display all the ideas for this combination of philosophers
13 ideas
| 18281 | In mathematics everything is algorithm and nothing is meaning [Wittgenstein] |
| Full Idea: In mathematics everything is algorithm and nothing is meaning; even when it doesn't look like that because we seem to be using words to talk about mathematical things. | |
| From: Ludwig Wittgenstein (Philosophical Grammar [1932], p.468), quoted by J. Alberto Coffa - The Semantic Tradition from Kant to Carnap 13 'Constr' | |
| A reaction: I would have thought that an algorithm needs some raw material to work with. This leads to the idea that meaning arises from rules of usage. |
| 18738 | We don't get 'nearer' to something by adding decimals to 1.1412... (root-2) [Wittgenstein] |
| Full Idea: We say we get nearer to root-2 by adding further figures after the decimal point: 1.1412.... This suggests there is something we can get nearer to, but the analogy is a false one. | |
| From: Ludwig Wittgenstein (Lectures 1930-32 (student notes) [1931], Notes) |
| 18708 | Infinity is not a number, so doesn't say how many; it is the property of a law [Wittgenstein] |
| Full Idea: 'Infinite' is not an answer to the question 'How many?', since the infinite is not a number. ...Infinity is the property of a law, not of an extension. | |
| From: Ludwig Wittgenstein (Lectures 1930-32 (student notes) [1931], A VII.2) |
| 16321 | The compactness theorem can prove nonstandard models of PA [Halbach] |
| Full Idea: Nonstandard models of Peano arithmetic are models of PA that are not isomorphic to the standard model. Their existence can be established with the compactness theorem or the adequacy theorem of first-order logic. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 8.3) |
| 16343 | The global reflection principle seems to express the soundness of Peano Arithmetic [Halbach] |
| Full Idea: The global reflection principle ∀x(Sent(x) ∧ Bew[PA](x) → Tx) …seems to be the full statement of the soundness claim for Peano arithmetic, as it expresses that all theorems of Peano arithmetic are true. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 22.1) | |
| A reaction: That is, an extra principle must be introduced to express the soundness. PA is, of course, not complete. |
| 18153 | A number is a repeated operation [Wittgenstein] |
| Full Idea: A number is the index of an operation. | |
| From: Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921], 6.021) | |
| A reaction: Roughly, this means that a number indicates how many times some basic operation has been performed. Bostock 2009:286 expounds the idea. |
| 18160 | The concept of number is just what all numbers have in common [Wittgenstein] |
| Full Idea: The concept of number is simply what is common to all numbers, the general form of number. The concept of number is the variable number. | |
| From: Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921], 6.022) |
| 16312 | To reduce PA to ZF, we represent the non-negative integers with von Neumann ordinals [Halbach] |
| Full Idea: For the reduction of Peano Arithmetic to ZF set theory, usually the set of finite von Neumann ordinals is used to represent the non-negative integers. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 6) | |
| A reaction: Halbach makes it clear that this is just one mode of reduction, relative interpretability. |
| 18161 | The theory of classes is superfluous in mathematics [Wittgenstein] |
| Full Idea: The theory of classes is completely superfluous in mathematics. This is connected with the fact that the generality required in mathematics is not accidental generality. | |
| From: Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921], 6.031) | |
| A reaction: This fits Russell's no-class theory, which rests everything instead on propositional functions. |
| 11073 | Two and one making three has the necessity of logical inference [Wittgenstein] |
| Full Idea: "But doesn't it follow with logical necessity that you get two when you add one to one, and three when you add one to two? and isn't this inexorability the same as that of logical inference? - Yes! it is the same. | |
| From: Ludwig Wittgenstein (Remarks on the Foundations of Mathematics [1938], p.38), quoted by Robert Hanna - Rationality and Logic 6 | |
| A reaction: This need not be a full commitment to logicism - only to the fact that the inferential procedures in mathematics are the same as those of logic. Mathematics could still have further non-logical ingredients. Indeed, I think it probably does. |
| 16308 | Set theory was liberated early from types, and recent truth-theories are exploring type-free [Halbach] |
| Full Idea: While set theory was liberated much earlier from type restrictions, interest in type-free theories of truth only developed more recently. | |
| From: Volker Halbach (Axiomatic Theories of Truth [2011], 4) | |
| A reaction: Tarski's theory of truth involves types (or hierarchies). |
| 6849 | Wittgenstein hated logicism, and described it as a cancerous growth [Wittgenstein, by Monk] |
| Full Idea: Wittgenstein didn't just have an arguments against logicism; he hated logicism, and described is as a cancerous growth. | |
| From: report of Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921]) by Ray Monk - Interview with Baggini and Stangroom p.12 | |
| A reaction: This appears to have been part of an inexplicable personal antipathy towards Russell. Wittgenstein appears to have developed a dislike of all reductionist ideas in philosophy. |
| 23509 | The logic of the world is shown by tautologies in logic, and by equations in mathematics [Wittgenstein] |
| Full Idea: The logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics. | |
| From: Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921], 6.22) | |
| A reaction: White observes that this is Wittgenstein distinguishing logic from mathematics, and thus distancing himself from logicism. But see T 6.2. |