206 ideas
| 19359 | Leibniz aims to give coherent rational support for empiricism [Leibniz, by Perkins] |
| Full Idea: Leibniz's philosophy largely serves to justify and enable a coherent empirical account of the world. | |
| From: report of Gottfried Leibniz (works [1690]) by Franklin Perkins - Leibniz: Guide for the Perplexed 4.I | |
| A reaction: A nice counter to the simplistic idea of Locke as empiricist and Leibniz as rationalist. Leibniz is explicit that science needs a separate 'metaphysics' to underpin it. Perkins says Locke constructs experience, and Leibniz analyses it. |
| 13086 | Metaphysics is a science of the intelligible nature of being [Leibniz, by Cover/O'Leary-Hawthorne] |
| Full Idea: For Leibniz, metaphysics is above all a science of the intelligible nature of being. | |
| From: report of Gottfried Leibniz (works [1690]) by Cover,J/O'Leary-Hawthorne,J - Substance and Individuation in Leibniz 4.3.1 | |
| A reaction: [Their footnote gives two quotes in support] I could take this as my motto. We are not studying the 'nature of being', because we can't. We are studying what is 'intelligible' about it; my thesis is that the need for intelligibility imposes an order. |
| 16710 | Leibniz tried to combine mechanistic physics with scholastic metaphysics [Leibniz, by Pasnau] |
| Full Idea: Leibniz made a sustained attempt to combine a mechanistic physics with something like a scholastic metaphysics. | |
| From: report of Gottfried Leibniz (works [1690]) by Robert Pasnau - Metaphysical Themes 1274-1671 20.1 | |
| A reaction: This seems to me clear enough, and a lot of current philosophers seem to underestimate how Aristotelian Leibniz was. |
| 12997 | Analysis is the art of finding the middle term [Leibniz] |
| Full Idea: The art of finding intermediate terms (the 'middle term') is the art of 'analysis'. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 4.02) | |
| A reaction: This proposal is straight out of Aristotle's 'Posterior Analytics'. Nowadays we would say there was much more to analysis, the finding of necessary and sufficient conditions being the most obvious way of putting it. |
| 16897 | Reason is the faculty for grasping apriori necessary truths [Leibniz, by Burge] |
| Full Idea: Leibniz actually characterises reason as the faculty for apprehending priori, necessary truths. | |
| From: report of Gottfried Leibniz (works [1690]) by Tyler Burge - Frege on Apriority (with ps) 2 | |
| A reaction: No wonder it is called the Age of Reason when the claims are this grandiose. |
| 13009 | A reason is a known truth which leads to assent to some further truth [Leibniz] |
| Full Idea: A reason is a known truth whose connection with some less well-known truth leads us to give our assent to the latter. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 4.17) | |
| A reaction: This is plainly false, because you can have a reason for believing something, but still not give your assent to it, presumably because of counter-reasons. And a false belief could also be a reason, even to believe a truth. Tut tut. |
| 3346 | For Leibniz rationality is based on non-contradiction and the principle of sufficient reason [Leibniz, by Benardete,JA] |
| Full Idea: Leibniz distinguished two fundamental principles of rationality - the principle of non-contradiction and the principle of sufficient reason. | |
| From: report of Gottfried Leibniz (works [1690]) by José A. Benardete - Metaphysics: the logical approach Ch.18 |
| 12963 | Opposing reason is opposing truth, since reason is a chain of truths [Leibniz] |
| Full Idea: To speak against reason is to speak against truth, for reason is a chain of truths. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.21) | |
| A reaction: Truth has a talismanic quality here (which it didn't always have). This is a lovely slogan for defenders of the Enlightenment. It forces modern critics of the Enlightenment (Adorno etc) to launch an attack on truth, which is a doomed line. |
| 19360 | General principles, even if unconscious, are indispensable for thinking [Leibniz] |
| Full Idea: General principles enter into our thoughts, serving as their inner core and their mortar. Even if we give no thought to them, they are necessary for thought, as muscles and tendons are for walking. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 1.01.20) | |
| A reaction: Famously, Leibniz identified sufficient reason and non-contradiction as the two foundational principles. Modern logicians seem less keen on this idea, but then they have less interest in how we actually think. |
| 3347 | Leibniz said the principle of sufficient reason is synthetic a priori, since its denial is not illogical [Leibniz, by Benardete,JA] |
| Full Idea: Leibniz assigns synthetic a priori status to the principle of sufficient reason, readily conceding that one can deny it without fear of inconsistency. | |
| From: report of Gottfried Leibniz (works [1690]) by José A. Benardete - Metaphysics: the logical approach Ch.18 |
| 12983 | A nominal definition is of the qualities, but the real definition is of the essential inner structure [Leibniz] |
| Full Idea: The essence of gold is what constitutes it and gives it the sensible qualities which let us recognize it and which make its nominal definition; but if we could explain this structure or inner constitution we would possess the real, causal definition. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.03) | |
| A reaction: This is the view which I am championing, particularly in the role of explanation in the whole game. Explanation and understanding are the hallmarks of the discovery of a real essence. However, a falsehood may explain things well. Tricky. |
| 12982 | One essence can be expressed by several definitions [Leibniz] |
| Full Idea: Although a thing has only one essence, this can be expressed by several definitions. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.03) | |
| A reaction: See Idea 12976 and Idea 12977 for a view which seems to conflict with this. He seemed to imply that once you identify the essence, the definitions converge, with multiple definitions being symptomatic of imperfect ideas of things. |
| 12976 | If our ideas of a thing are imperfect, the thing can have several unconnected definitions [Leibniz] |
| Full Idea: The result of having an imperfect idea of something is that the same subject admits of several mutually independent definitions: we shall sometimes be unable to derive one from another, or see in advance that they must belong to a single subject. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.31) | |
| A reaction: See Idea 12975 for imperfect ideas of things. Obviously the idea is that perfect knowledge will converge on a single definition, which will pinpoint the essence of a thing, and then all explanations will flow. A nice addition to the Aristotelian view. |
| 12984 | Real definitions, unlike nominal definitions, display possibilities [Leibniz] |
| Full Idea: The real definition displays the possibility of the definiendum, and the nominal definition does not. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.03) | |
| A reaction: This, I take it, is because the real definition indicates the actual powers of the thing, and not just the superficial characteristics. Is knowledge of powers identical with knowledge of possibilities? |
| 12980 | Genus and differentia might be swapped, and 'rational animal' become 'animable rational' [Leibniz] |
| Full Idea: The genus can very often be turned into the differentia, ...so that in place of saying that man is a 'reasonable animal' we could, if language permitted, say that man is an 'animable rational', a rational substance with animal nature. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.03) | |
| A reaction: This is a very telling point which rather undermines any dogmatic approach to what Aristotle says about these sorts of definitions. I don't find this account of definitions very helpful anyway. Leibniz links it to the order of cataloguing. |
| 8627 | Leibniz is inclined to regard all truths as provable [Leibniz, by Frege] |
| Full Idea: Leibniz has an inclination to regard all truths as provable. | |
| From: report of Gottfried Leibniz (works [1690]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §15 | |
| A reaction: Leibniz sounds like the epitome of Enlightenment optimism about the powers of reason. Could God prove every truth? It's a nice thought. |
| 13000 | Truth is correspondence between mental propositions and what they are about [Leibniz] |
| Full Idea: Let us be content with looking for truth in the correspondence between the propositions which are in the mind and the things which they are about. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 4.05) | |
| A reaction: In an age when the nature of truth was hardly debated, and theories such as coherence and pragmatism, never mind semantic accounts, were unthought of, it is interesting to see that correspondence seems obvious to Leibniz. Correct! |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| Full Idea: The notion of a function evolved gradually from wanting to see what curves can be represented as trigonometric series. The study of arbitrary functions led Cantor to the ordinal numbers, which led to set theory. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| Full Idea: Cantor's Theorem says that for any set x, its power set P(x) has more members than x. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| Full Idea: Cantor's diagonalisation argument generalises to show that any set has more subsets than it has members. | |
| From: report of George Cantor (works [1880]) by David Bostock - Philosophy of Mathematics 4.5 | |
| A reaction: Thus three members will generate seven subsets. This means that 'there is no end to the series of cardinal numbers' (Bostock p.106). |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| Full Idea: Cantor taught that a set is 'a many, which can be thought of as one'. ...After a time the unfortunate beginner student is told that some classes - the singletons - have only a single member. Here is a just cause for student protest, if ever there was one. | |
| From: report of George Cantor (works [1880]) by David Lewis - Parts of Classes 2.1 | |
| A reaction: There is a parallel question, almost lost in the mists of time, of whether 'one' is a number. 'Zero' is obviously dubious, but if numbers are for counting, that needs units, so the unit is the precondition of counting, not part of it. |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| Full Idea: Cantor's theories exhibited the contradictions others had claimed to derive from the supposition of infinite sets as confusions resulting from the failure to mark the necessary distinctions with sufficient clarity. | |
| From: report of George Cantor (works [1880]) by Michael Potter - Set Theory and Its Philosophy Intro 1 |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| Full Idea: Cantor discovered that the continuum is the powerset of the integers. While adding or multiplying infinities didn't move up a level of complexity, multiplying a number by itself an infinite number of times did. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| Full Idea: Cantor first stated the Union Axiom in a letter to Dedekind in 1899. It is nearly too obvious to deserve comment from most commentators. Justifications usually rest on 'limitation of size' or on the 'iterative conception'. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Surely someone can think of some way to challenge it! An opportunity to become notorious, and get invited to conferences. |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| Full Idea: Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack. | |
| From: report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro | |
| A reaction: As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets. |
| 12992 | Logic teaches us how to order and connect our thoughts [Leibniz] |
| Full Idea: Logic teaches us how to order and connect our thoughts. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.10) | |
| A reaction: Leibniz had a higher opinion of logic than contemporaries like Locke. The question is whether logic can actually teach us better order than we could otherwise manage, or whether it just describes what most thinkers do. |
| 10056 | At bottom eternal truths are all conditional [Leibniz] |
| Full Idea: At bottom eternal truths are all conditional, saying 'granted such a thing, such another thing is'. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 4.11.14), quoted by Alan Musgrave - Logicism Revisited §4 | |
| A reaction: Thus showing Leibniz to have sympathy with the if-thenist view. He cites geometry as his illustration. |
| 12974 | People who can't apply names usually don't understand the thing to which it applies [Leibniz] |
| Full Idea: Someone who goes wrong in relating an idea to a name will usually go wrong about the thing he wants the name to stand for. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.29) | |
| A reaction: This seems to give tentative support to a Millian account of names, whose only content is just the thing which is named. Leibniz's observation certainly seems to be right. |
| 13002 | It is always good to reduce the number of axioms [Leibniz] |
| Full Idea: To reduce the number of axioms is always something gained. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 4.06) | |
| A reaction: This is rather revealing about the nature of axioms. They don't have any huge metaphysical status - in fact one might say that their status is epistemological, or even pedagogic. They enable us to get out minds round things. |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'. | |
| From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3 | |
| A reaction: [Smith summarises the diagonal argument] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does. |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly? |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| Full Idea: Cantor believed he had discovered that between the finite and the 'Absolute', which is 'incomprehensible to the human understanding', there is a third category, which he called 'the transfinite'. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 13008 | Geometry, unlike sensation, lets us glimpse eternal truths and their necessity [Leibniz] |
| Full Idea: What I value most in geometry, considered as a contemplative study, is its letting us glimpse the true source of eternal truths and of the way in which we can come to grasp their necessity, which is something confused sensory images cannot reveal. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 4.12) | |
| A reaction: This is strikingly straight out of Plato. We should not underestimate this idea, though nowadays it is with us, but with geometry replaced by mathematical logic. |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| Full Idea: In 1878 Cantor published the unexpected result that one can put the points on a plane, or indeed any n-dimensional space, into one-to-one correspondence with the points on a line. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| Full Idea: Cantor took the ordinal numbers to be primary: in his generalization of the cardinals and ordinals into the transfinite, it is the ordinals that he calls 'numbers'. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind VI | |
| A reaction: [Tait says Dedekind also favours the ordinals] It is unclear how the matter might be settled. Humans cannot give the cardinality of large groups without counting up through the ordinals. A cardinal gets its meaning from its place in the ordinals? |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| Full Idea: Cantor taught us to regard the totality of natural numbers, which was formerly thought to be infinite, as really finite after all. | |
| From: report of George Cantor (works [1880]) by John Mayberry - What Required for Foundation for Maths? p.414-2 | |
| A reaction: I presume this is because they are (by definition) countable. |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| Full Idea: Cantor introduced the distinction between cardinal and ordinal numbers. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind Intro | |
| A reaction: This seems remarkably late for what looks like a very significant clarification. The two concepts coincide in finite cases, but come apart in infinite cases (Tait p.58). |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| Full Idea: Cantor's work revealed that the notion of an ordinal number is more fundamental than that of a cardinal number. | |
| From: report of George Cantor (works [1880]) by Michael Dummett - Frege philosophy of mathematics Ch.23 | |
| A reaction: Dummett makes it sound like a proof, which I find hard to believe. Is the notion that I have 'more' sheep than you logically prior to how many sheep we have? If I have one more, that implies the next number, whatever that number may be. Hm. |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| Full Idea: The cardinal number of M is the general idea which, by means of our active faculty of thought, is deduced from the collection M, by abstracting from the nature of its diverse elements and from the order in which they are given. | |
| From: George Cantor (works [1880]), quoted by Bertrand Russell - The Principles of Mathematics §284 | |
| A reaction: [Russell cites 'Math. Annalen, XLVI, §1'] See Fine 1998 on this. |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| Full Idea: Cantor said he could show that every infinite set of points on the line could be placed into one-to-one correspondence with either the natural numbers or the real numbers - with no intermediate possibilies (the Continuum hypothesis). His proof failed. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| Full Idea: Cantor's diagonal argument showed that all the infinite decimals between 0 and 1 cannot be written down even in a single never-ending list. | |
| From: report of George Cantor (works [1880]) by Stephen Read - Thinking About Logic Ch.6 |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| Full Idea: Cantor's theory of Cauchy sequences defines a real number to be associated with an infinite set of infinite sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6 | |
| A reaction: This sounds remarkably like the endless decimals we use when we try to write down an actual real number. |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| Full Idea: Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2 |
| 9147 | Number cannot be defined as addition of ones, since that needs the number; it is a single act of abstraction [Fine,K on Leibniz] |
| Full Idea: Leibniz's talk of the addition of ones cannot define number, since it cannot be specified how often they are added without using the number itself. Number must be an organic unity of ones, achieved by a single act of abstraction. | |
| From: comment on Gottfried Leibniz (works [1690]) by Kit Fine - Cantorian Abstraction: Recon. and Defence §1 | |
| A reaction: I doubt whether 'abstraction' is the right word for this part of the process. It seems more like a 'gestalt'. The first point is clearly right, that it is the wrong way round if you try to define number by means of addition. |
| 12956 | Only whole numbers are multitudes of units [Leibniz] |
| Full Idea: The definition of 'number' as a multitude of units is appropriate only for whole numbers. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.15) | |
| A reaction: One can also define rational numbers by making use of units, but the strategy breaks down with irrational numbers like root-2 and pi. I still say the concept of a unit is the basis of numbers. Without whole numbers, we wouldn't call the real 'numbers'. |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| Full Idea: From the very nature of an irrational number, it seems necessary to understand the mathematical infinite thoroughly before an adequate theory of irrationals is possible. Infinite classes are obvious in the Dedekind Cut, but have logical difficulties | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II Intro | |
| A reaction: Almost the whole theory of analysis (calculus) rested on the irrationals, so a theory of the infinite was suddenly (in the 1870s) vital for mathematics. Cantor wasn't just being eccentric or mystical. |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| Full Idea: Cantor's 1891 diagonal argument revealed there are infinitely many infinite powers. Indeed, it showed more: it shows that given any set there is another of greater power. Hence there is an infinite power strictly greater than that of the set of the reals. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.2 |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| Full Idea: What we might call 'Cantor's Thesis' is that there won't be a potential infinity of any sort unless there is an actual infinity of some sort. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: This idea is nicely calculated to stop Aristotle in his tracks. |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| Full Idea: Cantor showed that the complete totality of natural numbers cannot be mapped 1-1 onto the complete totality of the real numbers - so there are different sizes of infinity. | |
| From: report of George Cantor (works [1880]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.4 |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| Full Idea: Cantor's 'continuum hypothesis' is the assertion that there are no infinite cardinalities strictly between the size of the natural numbers and the size of the real numbers. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Thinking About Mathematics 2.4 | |
| A reaction: The tricky question is whether this hypothesis can be proved. |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| Full Idea: Cantor's Continuum Hypothesis (CH) says that for every infinite set X of reals there is either a one-to-one correspondence between X and the natural numbers, or between X and the real numbers. | |
| From: report of George Cantor (works [1880]) by Peter Koellner - On the Question of Absolute Undecidability 1.2 | |
| A reaction: Every single writer I read defines this differently, which drives me crazy, but is also helpfully illuminating. There is a moral there somewhere. |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| Full Idea: Cantor conjectured that there is no size between those of the naturals and the reals - called the 'continuum hypothesis'. The generalized version says that for no infinite set A is there a set larger than A but smaller than P(A). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: Thus there are gaps between infinite numbers, and the power set is the next size up from any infinity. Much discussion as ensued about whether these two can be proved. |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| Full Idea: Cantor's Continuum Hypothesis states that there are no sets which are too large for there to be a one-to-one correspondence between the set and the natural numbers, but too small for there to exist a one-to-one correspondence with the real numbers. | |
| From: report of George Cantor (works [1880]) by Leon Horsten - Philosophy of Mathematics §5.1 |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| Full Idea: Cantor's conjecture (the Continuum Hypothesis) is that there are no sets between N and P(N). The 'generalized' version replaces N with an arbitrary infinite set. | |
| From: report of George Cantor (works [1880]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2 | |
| A reaction: The initial impression is that there is a single gap in the numbers, like a hole in ozone layer, but the generalised version implies an infinity of gaps. How can there be gaps in the numbers? Weird. |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| Full Idea: Cantor's Continuum Hypothesis was that there is no cardinal number greater than aleph-null but less than the cardinality of the continuum. | |
| From: report of George Cantor (works [1880]) by Charles Chihara - A Structural Account of Mathematics 05.1 | |
| A reaction: I have no view on this (have you?), but the proposal that there are gaps in the number sequences has to excite all philosophers. |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| Full Idea: Cantor's second innovation was to extend the sequence of ordinal numbers into the transfinite, forming a handy scale for measuring infinite cardinalities. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: Struggling with this. The ordinals seem to locate the cardinals, but in what sense do they 'measure' them? |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| Full Idea: Cantor's set theory was not of collections in some familiar sense, but of collections that can be counted using the indexes - the finite and transfinite ordinal numbers. ..He treated infinite collections as if they were finite. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| Full Idea: Cantor's first innovation was to treat cardinality as strictly a matter of one-to-one correspondence, so that the question of whether two infinite sets are or aren't of the same size suddenly makes sense. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: It makes sense, except that all sets which are infinite but countable can be put into one-to-one correspondence with one another. What's that all about, then? |
| 19375 | The continuum is not divided like sand, but folded like paper [Leibniz, by Arthur,R] |
| Full Idea: Leibniz said the division of the continuum should not be conceived 'to be like the division of sand into grains, but like that of a tunic or a sheet of paper into folds'. | |
| From: report of Gottfried Leibniz (works [1690], A VI iii 555) by Richard T.W. Arthur - Leibniz | |
| A reaction: This from the man who invented calculus. This thought might apply well to the modern physicist's concept of a 'field'. |
| 18080 | A tangent is a line connecting two points on a curve that are infinitely close together [Leibniz] |
| Full Idea: We have only to keep in mind that to find a tangent means to draw a line that connects two points of a curve at an infinitely small distance. | |
| From: Gottfried Leibniz (works [1690]), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.1 | |
| A reaction: [The quote can be tracked through Kitcher's footnote] |
| 18081 | Nature uses the infinite everywhere [Leibniz] |
| Full Idea: Nature uses the infinite in everything it does. | |
| From: Gottfried Leibniz (works [1690]), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.1 | |
| A reaction: [The quote can be tracked through Kitcher's footnote] He seems to have had in mind the infinitely small. |
| 12937 | We shouldn't just accept Euclid's axioms, but try to demonstrate them [Leibniz] |
| Full Idea: Far from approving the acceptance of doubtful principles, I want to see an attempt to demonstrate even Euclid's axioms, as some of the ancients tried to do. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 1.02) | |
| A reaction: This is the old idea of axioms, as a bunch of basic self-evident truths, rather than the modern idea of an economical set of propositions from which to make deductions. Demonstration has to stop somewhere. |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| Full Idea: Cantor's theorem entails that there are more property extensions than objects. So there are not enough objects in any domain to serve as extensions for that domain. So Frege's view that numbers are objects led to the Caesar problem. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Philosophy of Mathematics 4.6 | |
| A reaction: So the possibility that Caesar might have to be a number arises because otherwise we are threatening to run out of numbers? Is that really the problem? |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| Full Idea: Pure mathematics ...according to my conception is nothing other than pure set theory. | |
| From: George Cantor (works [1880], I.1), quoted by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: [an unpublished paper of 1884] So right at the beginning of set theory this claim was being made, before it was axiomatised, and so on. Zermelo endorsed the view, and it flourished unchallenged until Benacerraf (1965). |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| Full Idea: Cantor calls mathematics an empirical science in so far as it begins with consideration of things in the external world; on his view, number originates only by abstraction from objects. | |
| From: report of George Cantor (works [1880]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §21 | |
| A reaction: Frege utterly opposed this view, and he seems to have won the day, but I am rather thrilled to find the great Cantor endorsing my own intuitions on the subject. The difficulty is to explain 'abstraction'. |
| 12932 | The idea of being must come from our own existence [Leibniz] |
| Full Idea: I would like to know how we could have the idea of being if we did not, as beings ourselves, find being within us. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 1.01) | |
| A reaction: I could envisage a creature with an entirely 'externa' mind, that just focused on environment, and took its own place in it unthinkingly for granted. I suppose he's right, though. |
| 7565 | Leibniz proposes monads, since there must be basic things, which are immaterial in order to have unity [Leibniz, by Jolley] |
| Full Idea: Leibniz believes in monads because it would be contrary to reason or divine wisdom if everything was compounds, down to infinity; there must be ultimate unified building-blocks; they cannot be material, for material things lack genuine unity. | |
| From: report of Gottfried Leibniz (works [1690]) by Nicholas Jolley - Leibniz Ch.3 | |
| A reaction: It is hard to discern the basis for the claim that only immaterial things can have unity. The Greeks proposed atoms, and we have no reason to think that electrons lack unity. |
| 12966 | Objects of ideas can be divided into abstract and concrete, and then further subdivided [Leibniz] |
| Full Idea: Divide terms, objects of ideas, into abstract and concrete, then the abstract into absolute and relational, the absolute into attributes and modifications, and those two into simple and composite; the concrete are substances and their substantial things. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.22) | |
| A reaction: Make your tree from this: ABSTRACT{absolute[attributes(simple)(composite)][modifications(simple)(composite)],relations}CONCRETE{[substance][substantial form]} |
| 12993 | Have five categories - substance, quantity, quality, action/passion, relation - and their combinations [Leibniz] |
| Full Idea: The scheme of categories is very useful, and it might be that all that is needed are five general headings for beings - namely substance, quantity, quality, action or passion, and relation - with any formed by composition from those. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.10) | |
| A reaction: 'Action or passion' as a single category sounds intriguing. He is very keen on active force in the world of physical objects, which presumably falls into this category. His plan sounds, initially, as good as any I have heard. |
| 12989 | Our true divisions of nature match reality, but are probably incomplete [Leibniz] |
| Full Idea: It can be said that whatever we truthfully distinguish or compare is also distinguished or made alike by nature, although nature has distinctions and comparisons which are unknown to us and which may be better than ours. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.06) | |
| A reaction: This seems to me to be correct, though it is more like the credo of the sensible realist than it is like any sort of argument. |
| 10419 | If relations can be reduced to, or supervene on, monadic properties of relata, they are not real [Leibniz, by Swoyer] |
| Full Idea: Leibniz argued that relations could be reduced to monadic properties and so were dispensable, and some still agree, saying relations supervene on monadic properties of the relata, and are not actually real. | |
| From: report of Gottfried Leibniz (works [1690]) by Chris Swoyer - Properties 7.4 | |
| A reaction: At the very least a background of space and/or time seem required, in addition to any properties the relata may have. y only becomes 'to the left of x' when x appears to its right, so the relation doesn't seem to be intrinsic to y. |
| 13078 | Relations aren't in any monad, so they are distributed, so they are not real [Leibniz] |
| Full Idea: The relations which connect two monads are not in either the one or the other, but equally in both at once; and therefore properly speaking, in neither. I do not think you would wish to posit an accident which would inhere simultaneously in two subjects. | |
| From: Gottfried Leibniz (works [1690], G II:517), quoted by Cover,J/O'Leary-Hawthorne,J - Substance and Individuation in Leibniz 2.4.3 | |
| A reaction: Where Russell affirms relations as universals, and scholastics make them properties of individuals, Leibniz denies their reality entirely. It seems obvious that once the objects and properties are there, the relations come for free. |
| 12959 | We discern active power from our minds, so mind must be involved in all active powers [Leibniz] |
| Full Idea: The clearest idea of active power comes to us from the mind. So active power occurs only in things which are analogous to minds, that is, in entelechies; for strictly matter exhibits only passive power. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.21) | |
| A reaction: If this is meant to be a precise argument, then 'so' and 'only' are blatantly unjustified. I guess that if it isn't analogous to a mind then he won't allow it to be a TRUE active power! I say mind arises from the entelechies of the physical brain. |
| 12967 | I use the word 'entelechy' for a power, to include endeavour, as well as mere aptitude [Leibniz] |
| Full Idea: If 'power' is the source of action, it means more than aptitude or ability. It also includes endeavour. It is in order to express this sense that I appropriate the term 'entelechy' to stand for power. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.22) | |
| A reaction: An 'entelechy' is, roughly, an instantiated thing, but I like what Leibniz is fishing for here - that we will never understand the world if we think of it as passive. |
| 12965 | All occurrence in the depth of a substance is spontaneous 'action' [Leibniz] |
| Full Idea: Anything which occurs in what is strictly a substance must be a case of 'action' in the metaphysically rigorous sense of something which occurs in the substance spontaneously, arising out of its own depths. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.21) | |
| A reaction: I love this idea, which fits in with scientific essentialism. The question is whether Leibniz has idenified the end point of all explanations. Cutting edge physics is trying to give further explanations for what seemed basic, such as mass and gravity. |
| 12999 | Substances are primary powers; their ways of being are the derivative powers [Leibniz] |
| Full Idea: Primary powers are what make up the substances themselves; derivative powers, or 'faculties' if you like, are merely 'ways of being' - and they must be derived from substances. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 4.03) | |
| A reaction: We might talk of 'deep' and 'surface' properties, or maybe 'powers' and 'qualities' is better. 'Primary' and 'derivative' only gives the logical relationship, but not the causal relationship. |
| 12713 | Forms have sensation and appetite, the latter being the ability to act on other bodies [Leibniz, by Garber] |
| Full Idea: Leibniz's form contains both sensation and appetite, and he seems to associate appetite with the ability a body has to act on another. | |
| From: report of Gottfried Leibniz (works [1690]) by Daniel Garber - Leibniz:Body,Substance,Monad 3 | |
| A reaction: It strikes me (you may be surprised to hear) that this concept is not unlike Nietzsche's all-mastering 'will to power'. I offer Idea 7140 in evidence. |
| 13087 | The essence of a thing is its real possibilities [Leibniz, by Cover/O'Leary-Hawthorne] |
| Full Idea: In Leibniz's view, the essence of a thing is fundamentally the real possibilities of that thing. | |
| From: report of Gottfried Leibniz (works [1690]) by Cover,J/O'Leary-Hawthorne,J - Substance and Individuation in Leibniz 4.3.3 | |
| A reaction: Note that the essences are individual. On the whole I would prefer Leibniz in his own words, but this is too good to lose (..but see Idea 12981). It is the aspect of Leibniz that fits perfectly with modern scientific essentialism. |
| 5056 | Material or immaterial substances cannot be conceived without their essential activity [Leibniz] |
| Full Idea: I maintain that substances, whether material or immaterial, cannot be conceived in their bare essence without any activity, activity being of the essence of substance in general. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], Pref) | |
| A reaction: Thus there could be no 'tabula rasa', because that would be an inactive mental substance. This strikes me as a nice question for modern physicists. Do they regard movement as essential, or an addition to bare particles? I'm with Leibniz. Essentialism. |
| 12969 | The active powers which are not essential to the substance are the 'real qualities' [Leibniz] |
| Full Idea: Powers which are not essential to substance, and which include not merely an aptitude but also a certain endeavour, are exactly what are or should be meant by 'real qualities'. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.23) | |
| A reaction: An important part of Leibniz's account. There are thus essential powers, in the 'depth' of the substance, and more peripheral powers, which also initiate action, and give rise to the qualities. The second must derive from the first? |
| 12941 | There cannot be power without action; the power is a disposition to act [Leibniz] |
| Full Idea: Where will one ever find in the world a faculty consisting in sheer power without performing an act? There is always a particular disposition to action, and towards one action rather than another. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.01) | |
| A reaction: This is muddled. Leibniz defends powers in the possibilities of things, but he must then accept that some possibilities may never be realised, as with two complex chemicals which never ever come into contact with one another. |
| 12990 | Real (non-logical) abstract terms are either essences or accidents [Leibniz] |
| Full Idea: Real (as opposed to logical) abstract terms, or at least those which are conceived as real, are either essences or parts of essences, or else accidents (i.e. beings added to a substance). | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.08) | |
| A reaction: Interesting to refer to accidents as 'beings'. This seems to fit abstraction by ignoring, since you can either ignore the accidents to get the essence, or ignore the essence to get the accidents. |
| 12939 | Wholly uniform things like space and numbers are mere abstractions [Leibniz] |
| Full Idea: Things which are uniform, containing no variety, are always mere abstractions: for instance, time, space, and the other entities of pure mathematics. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.01) | |
| A reaction: I presume that being 'mere abstractions' denies them ontological status, and makes them creations of thought. If so, I like this idea a lot. |
| 12701 | Leibniz moved from individuation by whole entity to individuation by substantial form [Leibniz, by Garber] |
| Full Idea: By 1680 Leibniz had clearly abandoned the 'whole entity' conception of individuation, for a conception grounded in substantial form alone. | |
| From: report of Gottfried Leibniz (works [1690]) by Daniel Garber - Leibniz:Body,Substance,Monad 2 | |
| A reaction: In other words, Leibniz became more of an Aristotelian, and more of an essentialist. |
| 12979 | The only way we can determine individuals is by keeping hold of them [Leibniz] |
| Full Idea: It is impossible for us to know individuals or to find any way of precisely determining the individuality of any thing except by keeping hold of the thing itself. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.03) | |
| A reaction: 'The tallest woman in London' seems to determine someone perfectly well, though only by cross-referencing universal concepts like 'tall'. |
| 12971 | If two individuals could be indistinguishable, there could be no principle of individuation [Leibniz] |
| Full Idea: If two individuals were perfectly similar and equal and, in short, indistinguishable in themselves, there would be no principle of individuation. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.27) | |
| A reaction: This seems to be the main motive for Leibniz's unusual claim that there cannot be two indiscernible individuals, but it looks suspiciously like an a priori claim made about what should be an a posteriori discovery. Are electrons distinguishable? |
| 13098 | We use things to distinguish places and times, not vice versa [Leibniz] |
| Full Idea: It is by means of things that we must distinguish one time and place from another, rather than vice versa. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.27) | |
| A reaction: This rests on Leibniz's relative view of space (as opposed to Newton's absolute view). If you need to re-identify a thing to individuate it, re-identifying the exact place or time seems impossible, but it is usually manageable with thing. |
| 13105 | The laws-of-the-series plays a haecceitist role [Leibniz, by Cover/O'Leary-Hawthorne] |
| Full Idea: Leibniz takes the laws-of-the-series to play a haecceitistic role. | |
| From: report of Gottfried Leibniz (works [1690]) by Cover,J/O'Leary-Hawthorne,J - Substance and Individuation in Leibniz 7.5 | |
| A reaction: Idea 13092 for law-in-the-series. He thinks that a law-in-a-series is unique to a substance, and so can individuate it. That is a pretty good proposal, if anything is going to do the job. Perhaps I do believe in haecceities, as unique bundles of powers? |
| 13075 | No two things are quite the same, so there must be an internal principle of distinction [Leibniz] |
| Full Idea: In addition to the difference of time or of place there must always be an internal principle of distinction: although there can be many things of the same kind, it is still the case that none of them are ever exactly alike. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.27) | |
| A reaction: This rests on Leibniz's unusual view that all things (even electrons) are qualitatively distinct. Personally I disagree with that, but agree with the idea. Things have time and place because they have identity, not the other way around. |
| 12953 | Fluidity is basic, and we divide into bodies according to our needs [Leibniz] |
| Full Idea: Fluidity is the fundamental condition, and the division into bodies is carried out - there being no obstacle to it - according to our need. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.13) | |
| A reaction: Leibniz is referring to what he usually calls 'aggregates', like piles of bricks, which are things lacking a unifying substance. There may be no true substances, in which fluidity is the order of the day. |
| 16513 | Identity of a substance is the law of its persistence [Leibniz] |
| Full Idea: For there to be a certain persisting law which involves the future states of that which we conceive as one and the same continuant, this is what I say constitute's a substance's identity. | |
| From: Gottfried Leibniz (works [1690], G II:264), quoted by David Wiggins - Sameness and Substance 3.1 | |
| A reaction: This is a key remark for those who thing 'persistence conditions' are basic to metaphysics. I'm not so sure. |
| 12035 | Leibniz bases pure primitive entities on conjunctions of qualitative properties [Leibniz, by Adams,RM] |
| Full Idea: Leibniz is committed with apparent consistency to both a purely qualitative character of all thisnesses, and to primitiveness of individual identity. He regards thisnesses as conjunctions of simpler, logically independent suchnesses. | |
| From: report of Gottfried Leibniz (works [1690]) by Robert Merrihew Adams - Primitive Thisness and Primitive Identity 5 | |
| A reaction: Hence Leibniz is held to say that all of the qualitative properties are 'essential' to the object, since all of them are needed to constitute its identity. Hence absolutely nothing about an object, even an electron, could be different, which is daft. |
| 12943 | Individuality is in the bond substance gives between past and future [Leibniz] |
| Full Idea: Within each substance there is a perfect bond between the future and the past, which is what creates the identity of the individual. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.01) | |
| A reaction: I'm not quite sure if this means anything, but the idea that a bond across time is a necessity for intrinsic identity is interesting. The 'bond' would, I take it, have to be a causal one. |
| 13091 | Leibnizian substances add concept, law, force, form and soul [Leibniz, by Cover/O'Leary-Hawthorne] |
| Full Idea: To the traditional idea of substance (independent, subjects of predication, active, persistent) Leibniz adds, distinctively, complete individual concept, law-of-the-series, active force, form and soul or entelechy. | |
| From: report of Gottfried Leibniz (works [1690]) by Cover,J/O'Leary-Hawthorne,J - Substance and Individuation in Leibniz 6.1.1 | |
| A reaction: 'Form' seems to be Aristotelian, and 'soul' seems ridiculous. I don't think the 'complete concept' is much help. However, the 'law-in-the-series' is very interesting (Idea 13079), if employed sensibly, and 'active force' is spot-on. Powers define reality. |
| 11855 | Substances cannot be bare, but have activity as their essence [Leibniz] |
| Full Idea: I maintain that substances (material or immaterial) cannot be conceived in their bare essence devoid of activity; that activity is of the essence of substance in general | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], Pref 65) | |
| A reaction: Leibniz liked the idea that God was the source of this activity, but this remark makes Leibniz a direct ancestor of modern scientific essentialism. |
| 7561 | Substances are essentially active [Leibniz, by Jolley] |
| Full Idea: For Leibniz, it is the very essence of substances to be sources of activity. | |
| From: report of Gottfried Leibniz (works [1690]) by Nicholas Jolley - Leibniz Ch.2 | |
| A reaction: This makes the views of Leibniz sympathetic to modern essentialism (of which I am a fan), because it places active power at the centre of what it is to exist, rather than action being imposed on matter which is otherwise passive. |
| 12970 | We can imagine two bodies interpenetrating, as two rays of light seem to [Leibniz] |
| Full Idea: We find that two shadows or two rays of light interpenetrate, and we could devise an imaginary world where bodies did the same. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.27) | |
| A reaction: I suspect this is a case of being able to imagine something when you don't fully understand it (like a bonfire on the Moon), but when you fully understand the modern physics of it, you see the necessity of separation between objects. |
| 12986 | The essence of baldness is vague and imperfect [Leibniz] |
| Full Idea: There are vague and imperfect essences, as in the question of how few hairs a man can have without being bald. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.05) | |
| A reaction: This example is much discussed in contemporary debate, but I now learn that it has a venerable history. The surprise here is the word 'essences', because I had taken Leibnizian essences to be 'perfect ideas', and hence precise. |
| 12715 | Leibniz strengthened hylomorphism by connecting it to force in physics [Leibniz, by Garber] |
| Full Idea: A standard criticism of the scholastic notions of matter and form is that they are obscure and unintelligible. But in Leibniz's system they are connected directly with notions of active and passive force that play an intelligible roles in his physics. | |
| From: report of Gottfried Leibniz (works [1690]) by Daniel Garber - Leibniz:Body,Substance,Monad 4 | |
| A reaction: This seems to me to be very appealing. Aristotle was clearly on the right lines, but just ran out of things to say, once he had pointed in the right direction. Maybe 'fields' and 'strings' can fill out the Aristotelian conception of form. |
| 12968 | A 'substratum' is just a metaphor for whatever supports several predicates [Leibniz] |
| Full Idea: From the beginning we conceive several predicates in a single subject, and that is all there is to these metaphorical words 'support' and 'substratum'. So I do not see why it is made out to involve a problem. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.23) | |
| A reaction: The question is whether the substratum is 'bare' if you remove all the predicates, and clearly Leibniz believes you are left with true essential substance (although the removal process is presumably only possible in thought, thanks to God). |
| 12931 | Particular truths are just instances of general truths [Leibniz] |
| Full Idea: The reasons for particular truths rest wholly on the more general ones of which they are mere instances. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 1.01) | |
| A reaction: Clearly particulars have their own distinctive truth, but the Leibniz case seems to be that a particular is a unique intersection for an array of general truths - and nothing else. Audrey Hepburn's smile has no generalities to it. |
| 12811 | We can't know individuals, or determine their exact individuality [Leibniz] |
| Full Idea: It is impossible for us to have knowledge of individuals and to find the means of determining exactly the individuality of everything. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.3) | |
| A reaction: Aristotle was clearly also tempted by this doubt (since universals are involved), though individuals are what he wanted to understand. I think they are wrong. Leibniz gives the bizarre reason that we can't know individuals as they each contain infinity. |
| 12981 | Essence is just the possibility of a thing [Leibniz] |
| Full Idea: Essence is fundamentally nothing but the possibility of the thing under consideration. Something which is thought possible is expressed by a definition. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.03) | |
| A reaction: It is unclear whether he means 'possible modes of existence' or 'possible actions of the thing'. Leibniz sees more clearly than Aristotle that essences extend beyond the actual thing, because Leibniz is more aware of the active powers. |
| 5057 | If you fully understand a subject and its qualities, you see how the second derive from the first [Leibniz] |
| Full Idea: Every time we find some quality in a subject, we ought to think that, if we understood the nature of this subject and of this quality, we should conceive how this quality could result from it. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], Pref) | |
| A reaction: Thus (in Kripke's analogy) God cannot make 'subjects' on Thursday and then add 'qualities' on Friday. Add the point that all subjects are physical, and I say you have the whole story. The physical entails the mental. The laws result from the qualities. |
| 11878 | Leibniz's view (that all properties are essential) is extreme essentialism, not its denial [Leibniz, by Mackie,P] |
| Full Idea: The view standardly attributed to Leibniz, that makes all an individual's properties essential to it should be regarded as an extreme version of essentialism, not a denial of essentialism. | |
| From: report of Gottfried Leibniz (works [1690]) by Penelope Mackie - How Things Might Have Been 1.1 | |
| A reaction: Wiggins disagrees, saying that Leibniz was not an essentialist, which is an interesting topic of research for those who are interested. I would take Leibniz to be not an essentialist, on that basis, as essentialism makes a distinction. See Quine on that. |
| 12987 | For some sorts, a member of it is necessarily a member [Leibniz] |
| Full Idea: There are sorts or species such that if an individual has ever been of such a sort or species it cannot (naturally, at least) stop being of it. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.06) | |
| A reaction: Note the thoughtful 'naturally, at least', which blocks genetic engineering. But natural selection is genetic engineering. Crucially, Leibniz is not attributing this to all sorts or species, and allows exceptions. |
| 12884 | The same whole ceases to exist if a part is lost [Leibniz] |
| Full Idea: We cannot say - with complete fidelity to the truth of things - that the same whole continues to exist if a part of it is lost. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.27.11) | |
| A reaction: This is the reference Simons 1987:319 gives when he claims that Leibniz accepts mereological essentialism. I think this is mereological necessity of identity, but not what I call 'essentialism'. That has to distinguish essential from non-essential. |
| 12975 | We have a distinct idea of gold, to define it, but not a perfect idea, to understand it [Leibniz] |
| Full Idea: That gold is a metal which resists cupellation and is insoluble in aquafortis is a distinct idea, for it gives us the criteria or definition of 'gold'. But it is not a perfect idea, because we know too little about cupellation and actions of aquafortis. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.31) | |
| A reaction: This connects the 'perfect idea' of something with knowing its active substance, and hence its essence. See Idea 12976 for the connection between perfect ideas and definitions. |
| 12805 | If two people apply a single term to different resemblances, they refer to two different things [Leibniz] |
| Full Idea: If one person applies the name 'avarice' to one resemblance, and some one else to another, there will be two different species designated by the same name. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 6.6.292), quoted by Nicholas Jolley - Leibniz and Locke on Essences p.199 | |
| A reaction: Part of Leibniz's sustained attack on Locke's nominal essences. There is clearly an uninteresting nominal essence, where a 'big brown bear' is necessarily brown, but in the interesting respects I think Leibniz is right. |
| 12806 | Locke needs many instances to show a natural kind, but why not a single instance? [Leibniz, by Jolley] |
| Full Idea: Leibniz points out that it is a concealed premise of Locke's argument that if a natural kind exists it must have many instances, but there seems no a priori objection to the idea of a species with just one member. | |
| From: report of Gottfried Leibniz (New Essays on Human Understanding [1704], 6.6.311) by Nicholas Jolley - Leibniz and Locke on Essences p.200 | |
| A reaction: I can't see this bothering Locke. Generally we formulate nominal essences by induction from bundles of ideas, but we can formulate a cautious first stab at it from one instance. If you see a new creature, is it a normal one, or a 'monster'? |
| 11862 | Leibniz was not an essentialist [Leibniz, by Wiggins] |
| Full Idea: Leibniz was not an essentialist. | |
| From: report of Gottfried Leibniz (works [1690]) by David Wiggins - Sameness and Substance Renewed 4.2 n4 | |
| A reaction: Assuming this is right, it is rather helpful, because you can read mountains of Leibniz without ever being quite sure. Mackie says he IS an extreme essentialist, treating all properties as essential. Wiggins makes more sense there. |
| 12972 | Bodies, like Theseus's ship, are only the same in appearance, and never strictly the same [Leibniz] |
| Full Idea: We must acknowledge that organic bodies as well as others remain 'the same' only in appearance, and not strictly speaking. It is rather like the river whose water is continually changing, or like Theseus's ship which Athenians constantly repaired. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.27) | |
| A reaction: This is Leibniz's standard view, that something only remains the same if it has a unifying substance, and so a collection of planks is just an aggregate, and doesn't have any identity to begin with. |
| 16504 | Two eggs can't be identical, because the same truths can't apply to both of them [Leibniz] |
| Full Idea: It isn't possible to have two particulars that are similar in all respects - for example two eggs - for it is necessary that some things can be said about one of them that cannot be said about the other, else they could be substituted for one another. | |
| From: Gottfried Leibniz (works [1690]), quoted by David Wiggins - Sameness and Substance 2.2 | |
| A reaction: [from a 'fragment' for which Wiggins gives a reference] This quotation doesn't rest the distinctness of the eggs on some intrinsic difference, but on the fact that we can say different things about the two eggs. |
| 5055 | No two things are totally identical [Leibniz] |
| Full Idea: By virtue of insensible variations, two individual things can never be perfectly alike. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], Pref) | |
| A reaction: This sounds a bit like the 'discernibility of non-identicals', except that he says that the differences may not be 'sensible'. He has to be talking of physical things, since I presume that, say, the symmetry of two circles is perfectly identical. |
| 8650 | Things are the same if one can be substituted for the other without loss of truth [Leibniz] |
| Full Idea: Leibniz's definition is as follows: Things are the same as each other, of which one can be substituted for the other without loss of truth ('salva veritate'). | |
| From: Gottfried Leibniz (works [1690]), quoted by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §65 | |
| A reaction: Frege doesn't give a reference. (Anyone know it?). This famous definition is impressive, but has problems when the items being substituted appear in contexts of belief. 'Oedipus believes Jocasta (his mother!) would make a good wife'. |
| 13828 | Necessary truths are those provable from identities by pure logic in finite steps [Leibniz, by Hacking] |
| Full Idea: Leibniz argued that the necessary truths are just those which can be proved from identities by pure logic in a finite number of steps. ...[232] this claim is vindicated by Gentzen's sequent calculus. | |
| From: report of Gottfried Leibniz (works [1690]) by Ian Hacking - What is Logic? §01 | |
| A reaction: This seems an odd idea, as if there were no necessary truths other than those for which a proof could be constructed. Sounds like intuitionism. |
| 13084 | How can things be incompatible, if all positive terms seem to be compatible? [Leibniz] |
| Full Idea: It is yet unknown to me what is the reason of the incompossibility of things, or how it is that different essences can be opposed to each other, seeing that all purely positive terms seem to be compatible. | |
| From: Gottfried Leibniz (works [1690], G VII:194), quoted by Cover,J/O'Leary-Hawthorne,J - Substance and Individuation in Leibniz 3.4.4 | |
| A reaction: Since 'heavy' seems straightforwardly opposed to 'light', we would have to ask what he means by 'positive'. The suspicion is that all things are compossible by definition, so it is not surprising that impossibilities are a bit puzzling. |
| 4307 | A reason must be given why contingent beings should exist rather than not exist [Leibniz] |
| Full Idea: A reason must be given why contingent beings should exist rather than not exist. | |
| From: Gottfried Leibniz (works [1690]) | |
| A reaction: Spinoza rejects all contingency, but this seems an interesting support for it, even though we may need a reason for something where God does not because it is self-evident. |
| 12978 | A perfect idea of an object shows that the object is possible [Leibniz] |
| Full Idea: One mark of a perfect idea is that it shows conclusively that the object is possible. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.31) | |
| A reaction: Subtle but nice. My favourite example would be that the perfect idea of a bonfire on the Moon shows that it is not possible. Essence reveals necessity, as Aristotle and Kit Fine claim. A perfect idea has a single definition. |
| 17079 | Proofs of necessity come from the understanding, where they have their source [Leibniz] |
| Full Idea: The fundamental proof of necessary truths comes from the understanding alone, and other truths come from experience or from observations of the senses. Our mind is capable of knowing truths of both sorts, but it is the source of the former. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 1.01) | |
| A reaction: Interesting because it not only spells out that necessary truths are known a priori, but also explicitly says that the understanding is the 'source' of the truths, or at least the source of their proofs. He also says possibilities derive from essences. |
| 15883 | Leibniz narrows down God's options to one, by non-contradiction, sufficient reason, indiscernibles, compossibility [Leibniz, by Harré] |
| Full Idea: Leibniz sets up increasingly stringent conditions possible worlds must meet. The weakest is non-contradiction, for truths of reason; then sufficient reason, for rational worlds; then identity of indiscernibles, for duplicates; then compossibility. | |
| From: report of Gottfried Leibniz (works [1690]) by Rom Harré - Laws of Nature 4 | |
| A reaction: [my summary of a very nice two pages by Harré] God is the source of the principles which do the narrowing down. |
| 18822 | Each monad expresses all its compatible monads; a possible world is the resulting equivalence class [Leibniz, by Rumfitt] |
| Full Idea: Leibniz argued that each monad mirrors or expresses every monad with which it is compossible. Hence compossibility is an equivalence relation among monads; possible worlds may then be identified as the corresponding equivalence classes. | |
| From: report of Gottfried Leibniz (works [1690]) by Ian Rumfitt - The Boundary Stones of Thought 6.1 | |
| A reaction: [Rumfitt cites Benson Mates 1986:IV.1 for this claim] There is an analogous world of all the human minds that are in communication with one another - something like a 'culture'. |
| 7837 | Leibniz proposed possible worlds, because they might be evil, where God would not create evil things [Leibniz, by Stewart,M] |
| Full Idea: In his early writings the principle of sufficient reason made it difficult for Leibniz to conceive of possible things;...raising this to possible worlds means God does not choose things that are evil, but chooses a world which must have evil in it. | |
| From: report of Gottfried Leibniz (works [1690]) by Matthew Stewart - The Courtier and the Heretic Ch.14 | |
| A reaction: Where we think of possible worlds as explanations for conditional and counterfactual truths (I take it), Leibniz developed the original idea as part of his huge effort to achieve a consistent theodicy. |
| 13080 | Leibniz has a counterpart view of de re counterfactuals [Leibniz, by Cover/O'Leary-Hawthorne] |
| Full Idea: When Leibniz has the grounds of de re counterfactuals in mind, a counterpart picture, we have argued, is at work. | |
| From: report of Gottfried Leibniz (works [1690]) by Cover,J/O'Leary-Hawthorne,J - Substance and Individuation in Leibniz 3.2.2 | |
| A reaction: If Leibniz were a 'superessentialist', then individuals would be totally worldbound (because their relations would be essential). Cover/Hawthorne argue that he is just a 'strong' essentialist, allowing possible counterparts. Quite persuasive. |
| 12998 | Understanding grasps the agreements and disagreements of ideas [Leibniz] |
| Full Idea: Ordinarily, when ideas are thoroughly understood, their agreements and disagreements are apparent. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 4.02) | |
| A reaction: In other words, there is a holistic aspect of understanding, which makes us ask how understanding ever gets off the ground. Is it not possible to understand a single idea in isolation? |
| 19332 | For Leibniz, divine understanding grasps every conceivable possibility [Leibniz, by Perkins] |
| Full Idea: For Leibniz, what is this understanding which God has? What does it contain? All possibilities in all possible combinations, that is, everything which can be conceived. | |
| From: report of Gottfried Leibniz (works [1690]) by Franklin Perkins - Leibniz: Guide for the Perplexed 2.III | |
| A reaction: I like this, because it strikes me as essential that understanding should embrace possibilities as well as actualities. Perkins points out that the possibilities are restricted by an awareness of the limitations imposed by combination. |
| 12960 | We understand things when they are distinct, and we can derive necessities from them [Leibniz] |
| Full Idea: We understand the things of which we are aware only when we have distinct ideas of them accompanied by the power to reflect and to derive necessary truths from those ideas. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.21) | |
| A reaction: A rather startling way of putting it, but we also say that good understanding brings the power to predict. What must you understand in order to predict? What has to happen next! |
| 13006 | Certainty is where practical doubt is insane, or at least blameworthy [Leibniz] |
| Full Idea: Certainty might be knowledge of a truth such that to doubt it in a practical way would be insane; and sometimes it is taken more broadly, to cover cases where doubt would be very blameworthy. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 4.11) | |
| A reaction: The normative aspect of the second half of this touches on a trend in recent epistemology. You have rights to believe, and duties to believe, and virtues for the justifying process. I prefer more neutral, value-free epistemology. |
| 12996 | I know more than I think, since I know I think A then B then C [Leibniz] |
| Full Idea: Not only is it immediately evident to me that I think, but it is just as evident that I think various thoughts: at one time I think about A and at another about B and so on. Thus the Cartesian principle is sound, but it is not the only one of its kind. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 4.02) | |
| A reaction: I don't suppose that Descartes would object to this, but he was aware that there didn't seem to be any actual introspective experience that united the various thoughts into a single thinker. Only logical connections between the thoughts does that. |
| 13003 | The Cogito doesn't prove existence, because 'I am thinking' already includes 'I am' [Leibniz] |
| Full Idea: To say 'I think therefore I am' is not really to prove existence from thought, since 'to think' and 'to be thinking' are one and the same, and to say 'I am thinking' [je suis pensant] is already to say 'I am' [je suis]. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 4.07) | |
| A reaction: This is the objection which was offered by A.J. Ayer, and I take it to the one of the two principle objections to the Cogito (i.e. that it may be a tautology), along with the objection about the assumption of the continuity of the same thinker. |
| 5509 | Leibniz said dualism of mind and body is illusion, and there is only mind [Leibniz, by Martin/Barresi] |
| Full Idea: Leibniz held that dualism of mind and body is an illusion and that both are really the same thing, and that this thing is mind. | |
| From: report of Gottfried Leibniz (works [1690]) by R Martin / J Barresi - Introduction to 'Personal Identity' p.22 | |
| A reaction: I am puzzled by this, as Leibniz is famous for the view that mind and body are parallel. See idea 5038, and also 2109 and 2596. Monads are, of course, entirely mental, and are the building blocks of reality. Clearly I (and you) must read more Leibniz. |
| 7568 | Leibniz is an idealist insofar as the basic components of his universe are all mental [Leibniz, by Jolley] |
| Full Idea: To say that Leibniz is an idealist is to say that simple substances, the basic building-blocks of the universe, are all mental or at least quasi-mental in nature | |
| From: report of Gottfried Leibniz (works [1690]) by Nicholas Jolley - Leibniz Ch.3 | |
| A reaction: This is a bit different from the Berkelian type of idealism, which says that reality consists entirely of events within thinking minds. Is a monad the thinker or the thought? |
| 21253 | Descartes needs to demonstrate how other people can attain his clear and distinct conceptions [Leibniz] |
| Full Idea: It is not sufficient for Descartes to claim that he perceives something in himself clearly and distinctly, for this is to not complete the demonstration, unless he shows the method through which others can attain the same experience. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], App X) | |
| A reaction: For the simplest rational insight this seems a rather tough requirement. If you say A>B, and B>C, so A>C, then once you have grasped the concept of 'greater than' I'm not sure there is a further possible demonstration. |
| 12933 | Arithmetic and geometry are implicitly innate, awaiting revelation [Leibniz] |
| Full Idea: I would name the propositions of arithmetic and geometry as innate. ...The actual knowledge of them is not innate. What is innate is what might be called the implicit knowledge of them, as the veins of marble outline a shape for the sculptor. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 1.01) | |
| A reaction: This seems to walk straight into the empiricist guns. The marble example shows the problem, because the 'veins' will hardly outline David in the block. Locke's challenge is to show that merely 'implicit' ideas have demonstrable reality. |
| 12991 | Children learn language fast, with little instruction and few definitions [Leibniz] |
| Full Idea: I have sometimes been amazed that children can learn languages so early, ...considering how little trouble is taken to instruct children in their native tongue, and how little thought adults give to getting sharp definitions. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.10) | |
| A reaction: A striking anticipation of the key observation on which Chomsky built his theories, from a philosopher who was equally concerned to defend innate ideas and innate knowledge. |
| 12929 | All of our thoughts come from within the soul, and not from the senses [Leibniz] |
| Full Idea: I always accepted the innate idea of God, but my new system says all the thoughts and actions of the soul come from its own depths and could not be given to it by the senses. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 1.01) | |
| A reaction: It is hard to adjudicate on this one. The counterexamples would be associations. I see a face in the crowd and think of my friend. But Leibniz could be right even about that. Who cares? Externalism is designed to bypass this problem. |
| 12940 | What is left of the 'blank page' if you remove the ideas? [Leibniz] |
| Full Idea: Those who hold forth about the 'blank page' cannot say what is left of it once the ideas have been taken away. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.01) | |
| A reaction: This is a decisive criticism of the total tabula rasa idea, but empiricists responded by developing associationism - that what remains is principles of association for incoming experience. Brain mechanisms, we might say. |
| 19358 | Colour and pain must express the nature of their stimuli, without exact resemblance [Leibniz] |
| Full Idea: Ideas such as those of colour and pain are not arbitrary. ...That is not God's way ...I would say there is a resemblance of a kind, not a perfect one, but a resemblance in which one thing expresses another through some orderly relationship between them. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.08) | |
| A reaction: The main point of Locke's idea of 'secondary' qualities is that (unlike the 'primary' ones) they bear no resemblance to their stimuli. It's not much of an argument from Leibniz, to say that is not God's way, but he has a vast system to support his claim. |
| 12948 | A pain doesn't resemble the movement of a pin, but it resembles the bodily movement pins cause [Leibniz] |
| Full Idea: It is true that pain does not resemble the movement of a pin; but it might thoroughly resemble the motions which the pin causes in our body, and it might resemble them in the soul; and I have not the least doubt that it does. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.08) | |
| A reaction: He may not have the least doubt, but the rest of us do, I should think. Try as I will, I cannot see any resemblance between pain and a motion. What feeling does a pendulum resemble? |
| 13005 | Truth arises among sensations from grounding reasons and from regularities [Leibniz] |
| Full Idea: The truth of sensible things is established by the links amongst them; these depend upon intellectual truths, grounded in reason, and upon observations of regularities among sensible things themselves, even when the reasons are not apparent. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 4.11) | |
| A reaction: It's not clear why regularities would establish truths, given that most hallucinations have regularities in them. I'm thinking that Leibniz is not sufficiently rationalist here, and that it is the rational coherence of experience which validates it. |
| 4302 | You may experience a universal truth, but only reason can tell you that it is always true [Leibniz] |
| Full Idea: However often one experiences instances of a universal truth, one could never know inductively that it would always hold unless one knew through reason that it was necessary. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 1.01) | |
| A reaction: The problem, though, is that as soon as we go beyond experience we are not very reliable, and are liable to arrogance, error and lack of imagination. |
| 12947 | We only believe in sensible things when reason helps the senses [Leibniz] |
| Full Idea: The senses could not convince us of the existence of sensible things without help from reason. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.07) | |
| A reaction: This nicely pinpoints the big difficulty which keeps besetting orthodox empiricism. I've been educated as an empiricist, but I prefer Leibniz to Berkeley or Hume, and even to the more sensible Locke. |
| 12930 | The senses are confused, and necessities come from distinct intellectual ideas [Leibniz] |
| Full Idea: Intellectual ideas, from which necessary truths arise, do not come from the senses. ...The ideas that come from the senses are confused; and so too, at least in part, are the truths which depend on them, whereas intellectual ideas are distinct. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 1.01) | |
| A reaction: One might compare Descartes' example of the chiliagon, which is only grasped clearly by the intellect. However, the problem of vagueness seems to intrude as much into intellectual ideas as it does into the senses. He was a mathematician... |
| 13001 | Our sensation of green is a confused idea, like objects blurred by movement [Leibniz] |
| Full Idea: The sensory idea of green (made of blue and yellow) is a confused idea, like the swift rotation of a cog-wheel which makes us perceive an artificial transparency, and we are not able to discern the cause, the idea of the teeth on the wheel. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 4.06) | |
| A reaction: This is one of Leibniz's less well-known objections to empiricism. He always says that intellectual ideas are capable of a clarity which is never found in sensory experience. |
| 12949 | Light takes time to reach us, so objects we see may now not exist [Leibniz] |
| Full Idea: Since rays of light need time - however little - to reach us, it is possible that the object should be destroyed during the interval and no longer exist when the light reaches the eye. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.09) | |
| A reaction: This is the well know 'time lag' argument. Leibniz is no sceptic, but he can hardly fail to accept the truth of this problem. It seems self-evident that stars we observe may no longer exist, although special relativity confuses that issue. |
| 5053 | The instances confirming a general truth are never enough to establish its necessity [Leibniz] |
| Full Idea: All the instances which confirm a general truth, however numerous they may be, are not sufficient to establish the universal necessity of this same truth. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], Pref) | |
| A reaction: This is Leibniz's standardly rationalist view of induction. We can either say that induction is therefore inadequate, or (a better option) that there isn't much evidence for claims of necessity, and they must be treated with caution. |
| 13092 | The essence of substance is the law of its changes, as in the series of numbers [Leibniz] |
| Full Idea: The essence of substance consists in ...the law of the sequence of changes, as in the nature of the series in numbers. | |
| From: Gottfried Leibniz (works [1690], A 6.3.326), quoted by Cover,J/O'Leary-Hawthorne,J - Substance and Individuation in Leibniz 6.1.2 | |
| A reaction: Thus we might say, in this spirit, that the essence of number is the successor operation, as defined by Dedekind and Peano (and perhaps their amenability to inductive proof). I like this. Metaphysicians rule - they penetrate the heart of nature. |
| 12977 | We will only connect our various definitions of gold when we understand it more deeply [Leibniz] |
| Full Idea: We can define gold as the heaviest metal, or by assaying procedures, but only when men have penetrated more deeply into the nature of things will they be able to see why one belongs with the other. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.31) | |
| A reaction: He mentions that geometry is different, because we do have perfect ideas of things. This is part of Leibniz's optimism about the future of science, in comparison with the surprising pessimism of the empiricists. See Idea 12976 and Idea 12975. |
| 5054 | Animal thought is a shadow of reasoning, connecting sequences of images by imagination [Leibniz] |
| Full Idea: The sequences of the brutes are but a shadow of reasoning, that is to say, they are but connexions of imagination, transitions from one image to another. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], Pref) | |
| A reaction: This account of animal thought cannot capture the fact that they are motivated by their images, and obviously make decisions based on them. Externally, there is usually an obvious reason why even an insect does something. |
| 19354 | Leibniz introduced the idea of degrees of consciousness, essential for his monads [Leibniz, by Perkins] |
| Full Idea: The designation of degrees of conscious awareness is one of Leibniz's most significant innovations, and it is fundamental to almost every aspect of his account of monads. | |
| From: report of Gottfried Leibniz (works [1690]) by Franklin Perkins - Leibniz: Guide for the Perplexed 4.I | |
| A reaction: A very important development, which seems to have been ignored by philosophers for three hundred years, since they usually treat consciousness as all-or-nothing. Introspection makes degrees obvious, and I suspect sparrows are down the scale. |
| 12944 | It is a serious mistake to think that we are aware of all of our perceptions [Leibniz] |
| Full Idea: Many errors can flow from the belief that the only perceptions in the soul are the ones of which it is aware. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.01) | |
| A reaction: What a perceptive remark, for its time! I took it that it was only modern neuroscience and psychology which had woken us up to how much non-conscious activity is central to the mind. |
| 12951 | Abstraction attends to the general, not the particular, and involves universal truths [Leibniz] |
| Full Idea: Beasts recognise whiteness, but this does not amount to abstraction, which requires attention to the general apart from the particular, and consequently involves knowledge of universal truths. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.11) | |
| A reaction: I'm not sure where 'truth' creeps into this. I might hallucinate pink elephants, and abstract the general notion of pink from them. Nevertheless, the features picked out in abstraction tend to be the shared features. |
| 19364 | Volition automatically endeavours to move towards what it sees as good (and away from bad) [Leibniz] |
| Full Idea: Volition is the effort or endeavour ('conatus') to move towards what one finds good and away from what one finds bad, the endeavour arising immediately out of one's awareness of those things. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.21) | |
| A reaction: Modern neuroscience seems to confirm that there is a chicken-and-egg problem here. Is the moment of perception as good or bad itself an act of volition, or is it neutral? |
| 12942 | Memory doesn't make identity; a man who relearned everything would still be the same man [Leibniz] |
| Full Idea: If a man were made young again, and learned everything anew - would that make him a different man? So it is not memory that makes the very same man. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.01) | |
| A reaction: Leibniz takes this as a foregone conclusion. If you flipped to a possible world where someone you know well, as a physical being, has been brought up entirely differently (new language, culture, ethics etc), is it really the same person? |
| 12973 | We know our own identity by psychological continuity, even if there are some gaps [Leibniz] |
| Full Idea: To discover one's own moral identity unaided, it is sufficient that between one state and a neighbouring (or just a nearby) one there be a mediating bond of consciousness, even if this has a jump or forgotten interval mixed into it. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.27) | |
| A reaction: Leibniz appears to accept the psychological continuity view of personal identity (which was probably a new problem to him), even though he rightly rejects the account based purely on memory. |
| 7841 | We think we are free because the causes of the will are unknown; determinism is a false problem [Leibniz] |
| Full Idea: The will has its causes, but since we are ignorant of them, we believe ourselves independent. It is this chimera of imaginary independence which revolts us against determinism, and which brings us to believe there are difficulties where there are none. | |
| From: Gottfried Leibniz (works [1690]), quoted by Matthew Stewart - The Courtier and the Heretic Ch.16 | |
| A reaction: It seems that in his notebooks Leibniz was actually a (Spinozan) determinist. So he should have been, given his view that we live in the best of all possible worlds, and his claim that mind and brain run like two clocks. (Ideas 2114 and 2596) |
| 19368 | The will determines action, by what is seen as good, but it does not necessitate it [Leibniz] |
| Full Idea: Choice, however much the will is determined to make it, should not be called absolutely and strictly necessary: a predominance of goods of which one is aware inclines without necessitating, though this is determining and never fails to have its effect. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.21) | |
| A reaction: Something like seeing that 7+5 equals 12 makes you say '12', but it doesn't actually necessitate your saying '12'? Certain facts seem determined by nature, but not necessitated. Or not necessarily necessitated? |
| 5510 | Leibniz has a panpsychist view that physical points are spiritual [Leibniz, by Martin/Barresi] |
| Full Idea: In Leibniz's panpsychism, the so-called 'physical' points are souls or spiritual 'monads'. | |
| From: report of Gottfried Leibniz (works [1690]) by R Martin / J Barresi - Introduction to 'Personal Identity' p.23 | |
| A reaction: I'm not convinced that 'panpsychism' is the right description for Leibniz's theory of monads. I take panpsychism to be either a dualist or a dual aspect (or property dualism) view. Leibniz seems to believe there is strictly one substance. |
| 7564 | Occasionalism give a false view of natural laws, miracles, and substances [Leibniz, by Jolley] |
| Full Idea: Leibniz's three objections to occasionalism are: it disturbs the concept of laws of nature used in physics; it introduces perpetual miracles; and it doesn't recognise activity of substances (leading to the Spinozan heresy that God is the only substance). | |
| From: report of Gottfried Leibniz (works [1690]) by Nicholas Jolley - Leibniz Ch.2 | |
| A reaction: I wonder what would happen if, within the viewpoint of occasionalism, God suddenly packed up and abandoned his job? Presumably the world wouldn't disappear, so there would still be substances, but passive ones, in chaos. |
| 12935 | Every feeling is the perception of a truth [Leibniz] |
| Full Idea: Every feeling is the perception of a truth. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 1.02) | |
| A reaction: I wouldn't say feelings ARE perceptions of truths, but I might say that every experience we have has an intellectual dimension as well as an emotional one. Our beliefs drift and solidify in just the same way that feelings do. |
| 12938 | An idea is an independent inner object, which expresses the qualities of things [Leibniz] |
| Full Idea: An idea is an immediate inner object, which expresses the nature or qualities of things, ..but since it is the object of thought it can exist before and after the thoughts. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.01) | |
| A reaction: This sounds something like Frege's 'third realm' between mind and world (Idea 7740). Notice that Leibniz is also using the word 'object' in this context. Leibniz doesn't make the mistake of confusing concepts and images, as many did. |
| 12950 | We must distinguish images from exact defined ideas [Leibniz] |
| Full Idea: It is essential to distinguish images from exact ideas which are composed of definitions. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.09) | |
| A reaction: See Idea 12615, which is attacking Descartes and Locke, I think, but fails to register that Spinoza and Leibniz had got the notion of an 'idea' much more clearly. |
| 12945 | Thoughts correspond to sensations, but ideas are independent of thoughts [Leibniz] |
| Full Idea: I distinguish ideas from thoughts. For we always have all our pure or distinct ideas independently of the senses, but thoughts always correspond to some sensation. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.01) | |
| A reaction: Leibniz's concept of an 'idea' is quite different from the empiricist notion of them, and strikes me as being much closer to Frege's notion of a concept. On the whole I like the Leibniz account best. |
| 19357 | The idea of green seems simple, but it must be compounded of the ideas of blue and yellow [Leibniz] |
| Full Idea: It is obvious that green comes from a mixture of blue and yellow; which makes it credible that the idea of green is composed of the ideas of those two colours, although the idea of green appears to us as simple as that of blue. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.07) | |
| A reaction: This shows the use of 'idea' at that time for non-verbal mental events and concepts. Ideas are not, then, just undestood as phenomena, but can be analysed and explained more deeply. |
| 12995 | The name 'gold' means what we know of gold, and also further facts about it which only others know [Leibniz] |
| Full Idea: The name 'gold' signifies not merely what the speaker knows of gold , but also what he does not know, which may be known by someone else: an inner constitution from which flow colour and weight, and generates other properties. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.11) | |
| A reaction: [compressed] Thus in the course of defending true essences of gold (against Locke's claim that we are stuck with the nominal essence), Leibniz drifts into an externalist account of meaning. He mentions experts, as so often does Putnam. |
| 12807 | The word 'gold' means a hidden constitution known to experts, and not just its appearances [Leibniz] |
| Full Idea: The name 'gold' signifies not only what he who pronounces it knows about it, for example, something very heavy and yellow, but also what he does not know, and that another can know about it, its internal constitution from which colour and weight flow. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 6.6.354), quoted by Nicholas Jolley - Leibniz and Locke on Essences | |
| A reaction: Leibniz goes on to use the word 'expert'. This isn't just a hint of Putnam's externalism about concepts like 'water' - it is a clear spelling out of the full idea. Locke would have been astounded by 'atomic number 79', and Leibniz would be, like, 'yeah'. |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| Full Idea: Cantor (in his exploration of infinities) pushed the bounds of conceivability further than anyone before him. To discover what is conceivable, we have to enquire into the concept. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics 6.5 | |
| A reaction: This remark comes during a discussion of Husserl's phenomenology. Intuitionists challenge Cantor's claim, and restrict what is conceivable to what is provable. Does possibility depend on conceivability? |
| 19372 | Concepts are ordered, and show eternal possibilities, deriving from God [Leibniz, by Arthur,R] |
| Full Idea: Leibniz understood concepts as corresponding to eternal possibilities, with both concepts and their ordering having their foundation in the divine mind. | |
| From: report of Gottfried Leibniz (works [1690]) by Richard T.W. Arthur - Leibniz 2 'Nominalism' | |
| A reaction: It is is no longer the fashion to think of concepts as 'ordered', and yet there is a multitude of dependence relations between them. |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| Full Idea: Cantor thought that we abstract a number as something common to all and only those sets any one of which has as many members as any other. ...However one wants to see the logic of the inference. The irony is that set theory lays out this logic. | |
| From: comment on George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: The logic Hart has in mind is the notion of an equivalence relation between sets. This idea sums up the older and more modern concepts of abstraction, the first as psychological, the second as logical (or trying very hard to be!). Cf Idea 9145. |
| 13467 | Leibniz was the first modern to focus on sentence-sized units (where empiricists preferred word-size) [Leibniz, by Hart,WD] |
| Full Idea: Leibniz seems to be the first modern philosopher to focus on sentence-sized units that he called propositions. The Empiricists among the moderns focused on word-sized units like ideas. | |
| From: report of Gottfried Leibniz (works [1690]) by William D. Hart - The Evolution of Logic 2 | |
| A reaction: Historically, the sentential logic of the Stoics has a claim to have started this one. I find my initial sympathies to be with the empiricists. |
| 12946 | The idea of the will includes the understanding [Leibniz] |
| Full Idea: The idea of the will includes that of the understanding. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.06) | |
| A reaction: A nice remark, although I am not sure that I agree with it. The understanding (if we allow such talk of faculties, with which I have no problem) sometimes lags behind and sometimes forges ahead of the will. What else is weakness of the will? |
| 19365 | Limited awareness leads to bad choices, and unconscious awareness makes us choose the bad [Leibniz, by Perkins] |
| Full Idea: For Leibniz, while the limits of our knowledge explain why we sometimes choose things we think are good but which turn out to be bad, the force of minute perceptions explains why we sometimes choose things that we know are bad. | |
| From: report of Gottfried Leibniz (works [1690]) by Franklin Perkins - Leibniz: Guide for the Perplexed 4.IV | |
| A reaction: To be overwhelmed by selfish greed doesn't sound like a 'minute perception'. Leibniz thinks all desires are reactions to perceptions. Observing our degrees of knowledge is an interesting response to the intellectualist view of weakness of will. |
| 12964 | If would be absurd not to disagree with someone's taste if it was a taste for poisons [Leibniz] |
| Full Idea: If someone acquired a taste for poisons which would kill him or make him wretched, it would be absurd to say that we ought not to argue with him about his tastes. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.21) | |
| A reaction: This seems to be a first step in any discussion of taste on which you would hope that sensible persons would agree. 'It is just a matter of taste' is definitely not the end of discussion. Aesthetic taste is important, just as values are important. |
| 8110 | Leibniz identified beauty with intellectual perfection [Leibniz, by Gardner] |
| Full Idea: Leibniz identified beauty with intellectual perfection. | |
| From: report of Gottfried Leibniz (works [1690]) by Sebastian Gardner - Aesthetics 1.2.1 | |
| A reaction: Well he would, wouldn't he? Swots like Leibniz are inclined to value things which only they can fully appreciate. There may be intellectual subject matter in the study of a rose, but I do not believe that it is needed to appreciate the beauty. |
| 7569 | Humans are moral, and capable of reward and punishment, because of memory and self-consciousness [Leibniz, by Jolley] |
| Full Idea: For Leibniz, it is by virtue of possessing memory and self-consciousness that human minds are moral beings, capable of reward and punishment. | |
| From: report of Gottfried Leibniz (works [1690]) by Nicholas Jolley - Leibniz Ch.4 | |
| A reaction: I like this because it makes no mention of free will (though Leibniz struggled to defend free will). I would add meta-thought (the ability to ponder and evaluate our own thinking), which makes a change of mind possible. |
| 12958 | Love is pleasure in the perfection, well-being or happiness of its object [Leibniz] |
| Full Idea: To love is to be disposed to take pleasure in the perfection, well-being or happiness of the object of one's love. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.20) | |
| A reaction: I suppose so, though one might love a pathetic little plant that hangs on in the corner of the garden, just for its fighting qualities. He goes on to deny that we can truly love something that is incapable of happiness. Hm. |
| 12957 | The good is the virtuous, the pleasing, or the useful [Leibniz] |
| Full Idea: The good is divided into the virtuous, the pleasing, and the useful. ..The good is either pleasing or useful; and virtue itself consists of a pleasure of the mind. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.20) | |
| A reaction: I presume that the useful could be reduced to the pleasing. It strikes me as quite bizarre to define virtue as merely a pleasure of the mind. Aristotle says true virtue must also please the mind, but that is a different idea. |
| 12962 | Pleasure is a sense of perfection [Leibniz] |
| Full Idea: Fundamentally, pleasure is a sense of perfection, and pain a sense of imperfection. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.21) | |
| A reaction: A bit odd, but I like the idea that there is an intellectual aspect to even the most visceral feelings. |
| 12934 | We can't want everyone to have more than their share, so a further standard is needed [Leibniz] |
| Full Idea: We would wish for more than our share if we had our own way; so do we also owe to others more than their share? If the rule applies only to a just will, the rule will need a standard. The rule means that to judge fairly we must adopt others' viewpoints. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 1.02) | |
| A reaction: The first part of this is moving towards Kant's rational overview of ethics. Leibniz is wholly right here. All ethics faces the problem that initial values are needed to get it off the ground. What's wrong with pain, or unfairness, or hatred? |
| 7574 | Natural law theory is found in Aquinas, in Leibniz, and at the Nuremberg trials [Leibniz, by Jolley] |
| Full Idea: Leibniz rejects Hobbes's legal positivism in favour of the older natural law theory associated with Aquinas (which says nothing can be a law unless it derives from natural justice). The older view was revived at Nuremberg, to prosecute Nazis. | |
| From: report of Gottfried Leibniz (works [1690]) by Nicholas Jolley - Leibniz Ch.7 | |
| A reaction: This seems to suggest that Hobbes and co were using Ockham's Razor to eliminate morality from the law, but that the Nuremberg situation (and modern trials in The Hague) show that there is a necessity for natural law in international situations. |
| 12936 | There are natural rewards and punishments, like illness after over-indulgence [Leibniz] |
| Full Idea: There can be natural rewards and punishments without a law-maker; intemperance, for instance, is punished by illness. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 1.02) | |
| A reaction: Hm. Maybe if it didn't result in illness it wouldn't be labelled as 'intemperance'. Why isn't studying philosophy all day long classed as intemperance? |
| 12728 | Leibniz rejected atoms, because they must be elastic, and hence have parts [Leibniz, by Garber] |
| Full Idea: Leibniz held that there can be no atoms in nature, nothing perfectly solid and hard, since elasticity entails the existence of smaller parts that can move with respect to one another. | |
| From: report of Gottfried Leibniz (works [1690]) by Daniel Garber - Leibniz:Body,Substance,Monad 5 | |
| A reaction: Thus, I suppose, we discover that atoms have mercurial electron shells. Are quarks or electrons elastic? The debate about true atoms is not over, and probably never will be. Leibniz's point is a good one. |
| 19374 | Microscopes and the continuum suggest that matter is endlessly divisible [Leibniz] |
| Full Idea: Micrographers observe qualities of larger things found in smaller things. And if this proceeds to infinity - which is possible since the continuum is divisible to infinity - any atom will be an infinite species, and there will be worlds within worlds. | |
| From: Gottfried Leibniz (works [1690], A VI ii 241) | |
| A reaction: [a work of the 1670s] The microscope had a huge impact on Leibniz, much more than the telescope. |
| 7560 | Leibniz struggled to reconcile bodies with a reality of purely soul-like entities [Jolley on Leibniz] |
| Full Idea: Leibniz seems never to have made up his mind completely on how to accommodate bodies within a metaphysic which recognises only soul-like entities as fully real. | |
| From: comment on Gottfried Leibniz (works [1690]) by Nicholas Jolley - Leibniz Intro | |
| A reaction: [The soul-like entities are his 'monads']. His choice must be to either say they are unreal, or that they are real and separate from the monads, or that they are a manifestation of the monads. His problem, not mine. |
| 16683 | Leibniz eventually said resistance, rather than extension, was the essence of body [Leibniz, by Pasnau] |
| Full Idea: Leibniz eventually rejected extension altogether as part of the essence of body, and replaced it with resistance. | |
| From: report of Gottfried Leibniz (works [1690]) by Robert Pasnau - Metaphysical Themes 1274-1671 15.5 | |
| A reaction: This makes body consist of active force, rather than mere geometry. Much better. |
| 11856 | Qualities should be predictable from the nature of the subject [Leibniz] |
| Full Idea: Whenever we find some quality in a subject, we ought to believe that if we understood the nature of both the subject and the quality we would conceive how the quality could arise from it. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], Pref 66) | |
| A reaction: This is the idea that powers are prior to properties, which seems right to me. I take essence to be something like the best explanation of qualities. |
| 12994 | Gold has a real essence, unknown to us, which produces its properties [Leibniz] |
| Full Idea: The complex idea of gold includes its being something which has a real essence whose detailed constitution is unknown to us, except for the fact that such qualities as malleability depend upon it. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.10) | |
| A reaction: This is precisely the view of modern scientific essentialism. The underlying idea I take to be the conception of essence as the thing which explains the properties. |
| 12808 | Part of our idea of gold is its real essence, which is not known to us in detail [Leibniz] |
| Full Idea: It is very true that it is part of the complex idea of gold that it is a thing which has a real essence, the constitution of which is not otherwise known to us in detail. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 6.6.345), quoted by Nicholas Jolley - Leibniz and Locke on Essences p.201 | |
| A reaction: See also Idea 12807. This is the clearest possible statement of Leibniz's clear-cut scientific essentialism, here presented in opposition to Locke (thought I take the latter to be only bothered by our inability to know the hidden constitution). |
| 12725 | Leibniz wanted to explain motion and its laws by the nature of body [Leibniz, by Garber] |
| Full Idea: Leibniz seeks the big picture: the nature of body as a grounding for an account of motion and its laws. | |
| From: report of Gottfried Leibniz (works [1690]) by Daniel Garber - Leibniz:Body,Substance,Monad 4 | |
| A reaction: Garber is contrasting this with Newton's approaches, who just pleads ignorance of the bigger picture. Essentialists must beware of inventing a bigger picture simply because they desperately want a bigger picture. |
| 16507 | The law within something fixes its persistence, and accords with general laws of nature [Leibniz] |
| Full Idea: Nothing is permanent in a substance except the law itself which determines the continuous succession of its states and accords within the individual substance with the laws of nature that govern the whole world. | |
| From: Gottfried Leibniz (works [1690], G II:263), quoted by David Wiggins - Sameness and Substance 3 epig | |
| A reaction: An interesting link between the law-of-series within a substance, and the broader concept of laws outside it. |
| 7859 | Leibniz had an unusual commitment to the causal completeness of physics [Leibniz, by Papineau] |
| Full Idea: Unlike most philosophers prior to the twentieth century, Leibniz was committed to the causal completeness of physics. | |
| From: report of Gottfried Leibniz (works [1690]) by David Papineau - Thinking about Consciousness 1.4 | |
| A reaction: It has been suggested that Leibniz was actually, in private, a determinist (see Idea 7841), which would fit. Leibniz is enigmatic, but he may have proposed the closure of physics to glorify God, only to find that God was beginning to look irrelevant. |
| 12985 | Maybe motion is definable as 'change of place' [Leibniz] |
| Full Idea: I believe 'motion' to be definable, and the definition which says that it is 'change of place' deserves respect. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.04) | |
| A reaction: This seems to be the 'at-at' view of motion, championed by Bertrand Russell. (At p1 at t1, at p2 at t2...). Leibniz's version only mentions space and not time, and it includes 'change', which would need definition without mentioning motion. |
| 15307 | Leibniz uses 'force' to mean both activity and potential [Leibniz] |
| Full Idea: At this early period exegetical problems abound, since Leibniz uses 'force' both for actually acting forces and for potentials or powers. | |
| From: Gottfried Leibniz (works [1690], 9.II), quoted by Harré,R./Madden,E.H. - Causal Powers 9.II.B | |
| A reaction: I take Leibniz to be a key figure in the development of the Aristotelian approach, because he connected Aristotelian potential and essence with 'force' in the new physics. This is helpful in reading him correctly. |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| Full Idea: Cantor proved that one-dimensional space has exactly the same number of points as does two dimensions, or our familiar three-dimensional space. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 12952 | Space is an order among actual and possible things [Leibniz] |
| Full Idea: Space is a relationship: an order, not only among existents, but also among possibles as though they existed. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.13) | |
| A reaction: The modal end to this idea is a bit puzzling. Would there be any space if there were only possibles, and nothing yet existed, as in God's mind the instant before he got to work? |
| 12955 | If there were duration without change, we could never establish its length [Leibniz] |
| Full Idea: If there were a vacuum in space, one could establish its size. But if there were a vacuum in time, i.e. a duration without change, it would be impossible to establish its length. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.15) | |
| A reaction: See Idea 4226 for Shoemaker's wonderful counterproposal to this apparently unanswerable claim. I suppose Leibniz is right, but it just might be possible to bring induction to bear on the problem. |
| 12954 | God's essence is the source of possibilities, and his will the source of existents [Leibniz] |
| Full Idea: God is the source of possibilities and of existents alike, the one by his essence and the other by his will. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.15) | |
| A reaction: Every now and then I rebel against metaphysics, and think 'how do these people know all this great things about which they make these dogmatic claims?' And this is one of those occasions. I get the idea, though... |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| Full Idea: Cantor said that only God is absolutely infinite. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: We are used to the austere 'God of the philosophers', but this gives us an even more austere 'God of the mathematicians'. |
| 12988 | The universe contains everything possible for its perfect harmony [Leibniz] |
| Full Idea: The universe contains everything that its perfect harmony could admit. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 3.06) | |
| A reaction: This sort of Leibnizian remark leaves most modern readers, including me, totally bewildered. The claim depends entirely on the perfect nature of God. |
| 1414 | A perfection is a simple quality, which is positive and absolute, and has no limit [Leibniz] |
| Full Idea: I call every simple quality which is positive and absolute, or expresses whatever it expresses without any limits, a perfection. But a quality of this sort, because it is simple, is therefore irresolvable or indefinable. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], App X) | |
| A reaction: I don't think this definition of perfections would have occurred to anyone who wasn't planning to prove that perfections cannot be incompatible (as Leibniz is about to do). |
| 21252 | Perfections must have overlapping parts if their incompatibility is to be proved [Leibniz] |
| Full Idea: If two propositions (about perfections) are incompatible, that cannot be demonstrated without a resolution of the terms, for otherwise their nature would not enter into the ratiocination. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], App X) | |
| A reaction: If God is omnipotent and wholly free, these appear to be fully separate perfections. But it is their implications (can God decide to do otherwise, given His foreknowledge?) which lead to a problem. So this analyis of contradiction is wrong. |
| 19328 | Without the principle of sufficient reason, God's existence could not be demonstrated [Leibniz] |
| Full Idea: There is a fundamental axiom that 'nothing happens without reason', without which the existence of God and other great truths cannot be properly demonstrated. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 2.21.13) | |
| A reaction: I'm rather drawn to the Principle of Sufficient Reason, but also to John Keats's 'negative capability'. Belief that there must be a reason in each case is not a justification for inventing a reason every time. There may be a reason for the universe.... |
| 3889 | God's existence is either necessary or impossible [Leibniz, by Scruton] |
| Full Idea: Leibniz said that the ontological argument does not prove God's existence, but only the God's existence is either necessary or impossible. | |
| From: report of Gottfried Leibniz (works [1690]) by Roger Scruton - Modern Philosophy:introduction and survey 13.5 |
| 7842 | Leibniz was closer than Spinoza to atheism [Leibniz, by Stewart,M] |
| Full Idea: Leibniz sailed closer to the winds of unbelief than Spinoza did. | |
| From: report of Gottfried Leibniz (works [1690]) by Matthew Stewart - The Courtier and the Heretic Ch.16 | |
| A reaction: This is an unusual view, but Stewart's view is that whereas Spinoza is always sincere in his writings, Leibniz is inclined to put a very conservative spin on his opinions. A key question for Leibniz is "Is God merely a monad?" |
| 5058 | Animals have thought and sensation, and indestructible immaterial souls [Leibniz] |
| Full Idea: I hold (against the Cartesians) that brutes also have thought, and hold that they have sensation, and souls which are, properly speaking, immaterial, and as incapable of perishing as the atoms of Democritus or Gassendi. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], Pref) | |
| A reaction: Insect heaven will be a bit crowded. I can never grasp why theologians would claim that souls are 'indestructible', when they are held to come into existence at a particular moment in space-time. Transmigration of souls is a much more rational belief. |