32 ideas
21474 | Metaphysics studies the inexplicable ends of explanation [Schopenhauer] |
Full Idea: The plummet touches the bottom of the sea now at a greater depth, now at a less, but is bound to reach it somewhere sooner or later; the study of this inexplicable devolves upon metaphysics. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], I:1) | |
A reaction: This definition of metapysics contains the germ of despair about the subject. Does he hope that metaphysicians can explain what nobody else can? |
17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
Full Idea: One feature of the Axiom of Choice that troubled many mathematicians was the so-called Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres the same size as the original. | |
From: Penelope Maddy (Defending the Axioms [2011], 1.3) | |
A reaction: (The key is that the parts are non-measurable). To an outsider it is puzzling that the Axiom has been universally accepted, even though it produces such a result. Someone can explain that, I'm sure. |
17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
Full Idea: If-thenism denies that mathematics is in the business of discovering truths about abstracta. ...[their opponents] obviously don't regard any starting point, even a consistent one, as equally worthy of investigation. | |
From: Penelope Maddy (Defending the Axioms [2011], 3.3) | |
A reaction: I have some sympathy with if-thenism, in that you can obviously study the implications of any 'if' you like, but deep down I agree with the critics. |
3299 | In logic identity involves reflexivity (x=x), symmetry (if x=y, then y=x) and transitivity (if x=y and y=z, then x=z) [Baillie] |
Full Idea: In logic identity is an equivalence relation, which involves reflexivity (x=x), symmetry (if x=y, then y=x), and transitivity (if x=y and y=z, then x=z). | |
From: James Baillie (Problems in Personal Identity [1993], Intr p.4) |
17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
Full Idea: At the end of the nineteenth century there was a renewed emphasis on rigor, the central tool of which was axiomatization, along the lines of Hilbert's axioms for geometry and Dedekind's axioms for real numbers. | |
From: Penelope Maddy (Defending the Axioms [2011], 1.3) |
17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
Full Idea: The fact that two apparently fruitful mathematical themes turn out to coincide makes it all the more likely that they're tracking a genuine strain of mathematical depth. | |
From: Penelope Maddy (Defending the Axioms [2011], 5.3ii) |
17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
Full Idea: One form of the Continuum Hypothesis is the claim that every infinite set of reals is either countable or of the same size as the full set of reals. | |
From: Penelope Maddy (Defending the Axioms [2011], 2.4 n40) |
17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
Full Idea: Our set-theoretic methods track the underlying contours of mathematical depth. ...What sets are, most fundamentally, is markers for these contours ...they are maximally effective trackers of certain trains of mathematical fruitfulness. | |
From: Penelope Maddy (Defending the Axioms [2011], 3.4) | |
A reaction: This seems to make it more like a map of mathematics than the actual essence of mathematics. |
17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
Full Idea: Ordinary perceptual cognition is most likely involved in our grasp of elementary arithmetic, but ...this connection to the physical world has long since been idealized away in the infinitary structures of contemporary pure mathematics. | |
From: Penelope Maddy (Defending the Axioms [2011], 2.3) | |
A reaction: Despite this, Maddy's quest is for a 'naturalistic' account of mathematics. She ends up defending 'objectivity' (and invoking Tyler Burge), rather than even modest realism. You can't 'idealise away' the counting of objects. I blame Cantor. |
21470 | For me the objective thing-in-itself is the will [Schopenhauer] |
Full Idea: Thing in itself signifies that which exists independently of our perception, that which actually is; …to Kant it was '= x'; to me it is will. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], IV:61) | |
A reaction: Does he mean his own will, which is plausible since he has direct experience of it, or is he referring will in general - whatever that is? |
21479 | Knowledge is not power! Ignorant people possess supreme authority [Schopenhauer] |
Full Idea: Knowledge is power. The devil it is! One man can have a great deal of knowledge without its giving him the least power, while another possesses supreme authority but next to no knowledge. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], III:43) | |
A reaction: He is referring to Bacon's famous adage. Bacon may be right about military affairs, but not about politics. |
21476 | A priori propositions are those we could never be seriously motivated to challenge [Schopenhauer] |
Full Idea: A dictate of reason is the name we give to certain propositions which we hold to be true without investigation, and of which we think ourselves so firmly convinced we should be incapable of seriously testing them even if we wanted to. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], I:12) | |
A reaction: This is closer to the cautious way modern thinkers are inclined to express the idea. Even Quine would be reasonably happy with this. |
21473 | All knowledge and explanation rests on the inexplicable [Schopenhauer] |
Full Idea: The fundament upon which all our knowledge and learning rests is the inexplicable. It is to this that every explanation, through few or many intermediary stages, leads. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], I:1) | |
A reaction: This is obviously true, and the only question is whether it is a necessary or a contingent truth. |
21478 | Half our thinking is unconscious, and we reach conclusions while unaware of premises [Schopenhauer] |
Full Idea: One might almost believe that half our thinking takes place unconsciously.. Usually we arrive at a conclusion without having clearly thought about the premises which lead to it. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], III:40) | |
A reaction: Schopenhauer was a major pioneer of this crucial idea. I'm beginning to think it is much greater than a half. |
21477 | We don't control our own thinking [Schopenhauer] |
Full Idea: Thoughts come not when we want but when they want. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], III:37) | |
A reaction: One of my favourite Nietzsche ideas originated in Schopenhauer! |
21475 | All of our concepts are borrowed from perceptual knowledge [Schopenhauer] |
Full Idea: The entire property of a concept consists in nothing more than what has been begged and borrowed from perceptual knowledge, which is the true and inexhaustible source of all insight. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], I:9) | |
A reaction: Schopenhauer is usually seen as a sort of idealist, but this is a full endorsement of the empirical view of concepts, to which I largely subscribe. Note that he talks of 'knowledge', rather than of 'experience'. |
21372 | Aesthetics concerns how we can take pleasure in an object, with no reference to the will [Schopenhauer] |
Full Idea: The central problem of aesthetics is how satisfaction with and pleasure in an object are possible without any reference thereof to our willing. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], II:415), quoted by Christopher Janaway - Schopenhauer 6 'Aesthetic' | |
A reaction: This does seem a good distinction. We can divide pleasures into willed and unwilled. Compare thinking that some remote stranger (in a photograph) is very beautiful, with falling in love with someone. |
21488 | The beautiful is a perception of Plato's Forms, which eliminates the will [Schopenhauer] |
Full Idea: In the beautiful we always perceive the intrinsic and primary form of animate and inanimate nature, that is to say Plato's Ideas thereof. …When an aesthetic perception occurs the will completely vanishes from consciousness. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], XIX:205) | |
A reaction: An essential Schopenauer idea. Iris Murdoch said something similar. |
21483 | Man is essentially a dreadful wild animal [Schopenhauer] |
Full Idea: Man is at bottom a dreadful wild animal. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], VIII:114) | |
A reaction: As an example he cites the slave owners in the United States. |
21466 | Pleasure is weaker, and pain stronger, than we expect [Schopenhauer] |
Full Idea: As a rule we find pleasure much less pleasurable, pain much more painful than we expected. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], XII:149) | |
A reaction: Never go on holiday with Schopenhauer. This is more accurate about pain, I think. |
21484 | A man's character can be learned from a single characteristic action [Schopenhauer] |
Full Idea: As a botanist can recognise the whole plant from one leaf, …so an accurate knowledge of a man's character can be arrived at from a single characteristic action. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], VIII:118) | |
A reaction: Very true. Great novelists specialise in such observations. One word can reveal a character, as well as one action. |
21482 | The five Chinese virtues: pity, justice, politeness, wisdom, honesty [Schopenhauer] |
Full Idea: The Chinese name five cardinal virtues: pity, justice, politeness, wisdom and honesty. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], VIII:110) | |
A reaction: I like politeness being on the list, though it seems rather superficial to be a virtue of character. Respect would be better. |
21481 | Buddhists wisely start with the cardinal vices [Schopenhauer] |
Full Idea: Because of their profounder ethical and metaphysical insight, the Buddhists start not with the cardinal virtues but with cardinal vices, …which are lust, sloth, wrath and avarice (and maybe hatred). | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], VIII:110) | |
A reaction: This may be right. Our lives are affected much more by the vices of others than by their virtues, and most virtuous behaviour aims at rectifying the bad effects of other people's vices. |
21480 | Boredom is only felt by those clever enough to need activity [Schopenhauer] |
Full Idea: Only in the cleverest animals such as dogs and apes does the need for activity, and with that boredom, make itself felt. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], III:50) | |
A reaction: But it is much more frequently young creatures, of almost any kind, that seek constant activity, and get continually restless. The most active adults need not be the cleverest. |
21469 | Human life is a mistake, shown by boredom, which is direct awareness of the fact [Schopenhauer] |
Full Idea: Human life must be some kind of mistake. ...Boredom is a direct proof that existence is in itself valueless, for boredom is nothing other than the sensation of the emptiness of existence. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], XI:146) | |
A reaction: I think it is a good advertisement for existentialism that it makes something more out of boredom than Schopenhauer does. |
21485 | The state only exists to defend citizens, from exterior threats, and from one another [Schopenhauer] |
Full Idea: The state is essentially no more than an institution for the protection of the whole against attacks from without, and the protection of its individual members from attacks by one another. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], IX:123) | |
A reaction: He then has a swipe at Hegel for his inflated idea of the importance of the state. Schopenhauer is close to Hobbes on this one. |
21486 | Poverty and slavery are virtually two words for the same thing [Schopenhauer] |
Full Idea: Poverty and slavery are only two forms - on might almost say two words for - the same thing, the essence of which is that a man's energies are expended for the most part not on his own behalf but on that of others. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], IX:125) | |
A reaction: The modern world is full of people who righteously despise slavery, but think only of the poor that it serves them right. |
21487 | The freedom of the press to sell poison outweighs its usefulness [Schopenhauer] |
Full Idea: Freedom of the press must be regarded as a permit to sell poison. …I very much fear, therefore, that the dangers of press freedom outweigh its usefulness. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], IX:127) | |
A reaction: On the whole the modern world disagrees with this view, but watching the popular press in Britain in the last twenty years has made me sympathise with Schopenhauer. |
21471 | If suicide was quick and easy, most people would have done it by now [Schopenhauer] |
Full Idea: Perhaps there is no one alive who would not already have put an end to his life if this end were something purely negative, a sudden cessation of existence. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], XIII:158) | |
A reaction: Nonsense, on the whole, but it is a nice question how many people would do it if it only took a painless instant. |
21467 | Would humanity still exist if sex wasn't both desired and pleasurable? [Schopenhauer] |
Full Idea: If the act of procreation were neither the outcome of a desire nor accompanied by feelings of pleasure, but a matter to be decided on the basis of purely rational considerations, is it likely the human race would still exist? | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], XII:156) | |
A reaction: This is almost certainly correct in the modern world. In tougher economic circumstances people seem desperate to have children who will help them survive. |
21472 | Only religion introduces serious issues to uneducated people [Schopenhauer] |
Full Idea: Religion is the only means of introducing some notion of the high significance of life into the uncultivated heads of the masses. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], XV:174) | |
A reaction: Cf Philip Larkin's poem 'Church Going'. On the whole Schopenhauer didn't actually believe that our lives had any 'high significance'. |
21468 | The Creator created the possibilities for worlds, so should have made a better one than this possible [Schopenhauer] |
Full Idea: The Creator created not only the world, but also created possibility itself; therefore he should have created the possibility of a better world than this one. | |
From: Arthur Schopenhauer (Parerga and Paralipomena [1851], XII:156) | |
A reaction: This is explicitly a response to Leibniz's claim that the Creator selected the best of all possible worlds from the available options. The Euthyphro Question hovers here: must the Creator accept what is possible (the platonic view), or create possibility? |