62 ideas
7490 | Because of Darwin, wisdom as a definite attainable state has faded [Watson] |
Full Idea: As well as killing the need for God, Darwin's legacy transformed the idea of wisdom, as some definite attainable state, however far off. | |
From: Peter Watson (Ideas [2005], Ch.31) | |
A reaction: Where does this leave philosophy, if it is still (as I like to think) the love of wisdom? The best we can hope for is wisdom as a special sort of journey - touring, rather than arriving. |
7461 | The three key ideas are the soul, Europe, and the experiment [Watson] |
Full Idea: The three key ideas that I have settled on in the history of ideas are: the soul, Europe, and the experiment. | |
From: Peter Watson (Ideas [2005], Intro) | |
A reaction: The soul is a nice choice (rather than God). 'Europe' seems rather vast and indeterminate to count as a key idea. |
7464 | The big idea: imitation, the soul, experiments, God, heliocentric universe, evolution? [Watson] |
Full Idea: Candidates for the most important idea in human history are: mimetic thinking (imitation), the soul, the experiment, the One True God, the heliocentric universe, and evolution. | |
From: Peter Watson (Ideas [2005], Ch.03) | |
A reaction: From this list I would choose the heliocentric universe, because it so dramatically downgraded the importance of our species (effectively we went from everything to nothing). We still haven't recovered from the shock. |
7465 | Babylonian thinking used analogy, rather than deduction or induction [Watson] |
Full Idea: In Babylon thought seems to have worked mainly by analogy, rather than by the deductive or inductive processes we use in the modern world. | |
From: Peter Watson (Ideas [2005], Ch.04) | |
A reaction: Analogy seems to be closely related to induction, if it is comparing instances of something. Given their developments in maths and astronomy, they can't have been complete strangers to the 'modern' way of thought. |
15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
Full Idea: Second-order set theory is just like first-order set-theory, except that we use the version of Replacement with a universal second-order quantifier over functions from set to sets. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VII.4) |
15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
Full Idea: A member m of M is an 'upper bound' of a subset N of M if m is not less than any member of N. A member m of M is a 'least upper bound' of N if m is an upper bound of N such that if l is any other upper bound of N, then m is less than l. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
A reaction: [if you don't follow that, you'll have to keep rereading it till you do] |
15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
Full Idea: Since combinatorial collections are enumerated, some multiplicities may be too large to be gathered into combinatorial collections. But the size of a multiplicity seems quite irrelevant to whether it forms a logical connection. | |
From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
Full Idea: Many of those who are skeptical about the existence of infinite combinatorial collections would want to doubt or deny the Axiom of Choice. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VI.2) |
15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
Full Idea: The Power Set is just he codification of the fact that the collection of functions from a mathematical collection to a mathematical collection is itself a mathematical collection that can serve as a domain of mathematical study. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) |
15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
Full Idea: The Axiom of Replacement (of Skolem and Fraenkel) was remarkable for its universal acceptance, though it seemed to have no consequences except for the properties of the higher reaches of the Cantorian infinite. | |
From: Shaughan Lavine (Understanding the Infinite [1994], I) |
15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
Full Idea: The Axiom of Foundation (Zermelo 1930) says 'Every (descending) chain in which each element is a member of the previous one is of finite length'. ..This forbids circles of membership, or ungrounded sets. ..The iterative conception gives this centre stage. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.4) |
15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
Full Idea: The controversy was not about Choice per se, but about the correct notion of function - between advocates of taking mathematics to be about arbitrary functions and advocates of taking it to be about functions given by rules. | |
From: Shaughan Lavine (Understanding the Infinite [1994], I) |
15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
Full Idea: Combinatorial collections (defined just by the members) obviously obey the Axiom of Choice, while it is at best dubious whether logical connections (defined by a rule) do. | |
From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
Full Idea: The Peano-Russell notion of class is the 'logical' notion, where each collection is associated with some kind of definition or rule that characterises the members of the collection. | |
From: Shaughan Lavine (Understanding the Infinite [1994], IV.1) |
15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
Full Idea: The iterative conception of set was not so much as suggested, let alone advocated by anyone, until 1947. | |
From: Shaughan Lavine (Understanding the Infinite [1994], I) |
15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
Full Idea: The iterative conception of sets does not tell us how far to iterate, and so we must start with an Axiom of Infinity. It also presupposes the notion of 'transfinite iteration'. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.5) |
15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
Full Idea: The iterative conception does not provide a conception that unifies the axioms of set theory, ...and it has had very little impact on what theorems can be proved. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.5) | |
A reaction: He says he would like to reject the iterative conception, but it may turn out that Foundation enables new proofs in mathematics (though it hasn't so far). |
15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
Full Idea: Limitation of Size has it that if a collection is the same size as a set, then it is a set. The Axiom of Replacement is characteristic of limitation of size. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.5) |
15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
Full Idea: A collection M is 'well-ordered' by a relation < if < linearly orders M with a least element, and every subset of M that has an upper bound not in it has an immediate successor. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.4) |
15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
Full Idea: The distinctive feature of second-order logic is that it presupposes that, given a domain, there is a fact of the matter about what the relations on it are, so that the range of the second-order quantifiers is fixed as soon as the domain is fixed. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.3) | |
A reaction: This sounds like a rather large assumption, which is open to challenge. I am not sure whether it was the basis of Quine's challenge to second-order logic. He seems to have disliked its vagueness, because it didn't stick with 'objects'. |
15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
Full Idea: The Law of Excluded Middle is (part of) the foundation of the mathematical practice of employing proofs by contradiction. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) | |
A reaction: This applies in a lot of logic, as well as in mathematics. Come to think of it, it applies in Sudoku. |
15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
Full Idea: Mathematics is today thought of as the study of abstract structure, not the study of quantity. That point of view arose directly out of the development of the set-theoretic notion of abstract structure. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.2) | |
A reaction: It sounds as if Structuralism, which is a controversial view in philosophy, is a fait accompli among mathematicians. |
15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
Full Idea: One reason to introduce the rational numbers is that it simplifes the theory of division, since every rational number is divisible by every nonzero rational number, while the analogous statement is false for the natural numbers. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VI.3) | |
A reaction: That is, with rations every division operation has an answer. |
15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
Full Idea: The chief importance of the Continuum Hypothesis for Cantor (I believe) was that it would show that the real numbers form a set, and hence that they were encompassed by his theory. | |
From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
Full Idea: The Cauchy convergence criterion for a sequence: the sequence S0,S1,... has a limit if |S(n+r) - S(n)| is less than any given quantity for every value of r and sufficiently large values of n. He proved this necessary, but not sufficient. | |
From: Shaughan Lavine (Understanding the Infinite [1994], 2.5) |
15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
Full Idea: Roughly speaking, the upper and lower parts of the Dedekind cut correspond to the commensurable ratios greater than and less than a given incommensurable ratio. | |
From: Shaughan Lavine (Understanding the Infinite [1994], II.6) | |
A reaction: Thus there is the problem of whether the contents of the gap are one unique thing, or many. |
7466 | Mesopotamian numbers applied to specific things, and then became abstract [Watson] |
Full Idea: To begin with, in Mesopotamia, counting systems applied to specific commodities (so the symbol for 'three sheep' applied only to sheep, and 'three cows' applied only to cows), but later words for abstract qualities emerged. | |
From: Peter Watson (Ideas [2005], Ch.04) | |
A reaction: It seems from this that we actually have a record of the discovery of true numbers. Delightful. I think the best way to describe what happened is that they began to spot patterns. |
15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
Full Idea: Counting a set produces a well-ordering of it. Conversely, if one has a well-ordering of a set, one can count it by following the well-ordering. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
A reaction: Cantor didn't mean that you could literally count the set, only in principle. |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
Full Idea: My proposal is that the concept of the infinite began with an extrapolation from the experience of indefinitely large size. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2) | |
A reaction: I think it might be better to talk of an 'abstraction' than an 'extrapolition', since the latter is just more of the same, which doesn't get you to concept. Lavine spends 100 pages working out his proposal. |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
Full Idea: The indiscernibility of indefinitely large sizes will be a critical part of the theory of indefinitely large sizes. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2) |
15940 | The intuitionist endorses only the potential infinite [Lavine] |
Full Idea: The intuitionist endorse the actual finite, but only the potential infinite. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VI.2) |
15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
Full Idea: The symbol 'aleph-nought' denotes the cardinal number of the set of natural numbers. The symbol 'aleph-one' denotes the next larger cardinal number. 'Aleph-omega' denotes the omega-th cardinal number. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.3) |
15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
Full Idea: The ordinals are basic because the transfinite sets are those that can be counted, or (equivalently for Cantor), those that can be numbered by an ordinal or are well-ordered. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
A reaction: Lavine observes (p.55) that for Cantor 'countable' meant 'countable by God'! |
15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
Full Idea: The paradox of the largest ordinal (the 'Burali-Forti') is that the class of all ordinal numbers is apparently well-ordered, and so it has an ordinal number as order type, which must be the largest ordinal - but all ordinals can be increased by one. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.5) |
15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
Full Idea: The paradox of the largest cardinal ('Cantor's Paradox') says the diagonal argument shows there is no largest cardinal, but the class of all individuals (including the classes) must be the largest cardinal number. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.5) |
15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
Full Idea: Every theorem of mathematics has a counterpart with set theory - ...but that theory cannot serve as a basis for the notion of proof. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.3) |
15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
Full Idea: In modern mathematics virtually all work is only up to isomorphism and no one cares what the numbers or points and lines 'really are'. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) | |
A reaction: At least that leaves the field open for philosophers, because we do care what things really are. So should everybody else, but there is no persuading some people. |
15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
Full Idea: Intuitionism in philosophy of mathematics rejects set-theoretic foundations. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.3 n33) |
7357 | People who control others with fluent language often end up being hated [Kongzi (Confucius)] |
Full Idea: Of what use is eloquence? He who engages in fluency of words to control men often finds himself hated by them. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], V.5) | |
A reaction: I don't recall Socrates making this very good point to any of the sophists (such as Gorgias). The idea that if you battle or connive your way to dominance over others then you are successful is false. Life is a much longer game than that. |
7358 | All men prefer outward appearance to true excellence [Kongzi (Confucius)] |
Full Idea: I have yet to meet a man as fond of excellence as he is of outward appearances. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], IX.18) | |
A reaction: Interestingly, this cynical view of the love of virtue is put by Plato into the mouths of Glaucon and Adeimantus (in Bk II of 'Republic', e.g. Idea 12), and not into the mouth of Socrates, who goes on to defend the possibility of true virtue. |
7362 | Humans are similar, but social conventions drive us apart (sages and idiots being the exceptions) [Kongzi (Confucius)] |
Full Idea: In our natures we approximate one another; habits put us further and further apart. The only ones who do not change are sages and idiots. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XVII.2) | |
A reaction: I find most of Confucius rather uninteresting, but this is a splendid remark about the influence of social conventions on human nature. Sages can achieve universal morality if they rise above social convention, and seek the true virtues of human nature. |
7360 | Do not do to others what you would not desire yourself [Kongzi (Confucius)] |
Full Idea: Do not do to others what you would not desire yourself. Then you will have no enemies, either in the state or in your home. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XII.2) | |
A reaction: The Golden Rule, but note the second sentence. Logically, it leads to the absurdity of not giving someone an Elvis record for Christmas because you yourself don't like Elvis. Kant (Idea 3733) and Nietzsche (Idea 4560) offer good criticisms. |
7359 | Excess and deficiency are equally at fault [Kongzi (Confucius)] |
Full Idea: Excess and deficiency are equally at fault. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XI.16) | |
A reaction: This is the sort of wisdom we admire in Aristotle (and in any sensible person), but it may also be the deepest motto of conservatism, and it is a long way from romantic philosophy, and the clarion call of Nietzsche to greater excitement in life. |
7363 | The virtues of the best people are humility, maganimity, sincerity, diligence, and graciousness [Kongzi (Confucius)] |
Full Idea: He who in this world can practise five things may indeed be considered Man-at-his-best: humility, maganimity, sincerity, diligence, and graciousness. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XVII.5) | |
A reaction: A very nice list. Who could resist working with a colleague who had such virtues? Who could go wrong if they married a person who had them? I can't think of anything important that is missing. |
7361 | Men of the highest calibre avoid political life completely [Kongzi (Confucius)] |
Full Idea: Men of the highest calibre avoid political life completely. | |
From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XIV.37) | |
A reaction: Plato notes that such people tend to avoid political life (and a left sheltering, as if from a wild storm!), but he thinks they should be dragged into the political arena for the common good. Confucius seems to approve of the avoidance. Plato is right. |
23393 | Confucianism assumes that all good developments have happened, and there is only one Way [Norden on Kongzi (Confucius)] |
Full Idea: The two major limitations of Confucianism are that it assumes that all worthwhile cultural, social and ethical innovation has already occurred, and that it does not recognise the plurality of worthwhile ways of life. | |
From: comment on Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE]) by Bryan van Norden - Intro to Classical Chinese Philosophy 3.III | |
A reaction: In modern liberal terms that is about as conservative as it is possible to get. We think of it as the state of mind of an old person who can only long for the way things were when they were young. But 'hold fast to that which is good'! |
7477 | Modern democracy is actually elective oligarchy [Watson] |
Full Idea: What we regard as democracy in the twenty-first century is actually elective oligarchy. | |
From: Peter Watson (Ideas [2005], Ch.06) | |
A reaction: Even dictatorships want to be called 'democracies'. The modern system is a bit of a concession to Plato, and he would probably have preferred it to his system, because at least the rulers tend to be more educated than the direct assembly. |
7478 | Greek philosophers invented the concept of 'nature' as their special subject [Watson] |
Full Idea: Greek philosophers may have invented the concept of 'nature' to underline their superiority over poets and religious leaders. | |
From: Peter Watson (Ideas [2005], Ch.06) | |
A reaction: Brilliant. They certainly wrote a lot of books entitled 'Peri Physis' (Concerning Nature), and it was the target of their expertise. A highly significant development, along with their rational methods. Presumably Socrates extends nature to include ethics. |
7462 | DNA mutation suggests humans and chimpanzees diverged 6.6 million years ago [Watson] |
Full Idea: The basic mutation rate in DNA is 0.71 percent per million years. Working back from the present difference between human and chimpanzee DNA, we arrive at 6.6 million years ago for their divergence. | |
From: Peter Watson (Ideas [2005], Ch.01) | |
A reaction: This database is committed to evolution (a reminder that even databases have commitments), and so facts of this kind are included, even though they are not strictly philosophical. All complaints should be inwardly digested and forgotten. |
7470 | During the rise of civilizations, the main gods changed from female to male [Watson] |
Full Idea: Around the time of the rise of the first great civilizations, the main gods changed sex, as the Great Goddess, or a raft of smaller goddesses, were demoted and male gods took their place. | |
From: Peter Watson (Ideas [2005], Ch.05) | |
A reaction: Why? War, perhaps? |
7474 | Hinduism has no founder, or prophet, or creed, or ecclesiastical structure [Watson] |
Full Idea: Traditional Hinduism has been described as more a way of living than a way of thought; it has no founder, no prophet, no creed and no ecclesiastical structure. | |
From: Peter Watson (Ideas [2005], Ch.05) | |
A reaction: This contrast strikingly with all later religions, which felt they had to follow the Jews in becoming a 'religion of the book', with a sacred text, and hence a special status for the author(s) of that text. |
7479 | Modern Judaism became stabilised in 200 CE [Watson] |
Full Idea: The Judaism we know today didn't become stabilized until roughly 200 CE. | |
From: Peter Watson (Ideas [2005], Ch.07) | |
A reaction: By that stage it would have been subject to the influences of Christianity, ancient Greek philosophy, and neo-Platonism. |
7481 | The Israelites may have asserted the uniqueness of Yahweh to justify land claims [Watson] |
Full Idea: Archaeology offers datable figures that seem to support the idea that the Israelites of the 'second exile' period converted Yahweh into a special, single God to justify their claims to the land. | |
From: Peter Watson (Ideas [2005], Ch.07) | |
A reaction: The implications for middle eastern politics of this wicked observation are beyond the remit of a philosophy database. |
7480 | Monotheism was a uniquely Israelite creation within the Middle East [Watson] |
Full Idea: No one questions the fact that monotheism was a uniquely Israelite creation within the Middle East. | |
From: Peter Watson (Ideas [2005], Ch.07) | |
A reaction: I take the Middle East to exclude Greece, where they were developing similar ideas. Who knows? |
7471 | The Gathas (hymns) of Zoroastrianism date from about 1000 BCE [Watson] |
Full Idea: The Gathas, the liturgical hymns that make up the 'Avesta', the Zoroastrian canon, are very similar in language to the oldest Sanskrit of Hinduism, so they are not much younger than 1200 BCE. | |
From: Peter Watson (Ideas [2005], Ch.05) | |
A reaction: This implies a big expansion of religion before the well-known expansion of the sixth century BCE. |
7473 | Zoroaster conceived the afterlife, judgement, heaven and hell, and the devil [Watson] |
Full Idea: Life after death, resurrection, judgement, heaven and paradise, were all Zoroastrian firsts, as were hell and the devil. | |
From: Peter Watson (Ideas [2005], Ch.05) | |
A reaction: He appears to be the first 'prophet'. |
7483 | Paul's early writings mention few striking episodes from Jesus' life [Watson] |
Full Idea: Paul's writings - letters mainly - predate the gospels and yet make no mention of many of the more striking episodes that make up Jesus' life. | |
From: Peter Watson (Ideas [2005], Ch.07) | |
A reaction: This is not proof of anything, but it seems very significant if we are trying to get at the facts about Jesus. |
7484 | Jesus never intended to start a new religion [Watson] |
Full Idea: Jesus never intended to start a new religion. | |
From: Peter Watson (Ideas [2005], Ch.08) | |
A reaction: An intriguing fact, which makes you wonder whether any of the prophets ever had such an intention. |
7475 | Confucius revered the spiritual world, but not the supernatural, or a personal god, or the afterlife [Watson] |
Full Idea: Confucius was deeply religious in a traditional sense, showing reverence towards heaven and an omnipresent spiritual world, but he was cool towards the supernatural, and does not seem to have believed in either a personal god or an afterlife. | |
From: Peter Watson (Ideas [2005], Ch.05) | |
A reaction: The implication is that the spiritual world was very remote from us, and beyond communication. Sounds like deism. |
7476 | Taoism aims at freedom from the world, the body, the mind, and nature [Watson] |
Full Idea: Underlying Taoism is a search for freedom - from the world, from the body, from the mind, from nature. | |
From: Peter Watson (Ideas [2005], Ch.05) | |
A reaction: Of all the world's religions, I think Taoism is the most ridiculouly misconceived. |
7463 | The three basic ingredients of religion are: the soul, seers or priests, and ritual [Watson] |
Full Idea: Anthropologist distinguish three requirements for religion: a non-physical soul which can survive death; individuals who can receive supernatural inspiration; and rituals which can cause changes in the present world. | |
From: Peter Watson (Ideas [2005], Ch.01) | |
A reaction: The latter two, of course, also imply belief in supernatural powers. |
7468 | In ancient Athens the souls of the dead are received by the 'upper air' [Watson] |
Full Idea: An official Athenian war monument of 432 BCE says the souls of the dead will be received by the aither (the 'upper air'), though their bodies remain on earth. | |
From: Peter Watson (Ideas [2005], Ch.05) | |
A reaction: Intriguing. Did they think anything happened when they got there? There are also ideas about Hades, and the Isles of the Blessed floating around. |