100 ideas
| 6814 | Instead of prayer and charity, sinners pursue vain disputes and want their own personal scripture [Mohammed] |
| Full Idea: The sinners will say 'we never prayed or fed the hungry. We engaged in vain disputes and denied the Day of Reckoning'. Indeed, each one of them demands a scripture of his own to be unrolled before him. | |
| From: Mohammed (The Koran [c.622], Ch.74) | |
| A reaction: The implication seems to be that most disputes are 'vain'. The charge that everyone wants a 'scripture of his own' is a nice challenge to the world of liberal education, where we are all enjoined to pursue our personalised routes to our own truth. |
| 224 | When questions are doubtful we should concentrate not on objects but on ideas of the intellect [Plato] |
| Full Idea: Doubtful questions should not be discussed in terms of visible objects or in relation to them, but only with reference to ideas conceived by the intellect. | |
| From: Plato (Parmenides [c.364 BCE], 135e) |
| 232 | Opposites are as unlike as possible [Plato] |
| Full Idea: Opposites are as unlike as possible. | |
| From: Plato (Parmenides [c.364 BCE], 159a) |
| 8937 | Plato's 'Parmenides' is the greatest artistic achievement of the ancient dialectic [Hegel on Plato] |
| Full Idea: Plato's 'Parmenides' is the greatest artistic achievement of the ancient dialectic. | |
| From: comment on Plato (Parmenides [c.364 BCE]) by Georg W.F.Hegel - Phenomenology of Spirit Pref 71 | |
| A reaction: It is a long way from the analytic tradition of philosophy to be singling out a classic text for its 'artistic' achievement. Eventually we may even look back on, say, Kripke's 'Naming and Necessity' and see it in that light. |
| 8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
| Full Idea: An 'impredicative' definition is one that uses the terms being defined in order to give the definition; in some way the definition is then circular. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], Glossary) | |
| A reaction: There has been a big controversy in the philosophy of mathematics over these. Shapiro gives the definition of 'village idiot' (which probably mentions 'village') as an example. |
| 8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
| Full Idea: In classical logic definitions are thought of as revealing our attempts to refer to objects, ...but for intuitionist or constructivist logics, if our definitions do not uniquely characterize an object, we are not entitled to discuss the object. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.4) | |
| A reaction: In defining a chess piece we are obviously creating. In defining a 'tree' we are trying to respond to fact, but the borderlines are vague. Philosophical life would be easier if we were allowed a mixture of creation and fact - so let's have that. |
| 3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
| Full Idea: Reductio ad absurdum arguments are ones that start by denying what one wants to prove. We then prove a contradiction from this 'denied' idea and more reasonable ideas in one's theory, showing that we were wrong in denying what we wanted to prove. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: This is a mathematical definition, which rests on logical contradiction, but in ordinary life (and philosophy) it would be enough to show that denial led to absurdity, rather than actual contradiction. |
| 8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
| Full Idea: For the anti-realist, truth belongs to us, it is our servant, and as such, it must be 'epistemically constrained'. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.1) | |
| A reaction: Put as clearly as this, it strikes me as being utterly and spectacularly wrong, a complete failure to grasp the elementary meaning of a concept etc. etc. If we aren't the servants of truth then we jolly we ought to be. Truth is above us. |
| 8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
| Full Idea: In the classical or realist view of logic the meaning of abstract symbols for logical connectives is given by the truth-tables for the symbol. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007]) | |
| A reaction: Presumably this is realist because it connects them to 'truth', but only if that involves a fairly 'realist' view of truth. You could, of course, translate 'true' and 'false' in the table to empty (formalist) symbols such a 0 and 1. Logic is electronics. |
| 8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
| Full Idea: In intuitionist logic, if we do not know that we do not know A, it does not follow that we know A, so the inference (and, in general, double negation elimination) is not intuitionistically valid. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2) | |
| A reaction: That inference had better not be valid in any logic! I am unaware of not knowing the birthday of someone I have never heard of. Propositional attitudes such as 'know' are notoriously difficult to explain in formal logic. |
| 8694 | Free logic was developed for fictional or non-existent objects [Friend] |
| Full Idea: Free logic is especially designed to help regiment our reasoning about fictional objects, or nonexistent objects of some sort. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 3.7) | |
| A reaction: This makes it sound marginal, but I wonder whether existential commitment shouldn't be eliminated from all logic. Why do fictional objects need a different logic? What logic should we use for Robin Hood, if we aren't sure whether or not he is real? |
| 8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
| Full Idea: A 'subset' of A is a set containing only members of A, and a 'proper subset' is one that does not contain all the members of A. Note that the empty set is a subset of every set, but it is not a member of every set. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Is it the same empty set in each case? 'No pens' is a subset of 'pens', but is it a subset of 'paper'? Idea 8219 should be borne in mind when discussing such things, though I am not saying I agree with it. |
| 8672 | A 'powerset' is all the subsets of a set [Friend] |
| Full Idea: The 'powerset' of a set is a set made up of all the subsets of a set. For example, the powerset of {3,7,9} is {null, {3}, {7}, {9}, {3,7}, {3,9}, {7,9}, {3,7,9}}. Taking the powerset of an infinite set gets us from one infinite cardinality to the next. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Note that the null (empty) set occurs once, but not in the combinations. I begin to have queasy sympathies with the constructivist view of mathematics at this point, since no one has the time, space or energy to 'take' an infinite powerset. |
| 8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
| Full Idea: As a realist choice of what is basic in mathematics, set theory is rather clever, because it only makes a very simple ontological claim: that, independent of us, there exists the empty set. The whole hierarchy of finite and infinite sets then follows. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: Even so, for non-logicians the existence of the empty set is rather counterintuitive. "There was nobody on the road, so I overtook him". See Ideas 7035 and 8322. You might work back to the empty set, but how do you start from it? |
| 8666 | Infinite sets correspond one-to-one with a subset [Friend] |
| Full Idea: Two sets are the same size if they can be placed in one-to-one correspondence. But even numbers have one-to-one correspondence with the natural numbers. So a set is infinite if it has one-one correspondence with a proper subset. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Dedekind's definition. We can match 1 with 2, 2 with 4, 3 with 6, 4 with 8, etc. Logicians seem happy to give as a definition anything which fixes the target uniquely, even if it doesn't give the essence. See Frege on 0 and 1, Ideas 8653/4. |
| 8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
| Full Idea: Zermelo-Fraenkel and Gödel-Bernays set theory differ over the notions of ordinal construction and over the notion of class, among other things. Then there are optional axioms which can be attached, such as the axiom of choice and the axiom of infinity. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.6) | |
| A reaction: This summarises the reasons why we cannot just talk about 'set theory' as if it was a single concept. The philosophical interest I would take to be found in disentangling the ontological commitments of each version. |
| 8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
| Full Idea: The law of excluded middle is purely syntactic: it says for any well-formed formula A, either A or not-A. It is not a semantic law; it does not say that either A is true or A is false. The semantic version (true or false) is the law of bivalence. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2) | |
| A reaction: No wonder these two are confusing, sufficiently so for a lot of professional philosophers to blur the distinction. Presumably the 'or' is exclusive. So A-and-not-A is a contradiction; but how do you explain a contradiction without mentioning truth? |
| 8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
| Full Idea: In the intuitionist version of quantification, the universal quantifier (normally read as "all") is understood as "we have a procedure for checking every" or "we have checked every". | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.5) | |
| A reaction: It seems better to describe this as 'verificationist' (or, as Dummett prefers, 'justificationist'). Intuition suggests an ability to 'see' beyond the evidence. It strikes me as bizarre to say that you can't discuss things you can't check. |
| 13986 | Plato found antinomies in ideas, Kant in space and time, and Bradley in relations [Plato, by Ryle] |
| Full Idea: Plato (in 'Parmenides') shows that the theory that 'Eide' are substances, and Kant that space and time are substances, and Bradley that relations are substances, all lead to aninomies. | |
| From: report of Plato (Parmenides [c.364 BCE]) by Gilbert Ryle - Are there propositions? 'Objections' |
| 14150 | Plato's 'Parmenides' is perhaps the best collection of antinomies ever made [Russell on Plato] |
| Full Idea: Plato's 'Parmenides' is perhaps the best collection of antinomies ever made. | |
| From: comment on Plato (Parmenides [c.364 BCE]) by Bertrand Russell - The Principles of Mathematics §337 |
| 8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
| Full Idea: The realist meets the Burali-Forti paradox by saying that all the ordinals are a 'class', not a set. A proper class is what we discuss when we say "all" the so-and-sos when they cannot be reached by normal set-construction. Grammar is their only limit. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: This strategy would be useful for Class Nominalism, which tries to define properties in terms of classes, but gets tangled in paradoxes. But why bother with strict sets if easy-going classes will do just as well? Descartes's Dream: everything is rational. |
| 8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
| Full Idea: The Burali-Forti paradox says that if ordinals are defined by 'gathering' all their predecessors with the empty set, then is the set of all ordinals an ordinal? It is created the same way, so it should be a further member of this 'complete' set! | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: This is an example (along with Russell's more famous paradox) of the problems that began to appear in set theory in the early twentieth century. See Idea 8675 for a modern solution. |
| 8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
| Full Idea: The set of 'integers' is all of the negative natural numbers, and zero, together with the positive natural numbers. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Zero always looks like a misfit at this party. Credit and debit explain positive and negative nicely, but what is the difference between having no money, and money being irrelevant? I can be 'broke', but can the North Pole be broke? |
| 8668 | The 'rational' numbers are those representable as fractions [Friend] |
| Full Idea: The 'rational' numbers are all those that can be represented in the form m/n (i.e. as fractions), where m and n are natural numbers different from zero. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: Pythagoreans needed numbers to stop there, in order to represent the whole of reality numerically. See irrational numbers for the ensuing disaster. How can a universe with a finite number of particles contain numbers that are not 'rational'? |
| 8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
| Full Idea: A number is 'irrational' just in case it cannot be represented as a fraction. An irrational number has an infinite non-repeating decimal expansion. Famous examples are pi and e. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: There must be an infinite number of irrational numbers. You could, for example, take the expansion of pi, and change just one digit to produce a new irrational number, and pi has an infinity of digits to tinker with. |
| 8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
| Full Idea: The natural numbers are quite primitive, and are what we first learn about. The order of objects (the 'ordinals') is one level of abstraction up from the natural numbers: we impose an order on objects. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4) | |
| A reaction: Note the talk of 'levels of abstraction'. So is there a first level of abstraction? Dedekind disagrees with Friend (Idea 7524). I would say that natural numbers are abstracted from something, but I'm not sure what. See Structuralism in maths. |
| 8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
| Full Idea: The 'cardinal' numbers answer the question 'How many?'; the order of presentation of the objects being counted as immaterial. Def: the cardinality of a set is the number of members of the set. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: If one asks whether cardinals or ordinals are logically prior (see Ideas 7524 and 8661), I am inclined to answer 'neither'. Presenting them as answers to the questions 'how many?' and 'which comes first?' is illuminating. |
| 8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
| Full Idea: The set of 'real' numbers, which consists of the rational numbers and the irrational numbers together, represents "the continuum", since it is like a smooth line which has no gaps (unlike the rational numbers, which have the irrationals missing). | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: The Continuum is the perfect abstract object, because a series of abstractions has arrived at a vast limit in its nature. It still has dizzying infinities contained within it, and at either end of the line. It makes you feel humble. |
| 8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
| Full Idea: After the multiples of omega, we can successively raise omega to powers of omega, and after that is done an infinite number of times we arrive at a new limit ordinal, which is called 'epsilon'. We have an infinite number of infinite ordinals. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4) | |
| A reaction: When most people are dumbstruck by the idea of a single infinity, Cantor unleashes an infinity of infinities, which must be the highest into the stratosphere of abstract thought that any human being has ever gone. |
| 8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
| Full Idea: The first 'limit ordinal' is called 'omega', which is ordinal because it is greater than other numbers, but it has no immediate predecessor. But it has successors, and after all of those we come to twice-omega, which is the next limit ordinal. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4) | |
| A reaction: This is the gateway to Cantor's paradise of infinities, which Hilbert loved and defended. Who could resist the pleasure of being totally boggled (like Aristotle) by a concept such as infinity, only to have someone draw a map of it? See 8663 for sequel. |
| 8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
| Full Idea: Since between any two rational numbers there is an infinite number of rational numbers, we could consider that we have infinity in three dimensions: positive numbers, negative numbers, and the 'depth' of infinite numbers between any rational numbers. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5) | |
| A reaction: This is before we even reach Cantor's staggering infinities (Ideas 8662 and 8663), which presumably reside at the outer reaches of all three of these dimensions of infinity. The 'deep' infinities come from fractions with huge denominators. |
| 8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
| Full Idea: Successful competing founding disciplines in mathematics include: the various set theories, type theory, category theory, model theory and topology. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: Or none of the above? Set theories are very popular. Type theory is, apparently, discredited. Shapiro has a version of structuralism based on model theory (which sound promising). Topology is the one that intrigues me... |
| 8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
| Full Idea: Most of mathematics can be faithfully redescribed by classical (realist) set theory. More precisely, we can translate other mathematical theories - such as group theory, analysis, calculus, arithmetic, geometry and so on - into the language of set theory. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3) | |
| A reaction: This is why most mathematicians seem to regard set theory as foundational. We could also translate football matches into the language of atomic physics. |
| 8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
| Full Idea: There is no interest for the mathematician in studying the number 8 in isolation from the other numbers. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |
| A reaction: This is a crucial and simple point (arising during a discussion of Shapiro's structuralism). Most things are interesting in themselves, as well as for their relationships, but mathematical 'objects' just are relationships. |
| 8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
| Full Idea: Structuralists give a historical account of why the 'same' number occupies different structures. Numbers are equivalent rather than identical. 8 is the immediate predecessor of 9 in the whole numbers, but in the rationals 9 has no predecessor. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |
| A reaction: I don't become a different person if I move from a detached house to a terraced house. This suggests that 8 can't be entirely defined by its relations, and yet it is hard to see what its intrinsic nature could be, apart from the units which compose it. |
| 8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
| Full Idea: Structuralists disagree over whether objects in structures are 'ante rem' (before reality, existing independently of whether the objects exist) or 'in re' (in reality, grounded in the real world, usually in our theories of physics). | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |
| A reaction: Shapiro holds the first view, Hellman and Resnik the second. The first view sounds too platonist and ontologically extravagant; the second sounds too contingent and limited. The correct account is somewhere in abstractions from the real. |
| 8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
| Full Idea: According to the structuralist, mathematicians study the concepts (objects of study) such as variable, greater, real, add, similar, infinite set, which are one level of abstraction up from prima facie base objects such as numbers, shapes and lines. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.1) | |
| A reaction: This still seems to imply an ontology in which numbers, shapes and lines exist. I would have thought you could eliminate the 'base objects', and just say that the concepts are one level of abstraction up from the physical world. |
| 8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
| Full Idea: Structuralism says we study whole structures: objects together with their predicates, relations that bear between them, and functions that take us from one domain of objects to a range of other objects. The objects can even be eliminated. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.1) | |
| A reaction: The unity of object and predicate is a Quinean idea. The idea that objects are inessential is the dramatic move. To me the proposal has very strong intuitive appeal. 'Eight' is meaningless out of context. Ordinality precedes cardinality? Ideas 7524/8661. |
| 8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
| Full Idea: In the 'in re' version of mathematical structuralism, pattern-spotting is the process of abstraction. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.4) | |
| A reaction: This might work for non-mathematical abstraction as well, if we are allowed to spot patterns within sensual experience, and patterns within abstractions. Properties are causal patterns in the world? No - properties cause patterns. |
| 16150 | One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato] |
| Full Idea: If one is, there must also necessarily be number - Necessarily - But if there is number, there would be many, and an unlimited multitude of beings. ..So if all partakes of being, each part of number would also partake of it. | |
| From: Plato (Parmenides [c.364 BCE], 144a) | |
| A reaction: This seems to commit to numbers having being, then to too many numbers, and hence to too much being - but without backing down and wondering whether numbers had being after all. Aristotle disagreed. |
| 8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
| Full Idea: The main philosophical problem with the position of platonism or realism is the epistemic problem: of explaining what perception or intuition consists in; how it is possible that we should accurately detect whatever it is we are realists about. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.5) | |
| A reaction: The best bet, I suppose, is that the mind directly perceives concepts just as eyes perceive the physical (see Idea 8679), but it strikes me as implausible. If we have to come up with a special mental faculty for an area of knowledge, we are in trouble. |
| 8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
| Full Idea: Central to naturalism about mathematics are 'indispensability arguments', to the effect that some part of mathematics is indispensable to our best physical theory, and therefore we ought to take that part of mathematics to be true. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 6.1) | |
| A reaction: Quine and Putnam hold this view; Field challenges it. It has the odd consequence that the dispensable parts (if they can be identified!) do not need to be treated as true (even though they might follow logically from the dispensable parts!). Wrong! |
| 8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
| Full Idea: There are not enough constraints in the Formalist view of mathematics, so there is no way to select a direction for trying to develop mathematics. There is no part of mathematics that is more important than another. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 6.6) | |
| A reaction: One might reply that an area of maths could be 'important' if lots of other areas depended on it, and big developments would ripple big changes through the interior of the subject. Formalism does, though, seem to reduce maths to a game. |
| 8706 | Constructivism rejects too much mathematics [Friend] |
| Full Idea: Too much of mathematics is rejected by the constructivist. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.1) | |
| A reaction: This was Hilbert's view. This seems to be generally true of verificationism. My favourite example is that legitimate speculations can be labelled as meaningless. |
| 8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
| Full Idea: An intuitionist typically retains bivalence, but rejects the law of excluded middle. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 5.2) | |
| A reaction: The idea would be to say that only T and F are available as truth-values, but failing to be T does not ensure being F, but merely not-T. 'Unproven' is not-T, but may not be F. |
| 229 | The one was and is and will be and was becoming and is becoming and will become [Plato] |
| Full Idea: The one was and is and will be and was becoming and is becoming and will become. | |
| From: Plato (Parmenides [c.364 BCE], 155d) |
| 21821 | Plato's Parmenides has a three-part theory, of Primal One, a One-Many, and a One-and-Many [Plato, by Plotinus] |
| Full Idea: The Platonic Parmenides is more exact [than Parmenides himself]; the distinction is made between the Primal One, a strictly pure Unity, and a secondary One which is a One-Many, and a third which is a One-and-Many. | |
| From: report of Plato (Parmenides [c.364 BCE]) by Plotinus - The Enneads 5.1.08 | |
| A reaction: Plotinus approves of this three-part theory. Parmenides has the problem that the highest Being contains no movement. By placing the One outside Being you can give it powers which an existent thing cannot have. Cf the concept of God. |
| 221 | Absolute ideas, such as the Good and the Beautiful, cannot be known by us [Plato] |
| Full Idea: The absolute good and the beautiful and all which we conceive to be absolute ideas are unknown to us. | |
| From: Plato (Parmenides [c.364 BCE], 134c) |
| 227 | You must always mean the same thing when you utter the same name [Plato] |
| Full Idea: You must always mean the same thing when you utter the same name. | |
| From: Plato (Parmenides [c.364 BCE], 147d) |
| 223 | If you deny that each thing always stays the same, you destroy the possibility of discussion [Plato] |
| Full Idea: If a person denies that the idea of each thing is always the same, he will utterly destroy the power of carrying on discussion. | |
| From: Plato (Parmenides [c.364 BCE], 135c) |
| 211 | If admirable things have Forms, maybe everything else does as well [Plato] |
| Full Idea: It is troubling that if admirable things have abstract ideas, then perhaps everything else must have ideas as well. | |
| From: Plato (Parmenides [c.364 BCE], 130d) |
| 219 | If absolute ideas existed in us, they would cease to be absolute [Plato] |
| Full Idea: None of the absolute ideas exists in us, because then it would no longer be absolute. | |
| From: Plato (Parmenides [c.364 BCE], 133c) |
| 228 | Greatness and smallness must exist, to be opposed to one another, and come into being in things [Plato] |
| Full Idea: These two ideas, greatness and smallness, exist, do they not? For if they did not exist, they could not be opposites of one another, and could not come into being in things. | |
| From: Plato (Parmenides [c.364 BCE], 149e) |
| 16151 | Plato moves from Forms to a theory of genera and principles in his later work [Plato, by Frede,M] |
| Full Idea: It seems to me that Plato in the later dialogues, beginning with the second half of 'Parmenides', wants to substitute a theory of genera and theory of principles that constitute these genera for the earlier theory of forms. | |
| From: report of Plato (Parmenides [c.364 BCE]) by Michael Frede - Title, Unity, Authenticity of the 'Categories' V | |
| A reaction: My theory is that the later Plato came under the influence of the brilliant young Aristotle, and this idea is a symptom of it. The theory of 'principles' sounds like hylomorphism to me. |
| 210 | It would be absurd to think there were abstract Forms for vile things like hair, mud and dirt [Plato] |
| Full Idea: Are there abstract ideas for such things as hair, mud and dirt, which are particularly vile and worthless? That would be quite absurd. | |
| From: Plato (Parmenides [c.364 BCE], 130d) |
| 220 | The concept of a master includes the concept of a slave [Plato] |
| Full Idea: Mastership in the abstract is mastership of slavery in the abstract. | |
| From: Plato (Parmenides [c.364 BCE], 133e) |
| 218 | Participation is not by means of similarity, so we are looking for some other method of participation [Plato] |
| Full Idea: Participation is not by means of likeness, so we must seek some other method of participation. | |
| From: Plato (Parmenides [c.364 BCE], 133a) |
| 213 | Each idea is in all its participants at once, just as daytime is a unity but in many separate places at once [Plato] |
| Full Idea: Just as day is in many places at once, but not separated from itself, so each idea might be in all its participants at once. | |
| From: Plato (Parmenides [c.364 BCE], 131b) |
| 216 | If things are made alike by participating in something, that thing will be the absolute idea [Plato] |
| Full Idea: That by participation in which like things are made like, will be the absolute idea, will it not? | |
| From: Plato (Parmenides [c.364 BCE], 132e) |
| 212 | The whole idea of each Form must be found in each thing which participates in it [Plato] |
| Full Idea: The whole idea of each form (of beauty, justice etc) must be found in each thing which participates in it. | |
| From: Plato (Parmenides [c.364 BCE], 131a) |
| 215 | If things partake of ideas, this implies either that everything thinks, or that everything actually is thought [Plato] |
| Full Idea: If all things partake of ideas, must either everything be made of thoughts and everything thinks, or everything is thought, and so can't think? | |
| From: Plato (Parmenides [c.364 BCE], 132c) |
| 214 | If absolute greatness and great things are seen as the same, another thing appears which makes them seem great [Plato] |
| Full Idea: If you regard the absolute great and the many great things in the same way, will not another appear beyond, by which all these must appear to be great? | |
| From: Plato (Parmenides [c.364 BCE], 132a) |
| 217 | Nothing can be like an absolute idea, because a third idea intervenes to make them alike (leading to a regress) [Plato] |
| Full Idea: It is impossible for anything to be like an absolute idea, because a third idea will appear to make them alike, and if that is like anything, it will lead to another idea, and so on. | |
| From: Plato (Parmenides [c.364 BCE], 133a) |
| 8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
| Full Idea: What the mathematician labels an 'object' in her discipline, is called 'a place in a structure' by the structuralist. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 4.5) | |
| A reaction: This is a strategy for dispersing the idea of an object in the world of thought, parallel to attempts to eliminate them from physical ontology (e.g. Idea 614). |
| 15851 | Parts must belong to a created thing with a distinct form [Plato] |
| Full Idea: The part would not be the part of many things or all, but of some one character ['ideas'] and of some one thing, which we call a 'whole', since it has come to be one complete [perfected] thing composed [created] of all. | |
| From: Plato (Parmenides [c.364 BCE], 157d) | |
| A reaction: A serious shot by Plato at what identity is. Harte quotes it (125) and shows that 'character' is Gk 'idea', and 'composed' will translate as 'created'. 'Form' links this Platonic passage to Aristotle's hylomorphism. |
| 15846 | In Parmenides, if composition is identity, a whole is nothing more than its parts [Plato, by Harte,V] |
| Full Idea: At the heart of the 'Parmenides' puzzles about composition is the thesis that composition is identity. Considered thus, a whole adds nothing to an ontology that already includes its parts | |
| From: report of Plato (Parmenides [c.364 BCE]) by Verity Harte - Plato on Parts and Wholes 2.5 | |
| A reaction: There has to be more to a unified identity that mere proximity of the parts. When do parts come together, and when do they actually 'compose' something? |
| 15849 | Plato says only a one has parts, and a many does not [Plato, by Harte,V] |
| Full Idea: In 'Parmenides' it is argued that a part cannot be part of a many, but must be part of something one. | |
| From: report of Plato (Parmenides [c.364 BCE], 157c) by Verity Harte - Plato on Parts and Wholes 3.2 | |
| A reaction: This looks like the right way to go with the term 'part'. We presuppose a unity before we even talk of its parts, so we can't get into contradictions and paradoxes about their relationships. |
| 15850 | Anything which has parts must be one thing, and parts are of a one, not of a many [Plato] |
| Full Idea: The whole of which the parts are parts must be one thing composed of many; for each of the parts must be part, not of a many, but of a whole. | |
| From: Plato (Parmenides [c.364 BCE], 157c) | |
| A reaction: This is a key move of metaphysics, and we should hang on to it. The other way madness lies. |
| 13259 | It seems that the One must be composed of parts, which contradicts its being one [Plato] |
| Full Idea: The One must be composed of parts, both being a whole and having parts. So on both grounds the One would thus be many and not one. But it must be not many, but one. So if the One will be one, it will neither be a whole, nor have parts. | |
| From: Plato (Parmenides [c.364 BCE], 137c09), quoted by Kathrin Koslicki - The Structure of Objects 5.2 | |
| A reaction: This is the starting point for Plato's metaphysical discussion of objects. It seems to begin a line of thought which is completed by Aristotle, surmising that only an essential structure can bestow identity on a bunch of parts. |
| 15847 | Two things relate either as same or different, or part of a whole, or the whole of the part [Plato] |
| Full Idea: Everything is surely related to everything as follows: either it is the same or different; or, if it is not the same or different, it would be related as part to whole or as whole to part. | |
| From: Plato (Parmenides [c.364 BCE], 146b) | |
| A reaction: This strikes me as a really helpful first step in trying to analyse the nature of identity. Two things are either two or (actually) one, or related mereologically. |
| 8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
| Full Idea: In the hierarchy of reduction, when we investigate questions in biology, we have to assume the laws of chemistry but not of economics. We could never find a law of biology that contradicted something in physics or in chemistry. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 3.1) | |
| A reaction: This spells out the idea that there is a direction of dependence between aspects of the world, though we should be cautious of talking about 'levels' (see Idea 7003). We cannot choose the direction in which reduction must go. |
| 8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
| Full Idea: The extensional presentation of a concept is just a list of the objects falling under the concept. In contrast, an intensional presentation of a concept gives a characterization of the concept, which allows us to pick out which objects fall under it. | |
| From: Michèle Friend (Introducing the Philosophy of Mathematics [2007], 3.4) | |
| A reaction: Logicians seem to favour the extensional view, because (in the standard view) sets are defined simply by their members, so concepts can be explained using sets. I take this to be a mistake. The intensional view seems obviously prior. |
| 6823 | Repay evil with good and your enemies will become friends (though this is hard) [Mohammed] |
| Full Idea: Requite evil with good, and he who is your enemy will become your dearest friend; but none will attain this save those who endure with fortitude and are greatly favoured by Allah. | |
| From: Mohammed (The Koran [c.622], Ch.41) | |
| A reaction: This seems opposed to some of the more vengeful remarks in the Koran. It strikes me as good common sense, since vengeance only seems to breed counter-vengeance. It doesn't carry the full altruistic commitment, though, of unrewarded love. |
| 6828 | You may break off a treaty if you fear treachery from your ally [Mohammed] |
| Full Idea: If you fear treachery from any of your allies, you may retaliate by breaking off your treaty with them; Allah does not love the treacherous. | |
| From: Mohammed (The Koran [c.622], Ch.8) | |
| A reaction: I do not think this is good advice. Everybody fears treachery, but if we all acted on that fear human relationships and society would immediately collapse. If anyone thought this was good advice, I would not want to make a treaty with them. |
| 6826 | Allah rewards those who are devout, sincere, patient, humble, charitable, chaste, and who fast [Mohammed] |
| Full Idea: Allah will bestow forgiveness and a rich reward on those, both men and women, who are devout, sincere, patient, humble, charitable and chaste; who fast and are ever mindful of Allah. | |
| From: Mohammed (The Koran [c.622], Ch.33) | |
| A reaction: Most people would still agree that all of these are virtues, though other lists will show interesting virtues that are not mentioned here, and many on this list seem overrated in the modern pantheon of virtues. |
| 6822 | Those who avenge themselves when wronged incur no guilt [Mohammed] |
| Full Idea: Those who avenge themselves when wronged incur no guilt. | |
| From: Mohammed (The Koran [c.622], Ch.42) | |
| A reaction: Compare Ideas 1659 (Protagoras), 346 (Socrates), 6288 (Jesus), and 4286 (Scruton!). In the light of those ideas, this comment in the Koran strikes me as coming from an older and less civilized world. |
| 6833 | Punish theft in men or women by cutting off their hands [Mohammed] |
| Full Idea: As for the man or woman who is guilty of theft, cut off their hands to punish them for their crimes. | |
| From: Mohammed (The Koran [c.622], Ch.5) | |
| A reaction: I find this shocking because it is irrevocable and offers no hope of redemption. It is particularly shocking that the text does not enjoin any caution about inflicting the punishment on the young, most of whom reform from thieving in later life. |
| 222 | Only a great person can understand the essence of things, and an even greater person can teach it [Plato] |
| Full Idea: Only a man of very great natural gifts will be able to understand that everything has a class and absolute essence, and an even more wonderful man can teach this. | |
| From: Plato (Parmenides [c.364 BCE], 135a) |
| 6825 | Do not kill except for a just cause [Mohammed] |
| Full Idea: Do not kill except for a just cause. | |
| From: Mohammed (The Koran [c.622], Ch.25) | |
| A reaction: Slippery slope! I can see that pleasure would not be a just cause, and ensuring the entry of all humanity to paradise might be one, but I find the area in between a little unclear. The Koran seems to allow you to decide for yourself. |
| 6832 | Killing a human, except as just punishment, is like killing all mankind [Mohammed] |
| Full Idea: We laid it down for the Israelites that whoever killed a human being, except as a punishment for murder or other wicked crimes, should be looked upon as though he had killed all mankind. | |
| From: Mohammed (The Koran [c.622], Ch.5) | |
| A reaction: It seems inconceivable that the Koran could be used to justify indiscriminate terrorism, in the light of remarks such as this. |
| 225 | The unlimited has no shape and is endless [Plato] |
| Full Idea: The unlimited partakes neither of the round nor of the straight, because it has no ends nor edges. | |
| From: Plato (Parmenides [c.364 BCE], 137e) |
| 233 | Some things do not partake of the One [Plato] |
| Full Idea: The others cannot partake of the one in any way; they can neither partake of it nor of the whole. | |
| From: Plato (Parmenides [c.364 BCE], 159d) | |
| A reaction: Compare Idea 231 |
| 2062 | The only movement possible for the One is in space or in alteration [Plato] |
| Full Idea: If the One moves it either moves spatially or it is altered, since these are the only motions. | |
| From: Plato (Parmenides [c.364 BCE], 138b) |
| 231 | Everything partakes of the One in some way [Plato] |
| Full Idea: The others are not altogether deprived of the one, for they partake of it in some way. | |
| From: Plato (Parmenides [c.364 BCE], 157c) | |
| A reaction: Compare Idea 233. |
| 6811 | Allah is lord of creation, compassionate, merciful, king of judgement-day [Mohammed] |
| Full Idea: Praise be to Allah, Lord of Creation, The Compassionate, the Merciful, King of Judgement-day! | |
| From: Mohammed (The Koran [c.622], Exord) | |
| A reaction: The Muslim concept of God confronts directly a clear theological difficulty, a difficulty faced by any judge: the conflict between mercy and justice. Christianity seems to emphasise mercy, and Islam emphasises justice. |
| 234 | We couldn't discuss the non-existence of the One without knowledge of it [Plato] |
| Full Idea: There must be knowledge of the one, or else not even the meaning of the words 'if the one does not exist' would be known. | |
| From: Plato (Parmenides [c.364 BCE], 160d) |
| 6819 | True believers see that Allah made the night for rest and the day to give light [Mohammed] |
| Full Idea: Do they not see how We have made the night for them to rest in and the day to give them light? Surely there are signs in this for true believers. | |
| From: Mohammed (The Koran [c.622], Ch.27) | |
| A reaction: The main traditional argument for God implied in the Koran is the design argument. It is clear from this that Islam will not be comfortable with Darwinian evolution, which implies we are 'designed' for the Earth, not the Earth for us. |
| 6818 | Allah cannot have begotten a son, as He is self-sufficient [Mohammed] |
| Full Idea: They say: 'Allah has begotten a son.' Allah forbid! Self-sufficient is He. | |
| From: Mohammed (The Koran [c.622], Ch.10) | |
| A reaction: This is quite persuasive, except that the point of Jesus is that he suffers a cruel death, and we are required to identify with God's parental feelings here, His involvement, which would not occur with the death of one of His prophets. |
| 6831 | He that kills a believer by design shall burn in Hell for ever [Mohammed] |
| Full Idea: He that kills a believer by design shall burn in Hell for ever. | |
| From: Mohammed (The Koran [c.622], Ch.4) | |
| A reaction: This would appear to make modern indiscriminate urban terrorism a damning sin for a Muslim. |
| 6829 | There shall be no compulsion in religion [Mohammed] |
| Full Idea: There shall be no compulsion in religion. | |
| From: Mohammed (The Koran [c.622], Ch.2) | |
| A reaction: This seems to contradict some of the more aggressive remarks in the Koran, such as Idea 6827. As I read it, the three non-compelling ideas that lead to true religion in the Koran are desire for paradise, fear of punishment, and worship of divine design. |
| 6834 | Unbelievers try to interpret the ambiguous parts of the Koran, simply to create dissension [Mohammed] |
| Full Idea: Some of the verses of the Koran are precise in meaning - they are the foundations of the Book - and others are ambiguous. Disbelievers follow the ambiguous part, to create dissension by seeking to explain it. No one knows its meaning except Allah. | |
| From: Mohammed (The Koran [c.622], Ch.3) | |
| A reaction: It is tempting to ask why some of the verses are ambiguous. The implication here is that they are a deliberate test for believers, like the apple in the garden of Eden. |
| 6817 | The Koran is certainly composed by Allah; no one could compose a chapter like it [Mohammed] |
| Full Idea: This Koran could not have been composed by any but Allah. It is beyond doubt from the Lord of the Creation. If they say: 'It is your own invention', say: 'Compose one chapter like it. Call on your false gods to help you!' | |
| From: Mohammed (The Koran [c.622], Ch.10) | |
| A reaction: I find this unpersuasive, firstly because I couldn't imitate the sonnets of Shakespeare either, and secondly because the authority of a text must be asserted outside the text, not within it. Scribble "this is a ten pound note" on a scrap of paper. |
| 6824 | Do not split into sects, exulting in separate beliefs [Mohammed] |
| Full Idea: Do not split up your religion into sects, each exulting in its own beliefs. | |
| From: Mohammed (The Koran [c.622], Ch.30) | |
| A reaction: This seems like good advice to a religion, but it is very difficult to retrace steps and reunite once it has happened. Which sect should make the greatest concessions? Must they both admit to being somewhat wrong? |
| 6821 | I created mankind that it might worship Me [Mohammed] |
| Full Idea: I created mankind and the jinn in order that they might worship Me. | |
| From: Mohammed (The Koran [c.622], Ch.51) | |
| A reaction: This seems to be a view common to all the monotheistic religions, with monasticism as its clearest (and most logical) outcome. Nietzsche is the most obvious opponent of this idea that the abasement of mankind is its highest ideal. |
| 6815 | Be patient with unbelievers, and leave them to the judgement of Allah [Mohammed] |
| Full Idea: Bear patiently with what the unbelievers say, and leave their company without recrimination; leave to Me those that deny the truth. | |
| From: Mohammed (The Koran [c.622], Ch.73) | |
| A reaction: This explicitly says Muslims should not attack infidels simply for their unbelief in Allah. |
| 6827 | Make war on the unbelievers until Allah's religion reigns supreme [Mohammed] |
| Full Idea: Make war on the unbelievers until idolatry is no more and Allah's religion reigns supreme. | |
| From: Mohammed (The Koran [c.622], Ch.8) | |
| A reaction: This should presumably be seen in context, as a war speech written during a conflict, like Churchill 'fighting them on the beaches', which does not apply to modern German tourists. However, one worries about how fundamentalists might read it. |
| 6820 | The righteous shall dwell on couches in gardens, wedded to dark-eyed houris [Mohammed] |
| Full Idea: In fair gardens the righteous shall dwell in bliss, rejoicing in what their Lord will give them. They shall recline on couches ranged in rows. To dark-eyed houris We shall wed them. | |
| From: Mohammed (The Koran [c.622], Ch.52) | |
| A reaction: What I find distressing about this is that we have gradually worked out how young men can recline on couches in gardens with dark-eyed houris before death, and the Koran seems to depict it as the highest form of living. |
| 6812 | Heaven will be reclining on couches, eating fruit, attended by virgins [Mohammed] |
| Full Idea: All who dwell in heaven shall recline on couches lined with thick brocade, and within their reach will hang the fruits of gardens; they shall dwell with bashful virgins whom neither men nor jinnee will have touched before. | |
| From: Mohammed (The Koran [c.622], Ch.55) | |
| A reaction: In the seventh century this was more impressive than it seems now. I still find it sad (though understandable) that paradise must always be depicted in terms of physical pleasure. Aristotle wouldn't have yearned for such an immortality. |
| 6830 | Unbelievers will have their skin repeatedly burned off in hell [Mohammed] |
| Full Idea: Those that deny Our revelations We will burn in Hell-fire. No sooner will their skins be consumed that We shall give them other skins, so that they may truly taste Our scourge. Allah is mighty and wise. | |
| From: Mohammed (The Koran [c.622], Ch.4) | |
| A reaction: Of all the accounts of hell in the Koran, this strikes me as the most alarming. I cannot think of a worse infliction, because here every nerve which can experience pain will suffer it (though the drinking of boiling water, Idea 6816, will make it worse). |
| 6816 | The unbelievers shall drink boiling water [Mohammed] |
| Full Idea: As for the unbelievers, they shall drink boiling water. | |
| From: Mohammed (The Koran [c.622], Ch.10) | |
| A reaction: This seems to be presented not only as a threat to unbelievers, but also as a satisfaction to believers. |