Leibniz on Infinite Numbers,Infinite Wholes,and Composite Substances

Leibniz claims that nature is actually infinite but rejects infinite number. Are his mathematical commitments out of step with his metaphysical ones? It is widely accepted that Leibniz has a viable response to this problem: there can be infinitely many created substances, but no infinite number of them. But there is a second problem that has not been satisfactorily resolved. It has been suggested that Leibniz's argument against the world soul relies on his rejection of infinite number, and, as such, Leibniz cannot assert that any body has a soul without also accepting infinite number, since any body has infinitely many parts. Previous attempts to address this concern have misunderstood the character of Leibniz's rejection of infinite number. I argue that Leibniz draws an important distinction between ‘wholes’ – collections of parts that can be thought of as a single thing – and ‘fictional wholes’ – collections of parts that cannot be thought of as a single thing, which allows us to make sense of his rejection of infinite number in a way that does not conflict either with his view that the world is actually infinite or that the bodies of substances have infinitely many parts.     

A Tale of Two Thinkers,One Meeting,and Three Degrees of Infinity: Leibniz and Spinoza (1675–8)

The article presents Leibniz's preoccupation (in 1675–6) with the difference between the notion of infinite number, which he regards as impossible, and that of the infinite being, which he regards as possible. I call this issue ‘Leibniz's Problem’ and examine Spinoza's solution to a similar problem that arises in the context of his philosophy. ‘Spinoza's solution’ is expounded in his letter on the infinite (Ep.12), which Leibniz read and annotated in April 1676. The gist of Spinoza's solution is to distinguish between three kinds of infinity and, in particular, between one that applies to substance, and one that applies to numbers, seen as auxiliaries of the imagination. The rest of the paper examines the extent to which Spinoza's solution solves Leibniz's problem. The main thesis I advance is that, when Spinoza and Leibniz say that the divine substance is infinite, in most contexts it is to be understood in non-numerical and non-quantitative terms. Instead, for Spinoza and Leibniz, a substance is said to be infinite in a qualitative sense stressing that it is complete, perfect and indivisible. I argue that this approach solves one strand of Leibniz's problem and leaves another unsolved.     

Characterizing Equivalential and Algebraizable Logics by the Leibniz Operator

In [14] we used the term finitely algebraizable for algebraizable logics in the sense of Blok and Pigozzi [2] and we introduced possibly infinitely algebraizable, for short, p.i.-algebraizable logics. In the present paper, we characterize the hierarchy of protoalgebraic, equivalential, finitely equivalential, p.i.-algebraizable, and finitely algebraizable logics by properties of the Leibniz operator. A Beth-style definability result yields that finitely equivalential and finitely algebraizable as well as equivalential and p.i.-algebraizable logics can be distinguished by injectivity of the Leibniz operator. Thus, from a characterization of equivalential logics we obtain a new short proof of the main result of [2] that a finitary logic is finitely algebraizable iff the Leibniz operator is injective and preserves unions of directed systems. It is generalized to nonfinitary logics. We characterize equivalential and, by adding injectivity, p.i.-algebraizable logics.     

Infinite Reasoning

Our relationship to the infinite is controversial. But it is widely agreed that our powers of reasoning are finite. I disagree with this consensus; I think that we can, and perhaps do, engage in infinite reasoning. Many think it is just obvious that we can't reason infinitely. This is mistaken. Infinite reasoning does not require constructing infinitely long proofs, nor would it gift us with non-recursive mental powers. To reason infinitely we only need an ability to perform infinite inferences. I argue that we have this ability. My argument looks to our best current theories of inference and considers examples of apparent infinite reasoning. My position is controversial, but if I'm right, our theories of truth, mathematics, and beyond could be transformed. And even if I'm wrong, a more careful consideration of infinite reasoning can only deepen our understanding of thinking and reasoning.     

视觉运动知觉脑机制的研究现状

视觉运动是探索脑功能的一个重要研究途径,该对视觉运动通路与色彩的关系、视觉运动知觉的产生,对于视觉信息的不同阶段的处理以及立体视觉等若干问题进行了回顾与讨论,并概述了其中若干理论及其争论,提出了未来视觉研究可能存在的问题与研究方向。     

The brothers james and john bernoulli on the parallelism between logic and algebra

A short seventeenth-century text, sometimes cited as one of the first essays in mathematical logic, is introduced, translated and evaluated. Although by no means sharing the depth and magnitude of the investigations by Leibniz being undertaken at the same time, and although in particular not yet applying algebraic symbolism to logical structures, the treatise is of historical interest as an early published attempt to trace out analogies between logical and mathematical form, and may be viewed as a preliminary step toward the formalization of logic.     

On Kant’s Knowledge of Leibniz’ Metaphysics—a Reply to Garber

Daniel Garber has put forward an argument that aims to show that Kant’s understanding of Leibniz’ metaphysics should be discounted because he could only have had access to a small and narrow sample of Leibniz’ works from around 1710–1714. In particular, Garber argues that as Kant could not have read Leibniz’ correspondence with Arnauld or his correspondence with Des Bosses he could not have had an adequate conception of Leibniz’ understanding of the relation between substance and body. I will show that Kant could have read some of the Arnauld correspondence and practically all of the Des Bosses correspondence, as well as a number of other related texts that are important for understanding Leibniz’ metaphysics, over a decade before writing the Critique of Pure Reason. Garber’s historical-textual argument for dismissing Kant’s account of Leibniz’ metaphysics is therefore seriously misleading.     

Never getting to zero: Elementary school students' understanding of the infinite divisibility of number and matter

Clinical interviews administered to third- to sixth-graders explored children's conceptualizations of rational number and of certain extensive physical quantities. We found within child consistency in reasoning about diverse aspects of rational number. Children's spontaneous acknowledgement of the existence of numbers between 0 and 1 was strongly related to their induction that numbers are infinitely divisible in the sense that they can be repeatedly divided without ever getting to zero. Their conceptualizing number as infinitely divisible was strongly related to their having a model of fraction notation based on division and to their successful judgment of the relative magnitudes of fractions and decimals. In addition, their understanding number as infinitely divisible was strongly related to their understanding physical quantities as infinitely divisible. These results support a conceptual change account of knowledge acquisition, involving two-way mappings between the domains of number and physical quantity.     

Definability of Leibniz Equality

Given a structure for a first-order language L, two objects of its domain can be indiscernible relative to the properties expressible in L, without using the equality symbol, and without actually being the same. It is this relation that interests us in this paper. It is called Leibniz equality. In the paper we study systematically the problem of its definibility mainly for classes of structures that are the models of some equality-free universal Horn class in an infinitary language L_κκ, where κ is an infinite regular cardinal. This revised version was published online in June 2006 with corrections to the Cover Date.     

Leibniz and Luther on the Non-Cognitive Component of Faith

Leibniz was a Lutheran. Yet, upon consideration of certain aspects of his philosophical theology, one might suspect that he was a Lutheran more in name than in intellectual practice. Clearly Leibniz was influenced by the Catholic tradition; this is beyond doubt. However, the extent to which Leibniz was influenced by his own Lutheran tradition—indeed, by Martin Luther himself—has yet to be satisfactorily explored. In this essay, the views of Luther and Leibniz on the non-cognitive component of faith are considered in some detail. According to Luther, the only non-cognitive aspect of faith worth favoring is trust (fiducia), since it is trust in God’s promise of mercy that warrants justification for the sinner. Leibniz, for his part, sides with the Thomistic tradition in emphasizing love (caritas) as the non-cognitive element of faith par excellence. I argue that Leibniz falls into a trap forewarned by Luther himself, even if Leibniz had systematic metaphysical reasons for his disagreement.     

The logic of turmoil: some epistemological and clinical considerations on emotional experience and the infinite

The idea of the infinite has its origins in the very beginnings of western philosophy and was developed significantly by modern philosophers such as Galileo and Leibniz. Freud discovered the Unconscious which does not respect the laws of classical logic, flouting its fundamental principle of non-contradiction. This opened the way to a new epistemology in which classical logic coexists with an aberrant logic of infinite affects. Matte Blanco reorganized this Freudian revolution in logic and introduced the concept of bi-logic, which is an intermingling of symmetric and Aristotelic logics. The authors explore some epistemological and clinical aspects of the functioning of the deep unconscious where the emergence of infinity threatens to overwhelm the containing function of thought, connecting this topic to some of Bion's propositions. They then suggest that bodily experiences can be considered a prime source of the logic of turmoil, and link a psychoanalytic consideration of the infinite to the mind-body relation. Emotional catastrophe is seen both as a defect-a breakdown of the unfolding function which translates unconscious material into conscious experience-and as the consequence of affective bodily pressures. These pressures function in turn as symmetrizing or infinitizing operators. Two clinical vignettes are presented to exemplify the hypotheses.     

Leibniz Filters Revisited

Leibniz filters play a prominent role in the theory of protoalgebraic logics. In [3] the problem of the definability of Leibniz filters is considered. Here we study the definability of Leibniz filters with parameters. The main result of the paper says that a protoalgebraic logic S has its strong version weakly algebraizable iff it has its Leibniz filters explicitly definable with parameters.     

The Astounding Assumption of Infinite Life

The multimillennial philosophical discussion about life after death has received a recent boost in the prospect of immortality attained via technologies. In this newer version, humans generally are considered mortal but may develop means of making themselves immortal. If “immortal” means not mortal, thus existing for infinity, and if the proposed infinite‐existing entity is material, it must inhabit an infinite material universe. If the proposed entity is not material, there must be means by which it can shed its material substance and exist nonmaterially. The article examines arguments for how an infinite life would be possible given current physical understanding. The paper considers a Pascalian‐style wager weighing the likelihood of adjusting to existence wholly within a finite universe versus betting on there being some way to construe the universe(s) as a viable medium for infinite beings. Conclusion: the case for a finite being to exist infinitely has little viable support.     

Existence Is Evidence of Immortality

The universe plausibly has an infinite future and an infinite past. Given unlimited time, every qualitative state that has ever occurred will occur again, infinitely many times. There will thus exist in the future persons arbitrarily similar to you, in any desired respects. A person sufficiently similar to you in the right respects will qualify as literally another incarnation of you. Some theories about the nature of persons rule this out; however, these theories also imply, given an infinite past, that your present existence is a probability‐zero event. Hence, your present existence is evidence against such theories of persons.     

On the Proof Theory of the Modal mu-Calculus

We study the proof-theoretic relationship between two deductive systems for the modal mu-calculus. First we recall an infinitary system which contains an omega rule allowing to derive the truth of a greatest fixed point from the truth of each of its (infinitely many) approximations. Then we recall a second infinitary calculus which is based on non-well-founded trees. In this system proofs are finitely branching but may contain infinite branches as long as some greatest fixed point is unfolded infinitely often along every branch. The main contribution of our paper is a translation from proofs in the first system to proofs in the second system. Completeness of the second system then follows from completeness of the first, and a new proof of the finite model property also follows as a corollary. Presented by Heinrich Wansing     

Remarks on Bolzano's Conception of Necessary Truth

This essay presents a new interpretation of Bolzano's account of necessary truth as set out in §182 of the Theory of Science. According to this interpretation, Bolzano's conception is closely related to that of Leibniz, with some important differences. In the first place, Bolzano's conception of necessary truth embraces not only what Leibniz called metaphysical or brute necessities but also moral necessities (truths grounded in God's choice of the best among all metaphysical possibilities). Second, in marked contrast to Leibniz, Bolzano maintains that there is still plenty of room for contingency even on this broader conception of necessity.     

The mind–body problem and the role of pain: cross-fire between Leibniz and his Cartesian readers

This article is about the exchanges between Leibniz, Arnauld, Bayle and Lamy on the subject of pain. The inability of Leibniz’s system to account for the phenomenon of pain is a recurring objection of Leibniz’s seventeenth-century Cartesian readers to his hypothesis of pre-established harmony: according to them, the spontaneity of the soul and its representative nature cannot account for the affective component of pain. Strikingly enough, this problem has almost never been addressed in Leibniz studies, or only incidentally, through the more general problem of evil. My purpose in this article is to clarify Leibniz’s psychophysical parallelism by opposing his representationist account of pain to the functionalist account endorsed by Arnauld, Bayle, Lamy and Malebranche.     

Spinoza’s Arguments for the Existence of God*

It is often thought that, although Spinoza develops a bold and distinctive conception of God (the unique substance, or Natura Naturans, in which all else inheres and which possesses infinitely many attributes, including extension), the arguments that he offers which purport to prove God’s existence contribute nothing new to natural theology. Rather, he is seen as just another participant in the seventeenth century revival of the ontological argument initiated by Descartes and taken up by Malebranche and Leibniz among others. That this is the case is both puzzling and unfortunate. It is puzzling because although Spinoza does offer an ontological proof for the existence of God, he also offers three other non‐ontological proofs. It is unfortunate because these other non‐ontological proofs are both more convincing and more interesting than his ontological proof. In this paper, I offer reconstructions and assessments of all of Spinoza’s arguments and argue that Spinoza’s metaphysical rationalism and his commitment to something like a Principle of Sufficient Reason are the driving force behind Spinoza’s non‐ontological arguments.     

The best of possible worlds: A testable claim of choice

Leibniz said that the universe, if God-created, would exist at a unique, conjoint, physical maximum: Of all possible worlds, it would be richest in phenomena, but its richness would arise from the simplest physical laws and initial conditions. Using concepts of “variety” and algorithmic informational complexity, Leibniz' claim can be reframed as a testable theory. This theory predicts that the laws and conditions of the actual universe should be simpler, and the universe richer in phenomena, than the presence of observers would require. Tegmark has shown that inhabitants of an infinite multiverse would likely observe simple laws and conditions, but also phenomenal richness just great enough to explain their existence. Empirical observations fit the claim of divine choice better than the claim of an infinite multiverse. The future of the universe, including its future information-processing capacity, is predicted to be endless.