148 ideas
| 21916 | Philosophers can't be religious, and don't need to be; philosophy is perilous but free [Schopenhauer] |
| Full Idea: No one who is religious attains to philosophy; he does not need it. No one who really philosophizes is religious; he walks without leading-strings, perilously but free. | |
| From: Arthur Schopenhauer (Manuscript remains [1855], II p.241-3), quoted by Peter B. Lewis - Schopenhauer 3 | |
| A reaction: This is a direct reply to the opposite view expressed by Schleiermacher (and quoted by Lewis). I would say that to be a philosopher one must give priority to the philosophy, ahead of any religious beliefs. Thinking must be free. |
| 4171 | Philosophy considers only the universal, in nature as everywhere else [Schopenhauer] |
| Full Idea: Philosophy considers only the universal, in nature as everywhere else. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], II 027) | |
| A reaction: I think what draws people to philosophy is an interest in whatever is timeless. Contingent reality is so frustrating and exhausting. Hence I agree. |
| 4186 | Everyone is conscious of all philosophical truths, but philosophers bring them to conceptual awareness [Schopenhauer] |
| Full Idea: Every person is conscious of all philosophical truths, but to bring them to conceptual awareness, to reflection, is the business of the philosopher. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], IV.68) | |
| A reaction: I like this. All human beings are philosophical. It seems unlikely, though, that we are all pre-conceptually conscious of the higher levels of philosophical logic. |
| 12171 | Absurdity is incongruity between correct and false points of view [Schopenhauer] |
| Full Idea: The more correct the subsumption of objects from one point of view, and the greater and more glaring the incongruity from another point of view, the greater is the ludicrous effect which is produced by this contrast. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], I 013), quoted by Roger Scruton - Laughter §5 | |
| A reaction: This accounts for ludicrous humour, but there seem to be plenty of other types. Exceptional stupidity is usually amusing without necessarily being incongruous. Though it is a departure from the sensible norm. |
| 21366 | Metaphysics must understand the world thoroughly, as a principal source of knowledge [Schopenhauer] |
| Full Idea: The task of metaphysics is not to pass over experience in which the world exists, but to understand it thoroughly, since inner and outer experience are certainly the principal source of all knowledge. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], I 428), quoted by Christopher Janaway - Schopenhauer 3 'Will' | |
| A reaction: I wonder to what extent he meant ordinary experience, and to what extent he was advocating the study if science? |
| 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? |
| 21918 | Sufficient Reason can't be proved, because all proof presupposes it [Schopenhauer, by Lewis,PB] |
| Full Idea: Schopenhauer said the principle of sufficient reason is not susceptible to proof for the simple reason that it is presupposed in any argument or proof. | |
| From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813], §14 p.32-3) by Peter B. Lewis - Schopenhauer 3 | |
| A reaction: I would have thought it might be disproved by a counterexample, such as the Gödel sentence of his incompleteness proof, or quantum effects which seem to elude causation. Personally I believe the principle, which I see as the first axiom of philosophy. |
| 4187 | 'There is nothing without a reason why it should be rather than not be' (a generalisation of 'Why?') [Schopenhauer] |
| Full Idea: The Principle may be stated as 'There is nothing without a reason why it should be rather than not be', which is a generalisation of the assumption which justifies the question 'Why?', which is the mother of all science. | |
| From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], Ch.I) | |
| A reaction: This faith is the core of philosophy, to be maintained against all defeatists like Wittgenstein and Colin McGinn. Reality must be rational, or we wouldn't be here to think about it. (Maybe!) |
| 18194 | 'Forcing' can produce new models of ZFC from old models [Maddy] |
| Full Idea: Cohen's method of 'forcing' produces a new model of ZFC from an old model by appending a carefully chosen 'generic' set. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.4) |
| 18195 | A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy [Maddy] |
| Full Idea: A possible axiom is the Large Cardinal Axiom, which asserts that there are more and more stages in the cumulative hierarchy. Infinity can be seen as the first of these stages, and Replacement pushes further in this direction. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.5) |
| 13011 | New axioms are being sought, to determine the size of the continuum [Maddy] |
| Full Idea: In current set theory, the search is on for new axioms to determine the size of the continuum. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §0) | |
| A reaction: This sounds the wrong way round. Presumably we seek axioms that fix everything else about set theory, and then check to see what continuum results. Otherwise we could just pick our continuum, by picking our axioms. |
| 13013 | The Axiom of Extensionality seems to be analytic [Maddy] |
| Full Idea: Most writers agree that if any sense can be made of the distinction between analytic and synthetic, then the Axiom of Extensionality should be counted as analytic. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.1) | |
| A reaction: [Boolos is the source of the idea] In other words Extensionality is not worth discussing, because it simply tells you what the world 'set' means, and there is no room for discussion about that. The set/class called 'humans' varies in size. |
| 13014 | Extensional sets are clearer, simpler, unique and expressive [Maddy] |
| Full Idea: The extensional view of sets is preferable because it is simpler, clearer, and more convenient, because it individuates uniquely, and because it can simulate intensional notions when the need arises. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.1) | |
| A reaction: [She cites Fraenkel, Bar-Hillet and Levy for this] The difficulty seems to be whether the extensional notion captures our ordinary intuitive notion of what constitutes a group of things, since that needs flexible size and some sort of unity. |
| 18191 | Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy] |
| Full Idea: The axiom of infinity: that there are infinite sets is to claim that completed infinite collections can be treated mathematically. In its standard contemporary form, the axioms assert the existence of the set of all finite ordinals. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.3) |
| 13021 | The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics [Maddy] |
| Full Idea: The Axiom of Infinity is a simple statement of Cantor's great breakthrough. His bold hypothesis that a collection of elements that had lurked in the background of mathematics could be infinite launched modern mathematics. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.5) | |
| A reaction: It also embodies one of those many points where mathematics seems to depart from common sense - but then most subjects depart from common sense when they get more sophisticated. Look what happened to art. |
| 13022 | Infinite sets are essential for giving an account of the real numbers [Maddy] |
| Full Idea: If one is interested in analysis then infinite sets are indispensable since even the notion of a real number cannot be developed by means of finite sets alone. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.5) | |
| A reaction: [Maddy is citing Fraenkel, Bar-Hillel and Levy] So Cantor's great breakthrough (Idea 13021) actually follows from the earlier acceptance of the real numbers, so that's where the departure from common sense started. |
| 13023 | The Power Set Axiom is needed for, and supported by, accounts of the continuum [Maddy] |
| Full Idea: The Power Set Axiom is indispensable for a set-theoretic account of the continuum, ...and in so far as those attempts are successful, then the power-set principle gains some confirmatory support. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.6) | |
| A reaction: The continuum is, of course, notoriously problematic. Have we created an extra problem in our attempts at solving the first one? |
| 18193 | The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy] |
| Full Idea: In the presence of other axioms, the Axiom of Foundation is equivalent to the claim that every set is a member of some Vα. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.3) |
| 13024 | Efforts to prove the Axiom of Choice have failed [Maddy] |
| Full Idea: Jordain made consistent and ill-starred efforts to prove the Axiom of Choice. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.7) | |
| A reaction: This would appear to be the fate of most axioms. You would presumably have to use a different system from the one you are engaged with to achieve your proof. |
| 13025 | Modern views say the Choice set exists, even if it can't be constructed [Maddy] |
| Full Idea: Resistance to the Axiom of Choice centred on opposition between existence and construction. Modern set theory thrives on a realistic approach which says the choice set exists, regardless of whether it can be defined, constructed, or given by a rule. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.7) | |
| A reaction: This seems to be a key case for the ontology that lies at the heart of theory. Choice seems to be an invaluable tool for proofs, so it won't go away, so admit it to the ontology. Hm. So the tools of thought have existence? |
| 13026 | A large array of theorems depend on the Axiom of Choice [Maddy] |
| Full Idea: Many theorems depend on the Axiom of Choice, including that a countable union of sets is countable, and results in analysis, topology, abstract algebra and mathematical logic. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.7) | |
| A reaction: The modern attitude seems to be to admit anything if it leads to interesting results. It makes you wonder about the modern approach of using mathematics and logic as the cutting edges of ontological thinking. |
| 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. |
| 18169 | Axiom of Reducibility: propositional functions are extensionally predicative [Maddy] |
| Full Idea: The Axiom of Reducibility states that every propositional function is extensionally equivalent to some predicative proposition function. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) |
| 13019 | The Iterative Conception says everything appears at a stage, derived from the preceding appearances [Maddy] |
| Full Idea: The Iterative Conception (Zermelo 1930) says everything appears at some stage. Given two objects a and b, let A and B be the stages at which they first appear. Suppose B is after A. Then the pair set of a and b appears at the immediate stage after B. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.3) | |
| A reaction: Presumably this all happens in 'logical time' (a nice phrase I have just invented!). I suppose we might say that the existence of the paired set is 'forced' by the preceding sets. No transcendental inferences in this story? |
| 13018 | Limitation of Size is a vague intuition that over-large sets may generate paradoxes [Maddy] |
| Full Idea: The 'limitation of size' is a vague intuition, based on the idea that being too large may generate the paradoxes. | |
| From: Penelope Maddy (Believing the Axioms I [1988], §1.3) | |
| A reaction: This is an intriguing idea to be found right at the centre of what is supposed to be an incredibly rigorous system. |
| 17824 | The master science is physical objects divided into sets [Maddy] |
| Full Idea: The master science can be thought of as the theory of sets with the entire range of physical objects as ur-elements. | |
| From: Penelope Maddy (Sets and Numbers [1981], II) | |
| A reaction: This sounds like Quine's view, since we have to add sets to our naturalistic ontology of objects. It seems to involve unrestricted mereology to create normal objects. |
| 8755 | Maddy replaces pure sets with just objects and perceived sets of objects [Maddy, by Shapiro] |
| Full Idea: Maddy dispenses with pure sets, by sketching a strong set theory in which everything is either a physical object or a set of sets of ...physical objects. Eventually a physiological story of perception will extend to sets of physical objects. | |
| From: report of Penelope Maddy (Realism in Mathematics [1990]) by Stewart Shapiro - Thinking About Mathematics 8.3 | |
| A reaction: This doesn't seem to find many supporters, but if we accept the perception of resemblances as innate (as in Hume and Quine), it is isn't adding much to see that we intrinsically see things in groups. |
| 10594 | Henkin semantics is more plausible for plural logic than for second-order logic [Maddy] |
| Full Idea: Henkin-style semantics seem to me more plausible for plural logic than for second-order logic. | |
| From: Penelope Maddy (Second Philosophy [2007], III.8 n1) | |
| A reaction: Henkin-style semantics are presented by Shapiro as the standard semantics for second-order logic. |
| 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. |
| 18168 | 'Propositional functions' are propositions with a variable as subject or predicate [Maddy] |
| Full Idea: A 'propositional function' is generated when one of the terms of the proposition is replaced by a variable, as in 'x is wise' or 'Socrates'. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) | |
| A reaction: This implies that you can only have a propositional function if it is derived from a complete proposition. Note that the variable can be in either subject or in predicate position. It extends Frege's account of a concept as 'x is F'. |
| 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) |
| 18190 | Completed infinities resulted from giving foundations to calculus [Maddy] |
| Full Idea: The line of development that finally led to a coherent foundation for the calculus also led to the explicit introduction of completed infinities: each real number is identified with an infinite collection of rationals. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.3) | |
| A reaction: Effectively, completed infinities just are the real numbers. |
| 18171 | Cantor and Dedekind brought completed infinities into mathematics [Maddy] |
| Full Idea: Both Cantor's real number (Cauchy sequences of rationals) and Dedekind's cuts involved regarding infinite items (sequences or sets) as completed and subject to further manipulation, bringing the completed infinite into mathematics unambiguously. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1 n39) | |
| A reaction: So it is the arrival of the real numbers which is the culprit for lumbering us with weird completed infinites, which can then be the subject of addition, multiplication and exponentiation. Maybe this was a silly mistake? |
| 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) |
| 18172 | Infinity has degrees, and large cardinals are the heart of set theory [Maddy] |
| Full Idea: The stunning discovery that infinity comes in different degrees led to the theory of infinite cardinal numbers, the heart of contemporary set theory. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) | |
| A reaction: It occurs to me that these huge cardinals only exist in set theory. If you took away that prop, they would vanish in a puff. |
| 18175 | For any cardinal there is always a larger one (so there is no set of all sets) [Maddy] |
| Full Idea: By the mid 1890s Cantor was aware that there could be no set of all sets, as its cardinal number would have to be the largest cardinal number, while his own theorem shows that for any cardinal there is a larger. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) | |
| A reaction: There is always a larger cardinal because of the power set axiom. Some people regard that with suspicion. |
| 18196 | An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy] |
| Full Idea: An 'inaccessible' cardinal is one that cannot be reached by taking unions of small collections of smaller sets or by taking power sets. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.5) | |
| A reaction: They were introduced by Hausdorff in 1908. |
| 18187 | Theorems about limits could only be proved once the real numbers were understood [Maddy] |
| Full Idea: Even the fundamental theorems about limits could not [at first] be proved because the reals themselves were not well understood. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: This refers to the period of about 1850 (Weierstrass) to 1880 (Dedekind and Cantor). |
| 18182 | The extension of concepts is not important to me [Maddy] |
| Full Idea: I attach no decisive importance even to bringing in the extension of the concepts at all. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], §107) | |
| A reaction: He almost seems to equate the concept with its extension, but that seems to raise all sorts of questions, about indeterminate and fluctuating extensions. |
| 18177 | In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy] |
| Full Idea: In the ZFC cumulative hierarchy, Frege's candidates for numbers do not exist. For example, new three-element sets are formed at every stage, so there is no stage at which the set of all three-element sets could he formed. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: Ah. This is a very important fact indeed if you are trying to understand contemporary discussions in philosophy of mathematics. |
| 18164 | Frege solves the Caesar problem by explicitly defining each number [Maddy] |
| Full Idea: To solve the Julius Caesar problem, Frege requires explicit definitions of the numbers, and he proposes his well-known solution: the number of Fs = the extension of the concept 'equinumerous with F' (based on one-one correspondence). | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) | |
| A reaction: Why do there have to be Fs before there can be the corresponding number? If there were no F for 523, would that mean that '523' didn't exist (even if 522 and 524 did exist)? |
| 17825 | Set theory (unlike the Peano postulates) can explain why multiplication is commutative [Maddy] |
| Full Idea: If you wonder why multiplication is commutative, you could prove it from the Peano postulates, but the proof offers little towards an answer. In set theory Cartesian products match 1-1, and n.m dots when turned on its side has m.n dots, which explains it. | |
| From: Penelope Maddy (Sets and Numbers [1981], II) | |
| A reaction: 'Turning on its side' sounds more fundamental than formal set theory. I'm a fan of explanation as taking you to the heart of the problem. I suspect the world, rather than set theory, explains the commutativity. |
| 17826 | Standardly, numbers are said to be sets, which is neat ontology and epistemology [Maddy] |
| Full Idea: The standard account of the relationship between numbers and sets is that numbers simply are certain sets. This has the advantage of ontological economy, and allows numbers to be brought within the epistemology of sets. | |
| From: Penelope Maddy (Sets and Numbers [1981], III) | |
| A reaction: Maddy votes for numbers being properties of sets, rather than the sets themselves. See Yourgrau's critique. |
| 17828 | Numbers are properties of sets, just as lengths are properties of physical objects [Maddy] |
| Full Idea: I propose that ...numbers are properties of sets, analogous, for example, to lengths, which are properties of physical objects. | |
| From: Penelope Maddy (Sets and Numbers [1981], III) | |
| A reaction: Are lengths properties of physical objects? A hole in the ground can have a length. A gap can have a length. Pure space seems to contain lengths. A set seems much more abstract than its members. |
| 10718 | A natural number is a property of sets [Maddy, by Oliver] |
| Full Idea: Maddy takes a natural number to be a certain property of sui generis sets, the property of having a certain number of members. | |
| From: report of Penelope Maddy (Realism in Mathematics [1990], 3 §2) by Alex Oliver - The Metaphysics of Properties | |
| A reaction: [I believe Maddy has shifted since then] Presumably this will make room for zero and infinities as natural numbers. Personally I want my natural numbers to count things. |
| 18184 | Making set theory foundational to mathematics leads to very fruitful axioms [Maddy] |
| Full Idea: The set theory axioms developed in producing foundations for mathematics also have strong consequences for existing fields, and produce a theory that is immensely fruitful in its own right. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: [compressed] Second of Maddy's three benefits of set theory. This benefit is more questionable than the first, because the axioms may be invented because of their nice fruit, instead of their accurate account of foundations. |
| 18185 | Unified set theory gives a final court of appeal for mathematics [Maddy] |
| Full Idea: The single unified area of set theory provides a court of final appeal for questions of mathematical existence and proof. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: Maddy's third benefit of set theory. 'Existence' means being modellable in sets, and 'proof' means being derivable from the axioms. The slightly ad hoc character of the axioms makes this a weaker defence. |
| 18183 | Set theory brings mathematics into one arena, where interrelations become clearer [Maddy] |
| Full Idea: Set theoretic foundations bring all mathematical objects and structures into one arena, allowing relations and interactions between them to be clearly displayed and investigated. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: The first of three benefits of set theory which Maddy lists. The advantages of the one arena seem to be indisputable. |
| 18186 | Identifying geometric points with real numbers revealed the power of set theory [Maddy] |
| Full Idea: The identification of geometric points with real numbers was among the first and most dramatic examples of the power of set theoretic foundations. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: Hence the clear definition of the reals by Dedekind and Cantor was the real trigger for launching set theory. |
| 18188 | The line of rationals has gaps, but set theory provided an ordered continuum [Maddy] |
| Full Idea: The structure of a geometric line by rational points left gaps, which were inconsistent with a continuous line. Set theory provided an ordering that contained no gaps. These reals are constructed from rationals, which come from integers and naturals. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: This completes the reduction of geometry to arithmetic and algebra, which was launch 250 years earlier by Descartes. |
| 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. |
| 18163 | Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy] |
| Full Idea: Our much loved mathematical knowledge rests on two supports: inexorable deductive logic (the stuff of proof), and the set theoretic axioms. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I Intro) |
| 17830 | Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy] |
| Full Idea: I am not suggesting a reduction of number theory to set theory ...There are only sets with number properties; number theory is part of the theory of finite sets. | |
| From: Penelope Maddy (Sets and Numbers [1981], V) |
| 17827 | Sets exist where their elements are, but numbers are more like universals [Maddy] |
| Full Idea: A set of things is located where the aggregate of those things is located, ...but a number is simultaneously located at many different places (10 in my hand, and a baseball team) ...so numbers seem more like universals than particulars. | |
| From: Penelope Maddy (Sets and Numbers [1981], III) | |
| A reaction: My gut feeling is that Maddy's master idea (of naturalising sets by building them from ur-elements of natural objects) won't work. Sets can work fine in total abstraction from nature. |
| 17823 | If mathematical objects exist, how can we know them, and which objects are they? [Maddy] |
| Full Idea: The popular challenges to platonism in philosophy of mathematics are epistemological (how are we able to interact with these objects in appropriate ways) and ontological (if numbers are sets, which sets are they). | |
| From: Penelope Maddy (Sets and Numbers [1981], I) | |
| A reaction: These objections refer to Benacerraf's two famous papers - 1965 for the ontology, and 1973 for the epistemology. Though he relied too much on causal accounts of knowledge in 1973, I'm with him all the way. |
| 8756 | Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro] |
| Full Idea: Maddy says that intuition alone does not support very much mathematics; more importantly, a naturalist cannot accept intuition at face value, but must ask why we are justified in relying on intuition. | |
| From: report of Penelope Maddy (Realism in Mathematics [1990]) by Stewart Shapiro - Thinking About Mathematics 8.3 | |
| A reaction: It depends what you mean by 'intuition', but I identify with her second objection, that every faculty must ultimately be subject to criticism, which seems to point to a fairly rationalist view of things. |
| 17733 | We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins] |
| Full Idea: Maddy proposes that we can know (some) mind-independent mathematical truths through knowing about sets, and that we can obtain knowledge of sets through experience. | |
| From: report of Penelope Maddy (Realism in Mathematics [1990]) by Carrie Jenkins - Grounding Concepts 6.5 | |
| A reaction: Maddy has since backed off from this, and now tries to merely defend 'objectivity' about sets (2011:114). My amateurish view is that she is overrating the importance of sets, which merely model mathematics. Look at category theory. |
| 18204 | Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy] |
| Full Idea: Crudely, the scientist posits only those entities without which she cannot account for observations, while the set theorist posits as many entities as she can, short of inconsistency. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], II.5) |
| 18207 | Maybe applications of continuum mathematics are all idealisations [Maddy] |
| Full Idea: It could turn out that all applications of continuum mathematics in natural sciences are actually instances of idealisation. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], II.6) |
| 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. |
| 17829 | Number words are unusual as adjectives; we don't say 'is five', and numbers always come first [Maddy] |
| Full Idea: Number words are not like normal adjectives. For example, number words don't occur in 'is (are)...' contexts except artificially, and they must appear before all other adjectives, and so on. | |
| From: Penelope Maddy (Sets and Numbers [1981], IV) | |
| A reaction: [She is citing Benacerraf's arguments] |
| 18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy] |
| Full Idea: Recent commentators have noted that Frege's versions of the basic propositions of arithmetic can be derived from Hume's Principle alone, that the fatal Law V is only needed to derive Hume's Principle itself from the definition of number. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) | |
| A reaction: Crispin Wright is the famous exponent of this modern view. Apparently Charles Parsons (1965) first floated the idea. |
| 4168 | Matter and intellect are inseparable correlatives which only exist relatively, and for each other [Schopenhauer] |
| Full Idea: In my system matter and intellect are inseparable correlatives which exist only for each other, and so exist only relatively. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], I.Supp) | |
| A reaction: A plausible picture, but built from dualist presuppositions. Personally I think intellect is built out of matter, so I am not going down Schopenhauer's road. |
| 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? |
| 21926 | Schopenhauer, unlike other idealists, says reality is irrational [Schopenhauer, by Lewis,PB] |
| Full Idea: Schopenhauer radically departs from his fellow idealists in his assertion of the irrational character of reality. | |
| From: report of Arthur Schopenhauer (The World as Will and Idea [1819]) by Peter B. Lewis - Schopenhauer 4 | |
| A reaction: This is the rejection of the original confidence about rationality of the stoics. And yet Schopenauer saw the principle of sufficient reason as axiomatic. Not sure how to reconcile those. Lewis identifies this idea as 'Romantic'. |
| 4167 | The knowing subject and the crude matter of the world are both in themselves unknowable [Schopenhauer] |
| Full Idea: The world has two poles - the knowing subject and crude matter, which are both completely unknowable, the former because it is the knower, the latter because without form and quality it cannot be perceived. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], I Supp) | |
| A reaction: A nice concept, that all of reality comes from their relationship, but the two components are intrinsically unknowable. Does God the Knower know his own mind? |
| 18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
| Full Idea: The case of atoms makes it clear that the indispensable appearance of an entity in our best scientific theory is not generally enough to convince scientists that it is real. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], II.6) | |
| A reaction: She refers to the period between Dalton and Einstein, when theories were full of atoms, but there was strong reluctance to actually say that they existed, until the direct evidence was incontrovertable. Nice point. |
| 21920 | No need for a priori categories, since sufficient reason shows the interrelations [Schopenhauer, by Lewis,PB] |
| Full Idea: Schopenhauer dispenses with Kant's a priori categories, since all interrelations between representations are given through the principle of sufficient reason. | |
| From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813]) by Peter B. Lewis - Schopenhauer 3 | |
| A reaction: I'm not sure how Schopenhauer manages this move. Is it the stoic idea that reality has a logical structure, which can be inferred? Sounds good to me. Further investigation required. |
| 16698 | Days exist, and yet they seem to be made up of parts which don't exist [Burley] |
| Full Idea: I grant that a successive being is composed out of non-beings, as is clear of a day, which is composed of non-entities. Some part of this day is past and some future, and yet this day is. | |
| From: Walter Burley (Commentary on 'Physics' [1325], III text 11,f.65rb), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 18.3 | |
| A reaction: The dilemma of Aristotle over time infected the scholastic attempt to give an account of successive entities. A day is a wonderfully elusive entity for a metaphysician. |
| 16690 | Unlike permanent things, successive things cannot exist all at once [Burley] |
| Full Idea: This is the difference between permanent and successive things: that a permanent thing exists all at once, or at least can exist all at once, whereas it is incompatible with a successive thing to exist all at once. | |
| From: Walter Burley (Commentary on 'Physics' [1325], III txt 11,f.65rb), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 18.1 | |
| A reaction: Permanent things sound like what are now called 'three-dimensional' objects, but scholastic 'entia successiva' are not the same as spacetime 'worms' or collections of temporal stages. |
| 21362 | Necessity is physical, logical, mathematical or moral [Schopenhauer, by Janaway] |
| Full Idea: For Schopenauer there are physical necessity, logical necessity, mathematical necessity and moral necessity. | |
| From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813]) by Christopher Janaway - Schopenhauer 2 'Fourfold' | |
| A reaction: These derive from four modes of explanation, by causes, by grounding in truths or facts, by mathematical reality, and by motives. Not clear why mathematics gets its own necessity. I like metaphysics derived from explanations, though. Necessity makers. |
| 4192 | All necessity arises from causation, which is conditioned; there is no absolute or unconditioned necessity [Schopenhauer] |
| Full Idea: Necessity has no meaning other than the irresistible sequence of the effect where the cause is given. All necessity is thus conditioned, and absolute or unconditioned necessity is a contradiction in terms. | |
| From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], Ch.VIII) | |
| A reaction: I.e. there is only natural necessity, and no such thing as metaphysical necessity. But what about logical necessity(e.g. 2+3=5)? I think there may be metaphysical necessity, but we can't know much about it, and we are over-confident in assessing it. |
| 4190 | All understanding is an immediate apprehension of the causal relation [Schopenhauer] |
| Full Idea: All understanding is an immediate apprehension of the causal relation. | |
| From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], Ch.IV) | |
| A reaction: Based, I take it, on Hume. Presumably he means a posteriori understanding, as it hardly fits an understanding of arithmetic. Understanding needs more than just causation. What aspects of causation? |
| 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. |
| 4165 | Descartes found the true beginning of philosophy with the Cogito, in the consciousness of the individual [Schopenhauer] |
| Full Idea: By taking Cogito Ergo Sum as the only certainty, and by his provisionally regarding the existence of the world as problematical, the essential starting point of all philosophy was found, and its true focus in the subjective, the individual consciousness. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], I Supp) | |
| A reaction: Some people think this was a disaster, not a triumph. Descartes could have doubted himself and accepted the world as his starting point. |
| 21361 | For Schopenhauer, material things would not exist without the mind [Schopenhauer, by Janaway] |
| Full Idea: Schopenhauer is not a realist about material things, but an idealist: that is, material things would not exist, for him, without the mind. | |
| From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813]) by Christopher Janaway - Schopenhauer 2 'Fourfold' | |
| A reaction: Janaway places his views as close to Kant's, but it is not clear that Kant would agree that no mind means no world. Did Schopenhauer believe in the noumenon? |
| 21923 | Schopenhauer can't use force/energy instead of 'will', because he is not a materialist [Lewis,PB on Schopenhauer] |
| Full Idea: Some say Schopenhauer would be less misunderstood if he had used 'force' or 'energy' rather than 'will' to characterise inner natures. But this would have steered his idealism towards materialism, of which he was an avowed opponent. | |
| From: comment on Arthur Schopenhauer (The World as Will and Idea [1819]) by Peter B. Lewis - Schopenhauer 4 | |
| A reaction: I presume therefore that Nietzsche's will to power is a commitment to materialism, since it occurs in material objects as well as minds. |
| 4162 | The world only exists in relation to something else, as an idea of the one who conceives it [Schopenhauer] |
| Full Idea: The world which surrounds man exists only as idea - that is, only in relation to something else, the one who conceives the idea, which is himself. If any truth can be enunciated a priori, it is this. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], I 001) | |
| A reaction: Yes, but the idea we have is of a real world. It is definitely not part of the idea that it is an idea (unlike my idea for a Christmas present). |
| 21922 | We know reality because we know our own bodies and actions [Schopenhauer] |
| Full Idea: The double knowledge of the nature and action of our own body is the key to the inner being of every phenomenon in nature. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], I 105), quoted by Peter B. Lewis - Schopenhauer 4 | |
| A reaction: Lewis calls this 'the heart of his philosophy'. Bodily awareness comes from acts of willing. So Lewis says 'the thing-in-itself is revealed to us in willing'. We experience Being and causation. Is he trying to combine idealism with the thing-in-itself? |
| 21913 | Kant rightly separates appearance and thing-in-itself [Schopenhauer] |
| Full Idea: Kant's greatest merit is the distinction of the phenomenon from the thing-in-itself. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], I 417 App), quoted by Peter B. Lewis - Schopenhauer 3 | |
| A reaction: This is Schopenhauer firmly opposing the Absolute Idealism of Kant's successors, who dismissed the 'thing-in-itsef'. |
| 21919 | Object for a subject and representation are the same thing [Schopenhauer] |
| Full Idea: To be object for a subject and to be representation is to be one and the same thing. All representations are objects for a subject, all objects for a subject are representations. | |
| From: Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813], §16 p.41-2), quoted by Peter B. Lewis - Schopenhauer 3 | |
| A reaction: This is pure idealism in early Schopenhauer, derived from Kant. Are being 'an object for a subject' and being an object 'in itself' two different things? Compare Idea 21914, written later. I think Nietzsche's 'perspective' representations helps here. |
| 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. |
| 16719 | The primary qualities are mixed to cause secondary qualities [Burley] |
| Full Idea: Secondary qualities are caused by a mixture of primary qualities. | |
| From: Walter Burley (De formis [1330], pars post p.65), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 21.2 | |
| A reaction: Like paint. He probably has in mind hot, cold, wet and dry as the primary qualities. |
| 4164 | Direct feeling of the senses are merely data; perception of the world comes with understanding causes [Schopenhauer] |
| Full Idea: What the eye, the ear, or the hand feels, is not perception, it is merely data. Only when the understanding passes from the effect to the cause does the world lie before us as perception extended in space. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], I 004) | |
| A reaction: These certainly seems to be a sense-data theory. Philosophers are much more ready to separate the data from the understanding than neuroscientists are. |
| 4163 | All perception is intellectual [Schopenhauer] |
| Full Idea: All perception is intellectual. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], I 004) | |
| A reaction: Even in slugs? I suspect that this is a tautology. Schopenhauer will only allow my vision or hearing to become 'perception' when an intellectual element enters into it. |
| 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. |
| 21917 | The four explanations: objects by causes, concepts by ground, maths by spacetime, ethics by motive [Schopenhauer, by Lewis,PB] |
| Full Idea: There are four forms of explanation, depending on their topic. Causes explain objects. Grounding explains concepts, Points and moments explain mathematics. Motives explain ethics. | |
| From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813], §43 p.214) by Peter B. Lewis - Schopenhauer 3 | |
| A reaction: [My compression of Lewis's summary. I'm particularly pleased with this. I have done Schopenhauer a huge favour, should anyone ever visit this website]. The quirky account of mathematics derives from Kant. I greatly admire this whole idea. |
| 4166 | A consciousness without an object is no consciousness [Schopenhauer] |
| Full Idea: A consciousness without an object is no consciousness. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], I Supp) | |
| A reaction: This hints at Hume's observations about the self. Certainly totally vacant consciousness seems inconceivable, but is that a necessary or a contingent truth? |
| 21369 | We have hidden and unadmitted desires and fears, suppressed because of vanity [Schopenhauer] |
| Full Idea: We often do not know what we desire or fear. For years we can have a desire without admitting it to ourselves ....because the intellect is not to know anything about it, since the good opinion we have of ourselves would inevitably suffer thereby. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], II 210), quoted by Christopher Janaway - Schopenhauer 5 'Will' | |
| A reaction: The idea of unconscious thought crept up well before Freud. It is in La Rochefoucauld, and important in Nietzsche. Neuroscience seems to give it a strong priority over the conscious mind, which is a revolutionary idea. |
| 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. |
| 18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
| Full Idea: In science we treat the earth's surface as flat, we assume the ocean to be infinitely deep, we use continuous functions for what we know to be quantised, and we take liquids to be continuous despite atomic theory. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], II.6) | |
| A reaction: If fussy people like scientists do this all the time, how much more so must the confused multitude be doing the same thing all day? |
| 4191 | What we know in ourselves is not a knower but a will [Schopenhauer] |
| Full Idea: What we know in ourselves is never what knows, but what wills, the will. | |
| From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], Ch.VII) | |
| A reaction: An interesting slant on Hume's scepticism about personal identity. Hume was hunting for a thing-which-experiences. If he had sought his will, he might have spotted it. |
| 21367 | I know both aspects of my body, as representation, and as will [Schopenhauer] |
| Full Idea: My body is the only object of which I know not merely the one side, that of the representation, but also the other, that is called 'will'. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], I 125), quoted by Christopher Janaway - Schopenhauer 3 'Will' | |
| A reaction: I'm not convinced that knowledge of the body through the will (and action, presumably) constitutes a different sort of knowledge. Philosophers are always trying to split the world in two (but not Nietzsche!). |
| 21368 | The knot of the world is the use of 'I' to refer to both willing and knowing [Schopenhauer] |
| Full Idea: The identity of the subject of willing with that of knowing by virtue whereof ...the word 'I' includes and indicates both, is the knot of the world, and hence inexplicable. | |
| From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], p.211-2), quoted by Christopher Janaway - Schopenhauer 4 'Self' | |
| A reaction: I'm struggling to see this as a deep mystery. If we look objectively at animals and ask 'what is their brain for?' the answer seems obvious. This may be a case of everything looking mysterious after a philosopher has stared at it for a while. |
| 4175 | It is as perverse to resent our individuality being replaced by others, as to resent the body renewing itself [Schopenhauer] |
| Full Idea: It is as perverse to desire the continuity of one's individuality which is being replaced by other individuals, as to desire the permanence of the body's substance which is always being replaced by new substance. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], IV.54) | |
| A reaction: If I let that go, what am I supposed to hang on to? Nothing? Non-existence is not an attractive condition to aspire to. |
| 4176 | We all regard ourselves a priori as free, but see from experience that character and motive compel us [Schopenhauer] |
| Full Idea: Everyone regards himself a priori as free in his individual actions, and only a posteriori sees that necessarily his actions follow from the coincidence of character with motives. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], IV.55) | |
| A reaction: I'm not sure what experience shows. Necessity seems more obvious when observing other people. Samuel Johnson said experience showed freedom, not necessity. |
| 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! |
| 4170 | Man's actions are not free, because they follow strictly from impact of motive on character [Schopenhauer] |
| Full Idea: Man's action have been interpreted as free, which they are not, for every individual action follows with strict necessity from the impact of motive on character. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], II 023) | |
| A reaction: If 'character is fate' (Heraclitus) then presumably motive must also be fate to complete the determinist picture. I shall spend the next year redesigning my motivation. |
| 21921 | Concepts are abstracted from perceptions [Schopenhauer, by Lewis,PB] |
| Full Idea: For Schopenhauer concepts are abstractions from perception, what he calls 'representations of representations', and are linked to the creation of language. | |
| From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813]) by Peter B. Lewis - Schopenhauer 3 | |
| A reaction: This is a traditional view which dates back to Aristotle, and which I personally think is entirely correct. These days I am in minority on that. This idea means that (contrary to Kant) perception is not conceptual. |
| 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'. |
| 4169 | Every true act of will is also at once and without exception a movement of the body [Schopenhauer] |
| Full Idea: Every true act of will is also at once and without exception a movement of the body. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], II 018) | |
| A reaction: The word 'act' seems to beg the question (as does 'true'!). I am no longer sure that I know what an act of will is. Hobbes says there is no such thing. |
| 7187 | Schopenhauer was caught in Christian ideals, because he didn't deify his 'will' [Nietzsche on Schopenhauer] |
| Full Idea: Schopenhauer's interpretation of the in-itself as will was an essential step: but he didn't know how to deify the will, and remained caught in the moral, Christian ideal | |
| From: comment on Arthur Schopenhauer (The World as Will and Idea [1819]) by Friedrich Nietzsche - Writings from Late Notebooks 9[42] | |
| A reaction: Intriguingly, this seems to suggest that Nietzsche consciously sought to replace the absence of God with the human will, which strikes me as an odd, and very nineteenth century, idea. Loss of religion bothered them a lot. |
| 21365 | Only the will is thing-in-itself, seen both in blind nature and in human action [Schopenhauer] |
| Full Idea: Only the will is thing-in-itself. ...It appears in every blindly acting force of nature, and also in the deliberate conduct of man, and the great difference between the two concerns only the degree of the manifestation. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], I 110), quoted by Christopher Janaway - Schopenhauer 3 'Will' | |
| A reaction: If will acts 'blindly' in forces of nature, then these seems to be the same concept as Nietzsche's 'will to power'. This seems to be heading towards Heidegger's Dasein, as a central and distinctive mode of being. |
| 21924 | As the subject of willing I am wretched, but absorption in knowledge is bliss [Schopenhauer] |
| Full Idea: As the subject of willing I am an exceedingly wretched being, and all our suffering consistd in willing, ...but as soon as I am absorbed in knowledge, I am blissfully happy and nothing can assail me. | |
| From: Arthur Schopenhauer (Manuscript remains [1855], I p.137), quoted by Peter B. Lewis - Schopenhauer 4 | |
| A reaction: So the source of his pessimism is subjection to his own will. However, since becoming absorbed in knowledge is an easy task for a scholar, he has little to grumble about. Nietzsche mocked the great pessimist for playing the flute every day. |
| 4173 | If we were essentially intellect rather than will, our moral worth would depend on imagined motives [Schopenhauer] |
| Full Idea: If, as all philosophers imagine, the intellect is our actual nature and the will is arrived at through knowledge, then only the motive from which we imagined we were acting would decide our moral worth. Imagined and true motive would be indistinguishable. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], II Supp) | |
| A reaction: A nice argument. If motive is morally decisive, it is certainly crucial to decide between real and imagined motive (especially since Freud). But uncontrollable motive seems morally irrelevant. |
| 21363 | Motivation is causality seen from within [Schopenhauer] |
| Full Idea: Motivation is causality seen from within. | |
| From: Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813], p.214), quoted by Christopher Janaway - Schopenhauer 2 'Fourfold' | |
| A reaction: This is more illuminating about causation than about motivation, since we can be motivated without actually doing anything. |
| 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. |
| 21370 | Schopenhauer is a chief proponent of aesthetic experience as 'disinterested' [Schopenhauer, by Janaway] |
| Full Idea: Schopenhauer belongs to a tradition which equates aesthetic experience with a 'disinterested' attitude towards its object, and is often cited as one of the chief proponents of such a view. | |
| From: report of Arthur Schopenhauer (The World as Will and Idea [1819]) by Christopher Janaway - Schopenhauer 6 'Aesthetic' | |
| A reaction: 'Disinterested' is quite a nice word for one's attitude to art, though you then have to capture why you are also involved in it. |
| 4182 | A principal pleasure of the beautiful is that it momentarily silences the will [Schopenhauer] |
| Full Idea: The momentary silencing of all volition …is one of the principal elements in our pleasure in the beautiful. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], IV.65) | |
| A reaction: Iris Murdoch sees moral value in beauty, because it overrides selfishness. The perception of beauty is certainly something deeper than just another nice feeling. There is a cognitive element to it. |
| 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. |
| 21928 | The Sublime fights for will-less knowing, when faced with a beautiful threat to humanity [Schopenhauer, by Lewis,PB] |
| Full Idea: Exaltation of the Sublime is the struggle to maintain will-less knowing in the face of a threat to the human will. The Sublime contains an awful beauty and a delightful terror because it includes a threat to human existence. | |
| From: report of Arthur Schopenhauer (The World as Will and Idea [1819], I 201-7) by Peter B. Lewis - Schopenhauer 5 | |
| A reaction: Can you experience the Sublime when looking down a microscope? Can a mere theory in cosmology be sublime? Can a supposed perception of the Sublime ever be incorrect? We no longer worry about these questions, it seems. |
| 21927 | Schopenhauer emphasises Ideas in art, unlike most romantics [Schopenhauer, by Lewis,PB] |
| Full Idea: The emphasis on the presentation of Platonic Ideas distinguishes Schopenhauer's theory of art from standard Romantic theories, which emphasize the expression of emotion and feeling. | |
| From: report of Arthur Schopenhauer (The World as Will and Idea [1819], II) by Peter B. Lewis - Schopenhauer 5 | |
| A reaction: Theories of art that neglect ideas, even if subliminally expressed, have gone badly wrong. |
| 8116 | The will-less contemplation of art brings a liberation from selfhood [Schopenhauer, by Gardner] |
| Full Idea: For Schopenhauer, the point of art lies in the metaphysical liberation from selfhood that will-less aesthetic contemplation induces. | |
| From: report of Arthur Schopenhauer (The World as Will and Idea [1819]) by Sebastian Gardner - Aesthetics 3.6.3 | |
| A reaction: I've never understood why anyone (Buddhists included) would want 'liberation from selfhood'. Certainly art can make us forget ourselves in a more objective view of things, but science can do that too. |
| 4174 | Man is more beautiful than anything else, and the loftiest purpose of art is to reveal his nature [Schopenhauer] |
| Full Idea: Man is more beautiful than anything else, and the loftiest purpose of art is to reveal his nature. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], III 41) | |
| A reaction: A bit of a shock, because it implies human vanity, but it fits the best works of art rather well. What else reveals humanity's beauty? Beautiful deeds must be recorded. |
| 21380 | The only aim of our existence is to grasp that non-existence would be better [Schopenhauer] |
| Full Idea: Nothing else can be stated as the aim of our existence except the knowledge that it would be better for us not to exist. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], II 605), quoted by Christopher Janaway - Schopenhauer 8 'Denial' | |
| A reaction: Nonsense on stilts. Nietzsche rebelled against this. If there is such 'knowledge' then it obviously has nothing to do with the aim of our existence. It is just a rejection of aims. The aim is making the best of existence, not spurning it. |
| 21374 | We should no more expect ethical theory to produce good people than aesthetics to produce artists [Schopenhauer] |
| Full Idea: We should be just as foolish to expect that our moral systems and ethics would create virtuous, noble and hold men, as that our aesthetics would produce poets, painterd and musicians. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], I 271), quoted by Christopher Janaway - Schopenhauer 7 'Against' | |
| A reaction: Presumably the aim of ethical theory is to understand the truths about ethics. That can't do any harm, can it? In every other area of life we think that understanding leads to improvement. Unless, of course, there are no truths of ethics.... |
| 21378 | We clearly feel responsible for our deeds, because we are quite certain that we did them [Schopenhauer] |
| Full Idea: Another fact of consciousness ...is the wholly clear and certain feeling of responsibility for what we do, of the accountability of our actions, which rests on the unshakable certainty that we ourselves are the doers of our deeds. | |
| From: Arthur Schopenhauer (On the Freedom of the Will [1841], p.93-4), quoted by Christopher Janaway - Schopenhauer 7 'Freedom' | |
| A reaction: The point is that we have this feeling even if we do not believe in free will. I am struck by the fact that responsibility is very obvious in our own case, even if it is not when we objectively consider other people. Even villains can feel guilty. |
| 21375 | Reason can be vicious, and great crimes have to be rational [Schopenhauer] |
| Full Idea: Reasonable and vicious are quite consistent with each other, in fact, only through their union are great and far-reaching crimes possible. | |
| From: Arthur Schopenhauer (On the Basis of Morality [1841], p.83), quoted by Christopher Janaway - Schopenhauer 7 'Against' | |
| A reaction: This is opposed to Kant, who always looks wildly optimistic in his hope that high rationality entails a morally good will. Good people seem to have a fairly irrational empathy with their fellow citizens. |
| 21915 | To deduce morality from reason is blasphemy, because it is holy, and far above reason [Schopenhauer] |
| Full Idea: To deduce from reason [Vernunft] the moral element in conduct is blasphemy. In this element there is expressed the better consciousness which lies far above all reason. expresses itself in conduct as holiness, and is the true salvation of the world. | |
| From: Arthur Schopenhauer (Manuscript remains [1855], I p.47), quoted by Peter B. Lewis - Schopenhauer 3 | |
| A reaction: Aimed at Kant. Only Plato could inspire a non-religious person to write about morality is such terms. Maybe also the stoic ideal of beautiful deeds (given the supreme value Schopenhauer placed on the arts). |
| 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. |
| 21379 | Man's three basic ethical incentives are egoism, malice and compassion [Schopenhauer] |
| Full Idea: Man's three fundamental ethical incentives, egoism, malice and compassion, are present in everyone in different and incredibly unequal proportions. In accordance with them, motives will operate on man and actions will ensue. | |
| From: Arthur Schopenhauer (On the Basis of Morality [1841], p.192), quoted by Christopher Janaway - Schopenhauer 7 'Egoism' | |
| A reaction: A well chosen trio. Kant would be shocked that he has left out duty, which is supposed to rise above such feelings. |
| 4181 | Every good is essentially relative, for it has its essential nature only in its relation to a desiring will [Schopenhauer] |
| Full Idea: Every good is essentially relative, for it has its essential nature only in its relation to a desiring will. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], IV.65) | |
| A reaction: A nice way of stating the core of moral relativism. To me, though, it just seems a rejection of morality. Conflicting wills bring moral paralysis. Might is right. |
| 5649 | Will casts aside each of its temporary fulfilments, so human life has no ultimate aim [Schopenhauer, by Scruton] |
| Full Idea: Since for Schopenhauer will has no intrinsic end, but breaks through all its temporary fulfilments and casts them aside as irrelevant once attained, it becomes impossible to assert that there is any ultimate aim to human activity. | |
| From: report of Arthur Schopenhauer (The World as Will and Idea [1819]) by Roger Scruton - Short History of Modern Philosophy Ch.13 | |
| A reaction: This sums up part of the modern anti-teleological view of life, with its notion of purposes which can only arise out of consciousnesses. Such a view leaves untouched the key question, which is "What should I will?" |
| 4177 | Most people would probably choose non-existence at the end of their life, rather than relive the whole thing [Schopenhauer] |
| Full Idea: Perhaps no one at the end of his life, if he gives the matter sober consideration and is, at the same time, frank, ever wishes to live it over again; he more readily chooses non-existence. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], IV.59) | |
| A reaction: Hence Nietzsche's doctrine of 'eternal return' (Gay Science §341, idea 2936). From Schopenhauer it is just bleak pessimism, but from Nietzsche it is a wonderful challenge to live, perhaps the best ever. |
| 4185 | Altruistic people make less distinction than usual between themselves and others [Schopenhauer] |
| Full Idea: If we observe an altruistic action the simplest explanation and the essential character of the person's conduct is that they make less distinction than is usually made between themselves and others. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], IV.66) | |
| A reaction: Obvious, really, but Schopenhauer is talking about the will. Is the effacement of the Self desirable, apart from the benefit it might bring to other people. I don't find it appealing. |
| 4183 | Only self-love can motivate morality, but that also makes it worthless [Schopenhauer] |
| Full Idea: A theory of morals which motivates can only do so by working on self-love, but what springs from this latter has no moral worth. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], IV.66) | |
| A reaction: I just don't believe this pessimism. Schopenhauer was an incipient social darwinist who needed a course in modern game theory. Or he just needed to be a nicer man. |
| 4172 | Happiness is the swift movement from desire to satisfaction, and then again on to desire [Schopenhauer] |
| Full Idea: We are fortunate if we keep up the game whereby desire passes into satisfaction, and satisfaction into new desire - if the pace of this is swift, it is called happiness, and if it is slow, sorrow. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], II 029) | |
| A reaction: This seems to be the dream of the addict, as Socrates points out with his example of the leaky jar in 'Gorgias'. Should we want more desires? |
| 21371 | We can never attain happiness while our will is pursuing desires [Schopenhauer] |
| Full Idea: So long as our consciousness is filled by our will, so long as we are given up to the throng of desires with its constant hopes and fears, so long as we are the subject of willing, we never attain lasting happiness or peace. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], I 196), quoted by Christopher Janaway - Schopenhauer 6 'Aesthetic' | |
| A reaction: I hate this idea. It obviously leads to his Buddhism, and the eastern idea that life is generally a bad idea and to be avoided. I think Nietzsche rebelled strongly against this attitude of Schopenhauer's. |
| 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. |
| 4184 | Virtue must spring from an intuitive recognition that other people are essentially like us [Schopenhauer] |
| Full Idea: Virtue must spring from that intuitive knowledge which recognises in the individuality of others the same essence as in our own. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], IV.66) | |
| A reaction: After all his pessimism, he arrives at a view similar to Hume's, that morality is built on natural empathy. But why built a moral theory on one base. Everything points us towards morality! Moral actions are more beautiful. |
| 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. |
| 21376 | Philosophy treats animals as exploitable things, ignoring the significance of their lives [Schopenhauer] |
| Full Idea: In philosophical morals animals are mere 'things', mere means to any end whatsoever. ...Shame on such a morality, that fails to recognise the eternal essence that lives in every living thing, and shines forth with inscrutable significance from all eyes. | |
| From: Arthur Schopenhauer (On the Basis of Morality [1841], p.96), quoted by Christopher Janaway - Schopenhauer 7 'Against' | |
| A reaction: Good. I find Kant's theoretical indifference to animals very creepy (despite his kind attitude to them). And I also think the utilitarians are wrong to only value animals for their pain, as if any animal could be shredded for fun, if it felt no pain. |
| 4179 | The essence of nature is the will to life itself [Schopenhauer] |
| Full Idea: The essence of nature is the will to life itself. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], IV.60) | |
| A reaction: How would he have responded to Darwin? The will to life is the product, there, of a different and more remote force, such as the 'energy' of the physicist (whatever that is!). |
| 4189 | Time may be defined as the possibility of mutually exclusive conditions of the same thing [Schopenhauer] |
| Full Idea: Time may be defined as the possibility of mutually exclusive conditions of the same thing. | |
| From: Arthur Schopenhauer (Abstract of 'The Fourfold Root' [1813], Ch.IV) | |
| A reaction: An off-beat philosophical view of the question. Sounds more like a consequence of time than its essential nature. |
| 4178 | Christianity is a pessimistic religion, in which the world is equated with evil [Schopenhauer] |
| Full Idea: Let no one think that Christianity is conducive to optimism; on the contrary, in the Gospels 'world' and 'evil' are used almost synonymously. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], IV.59) | |
| A reaction: The source of Nietzsche's 'Beyond Good and Evil'. Do any religions throw you vigorously back into the middle of life, with its conflict and creativity? |
| 4180 | Religion is the mythical clothing of the truth which is inaccessible to the crude human intellect [Schopenhauer] |
| Full Idea: All systems of religion are the mythical clothing of the truth which is inaccessible to the crude human intellect. | |
| From: Arthur Schopenhauer (The World as Will and Idea [1819], IV.63) | |
| A reaction: Is this a compliment? It seems to be, because at least the mysteries are identified and given an outward form. A nice thought. |
| 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? |