68 ideas
| 18290 | But what is the reasoning of the body, that it requires the wisdom you seek? [Nietzsche] |
| Full Idea: There is more reason in your body than in your best wisdom. For who knows for what purpose your body requires precisely your best wisdom? | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.05) | |
| A reaction: Lovely question. For years I've paid lip-service to wisdom as the rough aim of all philosophy. Not quite knowing what wisdom is doesn't bother me, but knowing why I want wisdom certainly does, especially after this idea. |
| 18303 | Reject wisdom that lacks laughter [Nietzsche] |
| Full Idea: Let that wisdom be false to us that brought no laughter with it! | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 3.12.23) |
| 18305 | To love truth, you must know how to lie [Nietzsche] |
| Full Idea: Inability to lie is far from being love of truth. ....He who cannot lie does not know what truth is. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 4.13.9) |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| Full Idea: By Gödel's First Incompleteness Theorem, there cannot be a negation-complete set theory. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 01.3) | |
| A reaction: This means that we can never prove all the truths of a system of set theory. |
| 10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
| Full Idea: Going second-order in arithmetic enables us to prove new first-order arithmetical sentences that we couldn't prove before. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 23.4) | |
| A reaction: The wages of Satan, perhaps. We can prove things about objects by proving things about their properties and sets and functions. Smith says this fact goes all the way up the hierarchy. |
| 10066 | Putnam coined the term 'if-thenism' [Putnam, by Musgrave] |
| Full Idea: Putnam coined the term 'if-thenism'. | |
| From: report of Hilary Putnam (The Thesis that Mathematics is Logic [1967]) by Alan Musgrave - Logicism Revisited §5 n |
| 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) |
| 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) |
| 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) |
| 10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
| Full Idea: A 'natural deduction system' will have no logical axioms but may rules of inference. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 09.1) | |
| A reaction: He contrasts this with 'Hilbert-style systems', which have many axioms but few rules. Natural deduction uses many assumptions which are then discharged, and so tree-systems are good for representing it. |
| 10613 | No nice theory can define truth for its own language [Smith,P] |
| Full Idea: No nice theory can define truth for its own language. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 21.5) | |
| A reaction: This leads on to Tarski's account of truth. |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
| Full Idea: An 'injective' function is 'one-to-one' - each element of the output set results from a different element of the original set. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) | |
| A reaction: That is, two different original elements cannot lead to the same output element. |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| Full Idea: A 'surjective' function is 'onto' - the whole of the output set results from the function being applied to elements of the original set. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| Full Idea: A 'bijective' function has 'one-to-one correspondence' - it is both surjective and injective, so that every element in each of the original and the output sets has a matching element in the other. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) | |
| A reaction: Note that 'injective' is also one-to-one, but only in the one direction. |
| 10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
| Full Idea: If everything that a theory proves must be true, then it is a 'sound' theory. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 01.1) |
| 10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
| Full Idea: Soundness is normally a matter of having true axioms and a truth-preserving proof system. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) | |
| A reaction: The only exception I can think of is if a theory consisted of nothing but the axioms. |
| 10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
| Full Idea: A theory is 'sound' iff every theorem of it is true (i.e. true on the interpretation built into its language). Soundness is normally a matter of having true axioms and a truth-preserving proof system. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) |
| 10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
| Full Idea: A theory is 'negation complete' if it decides every sentence of its language (either the sentence, or its negation). | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) |
| 10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
| Full Idea: There is an annoying double-use of 'complete': a logic may be semantically complete, but there may be an incomplete theory expressed in it. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) |
| 10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
| Full Idea: Logicians say that a theory T is '(negation) complete' if, for every sentence φ in the language of the theory, either φ or ¬φ is deducible in T's proof system. If this were the case, then truth could be equated with provability. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 01.1) | |
| A reaction: The word 'negation' seems to be a recent addition to the concept. Presumable it might be the case that φ can always be proved, but not ¬φ. |
| 10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
| Full Idea: There are two routes to Incompleteness results. One goes via the semantic assumption that we are dealing with sound theories, using a result about what they can express. The other uses the syntactic notion of consistency, with stronger notions of proof. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 18.1) |
| 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
| Full Idea: An 'effectively decidable' (or 'computable') algorithm will be step-by-small-step, with no need for intuition, or for independent sources, with no random methods, possible for a dumb computer, and terminates in finite steps. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.2) | |
| A reaction: [a compressed paragraph] |
| 10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
| Full Idea: A theory is 'decidable' iff there is a mechanical procedure for determining whether any sentence of its language can be proved. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) | |
| A reaction: Note that it doesn't actually have to be proved. The theorems of the theory are all effectively decidable. |
| 10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
| Full Idea: Any consistent, axiomatized, negation-complete formal theory is decidable. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 03.6) |
| 10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
| Full Idea: A set is 'enumerable' iff either the set is empty, or there is a surjective function to the set from the set of natural numbers, so that the set is in the range of that function. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.3) |
| 10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
| Full Idea: A set is 'effectively enumerable' if an (idealised) computer could be programmed to generate a list of its members such that any member will eventually be mentioned (even if the list is empty, or without end, or contains repetitions). | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.4) |
| 10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
| Full Idea: A finite set of finitely specifiable objects is always effectively enumerable (for example, the prime numbers). | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.4) |
| 10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
| Full Idea: The set of ordered pairs of natural numbers (i,j) is effectively enumerable, as proven by listing them in an array (across: <0,0>, <0,1>, <0,2> ..., and down: <0,0>, <1,0>, <2,0>...), and then zig-zagging. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 02.5) |
| 10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
| Full Idea: The theorems of any properly axiomatized theory can be effectively enumerated. However, the truths of any sufficiently expressive arithmetic can't be effectively enumerated. Hence the theorems and truths of arithmetic cannot be the same. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 05 Intro) |
| 10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
| Full Idea: Whether a property is 'expressible' in a given theory depends on the richness of the theory's language. Whether the property can be 'captured' (or 'represented') by the theory depends on the richness of the axioms and proof system. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 04.7) |
| 10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
| Full Idea: For prime numbers we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))). That is, the only way to multiply two numbers and a get a prime is if one of them is 1. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 04.5) |
| 10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
| Full Idea: It has been proved (by Tarski) that the real numbers R is a complete theory. But this means that while the real numbers contain the natural numbers, the pure theory of real numbers doesn't contain the theory of natural numbers. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 18.2) |
| 10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
| Full Idea: The truths of arithmetic are just the true equations involving particular numbers, and universally quantified versions of such equations. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 27.7) | |
| A reaction: Must each equation be universally quantified? Why can't we just universally quantify over the whole system? |
| 10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
| Full Idea: The number of Fs is the 'successor' of the number of Gs if there is an object which is an F, and the remaining things that are F but not identical to the object are equinumerous with the Gs. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 14.1) |
| 10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
| Full Idea: All numbers are related to zero by the ancestral of the successor relation. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 23.5) | |
| A reaction: The successor relation only ties a number to the previous one, not to the whole series. Ancestrals are a higher level of abstraction. |
| 10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
| Full Idea: Baby Arithmetic 'knows' the addition of particular numbers and multiplication, but can't express general facts about numbers, because it lacks quantification. It has a constant '0', a function 'S', and functions '+' and 'x', and identity and negation. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 08.1) |
| 10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
| Full Idea: Baby Arithmetic is negation complete, so it can prove every claim (or its negation) that it can express, but it is expressively extremely impoverished. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 08.3) |
| 10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
| Full Idea: We can beef up Baby Arithmetic into Robinson Arithmetic (referred to as 'Q'), by restoring quantifiers and variables. It has seven generalised axioms, plus standard first-order logic. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 08.3) |
| 10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
| Full Idea: Robinson Arithmetic (Q) is not negation complete | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 08.4) |
| 10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
| Full Idea: The sequence of natural numbers starts from zero, and each number has just one immediate successor; the sequence continues without end, never circling back on itself, and there are no 'stray' numbers, lurking outside the sequence. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 01.1) | |
| A reaction: These are the characteristics of the natural numbers which have to be pinned down by any axiom system, such as Peano's, or any more modern axiomatic structures. We are in the territory of Gödel's theorems. |
| 10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
| Full Idea: If the logic of arithmetic doesn't have second-order quantifiers to range over properties of numbers, how can it handle induction? | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 10.1) |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| Full Idea: Multiplication in itself isn't is intractable. In 1929 Skolem showed a complete theory for a first-order language with multiplication but lacking addition (or successor). Multiplication together with addition and successor produces incompleteness. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 10.7 n8) |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| Full Idea: Putting multiplication together with addition and successor in the language of arithmetic produces incompleteness. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 10.7) | |
| A reaction: His 'Baby Arithmetic' has all three and is complete, but lacks quantification (p.51) |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| Full Idea: The 'ancestral' of a relation is that relation which holds when there is an indefinitely long chain of things having the initial relation. | |
| From: Peter Smith (Intro to Gödel's Theorems [2007], 23.5) | |
| A reaction: The standard example is spotting the relation 'ancestor' from the receding relation 'parent'. This is a sort of abstraction derived from a relation which is not equivalent (parenthood being transitive but not reflexive). The idea originated with Frege. |
| 20757 | The powerful self behind your thoughts and feelings is your body [Nietzsche] |
| Full Idea: Behind your thoughts and feelings stands a powerful commander, an unknown wise man - he is called a self. He lives in your body; he is your body. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], I.4), quoted by Kevin Aho - Existentialism: an introduction 5 'Creature' | |
| A reaction: I find Nietzsche's view of the self very congenial, though I tend to see the self as certain central functions of the brain. The brain is enmeshed in the body (as in the location of pains). |
| 18289 | Forget the word 'I'; 'I' is performed by the intelligence of your body [Nietzsche] |
| Full Idea: You say 'I' and you are proud of this word. But greater than this - although you will not believe in it - is your body and its great intelligence, which does not say 'I' but performs 'I'. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.05) | |
| A reaction: I'm not sure if I understand this, but I offer it as a candidate for the most profound idea ever articulated about personal identity. |
| 18299 | The will is constantly frustrated by the past [Nietzsche] |
| Full Idea: Powerless against that which has been done, the will is an angry spectator of all things past. The will cannot will backwards; that it cannot break time and time's desire - that is the will's most lonely affliction. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 2.20) |
| 18297 | We created meanings, to maintain ourselves [Nietzsche] |
| Full Idea: Man first implanted values into things to maintain himself - he first created the meaning of things, a human meaning! | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.16) | |
| A reaction: It is certainly hard to see anything resembling values or meaning in the cosmos, if you remove the human beings. We should expect an evolutionary grounding in their explanation. |
| 18293 | The noble man wants new virtues; the good man preserves what is old [Nietzsche] |
| Full Idea: The noble man wants to create new things and a new virtue. The good man wants the old things and that the old things shall be preserved. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.09) | |
| A reaction: There is a limit to how many plausible virtues the noble men can come up with. We may already have run out. Are we going to have to re-run the Iliad? |
| 18301 | We only really love children and work [Nietzsche] |
| Full Idea: One loves from the very heart only one's child and one's work. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 3.03) | |
| A reaction: Very Nietzchean (and masculine?) to cite one's work. Rachmaninov said he was 85% musician and 15% human being, so I guess he loved music from the very heart. |
| 18307 | I want my work, not happiness! [Nietzsche] |
| Full Idea: Do I aspire after happiness? I aspire after my work! | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 4.20) | |
| A reaction: I empathise with aspiring to do something, rather than be something. But what do we wish for our children? Happiness first, then achievement? |
| 18291 | Virtues can destroy one another, through jealousy [Nietzsche] |
| Full Idea: Every virtue is jealous of the others, and jealousy is a terrible thing. Even virtues can be destroyed through jealousy. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.07) | |
| A reaction: How much more subtle and plausible than the picture of accumulating virtues, like medals! Zarathustra says it is best to have just one virtue. |
| 18287 | People now find both wealth and poverty too much of a burden [Nietzsche] |
| Full Idea: Nobody grows rich or poor any more: both are too much of a burden. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.01) | |
| A reaction: True. Most people I know are just puzzled by people who actually seem to want to be extremely wealthy. |
| 18295 | If you want friends, you must be a fighter [Nietzsche] |
| Full Idea: If you want a friend, you must be willing to wage war for him: and to wage war, you must be capable of being an enemy. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.15) |
| 18286 | The greatest experience possible is contempt for your own happiness, reason and virtue [Nietzsche] |
| Full Idea: What is the greatest thing you can experience? It is the hour of the great contempt. The hour in which even your happiness grows loathsome to you, and your reason and your virtue also. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.01) | |
| A reaction: This would be a transient state for Nietzsche, in which you realise the hollowness of those traditional ideas, and begin to seek something else. |
| 18296 | An enduring people needs its own individual values [Nietzsche] |
| Full Idea: No people could live without evaluating; but if it wishes to maintain itself it must not evaluate as its neighbour evaluates. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.16) | |
| A reaction: Political philosophers say plenty about a 'people', but little about what unifies them, or about what keeps one people distinct from another. Most people's are proud of their local values. |
| 18294 | The state coldly claims that it is the people, but that is a lie [Nietzsche] |
| Full Idea: The state is the coldest of all cold monsters. Coldly it lies, too; and this lie creeps from its mouth: 'I, the state, am the people'. It is a lie! | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.12) | |
| A reaction: This strikes me as just as true even after everyone gets the vote. Rulers can't help gradually forgetting about the people. |
| 18304 | Saints want to live as they desire, or not to live at all [Nietzsche] |
| Full Idea: 'To live as I desire to live or not to live at all': that is what I want, that is what the most saintly man wants. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 4.09) | |
| A reaction: [spoken by Zarathustra] |
| 18300 | Whenever we have seen suffering, we have wanted the revenge of punishment [Nietzsche] |
| Full Idea: The spirit of revenge: my friends, that, up to now, has been mankind's chief concern; and where there was suffering, there was always supposed to be punishment. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 2.20) |
| 18302 | Man and woman are deeply strange to one another! [Nietzsche] |
| Full Idea: Who has fully conceived how strange man and woman are to one another! | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 3.10.2) |
| 18292 | I can only believe in a God who can dance [Nietzsche] |
| Full Idea: I should believe only in a God who understood how to dance. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.08) |
| 18298 | Not being a god is insupportable, so there are no gods! [Nietzsche] |
| Full Idea: If there were gods, how could I endure not to be a god! Therefore there are no gods. ...For what would there to be create if gods - existed! | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 2.02) | |
| A reaction: [Zarathustra says this, not Nietzsche!] |
| 18288 | Heaven was invented by the sick and the dying [Nietzsche] |
| Full Idea: It was the sick and dying who despised the body and the earth and invented the things of heaven and the redeeming drops of blood. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 1.04) |
| 18306 | We don't want heaven; now that we are men, we want the kingdom of earth [Nietzsche] |
| Full Idea: We certainly do not want to enter into the kingdom of heaven: we have become men, so we want the kingdom of earth. | |
| From: Friedrich Nietzsche (Thus Spake Zarathustra [1884], 4.18.2) |