22 ideas
| 16123 | Whenever you perceive a community of things, you should also hunt out differences in the group [Plato] |
| Full Idea: The rule is that when one perceives first the community between the members of a group of many things, one should not desist until one sees in it all those differences that are located in classes. | |
| From: Plato (The Statesman [c.356 BCE], 285b) | |
| A reaction: He goes on to recommend the opposite as well - see community even when there appears to be nothing but differences. I take this to be analysis, just as much as modern linguistic approaches are. Analyse the world, not language. |
| 16125 | To reveal a nature, divide down, and strip away what it has in common with other things [Plato] |
| Full Idea: Let's take the kind posited and cut it in two, .then follow the righthand part of what we've cut, and hold onto things that the sophist is associated with until we strip away everything he has in common with other things, then display his peculiar nature. | |
| From: Plato (The Statesman [c.356 BCE], 264e) | |
| A reaction: This seems to be close to Aristotle's account of definition, when he is trying to get at what-it-is-to-be some thing. But if you strip away everything the definiendum has in common with other things, will anything remain? |
| 16124 | No one wants to define 'weaving' just for the sake of weaving [Plato] |
| Full Idea: I don't suppose that anyone with any sense would want to hunt down the definition of 'weaving' for the sake of weaving itself. | |
| From: Plato (The Statesman [c.356 BCE], 285d) | |
| A reaction: The point seems to be that the definition brings out the connections between weaving and other activities and objects, thus enlarging our understanding. |
| 18192 | Do the Replacement Axioms exceed the iterative conception of sets? [Boolos, by Maddy] |
| Full Idea: For Boolos, the Replacement Axioms go beyond the iterative conception. | |
| From: report of George Boolos (The iterative conception of Set [1971]) by Penelope Maddy - Naturalism in Mathematics I.3 |
| 10146 | Cantor's theories needed the Axiom of Choice, but it has led to great controversy [Feferman/Feferman] |
| Full Idea: The Axiom of Choice is a pure existence statement, without defining conditions. It was necessary to provide a foundation for Cantor's theory of transfinite cardinals and ordinal numbers, but its nonconstructive character engendered heated controversy. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) |
| 10147 | The Axiom of Choice is consistent with the other axioms of set theory [Feferman/Feferman] |
| Full Idea: In 1938 Gödel proved that the Axiom of Choice is consistent with the other axioms of set theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: Hence people now standardly accept ZFC, rather than just ZF. |
| 10148 | Axiom of Choice: a set exists which chooses just one element each of any set of sets [Feferman/Feferman] |
| Full Idea: Zermelo's Axiom of Choice asserts that for any set of non-empty sets that (pairwise) have no elements in common, then there is a set that 'simultaneously chooses' exactly one element from each set. Note that this is an existential claim. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The Axiom is now widely accepted, after much debate in the early years. Even critics of the Axiom turn out to be relying on it. |
| 10149 | Platonist will accept the Axiom of Choice, but others want criteria of selection or definition [Feferman/Feferman] |
| Full Idea: The Axiom of Choice seems clearly true from the Platonistic point of view, independently of how sets may be defined, but is rejected by those who think such existential claims must show how to pick out or define the object claimed to exist. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The typical critics are likely to be intuitionists or formalists, who seek for both rigour and a plausible epistemology in our theory. |
| 10150 | The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman] |
| Full Idea: The Trichotomy Principle (any number is less, equal to, or greater than, another number) turned out to be equivalent to the Axiom of Choice. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: [He credits Sierpinski (1918) with this discovery] |
| 10158 | A structure is a 'model' when the axioms are true. So which of the structures are models? [Feferman/Feferman] |
| Full Idea: A structure is said to be a 'model' of an axiom system if each of its axioms is true in the structure (e.g. Euclidean or non-Euclidean geometry). 'Model theory' concerns which structures are models of a given language and axiom system. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This strikes me as the most interesting aspect of mathematical logic, since it concerns the ways in which syntactic proof-systems actually connect with reality. Tarski is the central theoretician here, and his theory of truth is the key. |
| 10162 | Tarski and Vaught established the equivalence relations between first-order structures [Feferman/Feferman] |
| Full Idea: In the late 1950s Tarski and Vaught defined and established basic properties of the relation of elementary equivalence between two structures, which holds when they make true exactly the same first-order sentences. This is fundamental to model theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This is isomorphism, which clarifies what a model is by giving identity conditions between two models. Note that it is 'first-order', and presumably founded on classical logic. |
| 10160 | Löwenheim-Skolem says if the sentences are countable, so is the model [Feferman/Feferman] |
| Full Idea: The Löwenheim-Skolem Theorem, the earliest in model theory, states that if a countable set of sentences in a first-order language has a model, then it has a countable model. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: There are 'upward' (sentences-to-model) and 'downward' (model-to-sentences) versions of the theory. |
| 10159 | Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory [Feferman/Feferman] |
| Full Idea: Before Tarski's work in the 1930s, the main results in model theory were the Löwenheim-Skolem Theorem, and Gödel's establishment in 1929 of the completeness of the axioms and rules for the classical first-order predicate (or quantificational) calculus. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10161 | If a sentence holds in every model of a theory, then it is logically derivable from the theory [Feferman/Feferman] |
| Full Idea: Completeness is when, if a sentences holds in every model of a theory, then it is logically derivable from that theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10156 | 'Recursion theory' concerns what can be solved by computing machines [Feferman/Feferman] |
| Full Idea: 'Recursion theory' is the subject of what can and cannot be solved by computing machines | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Ch.9) | |
| A reaction: This because 'recursion' will grind out a result step-by-step, as long as the steps will 'halt' eventually. |
| 10155 | Both Principia Mathematica and Peano Arithmetic are undecidable [Feferman/Feferman] |
| Full Idea: In 1936 Church showed that Principia Mathematica is undecidable if it is ω-consistent, and a year later Rosser showed that Peano Arithmetic is undecidable, and any consistent extension of it. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int IV) |
| 5961 | The soul gets its goodness from god, and its evil from previous existence. [Plato] |
| Full Idea: From its composer the soul possesses all beautiful things, but from its former condition, everything that proves to be harsh and unjust in heaven. | |
| From: Plato (The Statesman [c.356 BCE], 273b) | |
| A reaction: A neat move to explain the origins of evil (or rather, to shift the problem of evil to a long long way from here). This view presumably traces back to the views of Empedocles on good and evil. Can the soul acquire evil in its current existence? |
| 283 | The question of whether or not to persuade comes before the science of persuasion [Plato] |
| Full Idea: The science of whether one must persuade or not must rule over the science capable of persuading. | |
| From: Plato (The Statesman [c.356 BCE], 304c) | |
| A reaction: Plato probably thinks that reason has to be top of the pyramid, but there is always the Nietzschean/romantic question of why we should place such a value on what is rational. |
| 282 | Non-physical beauty can only be shown clearly by speech [Plato] |
| Full Idea: The bodiless things, being the most beautiful and the greatest, are only shown with clarity by speech and nothing else. | |
| From: Plato (The Statesman [c.356 BCE], 286a) | |
| A reaction: Unfortunately this will be true of warped and ugly ideas as well. |
| 281 | The arts produce good and beautiful things by preserving the mean [Plato] |
| Full Idea: It is by preserving the mean that arts produce everything that is good and beautiful. | |
| From: Plato (The Statesman [c.356 BCE], 284b) |
| 22559 | Democracy is the worst of good constitutions, but the best of bad constitutions [Plato, by Aristotle] |
| Full Idea: Plato judged that when the constitution is decent, democracy is the worst of them, but when they are bad it is the best. | |
| From: report of Plato (The Statesman [c.356 BCE], 302e) by Aristotle - Politics 1289b07 | |
| A reaction: Aristotle denies that a good oligarchy is superior. What of technocracy? The challenge is to set up institutions which ensure the health of the democracy. The big modern problem is populists who lie. |
| 279 | Only divine things can always stay the same, and bodies are not like that [Plato] |
| Full Idea: It is fitting for only the most divine things of all to be always the same and in the same state and in the same respects, and the nature of body is not of this ordering. | |
| From: Plato (The Statesman [c.356 BCE], 269b) |