Combining Texts

All the ideas for 'Remarks on the definition and nature of mathematics', 'Introduction to 'Absolute Generality'' and 'Sentences'

unexpand these ideas     |    start again     |     specify just one area for these texts

---

10 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory

13451	The two best understood conceptions of set are the Iterative and the Limitation of Size [Rayo/Uzquiano]
	Full Idea: The two best understood conceptions of set are the Iterative Conception and the Limitation of Size Conception.
	From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.2.2)

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / m. Axiom of Separation

13452	Some set theories give up Separation in exchange for a universal set [Rayo/Uzquiano]
	Full Idea: There are set theories that countenance exceptions to the Principle of Separation in exchange for a universal set.
	From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.2.2)

5. Theory of Logic / A. Overview of Logic / 3. Value of Logic

16728	Logicians acknowledge too few things, while others acknowledge too many [Fitzralph]
	Full Idea: Those who have been well trained in logic err in recognising too few things, whereas others who are ignorant of logic ascribe to every statement a new entity, postulating more entities than God has ever established as real.
	From: Richard Fitzralph (Sentences [1328], II.1.2), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 22.3

5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic

17807	To study formal systems, look at the whole thing, and not just how it is constructed in steps [Curry]
	Full Idea: In the study of formal systems we do not confine ourselves to the derivation of elementary propositions step by step. Rather we take the system, defined by its primitive frame, as datum, and then study it by any means at our command.
	From: Haskell B. Curry (Remarks on the definition and nature of mathematics [1954], 'The formalist')
	A reaction: This is what may potentially lead to an essentialist view of such things. Focusing on bricks gives formalism, focusing on buildings gives essentialism.

5. Theory of Logic / G. Quantification / 2. Domain of Quantification

13449	We could have unrestricted quantification without having an all-inclusive domain [Rayo/Uzquiano]
	Full Idea: The possibility of unrestricted quantification does not immediately presuppose the existence of an all-inclusive domain. One could deny an all-inclusive domain but grant that some quantifications are sometimes unrestricted.
	From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.1)
	A reaction: Thus you can quantify over anything you like, but only from what is available. Eat what you like (in this restaurant).

13450	Absolute generality is impossible, if there are indefinitely extensible concepts like sets and ordinals [Rayo/Uzquiano]
	Full Idea: There are doubts about whether absolute generality is possible, if there are certain concepts which are indefinitely extensible, lacking definite extensions, and yielding an ever more inclusive hierarchy. Sets and ordinals are paradigm cases.
	From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.2.1)

5. Theory of Logic / G. Quantification / 5. Second-Order Quantification

13453	Perhaps second-order quantifications cover concepts of objects, rather than plain objects [Rayo/Uzquiano]
	Full Idea: If one thought of second-order quantification as quantification over first-level Fregean concepts [note: one under which only objects fall], talk of domains might be regimented as talk of first-level concepts, which are not objects.
	From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.2.2)
	A reaction: That is (I take it), don't quantify over objects, but quantify over concepts, but only those under which known objects fall. One might thus achieve naïve comprehension without paradoxes. Sound like fun.

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism

17806	It is untenable that mathematics is general physical truths, because it needs infinity [Curry]
	Full Idea: According to realism, mathematical propositions express the most general properties of our physical environment. This is the primitive view of mathematics, yet on account of the essential role played by infinity in mathematics, it is untenable today.
	From: Haskell B. Curry (Remarks on the definition and nature of mathematics [1954], 'The problem')
	A reaction: I resist this view, because Curry's view seems to imply a mad metaphysics. Hilbert resisted the role of the infinite in essential mathematics. If the physical world includes its possibilities, that might do the job. Hellman on structuralism?

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique

17808	Saying mathematics is logic is merely replacing one undefined term by another [Curry]
	Full Idea: To say that mathematics is logic is merely to replace one undefined term by another.
	From: Haskell B. Curry (Remarks on the definition and nature of mathematics [1954], 'Mathematics')

19. Language / F. Communication / 5. Pragmatics / a. Contextual meaning

13448	The domain of an assertion is restricted by context, either semantically or pragmatically [Rayo/Uzquiano]
	Full Idea: We generally take an assertion's domain of discourse to be implicitly restricted by context. [Note: the standard approach is that this restriction is a semantic phenomenon, but Kent Bach (2000) argues that it is a pragmatic phenomenon]
	From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.1)
	A reaction: I think Kent Bach is very very right about this. Follow any conversation, and ask what the domain is at any moment. The reference of a word like 'they' can drift across things, with no semantics to guide us, but only clues from context and common sense.