Marcus Giaquinto
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780199285945
- eISBN:
- 9780191713811
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199285945.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Visual thinking — visual imagination or perception of diagrams and symbol arrays, and mental operations on them — is omnipresent in mathematics. Is this visual thinking merely a psychological aid, ...
More
Visual thinking — visual imagination or perception of diagrams and symbol arrays, and mental operations on them — is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? This book argues that visual thinking in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. The book explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. It shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. This book reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences.Less
Visual thinking — visual imagination or perception of diagrams and symbol arrays, and mental operations on them — is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? This book argues that visual thinking in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. The book explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. It shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. This book reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences.
Michael Potter
- Published in print:
- 2002
- Published Online:
- May 2007
- ISBN:
- 9780199252619
- eISBN:
- 9780191712647
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199252619.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This book is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. It reassesses the brilliant ...
More
This book is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. It reassesses the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as the understanding of mathematics. The book argues that through the problem of arithmetic participates in the larger puzzle of the relationship between thought, language, experience, and the world, we can distinguish accounts that look to each of these to supply the content we require: those that involve the structure of our experience of the world; those that explicitly involve our grasp of a ‘third realm’ of abstract objects distinct from the concrete objects of the empirical world and the ideas of the author's private Gedankenwelt; those that appeal to something non-physical that is nevertheless an aspect of reality in harmony with which the physical aspect of the world is configured; and finally those that involve only our grasp of language.Less
This book is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. It reassesses the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as the understanding of mathematics. The book argues that through the problem of arithmetic participates in the larger puzzle of the relationship between thought, language, experience, and the world, we can distinguish accounts that look to each of these to supply the content we require: those that involve the structure of our experience of the world; those that explicitly involve our grasp of a ‘third realm’ of abstract objects distinct from the concrete objects of the empirical world and the ideas of the author's private Gedankenwelt; those that appeal to something non-physical that is nevertheless an aspect of reality in harmony with which the physical aspect of the world is configured; and finally those that involve only our grasp of language.
C. S. Jenkins
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199231577
- eISBN:
- 9780191716102
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199231577.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This book is a philosophical discussion of arithmetical knowledge. No extant account, it seems, is able to respect simultaneously these three strong pre-theoretic intuitions: (a) that arithmetic is ...
More
This book is a philosophical discussion of arithmetical knowledge. No extant account, it seems, is able to respect simultaneously these three strong pre-theoretic intuitions: (a) that arithmetic is an a priori discipline; (b) that arithmetical realism is correct, i.e.. that arithmetical claims are true independently of us; and (c) that empiricism is correct, i.e., that all knowledge of the independent world is obtained through the senses. This book investigates the possibility of a new kind of epistemology for arithmetic, one which will is specifically designed to respect all of (a)-(c). The book proposes that we could develop such an epistemology if we were prepared to accept three claims: (1) that arithmetical truths are known through an examination of our arithmetical concepts; (2) that (at least our basic) arithmetical concepts map the arithmetical structure of the independent world; and (3) that this mapping relationship obtains in virtue of the normal functioning of our sensory apparatus. Roughly speaking, the first of these claims protects a priorism, the second realism, and the third empiricism.Less
This book is a philosophical discussion of arithmetical knowledge. No extant account, it seems, is able to respect simultaneously these three strong pre-theoretic intuitions: (a) that arithmetic is an a priori discipline; (b) that arithmetical realism is correct, i.e.. that arithmetical claims are true independently of us; and (c) that empiricism is correct, i.e., that all knowledge of the independent world is obtained through the senses. This book investigates the possibility of a new kind of epistemology for arithmetic, one which will is specifically designed to respect all of (a)-(c). The book proposes that we could develop such an epistemology if we were prepared to accept three claims: (1) that arithmetical truths are known through an examination of our arithmetical concepts; (2) that (at least our basic) arithmetical concepts map the arithmetical structure of the independent world; and (3) that this mapping relationship obtains in virtue of the normal functioning of our sensory apparatus. Roughly speaking, the first of these claims protects a priorism, the second realism, and the third empiricism.
Michael Potter
- Published in print:
- 2004
- Published Online:
- September 2011
- ISBN:
- 9780199269730
- eISBN:
- 9780191699443
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199269730.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This book presents a philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. The book offers an ...
More
This book presents a philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. The book offers an account of cardinal and ordinal arithmetic, and the various axiom candidates. It discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. The book offers a simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. The book interweaves a presentation of the technical material with a philosophical critique. The book does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true.Less
This book presents a philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. The book offers an account of cardinal and ordinal arithmetic, and the various axiom candidates. It discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. The book offers a simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. The book interweaves a presentation of the technical material with a philosophical critique. The book does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true.
Patricia A. Blanchette
- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780199891610
- eISBN:
- 9780199933211
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199891610.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Frege’s Coneption of Logic explores the relationship between Frege’s understanding of conceptual analysis and his understanding of logic. It is argued that the fruitfulness of Frege’s ...
More
Frege’s Coneption of Logic explores the relationship between Frege’s understanding of conceptual analysis and his understanding of logic. It is argued that the fruitfulness of Frege’s conception of logic, and the illuminating differences between that conception and those more modern views that have largely supplanted it, are best understood against the backdrop of a clear account of the role of conceptual analysis in logical investigation. The first part of the book locates the role of conceptual analysis in Frege’s logicist project. It is argued that, despite a number of difficulties, Frege’s use of analysis in the service of logicism is a powerful and coherent tool. As a result of coming to grips with his use of that tool, we can see that there is, despite appearances, no conflict between Frege’s intention to demonstrate the grounds of ordinary arithmetic and the fact that the numerals of his derived sentences fail to co-refer with ordinary numerals. The second part of the book explores the resulting conception of logic itself, and some of the straightforward ways in which Frege’s conception differs from its now-familiar descendants. In particular, it’s argued that consistency, as Frege understands it, differs significantly from the kind of consistency demonstrable via the construction of models. To appreciate this difference is to appreciate the extent to which Frege was right in his debate with Hilbert over consistency- and independence-proofs in geometry. For similar reasons, modern results such as the completeness of formal systems and the categoricity of theories do not have for Frege the same importance they are commonly taken to have by his post-Tarskian descendants. These differences, together with the coherence of Frege’s position, provide reason for caution with respect to the appeal to formal systems and their properties in the treatment of fundamental logical properties and relations.Less
Frege’s Coneption of Logic explores the relationship between Frege’s understanding of conceptual analysis and his understanding of logic. It is argued that the fruitfulness of Frege’s conception of logic, and the illuminating differences between that conception and those more modern views that have largely supplanted it, are best understood against the backdrop of a clear account of the role of conceptual analysis in logical investigation. The first part of the book locates the role of conceptual analysis in Frege’s logicist project. It is argued that, despite a number of difficulties, Frege’s use of analysis in the service of logicism is a powerful and coherent tool. As a result of coming to grips with his use of that tool, we can see that there is, despite appearances, no conflict between Frege’s intention to demonstrate the grounds of ordinary arithmetic and the fact that the numerals of his derived sentences fail to co-refer with ordinary numerals. The second part of the book explores the resulting conception of logic itself, and some of the straightforward ways in which Frege’s conception differs from its now-familiar descendants. In particular, it’s argued that consistency, as Frege understands it, differs significantly from the kind of consistency demonstrable via the construction of models. To appreciate this difference is to appreciate the extent to which Frege was right in his debate with Hilbert over consistency- and independence-proofs in geometry. For similar reasons, modern results such as the completeness of formal systems and the categoricity of theories do not have for Frege the same importance they are commonly taken to have by his post-Tarskian descendants. These differences, together with the coherence of Frege’s position, provide reason for caution with respect to the appeal to formal systems and their properties in the treatment of fundamental logical properties and relations.
Ernest Hartmann
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199751778
- eISBN:
- 9780199863419
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199751778.003.0006
- Subject:
- Psychology, Cognitive Psychology, Clinical Psychology
This chapter shows that dreaming is hyper-connective. At the dreaming end of the continuum connections are made more easily than in waking, and connections are made more broadly and loosely. Dreaming ...
More
This chapter shows that dreaming is hyper-connective. At the dreaming end of the continuum connections are made more easily than in waking, and connections are made more broadly and loosely. Dreaming avoids tightly structured, over-learned material. Thus, we do not dream of the “three R's”—reading, writing, and arithmetic. The connections are not random, but guided by the dreamer's emotions.Less
This chapter shows that dreaming is hyper-connective. At the dreaming end of the continuum connections are made more easily than in waking, and connections are made more broadly and loosely. Dreaming avoids tightly structured, over-learned material. Thus, we do not dream of the “three R's”—reading, writing, and arithmetic. The connections are not random, but guided by the dreamer's emotions.
José Ferreirós
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691167510
- eISBN:
- 9781400874002
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691167510.003.0007
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This chapter considers the idea that we have certainty in our basic arithmetic knowledge. The claim that arithmetical knowledge enjoys certainty cannot be extended to a similar claim about number ...
More
This chapter considers the idea that we have certainty in our basic arithmetic knowledge. The claim that arithmetical knowledge enjoys certainty cannot be extended to a similar claim about number theory “as a whole.” It is thus necessary to distinguish between elementary number theory and other, more advanced, levels in the study of numbers: algebraic number theory, analytic number theory, and perhaps set-theoretic number theory. The chapter begins by arguing that the axioms of Peano Arithmetic are true of counting numbers and describing some elements found in counting practices. It then offers an account of basic arithmetic and its certainty before discussing a model theory of arithmetic and the logic of mathematics. Finally, it asks whether elementary arithmetic, built on top of the practice of counting, should be classical arithmetic or intuitionistic arithmetic.Less
This chapter considers the idea that we have certainty in our basic arithmetic knowledge. The claim that arithmetical knowledge enjoys certainty cannot be extended to a similar claim about number theory “as a whole.” It is thus necessary to distinguish between elementary number theory and other, more advanced, levels in the study of numbers: algebraic number theory, analytic number theory, and perhaps set-theoretic number theory. The chapter begins by arguing that the axioms of Peano Arithmetic are true of counting numbers and describing some elements found in counting practices. It then offers an account of basic arithmetic and its certainty before discussing a model theory of arithmetic and the logic of mathematics. Finally, it asks whether elementary arithmetic, built on top of the practice of counting, should be classical arithmetic or intuitionistic arithmetic.
Graham Priest
- Published in print:
- 2006
- Published Online:
- May 2007
- ISBN:
- 9780199263301
- eISBN:
- 9780191718823
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199263301.003.0018
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
The study of formal inconsistent arithmetics has already occasioned a number of interesting technical results, as well as philosophical spin-offs. This chapter looks at some of these. It shows how ...
More
The study of formal inconsistent arithmetics has already occasioned a number of interesting technical results, as well as philosophical spin-offs. This chapter looks at some of these. It shows how inconsistent arithmetics are delivered by certain kinds of inconsistent models, and what such models are like. It then turns to more philosophical issues. These concern, importantly, the way that inconsistent arithmetics relate to some of the limitative results of classical metamathematics.Less
The study of formal inconsistent arithmetics has already occasioned a number of interesting technical results, as well as philosophical spin-offs. This chapter looks at some of these. It shows how inconsistent arithmetics are delivered by certain kinds of inconsistent models, and what such models are like. It then turns to more philosophical issues. These concern, importantly, the way that inconsistent arithmetics relate to some of the limitative results of classical metamathematics.
Michael Potter
- Published in print:
- 2002
- Published Online:
- May 2007
- ISBN:
- 9780199252619
- eISBN:
- 9780191712647
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199252619.003.0010
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
By the time Hilbert's first series of published research on the foundations of mathematics came to an end around 1904, he had formulated but not solved the problem of finding — for a formal system of ...
More
By the time Hilbert's first series of published research on the foundations of mathematics came to an end around 1904, he had formulated but not solved the problem of finding — for a formal system of arithmetic such as that supplied by Peano — proof of consistency not reliant on the construction of a model. Even if he succeeded in obtaining such a non-semantic proof, there would remain two substantial objections to placing any philosophical significance on the result. First is Frege's objection that the consistency of a list of axioms does not in itself ensure the existence of anything satisfying them. Second is Poincaré's objection that the proof of consistency must make use of the principle of mathematical induction, and is therefore inherently circular. This chapter studies how Hilbert maintained the centrality to his project of finding a consistency proof for arithmetic while adopting a sort of formalism that transformed the significance of such a proof so as to address these two objections.Less
By the time Hilbert's first series of published research on the foundations of mathematics came to an end around 1904, he had formulated but not solved the problem of finding — for a formal system of arithmetic such as that supplied by Peano — proof of consistency not reliant on the construction of a model. Even if he succeeded in obtaining such a non-semantic proof, there would remain two substantial objections to placing any philosophical significance on the result. First is Frege's objection that the consistency of a list of axioms does not in itself ensure the existence of anything satisfying them. Second is Poincaré's objection that the proof of consistency must make use of the principle of mathematical induction, and is therefore inherently circular. This chapter studies how Hilbert maintained the centrality to his project of finding a consistency proof for arithmetic while adopting a sort of formalism that transformed the significance of such a proof so as to address these two objections.
Ted McCormick
- Published in print:
- 2009
- Published Online:
- September 2009
- ISBN:
- 9780199547890
- eISBN:
- 9780191720529
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199547890.003.0010
- Subject:
- History, British and Irish Early Modern History, Economic History
The conclusion draws together several of the main themes of the book, arguing that political arithmetic can only be understood properly from the manuscripts he circulated in his lifetime, and in the ...
More
The conclusion draws together several of the main themes of the book, arguing that political arithmetic can only be understood properly from the manuscripts he circulated in his lifetime, and in the context of his lifelong engagement with natural philosophy and his Baconian interpretation of the methods and purposes of science. Petty's project to transform government through the systematic manipulation of populations in the interests of the state undermines any retrospective distinction between his contribution to social science and his interest in social engineering; the concept of ‘biopolitics', though equally anachronistic, is more appropriate. By the same token, however, the massive expansion (in real terms) of demographic manipulation after Petty's death suggests that political arithmetic's ambitions were not abandoned but merely concealed.Less
The conclusion draws together several of the main themes of the book, arguing that political arithmetic can only be understood properly from the manuscripts he circulated in his lifetime, and in the context of his lifelong engagement with natural philosophy and his Baconian interpretation of the methods and purposes of science. Petty's project to transform government through the systematic manipulation of populations in the interests of the state undermines any retrospective distinction between his contribution to social science and his interest in social engineering; the concept of ‘biopolitics', though equally anachronistic, is more appropriate. By the same token, however, the massive expansion (in real terms) of demographic manipulation after Petty's death suggests that political arithmetic's ambitions were not abandoned but merely concealed.
A. H. Halsey
- Published in print:
- 2004
- Published Online:
- April 2004
- ISBN:
- 9780199266609
- eISBN:
- 9780191601019
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199266603.003.0003
- Subject:
- Political Science, UK Politics
The history of the search for a science of society is traced back to the political arithmeticians of the seventeenth century. Social survey is defined. What is right and what is wrong with sociology ...
More
The history of the search for a science of society is traced back to the political arithmeticians of the seventeenth century. Social survey is defined. What is right and what is wrong with sociology is discussed in the context of historical development. Claims for statistical originality are considered in respect of Graunt, Pascal, Quetelet, Florence Nightingale, Galton, Fisher, Lazersfeld, and Yule. Booth's biography is retailed. Social surveys and sampling multivariate analysis and log linear modelling, and the new English statistics are put historically into the context of a developing society and its concern with social policy.Less
The history of the search for a science of society is traced back to the political arithmeticians of the seventeenth century. Social survey is defined. What is right and what is wrong with sociology is discussed in the context of historical development. Claims for statistical originality are considered in respect of Graunt, Pascal, Quetelet, Florence Nightingale, Galton, Fisher, Lazersfeld, and Yule. Booth's biography is retailed. Social surveys and sampling multivariate analysis and log linear modelling, and the new English statistics are put historically into the context of a developing society and its concern with social policy.
Graham Priest
- Published in print:
- 2005
- Published Online:
- May 2006
- ISBN:
- 9780199263288
- eISBN:
- 9780191603631
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199263280.003.0011
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This chapter argues that logic is a theory, and can be revised as any other scientific theory. The comparison with geometry is helpful in this regard. It also discusses Quine’s views on the matter, ...
More
This chapter argues that logic is a theory, and can be revised as any other scientific theory. The comparison with geometry is helpful in this regard. It also discusses Quine’s views on the matter, particularly the claim that any changing of logic is a changing of subject.Less
This chapter argues that logic is a theory, and can be revised as any other scientific theory. The comparison with geometry is helpful in this regard. It also discusses Quine’s views on the matter, particularly the claim that any changing of logic is a changing of subject.
Philip Burton
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780199266227
- eISBN:
- 9780191709098
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199266227.003.0003
- Subject:
- Religion, Early Christian Studies
This chapter considers Augustine's use of Greek and Latin terminology in the technical register of the Seven Liberal Arts (grammar, rhetoric, dialect, arithmetic, geometry, music, and philosophy). It ...
More
This chapter considers Augustine's use of Greek and Latin terminology in the technical register of the Seven Liberal Arts (grammar, rhetoric, dialect, arithmetic, geometry, music, and philosophy). It is argued that despite his notional assent to the equality of all languages, Augustine in practice often uses Greek to express a liberal art when viewed negatively. At the same time, he takes advantage of the wider semantic range of the different Latin translations to demystify the arts, encouraging his reader to see them not as the prerogative of the educated few, but as something capable of being practised in the everyday life of everyone.Less
This chapter considers Augustine's use of Greek and Latin terminology in the technical register of the Seven Liberal Arts (grammar, rhetoric, dialect, arithmetic, geometry, music, and philosophy). It is argued that despite his notional assent to the equality of all languages, Augustine in practice often uses Greek to express a liberal art when viewed negatively. At the same time, he takes advantage of the wider semantic range of the different Latin translations to demystify the arts, encouraging his reader to see them not as the prerogative of the educated few, but as something capable of being practised in the everyday life of everyone.
Rein Taagepera
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199534661
- eISBN:
- 9780191715921
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199534661.003.0009
- Subject:
- Political Science, Comparative Politics, Political Economy
Geometric means are often more meaningful than arithmetic means, because they are closer to the central figure (median). When x and y can conceptually take only positive values, their distributions ...
More
Geometric means are often more meaningful than arithmetic means, because they are closer to the central figure (median). When x and y can conceptually take only positive values, their distributions cannot be normal and may be lognormal. When running a normal distribution yields a standard deviation larger than one-half of the mean, one should dump the normal fit and try a lognormal fit instead.Less
Geometric means are often more meaningful than arithmetic means, because they are closer to the central figure (median). When x and y can conceptually take only positive values, their distributions cannot be normal and may be lognormal. When running a normal distribution yields a standard deviation larger than one-half of the mean, one should dump the normal fit and try a lognormal fit instead.
Robert Hanna
- Published in print:
- 2006
- Published Online:
- January 2007
- ISBN:
- 9780199285549
- eISBN:
- 9780191713965
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199285549.003.0007
- Subject:
- Philosophy, History of Philosophy
This chapter examines Kant's much-criticized views on mathematics in general and arithmetic in particular. It makes a case for the claim that Kant's theory of arithmetic is not subject to the most ...
More
This chapter examines Kant's much-criticized views on mathematics in general and arithmetic in particular. It makes a case for the claim that Kant's theory of arithmetic is not subject to the most familiar and forceful objection against it, namely, that his doctrine of the dependence of arithmetic on time is plainly false, or even worse, simply unintelligible. It is argued that Kant's doctrine about time and arithmetic is highly original, fully intelligible, and with qualifications due to the inherent limitations of his conceptions of arithmetic and logic, to an important extent defensible. The most philosophically striking thing about Kant's doctrine is the fact that arithmetic turns out to be a paradigm of the exact sciences (exacten Naturwissenschaften) only by virtue of its ultimately being one of the human or moral sciences (Geisteswissenschaften).Less
This chapter examines Kant's much-criticized views on mathematics in general and arithmetic in particular. It makes a case for the claim that Kant's theory of arithmetic is not subject to the most familiar and forceful objection against it, namely, that his doctrine of the dependence of arithmetic on time is plainly false, or even worse, simply unintelligible. It is argued that Kant's doctrine about time and arithmetic is highly original, fully intelligible, and with qualifications due to the inherent limitations of his conceptions of arithmetic and logic, to an important extent defensible. The most philosophically striking thing about Kant's doctrine is the fact that arithmetic turns out to be a paradigm of the exact sciences (exacten Naturwissenschaften) only by virtue of its ultimately being one of the human or moral sciences (Geisteswissenschaften).
Roman Kossak and James H. Schmerl
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198568278
- eISBN:
- 9780191718199
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568278.003.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy
This introductory chapter covers a wide range of topics, from basic notational conventions and coding in arithmetic to important classical results. The well-known theorems, such as Gaifman's ...
More
This introductory chapter covers a wide range of topics, from basic notational conventions and coding in arithmetic to important classical results. The well-known theorems, such as Gaifman's splitting theorem, Arithmetized Completeness Theorem, or Friedman's Embedding Theorem are discussed without proofs. Proofs are given for less known facts, like Blass-Gaifman and Ehrenfeucht lemmas. The chapter also presents a systematic introduction to recursively and arithmetically saturated models, resplendent models, and satisfaction classes.Less
This introductory chapter covers a wide range of topics, from basic notational conventions and coding in arithmetic to important classical results. The well-known theorems, such as Gaifman's splitting theorem, Arithmetized Completeness Theorem, or Friedman's Embedding Theorem are discussed without proofs. Proofs are given for less known facts, like Blass-Gaifman and Ehrenfeucht lemmas. The chapter also presents a systematic introduction to recursively and arithmetically saturated models, resplendent models, and satisfaction classes.
Mark Green, Phillip A. Griffiths, and Matt Kerr
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691154244
- eISBN:
- 9781400842735
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691154244.001.0001
- Subject:
- Mathematics, Analysis
Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive ...
More
Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive exploration of Mumford-Tate groups and domains. Containing basic theory and a wealth of new views and results, it is an essential resource for graduate students and researchers. Although Mumford-Tate groups can be defined for general structures, their theory and use to date has mainly been in the classical case of abelian varieties. While the book does examine this area, it focuses on the nonclassical case. The general theory turns out to be very rich, such as in the unexpected connections of finite dimensional and infinite dimensional representation theory of real, semisimple Lie groups. The book gives the complete classification of Hodge representations, a topic that should become a standard in the finite-dimensional representation theory of noncompact, real, semisimple Lie groups. It also indicates that in the future, a connection seems ready to be made between Lie groups that admit discrete series representations and the study of automorphic cohomology on quotients of Mumford-Tate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject.Less
Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive exploration of Mumford-Tate groups and domains. Containing basic theory and a wealth of new views and results, it is an essential resource for graduate students and researchers. Although Mumford-Tate groups can be defined for general structures, their theory and use to date has mainly been in the classical case of abelian varieties. While the book does examine this area, it focuses on the nonclassical case. The general theory turns out to be very rich, such as in the unexpected connections of finite dimensional and infinite dimensional representation theory of real, semisimple Lie groups. The book gives the complete classification of Hodge representations, a topic that should become a standard in the finite-dimensional representation theory of noncompact, real, semisimple Lie groups. It also indicates that in the future, a connection seems ready to be made between Lie groups that admit discrete series representations and the study of automorphic cohomology on quotients of Mumford-Tate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject.
Paul Horwich
- Published in print:
- 2005
- Published Online:
- September 2006
- ISBN:
- 9780199251247
- eISBN:
- 9780191603983
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/019925124X.003.0006
- Subject:
- Philosophy, Philosophy of Language
Our beliefs and inferential transitions are subject to evaluation as rational or irrational by reference to general epistemic norms. But what could determine certain norms as the correct ones? This ...
More
Our beliefs and inferential transitions are subject to evaluation as rational or irrational by reference to general epistemic norms. But what could determine certain norms as the correct ones? This chapter explores and opposes the answer that certain patterns of belief formation (e.g., in arithmetic and in deductive logic) are justified by virtue of the fact that they constitute the relevant concepts or the meanings of the relevant words (e.g., ‘number’, ‘successor’, ‘every’, ‘not’, etc.). This ‘semantogenetic’ proposal goes back to Hilbert, Poincare, and the logical positivists, and was recently defended by Boghossian, Peackocke, Hale, and Wright. Among the arguments developed against it are that although it might account for the epistemic legitimacy of certain beliefs, it cannot explain why certain commitments are epistemically obligatory; and that the practices that provide words with their meanings are ‘conditionalized’ and therefore do not coincide (even approximately) with the practices recommended by our epistemic norms.Less
Our beliefs and inferential transitions are subject to evaluation as rational or irrational by reference to general epistemic norms. But what could determine certain norms as the correct ones? This chapter explores and opposes the answer that certain patterns of belief formation (e.g., in arithmetic and in deductive logic) are justified by virtue of the fact that they constitute the relevant concepts or the meanings of the relevant words (e.g., ‘number’, ‘successor’, ‘every’, ‘not’, etc.). This ‘semantogenetic’ proposal goes back to Hilbert, Poincare, and the logical positivists, and was recently defended by Boghossian, Peackocke, Hale, and Wright. Among the arguments developed against it are that although it might account for the epistemic legitimacy of certain beliefs, it cannot explain why certain commitments are epistemically obligatory; and that the practices that provide words with their meanings are ‘conditionalized’ and therefore do not coincide (even approximately) with the practices recommended by our epistemic norms.
Marcus Giaquinto
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780199285945
- eISBN:
- 9780191713811
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199285945.003.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This introductory chapter outlines the main themes of the book. It then describes visual thinking in mathematics during the 19th century. An overview of the chapters in the book is presented.
This introductory chapter outlines the main themes of the book. It then describes visual thinking in mathematics during the 19th century. An overview of the chapters in the book is presented.
Marcus Giaquinto
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780199285945
- eISBN:
- 9780191713811
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199285945.003.0007
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
This chapter describes the roles of visual thinking in calculation, with the aim of showing that in many cases it is not peripheral. It does this by setting the visual aspects of calculation within ...
More
This chapter describes the roles of visual thinking in calculation, with the aim of showing that in many cases it is not peripheral. It does this by setting the visual aspects of calculation within an account of the operations involved and the cognitive resources used. The chapter also considers the epistemology of basic arithmetic.Less
This chapter describes the roles of visual thinking in calculation, with the aim of showing that in many cases it is not peripheral. It does this by setting the visual aspects of calculation within an account of the operations involved and the cognitive resources used. The chapter also considers the epistemology of basic arithmetic.
