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Introduction to the philosophy of mathematics

Author: Hugh Lehman
Publisher: Totowa, N.J. : Rowman and Littlefield, 1979.
Series: APQ library of philosophy.
Edition/Format:   Print book : EnglishView all editions and formats
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Additional Physical Format: Online version:
Lehman, Hugh.
Introduction to the philosophy of mathematics.
Totowa, N.J. : Rowman and Littlefield, 1979
(OCoLC)654547070
Document Type: Book
All Authors / Contributors: Hugh Lehman
ISBN: 0847661091 9780847661091
OCLC Number: 4496683
Notes: Includes indexes.
Description: xi, 177 pages ; 22 cm.
Contents: 1. The aim of this work is to discuss ontological and epistemological issues --
2. It will avoid assuming extensive knowledge of mathematics or logic --
3. Concerning the ontological and epistemological issues and of the present approach in supporting these views --
4. Statements of mathematical theories carry ontological implications, for example, the axioms of real numbers --
5. Numbers are unobservable, neither physical nor mental and universals, i.e., numbers are queer entities --
6. Consideration of the substitutional interpretation as a way of accepting mathematical truth either queer entities --
7. Mathematical statements are alleged to be conditional. There are two versions of this view --
8. But mathematical statements are not material implications --
9. A second version of if-thenism is the view that mathematical statements are logical implications. Thief view is open to two sorts of criticisms. We may ask whether acceptance of logical truths involves making ontological commitments --
10. We may also ask whether mathematical statements are logical implications. Definitions of "real numbers" must be considered --
11. Someone may object to my claim because he subscribes to a more inclusive notion of logical truth. But I ask, how he distinguishes logical truth from non-logical truths. In my view a logical truth is one which is an instance of a formula which is "true in all possible worlds." --
12. One further way of defining "real number" is considered. --
13. Conclusion of chapter one --
14. The definition of real numbers considered in section 12 suggests a view of mathematics held (at one time) by Bertrand Russell and also by Henri Poincare, namely the view that mathematical statements assert only that theorems are consequences of certain assumptions. --
15. But, even if postulations is true with respect to pure mathematics, in applications of mathematics categorical mathematical assertions are made. Thus, postulations does not enable us to use mathematical knowledge while avoiding the implication that queer entities exist. --
16. But postulations does not give a correct description of the activity of the "pure" mathematician. Example regarding convexity. --
17. Nor does it correctly describe the class of mathematical truths, as was pointed out by Quine. --
18. Conclusion of chapter two. 19. Some philosophers have claimed that mathematical propositions are analytic. In particular they have asserted that mathematical truths are true by definition and lacking in factual import --
20. Is "3+2=5" true by virtue of definitions of '3', '2', '+', '5', and '='? We are not compelled to say that it is. But even if it is true by definition, the theory presupposed by these definitions has ontological implications. --
21. Defenders of the view that mathematical propositions are analytic have claimed that considerations of the ways in which such propositions are learned are irrelevant to epistemology. But this position is wrong. It leads to an absurd consequence as consideration of the notion learning shows. --
22. Defenders of the view that mathematical propositions are analytic have claimed that arithmetic truths cannot be refuted by observed counter-instances. But this claim is apparently untenable. --
23. Some philosophers, notably C.I. Lewis and E. Nagel have maintained that mathematical and logical propositions are prescriptions and therefore have no ontological import. We argue that while such propositions have a prescriptive role it does not follow that they have no ontological import. --
24. Conclusion of chapter three. --
25. In this section we try to explain some of the basic ideas of Catnap's theories. In particular we explain the notions of syntactical and semantical systems --
26. An incomplete example of a syntactical and semantical system is given. A complete system of the sort given constitutes, according to Carnap, the rules of ordinary logical reasoning. --
27. Here we consider some truths of mathematics, namely Peano's postulates. Carnap presented a syntactic system for these postulates, and also gave semantical rules. He also gave definitions so that statements of Peano's postulates could be obtained in the semantical system of ordinary logic. --
28. While the translations of Peano's postulates into the semantical system developed by Carnap are L-true, we may ask whether Carnap has shown that mathematical principles make no ontological commitments. For one thing we may ask whether Carnap's translations really express Peano's postulates. --
29. Consideration of predicative definitions, predicative functions, the vicious circle principle and the axiom of reducibility. --
30. Carnap has argued that there is a distinction between internal and external questions and that external questions regarding existence are meaningless. We criticize the distinction and argue that his conclusion that external questions regarding existence are meaningless is false. --
31. Consideration of the view of Hans Vaihinger that mathematical concepts are fictions. His theory that mathematical statements are all fictions or semi-fictions seems mistaken since it is incompatible with the fact that some people have mathematical knowledge. On our view a pragmatic theory of mathematical knowledge is correct. --
32. Consideration of the nature of mathematical proof. --
33. Sime mathematical principles must be known without proof, since there is mathematical knowledge through proof's and the number of premises of such knowledge is finite. --
34. Consideration of the view of Kurt Gödel that mathematical knowledge rests on principles known via intuition of mathematical objects. Objections to this view. --
35. Mathematical intuition and the casual theory of perception. Views of Mark Steiner. --
36. Conclusion of chapter five. ---
37. Explanation of intuitionist view on mathematical existence and knowledge. Consideration of Intuitionist and Wittgensteinian objections to the law of excluded middle. Their objections are not sound. --
39. Consideration of intuitionist real number theory (of intuitionist theory of the continuum). The Intuitionists cannot account for all of our mathematical knowledge. --
40. Criticism of the intuitionist view that mathematical knowledge rests on self-evident principles and of the intuitionist theory of the reference of mathematical terms. --
41. Intuition and learning mathematics --
42. Conclusion --
43. J.S. Mill's view of mathematical knowledge --
44. Criticisms of Mill's view by positivists. --
45. Gottlob Frege's criticisms of Mill. --
46. Hilbert's theory of mathematical knowledge. --
47. Criticisms of Hilbert's theory. --
48. Discussion and criticism of Haskell Curry's version of formalism. --
49. Critical discussion of the formalist theory of Abraham Robinson. Robinson avoids the objections directed against Hilbert's epistemology. But his view is essentially incomplete. --
50. Conclusion --
51. Skepticism revisited. Discussion of the view of Stepah Korner. --
52. Discussion of the significance of non-Euclidean geometries and of alternative set theories with geometries and alternative set theories with respect to the existence of mathematical knowledge. --
53. Brief explanation of our theory of mathematical knowledge. Mathematical principles are confirmed via hypothetic-deductive inferences. --
54. Consideration of an objection: It is alleged sometimes that mathematical knowledge is certain and so cannot be empirically confirmed. --
55. Could science dispense with real number theory and make do with rational number theory instead? --
56. Does confirmation of mathematical principles show that while we are warranted in using such principles in natural science we are not warranted in believing that they are true? --
57. Consideration of the view that knowledge of mathematical principles presupposes that (1) no impredicatice definitions occur in such principles, (2) there is no reference to infinite totalities in such principles, (3) there is no reference to the existence of sets in such principles or (4) that such principles be "constructive."
Series Title: APQ library of philosophy.
Responsibility: Hugh Lehman.

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Defenders of the view that mathematical propositions are analytic have claimed that considerations of the ways in which such propositions are learned are irrelevant to epistemology. But this position is wrong. It leads to an absurd consequence as consideration of the notion learning shows. -- 22. Defenders of the view that mathematical propositions are analytic have claimed that arithmetic truths cannot be refuted by observed counter-instances. But this claim is apparently untenable. -- 23. Some philosophers, notably C.I. Lewis and E. Nagel have maintained that mathematical and logical propositions are prescriptions and therefore have no ontological import. We argue that while such propositions have a prescriptive role it does not follow that they have no ontological import. -- 24. Conclusion of chapter three. -- 25. In this section we try to explain some of the basic ideas of Catnap's theories. In particular we explain the notions of syntactical and semantical systems -- 26. 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Discussion and criticism of Haskell Curry's version of formalism. -- 49. Critical discussion of the formalist theory of Abraham Robinson. Robinson avoids the objections directed against Hilbert's epistemology. But his view is essentially incomplete. -- 50. Conclusion -- 51. Skepticism revisited. Discussion of the view of Stepah Korner. -- 52. Discussion of the significance of non-Euclidean geometries and of alternative set theories with geometries and alternative set theories with respect to the existence of mathematical knowledge. -- 53. Brief explanation of our theory of mathematical knowledge. Mathematical principles are confirmed via hypothetic-deductive inferences. -- 54. Consideration of an objection: It is alleged sometimes that mathematical knowledge is certain and so cannot be empirically confirmed. -- 55. Could science dispense with real number theory and make do with rational number theory instead? -- 56. 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