This paper introduces the Theory of Spatial Materialization of Quantum Possibilities in an Infinite Space, proposing a novel perspective on the realization of quantum probabilities. Traditional interpretations of quantum mechanics, such as the Copenhagen interpretation and the Many-Worlds hypothesis, approach quantum probabilities as either collapsing into a single observable state or manifesting across parallel universes. This theory suggests an alternative: in an infinite space, quantum possibilities materialize simultaneously in distinct spatial regions, without requiring collapse or parallel universes. Building (...) on principles of infinite space and inevitable probabilities, the theory reinterprets phenomena such as superposition and cosmological configurations, offering a unified framework for understanding how quantum possibilities are distributed in space. The implications extend to questions of causality, time, and the nature of observable reality, while challenging traditional assumptions in quantum mechanics. Although speculative, this approach provides fertile ground for interdisciplinary exploration, bridging cosmology, philosophy of science, and quantum mechanics. (shrink)

Key Statements -/- A new definition of consciousness. Consciousness is the minimal capacity to detect = feel (a) difference(s), whether in environmental conditions or internal states. -/- Reality is posited as an infinite, one-dimensional sequence of informational states, fraying into fractal complexity. This continuum has no origin in time, no final endpoint, and no external boundary. -/- Localized “Islands of Meaning”: Not all configurations appear comprehensible to us. Certain stable pockets yield phenomena that we interpret as consistent physical (...) laws, living organisms, or advanced technologies; we inhabit one of these pockets. -/- Consciousness as a Lens: Though the cosmic code is static in total, consciousness imposes a reading order that births our sense of time and development. Consciousness does not create the code but highlights specific strings as “our reality.” -> Creation of time. -/- Near Kinship of Natural and Artificial Intelligence: Both spring from the same portion of the universal code where symbol manipulation can yield adaptive, learning-based behavior. The difference is one of substrate and historical path, not fundamental principle. -/- Tree of known Intelligences: Biology, memetics, and AI form layered ‘software’ with shared roots on a unified information island, all existing within a vast sea of interconnected data flows and influences. -/- Life - a Servant of Entropy: In our pocket of informational infinity, higher-order information systems, e.g., life — as defined above — can be described as catalysts toward a state of increased entropy. Higher-order information processes, such as (biological and so-called artificial or synthetic) life and other software agents, are governed by the natural law of generating as many microstates — the amount of uncertainty or information in a system — as possible. -/- Determinism Amid Perceived Freedom: All is predetermined at the highest level, yet locally, we experience genuine choice because we lack full knowledge of how the program extends. Moral and ethical considerations remain relevant to navigate these local experiences. (shrink)

In the spatialized Prisoner's Dilemma, players compete against their immediate neighbors and adopt a neighbor's strategy should it prove locally superior. Fields of strategies evolve in the manner of cellular automata (Nowak and May, 1993; Mar and St. Denis, 1993a,b; Grim 1995, 1996). Often a question arises as to what the eventual outcome of an initial spatial configuration of strategies will be: Will a single strategy prove triumphant in the sense of progressively conquering more and more territory without opposition, or (...) will an equilibrium of some small number of strategies emerge? Here it is shown, for finite configurations of Prisoner's Dilemma strategies embedded in a given infinite background, that such questions are formally undecidable: there is no algorithm or effective procedure which, given a specification of a finite configuration, will in all cases tell us whether that configuration will or will not result in progressive conquest by a single strategy when embedded in the given field. The proof introduces undecidability into decision theory in three steps: by (1) outlining a class of abstract machines with familiar undecidability results, by (2) modelling these machines within a particular family of cellular automata, carrying over undecidability results for these, and finally by (3) showing that spatial configurations of Prisoner's Dilemma strategies will take the form of such cellular automata. (shrink)

In this paper, I aim to show that a certain form of immortality, without the need for any intervention from a supernatural being, is almost inevitable for human beings. I take a physicalist starting point: I am a certain configuration of physical particles. Thus, if these particles were reassembled in the same configuration, I would necessarily come back into existence. I address a number of objections raised against this prospect by Eric T. Olson, who argues that the reassembly of such (...) particles following their dispersal after death would be simply impossible, and that if it did occur, it would necessarily be a mere replica rather than the real ‘me’. I suggest that the random redistribution of particles can be thought of as the throw of cosmic ‘dice’. With any specific throw, the likelihood that it yields the particular configuration that constitutes ‘me’ is vanishingly small. But over infinite spans of time, this likelihood increases until it becomes a near certainty. I show that even if this reconfiguration lacks the same causal features as those that gave rise to the original me, this cannot imply that the reassembled me is a mere replica. I acknowledge that my conjectured form of immortality may be unappealing to theists and non-theists alike. I also note that it rests on a linear conception of time, which may not harmonise with current thinking in physics. However, these issues notwithstanding, my conjectured version of immortality is at least as inexorable as many other things we take for granted, which also rest on potentially flawed beliefs about the nature of time and space. (shrink)

In this work, we present a vision of the universe as an eternal chessboard, where physical, chemical, and biological laws function as immutable rules, yet within this structured framework an almost infinite ludic freedom emerges. By integrating the philosophies of Parmenides and Heraclitus with the First and Second Laws of Thermodynamics, we propose a holistic model of reality-one that is simultaneously regulated and unpredictable, ordered and free. The chessboard metaphor serves as a bridge between determinism and indeterminacy, demonstrating how (...) the apparent tension between order and chaos dissolves into a cyclical, self-renewing structure of existence. This perspective challenges the conventional notion of a cosmic beginning or end, rejecting "nothingness" and "creation" as linguistic artifacts rather than ontological realities. Just as each chess game unfolds uniquely within a fixed set of rules, so too does the universe continuously reshuffle its fundamental elements in ways that, while constrained by necessity, remain boundlessly unpredictable. The vast combinatorial possibilities described by Shannon's number in chess-far exceeding the number of atoms in the observable universe-serve as a compelling analogy for the infinite potential configurations of Being. By reframing reality through this lens, this work offers a fresh philosophical, cosmological and metaphysical perspective, inviting deeper reflection on the fundamental nature of existence. (shrink)

Abstract: The Missing Phase in E=mc²—Plasma as the Foundational State of Energy-Mass Equivalence -/- 1. Problem Statement • E=mc² assumes an instantaneous energy-mass transition but lacks an intermediary stabilization state. • Mass should not be treated as a fundamental property but as an emergent resonance of structured energy. • Without a structured intermediary, mass formation remains incomplete, leaving gaps in quantum field theory and cosmology. -/- 2. Core Hypothesis – Plasma-First Theory (PFT) • Mass does not emerge directly from energy (...) but through a coherence-scored resonance process within a Quantum Coherence Field (QCF). • Plasma serves as a structured decoherence state, mediating energy-mass transitions via prime resonance constraints. • Gravity is not an independent force but a standing-wave remnant of plasma-field interactions. -/- 3. Key Implications • Mass formation follows structured prime resonance constraints rather than spontaneous emergence. • Dark matter may be a phase-locked plasma state, explaining its gravitational effects without direct interaction. • Plasma-first modeling predicts new methods for mass manipulation and energy extraction beyond fusion. • Black holes are extreme plasma-phase coherence transitions, not singularities. -/- 4. Experimental Validation • Prime-Based Plasma Spectroscopy → High-energy plasma emissions should exhibit prime-numbered coherence gaps. • Gravitational Resonance Quantization → LIGO data should reveal structured prime-frequency distortions. • Cosmic Spectral Analysis → Dark matter distributions should align with prime resonance constraints. -/- Technological Implications of Plasma-First Theory (PFT) -/- 1. Energy Generation & Storage • Beyond Fusion Power → Plasma resonance tuning could enable more efficient, controlled energy extraction beyond conventional fusion methods. • Resonance-Based Energy Harvesting → Coherence-scored plasma fields could allow direct conversion of structured energy states into usable power. • Zero-Waste Energy Systems → Understanding energy as a phase-state transition could lead to near-perfect energy recycling methods. -/- 2. Mass Manipulation & Material Science • Programmable Mass Formation → Using prime-locked plasma coherence, mass structures could be dynamically altered or synthesized at will. • New State of Matter Engineering → Plasma stabilization techniques may unlock previously unknown stable matter configurations. • Self-Organizing Materials → Materials could be designed to adaptively shift between energy and mass phases based on environmental conditions. -/- 3. Gravity & Inertia Control • Interference-Based Gravity Modulation → If gravity is a plasma standing wave effect, controlled resonance tuning could enable gravitational lensing or reduction of inertia. • Anti-Gravity Applications → Prime-frequency coherence interference might allow local mass fields to become inertia-free, revolutionizing transportation. • Resonant Spacecraft Propulsion → If mass emerges from structured resonance, spacecraft could manipulate their mass-energy phase to optimize thrust efficiency. -/- 4. Black Hole & Dark Matter Engineering • Black Hole Energy Extraction → If black holes are extreme plasma-phase transitions, they could be harnessed for structured energy retrieval instead of seen as one-way sinks. • Dark Matter Utilization → If dark matter is phase-locked plasma, unlocking its resonance signature could enable controlled interaction for energy extraction or shielding applications. -/- 5. Computing & Information Processing • Plasma-Phase Quantum Computing → If coherence-scored resonance governs energy structures, plasma-based quantum computing may outperform conventional qubit models. • Information Storage in Plasma Fields → Instead of silicon-based hardware, structured plasma coherence could encode and retrieve data with near-infinite density and efficiency. -/- 6. Medical & Biophysical Applications • Coherence-Based Healing Technologies → Structured energy fields could be tuned for cellular regeneration, accelerating healing and potentially reversing degenerative conditions. • Bio-Resonance Medicine → If biological structures follow prime-locked resonance principles, medical treatments could be optimized to match natural coherence states for maximum effectiveness. -/- 7. Universal Implications • Cosmological Engineering → Understanding how mass emerges from structured energy could allow future civilizations to modify space-time at will. • Resonance-Based AI → AI could be designed to operate on structured plasma coherence, allowing it to interact with fundamental physical principles instead of traditional computational limitations. In short, PFT could redefine energy, mass, gravity, and technology itself—bridging physics with engineering in a way that eliminates inefficiencies and unlocks entirely new paradigms. -/- This framework redefines mass, gravity, and matter formation, proposing that plasma is the missing intermediary state in energy-matter interactions. (shrink)

A natural extension of Heron's 2000 year old formula for the area of a triangle to the volume of a tetrahedron is presented. This gives the fourth power of the volume as a polynomial in six simple rational functions of the areas of its four faces and three medial parallelograms, which will be referred to herein as "interior faces." Geometrically, these rational functions are the areas of the triangles into which the exterior faces are divided by the points at which (...) the tetrahedron's in-sphere touches those faces. Part I presents an overview of these results and some necessary but little-known background in areal geometry. Part II derives the promised extension, and ends with a conjecture as to how the formula extends to n-dimensional simplices for all n>3. Part III explains how, for n=3, the zeros of the polynomial constitute a five-dimensional semi-algebraic variety consisting almost entirely of collinear tetrahedra with vertices separated by infinite distances, but with generically well-defined distance ratios; it further proves that these unconventional Euclidean configurations can be identified with a quotient of the Klein quadric by an action of a group of reflections isomorphic to ℤ42, wherein four-point configurations in the affine plane constitute a distinguished three-dimensional subvariety. Part IV consists of five appendices which show, among other things, that the algebraic structure of the zeros in the affine plane naturally defines the associated four-element, rank 3 chirotope, aka affine oriented matroid. (shrink)

Writers endorsing a general account of meaning as non-recuperable or non-coincidental from one instantiation to the next may nonetheless treat the heterogeneous contacts between instants of experience as transformations of fleeting forms, states, logics, structures, outlines, surfaces, presences, organizations, patterns, procedures, frames, standpoints. When thought as pattern, the structural- ranscendental moment of eventness upholds a certain logic of internal relation; the elements of the configuration mutually signify each other and the structure presents itself as a fleeting identity, a gathered field. (...) The particularity of eventness is not allowed to split the presumed (temporary) identity of the internal configuration that defines the structure as structure. History would be the endless reframing of a frame, the infinite shifting from paradigm to paradigm. Heidegger and Derrida argue instead that In their essence, Beings don’t HAVE structure or constitution. There is no such THING as a form, a structure, a state. There is no trans-formation but rather a trans-differentiation, (transformation without form, articulation as dislocation) What is being transcended is not form but difference. (shrink)

The quantum measurement problem resolves according to the twofold nature of time. Whereas the continuous evolution of the wave function reflects the fundamental nature of time as continuous presence, the collapse of the wave function indicates the subsidiary aspect of time as the projection of instantaneity from the ongoing present. Each instant irreversibly emerges from the reversible temporal continuum implicit in the smoothly propagating wave function. The basis of this emergence is periodic conflict between quantum systems, the definitive resolution of (...) which requires the momentary reduction of each system from the potentially infinite dimensions of configuration space to the three dimensions of classical space at an instant. (shrink)

What is the form of that which can hold, size, and measure what is happening? Are there some gestures that can, in some sense, size what is occurring in absolute fragility? We trace forms that, suddenly, sometimes, can grasp in a vanishing present what cannot be grasped by any other means. Certain art practices can sustain and give shelter, for a while, to an echo of that multiple and unending past. These practices inexist. These inexistences compose artistic configurations, of (...) which the artworks referred to in this book are outstanding examples. The axiomatic declaration of the inexistence of art operates as an act of flight, a subtraction. Undomiciled artistic practices are multiple, infinite, and emancipatory. From these practices, we can find the sub-stance for an infrapolitical anarchitecture that works in constructing a future yet to be imagined. Another world is possible, of course. We must continue to whisper that other worlds are possible because we are already participants in that memory to come. On Gestural Apparatuses works as a mechanism of delay that aims to indicate an operative movement within artistic thinking. (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)

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)

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)

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)

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)

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)

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 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)

In western philosophy and beyond, a tension between pluralism and monism has sparked many developments and debates. Pluralism of norms, of forms of knowledge, of aesthetic and moral values, of interests etc. has often been pitted against monism. Monism usually implies a hierarchical order of such norms etc. After having traced the origin of this tension between pluralism and monism in ancient tragedy and philosophy, I’m asking in this article whether a rejection of monism and embrace of pluralism necessarily raises (...) the specter of inconsistency and contradiction. The threat of inconsistency need not be associated with pluralism as even with regard to logic an argument can be made for a ‘multiplicity of logics’, as van Lambalgen and Stenning argue in several places. They refer to the varieties in reasoning that can be observed in humans and which are partly due to there being ‘dual systems’ of reasoning: System 1 processes information fast, automatized, and emotional, while System 2 is rather slow, more deliberative, and more rational. In contrast to a widely held view, the authors argue that System 1 processing is not without logic, even though it is a different form of logic from System 2. In addition, the multiplicity of logics they discuss is related to the multiple semantics required for distinct domains of reasoning. This perspective raises the subsequent question how this pluralism can be available while maintaining consistency, how should the available options be configured or related to each other? In the remainer of this chapter I’m addressing several such options for configurating pluralism like: non-foundational pluralism and foundational pluralism, moderate or temporary pluralism, antagonistic pluralism, incompatible pluralism, incommensurable pluralism, ‘Anything goes’ pluralism, complementary pluralism, integrative pluralism, and interactive pluralism. The discussion and examples of these options show that irrespective of its domain, the encounter with different configurations of pluralism shows how some forms of pluralism are likely to be productive, whereas others are less so - which is an important lesson given the prominent pluralism of pluralisms in philosophy and beyond. (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.

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)

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.

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)

This argument demonstrates that within an infinite framework of possibilities, the existence of God—a necessary, omnipotent, omniscient, and perfectly good being—is not only logically possible but metaphysically inevitable. By grounding its premises in logical coherence, probabilistic reasoning, and metaphysical necessity, the argument provides a profound exploration of existence, divinity, and humanity's role in infinite potential. It offers implications for ethical striving, collective responsibility, and the divine foundation of reality. -/- .

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.

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 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 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.

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)

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 article presents a conceptual framework for distinguishing different sorts of heterogeneous digital materials. The hypothesis is that a wide range of heterogeneous data resources can be characterized and classified due to their particular configurations of hypertext features such as scripts, links, interactive processes, and time scalings, and that the hypertext configuration is a major but not sole source of the messiness of big data. The notion of hypertext will be revalidated, placed at the center of the interpretation of (...) networked digital media, and used in the analysis of the fast-growing amounts of heterogeneous digital collections, assemblages, and corpora. The introduction summarizes the wider background of a fast-changing data landscape. (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)

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)

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)

The purpose of this note is to contrast a Cantorian outlook with a non-Cantorian one and to present a picture that provides support for the latter. In particular, I suggest that: i) infinite hyperreal numbers are the (actual, determined) infinite numbers, ii) ω is merely potentially infinite, and iii) infinitesimals should not be used in the di Finetti lottery. Though most Cantorians will likely maintain a Cantorian outlook, the picture is meant to motivate the obvious nature of (...) the non-Cantorian outlook. (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)

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.

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.

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)

I will explore the paradoxical nature of epistemic access. By critiquing the traditional conception of mental states that are labelled as ’knowledge’, I demonstrate the susceptibility of these states to an infinite regress, thus, challenging their existence and validity. I scrutinise the assumption that an epistemic agent can have complete epistemic access to all facts about a given object while simultaneously being ignorant of certain truths that impact the very knowledge claims about the object. I further analyse the implications (...) of this paradox when attempting to fine-grain the parameters of epistemic access, which happens to reveal the vacuity of such definitions and the inevitability of the regress. (shrink)

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)

Many have the intuition that human persons are both extremely and equally valuable. This seeming extremity and equality of vale is puzzling: if overall value is the sum of one’s final value and instrumental value, how could it be that persons share the same extreme value? One way that we can solve the Value Puzzle is by following Andrew Bailey and Josh Rasmussen. Philosophy and Phenomenological Research, 103, 264–277 (2020) and accepting that persons have infinite final value. But there (...) are some significant downsides to their way of thinking about values, which relies on the extended real numbers. We offer a different approach: if we model values using the hyperreal numbers, we can capture many of our intuitions about the extremity and relative equality of human value without incurring the substantial theoretical costs of using the extended real numbers. We also examine other ways of modeling the infinite value of persons and evaluate the strengths and weaknesses of these other accounts compared to ours. (shrink)

Some environmental ethicists and economists argue that attributing infinite value to the environment is a good way to represent an absolute obligation to protect it. Others argue against modelling the value of the environment in this way: the assignment of infinite value leads to immense technical and philosophical difficulties that undermine the environmentalist project. First, there is a problem of discrimination: saving a large region of habitat is better than saving a small region; yet if both outcomes have (...) infinite value, then decision theory prescribes indifference. Second, there is a problem of swamping probabilities: an act with a small but positive probability of saving an endangered species appears to be on par with an act that has a high probability of achieving this outcome, since both have infinite expected value. Our paper shows that a relative concept of infinite value can be meaningfully defined, and provides a good model for securing the priority of the natural environment while avoiding the failures noted by sceptics about infinite value. Our claim is not that the relative infinity utility model gets every detail correct, but rather that it provides a rigorous philosophical framework for thinking about decisions affecting the environment. (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)

There are infinitely many ways the world might be, and there may well be infinitely many people in it. These facts raise moral paradoxes. We explore a conflict between two highly attractive principles: a Pareto principle that says that what is better for everyone is better overall, and a statewise dominance principle that says that what is sure to turn out better is better on balance. We refine and generalize this paradox, showing that the problem is faced by many theories (...) of interpersonal aggregation besides utilitarianism, and by many decision theories besides expected value theory. Considering the range of consistent responses, we find all of them to be quite radical. (shrink)

Une preuve est pure si, en gros, elle ne réfère dans son développement qu’à ce qui est « proche » de, ou « intrinsèque » à l’énoncé à prouver. L’infinité des nombres premiers, un théorème classique de l’arithmétique, est un cas d’étude particulièrement riche pour les recherches philosophiques sur la pureté. Deux preuves différentes de ce résultat sont ici considérées, à savoir la preuve euclidienne classique et une preuve « topologique » plus récente proposée par Furstenberg. D’un point de vue (...) naïf, il semblerait que la première soit pure et la seconde impure. Des objections à cette vue naïve sont ici considérées et réfutées. Concernant la preuve euclidienne, la question relève de la logique, notamment de la définissabilité arithmétique de l’addition en termes de successeur et de divisibilité telle que démontrée par Julia Robinson, tandis qu’en ce qui concerne la preuve topologique, la question relève de la sémantique, notamment pour tout ce qui touche au problème de savoir ce qui est « inclus » dans le contenu d’énoncés particuliers.A proof is pure, roughly, if it draws only on what is « close » or « intrinsic » to the statement being proved. The infinitude of prime numbers, a classical theorem of arithmetic, is a rich case study for philosophical investigation of purity. Two different proofs of this result are considered, namely the classical Euclidean proof and a more recent « topological » proof by Furstenberg. Naively the former would seem to be pure and the latter to be impure. Objections to these naive views are considered and met. In the case of the former the issue rests on logical matters, specifically the arithmetic definability of addition in terms of successor and divisibility shown by Julia Robinson, while in the case of the latter the issue rests on semantic matters, specifically with respect to what is « contained » in the content of particular statements. (shrink)

Noether's Theorem produces the configuration (and symmetry) for everything in Nature.

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)

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)