### Ideas from 'Thinking About Mathematics' by Stewart Shapiro [2000], by Theme Structure

#### [found in 'Thinking About Mathematics' by Shapiro,Stewart [OUP 2000,0-19-289306-8]].

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###### 5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
 8729 Intuitionists deny excluded middle, because it is committed to transcendent truth or objects
 Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle.
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
 8763 The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex
 Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed.
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
 18249 Cauchy gave a formal definition of a converging sequence.
 Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) A reaction: The sequence is 'Cauchy' if N exists.
###### 6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
 8764 Categories are the best foundation for mathematics
 Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties.
###### 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
 8762 Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3
 Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them.
###### 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
 8760 Numbers do not exist independently; the essence of a number is its relations to other numbers
 Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate.
 8761 A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them
 Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts.
###### 6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
 8744 Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own
 Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism.
###### 6. Mathematics / C. Sources of Mathematics / 7. Formalism
 8749 Term Formalism says mathematics is just about symbols - but real numbers have no names
 Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up.
 8750 Game Formalism is just a matter of rules, like chess - but then why is it useful in science?
 Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare.
 8752 Deductivism says mathematics is logical consequences of uninterpreted axioms
 Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right.
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
 8753 Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions
 Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them.
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
 8731 Conceptualist are just realists or idealist or nominalists, depending on their view of concepts
 Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football.
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
 8730 'Impredicative' definitions refer to the thing being described
 Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise.
###### 12. Knowledge Sources / C. Rationalism / 1. Rationalism
 8725 Rationalism tries to apply mathematical methodology to all of knowledge
 Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach.