74 ideas
| 12926 | Wisdom is the science of happiness [Leibniz] |
| Full Idea: Wisdom is the science of happiness. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1690.03.23) | |
| A reaction: That probably comes down to common sense, or Aristotle's 'phronesis'. I take wisdom to involve understanding, as well as the quest for happiness. |
| 12903 | Wise people have fewer acts of will, because such acts are linked together [Leibniz] |
| Full Idea: The wiser one is, the fewer separate acts of will one has and the more one's views and acts of will are comprehensive and linked together. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.04.12) | |
| A reaction: [letter to Landgrave, about Arnauld] It is unusual to find a philosopher who actually tries to analyse the nature of wisdom, instead of just paying lipservice to it. I take Leibniz to be entirely right here. He equates wisdom with rational behaviour. |
| 12914 | Metaphysics is geometrical, resting on non-contradiction and sufficient reason [Leibniz] |
| Full Idea: I claim to give metaphysics geometric demonstrations, assuming only the principle of contradiction (or else all reasoning becomes futile), and that nothing exists without a reason, or that every truth has an a priori proof, from the concept of terms. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.07.4/14 XI) | |
| A reaction: For the last bit, see Idea 12910. This idea is the kind of huge optimism about metaphysic which got it a bad name after Kant, and in modern times. I'm optimistic about metaphysics, but certainly not about 'geometrical demonstrations' of it. |
| 12915 | Definitions can only be real if the item is possible [Leibniz] |
| Full Idea: Definitions to my mind are real, when one knows that the thing defined is possible; otherwise they are only nominal, and one must not rely on them. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.07.4/14 XI) | |
| A reaction: It is interesting that things do not have to actual to have real definitions. For Leibniz, what is possible will exist in the mind of God. For me what is possible will exist in the potentialities of the powers of what is actual. |
| 12910 | The predicate is in the subject of a true proposition [Leibniz] |
| Full Idea: In a true proposition the concept of the predicate is always present in the subject. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.07.4/14 X) | |
| A reaction: This sounds very like the Kantian notion of an analytic truth, but Leibniz is applying it to all truths. So Socrates must contain the predicate of running as part of his nature (or essence?), if 'Socrates runs' is to be true. |
| 19333 | A truth is just a proposition in which the predicate is contained within the subject [Leibniz] |
| Full Idea: In every true affirmative proposition, necessary or contingent, universal or particular, the concept of the predicate is in a sense included in that of the subject; the predicate is present in the subject; or else I do not know what truth is. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.07.4/14) | |
| A reaction: Why did he qualify this with "in a sense"? This is referred to as the 'concept containment theory of truth'. This is an odd view of the subject. If the truth is 'Peter fell down stairs', we don't usually think the concept of Peter contains such things. |
| 9572 | Realists about sets say there exists a null set in the real world, with no members [Chihara] |
| Full Idea: In the Gödelian realistic view of set theory the statement that there is a null set as the assertion of the existence in the real world of a set that has no members. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.6) | |
| A reaction: It seems to me obvious that such a claim is nonsense on stilts. 'In the beginning there was the null set'? |
| 9550 | We only know relational facts about the empty set, but nothing intrinsic [Chihara] |
| Full Idea: Everything we know about the empty set is relational; we know that nothing is the membership relation to it. But what do we know about its 'intrinsic properties'? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: Set theory seems to depend on the concept of the empty set. Modern theorists seem over-influenced by the Quine-Putnam view, that if science needs it, we must commit ourselves to its existence. |
| 9562 | In simple type theory there is a hierarchy of null sets [Chihara] |
| Full Idea: In simple type theory, there is a null set of type 1, a null set of type 2, a null set of type 3..... (Quine has expressed his distaste for this). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.4) | |
| A reaction: It is bad enough trying to individuate the unique null set, without whole gangs of them drifting indistinguishably through the logical fog. All rational beings should share Quine's distaste, even if Quine is wrong. |
| 9573 | The null set is a structural position which has no other position in membership relation [Chihara] |
| Full Idea: In the structuralist view of sets, in structures of a certain sort the null set is taken to be a position (or point) that will be such that no other position (or point) will be in the membership relation to it. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.6) | |
| A reaction: It would be hard to conceive of something having a place in a structure if nothing had a relation to it, so is the null set related to singeton sets but not there members. It will be hard to avoid Platonism here. Set theory needs the null set. |
| 9551 | What is special about Bill Clinton's unit set, in comparison with all the others? [Chihara] |
| Full Idea: What is it about the intrinsic properties of just that one unit set in virtue of which Bill Clinton is related to just it and not to any other unit sets in the set-theoretical universe? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: If we all kept pet woodlice, we had better not hold a wood louse rally, or we might go home with the wrong one. My singleton seems seems remarkably like yours. Could we, perhaps, swap, just for a change? |
| 9549 | The set theorist cannot tell us what 'membership' is [Chihara] |
| Full Idea: The set theorist cannot tell us anything about the true relationship of membership. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: If three unrelated objects suddenly became members of a set, it is hard to see how the world would have changed, except in the minds of those thinking about it. |
| 9571 | ZFU refers to the physical world, when it talks of 'urelements' [Chihara] |
| Full Idea: ZFU set theory talks about physical objects (the urelements), and hence is in some way about the physical world. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.5) | |
| A reaction: This sounds a bit surprising, given that the whole theory would appear to be quite unaffected if God announced that idealism is true and there are no physical objects. |
| 9563 | A pack of wolves doesn't cease when one member dies [Chihara] |
| Full Idea: A pack of wolves is not thought to go out of existence just because some member of the pack is killed. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.5) | |
| A reaction: The point is that the formal extensional notion of a set doesn't correspond to our common sense notion of a group or class. Even a highly scientific theory about wolves needs a loose notion of a wolf pack. |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
| 9561 | The mathematics of relations is entirely covered by ordered pairs [Chihara] |
| Full Idea: Everything one needs to do with relations in mathematics can be done by taking a relation to be a set of ordered pairs. (Ordered triples etc. can be defined as order pairs, so that <x,y,z> is <x,<y,z>>). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.2) | |
| A reaction: How do we distinguish 'I own my cat' from 'I love my cat'? Or 'I quite like my cat' from 'I adore my cat'? Nevertheless, this is an interesting starting point for a discussion of relations. |
| 9552 | Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced [Chihara] |
| Full Idea: In first-order logic a set of sentences is 'consistent' iff there is an interpretation (or structure) in which the set of sentences is true. ..For Frege, though, a set of sentences is consistent if it is not possible to deduce a contradiction from it. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 02.1) | |
| A reaction: The first approach seems positive, the second negative. Frege seems to have a higher standard, which is appealing, but the first one seems intuitively right. There is a possible world where this could work. |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
| A reaction: The sequence is 'Cauchy' if N exists. |
| 12920 | There is no multiplicity without true units [Leibniz] |
| Full Idea: There is no multiplicity without true units. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1687.04.30) | |
| A reaction: Hence real numbers do not embody 'multiplicity'. So either they don't 'embody' anything, or they embody 'magnitudes'. Does this give two entirely different notions, of measure of multiplicity and measures of magnitude? |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
| A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
| 9553 | Analytic geometry gave space a mathematical structure, which could then have axioms [Chihara] |
| Full Idea: With the invention of analytic geometry (by Fermat and then Descartes) physical space could be represented as having a mathematical structure, which could eventually lead to its axiomatization (by Hilbert). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 02.3) | |
| A reaction: The idea that space might have axioms seems to be pythagoreanism run riot. I wonder if there is some flaw at the heart of Einstein's General Theory because of this? |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
| 10192 | We can replace existence of sets with possibility of constructing token sentences [Chihara, by MacBride] |
| Full Idea: Chihara's 'constructability theory' is nominalist - mathematics is reducible to a simple theory of types. Instead of talk of sets {x:x is F}, we talk of open sentences Fx defining them. Existence claims become constructability of sentence tokens. | |
| From: report of Charles Chihara (A Structural Account of Mathematics [2004]) by Fraser MacBride - Review of Chihara's 'Structural Acc of Maths' p.81 | |
| A reaction: This seems to be approaching the problem in a Fregean way, by giving an account of the semantics. Chihara is trying to evade the Quinean idea that assertion is ontological commitment. But has Chihara retreated too far? How does he assert existence? |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
| A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
| A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
| A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
| A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
| A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
| A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
| 12319 | What is not truly one being is not truly a being either [Leibniz] |
| Full Idea: What is not truly one being is not truly a being either. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1687.04.30), quoted by Alain Badiou - Briefings on Existence 1 | |
| A reaction: Badiou quotes this as identifying Being with the One. I say Leibniz had no concept of 'gunk', and thought everything must have a 'this' identity in order to exist, which is just the sort of thing a logician would come up with. |
| 12922 | A thing 'expresses' another if they have a constant and fixed relationship [Leibniz] |
| Full Idea: One thing 'expresses' another (in my terminology) when there exists a constant and fixed relationship between what can be said of one and of the other. This is the way that a perspectival projection expresses its ground-plan. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1687.10.09) | |
| A reaction: Arnauld was puzzled by what Leibniz might mean by 'express', and it occurs to me that Leibniz was fishing for the modern concept of 'supervenience'. It also sounds a bit like the idea of 'covariance' between mind and world. Maybe he means 'function'. |
| 9559 | If a successful theory confirms mathematics, presumably a failed theory disconfirms it? [Chihara] |
| Full Idea: If mathematics shares whatever confirmation accrues to the theories using it, would it not be reasonable to suppose that mathematics shares whatever disconfirmation accrues to the theories using it? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 05.8) | |
| A reaction: Presumably Quine would bite the bullet here, although maths is much closer to the centre of his web of belief, and so far less likely to require adjustment. In practice, though, mathematics is not challenged whenever an experiment fails. |
| 9566 | No scientific explanation would collapse if mathematical objects were shown not to exist [Chihara] |
| Full Idea: Evidently, no scientific explanations of specific phenomena would collapse as a result of any hypothetical discovery that no mathematical objects exist. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 09.1) | |
| A reaction: It is inconceivable that anyone would challenge this claim. A good model seems to be drama; a play needs commitment from actors and audience, even when we know it is fiction. The point is that mathematics doesn't collapse either. |
| 13079 | A substance contains the laws of its operations, and its actions come from its own depth [Leibniz] |
| Full Idea: Each indivisible substance contains in its nature the law by which the series of its operations continues, and all that has happened and will happen to it. All its actions come from its own depths, except for dependence on God. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1688.01.4/14) | |
| A reaction: I take the combination of 'laws' and 'forces', which Leibniz attributes to Aristotelian essences, to be his distinctive contribution towards giving us an Aristotelian metaphysic which is suitable for modern science. |
| 12745 | Philosophy needs the precision of the unity given by substances [Leibniz] |
| Full Idea: Philosophy cannot be better reduced to something precise, than by recognising only substances or complete beings endowed with a true unity, with different states that succeed one another; all else is phenomena, abstractions or relations. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1687.04.30), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 7 | |
| A reaction: This idea bothers me. Has the whole of modern philosophy been distorted by this yearning for 'precision'? It has put mathematicians and logicians in the driving seat. Do we only attribute unity because it suits our thinking? |
| 12921 | Accidental unity has degrees, from a mob to a society to a machine or organism [Leibniz] |
| Full Idea: There are degrees of accidental unity, and an ordered society has more unity than a chaotic mob, and an organic body or a machine has more unity than a society. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1687.04.30) | |
| A reaction: This immediately invites questions about the extremes. Why does the very highest degree of 'accidental unity' not achieve 'true unity'? And why cannot a very ununified aggregate have a bit of unity (as in unrestricted mereological composition)? |
| 12746 | We find unity in reason, and unity in perception, but these are not true unity [Leibniz] |
| Full Idea: A pair of diamonds is merely an entity of reason, and even if one of them is brought close to another, it is an entity of imagination or perception, that is to say a phenomenon; contiguity, common movement and the same end don't make substantial unity. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1687.04.30), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 7 | |
| A reaction: This invites the question of what you have to do to two objects to give them substantial unity. The distinction between unity 'of reason' and unity 'of perception' is good. |
| 12916 | A body is a unified aggregate, unless it has an indivisible substance [Leibniz] |
| Full Idea: One will never find a body of which it may be said that it is truly one substance, ...because entities made up by aggregation have only as much reality as exists in the constituent parts. Hence the substance of a body must be indivisible. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.11) | |
| A reaction: Leibniz rejected atomism, and he evidently believed that pure materialists must deny the real existence of physical objects. Common sense suggests that causal bonds bestow a high degree of unity on bodies (if degrees are allowed). |
| 12919 | Unity needs an indestructible substance, to contain everything which will happen to it [Leibniz] |
| Full Idea: Substantial unity requires a complete, indivisible and naturally indestructible entity, since its concept embraces everything that is to happen to it, which cannot be found in shape or motion. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.11.28/12.8) | |
| A reaction: Hence if a tile is due to be broken in half (Arnauld's example), it cannot have had unity in the first place. To what do we refer when we say 'the tile was broken'? |
| 12923 | Every bodily substance must have a soul, or something analogous to a soul [Leibniz] |
| Full Idea: Every bodily substance must have a soul, or at least an entelechy which is analogous to the soul. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1687.10.09) | |
| A reaction: He routinely commits to a 'soul', and then pulls back and says it may only be an 'analogy'. He had deep doubts about his whole scheme, which emerged in the late correspondence with Des Bosses. This not monads, says Garber. |
| 12704 | Aggregates don’t reduce to points, or atoms, or illusion, so must reduce to substance [Leibniz] |
| Full Idea: In aggregates one must necessarily arrive either at mathematical points from which some make up extension, or at atoms (which I dismiss), or else no reality can be found in bodies, or finally one must recognises substances that possess a true unity. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1687.04.30), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 2 | |
| A reaction: Garber calls this Leibniz's Aggregate Argument. Leibniz is, of course, talking of physical aggregates which have unity. He consistently points out that a pile of logs has no unity at all. But is substance just that-which-provides-unity? |
| 13077 | Basic predicates give the complete concept, which then predicts all of the actions [Leibniz] |
| Full Idea: Apart from those that depend on others, one must only consider together all the basic predicates in order to form the complete concept of Adam adequate to deduce from it everything that is ever to happen to him, as much as is necessary to account for it. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.06) | |
| A reaction: This (implausibly) goes beyond mere prediction of properties. Eve's essence seems to be relevant to Adam's life. Note that the complete concept is not every predicate, but only those 'necessary' to predict the events. Cf Idea 13082. |
| 12908 | Essences exist in the divine understanding [Leibniz] |
| Full Idea: Essences exist in the divine understanding before one considers will. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.07.4/14 X) | |
| A reaction: This is a sort of religious neo-platonism. The great dream seems to be that of mind-reading God, and the result is either Pythagoras (it's numbers!), or Plato (it's pure ideas!), or this (it's essences!). See D.H.Lawrence's poem on geranium and mignottes. |
| 12706 | Bodies need a soul (or something like it) to avoid being mere phenomena [Leibniz] |
| Full Idea: Every substance is indivisible and consequently every corporeal substance must have a soul or at least an entelechy which is analogous to the soul, since otherwise bodies would be no more than phenomena. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], G II 121), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 2 | |
| A reaction: There is a large gap between having 'a soul' and having something 'analogous to a soul'. I take the analogy to be merely as originators of action. Leibniz wants to add appetite and sensation to the Aristotelian forms (but knows this is dubious!). |
| 12906 | Truths about species are eternal or necessary, but individual truths concern what exists [Leibniz] |
| Full Idea: The concept of a species contains only eternal or necessary truths, whereas the concept of an individual contains, regarded as possible, what in fact exists or what is related to the existence of things and to time. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.06) | |
| A reaction: This seems to be what is behind the preference some have for kind-essences rather than individual essences. But the individual must be explained, as well as the kind. Not all tigers are identical. The two are, of course, compatible. |
| 12904 | If varieties of myself can be conceived of as distinct from me, then they are not me [Leibniz] |
| Full Idea: I can as little conceive of different varieties of myself as of a circle whose diameters are not all of equal length. These variations would all be distinct one from another, and thus one of these varieties of myself would necessarily not be me. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.05.13) | |
| A reaction: This seems to be, at the very least, a rejection of any idea that I could have a 'counterpart'. It is unclear, though, where he would place a version of himself who learned a new language, or who might have had, but didn't have, a haircut. |
| 11981 | If someone's life went differently, then that would be another individual [Leibniz] |
| Full Idea: If the life of some person, or something went differently than it does, nothing would stop us from saying that it would be another person, or another possible universe which God had chosen. So truly it would be another individual. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.07.14) | |
| A reaction: Plantinga quotes this as an example of 'worldbound individuals'. This sort of remark leads to people saying that Leibniz believes all properties are essential, since they assume that his notion of essence is bound up with identity. But is it? |
| 12905 | I cannot think my non-existence, nor exist without being myself [Leibniz] |
| Full Idea: I am assured that as long as I think, I am myself. For I cannot think that I do not exist, nor exist so that I be not myself. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.05.13) | |
| A reaction: Elsewhere he qualifies the Cogito, but here he seems to straighforwardly endorse it. |
| 19334 | I can't just know myself to be a substance; I must distinguish myself from others, which is hard [Leibniz] |
| Full Idea: It is not enough for understanding the nature of myself, that I feel myself to be a thinking substance, one would have to form a distinct idea of what distinguishes me from all other possible minds; but of that I have only a confused experience. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.07.4/14) | |
| A reaction: Not a criticism I have encountered before. Does he mean that I might be two minds, or might be a multitude of minds? It seems to be Hume's problem, that you are aware of experiences, but not of the substance that unites them. |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
| A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
| 5033 | Nothing should be taken as certain without foundations [Leibniz] |
| Full Idea: Nothing should be taken as certain without foundations. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1687.04.30) | |
| A reaction: This might leave open the option, if you were a modern 'Fallibilist', that something might lack foundations, and so not be certain, and yet still qualify as 'knowledge'. That is my view. Knowledge resides somewhere between opinion and certainty. |
| 12913 | Nature is explained by mathematics and mechanism, but the laws rest on metaphysics [Leibniz] |
| Full Idea: One must always explain nature along mathematical and mechanical lines, provided one knows that the very principles or laws of mechanics or of force do not depend upon mathematical extension alone but upon certain metaphysical reasons. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.07.4/14 X) | |
| A reaction: I like this, and may even use it as the epigraph of my masterwork. Recently Stephen Hawking (physicist) has been denigrating philosophy, but I am with Leibniz on this one. |
| 13089 | To fully conceive the subject is to explain the resulting predicates and events [Leibniz] |
| Full Idea: Even in the most contingent truths, there is always something to be conceived in the subject which serves to explain why this predicate or event pertains to it, or why this has happened rather than not. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.06) | |
| A reaction: The last bit, about containing what has happened, seems absurd, but the rest of it makes sense. It is just the Aristotelian essentialist view, that a full understanding of the inner subject will both explain and predict the surface properties. |
| 5034 | Mind is a thinking substance which can know God and eternal truths [Leibniz] |
| Full Idea: Minds are substances which think, and are capable of knowing God and of discovering eternal truths. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1687.10.09) | |
| A reaction: 'God' is there because the ability to grasp the ontological argument is seen as basic. Note a firm commitment to substance-dualism, and a rationalist commitment to the spotting of necessary truths as basic. He is not totally wrong. |
| 5032 | It seems probable that animals have souls, but not consciousness [Leibniz] |
| Full Idea: It appears probable that the brutes have souls, though they are without consciousness. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.12.08) | |
| A reaction: This will be a response to Descartes, who allowed animals sensations, but not minds or souls. Personally I cannot make head or tail of Leibniz's claim. What makes it "apparent" to him? |
| 5031 | Everything which happens is not necessary, but is certain after God chooses this universe [Leibniz] |
| Full Idea: It is not the case that everything which happens is necessary; rather, everything which happens is certain after God made choice of this possible universe, whose notion contains this series of things. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.05) | |
| A reaction: I think this distinction is best captured as 'metaphysical necessity' (Leibniz's 'necessity'), and 'natural necessity' (his 'certainty'). 'Certainty' seems a bad word, as it is either certain de dicto or de re. Is God certain, or is the thing certain? |
| 12911 | Concepts are what unite a proposition [Leibniz] |
| Full Idea: There must always be some basis for the connexion between the terms of a proposition, and it is to be found in their concepts. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.07.4/14 X) | |
| A reaction: We face the problem that bothered Russell, of the unity of the proposition. We are also led to the question of HOW our concepts connect the parts of a proposition. Do concepts have valencies? Are they incomplete, as Frege suggests? |
| 9568 | I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes' [Chihara] |
| Full Idea: What I refer to as an 'equivalence class' (of line segments of a particular length) is an open sentence in my Constructibility Theory. I just use this terminology of the Platonist for didactic purposes. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 09.10) | |
| A reaction: This is because 'equivalence classes' is committed to the existence of classes, which is Quinean Platonism. I am with Chihara in wanting a story that avoids such things. Kit Fine is investigating similar notions of rules of construction. |
| 9547 | Mathematical entities are causally inert, so the causal theory of reference won't work for them [Chihara] |
| Full Idea: Causal theories of reference seem doomed to failure for the case of reference to mathematical entities, since such entities are evidently causally inert. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.3) | |
| A reaction: Presumably you could baptise a fictional entity such as 'Polonius', and initiate a social causal chain, with a tradition of reference. You could baptise a baby in absentia. |
| 12925 | Beauty increases with familiarity [Leibniz] |
| Full Idea: The more one is familiar with things, the more beautiful one finds them. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1688.01.4/14) | |
| A reaction: This is always the reply given to those who say that science kills our sense of beauty. The first step in aesthetic life is certainly to really really pay attention to things. |
| 12927 | Happiness is advancement towards perfection [Leibniz] |
| Full Idea: Happiness, or lasting contentment, consists of continual advancement towards a greater perfection. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1690.03.23) | |
| A reaction: To the modern mind this smacks of the sort of hubris to which only the religious mind can aspire, but it's still rather nice. The idea of grubby little mammals approaching perfection sounds wrong, but which other animal has even thought of perfection? |
| 15955 | I think the corpuscular theory, rather than forms or qualities, best explains particular phenomena [Leibniz] |
| Full Idea: I still subscribe fully to the corpuscular theory in the explanation of particular phenomena; in this sphere it is of no value to speak of forms or qualities. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 14.07.1686) | |
| A reaction: I am puzzled by Garber's summary in Idea 12728, and a bit unclear on Leibniz's views on atoms. More needed. |
| 12907 | Each possible world contains its own laws, reflected in the possible individuals of that world [Leibniz] |
| Full Idea: As there exist an infinite number of possible worlds, there exists also an infinite number of laws, some peculiar to one world, some to another, and each individual of any one world contains in the concept of him the laws of his world. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.06) | |
| A reaction: Since Leibniz's metaphysics is thoroughly God-driven, he will obviously allow God to create any laws He wishes, and hence scientific essentialism seems to be rejected, even though Leibniz is keen on essences. Unless the stuff is different... |
| 12924 | Motion alone is relative, but force is real, and establishes its subject [Leibniz] |
| Full Idea: Motion in itself separated from force is merely relative, and one cannot establish its subject. But force is something real and absolute. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1688.01.4/14) | |
| A reaction: The striking phrase here is that force enables us to 'establish its subject'. That is, force is at the heart of reality, and hence, through causal relations, individuates objects. That's how I read it. |
| 9574 | 'Gunk' is an individual possessing no parts that are atoms [Chihara] |
| Full Idea: An 'atomless gunk' is defined to be an individual possessing no parts that are atoms. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], App A) | |
| A reaction: [Lewis coined it] If you ask what are a-toms made of and what are ideas made of, the only answer we can offer is that the a-toms are made of gunk, and the ideas aren't made of anything, which is still bad news for the existence of ideas. |
| 12909 | Everything, even miracles, belongs to order [Leibniz] |
| Full Idea: Everything, even miracles, belongs to order. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.07.4/14 X) | |
| A reaction: This is very reminiscent of Plato, for whom there was no more deeply held belief than that the cosmos is essentially orderly. Coincidences are a nice problem, if they are events with no cause. |
| 5030 | Miracles are extraordinary operations by God, but are nevertheless part of his design [Leibniz] |
| Full Idea: Miracles, or the extraordinary operations of God, none the less belong within the general order; they are in conformity with the principal designs of God, and consequently are included in the notion of this universe, which is the result of those designs. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.05) | |
| A reaction: Some philosophers just make up things to suit themselves. What possible grounds can he have for claiming this? At best this is tautological, saying that, by definition, if anything at all happens, it must be part of God's design. Move on to Hume… |
| 12912 | Immortality without memory is useless [Leibniz] |
| Full Idea: Immortality without memory would be useless. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.07.4/14 X) | |
| A reaction: I would say that having a mind of any sort needs memory. The question for immortality is whether it extends back to human life. See 'Wuthering Heights' (c. p90) for someone who remembers Earth as so superior to paradise that they long to return there. |
| 12917 | The soul is indestructible and always self-aware [Leibniz] |
| Full Idea: Not only is the soul indestructible, but it always knows itself and remains self-conscious. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.11) | |
| A reaction: Personally I am not even self-aware during much of my sleeping hours, and I would say that I cease to be self-aware if I am totally absorbed in something on which I concentrate. |
| 12918 | Animals have souls, but lack consciousness [Leibniz] |
| Full Idea: It appears probable that animals have souls although they lack consciousness. | |
| From: Gottfried Leibniz (Letters to Antoine Arnauld [1686], 1686.11) | |
| A reaction: Personally I would say that they lack souls but have consciousness, but then I am in no better position to know the answer than Leibniz was. Arnauld asks what would happen to the souls of 100,000 silkworms if they caught fire! |