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What does it mean to say that logic is formal
, 2000
"... Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and ..."
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Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and contingent, analytic and synthetic—indeed, it is often invoked to explain these. Nor, it turns out, can it be explained by appeal to schematic inference patterns, syntactic rules, or grammar. What does it mean, then, to say that logic is distinctively formal? Three things: logic is said to be formal (or “topicneutral”) (1) in the sense that it provides constitutive norms for thought as such, (2) in the sense that it is indifferent to the particular identities of objects, and (3) in the sense that it abstracts entirely from the semantic content of thought. Though these three notions of formality are by no means equivalent, they are frequently run together. The reason, I argue, is that modern talk of the formality of logic has its source in Kant, and these three notions come together in the context of Kant’s transcendental philosophy. Outside of this context (e.g., in Frege), they can come apart. Attending to this
The scope of logic: deduction, abduction, analogy
"... The present form of mathematical logic originated in the twenties and early thirties from the partial merging of two different traditions, the algebra of logic and the logicist tradition (see [27], [41]). This resulted in a new form of logic in which several features of the two earlier traditions co ..."
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The present form of mathematical logic originated in the twenties and early thirties from the partial merging of two different traditions, the algebra of logic and the logicist tradition (see [27], [41]). This resulted in a new form of logic in which several features of the two earlier traditions coexist. Clearly neither the algebra of logic nor the logicist’s logic is identical to the present form of mathematical logic, yet some of their basic ideas can be distinctly recognized within it. One of such ideas is Boole’s view that logic is the study of the laws of thought. This is not to be meant in a psychologistic way. Frege himself states that the task of logic can be represented “as the investigation of the mind; [though] of the mind, not of minds” [17, p. 369]. Moreover Frege never charges Boole with being psychologistic and in a letter to Peano even distinguishes between the followers of Boole and “the psychological logicians ” [16, p. 108]. In fact for Boole the laws of thought which are the object of logic belong “to the domain of what is termed necessary truth ” [2, p. 404]. For him logic does not depend on psychology, on the contrary psychology depends on logic insofar as it is only through an investigation of logical operations that we could obtain “some probable
EDUCATION
"... This research reviewed literature on proof in mathematics education. Several views of proof classified and identified such as psychological approach, (Platonism, empiricism), structural approach, (logicism, formalism, intuitionism), social approach, (ontology, axiomatic systems). All these views ..."
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This research reviewed literature on proof in mathematics education. Several views of proof classified and identified such as psychological approach, (Platonism, empiricism), structural approach, (logicism, formalism, intuitionism), social approach, (ontology, axiomatic systems). All these views of proof are valuable in mathematics education society. The concept of proof can be found in the form of analytic knowledge not of constructive knowledge. Human beings developed their knowledge in the sequence of constructive knowledge to analytic knowledge. Therefore, in mathematics education, the curriculum of mathematics should involve the process of cognitive knowledge development.
The American Mathematical Monthly.
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
What is “formal logic”?
"... The subject of formal logic when treated by the method of setting up a formalized language is called symbolic logic, or mathematical logic or logistic. Church, 1956, p.56. Many people understand the expression “formal logic ” as meaning modern mathematical logic by opposition to traditional logic be ..."
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The subject of formal logic when treated by the method of setting up a formalized language is called symbolic logic, or mathematical logic or logistic. Church, 1956, p.56. Many people understand the expression “formal logic ” as meaning modern mathematical logic by opposition to traditional logic before the revolution that happened in the second part of the 19 th century with Boole, Frege and others. But in fact this expression was created by Kant (see Scholz
The Sources of Certainty in Computation and Formal Systems
, 1999
"... In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathemati ..."
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In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathematical denitions and inferences in such a way that they are unshakable." Hilbert's program relied explicitly on formal systems (equivalently, computational systems) to provide certainty in mathematics. The concepts of computation and formal system were not dened in his time, but Descartes' method may be understood as seeking certainty in essentially the same way. In this article, I explain formal systems as concrete artifacts, and investigate the way in which they provide a high level of certainty arguably the highest level achievable by rational discourse. The rich understanding of formal systems achieved by mathematical logic and computer science in this century illuminates the nature of programs,...
The Sources of Certainty in Computation and Formal Systems
"... In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathemati ..."
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In his Discourse on the Method of Rightly Conducting the Reason, and Seeking Truth in the Sciences, Rene Descartes sought \clear and certain knowledge of all that is useful in life." Almost three centuries later, in \The foundations of mathematics," David Hilbert tried to \recast mathematical denitions and inferences in such a way that they are unshakable." Hilbert's program relied explicitly on formal systems (equivalently, computational systems) to provide certainty in mathematics. The concepts of computation and formal system were not dened in his time, but Descartes' method may be understood as seeking certainty in essentially the same way. In this article, I explain formal systems as concrete artifacts, and investigate the way in which they provide a high level of certainty arguably the highest level achievable by rational discourse. The rich understanding of formal systems achieved by mathematical logic and computer science in this century illuminates the nature of programs,...
Curry’s Formalism as Structuralism∗
, 2005
"... In 1939, Curry proposed a philosophy of mathematics he called formalism. He made this proposal in two works originally written in 1939. These are the two philosophical works for which Curry is known, and they have left a false impression of his views. In this article, I propose to clarify Curry’s vi ..."
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In 1939, Curry proposed a philosophy of mathematics he called formalism. He made this proposal in two works originally written in 1939. These are the two philosophical works for which Curry is known, and they have left a false impression of his views. In this article, I propose to clarify Curry’s views by referring to some of his later writings on the subject. I claim that Curry’s philosophy was not what is now usually called formalism, but is really a form of structuralism. 1 Curry’s early philosophy of mathematics In his [1939], which is a shortened form of the original manuscript of his [1951], Curry proposed a philosophy of mathematics he called formalism. These two works, which represent Curry’s views in 1939, early in his career, are often the only works by which Curry’s philosophical ideas are known; see, for example, [Shapiro, 2000, Chapter 6, §5], where [Curry, 1951] is mistakenly identified as a mature work.