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70 ideas

1. Philosophy / C. History of Philosophy / 4. Later European Philosophy / c. Eighteenth century philosophy
My dogmatic slumber was first interrupted by David Hume [Kant]
     Full Idea: I freely admit that remembrance of David Hume was the very thing that many years ago first interrupted my dogmatic slumber.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 4:260), quoted by A.W. Moore - The Evolution of Modern Metaphysics 5.2
     A reaction: A famous declaration. He realised that he had the answer the many scepticisms of Hume, and accept his emphasis on the need for experience.
1. Philosophy / E. Nature of Metaphysics / 1. Nature of Metaphysics
Metaphysics is generating a priori knowledge by intuition and concepts, leading to the synthetic [Kant]
     Full Idea: The generation of knowledge a priori, both according to intuition and according to concepts, and finally the generation of synthetic propositions a priori in philosophical knowledge, constitutes the essential content of metaphysics.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 274)
     A reaction: By 'concepts' he implies mere analytic thought, so 'intuition' is where the exciting bit is, and that is rather vague.
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 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).
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
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 / 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
Mathematics cannot proceed just by the analysis of concepts [Kant]
     Full Idea: Mathematics cannot proceed analytically, namely by analysis of concepts, but only synthetically.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284)
     A reaction: I'm with Kant insofar as I take mathematics to be about the world, no matter how rarefied and 'abstract' it may become.
6. Mathematics / A. Nature of Mathematics / 2. Geometry
Geometry is not analytic, because a line's being 'straight' is a quality [Kant]
     Full Idea: No principle of pure geometry is analytic. That the straight line beween two points is the shortest is a synthetic proposition. For my concept of straight contains nothing of quantity but only of quality.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 269)
     A reaction: I'm not sure what his authority is for calling straightness a quality rather than a quantity, given that it can be expressed quantitatively. It is a very nice example for focusing our questions about the nature of geometry. I can't decide.
Geometry rests on our intuition of space [Kant]
     Full Idea: Geometry is grounded on the pure intuition of space.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284)
     A reaction: I have the impression that recent thinkers are coming round to this idea, having attempted purely algebraic or logical accounts of geometry.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
Numbers are formed by addition of units in time [Kant]
     Full Idea: Arithmetic forms its own concepts of numbers by successive addition of units in time.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284)
     A reaction: It is hard to imagine any modern philosopher of mathematics embracing this idea. It sounds as if Kant thinks counting is the foundation of arithmetic, which I quite like.
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'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
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
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 / 4. Using Numbers / f. Arithmetic
7+5 = 12 is not analytic, because no analysis of 7+5 will reveal the concept of 12 [Kant]
     Full Idea: The concept of twelve is in no way already thought by merely thinking the unification of seven and five, and though I analyse my concept of such a possible sum as long as I please, I shall never find twelve in it.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 269)
     A reaction: It might be more plausible to claim that an analysis of 12 would reveal the concept of 7+5. Doesn't the concept of two collections of objects contain the concept of their combined cardinality?
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
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'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
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.
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.
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.
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 / 2. Intuition of Mathematics
Mathematics can only start from an a priori intuition which is not empirical but pure [Kant]
     Full Idea: We find that all mathematical knowledge has this peculiarity, that it must first exhibit its concept in intuition, and do so a priori, in an intuition that is not empirical but pure.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 281)
     A reaction: Later thinkers had grave doubts about this Kantian 'intuition', even if they though maths was known a priori. Personally I am increasing fan of rational intuition, even if I am not sure how to discern whether it is rational on any occasion.
All necessary mathematical judgements are based on intuitions of space and time [Kant]
     Full Idea: Space and time are the two intuitions on which pure mathematics grounds all its cognitions and judgements that present themselves as at once apodictic and necessary.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284)
     A reaction: This unlikely proposal seems to be based on the idea that mathematics must arise from the basic categories of our intuition, and these two are the best candidates he can find. I would say that high-level generality is the basis of mathematics.
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'.
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
Mathematics cannot be empirical because it is necessary, and that has to be a priori [Kant]
     Full Idea: Mathematical propositions are always judgements a priori, and not empirical, because they carry with them necessity, which cannot be taken from experience.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 268)
     A reaction: Presumably there are necessities in the physical world, and we might discern them by generalising about that world, so that mathematics is (by a tortuous abstract route) a posteriori necessary? Just a thought…
9. Objects / B. Unity of Objects / 2. Substance / e. Substance critique
The substance, once the predicates are removed, remains unknown to us [Kant]
     Full Idea: It has long since been noticed that in all substances the subject proper, namely what is left over after all the accidents (as predicates) have been taken away and hence the 'substantial' itself, is unknown to us.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 333)
     A reaction: This is the terminus of the process of abstraction (though Wiggins says such removal of predicates is a myth). Kant is facing the problem of the bare substratum, or haecceity.
11. Knowledge Aims / A. Knowledge / 1. Knowledge
'Transcendental' concerns how we know, rather than what we know [Kant]
     Full Idea: The word 'transcendental' signifies not a relation of our cognition to things, but only to the faculty of cognition.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 4:293), quoted by A.W. Moore - The Evolution of Modern Metaphysics 5.4
     A reaction: This is the annoying abduction of a word which is very useful in metaphysical contexts.
11. Knowledge Aims / C. Knowing Reality / 3. Idealism / b. Transcendental idealism
I admit there are bodies outside us [Kant]
     Full Idea: I do indeed admit that there are bodies outside us.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 289 n.II)
     A reaction: This is the end of a passage in which Kant very explicitly denies being an idealist. Of course, he says we can only know the representations of things, and not how they are in themselves.
'Transcendental' is not beyond experience, but a prerequisite of experience [Kant]
     Full Idea: The word 'transcendental' does not mean something that goes beyond all experience, but something which, though it precedes (a priori) all experience, is destined only to make knowledge by experience possible.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 373 n)
     A reaction: One of two explanations by Kant of 'transcendental', picked out by Sebastian Gardner. I think the word 'prerequisite' covers the idea nicely, using a normal English word. Or am I missing something?
12. Knowledge Sources / A. A Priori Knowledge / 5. A Priori Synthetic
A priori synthetic knowledge is only of appearances, not of things in themselves [Kant]
     Full Idea: Through intuition we can only know objects as they appear to us (to our senses), not as they may be in themselves; and this presupposition is absolutely necessary if synthetic propositions a priori are to be granted as possible.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 283)
     A reaction: This idea is basic to understanding Kant, and especially his claim that arithmetic is a priori synthetic.
12. Knowledge Sources / A. A Priori Knowledge / 9. A Priori from Concepts
A priori intuitions can only concern the objects of our senses [Kant]
     Full Idea: Intuitions which are possible a priori can never concern any other things than objects of our senses.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 283)
     A reaction: Given the Kantian idea that what is known a priori will also be necessary, we might have had great hopes for big-time metaphysics, but this idea cuts it down to size. Personally, I don't think we are totally imprisoned in the phenomena.
12. Knowledge Sources / A. A Priori Knowledge / 10. A Priori as Subjective
A priori intuition of objects is only possible by containing the form of my sensibility [Kant]
     Full Idea: The only way for my intuition to precede the reality of the object and take place as knowledge a priori is if it contains nothing else than the form of sensibility which in me as subject precedes all real impressions through which I'm affected by objects.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 283)
     A reaction: This may be the single most famous idea in Kant. I'm not really a Kantian, but this is a powerful idea, the culmination of Descartes' proposal to start philosophy by looking at ourselves. No subsequent thinking can ignore the idea.
12. Knowledge Sources / B. Perception / 2. Qualities in Perception / d. Secondary qualities
I can make no sense of the red experience being similar to the quality in the object [Kant]
     Full Idea: I can make little sense of the assertion that the sensation of red is similar to the property of the vermilion [cinnabar] which excites this sensation in me.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 290)
     A reaction: A sensible remark. In Kant's case it is probably a part of his scepticism that his intuitions reveal anything directly about reality. Locke seems to have thought (reasonably enough) that the experience contains some sort of valid information.
12. Knowledge Sources / B. Perception / 2. Qualities in Perception / e. Primary/secondary critique
I count the primary features of things (as well as the secondary ones) as mere appearances [Kant]
     Full Idea: I also count as mere appearances, in addition to [heat, colour, taste], the remaining qualities of bodies which are called primariae, extension, place, and space in general, with all that depends on it (impenetrability or materiality, shape etc.).
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 289 n.II)
     A reaction: He sides with Berkeley and Hume against Locke and Boyle. He denies being an idealist (Idea 16923), so it seems to me that Kant might be described as a 'phenomenalist'.
12. Knowledge Sources / B. Perception / 3. Representation
I can't intuit a present thing in itself, because the properties can't enter my representations [Kant]
     Full Idea: It seems inconceivable how the intuition of a thing that is present should make me know it as it is in itself, for its properties cannot migrate into my faculty of representation.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 282)
     A reaction: One might compare this with Locke's distinction of primary and secondary, where the primary properties seem to 'migrate into my faculty of representation', but the secondary ones fail to do so. I think I prefer Locke. This idea threatens idealism.
12. Knowledge Sources / D. Empiricism / 4. Pro-Empiricism
Appearance gives truth, as long as it is only used within experience [Kant]
     Full Idea: Appearance brings forth truth so long as it is used in experience, but as soon as it goes beyond the boundary of experience and becomes transcendent, it brings forth nothing but illusion.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 292 n.III)
     A reaction: This is the nearest I have found to Kant declaring for empiricism. It sounds something like direct realism, if experience itself can bring forth truth.
12. Knowledge Sources / E. Direct Knowledge / 2. Intuition
Intuition is a representation that depends on the presence of the object [Kant]
     Full Idea: Intuition is a representation, such as would depend on the presence of the object.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 282)
     A reaction: This is a distinctively Kantian view of intuition, which arises through particulars, rather than the direct apprehension of generalities.
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 / D. Concepts / 2. Origin of Concepts / a. Origin of concepts
Some concepts can be made a priori, which are general thoughts of objects, like quantity or cause [Kant]
     Full Idea: Concepts are of such a nature that we can make some of them ourselves a priori, without standing in any immediate relation to the object; namely concepts that contain the thought of an object in general, such as quantity or cause.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 282)
     A reaction: 'Quantity' seems to be the scholastic idea, of something having a magnitude (a big pebble, not six pebbles).
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.
19. Language / E. Analyticity / 1. Analytic Propositions
Analytic judgements say clearly what was in the concept of the subject [Kant]
     Full Idea: Analytic judgements say nothing in the predicate that was not already thought in the concept of the subject, though not so clearly and with the same consciousness. If I say all bodies are extended, I have not amplified my concept of body in the least.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 266)
     A reaction: If I say all bodies are made of atoms, have I extended my concept of 'body'? It would come as a sensational revelation for Aristotle, but it now seems analytic.
Analytic judgement rests on contradiction, since the predicate cannot be denied of the subject [Kant]
     Full Idea: Analytic judgements rest wholly on the principle of contradiction, …because the predicate cannot be denied of the subject without contradiction.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 267)
     A reaction: So if I say 'gold has atomic number 79', that is a (Kantian) analytic statement? This is the view of sceptics about Kripke's a posteriori necessity. …a few lines later Kant gives 'gold is a yellow metal' as an example.
21. Aesthetics / C. Artistic Issues / 7. Art and Morality
Musical performance can reveal a range of virtues [Damon of Ath.]
     Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice.
     From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where?
27. Natural Reality / C. Space / 2. Space
Space must have three dimensions, because only three lines can meet at right angles [Kant]
     Full Idea: That complete space …has three dimensions, and that space in general cannot have more, is built on the proposition that not more than three lines can intersect at right angles in a point.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 285)
     A reaction: Modern geometry seems to move, via the algebra, to more than three dimensions, and then battles for an intuition of how that can be. I don't know how they would respond to Kant's challenge here.
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
27. Natural Reality / D. Time / 1. Nature of Time / a. Absolute time
If all empirical sensation of bodies is removed, space and time are still left [Kant]
     Full Idea: If everything empirical, namely what belongs to sensation, is taken away from the empirical intuition of bodies and their changes (motion), space and time are still left.
     From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284)
     A reaction: This is an exercise in psychological abstraction, which doesn't sound like good evidence, though it is an interesting claim. Physicists want to hijack this debate, but I like Kant's idea.
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'.