We show that infinitesimal probabilities are much too small for modeling the individual outcome of a countably infinite fair lottery.

We seek to elucidate the philosophical context in which one of the most important conceptual transformations of modern mathematics took place, namely the so-called revolution in rigor in infinitesimal calculus and mathematical analysis. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind,and Weierstrass. The dominant current of philosophy in Germany at the time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our (...) main thesis is that Marburg neo-Kantian philosophy formulated a sophisticated position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the “great triumvirate” of Cantor, Dedekind, and Weierstrass that declared infinitesimals conceptus nongrati in mathematical discourse. Rather, following Cohen’s lead, the Marburg philosophers sought to clarify Leibniz’s principle of continuity, and to exploit it in making sense of infinitesimals and related concepts. (shrink)

In this paper, I advance an original view of the structure of space called Infinitesimal Gunk. This view says that every region of space can be further divided and some regions have infinitesimal size, where infinitesimals are understood in the framework of Robinson’s nonstandard analysis. This view, I argue, provides a novel reply to the inconsistency arguments proposed by Arntzenius and Russell, which have troubled a more familiar gunky approach. Moreover, it has important advantages over the alternative views these (...) authors suggested. Unlike Arntzenius’s proposal, it does not introduce regions with no interior. It also has a much richer measure theory than Russell’s proposal and does not retreat to mere finite additivity. (shrink)

The primary purpose of this paper is to analyze the relationship between the familiar non-Archimedean field of hyperreals from Abraham Robinson’s nonstandard analysis and Paolo Giordano’s ring extension of the real numbers containing nilpotents. There is an interesting nontrivial homomorphism from the limited hyperreals into the Giordano ring, whereas the only nontrivial homomorphism from the Giordano ring to the hyperreals is the standard part function, namely, the function that maps a value to its real part. We interpret this asymmetry to (...) mean that the nilpotent infinitesimal values of Giordano’s ring are “smaller” than the hyperreal infinitesimals. By viewing things from the “point of view” of the hyperreals, all nilpotents are zero, whereas by viewing things from the “point of view” of Giordano’s ring, nonnilpotent, nonzero infinitesimals register as nonzero infinitesimals. This suggests that Giordano’s infinitesimals are more fine-grained. (shrink)

Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general.

It is natural to think that questions in the metaphysics of chance are independent of the mathematical representation of chance in probability theory. After all, chance is a feature of events that comes in degrees and the mathematical representation of chance concerns these degrees but leaves the nature of chance open. The mathematical representation of chance could thus, un-controversially, be taken to be what it is commonly taken to be: a probability measure satisfying Kolmogorov’s axioms. The metaphysical questions about chance (...) seem to be left open by all this. I argue that this is a mistake. The employment of real numbers as measures of chance in standard probability theory brings with it commitments in the metaphysics of (objective) chance that are not only substantial but also mistaken. To measure chance properly we need to employ extensions of the real numbers that contain infinitesimals: positive numbers that are infinitely small. But simply using infinitesimals alone is not enough, as a number of arguments show. Instead we need to put three ideas together: infinitesimals, the non-locality of chance and flexibility in measurement. Only those three together give us a coherent picture of chance and its mathematical representation. (shrink)

I propose a theory of space with infinitesimal regions called smooth infinitesimal geometry based on certain algebraic objects, which regiments a mode of reasoning heuristically used by geometricists and physicists. I argue that SIG has the following utilities. It provides a simple metaphysics of vector fields and tangent space that are otherwise perplexing. A tangent space can be considered an infinitesimal region of space. It generalizes a standard implementation of spacetime algebraicism called Einstein algebras. It solves the long-standing problem of (...) interpreting smooth infinitesimal analysis realistically, an alternative foundation of spacetime theories to real analysis, 277–392, 1980). SIA is formulated in intuitionistic logic and is thought to have no classical reformulations. Against this, I argue that SIG is such a reformulation. But SIG has an unorthodox mereology, in which the principle of supplementation fails. (shrink)

The 'best-system' analysis of lawhood [Lewis 1994] faces the 'zero-fit problem': that many systems of laws say that the chance of history going actually as it goes--the degree to which the theory 'fits' the actual course of history--is zero. Neither an appeal to infinitesimal probabilities nor a patch using standard measure theory avoids the difficulty. But there is a way to avoid it: replace the notion of 'fit' with the notion of a world being typical with respect to a theory.

Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general. _1_ Introduction _2_ The Limits of Classical Probability Theory _2.1_ Classical probability functions _2.2_ Limitations _2.3_ Infinitesimals to the rescue? _3_ NAP Theory _3.1_ First four axioms of NAP _3.2_ Continuity and conditional probability _3.3_ The final axiom of (...) NAP _3.4_ Infinite sums _3.5_ Definition of NAP functions via infinite sums _3.6_ Relation to numerosity theory _4_ Objections and Replies _4.1_ Cantor and the Archimedean property _4.2_ Ticket missing from an infinite lottery _4.3_ Williamson’s infinite sequence of coin tosses _4.4_ Point sets on a circle _4.5_ Easwaran and Pruss _5_ Dividends _5.1_ Measure and utility _5.2_ Regularity and uniformity _5.3_ Credence and chance _5.4_ Conditional probability _6_ General Considerations _6.1_ Non-uniqueness _6.2_ Invariance Appendix. (shrink)

We seek to elucidate the philosophical context in which the so-called revolution of rigor in inifinitesimal calculus and mathematical analysis took place. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind, and Weierstrass. The dominant current of philosophy in Germany at that time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our main thesis is that Marburg Neo-Kantian philosophy formulated a sophisticated (...) position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the "great triumvirate" of Cantor, Dedekind, and Weierstrass. Expressed in terms of modern mathematics, the Marburg philosophers saw the introduction of both infinitesimals and limits as completions whose prototype was Dedekind's of the rational number system resulting in the real numbers. At least partially,, this idea of "completions" can be captured in terms of a category-theoretical description of the conceptual development of modern mathematics. The feasibility of such a modern reformuation may be taken as evidence that the philosophical resources of Marburg neo-Kantianism may be of interest even for contemporary philosophy of mathematics. (shrink)

Before establishing his mature interpretation of infinitesimals as fictions, Gottfried Leibniz had advocated their existence as actually existing entities in the continuum. In this paper I trace the development of these early attempts, distinguishing three distinct phases in his interpretation of infinitesimals prior to his adopting a fictionalist interpretation: (i) (1669) the continuum consists of assignable points separated by unassignable gaps; (ii) (1670-71) the continuum is composed of an infinity of indivisible points, or parts smaller than any assignable, (...) with no gaps between them; (iii) (1672- 75) a continuous line is composed not of points but of infinitely many infinitesimal lines, each of which is divisible and proportional to a generating motion at an instant (conatus). In 1676, finally, Leibniz ceased to regard infinitesimals as actual, opting instead for an interpretation of them as fictitious entities which may be used as compendia loquendi to abbreviate mathematical reasonings. (shrink)

I compare three sorts of case in which philosophers have argued that we cannot assert the Law of Excluded Middle for statements of identity. Adherents of Smooth Infinitesimal Analysis deny that Excluded Middle holds for statements saying that an infinitesimal is identical with zero. Derek Parfit contended that, in certain sci-fi scenarios, the Law does not hold for some statements of personal identity. He also claimed that it fails for the statement ‘England in 1065 was the same nation as England (...) in 1067’. I argue that none of these cases poses a serious threat to Excluded Middle. My analysis of the last example casts doubt on the principle of the Determinacy of Distinctness. While David Wiggins's ‘conceptualist realism’ provides a metaphysics which can dispense with that principle, it leaves no house-room for infinitesimals. (shrink)

"The development of the calculus during the 17th century was successful in mathematical practice, but raised questions about the nature of infinitesimals: were they real or rather fictitious? This collection of essays, by scholars from Canada, the US, Germany, United Kingdom and Switzerland, gives a comprehensive study of the controversies over the nature and status of the infinitesimal. Aside from Leibniz, the scholars considered are Hobbes, Wallis, Newton, Bernoulli, Hermann, and Nieuwentijt. The collection also contains newly discovered marginalia of (...) Leibniz to the writings of Hobbes."--BOOK JACKET. (shrink)

It has been recently debated whether there exists a so-called “easy road” to nominalism. In this essay, I attempt to fill a lacuna in the debate by making a connection with the literature on infinite and infinitesimal idealization in science through an example from mathematical physics that has been largely ignored by philosophers. Specifically, by appealing to John Norton’s distinction between idealization and approximation, I argue that the phenomena of fractional quantum statistics bears negatively on Mary Leng’s proposed path to (...) easy road nominalism, thereby partially defending Mark Colyvan’s claim that there is no easy road to nominalism. (shrink)

ABSTRACTA probability model is underdetermined when there is no rational reason to assign a particular infinitesimal value as the probability of single events. Pruss claims that hyperreal probabilities are underdetermined. The claim is based upon external hyperreal-valued measures. We show that internal hyperfinite measures are not underdetermined. The importance of internality stems from the fact that Robinson’s transfer principle only applies to internal entities. We also evaluate the claim that transferless ordered fields may have advantages over hyperreals in probabilistic modeling. (...) We show that probabilities developed over such fields are less expressive than hyperreal probabilities. (shrink)

Péraire, Y., Infinitesimal approach to almost-automorphic functions, Annals of Pure and Applied Logic 63 283–297. Thanks to the use of ideal elements of several levels, we are able to give a compact topological characterization of almost-automorphic functions. This new characterization turns out to be equivalent to a geometrical one: the existence of a relatively dense group of “pointwise periods”. However, the more significant result obtained, in our opinion, is a very important lowering of the complexity in characterizations and proofs.

In the years following Bertrand Russell's visit in China, fragments from his work on mathematical logic and the foundations of mathematics started to enter the Chinese intellectual world. While up...

The goal of this paper is to investigate the relation between Cohen's approach to differential calculus and his doctrine of pure thinking. We claim that Cohen's logic of origin is firmly based on his interpretation of infinitesimal analysis. More precisely, the transcendental method, when applied to differential calculus, reveals the productive capacity of thinking expressed by the judgment of origin.

In this paper, I develop an original view of the structure of space—called infinitesimal atomism—as a reply to Zeno’s paradox of measure. According to this view, space is composed of ultimate parts with infinitesimal size, where infinitesimals are understood within the framework of Robinson’s nonstandard analysis. Notably, this view satisfies a version of additivity: for every region that has a size, its size is the sum of the sizes of its disjoint parts. In particular, the size of a finite (...) region is the sum of the sizes of its infinitesimal parts. Although this view is a coherent approach to Zeno’s paradox and is preferable to Skyrms’s infinitesimal approach, it faces both the main problem for the standard view and the main problem for finite atomism, leaving it with no clear advantage over these familiar alternatives. (shrink)

The notion of indivisibles and atoms arose in ancient Greece. The continuum—that is, the collection of points in a straight line segment, appeared to have paradoxical properties, arising from the ‘indivisibles’ that remain after a process of division has been carried out throughout the continuum. In the seventeenth century, Italian mathematicians were using new methods involving the notion of indivisibles, and the paradoxes of the continuum appeared in a new context. This cast doubt on the validity of the methods and (...) the reliability of mathematical knowledge which had been regarded as established by the axiomatic method in geometry expounded by Aristotle’s younger contemporary Euclid. The teaching of indivisibles was banned within the Society of Jesus, the Jesuits. In England, indivisibles were used by the mathematician John Wallis, and there was an acrimonious and extended feud between Wallis and the philosopher Thomas Hobbes over legitimate methods of argument in mathematics. Notions of the infinitesimal were used by Isaac Newton and Gottfried Leibniz, and were attacked by Bishop Berkeley for the vagueness of the concept and the illegitimate reasoning applied to it. This article discusses aspects of these events with reference to the book Infinitesimal by Amir Alexander and to other sources. Also discussed are wider issues arising from Alexander’s book including: the changes in cultural sensibility associated with the growth of new mathematical and scientific knowledge in the seventeenth century, the changes in language concomitant with these changes, what constitutes valid methods of enquiry in various contexts, and the question of authoritarianism in knowledge. More general aims of this article are to widen the immediate mathematical and historical contexts in Alexander’s book, to bridge a gap in conversations between mathematics and the humanities, and to relate mathematical ideas to wider human and contemporary issues. (shrink)

During the Enlightenment period the concept of the infinitesimal was developed as a means to solve the mathematical problem of the incommensurability between human reason and the movements of physical beings. In this essay, the author analyzes the metaphysical prejudices subtending Enlightenment Humanism through the lens of the infinitesimal calculus. One of the consequences of this analysis is the perception of a two-fold possibility occasioned by the infinitesimal. On the one hand, it occasions an extreme form of humanism, “transhumanism,” which (...) exhibits limitless confidence in the possibility of human science. On the other hand, the concept of the infinitesimal also contains within itself a source for a critical “posthumanism,” that is to say, a source which initiates the dissolution of the presuppositions of humanism while simultaneously announcing a different ontological organization. In, Tostoy’s novel takes up the problem of the relation between reason and motion and makes the two-fold possibility visible by presenting a contrast between its theoretical presentations and the lived experiences of the characters in the novel. Thus, is the setting in which the author has chosen to conduct this analysis. (shrink)

The infinitesimal methods commonly used in the 17th and 18th centuries to solve analytical problems had a great deal of elegance and intuitive appeal. But the notion of infinitesimal itself was flawed by contradictions. These arose as a result of attempting to representchange in terms ofstatic conceptions. Now, one may regard infinitesimals as the residual traces of change after the process of change has been terminated. The difficulty was that these residual traces could not logically coexist with the static (...) quantities traditionally employed by mathematics. The solution to this difficulty, as it turns out, is to regard these quantities asalso being subject to (a form of) change, for then they will have the same nature as the infinitesimals representing the residual traces of change, and will become,ipso facto, compatible with these latter.In fact, the category-theoretic models which realize the Principle of Infinitesimal Linearity may themselves be regarded as representations of a general concept of variation (cf. Bell (1986)). While the static set-theoretical models represent change or motion by making a detour through the actual (but static) infinite, the varying category-theoretic models enable such change to be representeddirectly, thus permitting the introduction of geometric infinitesimals and, as we have attempted to demonstrate in this paper, the virtually complete incorporation of the methods of the early calculus.It is surely a remarkable — even an ironic — fact that the contradiction between the flux of the objective world and the stasis of mathematical entities has found its resolution in category theory, a branch of mathematics commonly, and, as one now sees, mistakenly, regarded as the summit of gratuitous abstraction. (shrink)

We discuss various ways, which have been plainly justified in the secondhalf of the twentieth century, to introduce infinitesimals, and we considerthe new style of reasoning in mathematical analysis that they allow.

We discuss various ways, which have been plainly justified in the secondhalf of the twentieth century, to introduce infinitesimals, and we considerthe new style of reasoning in mathematical analysis that they allow.

Es werden zwei Bedeutungen von „Infinitesimal“ unterschieden und zwei Thesen verteidigt: (1) Leibniz glaubte, das Infinitesimale in einer der beiden Bedeutungen sei nicht nur eine nützliche Erdichtung, sondern es sei sogar notwendig fur die Differentialrechnung; (2) die moderne Nichtstand-Analysis rechtfertigt weder Leibniz's Griinde fur die Einführung des Infinitesimalen noch seinen Gebrauch desselben.

This is a commentary on the use of squares in elementary statistics. One sees an ubiquitous use of squares in statistics, and the analogy of "distance in a statistical sense" is teased out. We conjecture that elementary statistical theory has its roots in classical Calculus, and preserves the notion of two senses described in this paper. We claim that the senses of the differentials dx/dy hold between classical and modern infinitesimal Calculus and show how this sense becomes cashed out in (...) both models. We suggest that the commonalities of both accounts can be used to find a informational realist account of Calculus. This account involves infinitesimal Calculus as acting as an epistemic mediator that expands our conceptual space and enables us to create new measures. (shrink)

A number of philosophers have attempted to solve the problem of null-probability possible events in Bayesian epistemology by proposing that there are infinitesimal probabilities. Hájek and Easwaran have argued that because there is no way to specify a particular hyperreal extension of the real numbers, solutions to the regularity problem involving infinitesimals, or at least hyperreal infinitesimals, involve an unsatisfactory ineffability or arbitrariness. The arguments depend on the alleged impossibility of picking out a particular hyperreal extension of the (...) real numbers and/or of a particular value within such an extension due to the use of the Axiom of Choice. However, it is false that the Axiom of Choice precludes a specification of a hyperreal extension—such an extension can indeed be specified. Moreover, for all we know, it is possible to explicitly specify particular infinitesimals within such an extension. Nonetheless, I prove that because any regular probability measure that has infinitesimal values can be replaced by one that has all the same intuitive features but other infinitesimal values, the heart of the arbitrariness objection remains. (shrink)

The foundations of analysis offered by Cauchy and Riemann were not immediately welcomed by the mathematical community. Before 1870 the foundations of mathematics were considered more or less a national affair. In this paper, Dutch ideas of rigour in analysis between 1840 and 1870 will be discussed. These ideas show that Dutch mathematicians were aware of developments abroad but preferred the concept of infinitesimals as a foundation of mathematics.

This article studies Leibniz’s treatment of infinitesimals: their application to the calculus and his opinion that they did not exist. In partial agreement with Brunschvicg’s and Ishiguro’s commentaries on the paradoxical status of Leibniz´s infinitesimals, this study proposes a synthesis of both..

In this paper, we endeavour to give a historically accurate presentation of how Leibniz understood his infinitesimals, and how he justified their use. Some authors claim that when Leibniz called them “fictions” in response to the criticisms of the calculus by Rolle and others at the turn of the century, he had in mind a different meaning of “fiction” than in his earlier work, involving a commitment to their existence as non-Archimedean elements of the continuum. Against this, we show (...) that by 1676 Leibniz had already developed an interpretation from which he never wavered, according to which infinitesimals, like infinite wholes, cannot be regarded as existing because their concepts entail contradictions, even though they may be used as if they exist under certain specified conditions—a conception he later characterized as “syncategorematic”. Thus, one cannot infer the existence of infinitesimals from their successful use. By a detailed analysis of Leibniz’s arguments in his De quadratura of 1675–1676, we show that Leibniz had already presented there two strategies for presenting infinitesimalist methods, one in which one uses finite quantities that can be made as small as necessary in order for the error to be smaller than can be assigned, and thus zero; and another “direct” method in which the infinite and infinitely small are introduced by a fiction analogous to imaginary roots in algebra, and to points at infinity in projective geometry. We then show how in his mature papers the latter strategy, now articulated as based on the Law of Continuity, is presented to critics of the calculus as being equally constitutive for the foundations of algebra and geometry and also as being provably rigorous according to the accepted standards in keeping with the Archimedean axiom. (shrink)

A construction of the real number system based on almost homomorphisms of the integers $\mathbb {Z}$ was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On -saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently (...) by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG . In NBG , it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter). (shrink)

In contrast with some recent theories of infinitesimals as non-Archimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth (...) Infinitesimal Analysis, as propounded by John Bell. I find some salient differences, especially with regard to higher-order infinitesimals. I illustrate these differences by a consideration of how each approach might be applied to propositions of Newton’s Principia concerning the derivation of force laws for bodies orbiting in a circle and an ellipse. “If the Leibnizian calculus needs a rehabilitation because of too severe treatment by historians in the past half century, as Robinson suggests (1966, 250), I feel that the legitimate grounds for such a rehabilitation are to be found in the Leibnizian theory itself.”—(Bos 1974–1975, 82–83). (shrink)

Unveränderter Nachdruck der Originalausgabe von 1883.

Fictionalists believe that scientific models are about model systems that are imaginary. Michael Weisberg has claimed that fictionalism is indefensible because many scientific models are about model systems that are unimaginable. According to a certain account of imagination, what Weisberg says is plausible. According to another, more defensible account of imagination, it is not. I discuss these issues within the context of an allegedly unimaginable model system in ecology, but the conclusions I draw are more general. I then describe how (...) fictionalism should be recast in order to deal with Weisberg’s critique. (shrink)

A set of six publications have introduced, commented, criticized and defended Amir Alexander’s book on infinitesimals published in 2014. The aim of the following article is to bring the various arguments together.

By recent work on some conjectures of Pillay, each definably compact group G in a saturated o-minimal expansion of an ordered field has a normal "infinitesimal subgroup" G00 such that the quotient G/G00, equipped with the "logic topology", is a compact (real) Lie group. Our first result is that the functor G → G/G00 sends exact sequences of definably compact groups into exact sequences of Lie groups. We then study the connections between the Lie group G/G00 and the o-minimal spectrum (...) G̃ of G. We prove that G/G00 is a topological quotient of G̃. We thus obtain a natural homomorphism ψ* from the cohomology of G/G00 to the (Čech-)cohomology of G̃. We show that if G00 satisfies a suitable contractibility conjecture then $\widetilde{G^{00}}$ is acyclic in Čech cohomology and ψ* is an isomorphism. Finally we prove the conjecture in some special cases. (shrink)

In “The Jesuits and the Method of Indivisibles” David Sherry criticizes a central thesis of my book Infinitesimal: that in the seventeenth century the Jesuits sought to suppress the method of indivisibles because it undermined their efforts to establish a perfect rational and hierarchical order in the world, modeled on Euclidean Geometry. Sherry accepts that the Jesuits did indeed suppress the method, but offers two objections. First, that the book does not distinguish between indivisibles and infinitesimals, and that whereas (...) the Jesuits might have reason to object to the first, the second posed no problem for them. Second, seeking an alternative explanation for the Jesuits’ hostility to the method, he proposes that its implicit atomism conflicted with the Catholic doctrine of the sacrament of the Eucharist, and was therefore heretical. In response to Sherry’s first criticism I point out that the Jesuits objected to all forms of the method of indivisibles, and that the distinction between indivisibles and infinitesimals consequently cannot explain the fight over the method in the seventeenth century. With regards to the Eucharist, I agree with Sherry that the Jesuits were indeed concerned about the method’s affinity to atomism and materialism, though for a different reason: these doctrines were antithetical to their efforts to impose divine hierarchy and order on the world. In as much as the technical details of the miracle of the Eucharist mattered, they provided no grounds for objecting to a mathematical doctrine. (shrink)

During the first months of 1887, while completing the drafts of his Mitteilungen zur Lehre vom Transfiniten, Georg Cantor maintained a continuous correspondence with Benno Kerry. Their exchange essentially concerned two main topics in the philosophy of mathematics, namely, (a) the concept of natural number and (b) the infinitesimals. Cantor's and Kerry's positions turned out to be irreconcilable, mostly because of Kerry's irremediably psychologistic outlook, according to Cantor at least. In this study, I will examine and reconstruct the main (...) points in the discussion around (a) and (b) and stress some interesting aspects of the philosophical and mathematical thought of Benno Kerry. (shrink)

In the paper of Brown and Priest 2004, the authors developed the chunk and permeate method, which they described as a ?paraconsistent reasoning strategy?. There it is suggested that the method of chunk and permeate could apply to the historical infinitesimal calculus. However, no attempt was made to look at actual historical examples. In this paper, I show that the method of chunk and permeate can indeed apply, as a rational reconstruction, to certain of Isaac Newton's arguments that use (...) class='Hi'>infinitesimals. This rational reconstruction maintains and uses, rather than sidesteps, the apparent contradictions in Newton's arguments. The applicability of chunk and permeate to other historical arguments, e.g. of Leibniz/L'Hospital and Fermat, has also been investigated and will be communicated in future publications. (shrink)