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All the ideas for 'works', 'Individuals in Aristotle' and 'Monadology'

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62 ideas

2. Reason / B. Laws of Thought / 2. Sufficient Reason
No fact can be real and no proposition true unless there is a Sufficient Reason (even if we can't know it) [Leibniz]
     Full Idea: The principle of sufficient reason says no fact can be real or existing and no proposition can be true unless there is a sufficient reason why it should be thus and not otherwise, even though in most cases these reasons cannot be known to us.
     From: Gottfried Leibniz (Monadology [1716], §32)
     A reaction: I think of this as my earliest philosophical perception, a childish rebellion against being told that there was 'no reason' for something. My intuition tells me that it is correct, and the foundation of ontology and truth. Don't ask me to justify it!
3. Truth / D. Coherence Truth / 1. Coherence Truth
Everything in the universe is interconnected, so potentially a mind could know everything [Leibniz]
     Full Idea: Every body is sensitive to everything in the universe, so that one who saw everything could read in each body what is happening everywhere, and even what has happened and will happen.
     From: Gottfried Leibniz (Monadology [1716], §61)
4. Formal Logic / F. Set Theory ST / 1. Set Theory
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
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / c. Basic theorems of ST
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
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).
4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
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.
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
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
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
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
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.
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / b. Combinatorial sets
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.
5. Theory of Logic / D. Assumptions for Logic / 3. Contradiction
Falsehood involves a contradiction, and truth is contradictory of falsehood [Leibniz]
     Full Idea: We judge to be false that which involves a contradiction, and true that which is opposed or contradictory to the false.
     From: Gottfried Leibniz (Monadology [1716], §31)
5. Theory of Logic / K. Features of Logics / 8. Enumerability
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]
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / b. Cantor's paradox
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.
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / e. Mirimanoff's paradox
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?
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
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
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
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
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
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?
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
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.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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).
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.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
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.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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
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
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
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.
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
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
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.
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
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
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.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
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
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
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.
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.
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.
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
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.
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.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
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?
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
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
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?
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
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?
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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).
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
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'.
7. Existence / C. Structure of Existence / 6. Fundamentals / c. Monads
The monad idea incomprehensibly spiritualises matter, instead of materialising soul [La Mettrie on Leibniz]
     Full Idea: The Leibnizians with their monads have constructed an incomprehensible hypothesis. They have spiritualized matter rather than materialising the soul.
     From: comment on Gottfried Leibniz (Monadology [1716]) by Julien Offray de La Mettrie - Machine Man p.3
     A reaction: I agree with La Mettrie. This disagreement shows, I think, how important the problem of interaction between mind and body was in the century after Descartes. Drastic action seemed needed to bridge the gap, one way or the other.
He replaced Aristotelian continuants with monads [Leibniz, by Wiggins]
     Full Idea: In the end Leibniz dethroned Aristotelian continuants, seen as imperfect from his point of view, in favour of monads.
     From: report of Gottfried Leibniz (Monadology [1716]) by David Wiggins - Sameness and Substance Renewed 3.1
     A reaction: I take the 'continuants' to be either the 'ultimate subject of predication' (in 'Categories'), or 'essences' (in 'Metaphysics'). Since monads seem to be mental (presumably to explain the powers of things), this strikes me as a bit mad.
Is a drop of urine really an infinity of thinking monads? [Voltaire on Leibniz]
     Full Idea: Can you really maintain that a drop of urine is an infinity of monads, and that each one of these has ideas, however obscure, of the entire universe?
     From: comment on Gottfried Leibniz (Monadology [1716]) by Francois-Marie Voltaire - works Vol 22:434
     A reaction: Monads are a bit like Christian theology - if you meet them cold they seem totally ridiculous, but if you meet them after ten years of careful preliminary study they make (apparently) complete sense. Defenders of panpsychism presumably like them.
It is unclear in 'Monadology' how extended bodies relate to mind-like monads. [Garber on Leibniz]
     Full Idea: It is never clear in the 'Monadologie' how exactly the world of extended bodies is related to the world of simple substances, the world of non-extended and mind-like monads.
     From: comment on Gottfried Leibniz (Monadology [1716]) by Daniel Garber - Leibniz:Body,Substance,Monad 9
     A reaction: Leibniz was always going to hit the interaction problem, as soon as he started giving an increasingly spiritual account of what a substance, and hence marginalising the 'force' which had held centre-stage earlier on. Presumably they are 'parallel'.
Changes in a monad come from an internal principle, and the diversity within its substance [Leibniz]
     Full Idea: A monad's natural changes come from an internal principle, ...but there must be diversity in that which changes, which produces the specification and variety of substances.
     From: Gottfried Leibniz (Monadology [1716], §11-12)
     A reaction: You don't have to like monads to like this generalisation (and Perkins says Leibniz had a genius for generalisations). Metaphysics must give an account of change. Succeeding time-slices etc explain nothing. Principle and substance must meet.
A 'monad' has basic perception and appetite; a 'soul' has distinct perception and memory [Leibniz]
     Full Idea: The general name 'monad' or 'entelechy' may suffice for those substances which have nothing but perception and appetition; the name 'souls' may be reserved for those having perception that is more distinct and accompanied by memory.
     From: Gottfried Leibniz (Monadology [1716], §19)
     A reaction: It is basic to the study of Leibniz that you don't think monads are full-blown consciousnesses. He isn't really a panpsychist, because the level of mental activity is so minimal. There seem to be degrees of monadhood.
9. Objects / B. Unity of Objects / 2. Substance / e. Substance critique
If a substance is just a thing that has properties, it seems to be a characterless non-entity [Leibniz, by Macdonald,C]
     Full Idea: For Leibniz, to distinguish between a substance and its properties in order to provide a thing or entity in which properties can inhere leads necessarily to the absurd conclusion that the substance itself must be a truly characterless non-entity.
     From: report of Gottfried Leibniz (Monadology [1716]) by Cynthia Macdonald - Varieties of Things Ch.3
     A reaction: This is obviously one of the basic thoughts in any discussion of substances. It is why physicists ignore them, and Leibniz opted for a 'bundle' theory. But the alternative seems daft too - free-floating properties, hooked onto one another.
9. Objects / E. Objects over Time / 9. Ship of Theseus
Insurance on the original ship would hardly be paid out if the plank version was wrecked! [Frede,M]
     Full Idea: No insurance company, presented with a policy written for 'Theoris' [the original ship] would pay for damages suffered if the ship contructed from the old planks had been shipwrecked.
     From: Michael Frede (Individuals in Aristotle [1978])
     A reaction: A very nicely dramatic way of presenting what is taken to be the usual reading of the basic case - that the original identity tracks the continuity of the original structure, not the matter.
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
There must be some internal difference between any two beings in nature [Leibniz]
     Full Idea: There are never two beings in nature that are perfectly alike, two beings in which it is not possible to discover an internal difference, that is, one founded on an intrinsic denomination.
     From: Gottfried Leibniz (Monadology [1716], §09)
     A reaction: From this it follows that if two things really are indiscernible, then we must say that they are one thing. He says monads all differ from one another. People certainly do. Leibniz must say this of electrons. How can he know this?
10. Modality / D. Knowledge of Modality / 1. A Priori Necessary
Truths of reason are known by analysis, and are necessary; facts are contingent, and their opposites possible [Leibniz]
     Full Idea: There are two kinds of truths: of reasoning and of facts. Truths of reasoning are necessary and their opposites impossible. Facts are contingent and their opposites possible. A necessary truth is known by analysis.
     From: Gottfried Leibniz (Monadology [1716], §33)
12. Knowledge Sources / A. A Priori Knowledge / 4. A Priori as Necessities
Mathematical analysis ends in primitive principles, which cannot be and need not be demonstrated [Leibniz]
     Full Idea: At the end of the analytical method in mathematics there are simple ideas of which no definition can be given. Moreover there are axioms and postulates, in short, primitive principles, which cannot be demonstrated and do not need demonstration.
     From: Gottfried Leibniz (Monadology [1716], §35)
     A reaction: My view is that we do not know such principles when we apprehend them in isolation. I would call them 'intuitions'. They only ascend to the status of knowledge when the mathematics is extended and derived from them, and found to work.
12. Knowledge Sources / C. Rationalism / 1. Rationalism
We all expect the sun to rise tomorrow by experience, but astronomers expect it by reason [Leibniz]
     Full Idea: When we expect it to be day tomorrow, we all behave as empiricists, because until now it has always happened thus. The astronomer alone knows this by reason.
     From: Gottfried Leibniz (Monadology [1716], §28)
15. Nature of Minds / B. Features of Minds / 3. Privacy
Increase a conscious machine to the size of a mill - you still won't see perceptions in it [Leibniz]
     Full Idea: If a conscious machine were increased in size, one might enter it like a mill, but we should only see the parts impinging on one another; we should not see anything which would explain a perception.
     From: Gottfried Leibniz (Monadology [1716], §17)
     A reaction: A wonderful image for capturing a widely held intuition. It seems to motivate Colin McGinn's 'Mysterianism'. The trouble is Leibniz didn't think big/small enough. Down at the level of molecules it might become obvious what a perception is. 'Might'.
16. Persons / C. Self-Awareness / 2. Knowing the Self
We know the 'I' and its contents by abstraction from awareness of necessary truths [Leibniz]
     Full Idea: It is through the knowledge of necessary truths and through their abstraction that we rise to reflective acts, which enable us to think of that which is called "I" and enable us to consider that this or that is in us.
     From: Gottfried Leibniz (Monadology [1716], §30)
     A reaction: For Leibniz, necessary truth can only be known a priori. Sense experience won't reveal the self, as Hume observed. We evidently 'abstract' the idea of 'I' from the nature of a priori thought. Animals have no self (or morals) for this reason.
18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
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?
18. Thought / E. Abstraction / 2. Abstracta by Selection
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.
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / f. Ancient elements
The true elements are atomic monads [Leibniz]
     Full Idea: Monads are the true atoms of nature and, in brief, the elements of things.
     From: Gottfried Leibniz (Monadology [1716], (opening)), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 2
     A reaction: Thus in one sentence Leibniz gives us a theory of natural elements, and an account of atoms. This kind of speculation got metaphysics a bad name when science unravelled a more accurate picture. The bones must be picked out of Leibniz.
27. Natural Reality / C. Space / 3. Points in Space
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
28. God / A. Divine Nature / 2. Divine Nature
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'.
28. God / A. Divine Nature / 3. Divine Perfections
This is the most perfect possible universe, in its combination of variety with order [Leibniz]
     Full Idea: From all the possible universes God chooses this one to obtain as much variety as possible, but with the greatest order possible; that is, it is the means of obtaining the greatest perfection possible.
     From: Gottfried Leibniz (Monadology [1716], §58)
28. God / B. Proving God / 2. Proofs of Reason / a. Ontological Proof
God alone (the Necessary Being) has the privilege that He must exist if He is possible [Leibniz]
     Full Idea: God alone (or the Necessary Being) has the privilege that He must exist if He is possible.
     From: Gottfried Leibniz (Monadology [1716], §45)