In this paper I challenge Paolo Palmieri’s reading of the Mach-Vailati debate on Archimedes’s proof of the law of the lever. I argue that the actual import of the debate concerns the possible epistemic (as opposed to merely pragmatic) role of mathematical arguments in empirical physics, and that construed in this light Vailati carries the upper hand. This claim is defended by showing that Archimedes’s proof of the law of the lever is not a way of appealing to a non-empirical (...) source of information, but a way of explicating the mathematical structure that can represent the empirical information at our disposal in the most general way. (shrink)

Leonardo's fascination with Archimedes as well as with his mathematics is well known. There are three fairly extensive and eccentric comments in the surviving notebooks: on his military inventions; on his part in an Anglo-Spanish conflict and on his activities, death and burial at the siege of Syracuse. Reti has examined the first of the three, that about theArchitronitoor steam cannon, mainly considering the origin of the idea for the cannon and its attribution to Archimedes, but with comments on the (...) later influence of Leonardo's ideas. Marshall Clagett has produced the most comprehensive attempt to try to identify Leonardo's sources for the third. (shrink)

This paper is a contribution to our understanding of the technical concept of given in Greek mathematical texts. By working through mathematical arguments by Menaechmus, Euclid, Apollonius, Heron and Ptolemy, I elucidate the meaning of given in various mathematical practices. I next show how the concept of given is related to the terms discussed by Marinus in his philosophical discussion of Euclid’s Data. I will argue that what is given does not simply exist, but can be unproblematically assumed or produced (...) through some effective procedure. Arguments by givens are shown to be general claims about constructibility and computability. The claim that an object is given is related to our concept of an assignment—what is given is available in some uniquely determined, or determinable, way for future mathematical work. (shrink)

ArgumentThe aim of this paper is to employ the newly contextualized historiographical category of “premodern algebra” in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on “geometrical algebra.” Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related toElem.II.5 as (...) a case study, we discuss Euclid's geometrical proofs, the so-called “semi-algebraic” alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing “premodern algebra,” and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition. (shrink)

About a century ago, Ernst Mach argued that Archimedes’s deduction of the principle of the lever is invalid, since its premises contain the conclusion to be demonstrated. Subsequently, many scholars defended Archimedes, mostly on historical grounds, by raising objections to Mach’s reconstruction of Archimedes’s deduction. In the debate, the Italian philosopher and historian of science Giovanni Vailati stood out. Vailati responded to Mach with an analysis of Archimedes’s deduction which was later quoted and praised by Mach himself. In this paper, (...) my objective is to show that the debate can be further advanced, as Mach indicated, by reframing it in terms of the empirical vs. the logical dimensions of mechanics. In this way, I will suggest, the debate about Archimedes’s deduction can be resolved in Mach’s favour.Keywords: Equilibrium; Experience; Thought experiment; Archimedes; Ernst Mach; Giovanni Vailati. (shrink)

The philosophical problem that stems from the successful application of mathematics in the empirical sciences has recently attracted growing interest within philosophers of mathematics and philosophers of science. Nevertheless, little attention has been devoted to the converse applicability issue of how physical considerations find successful application in mathematics. In this article, focusing on some case studies, I address the latter issue and argue that some successful applications of physics to mathematics essentially depend on the use of conservation principles. I conclude (...) with a short discussion of how the successful interplay between conservation principles and mathematics can be accounted for. (shrink)

In this paper I present two cases, taken from the history of science, in which mathematics and physics successfully interplay. These cases provide, respectively, an example of the successful application of mathematics in astronomy and an example of the successful application of mechanics in mathematics. I claim that an illustration of these cases has a twofold value in the context of the applicability debate. First, it enriches the debate with an historical perspective which is largely omitted in the contemporary discussion. (...) Second, it reveals a component of the applicability problem that has received little attention. This component concerns the successful application of physical principles in mathematical practice. With the help of the two examples, in the final part of the paper I address the following question: are successful applications of mathematics to physics and successful applications of physics to mathematics two sides of the same problem? (shrink)

We consider the Geometria Practica of Christopher Clavius, S.J., a surprisingly eclectic and comprehensive practical geometry text, whose first edition appeared in 1604. Our focus is on four particular sections from Books IV and VI where Clavius has either used his sources in an interesting way or where he has been uncharacteristically reticent about them. These include the treatments of Heron’s Formula, Archimedes’ Measurement of the Circle, four methods for constructing two mean proportionals between two lines, and finally an algorithm (...) for computing nth roots of numbers. (shrink)

This paper tries to make sense of Kant's scattered remarks about conic sections to see what light they shed on his philosophy of mathematics. It proceeds by confronting his remarks with the source that seems to have informed his thinking about conic sections: the Conica of Apollonius. The paper raises questions about Kant's attitude towards mathematics and the way he understood the cognitive resources available to us to do mathematics.

This paper explains how mathematical computation can be constructed from weaker recursive patterns typical of natural languages. A thought experiment is used to describe the formalization of computational rules, or arithmetical axioms, using only orally-based natural language capabilities, and motivated by two accomplishments of ancient Indian mathematics and linguistics. One accomplishment is the expression of positional value using versified Sanskrit number words in addition to orthodox inscribed numerals. The second is Pāṇini’s invention, around the fifth century BCE, of a formal (...) grammar for spoken Sanskrit, expressed in oral verse extending ordinary Sanskrit, and using recursive methods rediscovered in the twentieth century. The Sanskrit positional number compounds and Pāṇini’s formal system are construed as linguistic grammaticalizations relying on tacit cognitive models of symbolic form. The thought experiment shows that universal computation can be constructed from natural language structure and skills, and shows why intentional capabilities needed for language use play a role in computation across all media. The evolution of writing and positional number systems in Mesopotamia is used to transfer the thought experiment of “oral arithmetic” to inscribed computation. The thought experiment and historical evidence combine to show how and why mathematical computation is a cognitive technology extending generic symbolic skills associated with language structure, usage, and change. (shrink)

The ArgumentWhen we actually perform an experiment, we do many different things simultaneously – some belonging to the realm of theory, some to the realms of methodology and technique; however, a great deal of what happens is expressible in terms of socially determined images of knowledge or in terms of concepts of reflectivity – second-order concepts – namely thoughts about thoughts.The emergence of experiment as a second-order concept in late antiquity exemplifies the historical development of second-order concepts; it is shown (...) to be rooted in the Sophists' cunning reason and is followed up in the work of Ptolemy, Copernicus and Galileo.Then, by way of epistemological explication, the three levels of representation of an experiment are shown to be analogous to Baxandall's three levels of representation of a picture.Finally it is shown that such an interpretation only makes sense in terms of two-tier thinking: realism, inside a conceptual framework which is chosen or arrived at, relativistically. (shrink)

This paper aims to provide an updated synthesis on the works of Archimedes and the fundamental impact these have had on subsequent mathematical practice. The influence his mathematical processes have had on modern mathematics and how these have helped develop the field is discussed in historical perspective. Some of the recent investigations into the Archimedes Palimpsest are discussed and synthesized, namely, how they alter our understanding of some of his earlier works, and how Archimedean principles are seen to have laid (...) the foundations of possible new branches of mathematics. (shrink)

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such (...) as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised. (shrink)

This paper focuses on the normative analysis—in the sense of the classic decision-theoretic formulation—of decision problems that arise in connection with forensic expert reporting. We distinguish this analytical account from other common types of decision analyses, such as descriptive approaches. While decision theory is, since several decades, an extensively discussed topic in legal literature, its use in forensic science is more recent, and with an emphasis on goals such as the analysis of the logical structure of forensic expert conclusions regarding, (...) for example, propositions of common source of evidential and known materials. Typical examples are so-called identification decisions, especially categorical conclusions according to which fingermarks come from a particular a person of interest. We will present and compare ways of stating forensic identification decisions in decision-theoretic terms and explain their underlying rationale. In particular, we will emphasize the importance of viewing this analysis as normative in the sense of providing a reflective rather than a prescriptive reference point against which people in charge of forensic identification decisions may compare their otherwise intuitive and informal reasoning, before acting. Normative decision analysis in forensic science thus provides a vector through which current practice can be articulated, scrutinized and rethought. (shrink)

Archimedes’ statics is considered as an example of ancient Greek applied mathematics; it is even seen as the beginning of mechanics. Wilbur Knorr made the case regarding this work, as other works by him or other mathematicians from ancient Greece, that it lacks references to the physical phenomena it is supposed to address. According to Knorr, this is understandable if we consider the propositions of the treatise in terms of purely mathematical elaborations suggested by quantitative aspects of the phenomena. In (...) this paper, we challenge Knorr’s view, and address propositions of Archimedes’ statics in their relation to physical phenomena. (shrink)

in this paper we return to Marshall Clagett’s view about the existence of an ancient Greek geometry of motion. It can be read in two ways. As a basic presentation of ancient Greek geometry of motion, followed by some aspects of its further development in landmark works by Galileo and Newton. Conversely, it can be read as a basic presentation of aspects of Galileo’s and Newton’s mathematics that can be considered as developments of a geometry of motion that was first (...) conceived by ancient Greek mathematicians. (shrink)