Combining Texts

All the ideas for 'Remarks on the definition and nature of mathematics', 'Aristotle and Kant on the Source of Value' and 'New Foundations for Mathematical Logic'

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9 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets

9879	NF has no models, but just blocks the comprehension axiom, to avoid contradictions [Quine, by Dummett]
	Full Idea: Quine's New Foundations system of set theory, devised with no model in mind, but on the basis of a hunch that a purely formal restriction on the comprehension axiom would block all contradictions.
	From: report of Willard Quine (New Foundations for Mathematical Logic [1937]) by Michael Dummett - Frege philosophy of mathematics Ch.18
	A reaction: The point is that Quine (who had an ontological preference for 'desert landscapes') attempted to do without an ontological commitment to objects (and their subsequent models), with a purely formal system. Quine's NF is not now highly regarded.

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.

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')

22. Metaethics / B. Value / 1. Nature of Value / f. Ultimate value

18228	An end can't be an ultimate value just because it is useless! [Korsgaard]
	Full Idea: If what is final is whatever is an end but never a means, ...why should something be more valuable just because it is useless?
	From: Christine M. Korsgaard (Aristotle and Kant on the Source of Value [1986], 8 'Finality')
	A reaction: Korsgaard is offering this as a bad reading of what Aristotle intends.

18225	If we can't reason about value, we can reason about the unconditional source of value [Korsgaard]
	Full Idea: If you can only know what is intrinsically valuable through intuition (as Moore claims), you can still argue about what is unconditionally valuable. There must be something unconditionally valuable because there must be a source of value.
	From: Christine M. Korsgaard (Aristotle and Kant on the Source of Value [1986], 8 'Three')
	A reaction: If you only grasped the values through intuition, does that give you enough information to infer the dependence relations between values?

22. Metaethics / C. The Good / 1. Goodness / b. Types of good

18224	Goodness is given either by a psychological state, or the attribution of a property [Korsgaard]
	Full Idea: 'Subjectivism' identifies good ends with or by reference to some psychological state. ...'Objectivism' says that something is good as an end if a property, intrinsic goodness, is attributed to it.
	From: Christine M. Korsgaard (Aristotle and Kant on the Source of Value [1986], 8 'Three')

23. Ethics / C. Virtue Theory / 3. Virtues / g. Contemplation

18233	Contemplation is final because it is an activity which is not a process [Korsgaard]
	Full Idea: It is because contemplation is an activity that is not also a process that Aristotle identifies it as the most final good.
	From: Christine M. Korsgaard (Aristotle and Kant on the Source of Value [1986], 8 'Activity')
	A reaction: Quite a helpful way of labelling what Aristotle has in mind. So should we not aspire to be involved in processes, except reluctantly? I take the mind itself to be a process, so that may be difficult!

18226	For Aristotle, contemplation consists purely of understanding [Korsgaard]
	Full Idea: Contemplation, as Aristotle understand it, is not research or inquiry, but an activity that ensues on these: an activity that consists in understanding.
	From: Christine M. Korsgaard (Aristotle and Kant on the Source of Value [1986], 8 'Aristotle')
	A reaction: Fairly obvious, when you read the last part of 'Ethics', but helpful in grasping Aristotle, because understanding is the objective of 'Posterior Analytics' and 'Metaphysics', so he tells you how to achieve the ideal moral state.