40 ideas
15924 | Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Zermelo, by Lavine] |
Full Idea: On Zermelo's view, predicative definitions are not only indispensable to mathematics, but they are unobjectionable since they do not create the objects they define, but merely distinguish them from other objects. | |
From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Shaughan Lavine - Understanding the Infinite V.1 | |
A reaction: This seems to have an underlying platonism, that there are hitherto undefined 'objects' lying around awaiting the honour of being defined. Hm. |
17608 | We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo] |
Full Idea: Starting from set theory as it is historically given ...we must, on the one hand, restrict these principles sufficiently to exclude as contradiction and, on the other, take them sufficiently wide to retain all that is valuable in this theory. | |
From: Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro) | |
A reaction: Maddy calls this the one-step-back-from-disaster rule of thumb. Zermelo explicitly mentions the 'Russell antinomy' that blocked Frege's approach to sets. |
17607 | Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo] |
Full Idea: Set theory is that branch whose task is to investigate mathematically the fundamental notions 'number', 'order', and 'function', taking them in their pristine, simple form, and to develop thereby the logical foundations of all of arithmetic and analysis. | |
From: Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro) | |
A reaction: At this point Zermelo seems to be a logicist. Right from the start set theory was meant to be foundational to mathematics, and not just a study of the logic of collections. |
10870 | ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Zermelo, by Clegg] |
Full Idea: Zermelo-Fraenkel axioms: Existence (at least one set); Extension (same elements, same set); Specification (a condition creates a new set); Pairing (two sets make a set); Unions; Powers (all subsets make a set); Infinity (set of successors); Choice | |
From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.15 |
13012 | Zermelo published his axioms in 1908, to secure a controversial proof [Zermelo, by Maddy] |
Full Idea: Zermelo proposed his listed of assumptions (including the controversial Axiom of Choice) in 1908, in order to secure his controversial proof of Cantor's claim that ' we can always bring any well-defined set into the form of a well-ordered set'. | |
From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1 | |
A reaction: This is interesting because it sometimes looks as if axiom systems are just a way of tidying things up. Presumably it is essential to get people to accept the axioms in their own right, the 'old-fashioned' approach that they be self-evident. |
17609 | Set theory can be reduced to a few definitions and seven independent axioms [Zermelo] |
Full Idea: I intend to show how the entire theory created by Cantor and Dedekind can be reduced to a few definitions and seven principles, or axioms, which appear to be mutually independent. | |
From: Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro) | |
A reaction: The number of axioms crept up to nine or ten in subsequent years. The point of axioms is maximum reduction and independence from one another. He says nothing about self-evidence (though Boolos claimed a degree of that). |
13017 | Zermelo introduced Pairing in 1930, and it seems fairly obvious [Zermelo, by Maddy] |
Full Idea: Zermelo's Pairing Axiom superseded (in 1930) his original 1908 Axiom of Elementary Sets. Like Union, its only justification seems to rest on 'limitations of size' and on the 'iterative conception'. | |
From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.3 | |
A reaction: Maddy says of this and Union, that they seem fairly obvious, but that their justification is of prime importance, if we are to understand what the axioms should be. |
13015 | Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy] |
Full Idea: Zermelo used a weak form of the Axiom of Foundation to block Russell's paradox in 1906, but in 1908 felt that the form of his Separation Axiom was enough by itself, and left the earlier axiom off his published list. | |
From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.2 | |
A reaction: Foundation turns out to be fairly controversial. Barwise actually proposes Anti-Foundation as an axiom. Foundation seems to be the rock upon which the iterative view of sets is built. Foundation blocks infinite descending chains of sets, and circularity. |
13486 | Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Zermelo, by Hart,WD] |
Full Idea: Zermelo assumes that not every predicate has an extension but rather that given a set we may separate out from it those of its members satisfying the predicate. This is called 'separation' (Aussonderung). | |
From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by William D. Hart - The Evolution of Logic 3 |
13020 | The Axiom of Separation requires set generation up to one step back from contradiction [Zermelo, by Maddy] |
Full Idea: The most characteristic Zermelo axiom is Separation, guided by a new rule of thumb: 'one step back from disaster' - principles of set generation should be as strong as possible short of contradiction. | |
From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.4 | |
A reaction: Why is there an underlying assumption that we must have as many sets as possible? We are then tempted to abolish axioms like Foundation, so that we can have even more sets! |
13487 | In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Zermelo, by Hart,WD] |
Full Idea: In Zermelo's set theory, the Burali-Forti Paradox becomes a proof that there is no set of all ordinals (so 'is an ordinal' has no extension). | |
From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by William D. Hart - The Evolution of Logic 3 |
18178 | For Zermelo the successor of n is {n} (rather than n U {n}) [Zermelo, by Maddy] |
Full Idea: For Zermelo the successor of n is {n} (rather than Von Neumann's successor, which is n U {n}). | |
From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Naturalism in Mathematics I.2 n8 | |
A reaction: I could ask some naive questions about the comparison of these two, but I am too shy about revealing my ignorance. |
13027 | Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy] |
Full Idea: Zermelo was a reductionist, and believed that theorems purportedly about numbers (cardinal or ordinal) are really about sets, and since Von Neumann's definitions of ordinals and cardinals as sets, this has become common doctrine. | |
From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.8 | |
A reaction: Frege has a more sophisticated take on this approach. It may just be an updating of the Greek idea that arithmetic is about treating many things as a unit. A set bestows an identity on a group, and that is all that is needed. |
9627 | Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR] |
Full Idea: In Zermelo's set-theoretic definition of number, 2 is a member of 3, but not a member of 4; in Von Neumann's definition every number is a member of every larger number. This means they have two different structures. | |
From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by James Robert Brown - Philosophy of Mathematics Ch. 4 | |
A reaction: This refers back to the dilemma highlighted by Benacerraf, which was supposed to be the motivation for structuralism. My intuition says that the best answer is that they are both wrong. In a pattern, the nodes aren't 'members' of one another. |
8143 | Self is the rider, intellect the charioteer, mind the reins, and body the chariot [Anon (Upan)] |
Full Idea: Know that the Self (Atman) is the rider, and the body the chariot; that the intellect is the charioteer, and the mind the reins. | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Katha') | |
A reaction: This strikes me as exactly right. Even my intellectual powers are servants of the self. This suggests the view of the mind as a tool, which does not seem to occur in modern discussions. |
8147 | We have an apparent and a true self; only the second one exists, and we must seek to know it [Anon (Upan)] |
Full Idea: There are two selves, the apparent self, and the real Self. Of these it is the real Self (Atman), and he alone, who must be felt as truly existing. To the man who has felt him as truly existing he reveals his innermost nature. | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Katha') | |
A reaction: A central Hindu doctrine against which Buddhism rebelled, by denying the self altogether. I prefer the Hindu view. A desire to abandon the self just seems to be a desire for death. Knowledge of our essential self is more interesting. But see Idea 2932! |
8155 | Without speech we cannot know right/wrong, true/false, good/bad, or pleasant/unpleasant [Anon (Upan)] |
Full Idea: If there were no speech, neither right nor wrong would be known, neither the true nor the false, neither the good nor the bad, neither the pleasant nor the unpleasant. | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Chandogya') | |
A reaction: This could stand as the epigraph for the whole of modern philosophy of language. However, the text goes on to say that mind is higher than speech. The test question is the mental capabilities of animals. Do they 'know' pleasure, or truth? |
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. |
8142 | The wise prefer good to pleasure; the foolish are drawn to pleasure by desire [Anon (Upan)] |
Full Idea: The wise prefer the good to the pleasant; the foolish, driven by fleshly desires, prefer the pleasant ot the good. | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Katha') | |
A reaction: If you consider appropriate diet, this is too obvious to be worth saying. The complication is that it is doubtful whether a life without pleasure is wholly good, and even the pleasure of food is not bad. Of two good foods, prefer the pleasant one. |
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'! |
8151 | Let your teacher be a god to you [Anon (Upan)] |
Full Idea: Let your teacher be a god to you. | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Taittiriya') | |
A reaction: Yes indeed. The problem in the west is that we are committed to encouraging a critical and questioning attitude. A high value for knowledge must precede a high value for a teacher. |
8153 | By knowing one piece of clay or gold, you know all of clay or gold [Anon (Upan)] |
Full Idea: By knowing one lump of clay, all things made of clay are known; by knowing a nugget of gold, all things made of gold are known. | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Chandogya') | |
A reaction: I can't think of a better basic definition of a natural kind. There is an inductive assumption, of course, which hits trouble when you meet fool's gold, or two different sorts of jade. But the concept of a natural kind is no more than this. |
8154 | Originally there must have been just Existence, which could not come from non-existence [Anon (Upan)] |
Full Idea: In the beginning there was Existence, One only, without a second. Some say that in the beginning there was non-existence only, and that out of that the universe was born. But how could such a thing be? How could existence be born of non-existence? | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Chandogya') | |
A reaction: A very rare instance of an argument in the Upanishads, arising out of a disagreement. The monotheistic religions have preferred to make God the eternal element, presumably because that raises his status, but is also explains the start as a decision. |
8148 | Brahma, supreme god and protector of the universe, arose from the ocean of existence [Anon (Upan)] |
Full Idea: Out of the infinite ocean of existence arose Brahma, first-born and foremost among the gods. From him sprang the universe, and he became its protector. | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Mundaka') | |
A reaction: Brahma does not have eternal (or necessary) existence. Could Brahma cease to exist? I suppose we cannot ask what caused the appearance of Brahma? Is it part of a plan, or just luck, or some sort of necessity? |
8144 | Brahman is the Uncaused Cause [Anon (Upan)] |
Full Idea: Brahman is the Uncaused Cause. | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Katha') | |
A reaction: This precedes Aquinas (Idea 1430) by over two thousand years. The theological trick is to admit one Uncaused Cause, but somehow exclude further instances, such as my bicycle getting a puncture. Does this undermine the Principle of Sufficient Reason? |
8152 | Earth, food, fire, sun are all forms of Brahman [Anon (Upan)] |
Full Idea: Earth, food, fire, sun - all these that you worship - are forms of Brahman. | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Chandogya') | |
A reaction: In 'Taittiriya' food is named as the "chief of all things". Pantheism seems to arise from a desire that one's god should have every conceivable good, so in addition to power and knowledge, your god must keep you warm and healthy. |
8159 | Damayata - be self-controlled! Datta - be charitable! Dayadhwam - be compassionate! [Anon (Upan)] |
Full Idea: The storm-clouds thunder: Da! Da! Da! Damayata - be self-controlled! Datta - be charitable! Dayadhwam - be compassionate! | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Brihadaranyaka') | |
A reaction: Compassion seems to imply charity, so it comes down to 'Be self-controlled and compassionate'. Only the wildest romantic could be against self-control. Only Nietzsche could be against compassion (Idea 4425). |
8156 | The gods are not worshipped for their own sake, but for the sake of the Self [Anon (Upan)] |
Full Idea: It is not for the sake of the gods, my beloved, that the gods are worshipped, but for the sake of the Self (Atman). | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Brihadaranyaka') | |
A reaction: There is an uneasy selfish streak in all religions, which conflicts with their exhorations to altruism, and to the love of the gods. It also occurs in the exhortation of Socrates to be virtuous. 'Pure' altruism seems only to arise in the 18th century. |
8157 | A man with desires is continually reborn, until his desires are stilled [Anon (Upan)] |
Full Idea: A man acts according to desires; after death he reaps the harvest of his deeds, and returns again to the world of action. Thus he who has desires continues subject to rebirth, but he in who desire is stilled suffers no rebirth. | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Brihadaranyaka') | |
A reaction: I greatly prefer the Stoic idea (Idea 3066) that we should live according to nature, to this perverse longing to completely destroy our own nature and become something we are not. Play the cards you are dealt, which include desires. |
8145 | Those ignorant of Atman return as animals or plants, according to their merits [Anon (Upan)] |
Full Idea: Of those ignorant of the Self (Atman), some enter into beings possessed of wombs, others enter into plants - according to their deeds and the growth of their intelligence. | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Katha') | |
A reaction: "I sigh and sigh, and wish I were a tree" wrote George Herbert. You probably need the snobbery of the Indian caste system to appreciate the horrors of low rebirth. I quite fancy being a dolphin, but a tulip would be all right. |
8149 | Charity and ritual observance distract from the highest good of religion [Anon (Upan)] |
Full Idea: Considering religion to be observance of rituals and performance of acts of charity, the deluded remain ignorant of the highest good. | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Mundaka') | |
A reaction: An important reminder. In all the great religious texts the exhortation to love and charity is a minor aspect. The point is to live on a spiritual plain, attempting to relate the world of God/the gods. Daily life is either secondary or irrelevant. |
8158 | Do not seek to know Brahman by arguments, for arguments are idle and vain [Anon (Upan)] |
Full Idea: Do not seek to know Brahman by arguments, for arguments are idle and vain. | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Brihadaranyaka') | |
A reaction: In the end all the religions seem to gravitate towards fideism and away from reasoned argument. The Catholic Church may be the last bastion of rational theology. Islam (10th cent), Protestantism (16th) and Judaism (17th) all rejected philosophy. |
8146 | The immortal in us is the part that never sleeps, and shapes our dreams [Anon (Upan)] |
Full Idea: That which is awake in us even while we sleep, shaping in dream the objects of our desire - that indeed is pure, that is Brahman, and that verily is called the Immortal. | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Katha') | |
A reaction: That is a more helpful view of what the soul might be than anything found in Christian theology. It makes it the essence of the everyday Self. It is left with the difficulty of lacking individuality, and being of limited interest to my wider Self. |
8150 | The immortal Self and the sad individual self are like two golden birds perched on one tree [Anon (Upan)] |
Full Idea: Like two birds of golden plumage, the individual self and the immortal Self perch on the branches of the same tree. The individual self, deluded by forgetfulness of his identity with the divine self, bewildered by his ego, grieves and is sad. | |
From: Anon (Upan) (The Upanishads [c.950 BCE], 'Mundaka') | |
A reaction: Hinduism gives a much clearer and bolder picture of the soul than Christianity does. I don't see much consolation in the immortality of the wonderful Self, if my individual self is doomed to misery and extinction. Which one is me? |