kind served a different purpose in Turing, namely providing a demonstrative simulation to elucidate the force and scope of his computational method, whose primary theoretical import lies within the realm of mathematics rather than cognitive (...) modelling. Secondly, it is argued that a certain bias in Turing’s computational reasoning towards formalism and methodological individualism contributed to systematically unwarranted interpretations of the role of the TT as a simulation of cognitive processes. On the basis of the conceptual distinction in biology between structural homology vs. functional analogy, a view towards alternate versions of the TT is presented that could function as investigative simulations into the emergence of communicative patterns oriented towards shared goals. Unlike the original TT, the purpose of these alternate versions would be co-ordinative rather than deceptive. On this level, genuine functional analogies between human and machine behaviour could arise in quasi-evolutionary fashion. (shrink)

This paper is on Descartes’ account of modality and, in particular, his account of the necessity of the laws of nature. He famously argues that the necessity of the “eternal truths” of logic and mathematics depends on God’s will. Here I suggest he has the same view about the necessity of the laws of nature. Further, I argue, this is a plausible theory of laws. For philosophers often talk about something being nomologically or physically necessary because (...) of the laws of nature, but this necessity is thought to be metaphysically contingent. However, they struggle to explain how the laws could be genuinely necessary while being metaphysically contingent. The chief advantage of Descartes view, I argue, is that God’s will can plausibly explain both the necessity of the laws and the contingency of the laws. So, Descartes’ theistic account of laws provides a plausible explanation, perhaps the best explanation, of the contingent-necessity of laws of nature. (shrink)

In this paper I challenge Paolo Palmieri’s reading of the Mach-Vailati debate on Archimedes’s proof of the law of the lever. I argue that the actual import of the debate concerns the possible epistemic (as opposed to merely pragmatic) role of mathematical arguments in empirical physics, and that construed in this light Vailati carries the upper hand. This claim is defended by showing that Archimedes’s proof of the law of the lever is not a way of appealing to a non-empirical (...) source of information, but a way of explicating the mathematical structure that can represent the empirical information at our disposal in the most general way. (shrink)

In their correspondence in 1902 and 1903, after discussing the Russell paradox, Russell and Frege discussed the paradox of propositions considered informally in Appendix B of Russell’s Principles of Mathematics. It seems that the proposition, p, stating the logical product of the class w, namely, the class of all propositions stating the logical product of a class they are not in, is in w if and only if it is not. Frege believed that this paradox was avoided within his (...) philosophy due to his distinction between sense (Sinn) and reference (Bedeutung). However, I show that while the paradox as Russell formulates it is ill-formed with Frege’s extant logical system, if Frege’s system is expanded to contain the commitments of his philosophy of language, an analogue of this paradox is formulable. This and other concerns in Fregean intensional logic are discussed, and it is discovered that Frege’s logical system, even without its naive class theory embodied in its infamous Basic Law V, leads to inconsistencies when the theory of sense and reference is axiomatized therein. (shrink)

One of the most disputed controversy over the priority of scientific discoveries is that of the law of universal gravitation, between Isaac Newton and Robert Hooke. Hooke accused Newton of plagiarism, of taking over his ideas expressed in previous works. In this paper I try to show, on the basis of previous analysis, that both scientists were wrong: Robert Hooke because his theory was basically only ideas that would never have materialized without Isaac Newton's mathematical support; and the latter was (...) wrong by not recognizing Hooke's ideas in drawing up the theory of gravity. Moreover, after Hooke's death and taking over the Royal Society presidency, Newton removed from the institution any trace of the former president Robert Hooke. For this, I detail the accusations and arguments of each of the parts, and how this dispute was perceived by the contemporaries of the two scientists. I finish the paper with the conclusions drawn from the contents. -/- Keywords: Isaac Newton, Robert Hooke, law of gravity, priority, plagiarism -/- CONTENTS -/- Abstract Introduction Robert Hooke's contribution to the law of universal gravitation Isaac Newton's contribution to the law of universal gravitation Robert Hooke's claim of his priority on the law of universal gravitation Newton's defense The controversy in the opinion of other contemporary scientists What the supporters of Isaac Newton say What the supporters of Robert Hooke say Conclusions Bibliography -/- DOI: 10.13140/RG.2.2.19370.26567 . (shrink)

The paper follows the track of a previous paper “Natural cybernetics of time” in relation to history in a research of the ways to be mathematized regardless of being a descriptive humanitarian science withal investigating unique events and thus rejecting any repeatability. The pathway of classical experimental science to be mathematized gradually and smoothly by more and more relevant mathematical models seems to be inapplicable. Anyway quantum mechanics suggests another pathway for mathematization; considering the historical reality as dual or “complimentary” (...) to its model. The historical reality by itself can be seen as mathematical if one considers it in Hegel’s manner as a specific interpretation of the totality being in a permanent self-movement due to being just the totality, i.e. by means of the “speculative dialectics” of history, however realized as a theory both mathematical and empirical and thus falsifiable as by logical contradictions within itself as emprical discrepancies to facts. Not less, a Husserlian kind of “historical phenomenology” is possible along with Hegel’s historical dialectics sharing the postulate of the totality (and thus, that of transcendentalism). One would be to suggest the transcendental counterpart: an “eternal”, i.e. atemporal and aspatial history to the usual, descriptive temporal history, and equating the real course of history as with its alternative, actually happened branches of the regions of the world as with only imaginable, counterfactual histories. That universal and transcendental history is properly mathematical by itself, even in a neo-Pythagorean model. It is only represented on the temporal screen of the standard historiography as a discrete series of unique events. An analogy to the readings of the apparatus in quantum mechanics can be useful. Even more, that analogy is considered rigorously and logically as implied by the mathematical transcendental history and sharing with it the same quantity of information as an invariant to all possible alternative or counterfactual histories. One can involve the hypothetical external viewpoint to history (as if outside of history or from “God’s viewpoint to it), to which all alternative or counterfactual histories can be granted as a class of equivalence sharing the same information (i.e. the number choices, but realized in different sequence or adding redundant ones in each branch) being similar and even mathematically isomorphic to Feynman trajectories in quantum mechanics. Particularly, a fundamental law of mathematical history, the law of least choice of the real historical pathway is deducible from the same approach. Its counterpart in physics is the well-known and confirmed law of least action as far as the quantity of action corresponds equivocally to the quantity of information or that of number elementary historical choices. (shrink)

A case study of quantum mechanics is investigated in the framework of the philosophical opposition “mathematical model – reality”. All classical science obeys the postulate about the fundamental difference of model and reality, and thus distinguishing epistemology from ontology fundamentally. The theorems about the absence of hidden variables in quantum mechanics imply for it to be “complete” (versus Einstein’s opinion). That consistent completeness (unlike arithmetic to set theory in the foundations of mathematics in Gödel’s opinion) can be interpreted furthermore (...) as the coincidence of model and reality. The paper discusses the option and fact of that coincidence it its base: the fundamental postulate formulated by Niels Bohr about what quantum mechanics studies (unlike all classical science). Quantum mechanics involves and develops further both identification and disjunctive distinction of the global space of the apparatus and the local space of the investigated quantum entity as complementary to each other. This results into the analogical complementarity of model and reality in quantum mechanics. The apparatus turns out to be both absolutely “transparent” and identically coinciding simultaneously with the reflected quantum reality. Thus, the coincidence of model and reality is postulated as necessary condition for cognition in quantum mechanics by Bohr’s postulate and further, embodied in its formalism of the separable complex Hilbert space, in turn, implying the theorems of the absence of hidden variables (or the equivalent to them “conservation of energy conservation” in quantum mechanics). What the apparatus and measured entity exchange cannot be energy (for the different exponents of energy), but quantum information (as a certain, unambiguously determined wave function) therefore a generalized law of conservation, from which the conservation of energy conservation is a corollary. Particularly, the local and global space (rigorously justified in the Standard model) share the complementarity isomorphic to that of model and reality in the foundation of quantum mechanics. On that background, one can think of the troubles of “quantum gravity” as fundamental, direct corollaries from the postulates of quantum mechanics. Gravity can be defined only as a relation or by a pair of non-orthogonal separable complex Hilbert space attachable whether to two “parts” or to a whole and its parts. On the contrary, all the three fundamental interactions in the Standard model are “flat” and only “properties”: they need only a single separable complex Hilbert space to be defined. (shrink)

Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view (...) appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)

Bertrand Russell once pointed out that modern science doesn’t deal with causal laws and that assuming otherwise is not only wrong but such thinking is erroneously thought to do no harm. However, looking into the scientific practice of simulation or experimentation reveals a general causal comprehension of physical processes. In this paper I trace causal experiences to the existence of innate causal capacity by which we organize sensory information. This capacity, I argue, is something we have got in virtue (...) of natural selection as can be seen from experiments with intelligent animals like crows and chimpanzees. So understanding the empirical world is impossible without the use of causal categories. The reason why Russell believed that modern science does not refer to causal laws is, I think, because he argued that the laws of mathematical physics give us a non-causal description of reality. In contrast to such a claim I hold that theoretical laws are prescriptive rules of description rather than descriptions themselves. (shrink)

We propose that quantum frequencies generate gravitational interactions.

We study a new formal logic LD introduced by Prof. Grzegorczyk. The logic is based on so-called descriptive equivalence, corresponding to the idea of shared meaning rather than shared truth value. We construct a semantics for LD based on a new type of algebras and prove its soundness and completeness. We further show several examples of classical laws that hold for LD as well as laws that fail. Finally, we list a number of open problems. -/- .

Investigation into the sequence structure of the genetic code by means of an informatic approach is a real success story. The features of human language are also the object of investigation within the realm of formal language theories. They focus on the common rules of a universal grammar that lies behind all languages and determine generation of syntactic structures. This universal grammar is a depiction of material reality, i.e., the hidden logical order of things and its relations determined by natural (...) laws. Therefore mathematics is viewed not only as an appropriate tool to investigate human language and genetic code structures through computer sciencebased formal language theory but is itself a depiction of material reality. This confusion between language as a scientific tool to describe observations/experiences within cognitive constructed models and formal language as a direct depiction of material reality occurs not only in current approaches but was the central focus of the philosophy of science debate in the twentieth century, with rather unexpected results. This article recalls these results and their implications for more recent mathematical approaches that also attempt to explain the evolution of human language. (shrink)

The preparedness and performance of Grade 10 students in prerequisites of STEM mathematics was the subject discussed in this study. General findings pointed to the fact that most students exhibited "Satisfactory" performance scores in key areas, with some readiness gaps seen in certain areas such as the Law of Exponents and Logarithmic and Exponential Functions rated "Low" to "Moderate". Readiness was highly associated with performance on some of the specific competencies - namely quadratic equation techniques and formula manipulation, but (...) not others; readiness cannot be inferred from performance alone. There is a suggestion to design strategic intervention materials that would close these gaps and refresh skills in critical areas of mathematics. The SIM should have activities that are structured, ways to motivate and support instruction towards the goal, so that there is a bridging between readiness and performance. Overall, it points out that what matters is a focused intervention for improving readiness among students in advanced STEM mathematics-prepared better for future academic and career challenges in STEM disciplines. (shrink)

This paper distinguishes between 3 meanings of reversal, all of which are mathematically equivalent in classical mechanics: velocity reversal, retrodiction, and time reversal. It then concludes that in order to have well defined velocities a primitive arrow of time must be included in every time slice. The paper briefly mentions that this arrow cannot come from the Second Law of thermodynamics, but this point is developed in more details elsewhere.

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. (...) The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines. (shrink)

Biological evolution is a complex blend of ever changing structural stability, variability and emergence of new phe- notypes, niches, ecosystems. We wish to argue that the evo- lution of life marks the end of a physics world view of law entailed dynamics. Our considerations depend upon dis- cussing the variability of the very ”contexts of life”: the in- teractions between organisms, biological niches and ecosys- tems. These are ever changing, intrinsically indeterminate and even unprestatable: we do not know ahead of (...) time the ”niches” which constitute the boundary conditions on selec- tion. More generally, by the mathematical unprestatability of the ”phase space”, no laws of mo- tion can be formulated for evolution. We call this radical emergence, from life to life. The purpose of this paper is the integration of variation and diversity in a sound concep- tual frame and situate unpredictability at a novel theoretical level, that of the very phase space. Our argument will be carried on in close comparisons with physics and the mathematical constructions of phase spaces in that discipline. The role of symmetries as invariant preserving transformations will allow us to under- stand the nature of physical phase spaces and to stress the differences required for a sound biological theoretizing. In this frame, we discuss the novel notion of ”enablement”. Life lives in a web of enablement and radical emergence. This will restrict causal analyses to differential cases. Mutations or other causal differ- ences will allow us to stress that ”non conservation princi- ples” are at the core of evolution, in contrast to physical dynamics, largely based on conservation principles as sym- metries. Critical transitions, the main locus of symmetry changes in physics, will be discussed, and lead to ”extended criticality” as a conceptual frame for a better understanding of the living state of matter. (shrink)

Many philosophers are baffled by necessity. Humeans, in particular, are deeply disturbed by the idea of necessary laws of nature. In this paper I offer a systematic yet down to earth explanation of necessity and laws in terms of invariance. The type of invariance I employ for this purpose generalizes an invariance used in meta-logic. The main idea is that properties and relations in general have certain degrees of invariance, and some properties/relations have a stronger degree of invariance (...) than others. The degrees of invariance of highly-invariant properties are associated with high degrees of necessity of laws governing/describing these properties, and this explains the necessity of such laws both in logic and in science. This non-mysterious explanation has rich ramifications for both fields, including the formality of logic and mathematics, the apparent conflict between the contingency of science and the necessity of its laws, the difference between logical-mathematical, physical, and biological laws/principles, the abstract character of laws, the applicability of logic and mathematics to science, scientific realism, and logical-mathematical realism. (shrink)

The paper discusses Hilbert mathematics, a kind of Pythagorean mathematics, to which the physical world is a particular case. The parameter of the “distance between finiteness and infinity” is crucial. Any nonzero finite value of it features the particular case in the frameworks of Hilbert mathematics where the physical world appears “ex nihilo” by virtue of an only mathematical necessity or quantum information conservation physically. One does not need the mythical Big Bang which serves to concentrate all (...) the violations of energy conservation in a “safe”, maximally remote point in the alleged “beginning of the universe”. On the contrary, an omnipresent and omnitemporal medium obeying quantum information conservation rather than energy conservation permanently generates action and thus the physical world. The utilization of that creation “ex nihilo” is accessible to humankind, at least theoretically, as long as one observes the physical laws, which admit it in their new and wider generalization. One can oppose Hilbert mathematics to Gödel mathematics, which can be identified as all the standard mathematics until now featureable by the Gödel dichotomy of arithmetic to set theory: and then, “dialectic”, “intuitionistic”, and “Gödelian” mathematics within the former, according to a negative, positive, or zero value of the distance between finiteness and infinity. A mapping of Hilbert mathematics into pseudo-Riemannian space corresponds, therefore allowing for gravitation to be interpreted purely mathematically and ontologically in a Pythagorean sense. Information and quantum information can be involved in the foundations of mathematics and linked to the axiom of choice or alternatively, to the field of all rational numbers, from which the pair of both dual and anti-isometric Peano arithmetics featuring Hilbert arithmetic are immediately inferable. Noether’s theorems (1918) imply quantum information conservation as the maximally possible generalization of the pair of the conservation of a physical quantity and the corresponding Lie group of its conjugate. Hilbert mathematics can be interpreted from their viewpoint also after an algebraic generalization of them. Following the ideas of Noether’s theorem (1918), locality and nonlocality can be realized both physically and mathematically. The “light phase of the universe” can be linked to the gap of mathematics and physics in the Cartesian organization of cognition in Modernity and then opposed to its “dark phase”, in which physics and mathematics are merged. All physical quantities can be deduced from only mathematical premises by the mediation of the most fundamental physical constants such as the speed of light in a vacuum, the Planck and gravitational constants once they have been interpreted by the relation of locality and nonlocality. (shrink)

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

This paper aims both to tackle the technical issue of deciphering Hobbes’s derivation of the sine law of refraction and to throw some light to the broader issue of Hobbes’s mechanical philosophy. I start by recapitulating the polemics between Hobbes and Descartes concerning Descartes’ optics. I argue that, first, Hobbes’s criticisms do expose certain shortcomings of Descartes’ optics which presupposes a twofold distinction between real motion and inclination to motion, and between motion itself and determination of motion; second, Hobbes’s optical (...) theory presented in Tractatus Opticus I constitutes a more economical alternative, which eliminates the twofold distinction and only admits actual local motion, and Hobbes’s derivation of the sine law presented therein, which I call “the early model” and which was retained in Tractatus Opticus II and First Draught, is mathematically consistent and physically meaningful. These two points give Hobbes’s early optics some theoretical advantage over that of Descartes. However, an issue that has baffled commentators is that, in De Corpore Hobbes’s derivation of the sine law seems to be completely different from that presented in his earlier works, furthermore, it does not make any intuitive sense. I argue that the derivation of the sine law in De Corpore does make sense mathematically if we read it as a simplification of the early model, and Hobbes has already hinted toward it in the last proposition of Tractatus Opticus I. But now the question becomes, why does Hobbes take himself to be entitled to present this simplified, seemingly question-begging form without having presented all the previous results? My conjecture is that the switch from the early model to the late model is symptomatic of Hobbes’s changing views on the relation between physics and mathematics. (shrink)

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

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as the Riemann hypothesis, have had to be considered in terms of the evidence for and against them. In recent decades, massive increases in computer power have permitted the gathering of huge amounts of numerical evidence, both for conjectures in pure (...) class='Hi'>mathematics and for the behavior of complex applied mathematical models and statistical algorithms. Mathematics has therefore become (among other things) an experimental science (though that has not diminished the importance of proof in the traditional style). We examine how the evaluation of evidence for conjectures works in mathematical practice. We explain the (objective) Bayesian view of probability, which gives a theoretical framework for unifying evidence evaluation in science and law as well as in mathematics. Numerical evidence in mathematics is related to the problem of induction; the occurrence of straightforward inductive reasoning in the purely logical material of pure mathematics casts light on the nature of induction as well as of mathematical reasoning. (shrink)

From The Cone of Perception, volume one of my collected works, you will remember that one of the main topics in that work was V-Curvature, also called, "phenomenological velocity." In that work, although a solution to the v - curvature variable was provided as well as many graphs that yielded numerous jewels of spiral formulations in exquisite 3D color formations, that method by which the solution was found was not iterated. This chapter begins by showing how it is possible to (...) solve for something that ideally ought cancel out with itself and how, although commutation between square roots is valid, there may be room here for an alternate route of accessing a hidden dimension - that dimension we call V-Curvature, or, "Phenomenological Velocity." Herein is provided the pathway for solving for V-Curvature in terms of Csc, which can be translated into Sin and Cos functionality. Furthermore, the processing these equations through WolframAlpha yielded other insights into limits, roots, and series that logically follow. How did the solutions to the, "velocity," v - variable curvature in the Lorentz coefficient, "manifest," when the Lorentz coefficient ought cancel out with itself? The step - by - step solution below illustrates the algebraic process by which a specific solution for something that ought cancel out with itself can be found. The Sphere of Realization is a groundbreaking work of philosophical and mathematical theory. The first volume of this epic collection examines the concept of V-Curvature, or Phenomenological Velocity, using algebraic equations to explain how this phenomenon can be solved. This book goes on to explore the implications of these equations, such as limits, roots, series, and introspection into infinity, providing an alternate route for accessing a 'hidden dimension' that connects physical laws with abstract philosophical ideas. By utilizing this understanding, the author creates a platform for introspection into life and nature, and considers the philosophical implications of this perception, from the exploration of the infinity of angles to theions of time, velocity, the emergence of real numbers, and the meanings of eternity, forever, infinity, and never ending. This landmark work of mathematical exploration and philosophical discovery will challenge and expand our understanding of the universe, and invite us into a journey of introspection and understanding. An exploration of the esoteric terms therein, and the pathways they provide to access new modes of thought, will give the reader the ability to gain a greater insight into the intricacies of reality. The Sphere of Realization provides a unique and comprehensive approach to a difficult concept. Through a blend of philosophical and mathematical exploration, the author delves into the concept of V-Curvature, or Phenomenological Velocity to unlock the mysteries of "hidden dimensions" within our reality. These equations and theories allow the reader to explore angles, time, velocity, roots, series, and other related concepts. In so doing, the book offers an exciting journey into the esoteric. The author also provides insights into the emergence of real numbers from infinity, introducing concepts such as eternity, forever, and never ending in the process. A comprehensive discussion of the applicable equations and transformations, immune to the construct of time, serves as a powerful tool for unlocking a meaningful understanding of the universe. The Sphere of Realization is an opportunity to gain a deeper appreciation of the reality of our world. (shrink)

Al final de su libro “La conciencia inexplicada”, Juan Arana señala que la nomología, explicación según las leyes de la naturaleza, requiere de una nomogonía, una consideración del origen de las leyes. Es decir, que el orden que observamos en el mundo natural requiere una instancia previa que ponga ese orden específico. Sabemos que desde la revolución científica la mejor manera de explicar dicha nomología ha sido mediante las matemáticas. Sin embargo, en las últimas décadas se han presentado algunas propuestas (...) basadas en modelos matemáticos que fundamentarían muchos aspectos de la realidad. Dos claros ejemplos provienen de Roger Penrose y Max Tegmark. Esto lleva a pensar en una posición no solo nomológica sino además nomogónica de la matemática. ¿Puede la Naturaleza estar fundada por las matemáticas como señalan algunos físico-matemáticos? Y en ese caso, ¿sería pertinente buscar una nomo-génesis de esta índole en la constitución de la conciencia? -/- At the end of his book “La conciencia inexplicada”, Juan Arana points out that nomology, explanation according to the laws of nature requires a nomogony, an account of the origin of the laws. This means that the order that we can observe in the natural World demands something prior to posit that specific order. Since the scientific revolution we know that the best way to explain that nomology has been through mathematics. Anyway, in recent decades a number of proposals based on mathematical models that found many aspects of reality has been offered. Two clear examples come from Roger Penrose and Max Tegmark. This drives us to think of a position of mathematics as not only nomological but also nomogonical. Can Nature be founded by mathematics as some physicists and mathematicians point out? And, in this case, would be relevant this kind of approach to search a nomo-genesis in the constitution of consciousness? (shrink)

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. -/- We then adopt (...) what may be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. -/- We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. -/- We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. -/- We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences. -/- . (shrink)

Metaphysicians play an important role in our understanding of the universe. In recent years, physicists have focussed on finding accurate mathematical formalisms of the evolution of our physical system - if a metaphysician can uncover the metaphysical underpinnings of these formalisms; that is, why these formalisms seem to consistently map the universe, then our understanding of the world and the things in it is greatly enhanced. Science, then, plays a very important role in our project, as the best scientific formalisms (...) provide us with what we, as metaphysicians, should be trying to interpret. In this thesis I examine existing metaphysical views of what a law is (both from a conceptual and from a metaphysical perspective), to show how closely causation is linked to laws, and to provide a priori arguments for and against each of these positions. Ultimately, I aim to provide an analysis of a number of metaphysics of natural laws and causation, apply these accounts to our best scientific theories, and see how these metaphysics fit in with our concepts of cause and law. Although I do not attempt a definitive metaphysical account myself, I conclude that any successful metaphysic will be a broadly Humean one, and furthermore that given the concepts of cause and law that shall be agreed upon, Humean theories allow for there to be causal sequences and laws (in line with our concepts) in the world. (shrink)

In his famous article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” Eugen Wigner argues for a unique tie between mathematics and physics, invoking even religious language: “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve”. The possible existence of such a unique match between mathematics and physics has been extensively discussed by philosophers and historians (...) of mathematics. Whatever the merits of this claim are, a further question can be posed with regard to mathematization in science more generally: What happens when we leave the area of theories and laws of physics and move over to the realm of mathematical modeling in interdisciplinary contexts? Namely, in modeling the phenomena specific to biology or economics, for instance, scientists often use methods that have their origin in physics. How is this kind of mathematical modeling justified? (shrink)

Finnis claims that his theory proceeds from seven basic principles of practical reason that are self-evidently true. While much has been written about the claim of self-evidence, this article considers it in relation to the rigorous claims of logic and mathematics. It argues that when considered in this light, Finnis equivocates in his use of the concept of self-evidence between the realist Thomistic conception and a purely formal, modern symbolic conception. Given his respect for the modern positivist separation of (...) fact and value, the realism of the Thomistic conception cannot be the foundation for the natural law as Finnis would reconstruct it. Nor can the purely formal modern conception of self-evidence provide a foundation for practical reason. This raises some doubt as to whether there can be self-evident principles of practical reason as Finnis suggests. The article contributes to the analysis of the Finnis reconstruction of the natural law by providing a rigorous analysis of his claim for self-evidence. It frames this analysis historically, suggesting the roles that certainty and mystery play in the apprehension of moral meaning. (shrink)

I am interested in the use Kant makes of the pure intuition of space, and of properties and principles of space and spaces (i.e. figures, like spheres and lines), in the special metaphysical project of MAN. This is a large topic, so I will focus here on an aspect of it: the role of these things in his treatment of some of the laws of matter treated in the Dynamics and Mechanics Chapters. In MAN and other texts, Kant speaks (...) of space as the “ground,” “condition,” and “basis” of various laws, including the inverse-square and inverse-cube laws of attractive and repulsive force, and the Third Law of Mechanics. Moreover, in his proofs of all the laws just mentioned, the language of “construction” figures prominently, which suggests that Kant’s proofs (somehow) rest on or involve mathematical construction in his technical sense. Such claims give rise to a number of questions. How do properties and principles of space and spaces serve to ground this particular set of laws? Which spatial properties and principles is Kant appealing to? What, if anything, does the spatial grounding of the inverse-square and inverse-cube laws of diffusion (treated in the Dynamics Chapter) have in common with that of the Third treated in the Mechanics Chapter)? What role—if any—does mathematical construction play in Kant’s proofs of these laws? Finally, how if at all, are Kant’s grounding claims consistent with his other commitments—for example, how are they consistent with his notorious denial in Prolegomena §38 that there are any laws that “lie in space” (Prol 4:321)? I offer answers to these questions. (shrink)

Abstractionism in the philosophy of mathematics aims at deriving large fragments of mathematics by combining abstraction principles (i.e. the abstract objects $\S e_1, \S e_2$, are identical if, and only if, an equivalence relation $Eq_\S$ holds between the entities $e_1, e_2$) with logic. Still, as highlighted in work on the semantics for relevant logics, there are different ways theories might be combined. In exactly what ways must logic and abstraction be combined in order to get interesting mathematics? (...) In this paper, we investigate the matter by deriving the axioms of second-order Peano Arithmetic from Frege's Basic Law V (the extension of $F$ is identical with the extension of $G$ if, and only if, $F$ and $G$ are extensionally equivalent) in the presence of a relevant higher-order logic. The results are interesting. Not only must we take on logic as true, and not only must we apply our logic to abstraction principles, but also we have to apply our theory of abstraction back to the logic in order to arrive at arithmetic. Thus, what Abstractionism gives us is not simply what we get from abstraction via logic, but also what we get from logic via abstraction. (shrink)

What are the probabilities that this universe is repeated exactly the same with you in it again? Is God invented by human imagination or is the result of human intuition? The intuition that the same laws/mechanisms (evolution, stability winning probability) that have created something like the human being capable of self-awareness and controlling its surroundings, could create a being capable of controlling all what it exists? Will be the characteristics of the next universes random or tend to something? All (...) these ques-tions that with different shapes (but the same essence) have been asked by human be-ings from the beginning of times will be developed in this paper. (shrink)

So-called ‘distinctively mathematical explanations’ (DMEs) are said to explain physical phenomena, not in terms of contingent causal laws, but rather in terms of mathematical necessities that constrain the physical system in question. Lange argues that the existence of four or more equilibrium positions of any double pendulum has a DME. Here we refute both Lange’s claim itself and a strengthened and extended version of the claim that would pertain to any n-tuple pendulum system on the ground that such explanations (...) are actually causal explanations in disguise and their associated modal conditionals are not general enough to explain the said features of such dynamical systems. We argue and show that if circumscribing the antecedent for a necessarily true conditional in such explanations involves making a causal analysis of the problem, then the resulting explanation is not distinctively mathematical or non-causal. Our argument generalises to other dynamical systems that may have purported DMEs analogous to the one proposed by Lange, and even to some other counterfactual accounts of non-causal explanation given by Reutlinger and Rice. (shrink)

Much recent philosophy of physics has investigated the process of symmetry breaking. Here, I critically assess the alleged symmetry restoration at the fundamental scale. I draw attention to the contingency that gauge symmetries exhibit, that is, the fact that they have been chosen from an infinite space of possibilities. I appeal to this feature of group theory to argue that any metaphysical account of fundamental laws that expects symmetry restoration up to the fundamental level is not fully satisfactory. This (...) is a symmetry argument in line with Curie’s first principle. Further, I argue that this same feature of group theory helps to explain the ‘unreasonable’ effectiveness of mathematics in physics, and that it reduces the philosophical significance that has been attributed to the objectivity of gauge symmetries. (shrink)

Given the sharp distinction that follows from Hume’s Fork, the proper epistemic status of propositions of mixed mathematics seems to be a mystery. On the one hand, mathematical propositions concern the relation of ideas. They are intuitive and demonstratively certain. On the other hand, propositions of mixed mathematics, such as in Hume’s own example, the law of conservation of momentum, are also matter of fact propositions. They concern causal relations between species of objects, and, in this sense, they (...) are not intuitive or demonstratively certain, but probable or provable. In this article, I argue that the epistemic status of propositions of mixed mathematics is that of matters of fact. I wish to show that their epistemic status is not a mystery. The reason for this is that the propositions of mixed mathematics are dependent on the Uniformity Principle, unlike the propositions of pure mathematics. (shrink)

Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously -- albeit not necessary at its face value. Because alongside his exquisite in beauty ordinal construction and his fundamental powerset description of the continuum, Cantor has also left to us his obsessive presumption that the universe of sets should be subjected to laws similar to those governing the set of natural numbers, including the universal principles of cardinal (...) comparability and well-ordering -- and implying an ordinal re-creation of the continuum. During the last hundred years, the mainstream set-theoretical research -- all insights and adjustments due to Kurt G\"odel's revolutionary insights and discoveries notwithstanding -- has compliantly centered its efforts on ad hoc axiomatizations of Cantor's intuitive transfinite design. We demonstrate here that the ontological and epistemic sustainability} of this design has been irremediably compromised by the underlying peremptory, Reductionist mindset of the XIXth century's ideology of science. (shrink)

Language suggestive of natural law ethics, similar to the Catholic understanding of ethical foundations, is prevalent in a number of disciplines. But it does not always issue in a full-blooded commitment to objective ethics, being undermined by relativist ethical currents. In law and politics, there is a robust conception of "human rights", but it has become somewhat detached from both the worth of persons in themselves and from duties. In education, talk of "values" imports ethical considerations but hints at a (...) subjectivist view of them. In the psychology and sociology of drug use, ethically thin concepts of "harm minimisation" and "selfimage" dominate discussion and distract attention from the virtue of temperance and the training of character. A more forceful assertion of an ethics based on the worth of persons in these cases would be most desirable. Arguments against objectivity in the fundamentals may be replied to by examining the parallel between ethics and the discipline whose objectivity has been least challenged by relativist arguments, mathematics. (shrink)

We give some arguments for why the Mathematical Universe Hypothesis (MUH) might be too restrictive in its assertions of what can exist, and that the universe/multiverse might be formed by more than what can be expressed mathematically. In particular, we show a thought experiment which indicates that the principle of materialism in general is an inadequate hypothesis of how consciousness appears. Instead we propose a novel approach to solving the problem of consciousness, which is to hypothesize that each universe might (...) have different laws of consciousness, just as they might have different laws of physics. We also show that MUH, without such consciousness laws, and without any other modification, leads to complete chaos for the observers in the multiverse, and therefore cannot hold as it is. We then go on to propose another theory of existence, which does not seem to lead to complete chaos. This theory hypothesizes that the multiverse is the consequence of what we might call the Logic of Everything (LoE) continuously deducing new truths about itself. Lastly, we briefly discuss what fundamental ethics can be derived from this theory, and others like it. (shrink)

When this article was first planned, writing was going to be exclusively about two things - the origin of life and human evolution. But it turned out to be out of the question for the author to restrict himself to these biological and anthropological topics. A proper understanding of them required answering questions like “What is the nature of the universe – the home of life – and how did it originate?”, “How can time travel be removed from fantasy and (...) science fiction, to be made scientific and practical?”, and “How can the proposed young age of genus Homo be made to actually be reasonable – when simply stating it would be solid ground for instant rejection and dismissal?” The result is that the article also talks about subjects like Artificial Intelligence, General Relativity, and cosmology. -/- From where did life originate? God? Evolution? Panspermia? If the tendency of humans and scientists to regard undiscovered science as pseudoscience can be overcome, Einstein gave another alternative to consider when he introduced General Relativity. Time isn’t linear – progressing in a straight line from past to present to future. That assumption ignores Relativity which states that space AND TIME are curved. Where did life and the genetic code come from? Can the answer build AI? -/- The first question can be answered by the section of this article titled SETI, Evolution, and Time which says life (possibly multicellular and intelligent) and the genetic code came from humans acquiring knowledge of these things over the centuries, then applying that knowledge – via terraforming, accumulation of raw materials like amino acids and nucleic acids, genetic engineering - to a time in the past when life didn’t exist. From that origin, life evolved through innumerable mutations and adaptations, with humans once again acquiring knowledge of it in cyclic (nonlinear) time. -/- The second question is answered by saying artificial intelligence (AI) as the product of life is only half of the equation. The other half refers to Relativity’s curved space-time and violation of the notion that time always travels from past to future. We have always lived in an artificially intelligent, non-probabilistic universe where everything in time and space is connected into one thing by quantum entanglement – making the brain and genes products of binary-digit activity or artificial intelligence (life is not merely dependent on biology’s “lock and key” mechanisms but also possesses AI). -/- The earliest documented representative of the genus Homo is Homo habilis, which evolved around 2.8 million years ago. Scientists used to believe there was a straight line from H. habilis to us, Homo sapiens. This article will use the “advanced” waves loved by Physics Nobel laureate Richard Feynman, view the history of science through the lens of Conic Sections applied to Relativity’s curved space-time, and incorporate the necessity of so-called imaginary time * – popularized by Prof. Stephen Hawking. While the evolutionary proposals are more in agreement with this early straight line than with modern theories, Albert Einstein’s General Relativity is used to transform the straight line into a curved line, ultimately concluding that Homo habilis (H. habilis) originated only (and unbelievably, as far as today’s science and technology is concerned) ~250,000 years ago. Other branches and dead ends of Homo – e.g. Neanderthals – are the result of mutations and adaptations, with the resultant modifications to anatomy and physiology. The surprisingly young age of H. habilis allows nearly 200,000 years for habilis, or one of its descendants, to reach Australia … if this country’s indigenous Aboriginal population did, as claimed, reach this “island continent” 60,000 years ago. -/- * The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, is a failure of classical physics to predict observed phenomena: it can be shown that a blackbody - a hypothetical perfect absorber and radiator of energy - would release an infinite amount of energy, contradicting the principles of conservation of energy and indicating that a new model for the behaviour of blackbodies was needed. At the start of the 20th century, physicist Max Planck derived the correct solution by making some strange (for the time) assumptions. In particular, Planck assumed that electromagnetic radiation can only be emitted or absorbed in discrete packets, called quanta. Albert Einstein postulated that Planck's quanta were real physical particles (what we now call photons), not just a mathematical fiction. From there, Einstein developed his explanation of the photoelectric effect (when quanta or photons of light shine on certain metals, electrons are released and can form an electric current). So it appears entirely possible that another supposed mathematical trickery (imaginary time and the y-axis of Wick rotation) will find practical application in the future. -/- The article includes mathematical references to cosmology (spoiler alert – you’ll read about things like Vector-Tensor-Scalar Geometry, topology, the “eternal present”, Einstein’s Unified Field, the inverse-square law, and there being no Big Bang and no multiverse - but there will also be no equations). -/- The other subheadings in this essay are – -/- NONLINEAR TIME AND ELECTRICAL ENGINEERING (about a 2009 electrical engineering experiment at America’s Yale University, and cosmic wormholes) -/- BITS AND TOPOLOGY (base-2 maths aka Binary digiTS, Mobius strips, and figure-8 Klein bottles) -/- WICK ROTATION, CAUSALITY, AND UNITING TIME (do past, present and future co-exist in an “eternal present”?) -/- DIGITAL BRAIN, DIGITAL UNIVERSE (if the brain and the universe are ultimately composed of binary digits, we'll someday be able to do the same things with the brain and universe that we now do with computers) -/- PROPOSAL: HUMAN AND ANIMAL INSTINCTS ARE THE RESULT OF THE UNIVERSE BEING UNIFIED BY BINARY DIGITS (AND TOPOLOGY) (If everything in the universe is ultimately composed of electronic BITS, then the universe must possess Artificial Intelligence - some prefer the term Cosmic Consciousness) -/- INFORMATION THEORY CONQUERS A RED GIANT (preserving Earth by keeping the Sun near today’s level of activity forever). 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The determinism-free will debate is perhaps as old as philosophy itself and has been engaged in from a great variety of points of view including those of scientific, theological, and logical character. This chapter focuses on two arguments from logic. First, there is an argument in support of determinism that dates back to Aristotle, if not farther. It rests on acceptance of the Law of Excluded Middle, according to which every proposition is either true or false, no matter whether the (...) proposition is about the past, present or future. In particular, the argument goes, whatever one does or does not do in the future is determined in the present by the truth or falsity of the corresponding proposition. The second argument coming from logic is much more modern and appeals to Gödel's incompleteness theorems to make the case against determinism and in favour of free will, insofar as that applies to the mathematical potentialities of human beings. The claim more precisely is that as a consequence of the incompleteness theorems, those potentialities cannot be exactly circumscribed by the output of any computing machine even allowing unlimited time and space for its work. The chapter concludes with some new considerations that may be in favour of a partial mechanist account of the mathematical mind. (shrink)

Lange argues that some natural phenomena can be explained by appeal to mathematical, rather than natural, facts. In these “distinctively mathematical” explanations, the core explanatory facts are either modally stronger than facts about ordinary causal law or understood to be constitutive of the physical task or arrangement at issue. Craver and Povich argue that Lange’s account of DME fails to exclude certain “reversals”. Lange has replied that his account can avoid these directionality charges. Specifically, Lange argues that in legitimate DMEs, (...) but not in their “reversals,” the empirical fact appealed to in the explanation is “understood to be constitutive of the physical task or arrangement at issue” in the explanandum. I argue that Lange’s reply is unsatisfactory because it leaves the crucial notion of being “understood to be constitutive of the physical task or arrangement” obscure in ways that fail to block “reversals” except by an apparent ad hoc stipulation or by abandoning the reliance on understanding and instead accepting a strong realism about essence. (shrink)

If there are fundamental laws of nature, can they fail to be exact? In this paper, I consider the possibility that some fundamental laws are vague. I call this phenomenon 'fundamental nomic vagueness.' I characterize fundamental nomic vagueness as the existence of borderline lawful worlds and the presence of several other accompanying features. Under certain assumptions, such vagueness prevents the fundamental physical theory from being completely expressible in the mathematical language. Moreover, I suggest that such vagueness can be (...) regarded as 'vagueness in the world.' For a case study, we turn to the Past Hypothesis, a postulate that (partially) explains the direction of time in our world. We have reasons to take it seriously as a candidate fundamental law of nature. Yet it is vague: it admits borderline (nomologically) possible worlds. An exact version would lead to an untraceable arbitrariness absent in any other fundamental laws. However, the dilemma between fundamental nomic vagueness and untraceable arbitrariness is dissolved in a new quantum theory of time's arrow. (shrink)

In this paper, I argue for a distinction between two scales of coordination in scientific inquiry, through which I reassess Georg Simon Ohm’s work on conductivity and resistance. Firstly, I propose to distinguish between measurement coordination, which refers to the specific problem of how to justify the attribution of values to a quantity by using a certain measurement procedure, and general coordination, which refers to the broader issue of justifying the representation of an empirical regularity by means of abstract mathematical (...) tools. Secondly, I argue that the development of Ohm’s measurement practice between the first and the second experimental phase of his work involved the change of the measurement coordination on which he relied to express his empirical results. By showing how Ohm relied on different calibration assumptions and practices across the two phases, I demonstrate that the concurrent change of both Ohm’s experimental apparatus and the variable that Ohm measured should be viewed based on the different form of measurement coordination. Finally, I argue that Ohm’s assumption that tension is equally distributed in the circuit is best understood as part of the general coordination between Ohm’s law and the empirical regularity that it expresses, rather than measurement coordination. (shrink)

We discuss the philosophical implications of formal results showing the con- sequences of adding the epsilon operator to intuitionistic predicate logic. These results are related to Diaconescu’s theorem, a result originating in topos theory that, translated to constructive set theory, says that the axiom of choice (an “existence principle”) implies the law of excluded middle (which purports to be a logical principle). As a logical choice principle, epsilon allows us to translate that result to a logical setting, where one can (...) get an analogue of Diaconescu’s result, but also can disentangle the roles of certain other assumptions that are hidden in mathematical presentations. It is our view that these results have not received the attention they deserve: logicians are unlikely to read a discussion because the results considered are “already well known,” while the results are simultaneously unknown to philosophers who do not specialize in what most philosophers will regard as esoteric logics. This is a problem, since these results have important implications for and promise signif i cant illumination of contem- porary debates in metaphysics. The point of this paper is to make the nature of the results clear in a way accessible to philosophers who do not specialize in logic, and in a way that makes clear their implications for contemporary philo- sophical discussions. To make the latter point, we will focus on Dummettian discussions of realism and anti-realism. Keywords: epsilon, axiom of choice, metaphysics, intuitionistic logic, Dummett, realism, antirealism. (shrink)

Analysis is given of the Omega Point cosmology, an extensively peer-reviewed proof (i.e., mathematical theorem) published in leading physics journals by professor of physics and mathematics Frank J. Tipler, which demonstrates that in order for the known laws of physics to be mutually consistent, the universe must diverge to infinite computational power as it collapses into a final cosmological singularity, termed the Omega Point. The theorem is an intrinsic component of the Feynman-DeWitt-Weinberg quantum gravity/Standard Model Theory of Everything (...) (TOE) describing and unifying all the forces in physics, of which itself is also required by the known physical laws. With infinite computational resources, the dead can be resurrected--never to die again--via perfect computer emulation of the multiverse from its start at the Big Bang. Miracles are also physically allowed via electroweak quantum tunneling controlled by the Omega Point cosmological singularity. The Omega Point is a different aspect of the Big Bang cosmological singularity--the first cause--and the Omega Point has all the haecceities claimed for God in the traditional religions. -/- From this analysis, conclusions are drawn regarding the social, ethical, economic and political implications of the Omega Point cosmology. (shrink)