25 ideas
| 7990 | Serene wisdom is freedom from ties, and indifference to fortune [Anon (Bhag)] |
| Full Idea: Who everywhere is free from all ties, who neither rejoices nor sorrows if fortune is good or is ill, his is a serene wisdom. | |
| From: Anon (Bhag) (The Bhagavad Gita [c.500 BCE], 2.57) | |
| A reaction: This is very similar to the 'apatheia' of the Stoics, though they are always more committed to rationality. This is quite a good strategy when times are hard, but as a general rule it offers a bogus state of 'wisdom' which is really half way to death. |
| 7989 | Seek salvation in the wisdom of reason [Anon (Bhag)] |
| Full Idea: Seek salvation in the wisdom of reason. | |
| From: Anon (Bhag) (The Bhagavad Gita [c.500 BCE], 2.49) | |
| A reaction: Quotations like this can usually be counterbalanced in eastern philosophy by wild irrationality, but they certainly felt to tug of reason. Only the Dhaoists seem really opposed to reason (e.g. Idea 7289). |
| 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] |
| 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) |
| 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) |
| 7996 | I am all the beauty and goodness of things, says Krishna [Anon (Bhag)] |
| Full Idea: I am the beauty of all things beautiful; ...I am the goodness of those who are good, says Krishna. | |
| From: Anon (Bhag) (The Bhagavad Gita [c.500 BCE], 10.36) | |
| A reaction: Another attempt to annexe everything which is admirable to the nature of God. This sounds strikingly Platonic (c.f. Idea 7992, which seems Aristotelian). One scholar dates the text to 150 BCE. I think there is influence, one way or the other. |
| 7995 | In all living beings I am the light of consciousness, says Krishna [Anon (Bhag)] |
| Full Idea: In all living beings I am the light of consciousness, says Krishna. | |
| From: Anon (Bhag) (The Bhagavad Gita [c.500 BCE], 10.22) | |
| A reaction: Everything grand seems to be claimed for God at this stage of culture, but I am not sure how coherent this view is, unless this is pantheism. In what sense could we possibly be Krishna, when none of us (except Arjuna) is aware of it? |
| 7999 | All actions come from: body, lower self, perception, means of action, or Fate [Anon (Bhag)] |
| Full Idea: Whatever a man does, good or bad, in thought, word or deed, has these five sources of action: the body, the lower 'I am', the means of perception, the means of action, and Fate. | |
| From: Anon (Bhag) (The Bhagavad Gita [c.500 BCE], 18.14/15) | |
| A reaction: The 'means of action' will presumably take care of anything we haven't thought of! Nothing quite matches the idea of 'the will' here. A twitch from the first, eating from the second, a startled jump from the third, struck by lightning from the fifth. |
| 7991 | Hate and lust have their roots in man's lower nature [Anon (Bhag)] |
| Full Idea: Hate and lust for things of nature have their roots in man's lower nature. | |
| From: Anon (Bhag) (The Bhagavad Gita [c.500 BCE], 3.34) | |
| A reaction: It seems outmoded now (since Freud) to label parts of human nature as 'higher' and 'lower'. I would defend the distinction, but it is not self-evident. The basis of morality is good citizenship, and parts of our nature are detrimental to that. |
| 7988 | There is no greater good for a warrior than to fight in a just war [Anon (Bhag)] |
| Full Idea: There is no greater good for a warrior than to fight in righteous war. | |
| From: Anon (Bhag) (The Bhagavad Gita [c.500 BCE], 2.31) | |
| A reaction: What worries me now is not the urging to fight, as long as a good cause can be found, but the idea that someone should see his social role as 'warrior'. The modern 'soldier' is ready to fight, but a traditional 'warrior' is obliged to fight. |
| 6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
| Full Idea: Hermarchus said that animal killing is justified by considerations of human safety and nourishment and by animals' inability to form contractual relations of justice with us. | |
| From: report of Hermarchus (fragments/reports [c.270 BCE]) by David A. Sedley - Hermarchus | |
| A reaction: Could the last argument be used to justify torturing animals? Or could we eat a human who was too brain-damaged to form contracts? |
| 7992 | The visible forms of nature are earth, water, fire, air, ether; mind, reason, and the sense of 'I' [Anon (Bhag)] |
| Full Idea: The visible forms of nature are eight: earth, water, fire, air, ether; the mind, reason, and the sense of 'I'. | |
| From: Anon (Bhag) (The Bhagavad Gita [c.500 BCE], 7.4) | |
| A reaction: Presumably there is an implication that there are also invisible forms. The Bhuddists launched an attack on 'I' as one of the categories. The first five appear to be Aristotle's, which must be of scholarly (and chronological) interest. |
| 7994 | Everything, including the gods, comes from me, says Krishna [Anon (Bhag)] |
| Full Idea: All the gods come from me, says Krishna. ...I am the one source of all | |
| From: Anon (Bhag) (The Bhagavad Gita [c.500 BCE], 10.2/8) | |
| A reaction: This seems very close to monotheism, and sounds very similar to the position that Zeus seems to occupy in later Greek religion, where he is shading off into a supreme and spiritual entity. |
| 7993 | Brahman is supreme, Atman his spirit in man, and Karma is the force of creation [Anon (Bhag)] |
| Full Idea: Brahman is supreme, the Eternal. Atman is his Spirit in man. Karma is the force of creation, wherefrom all things have their life. | |
| From: Anon (Bhag) (The Bhagavad Gita [c.500 BCE], 8.3) | |
| A reaction: I can't help wondering how they know all this stuff, but then I'm just a typical product of my culture. We seem to have a trinity here. Who's in charge? Is Atman just a servant? Is Karma totally under the control of Brahman? |
| 7997 | Only by love can men see me, know me, and come to me, says Krishna [Anon (Bhag)] |
| Full Idea: Only by love can men see me, and know me, and come unto me, says Krishna | |
| From: Anon (Bhag) (The Bhagavad Gita [c.500 BCE], 11.54) | |
| A reaction: There seems to be a paradox here, as it is unclear how you can love Krishna, if you have not already seen him in some way. This is another paradox of fideism - that faith cannot possibly be the first step in a religion, as faith needs a target. |
| 7998 | The three gates of hell are lust, anger and greed [Anon (Bhag)] |
| Full Idea: Three are the gates of this hell, the death of the soul: the gate of lust, the gate of wrath, and the gate of greed. Let a man shun the three. | |
| From: Anon (Bhag) (The Bhagavad Gita [c.500 BCE], 16.21) | |
| A reaction: Anyone who wishes to procreate, champion justice, and make a living, has to pursue all three. Wisdom consists of pursuing the three appropriately, not in shunning them. How did this bizarre puritanism ever come to grip the human race? |