35 ideas
| 6259 | Why can't a wise man doubt everything? [Montaigne] |
| Full Idea: Why cannot a wise man dare to doubt anything and everything? | |
| From: Michel de Montaigne (Apology for Raymond Sebond [1580], p.0562) | |
| A reaction: This question seems to be the start of the Enlightenment Project, of attempting to prove everything. MacIntyre warns of the dangers of this in ethical theory. The story of modern philosophy is the discovery of its impossibility. E.g. Davidson on truth. |
| 6263 | No wisdom could make us comfortably walk a wide beam if it was high in the air [Montaigne] |
| Full Idea: Take a beam wide enough to walk along: suspend it between two towers: there is no philosophical wisdom, however firm, which could make us walk along it just as we would if we were on the ground. | |
| From: Michel de Montaigne (Apology for Raymond Sebond [1580], p.0672) | |
| A reaction: This proposes great scepticism about the practical application of philosophical wisdom, but if we talk in terms of the wise assessment of risk in any undertaking, our caution on the raised beam makes perfectly good sense. |
| 23122 | Montaigne was the founding father of liberalism [Montaigne, by Gopnik] |
| Full Idea: The first liberal, the founding father if we have one, is the great sixteenth century French essayist Michel de Montaigne. | |
| From: report of Michel de Montaigne (On Cruelty [1580]) by Adam Gopnik - A Thousand Small Sanities 1 | |
| A reaction: He says this not on the basis of his politicies or achievements, but his general attitudes and values. It may be another hundred years before we can identify another obvious liberal (Locke?). |
| 6258 | Virtue is the distinctive mark of truth, and its greatest product [Montaigne] |
| Full Idea: The distinctive mark of the Truth we hold ought to be virtue, which is the most exacting mark of Truth, the closest one to heaven and the most worthy thing that Truth produces. | |
| From: Michel de Montaigne (Apology for Raymond Sebond [1580], p.0493) | |
| A reaction: A long way from Tarski and minimalist theories of truth! But not so far from pragmatism. Personally I think Montaigne is making an important claim, which virtue theorists should be attempting to incorporate into their theory. Aristotle would sympathise. |
| 10859 | A set is 'well-ordered' if every subset has a first element [Clegg] |
| Full Idea: For a set to be 'well-ordered' it is required that every subset of the set has a first element. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10857 | Set theory made a closer study of infinity possible [Clegg] |
| Full Idea: Set theory made a closer study of infinity possible. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10864 | Any set can always generate a larger set - its powerset, of subsets [Clegg] |
| Full Idea: The idea of the 'power set' means that it is always possible to generate a bigger one using only the elements of that set, namely the set of all its subsets. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.14) |
| 10872 | Extensionality: Two sets are equal if and only if they have the same elements [Clegg] |
| Full Idea: Axiom of Extension: Two sets are equal if and only if they have the same elements. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10875 | Pairing: For any two sets there exists a set to which they both belong [Clegg] |
| Full Idea: Axiom of Pairing: For any two sets there exists a set to which they both belong. So you can make a set out of two other sets. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10876 | Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg] |
| Full Idea: Axiom of Unions: For every collection of sets there exists a set that contains all the elements that belong to at least one of the sets in the collection. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10878 | Infinity: There exists a set of the empty set and the successor of each element [Clegg] |
| Full Idea: Axiom of Infinity: There exists a set containing the empty set and the successor of each of its elements. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: This is rather different from the other axioms because it contains the notion of 'successor', though that can be generated by an ordering procedure. |
| 10877 | Powers: All the subsets of a given set form their own new powerset [Clegg] |
| Full Idea: Axiom of Powers: For each set there exists a collection of sets that contains amongst its elements all the subsets of the given set. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: Obviously this must include the whole of the base set (i.e. not just 'proper' subsets), otherwise the new set would just be a duplicate of the base set. |
| 10879 | Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg] |
| Full Idea: Axiom of Choice: For every set we can provide a mechanism for choosing one member of any non-empty subset of the set. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: This axiom is unusual because it makes the bold claim that such a 'mechanism' can always be found. Cohen showed that this axiom is separate. The tricky bit is choosing from an infinite subset. |
| 10871 | Axiom of Existence: there exists at least one set [Clegg] |
| Full Idea: Axiom of Existence: there exists at least one set. This may be the empty set, but you need to start with something. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10874 | Specification: a condition applied to a set will always produce a new set [Clegg] |
| Full Idea: Axiom of Specification: For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true. So the concept 'number is even' produces a set from the integers. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: What if the condition won't apply to the set? 'Number is even' presumably won't produce a set if it is applied to a set of non-numbers. |
| 10880 | Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg] |
| Full Idea: Three views of mathematics: 'pure' mathematics, where it doesn't matter if it could ever have any application; 'real' mathematics, where every concept must be physically grounded; and 'applied' mathematics, using the non-real if the results are real. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.17) | |
| A reaction: Very helpful. No one can deny the activities of 'pure' mathematics, but I think it is undeniable that the origins of the subject are 'real' (rather than platonic). We do economics by pretending there are concepts like the 'average family'. |
| 10860 | An ordinal number is defined by the set that comes before it [Clegg] |
| Full Idea: You can think of an ordinal number as being defined by the set that comes before it, so, in the non-negative integers, ordinal 5 is defined as {0, 1, 2, 3, 4}. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10861 | Beyond infinity cardinals and ordinals can come apart [Clegg] |
| Full Idea: With ordinary finite numbers ordinals and cardinals are in effect the same, but beyond infinity it is possible for two sets to have the same cardinality but different ordinals. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10854 | Transcendental numbers can't be fitted to finite equations [Clegg] |
| Full Idea: The 'transcendental numbers' are those irrationals that can't be fitted to a suitable finite equation, of which π is far and away the best known. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |
| 10858 | By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg] |
| Full Idea: The realisation that brought 'i' into the toolkit of physicists and engineers was that you could extend the 'number line' into a new dimension, with an imaginary number axis at right angles to it. ...We now have a 'number plane'. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.12) |
| 10853 | Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg] |
| Full Idea: It is a chicken-and-egg problem, whether the lack of zero forced forced classical mathematicians to rely mostly on a geometric approach to mathematics, or the geometric approach made 0 a meaningless concept, but the two remain strongly tied together. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |
| 10866 | Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg] |
| Full Idea: As far as Kronecker was concerned, Cantor had built a whole structure on the irrational numbers, and so that structure had no foundation at all. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10869 | The Continuum Hypothesis is independent of the axioms of set theory [Clegg] |
| Full Idea: Paul Cohen showed that the Continuum Hypothesis is independent of the axioms of set theory. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10862 | The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg] |
| Full Idea: The 'continuum hypothesis' says that aleph-one is the cardinality of the rational and irrational numbers. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.14) |
| 6262 | We lack some sense or other, and hence objects may have hidden features [Montaigne] |
| Full Idea: We may all lack some sense or other; because of that defect, most of the features of objects may be concealed from us. | |
| From: Michel de Montaigne (Apology for Raymond Sebond [1580], p.0666) | |
| A reaction: This strikes me as simple, straightforward common sense, and right. I cannot make sense of the claim that reality really is just the way it appears. We do not have a built-in neutrino detector, for example. |
| 6260 | Sceptics say there is truth, but no means of making or testing lasting judgements [Montaigne] |
| Full Idea: Pyrrhonians say that truth and falsehood exist; within us we have means of looking for them, but not of making any lasting judgements: we have no touchstone. | |
| From: Michel de Montaigne (Apology for Raymond Sebond [1580], p.0564) | |
| A reaction: This states the key difference between sceptics and relativists. The latter are more extreme as they say there is no such thing as truth. The former concede truth, and their scepticism is about the abilities of human beings. I am an anti-relativist. |
| 6261 | The soul is in the brain, as shown by head injuries [Montaigne] |
| Full Idea: The seat of the powers of the soul is in the brain, as is clearly shown by the fact that wounds and accidents affecting the head immediately harm the faculties of the soul. | |
| From: Michel de Montaigne (Apology for Raymond Sebond [1580], p.0614) | |
| A reaction: At last someone has finally got the facts clear. It seems surprising that the Greeks never clearly grasped this piece of irrefutable evidence - even those Greeks who speculated that the brain was the key. Here we have a fixed fact of philosophy of mind. |
| 7496 | Rules and duties are based on the will, as that is all we control [Montaigne] |
| Full Idea: Since actions and performances are not wholly in our power and since nothing is really in our power but our will - it is on the will that all the rules and duties of Man are based and established. | |
| From: Michel de Montaigne (I.7 Our deeds are judged by intention [1580], p.0028) | |
| A reaction: This is almost Kant's claim that the only truly good thing is a good will (e.g. Idea 3711). Aristotle disagrees, because a virtuous person should also have good desires. We may will to have good desires, but virtue requires actually having them. |
| 7495 | Apart from the fear, dying is an easy duty [Montaigne] |
| Full Idea: If our fears did not lend it weight, dying would be one of our lighter duties. | |
| From: Michel de Montaigne (III.12 On physiognomy [1580], p.1191) | |
| A reaction: An Epicurean thought. 'Duties' is nice - presumably death qualifies as a duty, because Nature requires it of us (we each of us 'owe nature a death'). The remark appears to me to be true. |
| 22269 | We must fight fiercely to hang on to the few pleasures which survive into old age [Montaigne] |
| Full Idea: I am training and sharpening my appetite for those pleasures that are left. ...We must cling tooth and claw to the use of the pleasures of this life which the advancing years, one after another, rip from our grasp. | |
| From: Michel de Montaigne (I.39 On Solitude [1580], p.276) | |
| A reaction: That may be one of the most inspiring ideas I have read about pleasure. |
| 20482 | Virtue inspires Stoics, but I want a good temperament [Montaigne] |
| Full Idea: What Stoics did from virtue I teach myself to do from temperament. | |
| From: Michel de Montaigne (III.10 On Restraining your Will [1580], p.1153) | |
| A reaction: I take this to be an Aristotelian criticism of Stoicism. They venerate virtue above everything, but Aristotle says you must integrate virtue into your very being, so that right actions flow from you, with very little need for premeditation. |
| 20480 | There is not much point in only becoming good near the end of your life [Montaigne] |
| Full Idea: It is almost better never to become a good man at all than to do so tardily, understanding how to live when you have no life left. | |
| From: Michel de Montaigne (III.10 On Restraining your Will [1580], p.1142) | |
| A reaction: A very nice perspective, which I don't recall Aristotle mentioning. It does, though, reinforce Aristotle's belief that early training is essential. |
| 20481 | Nothing we say can be worse than unsaying it in the face of authority [Montaigne] |
| Full Idea: Nothing which a gentleman says can seem worse than the shame of his unsaying it under duress from authority. | |
| From: Michel de Montaigne (III.10 On Restraining your Will [1580], p.1153) | |
| A reaction: The point is that you have to fight every day for free speech, because no matter what the law says, there are always people in power who want to shut you up. |
| 20479 | People at home care far more than soldiers risking death about the outcome of wars [Montaigne] |
| Full Idea: How many soldiers put themselves at risk every day in wars which they care little about, rushing into danger in battles the loss of which will not make them lose a night's sleep. Meanwhile a man at home is more passionate about the war than the soldier. | |
| From: Michel de Montaigne (III.10 On Restraining your Will [1580], p.1139) | |
| A reaction: It depends whether you are a mercenary (which the majority probably were in 1680), and what are the implications of defeat. |
| 1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |
| Full Idea: The Egyptians were the first to claim that the soul of a human being is immortal, and that each time the body dies the soul enters another creature just as it is being born. | |
| From: Herodotus (The Histories [c.435 BCE], 2.123.2) |