We argue that C. Darwin and more recently W. Hennig worked at times under the simplifying assumption of an eternal biosphere. So motivated, we explicitly consider the consequences which follow mathematically from this assumption, and the infinite graphs it leads to. This assumption admits certain clusters of organisms which have some ideal theoretical properties of species, shining some light onto the species problem. We prove a dualization of a law of T.A. Knight and C. Darwin, and sketch a (...) decomposition result involving the internodons of D. Kornet, J. Metz and H. Schellinx. A further goal of this paper is to respond to B. Sturmfels’ question, “Can biology lead to new theorems?”. (shrink)

In this paper, a number of traditional models related to the percolation theory has been considered by means of new computational methodology that does not use Cantor’s ideas and describes infinite and infinitesimal numbers in accordance with the principle ‘The part is less than the whole’. It gives a possibility to work with finite, infinite, and infinitesimal quantities numerically by using a new kind of a compute - the Infinity Computer – introduced recently in [18]. The new approach (...) does not contradict Cantor. In contrast, it can be viewed as an evolution of his deep ideas regarding the existence of different infinite numbers in a more applied way. Site percolation and gradient percolation have been studied by applying the new computational tools. It has been established that in an infinite system the phase transition point is not really a point as with respect of traditional approach. In light of new arithmetic it appears as a critical interval, rather than a critical point. Depending on “microscope” we use this interval could be regarded as finite, infinite and infinitesimal short interval. Using new approach we observed that in vicinity of percolation threshold we have many different infinite clusters instead of one infinite cluster that appears in traditional consideration. (shrink)

The ancient Ouroboros symbolism (one who eats oneself) is here integrated into a simple birth-death clustering process that needed nothing but itself for a transition from indistinguishable phases to a sort of higher level ”conscious” phases. Birth and death coefficients are formulated in terms of odd and even exponentials used to represent a suitable form for conscious states via the internal transfer of information. This toy model may ideally quantify conscious states having inner causes via an Ouroboros index 0 < (...) Υα,Ω < 1. The value Υα,Ω = 0 lends to infinite loops and the limiting values Υα,Ω → 1 disclose transformation into awareness. Relationships with physics theories of consciousness and the use of the Oroborous index to discern about conscious-like states in Artificial Intelligence systems are discussed. Consciousness and freedom of will may go side-by-side in the model when Υα,Ω is extended to be a complex number modulus. (shrink)

We argue the thesis that if (1) a physical process is mathematically representable by a Cauchy sequence; and (2) we accept that there can be no infinite processes, i.e., nothing corresponding to infinite sequences, in natural phenomena; then (a) in the absence of an extraneous, evidence-based, proof of `closure' which determines the behaviour of the physical process in the limit as corresponding to a `Cauchy' limit; (b) the physical process must tend to a discontinuity (singularity) which has not (...) been reflected in the Cauchy sequence that seeks to describe the behaviour of the physical process. We support our thesis by mathematical models of the putative behaviours of (i) a virus cluster; (ii) an elastic string; and (iii) a Universe that recycles from Big Bang to Ultimate Implosion, in which parity and local time reversal violation, and the existence of `dark energy' in a multiverse, need not violate Einstein's equations and quantum theory. We suggest that the barriers to modelling such processes in a mathematical language that seeks unambiguous communication are illusory; they merely reflect an attempt to ask of the language chosen for such representation more than it is designed to deliver. (shrink)

The paper argues for a decompositionalist account of lexical concepts. In particular, it presents and argues for a cluster decompositionalism, a view that claims that the complexes a token of a word corresponds to on a given occasion are typically built out of a determinate set of basic concepts, most of which are present on most other occasions of use of the word. The first part of the paper discusses some explanatory virtues of decompositionalism in general. The second singles out (...) cluster decompositionalism as the best explanation of the variability of meaning. The third part is devoted to responding to some problems. (shrink)

People with the kind of preferences that give rise to the St. Petersburg paradox are problematic---but not because there is anything wrong with infinite utilities. Rather, such people cannot assign the St. Petersburg gamble any value that any kind of outcome could possibly have. Their preferences also violate an infinitary generalization of Savage's Sure Thing Principle, which we call the *Countable Sure Thing Principle*, as well as an infinitary generalization of von Neumann and Morgenstern's Independence axiom, which we call (...) *Countable Independence*. In violating these principles, they display foibles like those of people who deviate from standard expected utility theory in more mundane cases: they choose dominated strategies, pay to avoid information, and reject expert advice. We precisely characterize the preference relations that satisfy Countable Independence in several equivalent ways: a structural constraint on preferences, a representation theorem, and the principle we began with, that every prospect has a value that some outcome could have. (shrink)

Noah Rosenberg et al.'s 2002 article “Genetic Structure of Human Populations” reported that multivariate genomic analysis of a large cell line panel yielded reproducible groupings (clusters) suggestive of individuals' geographical origins. The paper has been repeatedly cited as evidence that traditional notions of race have a biological basis, a claim its authors do not make. Critics of this misinterpretation have often suggested that it follows from interpreters' personal biases skewing the reception of an objective piece of scientific writing. I (...) contend, however, that the article itself to some degree facilitates this misrepresentation. I analyze in detail several verbal and visual features of the original article that may predispose aspects of its racial interpretation; and, tracing the arguments of one philosopher and one popular science writer, I show how these features are absorbed, transformed into arguments for a biological basis of race, and re-attributed to the original. The essay demonstrates how even slight ambiguities can enable the misappropriation of scientific writing, unintentionally undermining the authors' stated circumspection on the relationship between cluster and race. (shrink)

The cluster account, one of the best attempts at art classification, is guilty of ahistoricism. While cluster theorists may be happy to limit themselves to accounting for what art is now rather than how the term was understood in the past, they cannot ignore the fact that people seem to apply different clusters when judging art from different times. This paper shows that while allowing for this kind of historical relativity may be necessary to save the account, doing so (...) could result in incorporating an essentially institutional component or making the theory extremely complex and virtually impossible to use. (shrink)

For aggregative theories of moral value, it is a challenge to rank worlds that each contain infinitely many valuable events. And, although there are several existing proposals for doing so, few provide a cardinal measure of each world's value. This raises the even greater challenge of ranking lotteries over such worlds—without a cardinal value for each world, we cannot apply expected value theory. How then can we compare such lotteries? To date, we have just one method for doing so (proposed (...) separately by Arntzenius, Bostrom, and Meacham), which is to compare the prospects for value at each individual location, and to then represent and compare lotteries by their expected values at each of those locations. But, as I show here, this approach violates several key principles of decision theory and generates some implausible verdicts. I propose an alternative—one which delivers plausible rankings of lotteries, which is implied by a plausible collection of axioms, and which can be applied alongside almost any ranking of infinite worlds. (shrink)

In formal ontology, infinite regresses are generally considered a bad sign. One debate where such regresses come into play is the debate about fundamentality. Arguments in favour of some type of fundamentalism are many, but they generally share the idea that infinite chains of ontological dependence must be ruled out. Some motivations for this view are assessed in this article, with the conclusion that such infinite chains may not always be vicious. Indeed, there may even be room (...) for a type of fundamentalism combined with infinite descent as long as this descent is “boring,” that is, the same structure repeats ad infinitum. A start is made in the article towards a systematic account of this type of infinite descent. The philosophical prospects and scientific tenability of the account are briefly evaluated using an example from physics. (shrink)

Consider an infinite series whose items are each explained by their immediate successor. Does such an infinite explanation explain the whole series or does it leave something to be explained? Hume arguably claimed that it does fully explain the whole series. Leibniz, however, designed a very telling objection against this claim, an objection involving an infinite series of book copies. In this paper, I argue that the Humean claim can, in certain cases, be saved from the Leibnizian (...) “infinite book copies” objection, and that this provides an interesting way to defuse some cosmological arguments for the existence of God and to give a non-theistic but complete explanation of the Universe. In the course of my argumentation, I also show that circular explanations can be “self-explanatory” as well: explaining two items by each other can explain the couple of items tout court. (shrink)

This book examines the philosophy of the nineteenth-century Indian mystic Sri Ramakrishna and brings him into dialogue with Western philosophers of religion, primarily in the recent analytic tradition. Sri Ramakrishna’s expansive conception of God as the impersonal-personal Infinite Reality, Maharaj argues, opens up an entirely new paradigm for addressing central topics in the philosophy of religion, including divine infinitude, religious diversity, the nature and epistemology of mystical experience, and the problem of evil.

This paper discusses an infinite regress that looms behind a certain kind of historical explanation. The movement of one barbarian group is often explained by the movement of others, but those movements in turn call for an explanation. While their explanation can again be the movement of yet another group of barbarians, if this sort of explanation does not stop somewhere we are left with an infinite regress of barbarians. While that regress would be vicious, it cannot be (...) accommodated by several general views about what viciousness in infinite regresses amounts to. This example is additional evidence that we should prefer a pluralist approach to infinite regresses. (shrink)

Although there once was a general consensus among race scholars that applying race categories to humans is biologically illegitimate, this consensus has been erased over the past decade. This is largely due to advances in population genetics that allow biologists to pick out genetic population clusters that approximate some of our common sense racial categories. In this paper, I argue that this new ability really ought not undermine our confidence in the biological illegitimacy of the human races. Unfortunately, the (...) claim that races are biologically legitimate is ambiguous—the sense I will be concerned with here holds that the human species can be divided into biological subspecies. Sesardic and Andreasen have both argued that the cluster results are evidence for subspecies within the human population. I show the cluster results are in fact evidentially inert relative to each author’s preferred subspecies concept. I then sketch the kinds of biological facts that could be used to adjudicate whether human subspecies have existed in the past. (shrink)

This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from non-standard analysis are brought to bear on the problem.

This paper responds to commentaries by Kaiserman and Magidor, and Hawthorne. The case of the infinite clowns can teach us several things.

A common argument for atheism runs as follows: God would not create a world worse than other worlds he could have created instead. However, if God exists, he could have created a better world than this one. Therefore, God does not exist. In this paper I challenge the second premise of this argument. I argue that if God exists, our world will continue without end, with God continuing to create value-bearers, and sustaining and perfecting the value-bearers he has already created. (...) Given this, if God exists, our world—considered on the whole—is infinitely valuable. I further contend that this theistic picture makes our world's value unsurpassable. In support of this contention, I consider proposals for how infinitely valuable worlds might be improved upon, focusing on two main ways—adding value-bearers and increasing the value in present value-bearers. I argue that neither of these can improve our world. Depending on how each method is understood, either it would not improve our world, or our world is unsurpassable with respect to it. I conclude by considering the implications of my argument for the problem of evil more generally conceived. (shrink)

This paper uses a schema for infinite regress arguments to provide a solution to the problem of the infinite regress of justification. The solution turns on the falsity of two claims: that a belief is justified only if some belief is a reason for it, and that the reason relation is transitive.

Our prominent definitions of cognition are too vague and lack empirical grounding. They have not kept up with recent developments, and cannot bear the weight placed on them across many different debates. I here articulate and defend a more adequate theory. On this theory, behaviors under the control of cognition tend to display a cluster of characteristic properties, a cluster which tends to be absent from behaviors produced by non-cognitive processes. This cluster is reverse-engineered from the empirical tests that comparative (...) psychologists use to determine whether a behavior was generated by a cognitive or a non-cognitive process. Cognition should be understood as the natural kind of psychological process that non-accidentally exhibits the properties assessed by these tests (as well as others we have not yet discovered). Finally, I review two plausible neural accounts of cognition's underlying mechanisms?one based in localization of function to particular brain regions and another based in the more recent distributed networks approach to neuroscience?which would explain why these properties non-accidentally cluster. While this notion of cognition may be useful for a number of debates, I here focus on its application to a recent crisis over the distinction between cognition and association in comparative psychology. (shrink)

We can have credences in an infinite number of propositions—that is, our opinion set can be infinite. Accuracy-first epistemologists have devoted themselves to evaluating credal states with the help of the concept of ‘accuracy’. Unfortunately, under several innocuous assumptions, infinite opinion sets yield several undesirable results, some of which are even fatal, to accuracy-first epistemology. Moreover, accuracy-first epistemologists cannot circumvent these difficulties in any standard way. In this regard, we will suggest a non-standard approach, called a relativistic (...) approach, to accuracy-first epistemology and show that such an approach can successfully circumvent undesirable results while having some advantages over the standard approach. (shrink)

This article offers a reconstruction of an argument against infinite regress formulated by Aristotle in Posterior Analytics I 22. I argue against the traditional interpretation of the chapter, according to which singular terms and summa genera, in virtue of having restrict logical roles, provide limits for predicative chains, preventing them from proceeding ad infinitum. As I intend to show, this traditional reading is at odds with some important aspects of Aristotle’s theory of demonstration. More importantly, it fails to explain (...) how his proof is connected to a defence of the existence of ultimate explanations, a connection that must be the case if I 19-22 is advancing a foundationalist way-out to a sceptical challenge raised in I 3. (shrink)

Once one accepts that certain things metaphysically depend upon, or are metaphysically explained by, other things, it is natural to begin to wonder whether these chains of dependence or explanation must come to an end. This essay surveys the work that has been done on this issue—the issue of grounding and infinite descent. I frame the discussion around two questions: (1) What is infinite descent of ground? and (2) Is infinite descent of ground possible? In addressing the (...) second question, I will consider a number of arguments that have been made for and against the possibility of infinite descent of ground. When relevant, I connect the discussion to two important views about the way reality can be structured by grounding: metaphysical foundationalism and metaphysical infinitism. (shrink)

The homeostatic property cluster theory is widely influential for its ability to account for many natural-kind terms in the life sciences. However, the notion of homeostatic mechanism has never been fully explicated. In 2009, Carl Craver interpreted the notion in the sense articulated in discussions on mechanistic explanation and pointed out that the HPC account equipped with such notion invites interest-relativity. In this paper, we analyze two recent refinements on HPC: one that avoids any reference to the causes of the (...) clustering of properties and one that replaces homeostatic mechanisms with causal networks represented by causal graphs. We argue that the former is too slender to account for some inductive inference in science and the latter, thicker account invites interest-relativity, as the original HPC does. This suggests that human interest will be an un-eliminative part of a satisfactory account of natural kindness. We conclude by discussing the implication of interest-relativity to the naturalness, reality, or objectivity of kinds and indicating an overlooked aspect of natural kinds that requires further studies. (shrink)

Supererogatory acts are those that lie “beyond the call of duty.” There are two standard ways to define this idea more precisely. Although the definitions are often seen as equivalent, I argue that they can diverge when options are infinite, or when there are cycles of better options; moreover, each definition is acceptable in only one case. I consider two ways out of this dilemma.

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)

Suppose we wish to decide which of a pair of actions has better consequences in a case in which both actions result in infinite utility. Peter Vallentyne and others have proposed that one action has better consequences than a second if there is a time after which the cumulative utility of the first action always outstrips the cumulative utility of the second. I argue against this principle, in particular I show how cases may arise in which up to any (...) point of time action a1 produces more utility than action a2, yet for each individual involved a2 produces more utility. (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)

I present and defend a novel version of the homeostatic property cluster account of natural kinds. The core of the proposal is a development of the notion of co-occurrence, central to the HPC account, along information-theoretic lines. The resulting theory retains all the appealing features of the original formulation, while increasing its explanatory power, and formal perspicuity. I showcase the theory by applying it to the problem of reconciling the thesis that biological species are natural kinds with the fact that (...) many such species are polymorphic. (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 this chapter I explain Spinoza's concept of "infinite modes". After some brief background on Spinoza's thoughts on infinity, I provide reasons to think that Immediate Infinite Modes are identical to the attributes, and that Mediate Infinite Modes are merely totalities of finite modes. I conclude with some considerations against the alternative view that infinite modes are laws of nature.

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)

According to one of Leibniz's theories of contingency a proposition is contingent if and only if it cannot be proved in a finite number of steps. It has been argued that this faces the Problem of Lucky Proof , namely that we could begin by analysing the concept ‘Peter’ by saying that ‘Peter is a denier of Christ and …’, thereby having proved the proposition ‘Peter denies Christ’ in a finite number of steps. It also faces a more general but (...) related problem that we dub the Problem of Guaranteed Proof . We argue that Leibniz has an answer to these problems since for him one has not proved that ‘Peter denies Christ’ unless one has also proved that ‘Peter’ is a consistent concept, an impossible task since it requires the full decomposition of the infinite concept ‘Peter’. We defend this view from objections found in the literature and maintain that for Leibniz all truths about created individual beings are contingent. (shrink)

Benardete paradoxes involve a beginningless set each member of which satisfies some predicate just in case no earlier member satisfies it. Such paradoxes have been wielded on behalf of arguments for the impossibility of an infinite past. These arguments often deploy patchwork principles in support of their key linking premise. Here I argue that patchwork principles fail to justify this key premise.

The infinite judgement has long been forgotten and yet, as I am about to demonstrate, it may be urgent to revive it for its critical and productive potential. An infinite judgement is neither analytic nor synthetic; it does not produce logical truths, nor true representations, but it establishes the genetic conditions of real objects and the concepts appropriate to them. It is through infinite judgements that we reach the principle of transcendental logic, in the depths of which (...) all reality can emerge in its material and sensible singularity, making possible all generalization and formal abstraction. (shrink)

Le passage spéculatif de la catégorie du mauvais infini dans le véritable infini reste l’un des plus importants dans la Science de la logique. Comme il est bien connu, ce passage est expliqué par Hegel à travers sa théorie de l’idéalité du fini. Pourtant, du fait de sa structure complexe, le surgissement du véritable infini au sein du fini par l’idéalisation peut être considéré comme un processus abstrait, consistant seulement à supprimer la dualité de l’infinité. Cet article se propose donc (...) d’examiner pourquoi l’idéalisation de la véritable infini ne signifie ni une simple neutralisation de la catégorie de la finité ni une infinitisation extérieure de celle-ci, mais un processus dynamique qui s’infinitise en supprimant l’opposition statique du fini et de l’infini. (shrink)

A dominant strand of philosophical thought holds that natural kinds are clusters of objects with shared properties. Cluster theories of natural kinds are often taken to be a late twentieth-century development, prompted by dissatisfaction with essentialism in philosophy of biology. I will argue here, however, that a cluster theory of kinds had actually been formulated by William Whewell (1794-1866) more than a century earlier. Cluster theories of kinds can be characterized in terms of three central commitments, all of which (...) are present in Whewell’s work on classification. Like contemporary cluster theorists, Whewell claims that kinds are united by similarity, that many kinds do not have essences, and that there are “gaps” between kinds. Moreover, Whewell advises taxonomists to look for consilience (roughly, convergence) between different classificatory schemes, a recommendation that reinforces the identification of natural classes with property clusters. Thus Whewell was not only an early cluster theorist, but one with important insights into what a cluster theory of kinds means for the practice of classification. (shrink)

It is possible that the world contains infinitely many agents that have positive and negative levels of well-being. Theories have been developed to ethically rank such worlds based on the well-being levels of the agents in those worlds or other qualitative properties of the worlds in question, such as the distribution of agents across spacetime. In this thesis I argue that such ethical rankings ought to be consistent with the Pareto principle, which says that if two worlds contain the same (...) agents and some agents are better off in the first world than they are in the second and no agents are worse off than they are in the second, then the first world is better than the second. I show that if we accept four axioms – the Pareto principle, transitivity, an axiom stating that populations of worlds can be permuted, and the claim that if the ‘at least as good as’ relation holds between two worlds then it holds between qualitative duplicates of this world pair – then we must conclude that there is ubiquitous incomparability between infinite worlds. I show that this is true even if the populations of infinite worlds are disjoint or overlapping, and that we cannot use any qualitative properties of world pairs to rank these worlds. Finally, I argue that this incomparability result generates puzzles for both consequentialist and non-consequentialist theories of objective and subjective permissibility. (shrink)

Part of a symposium devoted to ‘Prediction, Understanding, and Medicine’, in which Alex Broadbent argues that the nature of medicine is determined by its competences, i.e., which things it can do well. He argues that, although medicine cannot cure well, it can do a good job of enabling people not only to understand states of the human organism and of what has caused them, but also to predict future states of it. From this Broadbent concludes that medicine is (at least (...) in part) essentially a practice of understanding and predicting, not curing. In reply to this bold position, I mount two major criticisms. First, I maintain that the reasons Broadbent gives for doubting that medicine can cure provide comparable reason for doubting that medicine can provide an understanding; roughly, the best explanation of why medicine cannot reliably cure is that we still lack much understanding of health and disease. Second, I object to the claim that a practice is medical only if it facilitates understanding and prediction. Although Broadbent has brought to light certain desirable purposes of medicine that are under-appreciated, my conclusion is that he has not yet provided enough reason to think that understanding and prediction are essential to it. Instead of supposing that medicine has an essence, in fact, I suggest that its nature is best understood in terms of a property cluster. (shrink)

Infinity exists as a concept but has no existence in actuality. For infinity to have existence in actuality either time or space have to already be infinite. Unless something is already infinite, the only way to become infinite is by an 'infinity leap' in an infinitely small moment, and this is not possible. Neither does infinitely small have an existence since anything larger than zero is not infinitely small. Therefore infinity has no existence in actuality.

Molyneux’s question, whether the newly sighted might immediately recognize tactilely familiar shapes by sight alone, has produced an array of answers over three centuries of debate and discussion. I propose the first pluralist response: many different answers, both yes and no, are individually sufficient as an answer to the question as a whole. I argue that this is possible if we take the question to be cluster concept of sub-problems. This response opposes traditional answers that isolate specific perceptual features as (...) uniquely applicable to Molyneux’s question and grant viability to only one reply. Answering Molyneux’s question as a cluster concept may also serve as a methodology for resolving other philosophical problems. (shrink)

A comment on Paul Schoemaker's target article in Behavioral and Brain Sciences, 14 (1991), p. 205-215, "The Quest for Optimality: A Positive Heuristic of Science?" (https://doi.org/10.1017/S0140525X00066140). This comment argues that the optimizing model of decision leads to an infinite regress, once internal costs of decision (i.e., information and computation costs) are duly taken into account.

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 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)

Aggregative moral theories face a series of devastating problems when we apply them in a physically realistic setting. According to current physics, our universe is likely _infinitely large_, and will contain infinitely many morally valuable events. But standard aggregative theories are ill-equipped to compare outcomes containing infinite total value so, applied in a realistic setting, they cannot compare any outcomes a real-world agent must ever choose between. This problem has been discussed extensively, and non-standard aggregative theories proposed to overcome (...) it. This paper addresses a further problem of similar severity. Physics tells us that, in our universe, how remotely in time an event occurs is _relative_. But our most promising aggregative theories, designed to compare outcomes containing infinitely many valuable events, are sensitive to how remote in time those events are. As I show, the evaluations of those theories are then relative too. But this is absurd; evaluations of outcomes must be absolute. So we must reject such theories. Is this objection fatal for all aggregative theories, at least in a relativistic universe like ours? I demonstrate here that, by further modifying these theories to fit with the physics, we can overcome it. (shrink)

The threshold problem is the task of adequately answering the question: “Where does the threshold lie between knowledge and lack thereof?” I start this paper by articulating two conditions for solving it. The first is that the threshold be neither too high nor too low; the second is that the threshold accommodate the significance of knowledge. In addition to explaining these conditions, I also argue that it is plausible that they can be met. Next, I argue that many popular accounts (...) of knowledge cannot meet them. In particular, I lay out a number of problems that standard accounts of knowledge face in trying to meet these conditions. Finally, near the end of this paper, I argue that there is one sort of account that seems to evade these problems. This sort of account, which is called a cluster account of knowledge, says that knowledge is to be accounted for in terms of truth, belief and a cluster of epistemic properties and also that knowledge doesn’t require having all members of the cluster, but merely some subset. (shrink)

Context: The infinite has long been an area of philosophical and mathematical investigation. There are many puzzles and paradoxes that involve the infinite. Problem: The goal of this paper is to answer the question: Which objects are the infinite numbers (when order is taken into account)? Though not currently considered a problem, I believe that it is of primary importance to identify properly the infinite numbers. Method: The main method that I employ is conceptual analysis. In (...) particular, I argue that the infinite numbers should be as much like the finite numbers as possible. Results: Using finite numbers as our guide to the infinite numbers, it follows that infinite numbers are of the structure w + (w* + w) a + w*. This same structure also arises when a large finite number is under investigation. Implications: A first implication of the paper is that infinite numbers may be large finite numbers that have not been investigated fully. A second implication is that there is no number of finite numbers. Third, a number of paradoxes of the infinite are resolved. One change that should occur as a result of these findings is that “infinitely many” should refer to structures of the form w + (w* + w) a + w*; in contrast, there are “indefinitely many” natural numbers. Constructivist content: The constructivist perspective of the paper is a form of strict finitism. (shrink)

I analyze the role of infinite idealizations used in the renormalization group (RG hereafter) method in explaining universality across microscopically different physical systems in critical phenomena. I argue that despite the reference to infinite limit systems such as systems with infinite correlation lengths during the RG process, the key to explaining universality in critical phenomena need not involve infinite limit systems. I develop my argument by introducing what I regard as the explanatorily relevant property in RG (...) explanations: linearization* property; I then motivate and prove a proposition about the linearization* property in support of my view. As a result, infinite limit systems in RG explanations are dispensable. (shrink)

This paper aims to show that a proper understanding of what Leibniz meant by “hypercategorematic infinite” sheds light on some fundamental aspects of his conceptions of God and of the relationship between God and created simple substances or monads. After revisiting Leibniz’s distinction between (i) syncategorematic infinite, (ii) categorematic infinite, and (iii) actual infinite, I examine his claim that the hypercategorematic infinite is “God himself” in conjunction with other key statements about God. I then discuss (...) the issue of whether the hypercategorematic infinite is a “whole”, comparing the four kinds of infinite outlined by Leibniz in 1706 with the three degrees of infinity outlined in 1676. In the last section, I discuss the relationship between the hypercategorematic infinite and created simple substances. I conclude that, for Leibniz, only a being beyond all determinations but eminently embracing all determinations can enjoy the pure positivity of what is truly infinite while constituting the ontological grounding of all things. (shrink)

Given an infinite set of finite-sized spheres extending in all directions forever, a finite-sized (relative to the spheres inside the set) observer within the set would view the set as a space composed of discrete, finite-sized objects. A hypothetical infinite-sized (relative to the spheres inside the set) observer would view the set as a continuous space and would see no distinct elements within the set. Using this analogy, the mathematics of infinities, such as the assignment of a cardinality (...) to a set, depends on the reference frame of the observer thinking about them (the mind of the mathematician) relative to the infinite set. This reasoning may also relate to the differing views of space in relativity as continuous and in quantum mechanics as discrete. (shrink)

Despite myriad controversies about reasons, two theses are frequently taken for granted: (i) reasons are sources of normative support for actions, attitudes, etc; and (ii) reasons, at least in simple, paradigmatic cases, consist in atomic facts. Call this conjunction “the atomic view.” Against this, we advocate what we call “the cluster view,” on which even in the simplest cases, the normative support for an action or attitude is typically provided by a whole cluster of facts. Moreover, many of these facts (...) are on an explanatory par—they each play the same kind of role in explaining why there is normative support for the action or attitude in question. We show that presupposition of the atomic view generates unnecessary (psuedo-)puzzles⁠—about topics such as rationality, moral worth, and doxastic justification⁠—that can be largely (dis)solved with the cluster view in place. The cluster view’s ability to do this constitutes an argument in its favor, beyond intuitive considerations that we also adduce. (shrink)