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Rethinking geometrical exactness
 Historia Mathematica
"... A crucial concern of earlymodern geometry was that of fixing appropriate norms for deciding whether some objects, procedures, or arguments should or should not be allowed in it. According to Bos, this is the exactness concern. I argue that Descartes ’ way to respond to this concern was to suggest a ..."
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A crucial concern of earlymodern geometry was that of fixing appropriate norms for deciding whether some objects, procedures, or arguments should or should not be allowed in it. According to Bos, this is the exactness concern. I argue that Descartes ’ way to respond to this concern was to suggest an appropriate conservative extension of Euclid’s plane geometry (EPG). In section 1, I outline the exactness concern as, I think, it appeared to Descartes. In section 2, I account for Descartes ’ views on exactness and for his attitude towards the most common sorts of constructions in classical geometry. I also explain in which sense his geometry can be conceived as a conservative extension of EPG. I conclude by briefly discussing some structural similarities and differences between Descartes’ geometry and EPG. Une question cruciale pour la geométrie à l’âge classique fut celle de décider si certains objets, procédures ou arguments devaient ou non être admis au sein de ses limites. Selon Bos, c’est la question de l’exactitude. J’avance que Descartes répondit à cette question en suggérant une extension conservative de la géomètrie plane d’Euclide (EPG). Dans la section 1, je reconstruis la question de l’exactitude ainsi que, selon moi, elle se présentait d’abord aux yeux de Descartes. Dans la section 2, je rends compte des vues de Descartes sur la question de l’exactitude et de son attitude face au types de constructions plus communes dans la geométrie classique. Je montre aussi en quel sens sa geométrie peut se concevoir comme une extension conservative de EPG. Je conclue en discutant brièvement certaines analogies et différences structurales entre la geométrie de Desacrtes et EPG.
Conceptions of the Continuum
"... Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question ..."
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Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question the idea from current set theory that the continuum is somehow a uniquely determined concept. Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions. 1. What is the continuum? On the face of it, there are several distinct forms of the continuum as a mathematical concept: in geometry, as a straight line, in analysis as the real number system (characterized in one of several ways), and in set theory as the power set of the natural numbers and, alternatively, as the set of all infinite sequences of zeros and ones. Since it is common to refer to the continuum, in what sense are these all instances of the same concept? When one speaks of the continuum in current settheoretical
Second order logic, set theory and foundations of mathematics
"... The question, whether second order logic is a better foundation for mathematics than set theory, is addressed. The main difference between second order logic and set theory is that set theory builds up a transfinite cumulative hierarchy while second order logic stays within one application of the po ..."
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The question, whether second order logic is a better foundation for mathematics than set theory, is addressed. The main difference between second order logic and set theory is that set theory builds up a transfinite cumulative hierarchy while second order logic stays within one application of the power sets. It is argued that in many ways this difference is illusory. More importantly, it is argued that the often stated difference, that second order logic has categorical characterizations of relevant mathematical structures, while set theory has nonstandard models, amounts to no difference at all. Second order logic and set theory permit quite similar categoricity results on one hand, and similar nonstandard models on the other hand. 1
Only up to isomorphism? Category theory and the . . .
"... Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can ..."
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Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in other branches of contemporary mathematics. If such a specification suffices, then a categorytheoretical approach will be highly appropriate. But if sets have a richer ‘nature ’ than is preserved under isomorphism, then such an approach will be inadequate.
INTERPRETABILITY IN ROBINSON’S Q
"... Abstract. Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is an impassable barrier in the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Rapha ..."
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Abstract. Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is an impassable barrier in the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Raphael Robinson’s theory of arithmetic Q. In the shadow of this program, some very nice logical investigations and results were produced by a number of people, not only regarding what can be interpreted in Q but also what cannot be so interpreted. We explain some of these results and rely on them to discuss Nelson’s position. §1. Introduction. Let L be the firstorder language with equality whose nonlogical symbols are the constant 0, the unary function symbol S (for successor) and two binary function symbols + (for addition) and · (for multiplication). The following theory was introduced in [35] (see also the systematic [42]): Definition 1. Raphael Robinson’s theory Q is the theory in the language L
Our Knowledge of the External World (1914)
, 2013
"... Universal skepticism, though logically irrefutable, is practically barren; it can only, therefore, give a certain hesitancy to our beliefs, and cannot be used to substitute other beliefs for them. ..."
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Universal skepticism, though logically irrefutable, is practically barren; it can only, therefore, give a certain hesitancy to our beliefs, and cannot be used to substitute other beliefs for them.
Philos Stud DOI 10.1007/s1109801301604 Mathematical representation: playing a role
"... Abstract The primary justification for mathematical structuralism is its capacity to explain two observations about mathematical objects, typically natural numbers. Noneliminative structuralism attributes these features to the particular ontology of mathematics. I argue that attributing the feature ..."
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Abstract The primary justification for mathematical structuralism is its capacity to explain two observations about mathematical objects, typically natural numbers. Noneliminative structuralism attributes these features to the particular ontology of mathematics. I argue that attributing the features to an ontology of structural objects conflicts with claims often made by structuralists to the effect that their structuralist theses are versions of Quine’s ontological relativity or Putnam’s internal realism. I describe and argue for an alternative explanation for these features which instead explains the attributes them to the mathematical practice of representing numbers using more concrete tokens, such as sets, strokes and so on.
MATHEMATICAL TRUTH REGAINED
"... Benacerraf’s Dilemma (BD), as formulated by Paul Benacerraf in “Mathematical Truth, ” is about the apparent impossibility of reconciling a “standard ” (i.e., classical Platonic) semantics of mathematics with a “reasonable ” (i.e., causal, spatiotemporal) epistemology of cognizing true statements. In ..."
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Benacerraf’s Dilemma (BD), as formulated by Paul Benacerraf in “Mathematical Truth, ” is about the apparent impossibility of reconciling a “standard ” (i.e., classical Platonic) semantics of mathematics with a “reasonable ” (i.e., causal, spatiotemporal) epistemology of cognizing true statements. In this paper I spell out a new solution to BD. I call this new solution a positive Kantian phenomenological solution for three reasons: (1) It accepts Benacerraf’s preliminary philosophical assumptions about the nature of semantics and knowledge, as well as all the basic steps of BD, and then shows how we can, consistently with those very assumptions and premises, still reject the skeptical conclusion of BD and also adequately explain mathematical knowledge. (2) The standard semantics of mathematically necessary truth that I offer is based on Kant’s philosophy of arithmetic, as interpreted by Charles Parsons and by me. (3) The reasonable epistemology of mathematical knowledge that I offer is based on the phenomenology of logical and mathematical selfevidence developed by early Husserl in Logical Investigations and
Full text: 23,000 From Numerical Concepts to Concepts of Number
"... Abstract (Short version): Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe these abilities set the stage for later mathematics: the natural numbers and arithmetic. But the connection between these early and later skills is far from obv ..."
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Abstract (Short version): Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe these abilities set the stage for later mathematics: the natural numbers and arithmetic. But the connection between these early and later skills is far from obvious. We sketch what we think is the most likely model for infant abilities and argue that children could not extrapolate mature math concepts from these beginnings. We suggest instead that children may arrive at natural numbers by constructing mathematical schemas on the basis of innate abilities and math principles. Abstract (Long version): Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe these abilities set the stage for later mathematics: the natural numbers and arithmetic. But the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (a) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children’s understanding of number terms do not necessarily tap these concepts. (b) True