We prove a representation theorem for preference relations over countably infinite lotteries that satisfy a generalized form of the Independence axiom, without assuming Continuity. The representing space consists of lexicographically ordered transfinite sequences of bounded real numbers. This result is generalized to preference orders on abstract superconvex spaces.

We propose an alternative approach to probability theory closely related to the framework of numerosity theory: non-Archimedean probability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and only the empty set gets assigned probability zero (in other words: the probability functions are regular). We use a non-Archimedean field as the range of the probability function. As a result, the property of countable additivity in Kolmogorov’s axiomatization of probability is (...) replaced by a different type of infinite additivity. (shrink)

In this paper the non-Archimedean multiple-validity is proposed for basic fuzzy logic BL∀∞ that is built as an ω-order extension of the logic BL∀. Probabilities are defined on the class of fuzzy subsets and, as a result, for the first time the non-Archimedean valued probability logic is constructed on the base of BL∀∞.

Non-Archimedean population axiologies – also known as lexical views – claim (i) that a sufficient number of lives at a very high positive welfare level would be better than any number of lives at a very low positive welfare level and/or (ii) that a sufficient number of lives at a very low negative welfare level would be worse than any number of lives at a very high negative welfare level. Such axiologies are popular because they can avoid the (Negative) (...) Repugnant Conclusion and satisfy the adequacy conditions given in the central impossibility result in population axiology due to Gustaf Arrhenius. I provide a novel argument against them which appeals to the way that good and bad lives can intuitively outweigh one other. (shrink)

We relate Popper functions to regular and perfectly additive such non-Archimedean probability functions by means of a representation theorem: every such non-Archimedean probability function is infinitesimally close to some Popper function, and vice versa. We also show that regular and perfectly additive non-Archimedean probability functions can be given a lexicographic representation. Thus Popper functions, a specific kind of non-Archimedean probability functions, and lexicographic probability functions triangulate to the same place: they are in a good sense interchangeable.

We study the impact of a kind of non‐archimedean stratifications (t‐stratifications) on tangent cones of definable sets in real closed fields. We prove that such stratifications induce stratifications of the same nature on the tangent cone of a definable set at a fixed point. As a consequence, the archimedean counterpart of a t‐stratification is shown to induce Whitney stratifications on the tangent cones of a semi‐algebraic set. Extensions of these results are proposed for real closed fields with further (...) structure. (shrink)

Application of recently developed non-Archimedean algebra to a flat and finite universe of total mass M 0 and radius R 0 is described. In this universe, mass m of a body and distance R between two points are bounded from above, i.e., 0≤m≤M 0, 0≤R≤R 0. The universe is characterized by an event horizon at R 0 (there is nothing beyond it, not even space). The radial distance metric is compressed toward horizon, which is shown to cause the phenomenon (...) of red shift. The corresponding modified Minkowski's metric and Lorentz transforms are obtained. Applications to Newtonian gravity shows a weakening at large scales (R→R 0) and a regular behavior as R→0. (shrink)

In this article, we introduce a hierarchy on the class of non-archimedean Polish groups that admit a compatible complete left-invariant metric. We denote this hierarchy by $\alpha $ -CLI and L- $\alpha $ -CLI where $\alpha $ is a countable ordinal. We establish three results: (1) G is $0$ -CLI iff $G=\{1_G\}$ ; (2) G is $1$ -CLI iff G admits a compatible complete two-sided invariant metric; and (3) G is L- $\alpha $ -CLI iff G is locally $\alpha (...) $ -CLI, i.e., G contains an open subgroup that is $\alpha $ -CLI. Subsequently, we show this hierarchy is proper by constructing non-archimedean CLI Polish groups $G_\alpha $ and $H_\alpha $ for $\alpha <\omega _1$, such that: (1) $H_\alpha $ is $\alpha $ -CLI but not L- $\beta $ -CLI for $\beta <\alpha $ ; and (2) $G_\alpha $ is $(\alpha +1)$ -CLI but not L- $\alpha $ -CLI. (shrink)

Many philosophers have claimed that extensive or additive measurement is incompatible with the existence of "higher values", any amount of which is better than any amount of some other value. In this paper, it is shown that higher values can be incorporated in a non-standard model of extensive measurement, with values represented by sets of ordered pairs of real numbers, rather than by single reals. The suggested model is mathematically fairly simple, and it applies to structures including negative as well (...) as positive values. (shrink)

In this contribution, we focus on probabilistic problems with a denumerably or non-denumerably infinite number of possible outcomes. Kolmogorov (1933) provided an axiomatic basis for probability theory, presented as a part of measure theory, which is a branch of standard analysis or calculus. Since standard analysis does not allow for non-Archimedean quantities (i.e. infinitesimals), we may call Kolmogorov's approach "Archimedean probability theory". We show that allowing non-Archimedean probability values may have considerable epistemological advantages in the infinite case. (...) The current paper focuses on the motivation for our new axiomatization. (shrink)

We develop a dimension theory for D-semianalytic sets over an arbitrary non-Archimedean complete field. Our main results are the equivalence of several notions of dimension and a theorem on additivity of dimensions of projections and fibers in characteristic 0. We also prove a parameterized version of normalization for D-semianalytic sets.

We study the notion of weak amalgamation in the context of diagonal conjugacy classes. Generalizing results of Kechris and Rosendal, we prove that for every countable structure M, Polish group G of permutations of M, and \, G has a comeager n-diagonal conjugacy class iff the family of all n-tuples of G-extendable bijections between finitely generated substructures of M, has the joint embedding property and the weak amalgamation property. We characterize limits of weak Fraïssé classes that are not homogenizable. Finally, (...) we investigate 1- and 2-diagonal conjugacy classes in groups of ball-preserving bijections of certain ordered ultrametric spaces. (shrink)

After generalizing the Archimedean property of real numbers in such a way as to make it adaptable to non-numeric structures, we demonstrate that the real numbers cannot be used to accurately measure non-Archimedean structures. We argue that, since an agent with Artificial General Intelligence (AGI) should have no problem engaging in tasks that inherently involve non-Archimedean rewards, and since traditional reinforcement learning rewards are real numbers, therefore traditional reinforcement learning probably will not lead to AGI. We indicate (...) two possible ways traditional reinforcement learning could be altered to remove this roadblock. (shrink)

One notable line of argument for epistemic relativism appeals to considerations to do with non-neutrality: in certain dialectical contexts—take for instance the famous dispute between Galileo and Cardinal Bellarmine concerning geocentrism—it seems as though a lack of suitably neutral epistemic standards that either side could appeal to in order to resolve their first-order dispute is itself—as Rorty influentially thought—evidence for epistemic relativism. In this essay, my aim is first to present a more charitable reformulation of this line of reasoning, one (...) that is framed not merely in terms of the availability of epistemic norms that are suitably neutral between interlocutors, but in terms of the availability of what I call Archimedean metanorms. Once this more charitable line of argument is developed, I show how, even though it avoids problems that face ‘non-neutrality’ versions of the argument, it nonetheless runs into various other problems that appear ultimately intractable and, further, that the strategy in question gives us no decisive reason to draw the relativist’s conclusion rather than the Pyrrhonian sceptic’s. (shrink)

Metaethics is traditionally understood as a non-moral discipline that examines moral judgements from a standpoint outside of ethics. This orthodox understanding has recently come under pressure from anti-Archimedeans, such as Ronald Dworkin and Matthew Kramer, who proclaim that rather than assessing morality from an external perspective, metaethical theses are themselves substantive moral claims. In this paper, I scrutinise this anti-Archimedean challenge as applied to the metaethical position of expressivism. More precisely, I examine the claim that expressivists do not avoid (...) moral commitments when accounting for moral thought, but instead presuppose them; they do not look at ethics from the outside, but operate from within ethics. This paper defends the non-moral status of expressivism against anti-Archimedeanism by rejecting a new anti-Archimedean challenge which, on the basis of Hume’s Law, aims to exploit expressivist explanations of supervenience in order to show that expressivism is a substantive moral position. (shrink)

The quantum transition probability assignment is an equiareal transformation from the annulus of symplectic spinorial amplitudes to the disk of complex state vectors, which makes it equivalent to the equiareal projection of Archimedes. The latter corresponds to a symplectic synchronization method, which applies to the quantum phase space in view of Weyl’s quantization approach involving an Abelian group of unitary ray rotations. We show that Archimedes’ method of synchronization, in terms of a measure-preserving transformation to an equiareal disk, imposes the (...) integrality of the quantum of action, and requires the extension of the classical moment map from the real line to the circle. Additionally, the same synchronization method is encoded in the structure of the Heisenberg group, viewed as a principal bundle with a connection, whose curvature and anholonomy is expressed in terms of area bounding loops in relation to the underlying Abelian shadow on a symplectic plane. In this manner, we show that the geometric phase pertains to the minimal synchronized area \ of the 2-d symplectic Abelian shadow of the symplectic ball, modulo \. The integrality condition naturally leads to the consideration of modular commutative observables pertaining to the role of the discrete Heisenberg group. We prove that the structural transition from non-commutativity to modular commutativity in accordance to Weyl’s group-theoretic commutation relations takes place via universal factorization through the discrete Heisenberg group. In this way, we derive a homology-theoretic formulation of the synchronization method in terms of the area-bounding cells of the modular lattice \ in relation to any Abelian symplectic shadow. Thus, we finally obtain the physical interpretation of the analytic representation of quantum states as theta functions corresponding to the sections of a complex line bundle with an integral symplectic structure. (shrink)

We show that a solution to Hilbert's Tenth Problem in the rings of algebraic integers and bigger subrings of number fields where it is currently not known, is equivalent to a problem of bounding archimedean valuations over non-real number fields.

This paper is divided into two parts. In the first one I distinguish between weak and strong Anti-Archimedeanisms, the latter being the view that metaethics, just as any other discipline attempting to work out a second-order conceptual, metaphysical non-committed discourse about the first-order discourse composing normative practices, is conceptually impossible or otherwise incoherent. I deal in particular with Ronald Dworkin’s famous exposition of the view. I argue that strong Anti-Archimedeanism constitutes an untenable philosophical stance, therefore making logical space for the (...) practice of a discipline such as metaethics—conceived as ethically neutral. This makes space, concurrently, for neutral conceptual jurisprudence. In the second part of the article, I attempt to show two things. On the one hand, that Dworkin’s widely discussed ‘challenge of disagreements’ to legal positivism is founded upon strong Anti-Archimedeanism. On the other hand, that having rejected strong Anti-Archimedeanism we should consequently reject the challenge as a serious challenge to positivism. This move, of course, does not thereby imply that accounting for legal disagreements is not an important jurisprudential task. But it marks—contra Dworkin—that there is no principled or a priori impossibility of doing so within a positivist framework. (shrink)

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to (...) teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms. (shrink)

We study the representation of sets of desirable gambles by sets of probability mass functions. Sets of desirable gambles are a very general uncertainty model, that may be non-Archimedean, and therefore not representable by a set of probability mass functions. Recently, Cozman (2018) has shown that imposing the additional requirement of even convexity on sets of desirable gambles guarantees that they are representable by a set of probability mass functions. Already more that 20 years earlier, Seidenfeld et al. (1995) (...) gave an axiomatisation of binary preferences—on horse lotteries, rather than on gambles—that also leads to a unique representation in terms of sets of probability mass functions. To reach this goal, they use two devices, which we will call ‘SSK–Archimedeanity’ and ‘SSK–extension’. In this paper, we will make the arguments of Seidenfeld et al. (1995) explicit in the language of gambles, and show how their ideas imply even convexity and allow for conservative reasoning with evenly convex sets of desirable gambles, by deriving an equivalence between the SSK–Archimedean natural extension, the SSK–extension, and the evenly convex natural extension. (shrink)

This paper presents an analysis of the sorites paradox for collective nouns and gradable adjectives within the framework of classical logic. The paradox is explained by distinguishing between qualitative and quantitative representations. This distinction is formally represented by the use of a different mathematical model for each type of representation. Quantitative representations induce Archimedean models, but qualitative representations induce non-Archimedean models. By using a non-standard model of \ called \, which contains infinite and infinitesimal numbers, the two paradoxes (...) are shown to have distinct structures. The sorites paradox for collective nouns arises from the use of infinite numbers, whereas the sorites paradox for gradable adjectives arises from the use of infinitesimal numbers. Each paradox can be traced to a different source of vagueness. The sorites paradox for collective nouns is caused by \, and the sorites paradox for gradable adjectives is caused by \ \. If correct, this analysis implies that infinite and infinitesimal numbers are cognitively real, and that they play a role in the semantic interpretation of natural language. (shrink)

Buddhist concepts of personal life and responsibilities.

In the paper we build up the ontology of Leśniewski’s type for formalizing synthetic propositions. We claim that for these propositions an unconventional square of opposition holds, where a, i are contrary, a, o (resp. e, i) are contradictory, e, o are subcontrary, a, e (resp. i, o) are said to stand in the subalternation. Further, we construct a non-Archimedean extension of Boolean algebra and show that in this algebra just two squares of opposition are formalized: conventional and the (...) square that we invented. As a result, we can claim that there are only two basic squares of opposition. All basic constructions of the paper (the new square of opposition, the formalization of synthetic propositions within ontology of Leśniewski’s type, the non-Archimedean explanation of square of opposition) are introduced for the first time. (shrink)

I introduce a mathematical account of expectation based on a qualitative criterion of coherence for qualitative comparisons between gambles (or random quantities). The qualitative comparisons may be interpreted as an agent’s comparative preference judgments over options or more directly as an agent’s comparative expectation judgments over random quantities. The criterion of coherence is reminiscent of de Finetti’s quantitative criterion of coherence for betting, yet it does not impose an Archimedean condition on an agent’s comparative judgments, it does not require (...) the binary relation reflecting an agent’s comparative judgments to be reflexive, complete or even transitive, and it applies to an absolutely arbitrary collection of gambles, free of structural conditions (e.g., closure, measurability, etc.). Moreover, unlike de Finetti’s criterion of coherence, the qualitative criterion respects the principle of weak dominance, a standard of rational decision making that obliges an agent to reject a gamble that is possibly worse and certainly no better than another gamble available for choice. Despite these weak assumptions, I establish a qualitative analogue of de Finetti’s Fundamental Theorem of Prevision, from which it follows that any coherent system of comparative expectations can be extended to a weakly ordered coherent system of comparative expectations over any collection of gambles containing the initial set of gambles of interest. The extended weakly ordered coherent system of comparative expectations satisfies familiar additivity and scale invariance postulates (i.e., independence) when the extended collection forms a linear space. In the course of these developments, I recast de Finetti’s quantitative account of coherent prevision in the qualitative framework adopted in this article. I show that comparative previsions satisfy qualitative analogues of de Finetti’s famous bookmaking theorem and his Fundamental Theorem of Prevision.The results of this article complement those of another article (Pedersen, Strictly coherent preferences, no holds barred, Manuscript, 2013). I explain how those results entail that any coherent weakly ordered system of comparative expectations over a unital linear space can be represented by an expectation function taking values in a (possibly non-Archimedean) totally ordered field extension of the system of real numbers. The ordered field extension consists of formal power series in a single infinitesimal, a natural and economical representation that provides a relief map tracing numerical non-Archimedean features to qualitative non-Archimedean features. (shrink)

Recently, Cluckers et al. developed a new axiomatic framework for tame non-Archimedean geometry, called Hensel minimality. It was extended to mixed characteristic together with the author. Hensel minimality aims to mimic o-minimality in both strong consequences and wide applicability. In this paper, we continue the study of Hensel minimality, in particular focusing on [Formula: see text]-h-minimality and [Formula: see text]-h-minimality, for [Formula: see text] a positive integer. Our main results include an analytic criterion for [Formula: see text]-h-minimality, preservation of (...) [Formula: see text]-h-minimality under coarsening of the valuation and [Formula: see text]-dimensional geometry. (shrink)

The philosophy of mathematics has been accused of paying insufficient attention to mathematical practice: one way to cope with the problem, the one we will follow in this paper on extensive magnitudes, is to combine the `history of ideas' and the `philosophy of models' in a logical and epistemological perspective. The history of ideas allows the reconstruction of the theory of extensive magnitudes as a theory of ordered algebraic structures; the philosophy of models allows an investigation into the way epistemology (...) might affect relevant mathematical notions. The article takes two historical examples as a starting point for the investigation of the role of numerical models in the construction of a system of non-Archimedean magnitudes. A brief exposition of the theories developed by Giuseppe Veronese and by Rodolfo Bettazzi at the end of the 19th century will throw new light on the role played by magnitudes and numbers in the development of the concept of a non-Archimedean order. Different ways of introducing non-Archimedean models will be compared and the influence of epistemological models will be evaluated. Particular attention will be devoted to the comparison between the models that oriented Veronese's and Bettazzi's works and the mathematical theories they developed, but also to the analysis of the way epistemological beliefs affected the concepts of continuity and measurement. (shrink)

QUALITY AND QUANTITY are fairly generally regarded as complementary terms, although the profound reason for their comple- mentarism is often far from being understood, this reason lying in the 'polar' correspondence referred to toward ...

A study of the constitution and development of the human being from the metaphysical point of view, with special reference to Vedantic doctrine.

The Reign of Quantity gives a concise but comprehensive view of the present state of affairs in the world, as it appears from the point of view of the 'ancient wisdom', formerly common both to the East and to the West, but now almost entirely lost sight of. The author indicates with his fabled clarity and directness the precise nature of the modern deviation, and devotes special attention to the development of modern philosophy and science, and to the part played (...) by them, with their accompanying notions of progress and evolution, in the formation of the industrial and democratic society which we now regard as 'normal'. Guénon sees history as a descent from Form (or Quality) toward Matter (or Quantity); but after the Reign of Quantity-modern materialism and the 'rise of the masses'-Guénon predicts a reign of 'inverted quality' just before the end of the age: the triumph of the 'counter-initiation', the kingdom of Antichrist. This text is considered the magnum opus among Guénon's texts of civilizational criticism, as is Symbols of Sacred Science among his studies on symbols and cosmology, and Man and His Becoming according to the Vedanta among his more purely metaphysical works. (shrink)

Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general.

The concluding chapter lays down the essential conditions for any genuine understanding between East and West, which can only come through the work of those who have attained, at least in some degree, to the realization of 'wisdom uncreate' ...

Description: Contents: Preface 1. General Remarks on the Vedanta 2. Fundamental Distinction Between The Self and the Ego 3. The Vital Centre of the Human Being, Seat of Brahma 4. Purusha and Prakriti 5. Purusha Unaffected by Individual Modifications 6. The Degrees of Individual Manifestation 7. Buddhi or the Higher Intellect 8. Manas or the Inward Sense : The Ten External Faculties of Sensation and Action 9. The Envelopes of the Self ; The Five Vayus or Vital Functions 10. The (...) Essential Unity and Identity of the Self in all the States of the Being 11. The Different Conditions of Atma in the Human Being 12. The Waking State or the Condition of Vaishwanara 13. The Dream State or the Condition of Taijasa 14. The State of Deep Sleep or the Condition of Prajna 15. The Unconditioned State of Atma 16. The Symbolical Representation of Atma and its Conditions by the Sacred Monosyllable Om 17. The Posthumous Evolution of the Human Being 18. The Reabsorption of the Individual Faculties 19. Differences in the Posthumous Conditions According to the Degrees of Knowledge 20. The Coronal Artery and the Solar Ray 21. The Divine Journey of the Being on the Path of Liberation 22. Final Deliverance 23. Videha-mukti and Jivana-mukti 24. The Spiritual State of the Yogi : The Supreme Identity. (shrink)