23 ideas
5896 | Speak the truth, for this alone deifies man [Pythagoras, by Porphyry] |
Full Idea: Pythagoras advised above all things to speak the truth, for this alone deifies man. | |
From: report of Pythagoras (reports [c.530 BCE]) by Porphyry - Life of Pythagoras §41 | |
A reaction: Idea 4421 (of Nietzsche) stands in contrast to this. I am not quite sure why speaking the truth has such a high value. I am inclined to a minimalist view, which is just that philosophy is an attempt to speak the truth, as fishermen try to catch fish. |
3051 | Pythagoras discovered the numerical relation of sounds on a string [Pythagoras, by Diog. Laertius] |
Full Idea: Pythagoras discovered the numerical relation of sounds on a string. | |
From: report of Pythagoras (reports [c.530 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 08.1.11 |
17833 | The first-order ZF axiomatisation is highly non-categorical [Hallett,M] |
Full Idea: The first-order Sermelo-Fraenkel axiomatisation is highly non-categorical. | |
From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1213) |
17834 | Non-categoricity reveals a sort of incompleteness, with sets existing that the axioms don't reveal [Hallett,M] |
Full Idea: The non-categoricity of the axioms which Zermelo demonstrates reveals an incompleteness of a sort, ....for this seems to show that there will always be a set (indeed, an unending sequence) that the basic axioms are incapable of revealing to be sets. | |
From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1215) | |
A reaction: Hallett says the incompleteness concerning Zermelo was the (transfinitely) indefinite iterability of the power set operation (which is what drives the 'iterative conception' of sets). |
17837 | Zermelo allows ur-elements, to enable the widespread application of set-theory [Hallett,M] |
Full Idea: Unlike earlier writers (such as Fraenkel), Zermelo clearly allows that there might be ur-elements (that is, objects other than the empty set, which have no members). Indeed he sees in this the possibility of widespread application of set-theory. | |
From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1217) |
10282 | Logic is the study of sound argument, or of certain artificial languages (or applying the latter to the former) [Hodges,W] |
Full Idea: A logic is a collection of closely related artificial languages, and its older meaning is the study of the rules of sound argument. The languages can be used as a framework for studying rules of argument. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.1) | |
A reaction: [Hodges then says he will stick to the languages] The suspicion is that one might confine the subject to the artificial languages simply because it is easier, and avoids the tricky philosophical questions. That approximates to computer programming. |
10283 | A formula needs an 'interpretation' of its constants, and a 'valuation' of its variables [Hodges,W] |
Full Idea: To have a truth-value, a first-order formula needs an 'interpretation' (I) of its constants, and a 'valuation' (ν) of its variables. Something in the world is attached to the constants; objects are attached to variables. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.3) |
10285 | I |= φ means that the formula φ is true in the interpretation I [Hodges,W] |
Full Idea: I |= φ means that the formula φ is true in the interpretation I. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.5) | |
A reaction: [There should be no space between the vertical and the two horizontals!] This contrasts with |-, which means 'is proved in'. That is a syntactic or proof-theoretic symbol, whereas |= is a semantic symbol (involving truth). |
10284 | There are three different standard presentations of semantics [Hodges,W] |
Full Idea: Semantic rules can be presented in 'Tarski style', where the interpretation-plus-valuation is reduced to the same question for simpler formulas, or the 'Henkin-Hintikka style' in terms of games, or the 'Barwise-Etchemendy style' for computers. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.3) | |
A reaction: I haven't yet got the hang of the latter two, but I note them to map the territory. |
10289 | Up Löwenheim-Skolem: if infinite models, then arbitrarily large models [Hodges,W] |
Full Idea: Upward Löwenheim-Skolem: every first-order theory with infinite models has arbitrarily large models. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.10) |
10288 | Down Löwenheim-Skolem: if a countable language has a consistent theory, that has a countable model [Hodges,W] |
Full Idea: Downward Löwenheim-Skolem (the weakest form): If L is a first-order language with at most countably many formulas, and T is a consistent theory in L. Then T has a model with at most countably many elements. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.10) |
10287 | If a first-order theory entails a sentence, there is a finite subset of the theory which entails it [Hodges,W] |
Full Idea: Compactness Theorem: suppose T is a first-order theory, ψ is a first-order sentence, and T entails ψ. Then there is a finite subset U of T such that U entails ψ. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.10) | |
A reaction: If entailment is possible, it can be done finitely. |
7485 | For Pythagoreans 'one' is not a number, but the foundation of numbers [Pythagoras, by Watson] |
Full Idea: For Pythagoreans, one, 1, is not a true number but the 'essence' of number, out of which the number system emerges. | |
From: report of Pythagoras (reports [c.530 BCE], Ch.8) by Peter Watson - Ideas Ch.8 | |
A reaction: I think this is right! Counting and numbers only arise once the concept of individuality and identity have arisen. Counting to one is no more than observing the law of identity. 'Two' is the big adventure. |
17836 | The General Continuum Hypothesis and its negation are both consistent with ZF [Hallett,M] |
Full Idea: In 1938, Gödel showed that ZF plus the General Continuum Hypothesis is consistent if ZF is. Cohen showed that ZF and not-GCH is also consistent if ZF is, which finally shows that neither GCH nor ¬GCH can be proved from ZF itself. | |
From: Michael Hallett (Introduction to Zermelo's 1930 paper [1996], p.1217) |
10286 | A 'set' is a mathematically well-behaved class [Hodges,W] |
Full Idea: A 'set' is a mathematically well-behaved class. | |
From: Wilfrid Hodges (First-Order Logic [2001], 1.6) |
3053 | Pythagoras taught that virtue is harmony, and health, and universal good, and God [Pythagoras, by Diog. Laertius] |
Full Idea: Pythagoras taught that virtue is harmony, and health, and universal good, and God. | |
From: report of Pythagoras (reports [c.530 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 08.1.19 | |
A reaction: I like the link with health, because I consider that a bridge over the supposed fact-value gap. Very Pythagorean to think that virtue is harmony. Plato liked that thought. |
5244 | For Pythagoreans, justice is simply treating all people the same [Pythagoras, by Aristotle] |
Full Idea: Some even think that what is just is simple reciprocity, as the Pythagoreans maintained, because they defined justice simply as having done to one what one has done to another. | |
From: report of Pythagoras (reports [c.530 BCE], 28) by Aristotle - Nicomachean Ethics 1132b22 | |
A reaction: One wonders what Pythagoreans made of slavery. Aristotle argues that officials, for example, have superior rights. The Pythagorean idea makes fairness the central aspect of justice, and that must at least be partly right. |
638 | Pythagoreans define timeliness, justice and marriage in terms of numbers [Pythagoras, by Aristotle] |
Full Idea: The Pythagoreans offered definitions of a limited range of things on the basis of numbers; examples are timeliness, justice and marriage. | |
From: report of Pythagoras (reports [c.530 BCE]) by Aristotle - Metaphysics 1078b |
553 | Pythagoreans think mathematical principles are the principles of all of nature [Pythagoras, by Aristotle] |
Full Idea: The Pythagoreans thought that the principles of mathematical entities were the principles of all entities. | |
From: report of Pythagoras (reports [c.530 BCE]) by Aristotle - Metaphysics 985b |
375 | When musical harmony and rhythm were discovered, similar features were seen in bodily movement [Pythagoras, by Plato] |
Full Idea: When our predecessors discovered musical scales, they also discovered similar features in bodily movement, which should also be measured numerically, and called 'tempos' and 'measures'. | |
From: report of Pythagoras (reports [c.530 BCE]) by Plato - Philebus 17d |
644 | For Pythagoreans the entire universe is made of numbers [Pythagoras, by Aristotle] |
Full Idea: For Pythagoreans the entire universe is constructed of numbers. | |
From: report of Pythagoras (reports [c.530 BCE]) by Aristotle - Metaphysics 1080b |
554 | Pythagoreans say things imitate numbers, but Plato says things participate in numbers [Pythagoras, by Aristotle] |
Full Idea: Pythagoreans said that entities existed by imitation of the numbers, whereas Plato said that it was by participation. | |
From: report of Pythagoras (reports [c.530 BCE]) by Aristotle - Metaphysics 987b |
7467 | The modern idea of an immortal soul was largely created by Pythagoras [Pythagoras, by Watson] |
Full Idea: The modern concept of the immortal soul is a Greek idea, which owes much to Pythagoras. | |
From: report of Pythagoras (reports [c.530 BCE]) by Peter Watson - Ideas Ch.5 | |
A reaction: You can see why it caught on - it is a very appealing idea. Watson connects the 'modern' view with the ideas of heaven and hell. Obviously the idea of an afterlife goes a long way back (judging from the contents of ancient graves). |