There are possible worlds in which time is circular and finite in duration, forming a loop of, say, 12,000 years. There are also possible worlds in which time is linear and infinite in both directions and in which history is repetitive, consisting of infinitely many 12,000-year epochs, each two of which are exactly alike with respect to all intrinsic, purely qualitative properties. Could one ever have empirical evidence that one inhabits a world of the first kind rather than (...) a world of the second kind? We argue for the affirmative answer, contra Quine, Newton-Smith, and Bergström. Our argument for that conclusion differs from an argument for the same conclusion due to Weir. Weir’s argument is probabilistic and explicitly requires having evidence against determinism. Our argument is a direct appeal to the simplicity of laws, and it involves no probabilistic component. It is modeled on Shoemaker’s argument that one could have evidence of time without change. (shrink)

The nature of time is yet to be fully grasped and finally agreed upon among physicists, philosophers, psychologists and scholars from various disciplines. Present paper takes clue from the known assumptions of time as - movement, change, becoming - and the nature of time will be thoroughly discussed. -/- The real and unreal existences of time will be pointed out and presented. The complex number notation of nature of time will be put forward. Natural scientific systems and various cosmic processes (...) will be identified as constructing physical form of time and the physical existence of time will be designed. -/- The finite and infinite forms of physical time and classical, quantum and cosmic times will be delineated and their mathematical constructions and loci will be narrated. -/- Thus the physics behind time-construction, time creation and time-measurement will be given. -/- Based on these developments the physics of Timelessness will be developed and presented. -/- . (shrink)

Information can be considered as the most fundamental, philosophical, physical and mathematical concept originating from the totality by means of physical and mathematical transcendentalism (the counterpart of philosophical transcendentalism). Classical and quantum information, particularly by their units, bit and qubit, correspond and unify the finite and infinite. As classical information is relevant to finite series and sets, as quantum information, to infinite ones. A fundamental joint relativity of the finite and infinite, of the external (...) and internal is to be investigated. The corresponding invariance is able to define physical action and its quantity only on the basis of information and especially: on the relativity of classical and quantum information. The concept of transcendental time, an epoché in relation to the direction of time arrow can be defined. Its correlate is that information invariant to the finite and infinite, therefore unifying both classical and quantum information. (shrink)

On the rise over the past 20 years has been ‘moderate supernaturalism’, the view that while a meaningful life is possible in a world without God or a soul, a much greater meaning would be possible only in a world with them. William Lane Craig can be read as providing an important argument for a version of this view, according to which only with God and a soul could our lives have an eternal, as opposed to temporally limited, significance, by (...) virtue of our moral choices then making an ultimate difference. I present two major objections to this position. On the one hand, I contend that if God existed and we had souls that lived forever, then, in fact, all our lives would turn out the same. On the other hand, I maintain that, if this objection is wrong, so that our moral choices would indeed make an ultimate difference and thereby confer an eternal significance on our lives (only) in a supernatural realm, then Craig could not capture the view, aptly held by moderate supernaturalists, that a meaningful life is possible in a purely natural world. An eternal significance would be ‘too big’, reducing any meaning possible during an earthly life to nothing by comparison. I conclude that moderate supernaturalists would be wise to avoid appealing to eternal (or infinite) meaning when spelling out the way God and a soul could alone impart a great meaning to our lives; some other notion of greatness should be considered, perhaps one involving a temporary afterlife. (shrink)

This note briefly discusses the observation of elapsed time in a flat universe while exploring the argument of past-eternal time versus emergent time in cosmology. A flat universe with an incomplete past forever has a finite age. Despite an infinite number of Planck time coordinates independent of phenomena and endless expansion, a flat universe never develops an age with an infinite number of Planck times. This observation indicates the impossibility of infinitely elapsed time in the future (...) or past, which limits acceptable scientific models of cosmology. (shrink)

Infinite machines (IMs) can do supertasks. A supertask is an infinite series of operations done in some finite time. Whether or not our universe contains any IMs, they are worthy of study as upper bounds on finite machines. We introduce IMs and describe some of their physical and psychological aspects. An accelerating Turing machine (an ATM) is a Turing machine that performs every next operation twice as fast. It can carry out infinitely many operations in (...) class='Hi'>finite time. Many ATMs can be connected together to form networks of infinitely powerful agents. A network of ATMs can also be thought of as the control system for an infinitely complex robot. We describe a robot with a dense network of ATMs for its retinas, its brain, and its motor controllers. Such a robot can perform psychological supertasks - it can perceive infinitely detailed objects in all their detail; it can formulate infinite plans; it can make infinitely precise movements. An endless hierarchy of IMs might realize a deep notion of intelligent computing everywhere. (shrink)

There are few notions in philosophy seen more clearly, and in parallel so laden with confusion, than that of space and time. The subjective nature of analyses is most likely to blame. As it stands, a universal agreement has not yet been reached. My position is simply that the mind, when passive, has no qualms with space and time itself, nor is it concerned with its principles. It is only when our passions are ignited, and our judgment is utilized, i.e. (...) when we begin to actively think about space and time, that the notion becomes confounded. Any further digression into semantics of the issue by trying to give a ‘say all end all’ explanation will always lead to a dead end. Both space and time are ideas of an infinite nature and can never be visited manifestly since the mind itself is undeniably finite. (shrink)

The quantum information introduced by quantum mechanics is equivalent to a certain generalization of classical information: from finite to infinite series or collections. The quantity of information is the quantity of choices measured in the units of elementary choice. The “qubit”, can be interpreted as that generalization of “bit”, which is a choice among a continuum of alternatives. The axiom of choice is necessary for quantum information. The coherent state is transformed into a well-ordered series of results in (...) time after measurement. The quantity of quantum information is the transfinite ordinal number corresponding to the infinity series in question. The transfinite ordinal numbers can be defined as ambiguously corresponding “transfinite natural numbers” generalizing the natural numbers of Peano arithmetic to “Hilbert arithmetic” allowing for the unification of the foundations of mathematics and quantum mechanics. (shrink)

Discusses Frege's formal definitions and characterizations of infinite and finite sets. Speculates that Frege might have discovered the "oddity" in Dedekind's famous proof that all infinite sets are Dedekind infinite and, in doing so, stumbled across an axiom of countable choice.

I critically examine Nietzsche’s argument in The Will to Power that all the detailed events of the world are repeating infinite times (on account of the merely finite possible arrangements of forces that constitute the world and the inevitability with which any arrangement of force must bring about its successors). Nietzsche celebrated this recurrence because of the power of belief in it to bring about a revaluation of values focused wholly on the value of one’s endlessly repeating (...) life. Belief in recurrence would bring joy to those who had achieved excellence in this life and crush those who had not. I point out, however, that this significance of recurrence must be spoiled by the consideration that within each of the long cycles of recurrence there would have to be, as well, countless variations of this life. And I consider the issue of personal identity within such recurrences. Is it the same person or merely resembling but distinct persons in the recurrences or the variations? If it is the same person in precise recurrences, should the subjectively identical experience be thought of as really additional for that person? (shrink)

The quantum information introduced by quantum mechanics is equivalent to that generalization of the classical information from finite to infinite series or collections. The quantity of information is the quantity of choices measured in the units of elementary choice. The qubit can be interpreted as that generalization of bit, which is a choice among a continuum of alternatives. The axiom of choice is necessary for quantum information. The coherent state is transformed into a well-ordered series of results in (...) time after measurement. The quantity of quantum information is the ordinal corresponding to the infinity series in question. Number and being (by the meditation of time), the natural and artificial turn out to be not more than different hypostases of a single common essence. This implies some kind of neo-Pythagorean ontology making related mathematics, physics, and technics immediately, by an explicit mathematical structure. (shrink)

This Essay presents law’s evolution as a paradoxical union of the finite and the infinite. At any given point in time, law is a finite body of norms, which can be identified. At the same time, law’s evolution is infinite because rule-mutations that alter those norms are indeterminable. In modern legal systems, law’s evolution occurs under the constraining influence of master texts, which provide normative durability by enshrining the fundamental norms of a legal system and fortifying (...) them against change. Despite this stabilizing role, however, master texts are themselves open to mutations. Therefore, law’s evolution under the constraints of a master text mediates the paradoxical union of law’s finite and infinite nature. (shrink)

Quantum information is discussed as the universal substance of the world. It is interpreted as that generalization of classical information, which includes both finite and transfinite ordinal numbers. On the other hand, any wave function and thus any state of any quantum system is just one value of quantum information. Information and its generalization as quantum information are considered as quantities of elementary choices. Their units are correspondingly a bit and a qubit. The course of time is what generates (...) choices by itself, thus quantum information and any item in the world in final analysis. The course of time generates necessarily choices so: The future is absolutely unorderable in principle while the past is always well-ordered and thus unchangeable. The present as the mediation between them needs the well-ordered theorem equivalent to the axiom of choice. The latter guarantees the choice even among the elements of an infinite set, which is the case of quantum information. The concrete and abstract objects share information as their common base, which is quantum as to the formers and classical as to the latters. The general quantities of matter in physics, mass and energy can be considered as particular cases of quantum information. The link between choice and abstraction in set theory allows of “Hume’s principle” to be interpreted in terms of quantum mechanics as equivalence of “many” and “much” underlying quantum information. Quantum information as the universal substance of the world calls for the unity of physics and mathematics rather than that of the concrete and abstract objects and thus for a form of quantum neo-Pythagoreanism in final analysis. (shrink)

It is commonly assumed that Aristotle denies any real existence to infinity. Nothing is actually infinite. If, in order to resolve Zeno’s paradoxes, Aristotle must talk of infinity, it is only in the sense of a potentiality that can never be actualized. Aristotle’s solution has been both praised for its subtlety and blamed for entailing a limitation of mathematic. His understanding of the infinite as simply indefinite (the “bad infinite” that fails to reach its accomplishment), his conception (...) of the cosmos and even its prime mover as finite (in the sense of autarchic/self-contained) have been contrasted with the subsequent claim of God as ens infinitum (understood as a “positive” infinity). The goal of this essay is to reexamine the major texts (notably De caelo) and to demonstrate that (1) Aristotle’s claim according to which there is no actual infinite concerns only substances, not processes. (2) That Aristotle does not deny an actual infinite as such. (3) That when considering time and God (qua eternal) Aristotle acknowledges an actual infinite. (shrink)

This chapter contains sections titled: * Brief History * How We Talk * Science and Infinity * Religion and Infinity * Concluding Remarks * Notes * References * Further Reading.

The concept of formal transcendentalism is utilized. The fundamental and definitive property of the totality suggests for “the totality to be all”, thus, its externality (unlike any other entity) is contained within it. This generates a fundamental (or philosophical) “doubling” of anything being referred to the totality, i.e. considered philosophically. Thus, that doubling as well as transcendentalism underlying it can be interpreted formally as an elementary choice such as a bit of information and a quantity corresponding to the number of (...) elementary choices to be defined. This is the quantity of information defined both transcendentally and formally and thus, philosophically and mathematically. If one defines information specifically, as an elementary choice between finiteness (or mathematically, as any natural number of Peano arithmetic) and infinity (i.e. an actually infinite set in the meaning of set theory), the quantity of quantum information is defined. One can demonstrate that the so-defined quantum information and quantum information standardly defined by quantum mechanics are equivalent to each other. The equivalence of the axiom of choice and the well-ordering “theorem” is involved. It can be justified transcendentally as well, in virtue of transcendental equivalence implied by the totality. Thus, all can be considered as temporal as far anything possesses such a temporal counterpart necessarily. Formally defined, the frontier of time is the current choice now, a bit of information, furthermore interpretable as a qubit of quantum information. (shrink)

Many philosophers have argued that the past must be finite in duration because otherwise reaching the present moment would have involved something impossible, namely, the sequential occurrence of an actual infinity of events. In reply, some philosophers have objected that there can be nothing amiss in such an occurrence, since actually infinite sequences are ‘traversed’ all the time in nature, for example, whenever an object moves from one location in space to another. This essay focuses on one of (...) the two available replies to this objection, namely, the claim that actual infinities are not traversed in nature because space, time, and other continuous wholes divide into parts only in so far as we divide them in thought, and thus divide into only a finite number of parts. I grant that this reply succeeds in blunting the anti-finitist objection, but I argue that it also subverts the very argument against an eternal past that it was intended to save. (shrink)

In this article, I argue that it is impossible to complete infinitely many tasks in a finite time. A key premise in my argument is that the only way to get to 0 tasks remaining is from 1 task remaining, when tasks are done 1-by-1. I suggest that the only way to deny this premise is by begging the question, that is, by assuming that supertasks are possible. I go on to present one reason why this conclusion (that supertasks (...) are impossible) is important, namely that it implies a new verdict on a decision puzzle propounded by Jeffrey Barrett and Frank Arntzenius. (shrink)

Benardete paradoxes involve infinite collections of Grim Reapers, assassins, demons, deafening peals, or even sentences. These paradoxes have recently been used in arguments for finitist metaphysical theses such as temporal finitism, causal finitism, and discrete views of time. Here we develop a new finite Benardete-like paradox. We then use this paradox to defend a companions in guilt argument that challenges recent applications of patchwork principles on behalf of the aforementioned finitist arguments. Finally, we develop another problem for those (...) applications by examining the notion of exact duplication. (shrink)

Quantum information is discussed as the universal substance of the world. It is interpreted as that generalization of classical information, which includes both finite and transfinite ordinal numbers. On the other hand, any wave function and thus any state of any quantum system is just one value of quantum information. Information and its generalization as quantum information are considered as quantities of elementary choices. Their units are correspondingly a bit and a qubit. The course of time is what generates (...) choices by itself, thus quantum information and any item in the world in final analysis. The course of time generates necessarily choices so: The future is absolutely unorderable in principle while the past is always well-ordered and thus unchangeable. The present as the mediation between them needs the well-ordered theorem equivalent to the axiom of choice. The latter guarantees the choice even among the elements of an infinite set, which is the case of quantum information. The concrete and abstract objects share information as their common base, which is quantum as to the formers and classical as to the latter. The general quantities of matter in physics, mass and energy can be considered as particular cases of quantum information. The link between choice and abstraction in set theory allows of “Hume’s principle” to be interpreted in terms of quantum mechanics as equivalence of “many” and “much” underlying quantum information. Quantum information as the universal substance of the world calls for the unity of physics and mathematics rather than that of the concrete and abstract objects and thus for a form of quantum neo-Pythagoreanism in final analysis. (shrink)

W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due in (...) large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. -/- For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. -/- In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. -/- Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here. (shrink)

Quantum mechanics involves a generalized form of information, that of quantum information. It is the transfinite generalization of information and re-presentable by transfinite ordinals. The physical world being in the current of time shares the quality of “choice”. Thus quantum information can be seen as the universal substance of the world serving to describe uniformly future, past, and thus the present as the frontier of time. Future is represented as a coherent whole, present as a choice among infinitely many alternatives, (...) and past as a well-ordering obtained as a result of a series of choices. The concept of quantum information describes the frontier of time, that “now”, which transforms future into past. Quantum information generalizes information from finite to infinite series or collections. The concept of quantum information allows of any physical entity to be interpreted as some nonzero quantity of quantum information. The fundament of quantum information is the concept of ‘quantum bit’, “qubit”. A qubit is a choice among an infinite set of alternatives. It generalizes the unit of classical information, a bit, which refer to a finite set of alternatives. The qubit is also isomorphic to a ball in Euclidean space, in which two points are chosen. (shrink)

“How can I, who am thinking about the entire, centerless universe, be anything so specific as this: this measly creature existing in a tiny morsel of space and time?” This metaphysically self-deprecating question, posed by Thomas Nagel, holds an insight into the nature of personhood and the ordinary ways we value it, in others and in ourselves. I articulate that insight and apply it to the phenomena of friendship, companionship, sexuality, solitude, and love. Although love comes in many forms, I (...) say, it always involves a sense of wonder at a finite creature thinking infinite thoughts. (shrink)

We address the question whether there is an explanation for the fact that as Fodor put it the micro-level “converges on stable macro-level properties”, and whether there are lessons from this explanation for other issues in the vicinity. We argue that stability in large systems can be understood in terms of statistical limit theorems. In the thermodynamic limit of infinite system size N → ∞ systems will have strictly stable macroscopic properties in the sense that transitions between different macroscopic (...) phases of matter (if there are any) will not occur in finite time. Indeed stability in this sense is a consequence of the absence of fluctuations, as (large) fluctuations would be required to induce such macroscopic transformations. These properties can be understood in terms of coarse-grained descriptions, and the statistical limit theorems for independent or weakly dependent random variable describing the behaviour averages and the statistics of fluctuations in the large system limit. We argue that RNG analyses applied to off-critical systems can provide a rationalization for the applicability of these limit theorems. Furthermore we discuss some related issues as, for example, the role of the infinite-system idealization. (shrink)

We introduce a ranking of multidimensional alternatives, including uncertain prospects as a particular case, when these objects can be given a matrix form. This ranking is separable in terms of rows and columns, and continuous and monotonic in the basic quantities. Owing to the theory of additive separability developed here, we derive very precise numerical representations over a large class of domains (i.e., typically notof the Cartesian product form). We apply these representationsto (1)streams of commodity baskets through time, (2)uncertain social (...) prospects, (3)uncertain individual prospects. Concerning(1), we propose a finite horizon variant of Koopmans’s (1960) axiomatization of infinite discounted utility sums. The main results concern(2). We push the classic comparison between the exanteand expostsocial welfare criteria one step further by avoiding any expected utility assumptions, and as a consequence obtain what appears to be the strongest existing form of Harsanyi’s (1955) Aggregation Theorem. Concerning(3), we derive a subjective probability for Anscombe and Aumann’s (1963) finite case by merely assuming that there are two epistemically independent sources of uncertainty. (shrink)

This paper investigates Descartes’s understanding of the imago Dei, that it is above all in virtue of the will that we bear the image and likeness of God. I challenge the key assumption of arguments that hold that Descartes’s comparison between the human will and the divine will is problematic—that in his conception of the imago Dei Descartes is alluding to Scholastic conceptions of analogy available to him at the time, which would place particular constraints on the legitimacy of the (...) comparison. I argue instead that Descartes is evoking a different tradition regarding the nature of image and imitation, stemming from Augustine and Aquinas, and thus, that those constraints do not apply. I then argue that Descartes thinks the likeness between the human will and God’s will is that both are infinite in “extent.” This means that human will can “extend itself” not only to things that can be the object of some other will, but to things that can be the object of God’s will. This is notable because Descartes famously thinks that absolutely anything can be the object of God’s will. I explain why this interpretation is not implausible, contrary to first appearances. (shrink)

According to Leibniz’s infinite-analysis account of contingency, any derivative truth is contingent if and only if it does not admit of a finite proof. Following a tradition that goes back at least as far as Bertrand Russell, several interpreters have been tempted to explain this biconditional in terms of two other principles: first, that a derivative truth is contingent if and only if it contains infinitely complex concepts and, second, that a derivative truth contains infinitely complex concepts if (...) and only if it does not admit of a finite proof. A consequence of this interpretation is that Leibniz’s infinite-analysis account of contingency falls prey to Robert Adams’s Problem of Lucky Proof. I will argue that this interpretation is mistaken and that, once it is properly understood how the idea of an infinite proof fits into Leibniz’s circle of modal notions, the problem of lucky proof simply disappears. (shrink)

David Furley's work on the cosmologies of classical antiquity is structured around what he calls "two pictures of the world." The first picture, defended by both Plato and Aristotle, portrays the universe, or all that there is (to pan), as identical with our particular ordered world-system. Thus, the adherents of this view claim that the universe is finite and unique. The second system, defended by Leucippus and Democritus, portrays an infinite universe within which our particular kosmos is only (...) one of countless kosmoi. Aristotle's argument in De caelo I.9 that the world is necessarily unique is an important contribution to this debate. This argument holds interest because it shows Aristotle wrestling with an apparent inconsistency in his own philosophy, as deeply-held convictions within his cosmology collide with an equally deeply-held conviction within his metaphysics. The following three principles, each of which Aristotle appears committed to, are inconsistent: -/- The cosmic uniqueness principle. The world is necessarily unique. The cosmic form principle. The world is an ordered, structured unity. As such, the world has a form. The possibility of multiple instantiation principle. For all F, if F is a form, it is possible that there exist multiple Fs. In De caelo I.9, Aristotle argues that we can establish the uniqueness of the universe, reject the multiple instantiation principle, yet still retain the distinction between 'this world' and 'world in general,' if the following is true (as it is): the world takes up all the matter that exists. Aristotle illustrates this argument with one of the stranger analogies in his corpus: imagine an aquiline nose that takes up all the flesh in the universe. If this were so, then there could not exist any other aquiline objects whatsoever. (For this reason, we dub the De caelo I.9 argument the 'Cosmic Nose argument.') This paper is an interpretation of how this argument is supposed to proceed and an assessment of its success. The first section states the problem Aristotle is confronted with, sorts through Aristotle's various statements of the Cosmic Nose argument, which exhibit some sloppiness, and reconstructs charitably a single argument. We also spend some time examining the significance of Aristotle's example of a gigantic aquiline nose. We argue that, even charitably reconstructed, the argument appears to commit a serious modal fallacy. The remainder of the paper explores whether this modal fallacy can be overcome. We conclude that, although not a cogent argument for the uniqueness of the world (as this would require a significant revision of our current astronomy), the Cosmic Nose argument does succeed on its own terms. However, it should not be regarded as a free-standing argument for the uniqueness of the world. Instead, it depends crucially on the earlier argument in De caelo I.8 for the universe's uniqueness; De caelo I.9 should be viewed as an attempt to extend the conclusion of De caelo I.8 and to show how this conclusion can be made consistent with Aristotle's metaphysical principles about the nature of form. (shrink)

We start by asking the question of ‘why there is something rather than nothing’ and change this to the question of ‘what are the weakest assumptions for existence’ Eagle [1]. Then we give a kind of Fragmental Perspectivalism. Within this Fragmentalist interpretation of quantum mechanics (each quantum mechanical system forms a fragment) Merriam [2], it turns out McTaggart’s [3] A-series of time (the A-series is future to the present to the past) has a kind of perspectivalism. We then use McTaggart’s (...) A-series and the B-series (the B-series is earlier times to later times) of time to differentiate between how far in the past the big bang was vs. how much earlier than now the big bang was. In one example model, the former goes infinitely far into the past while the latter stays finitely earlier-than. In this model the number of quantum interactions per unit 4-volume goes up to infinity as the big bang is approached from the present epoch. (shrink)

The human eyes and brain, which have finite boundaries, create a ‘‘virtual’’ space within our central nervous system that interprets and perceives a space that appears boundless and infinite. Using insights from studies on the visual system, we propose a novel fast processing mechanism involving the eyes, visual pathways, and cortex where external vision is imperceptibly processed in our brain in real time creating an internal representation of external space that appears as an external view. We introduce the (...) existence of a three-dimension default space consisting of intrapersonal body space that serves as the framework where visual and non-visual sensory information is sensed and experienced. We propose that the thalamus integrates processed information from corticothalamic feedback loops and fills-in the neural component of 3D default space with an internal visual representation of external space, leading to the experience of visual consciousness. This visual space inherently evades perception so we have introduced three easy clinical tests that can assist in experiencing this visual space. We also review visual neuroanatomical pathways, binocular vision, neurological disorders, and visual phenomenon to elucidate how the representation of external visible space is recreated within the mind. (shrink)

It is well-known that the global structure of every space-time model for relativistic cosmology is observationally underdetermined. In order to alleviate the severity of this underdetermination, it has been proposed that we adopt the Cosmological Principle because the Principle restricts our attention to a distinguished class of space-time models (spatially homogeneous and isotropic models). I argue that, even assuming the Cosmological Principle, the topology of space remains observationally underdetermined. Nonetheless, I argue that we can muster reasons to prefer various topological (...) properties over others. In particular, I favor the adoption of multiply connected universe models on grounds of (i) simplicity, (ii) Machian considerations, and (iii) explanatory power. We are able to appeal to such grounds because multiply connected topologies open up the possibility of finite universe models (consistent with our best data), which in turn avoid thorny issues concerning the postulation of an actually infinite universe. (shrink)

The seventeenth century was an important period in the conceptual development of the notion of the infinite. In 1643, Evangelista Torricelli (1608-1647)—Galileo’s successor in the chair of mathematics in Florence—communicated his proof of a solid of infinite length but finite volume. Many of the leading metaphysicians of the time, notably Spinoza and Leibniz, came out in defense of actual infinity, rejecting the Aristotelian ban on it, which had been almost universally accepted for two millennia. Though it would (...) be another two centuries before the notion of the actually infinite was rehabilitated in mathematics by Dedekind and Cantor (Cauchy and Weierstrass still considered it mere paradox), their impenitent advocacy of the concept had significant reverberations in both philosophy and mathematics. In this essay, I will attempt to clarify one thread in the development of the notion of the infinite. In the first part, I study Spinoza’s discussion and endorsement, in the Letter on the Infinite (Ep. 12), of Hasdai Crescas’ (c. 1340-1410/11) crucial amendment to a traditional proof of the existence of God (“the cosmological proof” ), in which he insightfully points out that the proof does not require the Aristotelian ban on actual infinity. In the second and last part, I examine the claim, advanced by Crescas and Spinoza, that God has infinitely many attributes, and explore the reasoning that motivated both philosophers to make such a claim. Similarities between Spinoza and Crescas, which suggest the latter’s influence on the former, can be discerned in several other important issues, such as necessitarianism, the view that we are compelled to assert or reject a belief by its representational content, the enigmatic notion of amor Dei intellectualis, and the view of punishment as a natural consequent of sin. Here, I will restrict myself to the issue of the infinite, clearly a substantial topic in itself. (shrink)

A multiverse is comprised of many universes, which quickly leads to the question: How many universes? There are either finitely many or infinitely many universes. The purpose of this paper is to discuss two conceptions of infinite number and their relationship to multiverses. The first conception is the standard Cantorian view. But recent work has suggested a second conception of infinite number, on which infinite numbers behave very much like finite numbers. I will argue that that (...) this second conception of infinite number is the correct one, and analyze what this means for multiverses. (shrink)

In this article, I consider the importance of philosophy in the dialogue between religious believers and non-believers. I begin by arguing that a new epistemology of epistemic peer disagreement is required if the dialogue is to progress. Rather than viewing the differences between the positions as due to a deficit of understanding, I argue that differences result from the existential anchoring of such enquiries in life projects and the under-determination of interpretations by experience. I then explore a central issue which (...) is often implicit in these dialogues, namely the ontological status of God-world relations. Drawing on the reflections of Hegel on the infinite and the finite, I argue that his version of panentheism provides an insightful way to conceptualise God-world relations that avoids both dualistic and monistic approaches and helps to explicate a holistic ontology of transcendence from within the world of experience. (shrink)

Very often traditional approaches studying dynamics of self-similarity processes are not able to give their quantitative characteristics at infinity and, as a consequence, use limits to overcome this difficulty. For example, it is well know that the limit area of Sierpinski’s carpet and volume of Menger’s sponge are equal to zero. It is shown in this paper that recently introduced infinite and infinitesimal numbers allow us to use exact expressions instead of limits and to calculate exact infinitesimal values of (...) areas and volumes at various points at infinity even if the chosen moment of the observation is infinitely faraway on the time axis from the starting point. It is interesting that traditional results that can be obtained without the usage of infinite and infinitesimal numbers can be produced just as finite approximations of the new ones. (shrink)

In this paper, I suggest that infinite numbers are large finite numbers, and that infinite numbers, properly understood, are 1) of the structure omega + (omega* + omega)Ө + omega*, and 2) the part is smaller than the whole. I present an explanation of these claims in terms of epistemic limitations. I then consider the importance, part of which is demonstrating the contradiction that lies at the heart of Cantorian set theory: the natural numbers are too large (...) to be counted by any finite number, but too small to be counted by any infinite number – there is no number of natural numbers. (shrink)

In 'Forever Finite: The Case Against Infinity' (Rond Books, 2023), the author argues that, despite its cultural popularity, infinity is not a logical concept and consequently cannot be a property of anything that exists in the real world. This article summarizes the main points in 'Forever Finite', including its overview of what debunking infinity entails for conceptual thought in philosophy, mathematics, science, cosmology, and theology.

The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses recently introduced infinite and infinitesimal numbers being in accordance with the principle ‘The part is less than the whole’ observed in the physical world around us. These numbers have a strong practical advantage with respect to traditional approaches: they are representable at a (...) new kind of a computer – the Infinity Computer – able to work numerically with all of them. An introduction to the theory of physical and mathematical continuity and differentiation (including subdifferentials) for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains is developed in the paper. This theory allows one to work with derivatives that can assume not only finite but infinite and infinitesimal values, as well. It is emphasized that the newly introduced notion of the physical continuity allows one to see the same mathematical object as a continuous or a discrete one, in dependence on the wish of the researcher, i.e., as it happens in the physical world where the same object can be viewed as a continuous or a discrete in dependence on the instrument of the observation used by the researcher. Connections between pure mathematical concepts and their computational realizations are continuously emphasized through the text. Numerous examples are given. (shrink)

The best accuracy arguments for probabilism apply only to credence functions with finite domains, that is, credence functions that assign credence to at most finitely many propositions. This is a significant limitation. It reveals that the support for the accuracy-first program in epistemology is a lot weaker than it seems at first glance, and it means that accuracy arguments cannot yet accomplish everything that their competitors, the pragmatic (Dutch book) arguments, can. In this paper, I investigate the extent to (...) which this limitation can be overcome. Building on the best arguments in finite domains, I present two accuracy arguments for probabilism that are perfectly general—they apply to credence functions with arbitrary domains. I then discuss how the arguments’ premises can be challenged. We will see that it is particularly difficult to characterize admissible accuracy measures in infinite domains. (shrink)

The Koch snowflake is one of the first fractals that were mathematically described. It is interesting because it has an infinite perimeter in the limit but its limit area is finite. In this paper, a recently proposed computational methodology allowing one to execute numerical computations with infinities and infinitesimals is applied to study the Koch snowflake at infinity. Numerical computations with actual infinite and infinitesimal numbers can be executed on the Infinity Computer being a new supercomputer patented (...) in USA and EU. It is revealed in the paper that at infinity the snowflake is not unique, i.e., different snowflakes can be distinguished for different infinite numbers of steps executed during the process of their generation. It is then shown that for any given infinite number n of steps it becomes possible to calculate the exact infinite number, Nn, of sides of the snowflake, the exact infinitesimal length, Ln, of each side and the exact infinite perimeter, Pn, of the Koch snowflake as the result of multiplication of the infinite Nn by the infinitesimal Ln. It is established that for different infinite n and k the infinite perimeters Pn and Pk are also different and the difference can be infinite. It is shown that the finite areas An and Ak of the snowflakes can be also calculated exactly (up to infinitesimals) for different infinite n and k and the difference An − Ak results to be infinitesimal. Finally, snowflakes constructed starting from different initial conditions are also studied and their quantitative characteristics at infinity are computed. (shrink)

In this paper, I present a puzzle involving special relativity and the random selection of real numbers. In a manner to be specified, darts thrown later hit reals further into a fixed well-ordering than darts thrown earlier. Special relativity is then invoked to create a puzzle. I consider four ways of responding to this puzzle which, I suggest, fail. I then propose a resolution to the puzzle, which relies on the distinction between the potential infinite and the actual (...) class='Hi'>infinite. I suggest that certain structures, such as a well-ordering of the reals, or the natural numbers, are examples of the potential infinite, whereas infinite integers in a nonstandard model of arithmetic are examples of the actual infinite. (shrink)

A new computational methodology for executing calculations with infinite and infinitesimal quantities is described in this paper. It is based on the principle ‘The part is less than the whole’ introduced by Ancient Greeks and applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). It is shown that it becomes possible to write down finite, infinite, and infinitesimal numbers by a finite number of symbols (...) as particular cases of a unique framework. The new methodology has allowed us to introduce the Infinity Computer working with such numbers (its simulator has already been realized). Examples dealing with divergent series, infinite sets, and limits are given. (shrink)

Throughout history, almost all mathematicians, physicists and philosophers have been of the opinion that space and time are infinitely divisible. That is, it is usually believed that space and time do not consist of atoms, but that any piece of space and time of non-zero size, however small, can itself be divided into still smaller parts. This assumption is included in geometry, as in Euclid, and also in the Euclidean and non- Euclidean geometries used in modern physics. Of the few (...) who have denied that space and time are infinitely divisible, the most notable are the ancient atomists, and Berkeley and Hume. All of these assert not only that space and time might be atomic, but that they must be. Infinite divisibility is, they say, impossible on purely conceptual grounds. (shrink)

There is a well-known equivalence between avoiding accuracy dominance and having probabilistically coherent credences (see, e.g., de Finetti 1974, Joyce 2009, Predd et al. 2009, Pettigrew 2016). However, this equivalence has been established only when the set of propositions on which credence functions are defined is finite. In this paper, I establish connections between accuracy dominance and coherence when credence functions are defined on an infinite set of propositions. In particular, I establish the necessary results to extend the (...) classic accuracy argument for probabilism to certain classes of infinite sets of propositions including countably infinite partitions. (shrink)

This paper suggests that time could have a much richer mathematical structure than that of the real numbers. Clark & Read (1984) argue that a hypertask (uncountably many tasks done in a finite length of time) cannot be performed. Assuming that time takes values in the real numbers, we give a trivial proof of this. If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an ultratask (a sequence (...) which includes one task done for each ordinal number—thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible. (shrink)

Causal finitism, the view that the causal history of any event must be finite, has garnered much philosophical interest recently—especially because of its applicability to the Kalām cosmological argument. The most prominent argument for causal finitism is the Grim Reaper argument, which attempts to show that, if infinite causal histories are possible, then other paradoxical states of affairs must also be possible. However, this style of argument has been criticized on the grounds of (i) relying on controversial modal (...) principles, and (ii) providing a false diagnosis of the paradoxes involved. In this paper, I develop a new kind of Grim Reaper argument immune to these criticisms. I show that, by using insights from the literature on time travel, causal finitists should instead argue that infinite causal histories are problematically inexplicable, as they entail the possibility of unexplained foiling mechanisms. The fruits of this paper are that (i) a novel supporting argument for the Kalām is developed, and (ii) along the way of building this argument, it is shown that the literatures on time travel and causal finitism are deeply and intimately connected. (shrink)

My two principal aims in this essay are interconnected. One aim is to provide a new interpretation of the ‘infinite modes’ in Spinoza’s Ethics. I argue that for Spinoza, God, conceived as the one infinite and eternal substance, is not to be understood as causing two kinds of modes, some infinite and eternal and the rest finite and non-eternal. That there cannot be such a bifurcation of divine effects is what I take the ‘infinite mode’ (...) propositions, E1p21–23, to establish; E1p21–23 show that each and every one of the immanent effects of an infinite and eternal God is an infinite and eternal mode. The other aim is to show that these propositions can be understood as part of an extended critical response to Descartes’s infamous doctrine that God creates eternal truths and true and immutable natures. If we have the correct (Spinozan) conceptions of what God is and how God works, we see that an eternal and infinite God can only be understood to cause ‘eternal truths,’ and that these eternal truths are infinite and eternal modes of God. (shrink)

Draft for presentation at the 14th International Kant-Congress, September 2024. -/- Abstract: Kant claims we intuit infinite space. There’s a problem: Kant thinks full awareness of infinite space requires synthesis—the act of putting representations together and comprehending them as one. But our ability to synthesize is finite. Tobias Rosefeldt has argued in a recent paper that Kant’s notion of decomposing synthesis offers a solution. This talk criticizes Rosefeldt’s approach. First, Rosefeldt is committed to nonconceptual yet determinate awareness (...) of (potentially) infinite space, but I argue that such awareness is (1) prima facie implausible, as well as contravened by textual evidence from Kant’s (2a) definitions of ‘infinity’ and (2b) account of geometrical construction. Second, scrutiny of how Kant thinks awareness of potential infinity is afforded by geometrical construction (e.g., extending a line segment ad indefinitum) indicates no essential role for decomposing synthesis. Familiar synthesis from parts to wholes is sufficient. (shrink)

Drawing upon the practice of caregiving and the insights of feminist care ethics, I offer a phenomenology of caregiving through the work of Eva Feder Kittay and Emmanuel Lévinas. I argue that caregiving is a material dialectic of embodied response involving moments of leveling, attention, and interruption. In this light, the Levinasian opposition between responding to another's singularity and leveling it via parity-based principles is belied in the experience of care. Contra much of response ethics’ and care ethics’ respective literatures, (...) this dialectic suggests that they are complementary in ways that productively illuminate themes of each. I conclude by suggesting that when response and care ethics are thought together through the experience of caregiving, such labors produce finite responsibility with infinite hope. (shrink)