Richard T. W. Arthur

In lively and readable prose, Arthur presents a new approach to the study of logic, one that seeks to integrate methods of argument analysis developed in modern “informal logic” with natural deduction techniques. The dry bones of logic are given flesh by unusual attention to the history of the subject, from Pythagoras, the Stoics, and Indian Buddhist logic, through Lewis Carroll, Venn, and Boole, to Russell, Frege, and Monty Python. A previous edition of this book appeared under the title _Natural Deduction_. This new edition adds clarifications of the notions of explanation, validity and formal validity, a more detailed discussion of derivation strategies, and another rule of inference, Reiteration.

In this essay I explore Leibniz’s changing views on the relation of substance to the continuum, with special attention to his calling the fundamental units of reality “metaphysical points”. I trace the development of his thought on this question, and on his notions of “physical” and “mathematical” points, from the early 1670s through to the end. I note certain enigmas on the way; namely, his notions of “transcreation” and “indistant” points, his peculiar characterization of contiguity, and the apparent violation of the Law of Continuity in texts of 1676 and 1705. I then argue that considerable light is thrown on these enigmas, and on both his earlier and later views, by manuscripts dating from the 1690s and later that have only recently been published.

"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.

In lively and readable prose, Arthur presents a new approach to the study of logic, one that seeks to integrate methods of argument analysis developed in modern “informal logic” with natural deduction techniques. The dry bones of logic are given flesh by unusual attention to the history of the subject, from Pythagoras, the Stoics, and Indian Buddhist logic, through Lewis Carroll, Venn, and Boole, to Russell, Frege, and Monty Python. A previous edition of this book appeared under the title _Natural Deduction_. This new edition adds clarifications of the notions of explanation, validity and formal validity, a more detailed discussion of derivation strategies, and another rule of inference, Reiteration.

In lively and readable prose, Arthur presents a new approach to the study of logic, one that seeks to integrate methods of argument analysis developed in modern “informal logic” with natural deduction techniques. The dry bones of logic are given flesh by unusual attention to the history of the subject, from Pythagoras, the Stoics, and Indian Buddhist logic, through Lewis Carroll, Venn, and Boole, to Russell, Frege, and Monty Python. A previous edition of this book appeared under the title _Natural Deduction_. This new edition adds clarifications of the notions of explanation, validity and formal validity, a more detailed discussion of derivation strategies, and another rule of inference, Reiteration.

In this chapter we give a detailed examination of the substantial treatise Leibniz composed in Paris in 1675-76, the De Quadratura Arithmetica, showing how the techniques that he developed in Proposition 6 of that treatise went beyond the traditional methods of quadrature using exhaustion or indivisibles, and relies upon what we have termed the “Principle of Unassignable Difference”. Although this is a “direct” proof, it does not depend on infinitesimals and infinites, which Leibniz does not introduce until Proposition 8. We show that in the intermediate Proposition 7, Leibniz provided an indirect proof by reductio of the legitimacy of the procedure adopted in Proposition 6. In Proposition 8 he shows how a “direct” proof relying on a continuity argument together with the fiction of infinities and the infinitely small can also be proved to be equivalent to the reductio proof of Proposition 7, thus establishing the inter-translatability between such proofs and direct proofs using the fiction. We then provide convincing evidence that Leibiz continued to value the methods of the DQA after developing his differential calculus, and show the intimate relationship between the two methods.

In this chapter we show that the characterization of certain mathematical entities as fictions, far from being an invention of Leibniz’s to deflect criticisms of his use of infinitesimals, was in fact part of a well-established tradition in the mathematics of his time. After documenting this tradition, we show how Leibniz’s use of the terms ‘fiction’ and its cognate ‘feign’ are consonant with this tradition. We then discuss the status of these fictions in his work with respect to possibility, and in particular the issue of entities such as imaginary roots of equations being “accidentally impossible” (impossibile per accidens).

In this chapter we summarize our main conclusions: that Leibniz treated the infinite and infinitely small as fictions beginning with his work on infinite series in Paris in the mid-1670s, and not as a result of later criticisms; that this did not in itself commit him to the existence of such fictions, and instead relied only on the Principle of Unassignable Difference; that he justified this principle in the DQA by arguments he continued to value, and which formed the basis of his later strategies for justifying the use of infinitesimals and infinities; that he applied himself to the question of the existence of infinitesimals after devising the differential algorithm, and decided it in the negative; that the notion of comparability underlying his “lemmas on incomparables” is a relational notion, and that taking incomparables as actual elements in the continuum is incompatible with Leibniz’s definitions of quantity and number; that Leibniz provided three different strategies for defending his calculus from criticism in the early 1700s, one depending on the Lemmas on Incomparables supported by a reductio argument, a second strategy appealing to the Law of Continuity that is assumed in the ordinary algebraic calculus; and a third that shows that even in these cases where one appeals to the fiction of infinitely small quantities, one can give a justification where these infinitely small quantities are understood syncategorematically: that is as standing for quantities that may be made as small as is needed for the error to be proven null.

In this chapter we examine aspects of Leibniz’s changing views on infinitesimals prior to his invention of the differential algorithm in October 1675, and in the period immediately afterwards. We show how he moved from regarding infinitesimals as actual elements of the continuum, to a point of view where the bounded infinite and the infinitely small were treated as mathematical fictions, and the question of their existence was postponed. When Leibniz resumes the latter question in the early to mid-1676, he comes to reject their existence; they may still be used as fictions, but they are not possible objects.

In this chapter we outline the main issues that have arisen in the interpretation of the status of infinitesimals in Leibniz’s Differential Calculus, and difficulties affecting their resolution. Then we give an overview of our interpretation, based on a thorough consideration of Leibniz’s writings, many of them unknown to previous interpreters. This is based on a firm distinction between the issues of the existence of infinitesimals, their use in mathematics, and the justification of the differential algorithm itself.

In this chapter we examine Leibniz’s publications of the differential calculus in the Nova Methodus and the Tentamen, his famous letter to Malebranche detailing his Law of Continuity, and also his Observatio quod rationes. We analyze his definitions of quantity, number, and homogeneity, and show how he justifies the rectification and quadrature of curves using infinites and infinitesimals by means of his novel conceptions of quasi-minima and quasi-transformations. We argue that this justification depends on taking an infinitesimal dx not to stand for a fixed infinitely small quantity, but for a finite variable quantity that one can take as small as one wishes, and then detail his arguments against taking infinities and infinitesimals as existing elements of the continuum.

In this chapter we discuss Leibniz’s appropriation of the Scholastic term “syncategorematic” to characterize any actually infinite multiplicity of things, taken distributively, not as a collection. We show how this conception coheres with his prior denial of infinite number, and does not connote a merely potential infinite (which Leibniz never advocates). We then discuss Leibniz’s various strategies in his mature writings for justifying his use of infinitesimals and infinites, and, finally, his justification for the rules of his differential algorithm itself.

I describe how Einstein constructed General Relativity, discussing the warping of spacetime, gravitational time dilation and the distortions of time near singularities. Contrary to Einstein’s aims, GR does not evidence a complete physical equivalence of reference frames, with time simply relative to the observer’s coordinate system. I argue that the principle of local becoming is embodied in the geodesic principle of GR, which guarantees the same connection of time with inertia as was ensconced in Newton’s physics and preserved in SR.