58 ideas
19335 | Reasonings have a natural ordering in God's understanding, but only a temporal order in ours [Leibniz] |
Full Idea: All reasonings are eminent in God, and they preserve an order among themselves in his understanding as well as in ours; but for him this is just an order and a priority of nature, whereas for us there is a priority of time. | |
From: Gottfried Leibniz (The Theodicy [1710], p.192), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.III | |
A reaction: This view is found in Frege, and seems to be the hallmark of rationalist philosophy. There is an apriori assumption that reality has a rational order, so that pure reason is a tool for grasping it. Lewis's 'mosaic' of experiences has no order. |
9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
Full Idea: A contextual definition shows how to analyse an expression in situ, by replacing a complete sentence (of a particular form) in which the expression occurs by another in which it does not. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.2) | |
A reaction: This is a controversial procedure, which (according to Dummett) Frege originally accepted, and later rejected. It might not be the perfect definition that replacing just the expression would give you, but it is a promising step. |
10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
Full Idea: When a definition contains a quantifier whose range includes the very entity being defined, the definition is said to be 'impredicative'. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.2) | |
A reaction: Presumably they are 'impredicative' because they do not predicate a new quality in the definiens, but make use of the qualities already known. |
10098 | The 'power set' of A is all the subsets of A [George/Velleman] |
Full Idea: The 'power set' of A is all the subsets of A. P(A) = {B : B ⊆ A}. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) |
10101 | Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B [George/Velleman] |
Full Idea: The 'Cartesian Product' of any two sets A and B is the set of all ordered pairs <a, b> in which a ∈ A and b ∈ B, and it is denoted as A x B. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) |
10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman] |
Full Idea: The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}}. The existence of this set is guaranteed by three applications of the Axiom of Pairing. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: See Idea 10100 for the Axiom of Pairing. |
10103 | Grouping by property is common in mathematics, usually using equivalence [George/Velleman] |
Full Idea: The idea of grouping together objects that share some property is a common one in mathematics, ...and the technique most often involves the use of equivalence relations. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) |
10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman] |
Full Idea: A relation is an equivalence relation if it is reflexive, symmetric and transitive. The 'same first letter' is an equivalence relation on the set of English words. Any relation that puts a partition into clusters will be equivalence - and vice versa. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: This is a key concept in the Fregean strategy for defining numbers. |
10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman] |
Full Idea: ZFC is a theory concerned only with sets. Even the elements of all of the sets studied in ZFC are also sets (whose elements are also sets, and so on). This rests on one clearly pure set, the empty set Φ. ..Mathematics only needs pure sets. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: This makes ZFC a much more metaphysically comfortable way to think about sets, because it can be viewed entirely formally. It is rather hard to disentangle a chair from the singleton set of that chair. |
10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman] |
Full Idea: The Axiom of Extensionality says that for all sets x and y, if x and y have the same elements then x = y. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: This seems fine in pure set theory, but hits the problem of renates and cordates in the real world. The elements coincide, but the axiom can't tell you why they coincide. |
10100 | Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman] |
Full Idea: The Axiom of Pairing says that for all sets x and y, there is a set z containing x and y, and nothing else. In symbols: ∀x∀y∃z∀w(w ∈ z ↔ (w = x ∨ w = y)). | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: See Idea 10099 for an application of this axiom. |
17900 | The Axiom of Reducibility made impredicative definitions possible [George/Velleman] |
Full Idea: The Axiom of Reducibility ...had the effect of making impredicative definitions possible. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) |
10109 | ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman] |
Full Idea: Sets, unlike extensions, fail to correspond to all concepts. We can prove in ZFC that there is no set corresponding to the concept 'set' - that is, there is no set of all sets. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4) | |
A reaction: This is rather an important point for Frege. However, all concepts have extensions, but they may be proper classes, rather than precisely defined sets. |
10108 | As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman] |
Full Idea: The problem with reducing arithmetic to ZFC is not that this theory is inconsistent (as far as we know it is not), but rather that is not completely general, and for this reason not logical. For example, it asserts the existence of sets. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4) | |
A reaction: Note that ZFC has not been proved consistent. |
10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
Full Idea: A hallmark of our realist stance towards the natural world is that we are prepared to assert the Law of Excluded Middle for all statements about it. For all statements S, either S is true, or not-S is true. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4) | |
A reaction: Personally I firmly subscribe to realism, so I suppose I must subscribe to Excluded Middle. ...Provided the statement is properly formulated. Or does liking excluded middle lead me to realism? |
10129 | A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman] |
Full Idea: A 'model' of a theory is an assignment of meanings to the symbols of its language which makes all of its axioms come out true. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7) | |
A reaction: If the axioms are all true, and the theory is sound, then all of the theorems will also come out true. |
10105 | Differences between isomorphic structures seem unimportant [George/Velleman] |
Full Idea: Mathematicians tend to regard the differences between isomorphic mathematical structures as unimportant. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: This seems to be a pointer towards Structuralism as the underlying story in mathematics. The intrinsic character of so-called 'objects' seems unimportant. How theories map onto one another (and onto the world?) is all that matters? |
10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman] |
Full Idea: Consistency is a purely syntactic property, unlike the semantic property of soundness. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6) |
10126 | A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman] |
Full Idea: If there is a sentence such that both the sentence and its negation are theorems of a theory, then the theory is 'inconsistent'. Otherwise it is 'consistent'. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7) |
10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman] |
Full Idea: Soundness is a semantic property, unlike the purely syntactic property of consistency. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6) |
10127 | A 'complete' theory contains either any sentence or its negation [George/Velleman] |
Full Idea: If there is a sentence such that neither the sentence nor its negation are theorems of a theory, then the theory is 'incomplete'. Otherwise it is 'complete'. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7) | |
A reaction: Interesting questions are raised about undecidable sentences, irrelevant sentences, unknown sentences.... |
10106 | Rational numbers give answers to division problems with integers [George/Velleman] |
Full Idea: We can think of rational numbers as providing answers to division problems involving integers. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: Cf. Idea 10102. |
10102 | The integers are answers to subtraction problems involving natural numbers [George/Velleman] |
Full Idea: In defining the integers in set theory, our definition will be motivated by thinking of the integers as answers to subtraction problems involving natural numbers. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: Typical of how all of the families of numbers came into existence; they are 'invented' so that we can have answers to problems, even if we can't interpret the answers. It it is money, we may say the minus-number is a 'debt', but is it? Cf Idea 10106. |
10107 | Real numbers provide answers to square root problems [George/Velleman] |
Full Idea: One reason for introducing the real numbers is to provide answers to square root problems. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: Presumably the other main reasons is to deal with problems of exact measurement. It is interesting that there seem to be two quite distinct reasons for introducing the reals. Cf. Ideas 10102 and 10106. |
9946 | Logicists say mathematics is applicable because it is totally general [George/Velleman] |
Full Idea: The logicist idea is that if mathematics is logic, and logic is the most general of disciplines, one that applies to all rational thought regardless of its content, then it is not surprising that mathematics is widely applicable. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.2) | |
A reaction: Frege was keen to emphasise this. You are left wondering why pure logic is applicable to the physical world. The only account I can give is big-time Platonism, or Pythagoreanism. Logic reveals the engine-room of nature, where the design is done. |
10125 | The classical mathematician believes the real numbers form an actual set [George/Velleman] |
Full Idea: Unlike the intuitionist, the classical mathematician believes in an actual set that contains all the real numbers. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6) |
17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman] |
Full Idea: The first-order version of the induction axiom is weaker than the second-order, because the latter applies to all concepts, but the first-order applies only to concepts definable by a formula in the first-order language of number theory. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7 n7) |
10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman] |
Full Idea: The idea behind the proofs of the Incompleteness Theorems is to use the language of Peano Arithmetic to talk about the formal system of Peano Arithmetic itself. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7) | |
A reaction: The mechanism used is to assign a Gödel Number to every possible formula, so that all reasonings become instances of arithmetic. |
17902 | A successor is the union of a set with its singleton [George/Velleman] |
Full Idea: For any set x, we define the 'successor' of x to be the set S(x) = x U {x}. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: This is the Fregean approach to successor, where the Dedekind approach takes 'successor' to be a primitive. Frege 1884:§76. |
10133 | Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman] |
Full Idea: The derivability of Peano's Postulates from Hume's Principle in second-order logic has been dubbed 'Frege's Theorem', (though Frege would not have been interested, because he didn't think Hume's Principle gave an adequate definition of numebrs). | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.8 n1) | |
A reaction: Frege said the numbers were the sets which were the extensions of the sets created by Hume's Principle. |
10130 | Set theory can prove the Peano Postulates [George/Velleman] |
Full Idea: The Peano Postulates can be proven in ZFC. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.7) |
10089 | Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman] |
Full Idea: One might well wonder whether talk of abstract entities is less a solution to the empiricist's problem of how a priori knowledge is possible than it is a label for the problem. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Intro) | |
A reaction: This pinpoints my view nicely. What the platonist postulates is remote, bewildering, implausible and useless! |
10131 | If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman] |
Full Idea: As, in the logicist view, mathematics is about nothing particular, it is little wonder that nothing in particular needs to be observed in order to acquire mathematical knowledge. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002]) | |
A reaction: At the very least we can say that no one would have even dreamt of the general system of arithmetic is they hadn't had experience of the particulars. Frege thought generality ensured applicability, but extreme generality might entail irrelevance. |
10092 | In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman] |
Full Idea: In the unramified theory of types, all objects are classified into a hierarchy of types. The lowest level has individual objects that are not sets. Next come sets whose elements are individuals, then sets of sets, etc. Variables are confined to types. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: The objects are Type 0, the basic sets Type 1, etc. |
10094 | The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman] |
Full Idea: The theory of types seems to rule out harmless sets as well as paradoxical ones. If a is an individual and b is a set of individuals, then in type theory we cannot talk about the set {a,b}. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: Since we cheerfully talk about 'Cicero and other Romans', this sounds like a rather disasterous weakness. |
10095 | Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman] |
Full Idea: A problem with type theory is that there are only finitely many individuals, and finitely many sets of individuals, and so on. The hierarchy may be infinite, but each level is finite. Mathematics required an axiom asserting infinitely many individuals. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) | |
A reaction: Most accounts of mathematics founder when it comes to infinities. Perhaps we should just reject them? |
17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman] |
Full Idea: If a is an individual and b is a set of individuals, then in the theory of types we cannot talk about the set {a,b}, since it is not an individual or a set of individuals, ...but it is hard to see what harm can come from it. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3) |
10134 | Much infinite mathematics can still be justified finitely [George/Velleman] |
Full Idea: It is possible to use finitary reasoning to justify a significant part of infinitary mathematics. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.8) | |
A reaction: This might save Hilbert's project, by gradually accepting into the fold all the parts which have been giving a finitist justification. |
10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman] |
Full Idea: In the first instance all bounded quantifications are finitary, for they can be viewed as abbreviations for conjunctions and disjunctions. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6) | |
A reaction: This strikes me as quite good support for finitism. The origin of a concept gives a good guide to what it really means (not a popular view, I admit). When Aristotle started quantifying, I suspect of he thought of lists, not totalities. |
10123 | The intuitionists are the idealists of mathematics [George/Velleman] |
Full Idea: The intuitionists are the idealists of mathematics. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6) |
10124 | Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman] |
Full Idea: For intuitionists, truth is not independent of proof, but this independence is precisely what seems to be suggested by Gödel's First Incompleteness Theorem. | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.8) | |
A reaction: Thus Gödel was worse news for the Intuitionists than he was for Hilbert's Programme. Gödel himself responded by becoming a platonist about his unprovable truths. |
15034 | Are genera and species real or conceptual? bodies or incorporeal? in sensibles or separate from them? [Porphyry] |
Full Idea: I shall beg off talking of a) whether genera and species are real or situated in bare thoughts alone, b) whether as real they are bodies or incorporeals, and c) whether they are separated or in sensibles and have their reality in connection with them. | |
From: Porphyry (Isagoge ('Introduction') [c.295], (2)) | |
A reaction: This passage, picking up on Aristotle, seems to be the original source that grew into the medievel debate about universals. It seems to rather neatly lay out the agenda for the universals debate which is still with us. |
19367 | Saying we must will whatever we decide to will leads to an infinite regress [Leibniz] |
Full Idea: As for volition itself, to say that it is the object of free will is incorrect. We will to act, strictly speaking, and we do not will to will, else we should still say we will to have the will to will, and that would go on to infinity. | |
From: Gottfried Leibniz (The Theodicy [1710], p.151), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 4.IV | |
A reaction: This strikes me as an elementary difficulty which most fans of free will appear to evade. Thoughts just arise in us, and some of them are volitions. We can say there is then a 'gap' (Searle) where we choose, but what happens in the gap? |
19351 | Perfections of soul subordinate the body, but imperfections of soul submit to the body [Leibniz] |
Full Idea: Insofar as the soul has perfection ...God has accommodated the body to the soul, and has arranged beforehand that the body is impelled to execute its orders. Insofar as it is imperfect and confused, God accommodates soul to body, swayed by passions. | |
From: Gottfried Leibniz (The Theodicy [1710], p.159), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 3.IV | |
A reaction: Perkins says this is the nearest Leibniz gets to the idea of interaction between body and soul. Perfection and confusion are on a continuum for Leibniz. With such speculations I always wonder how these things can be known. How perfect is my mind? |
10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
Full Idea: Corresponding to every concept there is a class (some classes will be sets, the others proper classes). | |
From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4) |
19331 | Will is an inclination to pursue something good [Leibniz] |
Full Idea: One may say that 'will' consists in the inclination to do something in proportion to the good it contains. | |
From: Gottfried Leibniz (The Theodicy [1710], p.136), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.III | |
A reaction: This emphasises that the will is faced with options, rather than generating the options. The context is a discussion of the nature of God's will. I think 'will' is a really useful concept, and dislike the Hobbesian rejection of will. |
19346 | Most people facing death would happily re-live a similar life, with just a bit of variety [Leibniz] |
Full Idea: I believe there would be few persons who, being at the point of death, were not content to take up life again, on condition of passing through the same amount of good and evil, provided that it were not the same kind. | |
From: Gottfried Leibniz (The Theodicy [1710], p.130), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.IV | |
A reaction: Nice challenge. People who refuse the offer are not necessarily suicidal. He's probably right, but Leibniz doesn't recognise the factor of boredom. Look up the suicide note of the actor George Sanders! One life may be enough. |
19340 | Metaphysical evil is imperfection; physical evil is suffering; moral evil is sin [Leibniz] |
Full Idea: Evil may be taken metaphysically, physically, and morally. Metaphysical evil consists in mere imperfection, physical evil is suffering, and moral evil is sin. | |
From: Gottfried Leibniz (The Theodicy [1710], p.136), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.IV | |
A reaction: There seem to be plenty of imperfections in the world which don't look like evil. Or do you only declare it to be an imperfection because it seems to be evil (by some other standard)? Human evil comes from ignorance, so metaphysical explains moral. |
19366 | You can't assess moral actions without referring to the qualities of character that produce them [Leibniz] |
Full Idea: One is more worthy of praise when one owes the action to one's good qualities, and more culpable in proportion as one has been impelled by one's evil qualities; assessing actions without weighing the qualities whence they spring is to talk at random. | |
From: Gottfried Leibniz (The Theodicy [1710], p.426), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 4.IV | |
A reaction: Mill tries to separate judgement of the agent from judgement of the consequences of the action, but I think Leibniz has spotted that just judging outcomes ceases to be a 'moral' judgement. |
19326 | God must be intelligible, to select the actual world from the possibilities [Leibniz] |
Full Idea: The cause of the world must be intelligent: for this existing world being contingent and an infinity of worlds being equally possible, with equal claim to existence, the cause of the world must have regarded all of these worlds to fix on one of them. | |
From: Gottfried Leibniz (The Theodicy [1710], p.127), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.II | |
A reaction: A wonderfully Leibnizian way of putting what looks like the design argument. |
19327 | The intelligent cause must be unique and all-perfect, to handle all the interconnected possibilities [Leibniz] |
Full Idea: The intelligent cause ought to be infinite in all ways, and absolutely perfect in power, in wisdom, and in goodness, since it relates to all that which is possible. Also, since all is connected together, there is no ground for admitting more than one. | |
From: Gottfried Leibniz (The Theodicy [1710], p.128), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.II | |
A reaction: Notice that Leibniz's possible worlds seem to be all connected together, unlike David Lewis's worlds, which are discrete. Personally I suspect that all perfections will lead to contradiction, though Leibniz strongly argues against it. |
19344 | God prefers men to lions, but might not exterminate lions to save one man [Leibniz] |
Full Idea: It is certain that God sets greater store by a man than a lion; nevertheless it can hardly be said with certainty that God prefers a single man in all respects to the whole of lion-kind. | |
From: Gottfried Leibniz (The Theodicy [1710], p.189), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.IV | |
A reaction: Lovely problems arise when you guess at God's values! We have the same problem. Would you kill a poacher who was wiping out the last remaining lions? How many lions would you kill to save a human? |
19330 | If justice is arbitrary, or fixed but not observed, or not human justice, this undermines God [Leibniz] |
Full Idea: The three dogmas (1) that the nature of justice is arbitrary, (2) it is fixed, but not certain God will observe it, or (3) the justice we know is not that which God observes, destroy our confidence in the love of God. | |
From: Gottfried Leibniz (The Theodicy [1710], p.237), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.III | |
A reaction: Leibniz proceeds to carefully refute these three responses to the dilemma about how justice relates to God. |
19325 | God is the first reason of things; our experiences are contingent, and contain no necessity [Leibniz] |
Full Idea: God is the first reason of things: all that we see and experience is contingent and nothing in them renders their existence necessary. | |
From: Gottfried Leibniz (The Theodicy [1710], p.127), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.II | |
A reaction: Perkins presents this as the first step in one of Leibniz's arguments for God. They all seem to be variants of the ontological argument. [His 'Theodicy' is the Huggard translation, 1985] This resembles Aquinas's Third Way. |
19329 | The laws of physics are wonderful evidence of an intelligent and free being [Leibniz] |
Full Idea: These admirable laws [of physics] are wonderful evidence of an intelligent and free being, as opposed to the system of absolute and brute necessity, advocated by Strato and Spinoza. | |
From: Gottfried Leibniz (The Theodicy [1710], p.332), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.II | |
A reaction: Note the swipe at Spinoza. Leibniz defends the absolute necessities residing in God, but is too polite to call those 'brute', though personally I can't see the difference. But he says the laws arise from 'perfection and order', not from God's necessity. |
19437 | Prayers are useful, because God foresaw them in his great plan [Leibniz] |
Full Idea: Not only cares and labours but also prayers are useful; God having had these prayers in view before he regulated things. | |
From: Gottfried Leibniz (The Theodicy [1710], Abridge III) | |
A reaction: Hm. I'm struggling with this one. So I can't skip prayers today, because God has foreseen them and included them in his great plan? Hard to motivate yourself, like starting a game of chess after you've already been declared the winner. |
19337 | How can an all-good, wise and powerful being allow evil, sin and apparent injustice? [Leibniz] |
Full Idea: There is this question of natural theology, how a sole Principle, all-good, all-wise and all-powerful, has been able to admit evil, and especially to permit sin, and how it could resolve to make the wicked often happy and the good unhappy? | |
From: Gottfried Leibniz (The Theodicy [1710], p.098), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.IV | |
A reaction: His answer is, roughly, that there is an unavoidable trade-off, which humans cannot fully understand. Personally I would say that if there is a God, the evidence for his benevolence towards humanity is not encouraging. |
19345 | Being confident of God's goodness, we disregard the apparent local evils in the visible world [Leibniz] |
Full Idea: Being made confident by demonstrations of the goodness and the justice of God, we disregard the appearances of harshness and justice which we see in this small portion of his Kingdom that is exposed to our gaze. | |
From: Gottfried Leibniz (The Theodicy [1710], p.120), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.IV | |
A reaction: Hm. If this locality is full of evils, and the rest of it is much better, how come we are stuck in this miserable corner of things? God is obliged to compromise, but did he select us to get the worst of it? |