The ‘Pierre Duhem Thesis.’ A Reappraisal of Duhem’s Discovery of the Physics of the Middle Ages

Pierre Duhem is the discoverer of the physics of the Middle Ages. The discovery that there existed a physics of the Middle Ages was a surprise primarily for Duhem himself. This discovery completely changed the way he saw the evolution of physics, bringing him to formulate a complex argument for the growth and continuity of scientific knowledge, which I call the ‘Pierre Duhem Thesis’ (not to be confused either with what Roger Ariew called the ‘true Duhem thesis’ as opposed to the Quine-Duhem thesis, which he persuasively argued is not Duhem’s, or with the famous ‘Quine-Duhem Thesis’ itself). The ‘Pierre Duhem Thesis’ consists of five sub-theses (some transcendental in nature, some other causal, factual, or descriptive), which are not independent, as they do not work separately (but only as a system) and do not relate to reality separately (but only simultaneously). The famous and disputed ‘continuity thesis’ is part, as a sub-thesis, from this larger argument. I argue that the ‘Pierre Duhem Thesis’ wraps up all of Duhem’s discoveries in the history of science and as a whole represents his main contribution to the historiography of science. The ‘Pierre Duhem Thesis’ is the central argument of Pierre Duhem's work as historian of science

The Basic Laws of Cardinal Number

An overview of what Frege accomplishes in Part II of Grundgesetze, which contains proofs of axioms for arithmetic and several additional results concerning the finite, the infinite, and the relationship between these notions. One might think of this paper as an extremely compressed form of Part II of my book Reading Frege's Grundgesetze

The road to modern logic - an interpretation

This paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the emergence of modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplars (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalences in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatic, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which First-Order Logic matured will be seen to have played a primary role in its shaping

Mathematical Abstraction, Conceptual Variation and Identity

One of the key features of modern mathematics is the adoption of the abstract method. Our goal in this paper is to propose an explication of that method that is rooted in the history of the subject

Mathematical Abstraction, Conceptual Variation and Identity

One of the key features of modern mathematics is the adoption of the abstract method. Our goal in this paper is to propose an explication of that method that is rooted in the history of the subject

Discourse on method: Questions on Polo’s method of the abandonment of the limit

Journal article published in Miscelanea PolianaAfter studying a few authors of the 'System Philosophies' —the family of views that draw inspiration from Descartes— an aspiring young philosopher remarks: why are these people obsessed with the theory of knowledge instead of tackling the real issues? The youngster could have been wrong in his observation, yet all agree that the obsession for the method over thematic questions is the hallmark of the modern thinkers...with Kant marking the so-called critique/dogmatic divide… Is it any wonder then that they seem to be quickly running out of line? Consider the following example: after seeing a slithering cobra coiled up in a corner of your tent on waking up in the morning in a camping expedition would you first stop to think of whether the eyes are reliable enough to be taken on their face value? Would you not rather be more inclined to think that the matter in hand is weightier than a consideration of the conditions for the possibility of seeing it? What is more important: the disease causing organism under observation or the electron microscope the researcher is using to observe it? Why the obsession for method with the consequent relegation of the real topics to a distant second place? We know that thinking is important but should we stay the course of our inquiry just in the thought process? Would the following expose provide an answer to this puzzle?After studying a few authors of the 'System Philosophies' —the family of views that draw inspiration from Descartes— an aspiring young philosopher remarks: why are these people obsessed with the theory of knowledge instead of tackling the real issues? The youngster could have been wrong in his observation, yet all agree that the obsession for the method over thematic questions is the hallmark of the modern thinkers...with Kant marking the so-called critique/dogmatic divide… Is it any wonder then that they seem to be quickly running out of line? Consider the following example: after seeing a slithering cobra coiled up in a corner of your tent on waking up in the morning in a camping expedition would you first stop to think of whether the eyes are reliable enough to be taken on their face value? Would you not rather be more inclined to think that the matter in hand is weightier than a consideration of the conditions for the possibility of seeing it? What is more important: the disease causing organism under observation or the electron microscope the researcher is using to observe it? Why the obsession for method with the consequent relegation of the real topics to a distant second place? We know that thinking is important but should we stay the course of our inquiry just in the thought process? Would the following expose provide an answer to this puzzle

Philosophical Method and Galileo's Paradox of Infinity

We consider an approach to some philosophical problems that I call the Method of Conceptual Articulation: to recognize that a question may lack any determinate answer, and to re-engineer concepts so that the question acquires a definite answer in such a way as to serve the epistemic motivations behind the question. As a case study we examine “Galileo’s Paradox”, that the perfect square numbers seem to be at once as numerous as the whole numbers, by one-to-one correspondence, and yet less numerous, being a proper subset. I argue that Cantor resolved this paradox by a method at least close to that proposed—not by discovering the true nature of cardinal number, but by articulating several useful and appealing extensions of number to the infinite. Galileo was right to suggest that the concept of relative size did not apply to the infinite, for the concept he possessed did not. Nor was Bolzano simply wrong to reject Hume’s Principle (that one-to-one correspondence implies equal number) in the infinitary case, in favor of Euclid’s Common Notion 5 (that the whole is greater than the part), for the concept of cardinal number (in the sense of “number of elements”) was not clearly defined for infinite collections. Order extension theorems now suggest that a theory of cardinality upholding Euclid’s principle instead of Hume’s is possible. Cantor’s refinements of number are not the only ones possible, and they appear to have been shaped by motivations and fruitfulness, for they evolved in discernible stages correlated with emerging applications and results. Galileo, Bolzano, and Cantor shared interests in the particulate analysis of the continuum and in physical applications. Cantor’s concepts proved fruitful for those pursuits. Finally, Gödel was mistaken to claim that Cantor’s concept of cardinality is forced on us; though Gödel gives an intuitively compelling argument, he ignores the fact that Euclid’s Common Notion is also intuitively compelling, and we are therefore forced to make a choice. The success of Cantor’s concept of cardinality lies not in its truth (for concepts are not true or false), nor its uniqueness (for it is not the only extension of number possible), but in its intuitive appeal, and most of all, its usefulness to the understanding

Philosophical Method and Galileo's Paradox of Infinity

We consider an approach to some philosophical problems that I call the Method of Conceptual Articulation: to recognize that a question may lack any determinate answer, and to re-engineer concepts so that the question acquires a definite answer in such a way as to serve the epistemic motivations behind the question. As a case study we examine “Galileo’s Paradox”, that the perfect square numbers seem to be at once as numerous as the whole numbers, by one-to-one correspondence, and yet less numerous, being a proper subset. I argue that Cantor resolved this paradox by a method at least close to that proposed—not by discovering the true nature of cardinal number, but by articulating several useful and appealing extensions of number to the infinite. Galileo was right to suggest that the concept of relative size did not apply to the infinite, for the concept he possessed did not. Nor was Bolzano simply wrong to reject Hume’s Principle (that one-to-one correspondence implies equal number) in the infinitary case, in favor of Euclid’s Common Notion 5 (that the whole is greater than the part), for the concept of cardinal number (in the sense of “number of elements”) was not clearly defined for infinite collections. Order extension theorems now suggest that a theory of cardinality upholding Euclid’s principle instead of Hume’s is possible. Cantor’s refinements of number are not the only ones possible, and they appear to have been shaped by motivations and fruitfulness, for they evolved in discernible stages correlated with emerging applications and results. Galileo, Bolzano, and Cantor shared interests in the particulate analysis of the continuum and in physical applications. Cantor’s concepts proved fruitful for those pursuits. Finally, Gödel was mistaken to claim that Cantor’s concept of cardinality is forced on us; though Gödel gives an intuitively compelling argument, he ignores the fact that Euclid’s Common Notion is also intuitively compelling, and we are therefore forced to make a choice. The success of Cantor’s concept of cardinality lies not in its truth (for concepts are not true or false), nor its uniqueness (for it is not the only extension of number possible), but in its intuitive appeal, and most of all, its usefulness to the understanding