Aggregative consequentialism and several other popular moral theories are threatened with paralysis: when coupled with some plausible assumptions, they seem to imply that it is always ethically indifferent what you do. Modern cosmology teaches that the world might well contain an infinite number of happy and sad people and other candidate value-bearing locations. Aggregative ethics implies that such a world contains an infinite amount of positive value and an infinite amount of negative value. You can affect only a finite amount (...) of good or bad. In standard cardinal arithmetic, an infinite quantity is unchanged by the addition or subtraction of any finite quantity. So it appears you cannot change the value of the world. Modifications of aggregationism aimed at resolving the paralysis are only partially effective and cause severe side effects, including problems of “fanaticism”, “distortion”, and erosion of the intuitions that originally motivated the theory. Is the infinitarian challenge fatal? (shrink)

The attempt to formulate a viable empiricist and non-foundationalist epistemology of science faces four problems here confronted. The first is an apparent loss of objectivity in science, in the conditions of use of models in applied science. The second derives from the theory-infection of scientific language, with an apparent loss of objective conditions of truth and reference. The third, often cited as objection to The Scientific Image, is the apparent theory-dependence of the distinction between what is and is not observable. (...) The fourth and last is the loss of the possibility of objective evaluation of rationality in scientific methodology. It is argued that each of these problems is illusory. (shrink)

_Winner of the 1974 National Book Award_ The product of many years of reflection on phenomenology, this book is a comprehensive and creative introduction to the philosophy of Edmund Husserl. Natanson uses Husserl's later work as a clue to the meaning of his entire intellectual career, showing how his earlier methodological work evolved into the search for transcendental roots and developed into a philosophy of the life-world. Phenomenology, for Natanson, emerges as a philosophy of origin, a transcendental discipline concerned with (...) consciousness, history, and world rather than with introspection and traditional metaphysical warfare. (shrink)

Thermodynamics and Statistical Mechanics are related to one another through the so-called "thermodynamic limit'' in which, roughly speaking the number of particles becomes infinite. At critical points (places of physical discontinuity) this limit fails to be regular. As a result, the "reduction'' of Thermodynamics to Statistical Mechanics fails to hold at such critical phases. This fact is key to understanding an argument due to Craig Callender to the effect that the thermodynamic limit leads to mistakes in Statistical Mechanics. I discuss (...) this argument and argue that the conclusion is misguided. In addition, I discuss an analogous example where a genuine physical discontinuity---the breaking of drops---requires the use of infinite idealizations. (shrink)

This paper presents examples of infinite diagrams whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized, and a “pre” form of this thesis that every proof can be presented in everyday statements-only form.

With respect to the metaphysics of infinity, the tendency of standard debates is to either endorse or to deny the reality of ‘the infinite’. But how should we understand the notion of ‘reality’ employed in stating these options? Wittgenstein’s critical strategy shows that the notion is grounded in a confusion: talk of infinity naturally takes hold of one’s imagination due to the sway of verbal pictures and analogies suggested by our words. This is the source of various philosophical pictures that (...) in turn give rise to the standard metaphysical debates: that the mathematics of infinity corresponds to a special realm of infinite objects, that the infinite is profoundly huge or vast, or that the ability to think about infinity reveals mysterious powers in human beings. First, I explain Wittgenstein’s general strategy for undermining philosophical pictures of ‘the infinite’ – as he describes it in Zettel; and then show how that critical strategy is applied to Cantor’s diagonalization proof in Remarks on the Foundations of Mathematics II. (shrink)

Pierre de Fermat is known as the inventor of modern number theory. He invented–improved many methods useful in this discipline. Fermat often claimed to have proved his most difficult theorems thanks to a method of his own invention: the infinite descent. He wrote of numerous applications of this procedure. Unfortunately, he left only one almost complete demonstration and an outline of another demonstration. The outline concerns the theorem that every prime number of the form 4n + 1 is the sum (...) of two squares. In this paper, we analyse a recent proof of this theorem. It is interesting because: it follows all the elements of which Fermat wrote in his outline; it represents a good introduction to all logical nuances and mathematical variants concerning this method of which Fermat spoke. The assertions by Fermat will also be framed inside their historical context. Therefore, the aims of this paper are related to the history of mathematics and to the logic of proof-methods. (shrink)

In this essay, I provide an overview of the debate on infinite and essential idealizations in physics. I will first present two ostensible examples: phase transitions and the Aharonov–Bohm effect. Then, I will describe the literature on the topic as a debate between two positions: Essentialists claim that idealizations are essential or indispensable for scientific accounts of certain physical phenomena, while dispensabilists maintain that idealizations are dispensable from mature scientific theory. I will also identify some attempts at finding a middle (...) ground between the essentialists and dispensabilists camps. Finally, I will raise questions for future research on essential and infinite idealizations via the notion of exploration. (shrink)

Popper functions allow one to take conditional probabilities as primitive instead of deriving them from unconditional probabilities via the ratio formula P=P/P. A major advantage of this approach is it allows one to condition on events of zero probability. I will show that under plausible symmetry conditions, Popper functions often fail to do what they were supposed to do. For instance, suppose we want to define the Popper function for an isometrically invariant case in two dimensions and hence require the (...) Popper function to be rotationally invariant and defined on pairs of sets from some algebra that contains at least all countable subsets. Then it turns out that the Popper function trivializes for all finite sets: P=1 for all A ) if B is finite. Likewise, Popper functions invariant under all sequence reflections can’t be defined in a way that models a bidirectionally infinite sequence of independent coin tosses. (shrink)

One argument that Leibniz employed to rule out the possibility of a world soul appears to turn on the assumption that the very notion of an infinite number or of an infinite whole is inconsistent. This argument was considered in a series of three papers published in The Leibniz Review: in the first, by Laurence Carlin, the argument was delineated and analyzed; in the second, by myself, the argument was criticized and rejected; in the third, by Richard Arthur, an attempt (...) was made to defend Leibniz’s argument against my criticisms. In the present paper, I take up the matter again in an attempt to clarify the issues involved and to defend my original criticisms of the argument against the objections raised by Arthur. (shrink)

In the Transcendental Ideal Kant discusses the principle of complete determination: for every object and every predicate A, the object is either determinately A or not-A. He claims this principle is synthetic, but it appears to follow from the principle of excluded middle, which is analytic. He also makes a puzzling claim in support of its syntheticity: that it represents individual objects as deriving their possibility from the whole of possibility. This raises a puzzle about why Kant regarded it as (...) synthetic, and what his explanatory claim means. I argue that the principle of complete determination does not follow from the principle of excluded middle because the externally negated or ?negative? judgement ?Not (S is P)? does not entail the internally negated or ?infinite? judgement ?S is not-P.? Kant's puzzling explanatory claim means that empirical objects are determined by the content of the totality of experience. This entails that empirical objects are completely determinate if and only if the totality of experience has a completely determinate content. I argue that it is not a priori whether experience has such a completely determinate content and thus not analytic that objects obey the principle of complete determination. (shrink)

We study the relationship between the cofinality $c(Sym(\omega))$ of the infinite symmetric group and the cardinal invariants $\frak{u}$ and $\frak{g}$ . In particular, we prove the following two results. Theorem 0.1 It is consistent with ZFC that there exists a simple $P_{\omega_{1}}$ -point and that $c(Sym(\omega)) = \omega_{2} = 2^{\omega}$ . Theorem 0.2 If there exist both a simple $P_{\omega_{1}}$ -point and a $P_{\omega_{2}}$ -point, then $c(Sym(\omega)) = \omega_{1}$.

Es hat sich als ausgesprochen schwierig erwiesen, für Leibniz' Auffassung, daß kontingente Wahrheiten unendlich komplex sind, eine Interpretation zu finden, die diese Auffassung kohärent erscheinen läßt. Dies liegt daran, daß seine Kommentatoren dazu neigen, sich für die unendliche Analyse einer kontingenten Wahrheit am Vorbild eines nicht endenden logischen Beweises zu orientieren. Ich versuche hingegen zu zeigen, daß unendliche Analysen als unendliche Reihe immer komplexerer und detaillierterer physischer Argumente aufgefaßt werden sollten. Ferner versuche ich zu zeigen, daß die Theorie der unendlichen (...) Analyse speziell für die Wahrung dessen entworfen wurde, was man ,innerweltliche' Kontingenz nennen könnte, d.h. einer Kontingenz, die aus internen Merkmalen einer gegebenen moglichen Welt hervorgeht, im Gegensatz zu einer Kontingenz, die aus Überlegungen dariiber entsteht, welche anderen Welten Gott geschaffen haben konnte. Um innerweltliche Köntingenz geht es, wenn Leibniz scheinbare Gefahren für die Freiheit betrachtet, die aus seiner Erklärung von Wahrheit als einem begrifflichen Enthaltensein entstehen. (shrink)

This article builds on a recent paper coauthored by the present author, H. Hosni and F. Montagna. It is meant to contribute to the logical foundations of probability theory on many-valued events and, specifically, to a deeper understanding of the notion of strict coherence. In particular, we will make use of geometrical, measure-theoretical and logical methods to provide three characterizations of strict coherence on formulas of infinite-valued Łukasiewicz logic.

The interpretation of propositions in Lukasiewicz's infinite-valued calculus as answers in Ulam's game with lies--the Boolean case corresponding to the traditional Twenty Questions game--gives added interest to the completeness theorem. The literature contains several different proofs, but they invariably require technical prerequisites from such areas as model-theory, algebraic geometry, or the theory of ordered groups. The aim of this paper is to provide a self-contained proof, only requiring the rudiments of algebra and convexity in finite-dimensional vector spaces.

Descartes’s metaphysics posits a sharp distinction between two types of non-finitude, or unlimitedness: whereas God alone is infinite, numbers, space, and time are indefinite. The distinction has proven difficult to interpret in a way that abides by the textual evidence and conserves the theoretical roles that the distinction plays in Descartes’s philosophy—in particular, the important role it plays in the causal proof for God’s existence in the Meditations. After formulating the interpretive task, I criticize extant interpretations of the distinction. I (...) then propose an alternative at whose core is the idea that whereas the indefinite is a structural, iterative notion, designating the absence of an upper bound, the infinite is an ontic notion, signifying being in general, or what is, without qualification. (shrink)

If there is one question that has perplexed the best minds in every society, it is how to raise the individuals from their present state to a higher state of existence and perfection? The answers have been tried using different formulations in history: religious, scientific and political. The common factor in all these historical formulations was that they were designed in opposition to each other and therefore left many things unaccounted. The aim of this paper is to explore the idea (...) of self-perfection as a continuous streak of infinite vertical ascension. The paper has elaborated the idea of ‘Verticality’, propounded by Peter Sloterdijk for looking in to the domains of self-perfection and the stages of vertical ascension. For our purpose, we have relied on the example of discipline of Management that has progressively theorized about different aspects and stages of existence. (shrink)

David Bostock; III*—Aristotle, Zeno, and the Potential Infinite, Proceedings of the Aristotelian Society, Volume 73, Issue 1, 1 June 1973, Pages 37–52, https://.

We give a constructive proof of McNaughton's theorem stating that every piecewise linear function with integral coefficients is representable by some sentence in the infinite-valued calculus of Lukasiewicz. For the proof we only use Minkowski's convex body theorem and the rudiments of piecewise linear topology.

This paper deals with stochastically globally exponential stability for stochastic impulsive differential systems with discrete delays and infinite distributed delays. By using vector Lyapunov function and average dwell-time condition, we investigate the unstable impulsive dynamics and stable impulsive dynamics of the suggested system, and some novel stability criteria are obtained for SIDSs with DDs and IDDs. Moreover, our results allow the discrete delay term to be coupled with the nondelay term, and the infinite distributed delay term to be coupled with (...) the nondelay term. Finally, two examples are given to verify the effectiveness of our theories. (shrink)

In this paper, the author turns to Hans-Georg Gadamer’s philosophical hermeneutics to examine the experience of grieving. Specifically, the author argues that grieving may be grasped as a limit situation of memory. This approach suggests that grieving cannot be adequately captured by a stage model theory but, instead, poses an infinite task that is fraught with difficulty and ethical demands. The author develops this approach in reference not only to Hans-Georg Gadamer but recent research by Nancy Moules and Kate Beamer.

We present a general schema of easy normalization proofs for finite systems S like first-order arithmetic or subsystems of analysis, which have good infinitary counterparts S ∞ . We consider a new system S ∞ + with essentially the same rules as S ∞ but different derivable objects: a derivation d∈S ∞ + of a sequent Γ contains a derivation Φ∈S of Γ . Three simple conditions on Φ including a normal form theorem for S ∞ + easily imply a (...) weak normalization theorem for S . We give three examples of application of this schema. First, we take S≡PA but restrict the attention to derivations of Σ 1 0 -sentences. In this case it is possible to take S ∞ + to be essentially standard formulation of PA ∞ . Next, we illustrate extension to subsystems of analysis and consider the system BI 1 ∗ of W. Buchholz having the strength of ID 1 , again for derivations of Σ 1 0 -sentences. Finally, we return to the first-order arithmetic to illustrate changes needed to treat derivations of arbitrary formulas. (shrink)

Loveland and Meyer have studied necessary and sufficient conditions for an infinite binary string x to be recursive in terms of the programsize complexity relative to n of its n-bit prefixes xn. Meyer has shown that x is recursive iff ∃c, ∀n, K(xn/n) ≤ c, and Loveland has shown that this is false if one merely stipulates that K(xn/n) ≤ c for infinitely..

By around the age of 5½, many children in the United States judge that numbers never end, and that it is always possible to add 1 to a set. These same children also generally perform well when asked to label the quantity of a set after one object is added (e.g., judging that a set labeled “five” should now be “six”). These findings suggest that children have implicit knowledge of the “successor function”: Every natural number, n, has a successor, n (...) + 1. Here, we explored how children discover this recursive function, and whether it might be related to discovering productive morphological rules that govern language‐specific counting routines (e.g., the rules in English that represent base‐10 structure). We tested 4‐ and 5‐year‐old children’s knowledge of counting with three tasks, which we then related to (a) children’s belief that 1 can always be added to any number (the successor function) and (b) their belief that numbers never end (infinity). Children who exhibited knowledge of a productive counting rule were significantly more likely to believe that numbers are infinite (i.e., there is no largest number), though such counting knowledge was not directly linked to knowledge of the successor function, per se. Also, our findings suggest that children as young as 4 years of age are able to implement rules defined over their verbal count list to generate number words beyond their spontaneous counting range, an insight which may support reasoning over their acquired verbal count sequence to infer that numbers never end. (shrink)

In classical propositional logic, a theory T is prime iff it is complete. In Łukasiewicz infinite-valued logic the two notions split, completeness being stronger than primeness. Using toric desingularization algorithms and the fine structure of prime ideal spaces of free ℓ -groups, in this paper we shall characterize prime theories in infinite-valued logic. We will show that recursively enumerable prime theories over a finite number of variables are decidable, and we will exhibit an example of an undecidable r.e. prime theory (...) over countably many variables. (shrink)

In a satiric chapter of David Foster Wallace’s novel Infinite Jest, a mock media expert reports how American consumers of the near future recoil from a new communication device known as “videophony” and return to the voice-only telephone of the Bell Era. This article explores the said chapter in the framework of media theories reading the telephone as a “synecdoche of technology,” considering Wallace’s vision of videophony’s rise and fall in a future society from two angles: It discusses the antitechnological (...) bent of the chapter against the background of other visions of future media in 1990s American fiction and it foregrounds the segment’s emphasis on solipsism as a destructive force in all forms of communication. The article argues that Infinite Jest moves beyond questions relevant only to the (video) telephone itself. Rather, Wallace’s novel contrasts literary communication—complex, multilayered, at times monumental—with discourses and practices overly optimistic with regard to the facility of mediated interactions per se. (shrink)

Given a countable graph, we say a set A of its vertices is universal if it contains every countable graph as an induced subgraph, and A is weakly universal if it contains every finite graph as an induced subgraph. We show that, for almost every graph on, (1) every set of positive upper density is universal, and (2) every set with divergent reciprocal sums is weakly universal. We show that the second result is sharp (i.e., a random graph on will (...) almost surely contain non‐universal sets with divergent reciprocal sums) and, more generally, that neither of these two results holds for a large class of partition regular families. (shrink)

On Malcolm Schofield’s highly influential reading of the Similarity Regress in Part I of the Parmenides, the problem that the Regress poses is explanatory. Socrates posited the Similarity Form in order to explain why similar things are similar: similar things are similar because they participate in the Form Similarity as copies of the same original. Yet, the Similarity Regress generates an infinite series of Similarity Forms such that explanation is deferred ad infinitum. Schofield provides a philosophical incentive for adopting his (...) reading. He argues that the treatment of similarity in Part II of the dialogue yields a complete explanation of similarity. If we adopt this account, we can avoid the Similarity Regress altogether since a Form of Similarity is not needed in order to explain why similar things are similar. Thus, his interpretation has a hugely important philosophical pay-off. However, there is a different way to read the argument. Socrates claims that each Form is only one. Yet, the Similarity Regress is an argument that generates an infinite series of Similarity Forms. This results in a violation of the principle of non-contradiction: there is both only one Similarity Form and infinitely many Similarity Forms. Yet, anything that is incompatible with the principle of non-contradiction is surely absurd. Nobody, as far as I am aware, has explored whether this reading also has philosophical pay-off if we look at it together with similarity in Part II. However, should this interpretation pay off, it would be a viable alternative to Schofield’s. In this paper, I explore both views in the context of the treatment of similarity in Part II of the Parmenides. I argue for an account of similarity that differs from Schofield’s. Although my account is not wholly dismissive of Schofield’s, it renders the pay-off of Schofield’s account less appealing than he suggests. Furthermore, I show that the account of similarity in Part II also yields important lessons for the proponent of the alternative reading of the Similarity Regress: similarity as treated in Part II simply leads to further infinite regresses, thereby pushing us to consider rejecting the account of similarity in Part II too and to look for some other account intead. (shrink)

In this article, we will describe two theistic modes of “paradoxical detachment” from the Presence of the Infinite, implying the coexistence of attachment and detachment. We will analyze two forms of Christianity-based paradoxical detachment: being dependent on the Ground of soul, while being detached from the representations of the Infinite ; being absolutely dependent on the Infinite, while being detached from any religious morality. The nontheistic mode of detachment from the Presence of the Infinite requires an absolute detachment. We will (...) examine two forms of absolute detachment towards the Presence of the Infinite: on one hand, the all-encompassing emptiness in the Kagyü and Gelug lineages of Tibetan Buddhism; on the other hand, the Heideggerian notion of “groundless abyss.” In the Kagyü and Gelug lineages of Tibetan Buddhism, being absolutely detached is searching for the Enlightenment, while being detached from all concepts. Heideggerian notions of “groundless abyss” and “de-hominization” allow us to reach absolute detachment, while remaining in a non-theistic way of thinking. (shrink)

One argument that Leibniz employed to rule out the possibility of a world soul appears to turn on the assumption that the very notion of an infinite number or of an infinite whole is inconsistent. This argument was considered in a series of three papers published in The Leibniz Review: in the first, by Laurence Carlin, the argument was delineated and analyzed; in the second, by myself, the argument was criticized and rejected; in the third, by Richard Arthur, an attempt (...) was made to defend Leibniz’s argument against my criticisms. In the present paper, I take up the matter again in an attempt to clarify the issues involved and to defend my original criticisms of the argument against the objections raised by Arthur. (shrink)