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)

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)

Abstract: We sometimes say our moral claims are "objectively true," or are "right, even if nobody believes it." These additional claims are often taken to be staking out metaethical positions, representative of a certain kind of theorizing about morality that "steps outside" the practice in order to comment on its status. Ronald Dworkin has argued that skepticism about these claims so understood is not tenable because it is impossible to step outside such practices. I show that externally skeptical metaethical theory (...) can withstand his attacks, thereby defending the possibility of this kind of metatheoretical method and showing that the additional objectivity claims still make sense as external claims. Four interpretations of the additional objectivity claims can still be understood externally: as secondary properties, as arguing for some form of causal correspondence, as explaining error, and under Blackburn's expressivism. In the end, Dworkin's argument can be turned against itself. (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)

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.

This paper investigates a quasi-variety of representable integral commutative residuated lattices axiomatized by the quasi-identity resulting from the well-known Wajsberg identity → q ≤ → p if it is written as a quasi-identity, i. e., → q ≈ 1 ⇒ → p ≈ 1. We prove that this quasi-identity is strictly weaker than the corresponding identity. On the other hand, we show that the resulting quasi-variety is in fact a variety and provide an axiomatization. The obtained results shed some light (...) on the structure of Archimedean integral commutative residuated chains. Further, they can be applied to various subvarieties of MTL-algebras, for instance we answer negatively Hájek's question asking whether the variety of ΠMTL-algebras is generated by its Archimedean members. (shrink)

Biton’s Construction of Machines of War and Catapults describes six machines by five engineers or inventors; the fourth machine is a rolling elevatable scaling ladder, named sambukē, designed by one Damis of Kolophōn. The first sambukē was invented by Herakleides of Taras, in 214 BCE, for the Roman siege of Syracuse. Biton is often dismissed as incomprehensible or preposterous. I here argue that the account of Damis’ device is largely coherent and shows that Biton understood that Damis had built a (...) machine that embodied Archimedean principles. The machine embodies three such principles: (1) the proportionate balancing of the torques on a lever (from Plane Equilibria, an early work); (2) the concept of specific gravity or density (from Floating Bodies, a late work); and (3) the κοχλίας, i.e., a worm drive (invented ca 240 BCE), with the toothed wheel functioning as the horizontal axis of rotation of the elevated ladder. Moreover, the stone-thrower of Isidoros of Abydos (the second machine in Biton) also embodies the κοχλίας. (shrink)

The Archimedean Axiom is often held to be an intuitively obvious truth about the geometry of physical space. After a general discussion of the varieties of geometrical intuition that have been proposed, I single out one variety which we can plausibly be held to have and then argue that it does not underwrite the intuitive obviousness of the Archimedean Axiom. Generalizing that result, I conclude that the Axiom is not intuitively obvious.

According to the orthodox account of meaning and translation in the literature, meaning is a property of expressions of a language, and translation is a matching of synonymous expressions across languages. This linguistic account of translation gives rise to well-known skeptical conclusions about translation, objectivity, meaning and truth, but it does not conform to our best translational practices. In contrast, I argue for a textual account of meaning based on the concept of a TEXT-TYPE that does conform to our best (...) translational practices. With their semantic function in view, text-types are Archimedean points for their respective disciplines. The text-type of philosophy is no exception. Culture-transcendent conceptual analysis can proceed on firm footing without having to deny the reality of radical cultural and linguistic difference by treating components of text-types as the concepts to be analyzed. Analyses of central philosophical concepts are provided as a means of adjudicating philosophical controversy. (shrink)

There is no authoritative biography of Archimedes, but there are moments from his life which, apocryphal or not, have become the stuff of legend. These include accounts of Archimedes running naked through the streets after realizing that his body displaces water in the bath , how he sat musing over diagrams in the sand as sword-bearing Romans descended upon him during a siege of Syracuse, and of course, his mechanically-informed claim that a firm resting place is all he would need (...) to dislodge the world from its axis. This is the birth of the Archimedean point, which latter-day interpretations craft into a figure of thought that works according to the mechanical principle it describes: effortlessly, it .. (shrink)

Generalizations of Boolean elements of a BL-algebra L are studied. By utilizing the MV-center MV(L) of L, it is reproved that an element x L is Boolean iff x x * = 1. L is called semi-Boolean if for all x L, x * is Boolean. An MV-algebra L is semi-Boolean iff L is a Boolean algebra. A BL-algebra L is semi-Boolean iff L is an SBL-algebra. A BL-algebra L is called hyper-Archimedean if for all x L, xn is Boolean (...) for some finite n 1. It is proved that hyper-Archimedean BL-algebras are MV-algebras. The study has application in mathematical fuzzy logics whose Lindenbaum algebras are MV-algebras or BL-algebras. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

This paper aims to provide an updated synthesis on the works of Archimedes and the fundamental impact these have had on subsequent mathematical practice. The influence his mathematical processes have had on modern mathematics and how these have helped develop the field is discussed in historical perspective. Some of the recent investigations into the Archimedes Palimpsest are discussed and synthesized, namely, how they alter our understanding of some of his earlier works, and how Archimedean principles are seen to have laid (...) the foundations of possible new branches of mathematics. (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)

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)

Ronald Dworkin famously argued that many putatively nonmoral metaethical theories can only be understood as being internal to the moral domain. If correct, this position, referred to as anti-archimedeanism, has profound implications for the methodology of metaethics. This is particularly true for skeptical metaethical theories. This article defends a version of anti-archimedeanism that is true to the spirit rather than the letter of Dworkin's original thesis from several recent objections. First, it addresses Kenneth Ehrenberg's recent attempt to demonstrate (...) how certain metaethical theories can be understood in a morally neutral manner. It then discusses Charles Pigden's claim that Dworkin begs the question against error theorists and nihilists by assuming a conceptual space that error theorists and nihilists would reject. It concludes that the anti-archimedean methodology originally proposed by Dworkin is defensible, and can be used to support a robust form of moral realism. (shrink)

In A Theory of Justice, John Rawls defines a hypothetical contract situation and argues rational people will agree on reflection it is fair to contractors. He solves the rational choice problem it poses by deriving two lexically-ordered principles of justice and suggests the derivation justifies the principles. Its soundness aside, just what justificatory force does such a derivation have?On one view, there is no justificatory force because the contract is rigged specifically to yield principles which match our pre-contract moral judgments. (...) Rawls provides ammunition for this claim: “By going back and forth, sometimes altering the conditions of the contractual circumstances, at others withdrawing our judgments [about what is just] and conforming them to principle, I assume that eventually we shall find a description of the initial situation that both expresses reasonable conditions and yields principles which match our considered judgments duly pruned and adjusted.”. (shrink)

Axiomatizations of measurement systems usually require an axiom--called an Archimedean axiom--that allows quantities to be compared. This type of axiom has a different form from the other measurement axioms, and cannot--except in the most trivial cases--be empirically verified. In this paper, representation theorems for extensive measurement structures without Archimedean axioms are given. Such structures are represented in measurement spaces that are generalizations of the real number system. Furthermore, a precise description of "Archimedean axioms" is given and it is shown that (...) in all interesting cases "Archimedean axioms" are independent of other measurement axioms. (shrink)

Johann Gottlieb Fichte’s first treatise, “Concerning the Concept of the ‘Wissenschaftslehre’” , is a text built in no small measure upon metaphors. Unlike the increasingly abstract lectures on the theory of knowledge, the 1794 treatise is organized around two fields of imagery: construction and cosmology. These two fields overlap in that the Earth doubles as a planet in the universe and the foundation upon which people raise buildings. Both metaphorical fields come together most prominently when Fichte draws on the anecdote (...) about Archimedes’ promise to displace the Earth if could he just find a long enough lever and a firm point .. (shrink)

In this contribution I discuss Hannah Arendt's philosophy of culture in three rounds. First I give an account of my view on Hannah Arendt's main work The Human Condition. In this frame of reference I distance myself from the importance attached to Hannah Arendt as a political philosopher and hold a warm plea for her as a philosopher of culture (I and II). Second I pay attention to her view on science and technology in their cultural meaning, expressed in the (...) last chapter of The Human Condition. This part consists in a summary of her thoughts as I read them (III, IV, and V). After these two rounds I make some critical remarks on Hannah Arendt's interpretation of science and technology. The viewpoint of ?eccentricity? will be discussed as a frame of reference for her philosophy of culture (VI). (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.

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)

This article consists of an original translation of Ernst Bloch’s 1923 review of Lukács History and Class Consciousness, preceded by a translator’s introduction contextualising Bloch’s review and interpreting what it tells us about the intellectual and personal relationship between Bloch and Lukács. I argue that Bloch’s review highlights some of the key differences and points of intersection between their thinking. Written when their personal relationship had already soured for both political and intellectual reasons, Bloch’s review makes clear his ongoing commitment (...) to a form of utopianism Lukács had long abandoned. Unlike Lukács, who in History and Class Consciousness argued that the dialectical method could only be applied to the social realm, Bloch followed Engels in developing a dialectics of nature. However, even if Bloch was less willing to become involved in party politics than his erstwhile friend, as the review reveals, both men ultimately emphasised the present moment as the privileged locus of urgent political action. (shrink)

“The Archimedean point for moral life” discussed in this article refers to the starting point of one's moral reasoning and what ultimately makes moral life possible. The article intends to show that Mengzi's doctrine of the Four Beginnings may throw some light on our search for such an Archimedean point. More specifically, it argues for the following: Mengzi's doctrine of the Four Beginnings actually takes moral sentiments as the Archimedean point for moral life; Mengzi's view of the starting point of (...) moral reasoning and the ultimate ground for moral life not only can be empirically supported to a great extent, but also logically plausible. (shrink)

Face perception and emotion categorization are widely investigated under laboratory conditions that are devoid of real social interaction. Using mobile eye-tracking glasses in a standardized diagnostic setting while applying the Autism Diagnostic Observation Schedule, we had the opportunity to record gaze behavior of children and adolescents with and without Autism Spectrum Conditions during social interaction. The objective was to investigate differences in eye-gaze behavior between three groups of children and adolescents either with ASC or with unconfirmed diagnosis of ASC or (...) with neurotypical development during social interaction with an adult interviewer in a diagnostic standard situation using the ADOS-2. In a case control study, we used mobile eye-tracking glasses in an ecologically valid and highly standardized diagnostic interview to investigate suspected cases of ASC. After completion of the ASC diagnostic gold standard including the ADOS-2, the participants were assigned to two groups based on their diagnosis and compared with a matched group of neurotypically developed controls. The primary outcome measure is the percentage of total dwell times assessed for different areas of interest with regard to the face and body of a diagnostic interviewer and the surrounding space. Overall, 65 children and adolescents within an age range of 8.3–17.9 years were included in the study. The data revealed significant group differences, especially in the central-face area. Previous investigations under laboratory conditions gave preferential attention to the eye region during face perception to describe differences between ASC and NTD. In this study – using an ecologically valid setting within a standard diagnostic procedure – the results indicate that neurotypically developed controls seem to process faces and facial expressions in a holistic manner originating from the central-face region. Conversely, participants on the Autism Spectrum seem to avoid the central-face region and show unsystematic gaze behavior, not using the preferred landing position in the central-face region as the Archimedean point of face perception. This study uses a new approach, and it will be important to replicate these preliminary findings in future research. (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)

The question of how our conception of the world could differ so widely from the disclosed nature of the world will with perfect equanimity be relinquished to the physiology and history of the evolution of organisms and concepts.In an interview conducted by the Italian literary journal Alfabeta in April of 1987,1 Niklas Luhmann was asked if sociology, in particular its systems-theoretical variant, could replace the privileged position that art, religion, philosophy, and politics had lost, and provide an Archimedean point from (...) which to describe society as a whole. Luhmann responded that today it is no longer possible to imagine such an outside position for the observation of the whole, sociology being no exception. .. (shrink)

Robinson and Zakon gave necessary and sufficient conditions for an abelian ordered group to satisfy the same first‐order sentences as an archimedean abelian ordered group (i.e., which embeds in the group of real numbers). The present paper generalizes their work to obtain similar results for infinite subgroups of the group of unimodular complex numbers. Furthermore, the groups which satisfy the same first‐order sentences as ultraproducts of finite cyclic groups are characterized.

Lexical views in population axiology can avoid the Repugnant Conclusion without violating Transitivity or Separability. However, they imply a dilemma: either some good life is better than any number of slightly worse lives, or else the ‘at least as good as’ relation on populations is radically incomplete. In this paper, I argue that Archimedean views face an analogous dilemma. I thus conclude that the lexical dilemma gives us little reason to prefer Archimedean views. Even if we give up on lexicality, (...) problems of the same kind remain. (shrink)

We study whether it is possible to generalise Seidenfeld et al.’s representation result for coherent choice functions in terms of sets of probability/utility pairs when we let go of Archimedeanity. We show that the convexity property is necessary but not sufficient for a choice function to be an infimum of a class of lexicographic ones. For the special case of two-dimensional option spaces, we determine the necessary and sufficient conditions by weakening the Archimedean axiom.

Archimedes of Syracuse has long provided a touchstone for considering how we make and acquire knowledge. Since the early Roman chroniclers of Archimedes’ life, and especially intensively since Descartes, scholars have described, sought, or derided the Archimedean point, defining and redefining its epistemic role. “Knowledge,” at least within modernity, is rhetorically tied to the figure of the Archimedean point, a place somewhere outside a regular and constrained world of experience. If this figure still leads to useful ways of thinking about (...) knowing, we are left with the question of how different modes of making knowledge approach their “Archimedean” points. The question is especially important today as a renewed .. (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)