95 ideas
| 13773 | For the truth you need Prodicus's fifty-drachma course, not his one-drachma course [Socrates] |
| Full Idea: Socrates: If I'd attended Prodicus's fifty-drachma course, I could tell you the truth about names straightway, but as I've only heard the one-drachma course, I don't know the truth about it. | |
| From: Socrates (reports of career [c.420 BCE]), quoted by Plato - Cratylus 384b |
| 343 | The unexamined life is not worth living for men [Socrates] |
| Full Idea: The unexamined life is not worth living for men. | |
| From: Socrates (reports of last days [c.399 BCE]), quoted by Plato - The Apology 38a | |
| A reaction: I wonder why? I can see Nietzsche offering aristocratic heroes and dancers as counterexamples. Compare Idea 3798. |
| 7421 | A philosopher is one who cares about what other people care about [Socrates, by Foucault] |
| Full Idea: Socrates asks people 'Are you caring for yourself?' He is the man who cares about the care of others; this is the particular position of the philosopher. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Michel Foucault - Ethics of the Concern for Self as Freedom p.287 | |
| A reaction: Priests, politicians and psychiatrists also care quite intensely about the concerns of other people. Someone who was intensely self-absorbed with the critical task of getting their own beliefs right would count for me as a philosopher. |
| 1649 | Socrates opened philosophy to all, but Plato confined moral enquiry to a tiny elite [Vlastos on Socrates] |
| Full Idea: To confine, as Plato does in 'Republic' IV-VII, moral inquiry to a tiny elite, is to obliterate the Socratic vision which opens up the philosophic life to all. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.18 | |
| A reaction: This doesn't mean that Plato is necessarily 'elitist'. It isn't elitist to point out that an activity is very difficult. |
| 5842 | Philosophical discussion involves dividing subject-matter into categories [Socrates, by Xenophon] |
| Full Idea: Self-discipline and avoidance of pleasure makes people most capable of philosophical discussion, which is called 'discussion' (dialegesthai - sort out) because people divide their subject-matter into categories. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 4.5.12 | |
| A reaction: This could be the original slogan for analytical philosophy, as far as I am concerned. I don't think philosophy aims at complete and successful analysis (cf. Idea 2958), but at revealing the structure and interconnection of ideas. This is wisdom. |
| 648 | Socrates began the quest for something universal with his definitions, but he didn't make them separate [Socrates, by Aristotle] |
| Full Idea: Socrates began the quest for something universal in addition to the radical flux of perceptible particulars, with his definitions. But he rightly understood that universals cannot be separated from particulars. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Aristotle - Metaphysics 1086b |
| 164 | It is legitimate to play the devil's advocate [Socrates] |
| Full Idea: It is legitimate to play the devil's advocate. | |
| From: Socrates (reports of career [c.420 BCE]), quoted by Plato - Phaedrus 272c |
| 1647 | In Socratic dialogue you must say what you believe, so unasserted premises are not debated [Vlastos on Socrates] |
| Full Idea: Socrates' rule of "say only what you believe"….excluded debate on unasserted premises, thereby distinguishing Socratic from Zenonian and earlier dialectics. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.14 |
| 115 | Socrates was pleased if his mistakes were proved wrong [Socrates] |
| Full Idea: Socrates: I'm happy to have a mistaken idea of mine proved wrong. | |
| From: Socrates (reports of career [c.420 BCE]), quoted by Plato - Gorgias 458a |
| 22099 | The method of Socrates shows the student is discovering the truth within himself [Socrates, by Carlisle] |
| Full Idea: Socrates tended to prefer the method of questioning, for this made it clear that the student was discovering the truth within himself. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Clare Carlisle - Kierkegaard: a guide for the perplexed 7 | |
| A reaction: Sounds like it will only facilitate conceptual analysis, and excludes empirical knowledge. Can you say to Socrates 'I'll just google that'? |
| 5844 | Socrates always proceeded in argument by general agreement at each stage [Socrates, by Xenophon] |
| Full Idea: When Socrates was setting out a detailed argument, he used to proceed by such stages as were generally agreed, because he thought that this was the infallible method of argument. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 4.6.16 | |
| A reaction: This sounds right, and shows how strongly Socrates perceived philosophy to be a group activity, of which I approve. It seems to me that philosophy is clearly a spoken subject before it is a written one. The lonely speculator comes much later. |
| 11389 | Socrates sought essences, which are the basis of formal logic [Socrates, by Aristotle] |
| Full Idea: It is not surprising that Socrates sought essences. His project was to establish formal reasoning, of whose syllogisms essences are the foundations. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Aristotle - Metaphysics 1078b22 | |
| A reaction: This seems to reinforce the definitional view of essences, since definitions seem to be at the centre of most of Socrates's quests. |
| 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. |
| 639 | Socrates developed definitions as the basis of syllogisms, and also inductive arguments [Socrates, by Aristotle] |
| Full Idea: Socrates aimed to establish formal logic, of whose syllogisms essences are the foundations. He developed inductive arguments and also general definitions. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Aristotle - Metaphysics 1078b |
| 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. |
| 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. |
| 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. |
| 1652 | Socrates did not consider universals or definitions as having separate existence, but Plato made Forms of them [Socrates, by Aristotle] |
| Full Idea: Socrates did not regard the universals or the objects of definitions as separate existents, while Plato did separate them, and called this sort of entity ideas/forms. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Aristotle - Metaphysics 1078b30 |
| 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). |
| 1650 | For Socrates our soul, though hard to define, is our self [Vlastos on Socrates] |
| Full Idea: For Socrates our soul is our self - whatever that might turn out to be. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.55 | |
| A reaction: The problem with any broad claim like this is that we seem to be able to distinguish between essential and non-essential aspects of the self or of the soul. |
| 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. |
| 23252 | Socrates first proposed that we are run by mind or reason [Socrates, by Frede,M] |
| Full Idea: It would seem that historically the decisive step was taken by Socrates in conceiving of human beings as being run by a mind or reason.. …He postulated an entity whose precision nature and function then was a matter of considerable debate. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Michael Frede - Intro to 'Rationality in Greek Thought' p.19 | |
| A reaction: This is, for me, a rather revelatory idea. I am keen on the fact the animals make judgements which are true and false, and also that we exhibit rationality when walking across uneven ground. So pure rationality is a cultural construct! |
| 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. |
| 199 | The common belief is that people can know the best without acting on it [Socrates] |
| Full Idea: Most people think there are many who recognise the best but are unwilling to act on it. | |
| From: Socrates (reports of career [c.420 BCE]), quoted by Plato - Protagoras 352d |
| 195 | No one willingly commits an evil or base act [Socrates] |
| Full Idea: I am fairly certain that no wise man believes anyone sins willingly or willingly perpetrates any evil or base act. | |
| From: Socrates (reports of career [c.420 BCE]), quoted by Plato - Protagoras 345e |
| 1653 | Socrates did not accept the tripartite soul (which permits akrasia) [Vlastos on Socrates] |
| Full Idea: Xenophon indirectly indicates that he does not associate Socrates in any way with the tripartite psychology of the 'Republic', for within that theory akrasia would be all too possible. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.102 |
| 5843 | People do what they think they should do, and only ever do what they think they should do [Socrates, by Xenophon] |
| Full Idea: There is no one who knows what they ought to do, but thinks that they ought not to do it, and no one does anything other than what they think they ought to do. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 4.6.6 | |
| A reaction: This is Socrates' well-known rejection of the possibility of weakness of will (akrasia - lit. 'lack of control'). Aristotle disagreed, and so does almost everyone else. Modern smokers seem to exhibit akrasia. I have some sympathy with Socrates. |
| 5253 | Socrates was shocked by the idea of akrasia, but observation shows that it happens [Aristotle on Socrates] |
| Full Idea: Socrates thought it a shocking idea that when a man actually has knowledge in him something else should overmaster it, ..but this is glaringly inconsistent with the observed facts. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Aristotle - Nicomachean Ethics 1145b24 | |
| A reaction: Aristotle seems very confident, but it is not at all clear (even to the agent) what is going on when apparent weakness of will occurs (e.g. breaking a diet). What exactly does the agent believe at the moment of weakness? |
| 5839 | For Socrates, wisdom and prudence were the same thing [Socrates, by Xenophon] |
| Full Idea: Socrates did not distinguish wisdom from prudence, but judged that the man who recognises and puts into practice what is truly good, and the man who knows and guards against what is disgraceful, are both wise and prudent. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 3.9.3 | |
| A reaction: Compare Aristotle, who separates them, claiming that prudence is essential for moral virtue, but wisdom is pursued at a different level, closer to the gods than to society. |
| 5867 | For Socrates, virtues are forms of knowledge, so knowing justice produces justice [Socrates, by Aristotle] |
| Full Idea: Socrates thought that the virtues were all forms of knowledge, and therefore once a man knew justice, he would be a just man. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Aristotle - Eudemian Ethics 1216b07 | |
| A reaction: The clearest possible statement of Socrates' intellectualism. Aristotle rejected the Socrates view, but I find it sympathetic. Smokers who don't want to die seem to be in denial. To see the victims is to condemn the crime. |
| 5069 | Socrates was the first to base ethics upon reason, and use reason to explain it [Taylor,R on Socrates] |
| Full Idea: Socrates was the first significant thinker to try basing ethics upon reason, and to try uncovering its natural principles solely by the use of reason. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Richard Taylor - Virtue Ethics: an Introduction Ch.7 | |
| A reaction: Interesting. It seems to me that Socrates overemphasised reason, presumably because it was a novelty. Hence his view that akrasia is impossible, and that virtue is simply knowledge. Maybe action is not just rational, but moral action is. |
| 5836 | All human virtues are increased by study and practice [Socrates, by Xenophon] |
| Full Idea: If you consider the virtues that are recognised among human beings, you will find that they are all increased by study and practice. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 2.6.41 | |
| A reaction: 'Study' is the intellectualist part of this remark; the reference to 'practice' fits with Aristotle view that virtue is largely a matter of good habits. The next question would be how theoretical the studies should be. Philosophy, or newspapers? |
| 5840 | The wise perform good actions, and people fail to be good without wisdom [Socrates, by Xenophon] |
| Full Idea: It is the wise who perform truly good actions, and those who are not wise cannot, and, if they try to, fail. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 3.9.6 | |
| A reaction: The essence of Socrates' intellectualism, with which Aristotle firmly disagreed (when he assert that only practical reason was needed for virtuous actions, rather than wisdom or theory). Personally I side more with Socrates than with Aristotle on this. |
| 185 | Socrates despised good looks [Socrates, by Plato] |
| Full Idea: Socrates despises good looks to an almost inconceivable extent. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Plato - The Symposium 216e |
| 5070 | Socrates conservatively assumed that Athenian conventions were natural and true [Taylor,R on Socrates] |
| Full Idea: Socrates' moral philosophy was essentially conservative. He assumed that the principles the Athenians honoured were true and natural, so there was little possibility of conflict between nature and convention in his thinking. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Richard Taylor - Virtue Ethics: an Introduction Ch.8 | |
| A reaction: Taylor contrasts Socrates with Callicles, who claims that conventions oppose nature. This fits with Nietzsche's discontent with Socrates, as the person who endorses conventional good and evil, thus constraining the possibilities of human nature. |
| 5838 | A well-made dung basket is fine, and a badly-made gold shield is base, because of function [Socrates, by Xenophon] |
| Full Idea: A dung-basket is fine, and a golden shield contemptible, if the one is finely and the other badly constructed for carrying out its function. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 3.8.6 | |
| A reaction: This is the basis of a key idea in Aristotle, that virtue (or excellence) arises directly from function. I think it is the most important idea in virtue theory, and seems to have struck most Greeks as being self-evident. |
| 344 | If death is like a night of dreamless sleep, such nights are very pleasant [Socrates] |
| Full Idea: If death is like a night of dreamless sleep it is an advantage, for such nights are very pleasant, and eternity would seem like a single night. | |
| From: Socrates (reports of last days [c.399 BCE]), quoted by Plato - The Apology 40d | |
| A reaction: Dreamless sleep is only pleasant if being awake is unpleasant. Very quiet days are only pleasant if the active days are horrible. A desire for a totally quiet life is absurd. |
| 339 | Men fear death as a great evil when it may be a great blessing [Socrates] |
| Full Idea: No one knows whether death may not be the greatest of all blessings for a man, yet men fear it as if they knew that it is the greatest of evils. | |
| From: Socrates (reports of last days [c.399 BCE]), quoted by Plato - The Apology 29a | |
| A reaction: As a neutral observer, I see little sign of it being a blessing, except as a relief from misery. It seem wrong to view such a natural thing as evil, but it is the thing most of us least desire. |
| 5837 | Things are both good and fine by the same standard [Socrates, by Xenophon] |
| Full Idea: Things are always both good and fine by the same standard. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 3.8.5 | |
| A reaction: This begs many questions, but perhaps it leads to what we call intuitionism, which is an instant ability is perceive a fine action (even in an enemy). This leads to the rather decadent view that the aim of life is the production of beauty. |
| 3017 | The only good is knowledge, and the only evil is ignorance [Socrates, by Diog. Laertius] |
| Full Idea: There is only one good, namely knowledge, and there is only one evil, namely ignorance. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.4.14 | |
| A reaction: Ignorance of how to commit evil sounds quite good. |
| 1646 | Socrates was the first to put 'eudaimonia' at the centre of ethics [Socrates, by Vlastos] |
| Full Idea: Socrates' true place in the development of Greek thought is that he is the first to establish the eudaimonist foundation of ethical theory, which became the foundation of the schools which sprang up around him. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.10 | |
| A reaction: I suspect that he was the first to fully articulate a widely held Greek belief. The only ethical question that they asked was about the nature of a good human life. |
| 2 | We should not even harm someone who harms us [Socrates] |
| Full Idea: One should never return an injustice nor harm another human being no matter what one suffers at their hands. | |
| From: Socrates (reports of last days [c.399 BCE]), quoted by Plato - Crito 49c | |
| A reaction: Jesus of Nazareth was not the first person to make this suggestion. |
| 1663 | By 'areté' Socrates means just what we mean by moral virtue [Vlastos on Socrates] |
| Full Idea: Socrates uses the word 'areté' to mean precisely what we mean by moral virtue. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.200 |
| 345 | A good man cannot be harmed, either in life or in death [Socrates] |
| Full Idea: A good man cannot be harmed, either in life or in death. | |
| From: Socrates (reports of last days [c.399 BCE]), quoted by Plato - The Apology 41d |
| 4323 | Socrates is torn between intellectual virtue, which is united and teachable, and natural virtue, which isn't [PG on Socrates] |
| Full Idea: Socrates worries about the unity and teachability of virtue because he is torn between virtue as intellectual (unified and teachable) and virtue as natural (plural and unteachable). | |
| From: comment on Socrates (reports of career [c.420 BCE]) by PG - Db (ideas) | |
| A reaction: Admittedly virtue could be natural but still unified and teachable, but Socrates clearly had a dilemma, and this seems to make sense of it. |
| 8003 | Socrates agrees that virtue is teachable, but then denies that there are teachers [Socrates, by MacIntyre] |
| Full Idea: Socrates' great point of agreement with the sophists is his acceptance of the thesis that areté is teachable. But paradoxically he denies that there are teachers. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Alasdair MacIntyre - A Short History of Ethics Ch.3 | |
| A reaction: This is part of Socrates's presentation of himself as 'not worthy'. Virtue would be teachable, if only anyone knew what it was. He's wrong. Lots of people have a pretty good idea of virtue, and could teach it. The problem is in the pupils. |
| 126 | We should ask what sort of people we want to be [Socrates] |
| Full Idea: Socrates: What sort of person should one be? | |
| From: Socrates (reports of career [c.420 BCE]), quoted by Plato - Gorgias 487e |
| 4111 | Socrates believed that basically there is only one virtue, the power of right judgement [Socrates, by Williams,B] |
| Full Idea: Socrates believed that basically there is only one virtue, the power of right judgement. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Bernard Williams - Ethics and the Limits of Philosophy Ch.1 | |
| A reaction: Which links with Aristotle's high place for 'phronesis' (prudence?). The essence of Socrates' intellectualism. Robots and saints make very different judgements, though. |
| 7808 | Socrates made the civic values of justice and friendship paramount [Socrates, by Grayling] |
| Full Idea: In Socrates' thought, the expressly civic values of justice and friendship became paramount. | |
| From: report of Socrates (reports of career [c.420 BCE]) by A.C. Grayling - What is Good? Ch.2 | |
| A reaction: This is the key move in ancient ethics, away from heroism, and towards the standard Aristotelian social virtues. I say this is the essence of what we call morality, and the only one which can be given a decent foundational justification (social health). |
| 346 | One ought not to return a wrong or injury to any person, whatever the provocation [Socrates] |
| Full Idea: One ought not to return a wrong or an injury to any person, whatever the provocation is. | |
| From: Socrates (reports of last days [c.399 BCE]), quoted by Plato - Crito 49b | |
| A reaction: The same as the essential moral teachings of Jesus (see Idea 6288) and Lao Tzu (Idea 6324). The big target is not to be corrupted by the evil of other people. |
| 23907 | Courage is scientific knowledge [Socrates, by Aristotle] |
| Full Idea: Socrates thought that courage is scientific knowledge. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Aristotle - Eudemian Ethics 1230a06 | |
| A reaction: Aristotle himself says that reason produces courage, but he also says it arises from natural youthful spirits. I favour the view that there is a strong rational component in true courage. |
| 341 | Wealth is good if it is accompanied by virtue [Socrates] |
| Full Idea: Wealth does not bring about excellence, but excellence makes wealth and everything else good for men. | |
| From: Socrates (reports of last days [c.399 BCE]), quoted by Plato - The Apology 30b |
| 7585 | Socrates emphasises that the knower is an existing individual, with existence his main task [Socrates, by Kierkegaard] |
| Full Idea: The infinite merit of the Socratic position was precisely to accentuate the fact that the knower is an existing individual, and that the task of existing is his essential task. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Søren Kierkegaard - Concluding Unscientific Postscript 'Inwardness' | |
| A reaction: Always claim Socrates as the first spokesman for your movement! It is true that Socrates is always demanding the views of his interlocutors, and not just abstract theories. See Idea 1647. |
| 5841 | Obedience to the law gives the best life, and success in war [Socrates, by Xenophon] |
| Full Idea: A city in which the people are most obedient to the laws has the best life in time of peace and is irresistible in war. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Memorabilia of Socrates 4.4.15 | |
| A reaction: This is a conservative view, with the obvious problem case of bad laws, but in general it seems to me clearly right. This is why it is so vital that nothing should be done to bring the law into disrepute, such as petty legislation or prosecution. |
| 347 | Will I stand up against the law, simply because I have been unjustly judged? [Socrates] |
| Full Idea: Do I intend to destroy the laws, because the state wronged me by passing a faulty judgement at my trial? | |
| From: Socrates (reports of last days [c.399 BCE]), quoted by Plato - Crito 50c |
| 1661 | Socrates was the first to grasp that a cruelty is not justified by another cruelty [Vlastos on Socrates] |
| Full Idea: Socrates was the first Greek to grasp the truth that if someone has done a nasty thing to me, this does not give the slightest moral justification for doing anything nasty to him. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.190 |
| 5846 | A lover using force is a villain, but a seducer is much worse, because he corrupts character [Socrates, by Xenophon] |
| Full Idea: The fact that a lover uses not force but persuasion makes him more detestable, because a lover who uses force proves himself a villain, but one who uses persuasion ruins the character of the one who consents. | |
| From: report of Socrates (reports of career [c.420 BCE]) by Xenophon - Symposium 8.20 | |
| A reaction: A footnote says that this distinction was enshrined in Athenian law, where seduction was worse than rape. This is a startling and interest contrast to the modern view, which enshrines rights and freedoms, and says seduction is usually no crime at all. |
| 6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
| Full Idea: Hermarchus said that animal killing is justified by considerations of human safety and nourishment and by animals' inability to form contractual relations of justice with us. | |
| From: report of Hermarchus (fragments/reports [c.270 BCE]) by David A. Sedley - Hermarchus | |
| A reaction: Could the last argument be used to justify torturing animals? Or could we eat a human who was too brain-damaged to form contracts? |
| 1657 | Socrates holds that right reason entails virtue, and this must also apply to the gods [Vlastos on Socrates] |
| Full Idea: It is essential to Socrates' rationalist programme in theology to assume that the entailment of virtue by wisdom binds gods no less than men. He would not tolerate one moral standard for me and another for gods. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.164 |
| 1662 | A new concept of God as unswerving goodness emerges from Socrates' commitment to virtue [Vlastos on Socrates] |
| Full Idea: Undeviating beneficent goodness guides Socrates' thought so deeply that he applies it even to the deity; he projects a new concept of god as a being that can cause only good, never evil. | |
| From: comment on Socrates (reports of career [c.420 BCE]) by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.197 |
| 338 | Socrates is accused of denying the gods, saying sun is stone and moon is earth [Socrates, by Plato] |
| Full Idea: Socrates denies the gods, because he says the sun is stone and the moon is earth. | |
| From: report of Socrates (reports of last days [c.399 BCE]) by Plato - The Apology 26d |