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Ideas for 'Mathematical Methods in Philosophy', 'How the Laws of Physics Lie' and 'Intro to Gdel's Theorems'

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8 ideas

5. Theory of Logic / E. Structures of Logic / 1. Logical Form

18741	Logical formalization makes concepts precise, and also shows their interrelation [Horsten/Pettigrew]
	Full Idea: Logical formalization forces the investigator to make the central philosophical concepts precise. It can also show how some philosophical concepts and objects can be defined in terms of others.
	From: Horsten,L/Pettigrew,R (Mathematical Methods in Philosophy [2014], 2)
	A reaction: This is the main rationale of the highly formal and mathematical approach to such things. The downside is when you impose 'precision' on language that was never intended to be precise.

5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic

10076	The 'range' of a function is the set of elements in the output set created by the function [Smith,P]
	Full Idea: The 'range' of a function is the set of elements in the output set that are values of the function for elements in the original set.
	From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1)
	A reaction: In other words, the range is the set of values that were created by the function.

10605	Two functions are the same if they have the same extension [Smith,P]
	Full Idea: We count two functions as being the same if they have the same extension, i.e. if they pair up arguments with values in the same way.
	From: Peter Smith (Intro to Gödel's Theorems [2007], 11.3)
	A reaction: So there's only one way to skin a cat in mathematical logic.

10075	A 'partial function' maps only some elements to another set [Smith,P]
	Full Idea: A 'partial function' is one which maps only some elements of a domain to elements in another set. For example, the reciprocal function 1/x is not defined for x=0.
	From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1 n1)

10074	A 'total function' maps every element to one element in another set [Smith,P]
	Full Idea: A 'total function' is one which maps every element of a domain to exactly one corresponding value in another set.
	From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1)

10612	An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P]
	Full Idea: If a function f maps the argument a back to a itself, so that f(a) = a, then a is said to be a 'fixed point' for f.
	From: Peter Smith (Intro to Gödel's Theorems [2007], 20.5)

5. Theory of Logic / E. Structures of Logic / 7. Predicates in Logic

10615	The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P]
	Full Idea: The so-called Comprehension Schema ∃X∀x(Xx ↔ φ(x)) says that there is a property which is had by just those things which satisfy the condition φ.
	From: Peter Smith (Intro to Gödel's Theorems [2007], 22.3)

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

10595	A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P]
	Full Idea: 'Theorem': given a derivation of the sentence φ from the axioms of the theory T using the background logical proof system, we will say that φ is a 'theorem' of the theory. Standard abbreviation is T |- φ.
	From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4)