Mathematical Realism

This work proposes a rethinking of Goedel’s incomplete results; Specifically, rather than some true statements remaining unreachable for verification on a true-false basis in any axiomatic system that is strong enough to express basic arithmetic, the structure itself ensures that no statement remains unverified on a true-false basis, which is consistent with both Goedel’s results as well as Hilbert’s assertion that such algorithm does exist, albeit not one that could be derived within the separately given axiomatic system itself. A statement defined here as phenomena bridges that divide. Phenomena, defined as non-axiomatic, non-tautological, and non-contradictory statements exist invariably in any such axiomatic system, regardless of the possibility to construct them within the axiomatic system. All other non-axiomatic, non-tautological and non-contradictory statements are also defined as phenomena; Phenomena allow for incorporation of Peano arithmetic, Euclidean geometry and Real analysis into ZFT; Axiom of choice would hence be reformulated and incorporated as a theorem. Reasoning provided on why Cantor’s paradox should be included into the ZFT as an axiom. Proof of Riemann conjecture is provided through the application of phenomena.

The Universal Model of Physics proposes that prime numbers, fractal geometry, and modular arithmetic serve as the mathematical code underpinning the cosmos. By distinguishing physical dimensions (space-time) from conceptual (prime-based) frameworks, this model seeks to unify quantum mechanics, relativity, and cosmology without relying on additional spatial dimensions. Prime distributions guide fractal self-similarity, shaping phenomena from wavefront interference to large-scale cosmic clustering. This paper presents the theoretical underpinnings of the model—emphasizing prime periodicity, fractal geometry, and their implications for nuclear stability, dark matter, and quantum behavior—and outlines testable predictions in areas such as the Cosmic Microwave Background (CMB), quantum interference, and gravitational lensing. Ultimately, the Universal Model interprets matter, energy, and even “dark” phenomena as emergent from an elegant prime-driven fractal structure, challenging materialist assumptions and inviting further empirical exploration of a timeless mathematical blueprint.

Forbindelsen mellem den synlige og den usynlige verden, Guds engles tjeneste og onde ånders virksomhed er tydeligt åbenbaret i Skrifterne og uadskilleligt sammenflettet med menneskets historie. Der er en voksende tendens til ikke at tro på eksistensen af onde ånder, mens de hellige engle, som er “tjenende ånder, der sendes for at hjælpe dem, som skal arve frelse”, betragtes som de dødes ånder. Bibelen lærer ikke blot om eksistensen af engle, gode såvel som onde, men præsenterer ubestrideligt bevis på, at disse ikke er frigjorte ånder fra døde mennesker. Der fandtes engle, før menneskets skabelse, for “alle morgenstjerner jublede, og alle gudssønner råbte af fryd,” da jordens grundvold blev lagt. Efter syndefaldet blev engle sat til at vogte livets træ, og det var, før noget menneske var afgået ved døden. Englenes natur er mere ophøjet end menneskenes. Salmisten siger, at mennesket blev skabt “lidt ringere end englene.” Onde ånder, oprindelig skabt syndfrie, var i natur, magt og herlighed jævnbyrdige med de hellige væsener, der nu er Guds sendebud...

Forbindelsen mellem den synlige og den usynlige verden, Guds engles tjeneste og onde ånders virksomhed er tydeligt åbenbaret i Skrifterne og uadskilleligt sammenflettet med menneskets historie. Der er en voksende tendens til ikke at tro på eksistensen af onde ånder, mens de hellige engle, som er “tjenende ånder, der sendes for at hjælpe dem, som skal arve frelse”, betragtes som de dødes ånder. Bibelen lærer ikke blot om eksistensen af engle, gode såvel som onde, men præsenterer ubestrideligt bevis på, at disse ikke er frigjorte ånder fra døde mennesker. Der fandtes engle, før menneskets skabelse, for “alle morgenstjerner jublede, og alle gudssønner råbte af fryd,” da jordens grundvold blev lagt. Efter syndefaldet blev engle sat til at vogte livets træ, og det var, før noget menneske var afgået ved døden. Englenes natur er mere ophøjet end menneskenes. Salmisten siger, at mennesket blev skabt “lidt ringere end englene.” Onde ånder, oprindelig skabt syndfrie, var i natur, magt og herlighed jævnbyrdige med de hellige væsener, der nu er Guds sendebud...

Forbindelsen mellem den synlige og den usynlige verden, Guds engles tjeneste og onde ånders virksomhed er tydeligt åbenbaret i Skrifterne og uadskilleligt sammenflettet med menneskets historie. Der er en voksende tendens til ikke at tro på eksistensen af onde ånder, mens de hellige engle, som er “tjenende ånder, der sendes for at hjælpe dem, som skal arve frelse”, betragtes som de dødes ånder. Bibelen lærer ikke blot om eksistensen af engle, gode såvel som onde, men præsenterer ubestrideligt bevis på, at disse ikke er frigjorte ånder fra døde mennesker. Der fandtes engle, før menneskets skabelse, for “alle morgenstjerner jublede, og alle gudssønner råbte af fryd,” da jordens grundvold blev lagt. Efter syndefaldet blev engle sat til at vogte livets træ, og det var, før noget menneske var afgået ved døden. Englenes natur er mere ophøjet end menneskenes. Salmisten siger, at mennesket blev skabt “lidt ringere end englene.” Onde ånder, oprindelig skabt syndfrie, var i natur, magt og herlighed jævnbyrdige med de hellige væsener, der nu er Guds sendebud...

Nøgleord:

engle, ånder, dæmoner, vinger, himmel, helvede, spiritualitet, spiritisme, usynlig, overnaturlig, bøn, visioner, mirakler, paranormal, bibel, lucifer, michael, ærkeengel, gabriel, magt, herlighed, religion, teologi, satan

Felsefe tarihinde en çok tartışılan konuların başında kuşkusuz hakikat (truth) gelmektedir. Bu girift konuyla ilgili tartışmaları karakterize eden iki ana problem vardır. Bunların ilki, kendinde bir hakikatin olup olmadığı; ikincisi, bir hakikat varsa bile bu hakikate dair bilginin mümkün olup olmadığıdır. Bu iki ana unsur çerçevesinde tartışmalar oldukça çeşitlenmiştir. Özellikle mantık, matematik ve bilim alanında meydana gelen gelişmeler, bu alanlar arasındaki etkileşimi artırarak tartışmaların boyutlarını genişletmiştir. Söz konusu etkileşimin ortaya çıkardığı geniş problem alanı, konuları ele almayı zorlaştırıyor görünse de meselenin temelde bir “hakikat” meselesi olduğu fark edildiğinde tartışmaların takibi nispeten kolaylaşabilmektedir. Bu takip yapıldığında hakikat tartışmalarında iki ana akımın belirgin etkisi açıkça fark edilir. Bunlar realizm ve natüralizmdir. Bu çalışmada, yalnızca realizm ile eleştirilerine yer verilecektir. Bu bağlamda literatürde oldukça etkili olmuş Paul Benacerraf ile Penelope Maddy’nin görüşleri merkeze alınacaktır. Bu çerçevede, Benacerraf’ın sayı üzerinden realizme yönettiği epistemolojik ve ontolojik itirazlar ile Maddy’nin bu itirazlara verdiği cevaplar ele alınacaktır.

Præsentation 1: Abraham, Isak & Kristus - “Din eneste søn.”
”I tro bragte Abraham Isak som offer, da han blev sat på prøve, og var rede til at ofre sin eneste søn, skønt han havde fået løfterne, …” Hebr 11,17. ” Da Abraham så op, fik han øje på en vædder, som hang fast med hornene i det tætte krat bagved. Abraham gik hen og tog vædderen og bragte den som brændoffer i stedet for sin søn.” 1 Mos 22,13

Præsentation 2: Jesu offer – ”Han trådte i synderes sted.”
”Kristi offer som soning for synd er den store sandhed, omkring hvilken alle andre sandheder klynger sig. For at blive rigtigt forstået og værdsat, må enhver sandhed i Guds Ord, fra Første Mosebog til Åbenbaringen, studeres i det lys, der strømmer fra korset på Golgata. Jeg præsenterer for jer det store, storslåede monument af barmhjertighed og fornyelse, frelse og forløsning – Guds Søn opløftet på korset. Dette skal være grundlaget for enhver tale, der holdes af vores prædikanter.” Ellen White, Gospel Workers (1915), 315.

Præsentation 3: Herrens store dag – ”Da skal jeg kende fuldt ud, ligesom jeg selv er kendt fuldt ud.”
”Retten blev sat. Bøgerne blev åbnet.” Dan 7,10. ”Han svarede mig: ’2300 aftener og morgener! Derefter skal helligdommen få sin ret tilbage.” Dan 8,14. ”Den rette forståelse af tjenesten i den himmelske helligdom er grundlaget for vores tro.” Ellen White, Evangelism (1946), 221.

Temadag, Østervrå Adventistkirke, november 2024.
Vintermøde for de fynske adventistmenigheder, februar 2024.
Temadag, Faxe Adventistkirke, januar 2024.

Opdateret ppt 11. marts, 2025.

The standard semantic definition of consequence with respect to a selected set X of symbols, in terms of truth preservation under replacement (Bolzano) or reinterpretation (Tarski) of symbols outside X, yields a function mapping X to a consequence relation ⇒X. We investigate a function going in the other direction, thus extracting the constants of a given consequence relation, and we show that this function (a) retrieves the usual logical constants from the usual logical consequence relations, and (b) is an inverse to-more precisely, forms a Galois connection with-the Bolzano-Tarski function.

Det tyske kejserrige var en sammenslutning af forskellige stater med en kejser i spidsen. Hver stat udøvede den højeste myndighed inden for sit eget territorium. Den kejserlige rigsdag, som bestod af alle fyrsterne eller de suveræne stater, lovgav for hele den germanske krop. Det var kejserens pligt at ratificere og anvende forsamlingens love, dekreter eller beslutninger. Ligesom Judæa, da kristendommen opstod, var i centrum af den antikke verden, var Tyskland i centrum af kristendommen og så på én gang mod Nederlandene, England, Frankrig, Schweiz, Italien, Ungarn, Bøhmen, Polen, Danmark og hele Norden. I Europas hjerte skulle livets princip udvikles. Dette hjertes slag skulle gennem alle kroppens arterier cirkulere det ædle blod, som skulle give liv til alle dets medlemmer.

It surely goes without saying that Charles Parsons is one of the most important philosophers of mathematics in our generation. Through his many publications and his teaching and lecturing, he has spawned and influenced a large body of historically sensitive work in the philosophy of mathematics and an equally large body of philosophically sensitive work in the history of the philosophy of mathematics. This is Parsons' fourth book. It and From Kant to Husserl (Harvard University Press, 2012) collect together most of his essays on other philosophers, with the exception of those essays reprinted in his earlier collection, Mathematics in Philosophy (Ithaca, New York, Cornell University Press, 1983). As indicated by its title, the essays in the present book are on philosophers whose work appeared primarily, or entirely, in the twentieth century. All but one of the essays have been published elsewhere. It is good to have them collected here, for at least two reasons. First, some of the essays appear in collections that are not readily available, including volumes honoring some of the authors discussed, which are sources that are notoriously hard to acquire. Second, and more important, it is interesting and helpful to see similar ideas and themes developed in different contexts. Four of the essays have new postscripts.

Is second-order logic logic? Famously Quine argued second-order logic wasn't logic but his arguments have been the subject of influential criticisms. In the early sections of this paper, I develop a deeper perspective upon Quine's philosophy of logic by exploring his positive conception of what logic is for and hence what logic is. Seen from this perspective, I argue that many of the criticisms of his case against second-order logic miss their mark. Then, in the later sections, I go beyond Quine to develop a novel case that quantification into polyadic predicate position, understood as requiring quantifiers to range over relations, isn't intelligible.

Truth is a fundamental concept in philosophy, mathematics, and logic. This paper aims to provide a comprehensive examination of truth using first-order logic (FOL). We explore the philosophical underpinnings of truth, formalize its definition, and construct a framework to prove the truth of statements within a logical system. Our analysis begins with a review of the linguistic and philosophical aspects of truth, followed by a detailed formalization using FOL. We then extend this framework to more complex scenarios and demonstrate the robustness of our approach through various examples.

I will argue that the standard formulation of non-factualism in terms of a denial of truth-aptness is consistent with a version of de ationsim. My line of argument assumes the use conception of meaning. is brings out an interesting consequence since mostly the philosophers who endorse the use conception of meaning, e.g. Paul Horwich, hold that de ationism is inconsistent with the strategy of implementing non-factualism in terms of a denial of truth-aptness and thereby urge a reformulation of non-factualism

This book serves as a concise introduction to some main topics in modern formal logic for undergraduates who already have some familiarity with formal languages. There are chapters on sentential and quantificational logic, modal logic, elementary set theory, a brief introduction to the incompleteness theorem, and a modern development of traditional Aristotelian Logic: the "term logic" of Sommers (1982) and Englebretsen (1996). Most of the book provides compact introductions to the syntax and semantics of various familiar formal systems. Here and there, the authors briefly indicate how intuitionist logic diverges from the classical treatments that are more fully explored. The book is appropriate for an undergraduate-level second course in logic that will approach the topic from a philosophical (rather than primarily mathematical) perspective. Philosophical topics (sometimes very briefly) touched upon in the book include: intuitionist logic, substitutional quantification, the nature of logical consequence, deontic logic, the incompleteness theorem, and the interaction of quantification and modality. The book provides an effective jumping-off point for several of these topics. In particular, the presentation of the intuitive idea of the

In this paper we isolate a notion that we call "formalism freeness" from Gödel's 1946 Princeton Bicentennial Lecture, which asks for a transfer of the Turing analysis of computability to the cases of definability and provability. We suggest an implementation of Gödel's idea in the case of definability, via versions of the constructible hierarchy based on fragments of second order logic. We also trace the notion of formalism freeness in the very wide context of developments in mathematical logic in the 20th century. §1. Introduction. That mathematics is practiced in what one might call a formalism free manner has always been the case-and remains the case. Of

to be indeed indestructible by which all this is pervaded. None can work the destruction of this immutable Being.-Maharishi, Gita, verse II.17 Research Program: Enliven wholeness in the foundation of mathematics to align the mathematical enterprise more fully with Natural Law

As Bernays maintained in his famous 1935 article "On Platonism in Mathematics", in mathematical practice objects, functions, relations, properties, structures, etc. are treated as entities that exist independently of our discourse and of our constructions. Bernays maintains that this form of platonism is essentially a manner of speaking, which does not involve a commitment to a strict form of platonism. In this note I defend Bernays' position and propose a more systematic formulation for this form of platonism, combining it with ideas of Frege and of Gödel.

In this thesis, I explore Isaacson’s thesis and Wilkie’s theorem, providing philosophical and formal results on how they relate to each other. At a first approximation, Isaacson’s thesis claims that Peano arithmetic is sound and complete with respect to genuinely arithmetical statements. Using internalist notions familiar from recent work on internal categoricity theorems, I provide a formal definition of genuinely arithmetical statements. As for Wilkie’s theorem, it roughly says that, from an external perspective, Peano arithmetic is minimal, in that it is entailed by all categorical axiomatisations of the natural numbers satisfying a certain syntactic restriction. After expositing Wilkie’s theorem and the relation of its proof to other known techniques, I discuss its relation to Isaacson’s thesis, in particular whether Peano arithmetic is a maximal theory obtained from the categorical characterisation.

To all these institutions we express our warm gratitude. We are also grateful to the members of the PILM scientific committee for their invaluable help in preparing the program and reporting on so many lectures, as well as to the staff of the Poincaré Archives for their help in the preparation of the conference and their expert assistance in various ways. We particularly express our gratitude to Dr. Prosper Doh (Poincaré Archives) who has taken on the job of technical editor for this book and who has realized the index and the camera-ready copy. Finally, the editors are indebted to the editorial board of "Logic, Epistemology, and the Unity of Science" for accepting this volume in their series. They would also like to thank Springer Publishers and, in particular, Floor Oosting.

To all these institutions we express our warm gratitude. We are also grateful to the members of the PILM scientific committee for their invaluable help in preparing the program and reporting on so many lectures, as well as to the staff of the Poincaré Archives for their help in the preparation of the conference and their expert assistance in various ways. We particularly express our gratitude to Dr. Prosper Doh (Poincaré Archives) who has taken on the job of technical editor for this book and who has realized the index and the camera-ready copy. Finally, the editors are indebted to the editorial board of "Logic, Epistemology, and the Unity of Science" for accepting this volume in their series. They would also like to thank Springer Publishers and, in particular, Floor Oosting.

From the Platonistic standpoint, mathematical edifices form an immaterial, unchanging, and eternal world that exists independently of human thought. By extension, "scientific Platonism" says that directly mathematizable physical phenomena-in other terms, the research field of physics-are governed by entities belonging to this objectively existing mathematical world. Platonism is a metaphysical theory. But since metaphysical theories, by definition, are neither provable nor refutable, anti-Platonistic approaches cannot be less metaphysical than Platonism itself. In other words, anti-Platonism is not "more scientifical" than Platonism. All we can do is to compare Platonism and its negations under epistemological criteria such as simplicity, economy of hypotheses, or consistency with regard to their respective consequences. In this paper I intend to show that anti-Platonism claiming in a first approximation (i) that mathematical edifices consist of meaningless signs assembled according to arbitrary rules, and (ii) that the adequacy of mathematical entities and phenomena covered by physics results from idealization of these phenomena, is based as much as Platonism on metaphysical presuppositions. Thereafter, without directly taking position, I try to launch a debate focusing on the following questions: (i) To maintain its coherence, is anti-Platonism not constrained to adopt extremely complex assumptions, difficult to defend, and not always consistent with current realities or practices of scientific knowledge? (ii) Instead of supporting anti-Platonism whatever the cost, in particular by the formulation of implausible hypotheses, would it not be more adequate to accept the idea of a mathematical world existing objectively and governing certain aspects of the material world, just as we note the existence of the material world which could also not exist?

evolutions will tend to drive almost any initial distribution to approximate 1 Y I * when coarse-grained, and also to suggest some probabilistic modifications of the dynamics that would have the same effect. Bohm and Hiley provide searching examinations of rival interpretations, from Bohr's view to the 'many-worlds' theories of Everett and DeWitt, the spontaneous collapse theory of Ghirardi, Rimini and Weber, and the recent decoherence accounts. They also permit themselves some fairly unconstrained speculation about the possible course of future physics. Readers unfamiliar with the theory should bear in mind a few mild caveats. One is that the use of the quantum potential, while sometimes heuristically useful, is superfluous. Bohm's 'guidance equation' is the dynamical heart of the theory, determining the evolution of the particles. The quantum potential allows one, in some circumstances, to interpret the dynamics in terms of Newtonian mechanics, but even the authors admit that for the many-body problem and for spin this expedient fails. The other is that unwarranted speculation occasionally intrudes, such as the suggestion that the electron may have an inner structure "comparable to that of a radio" (p. 37) in order to respond to the wave function. The theory contains an explicit dynamics that demands, and suggests, no such internal complexity. Philosophers of physics should nonetheless consider this book mandatory reading. It is clear, concise, and detailed. Whether one is converted to the ontological interpretation or not, The Undivided Universe sets a standard of rigour, candour, breadth and precision that is unmatched among competing interpretations of quantum mechanics.

Since 1995, when Andrew Wiles finally proved Fermat's Last Theorem, number theory has enjoyed a higher profile in the world's imagination. And it has brought in its train a number of philosophical questions that once were the province of philosophers of mathematics alone, but are now felt by those who, under normal circumstances, might never have worried about such matters; questions such as: what are numbers? How ought we to conceive of them, ontologically, and perhaps even more pressingly, epistemologically? And how do they fit into our total scientific world view? Of course this issue is made vastly more complicated by the way in which mathematics has embraced number systems that go beyond the natural numbers or integers. The negative numbers and zero may once have been regarded with suspicion (and Cardano in the sixteenth century called them fictae), but we have surely adjusted to them. Schoolchildren learn to think of negative numbers as either debts or points on a line on the other side of zero. And it surely helps them to think of them as having some kind of equal status with positive numbers that the zero point can be entered at an arbitrary point on the graph paper: so positive and negative begins to look as arbitrary as 'right and left of here'. But people in large numbers still baulk at imaginary numbers like Ö)1: schoolchildren, philosophers, and even mathematicians.

We simplify and slightly modify the theory of types that Church provided with semantic primitive predicates. Two goals are pursued. The first goal is to present a simple application of Church's approach to paradoxes and to point out some aspects of this ap- proach. The second, perhaps more interesting, goal is to show that when type distinctions are removed some basic Churchian principles need to be restricted and different restrictions correspond to Tarski's and Kripke's different approaches to truth. Finally, we briefly hint at how to move in the direction of Field's recent approach to truth by giving up some specific essential points of the Churchian frame- work.

I argue on the basis of an example, Fourier theory applied to the problem of vibration, that Field's program for nominalizing science is unlikely to succeed generally, since no nominalistic variant will provide us with the kind of physical insight into the phenomena that the standard theory supplies. Consideration of the same example also shows, I argue, that some of the motivation for mathematical fictionalism, particularly the alleged problem of cognitive access, is more apparent than real.

The material of this book is divided into six chapters and a short appendix. Chapter 1 is an introduction. It consists of preliminary discussions of substantive and deflationary accounts of truth, correspondence and disquotation conceptions of truth, truth theory and its structure, and the bearers of truth. The second chapter is a survey of the main issues and tasks of a correspondence conception of truth. The third is a brief discussion of the main motivations for deflationism. The fourth is a detailed analysis of the thesis of disquotationalism and its appropriate formulation. The fifth, which is the longest chapter (about 80 pages), is an elaborate critique of disquotationalism. Chapter 6 is a two-page review of the book's central conclusion: disquotationalism fails because it has too many absurd consequences, and a correspondence conception of truth seems to be the only feasible alternative for someone who wishes to hold on to the basic intuition that 'Snow is white is true' if and only if snow is white. The appendix shows that the liar paradox is formalizable in a language whose truth predicate is defined disquotationally, and hence the liar and liar-like phenomena present similar challenges to both conceptions of truth, correspondence and disquotation. David's book is really a critical essay on disquotationalism. It is very likely the most elaborate study of disquotationalism available in the philosophical literature to date. Out of the 188 pages that make the philosophical text of the book, 155 pages are almost entirely devoted to discussing disquotationalism. The one chapter allocated to the exposition of correspondence accounts of truth is included mostly for the sake of motivating disquotationalism. The latter, being a radically deflationist account of truth, is best seen when contrasted with the theory that it seeks to deflate. 2 Theories of truth are of two kinds, substantive and deflationary. Proponents of the first kind believe that the concept of truth has a deep nature that requires an ideologically sophisticated and ontologically rich philosophical account. Advocates of the

We are physical objects in a physical world; our bodies, collections of molecules, move, and among the myriad products of these movements are marks and sounds. These physical phenomena have physical explanations, forthcoming, in principle, from the physical sciences, from physics, at the most fundamental level, to the neuro-biological sciences at more specialized levels. So much for the unassailable. At the same time, we are apt, pretheoretically, to suppose that some of these marks and sounds have semantical properties, and that those who produce them have psychological states, notably beliefs, desires, and intentions. The sequence of marks 'Mitterand defeated Giscard', for example, is a sentence of a language, it has meaning, viz., that Mitterand defeated Giscard, it is true, it contains names that refer to people, and a predicate that is true of pairs of things. One producing this sequence of marks is not unlikely to believe that Mitterand defeated Giscard, and to intend, in producing those marks, to instill the same belief in another. The subject matter of the philosophy of language, if it has one, is the nature of the semantical properties of linguistic items. But no complete account of those properties will leave unanswered these two questions: (1) How is the semantic related to the psychological? (2) How are the semantic and the psychological related to the physical? I believe, for familiar reasons, later briefly to be touched on, that (2) is the urgent question, in this sense: that we should not be prepared to maintain that there are semantic or psychological facts unless we are prepared to maintain that such facts are completely determined by, are nothing over and above, physical facts. I am also inclined towards a reductionist response to *I am grateful to Brian Loar for running conversations on these topics.

A short description of several views on the nature of math, including platonism, nominalism, and fictionalism, and my claim that platonism matches mathematics the best (math is discovered), by discussing thought, science, and the complexity and practicality of mathematics.

[See https://journals.publishing.umich.edu/phimp/article/id/1872/ for the final version]. It's not correspondence, and not deflation. His view is largely Tarski’s, and at its core is what I shall call the generalising function of truth. After stating Quine’s actual doctrine non-figuratively and with care, I turn to a most interesting piece by Lars Bergström which examines various related Quinean philosophical projects and commitments, with the implication that Quine must accept a richer conception of truth than the comparatively austere Tarskian conception. I find against Bergström. But the reasoning serves to illuminate several interconnected but wide-ranging aspects of Quine’s philosophy, including such seemingly disparate items as the place it accords to logic, the place of value, what is meant by the ‘Pursuit of Truth’, the take on realism, and the status of metaphysics. To see the point is to appreciate how thoroughgoing and powerful the notion of truth is, despite its being, 'in itself', only a device of sentential generalization. This dovetails with Quine’s famous naturalism, a connection which I bring out at the end.

Mathematical Platonism is the view that mathematicians study objects that exist in a nonphysical realm independently of our awareness of them. The process of mathematical discovery provides a compelling argument for this view. I consider the significance for mathematical Platonism of the fact that mathematicians sometimes discover that an item they are studying is inconsistent.

This paper discusses the consequences of the latest PUEBI EYD V regulations for scientific ontological theorization through analyzing the semantical metaphysical commitment it reflects when we write formal mathematical statements using purely mathematical symbol (e.g., “there are 22 aardvarks”). This paper shows that PUEBI EYD V commits to mathematical Platonism metaphysically. This commitment brings harm to observable entities ontological nature in scientific theorization as shown in nominalism projects of philosophy of mathematics. Scientific theories - and even mathematical theories - should always reject the existence of independent objects for there exists only structures (as truth-value). Authors use nominalism stance as a methodology to reject the metaphysical commitment of PUEBI and defend the formal usage of writing mathematical statement without number symbols (“there are twenty-two aardvarks”, etc.).

PREFACE The axiomatic method counts two thousand and three hundred years circa. Suppes [61] has proposed the category of Euclidean-Archimedean tradition to refer to the axiomatic theories that have been developed before the inven-tion/discovery of the non-Euclidean geometries. Among these theories the first axiomatic system that we know is Euclid's Elements [16], a mathematical tractate consisting of thirteen books in which three centuries of Greek mathematical knowledge were given an order and were presented as a unified theory. 1 Euclid produced another axiomatic theory, the Optics [15]. This represents a theory of vision in Euclidean perspective rather than a tractate on physical optics. It is interesting that Archimedes's Treatise [12], probably the first book on mathematical physics, is an axiomatic theory. The axiomatic method in the Euclidean-Aristotelian tradition was transmitted during the medieval age and scholarship in history of science has established the use of...

Den ondes mørke omslutter dem, som forsømmer at bede. Fjendens hviskede fristelser får dem til at synde; og det er alt sammen, fordi de ikke gør brug af det privilegium, som Gud har givet dem i det guddommelige redskab — bønnen. Hvorfor er Guds sønner og døtre modstræbende med at bede, når bøn er nøglen i troens hånd til at åbne himlens lager, hvor almagtens endeløse ressourcer ligger gemt? Uden uophørlig bøn og flittig årvågenhed er vi i fare for at blive ligegyldige og vige fra den rette vej. Modstanderen forsøger til stadighed at blokere vejen til nådens trone, så vi ikke i oprigtig bønfaldelse og tro opnår nåde og magt til at modstå fristelse.. Bønnen har en vældig magt. Vor store modstander søger bestandig at holde den ængstede sjæl borte fra Gud. En henvendelse til Himmelen fra den ringeste hellige indgyder Satan mere frygt end regeringsafgørelser eller kongelige befalinger.…

Eugeniusz Grodzinski argues '... that the Fregean theory of truth (if one can say that such a theory exists) is detrimental to science because it is incoherent and full of contradictions' (Grodzinski 1993: 352). This article argues, among other things, that (1) Frege does not have a theory of truth and (2) that his remarks on truth are neither incoherent nor contradictory. Indeed, it will be established that Grodzinski's attack against Frege is grossly misplaced since it fails to examine Frege's deflationist pronouncements on the issue of truth. As a matter of fact, Frege's deflationist approach to truth precludes having a theory of truth. The case against Grodzinski proceeds as follows: (1) a presentation of each of Grodzinski's criticisms of Frege; and (2) an explanation of why the criticism in question is misplaced. Frege's own deflationist stance on truth will be inserted within the discussions of Grodzinski's attack against Frege. Grodzinski compliments Frege for having assigned truth a significant role in science, and particularly for defending the idea that the goal of logic is to discover the laws of truth, thus making logic the science of truth. Grodzinski states that the preceding claim represents e ... a rational seed in Frege's theory of truth' (Grodzinski 1993: 352). But no sooner has he acknowledged this positive element in Frege's position than he quickly turns and castigates him without adequately developing or considering the consequence of Frege's claim that logic is the science of truth. One of Frege's main theses, according to Grodzinski, is the claim that the addition of the expression 'it is true' to a sentence does not add new information to the sentence. For example, Frege would consider the following sentences identical: (1) Bill Clinton is the president of the United States. (2) It is true that Bill Clinton is the president of the United States. Frege would state that the addition of the expression £ it is true' in (2) is redundant; Grodzinski, on the other hand, would strongly disagree.