37 ideas
| 19635 | Hegel produced modern optimism; he failed to grasp that consciousness never progresses [Hegel, by Cioran] |
| Full Idea: Hegel is chiefly responsible for modern optimism. How could he have failed to see that consciousness changes only its forms and modalities, but never progresses. | |
| From: report of Georg W.F.Hegel (works [1812]) by E.M. Cioran - A Short History of Decay 5 |
| 8215 | Hegel was the last philosopher of the Book [Hegel, by Derrida] |
| Full Idea: Hegel was the last philosopher of the Book. | |
| From: report of Georg W.F.Hegel (works [1812]) by Jacques Derrida - Positions p.64 | |
| A reaction: Reference to 'the Book' connects this to the great religions which rely on one holy text. The implication is that Hegel was proposing one big solution to all problems. It is doubtful if many philosophers before Hegel dreamt of that either. |
| 16011 | Hegel doesn't storm the heavens like the giants, but works his way up by syllogisms [Kierkegaard on Hegel] |
| Full Idea: Hegel is a Johannes Climacus who does not storm the heavens, like the giants, by putting mountain upon mountain, but climbs aboard them by way of his syllogisms. | |
| From: comment on Georg W.F.Hegel (works [1812]) by Søren Kierkegaard - The Journals of Kierkegaard 2A | |
| A reaction: [Idea from SY] This appears to be an attempt at insulting Hegel for his timidity, but it seems to be describing the cautious approach which most modern philosophers take to be correct. [PG] |
| 5433 | For Hegel, things are incomplete, and contain external references in their own nature [Hegel, by Russell] |
| Full Idea: The basis of Hegel's system is that what is incomplete must not be self-subsistent, and needs the support of other things; whatever has relations to things outside itself must contain some reference to those outside things in its own nature. | |
| From: report of Georg W.F.Hegel (works [1812]) by Bertrand Russell - Problems of Philosophy Ch.14 | |
| A reaction: This leads to the idealist doctrine of 'internal relations'. It has some plausibility if you think about the physicist's definition of mass, which has to refer to forces etc. Presumably there is one essence for all of reality, instead of separate ones. |
| 3301 | On the continent it is generally believed that metaphysics died with Hegel [Benardete,JA on Hegel] |
| Full Idea: In continental Europe it is widely believed that the metaphysical game was played out in Hegel. | |
| From: comment on Georg W.F.Hegel (works [1812]) by José A. Benardete - Metaphysics: the logical approach Intro |
| 19661 | Making sufficient reason an absolute devalues the principle of non-contradiction [Hegel, by Meillassoux] |
| Full Idea: Hegel saw that the absolutization of the principle of sufficient reason (which marked the culmination of the belief in the necessity of what is) required the devaluation of the principle of non-contradiction. | |
| From: report of Georg W.F.Hegel (works [1812], 3) by Quentin Meillassoux - After Finitude; the necessity of contingency 3 | |
| A reaction: I pass this on without understanding it, though a joint study of my collection of ideas on sufficient reason and non-contradiction might make it clear. [Let me know if you can explain it!] |
| 20952 | Rather than in three stages, Hegel presented his dialectic as 'negation of the negation' [Hegel, by Bowie] |
| Full Idea: Hegel's 'dialectic' is often characterised in terms of the triad of thesis, antithesis and synthesis. This is, however, not the way he presents it. The core of the dialectic is rather what Hegel terms the 'negation of the negation'. | |
| From: report of Georg W.F.Hegel (works [1812]) by Andrew Bowie - Introduction to German Philosophy | |
| A reaction: Interestingly, this connects it to debates about intuitionist logic, which denies that double-negation necessarily makes a positive. Presumably Marx emphasised the first reading. |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
| 21777 | Negation of negation doubles back into a self-relationship [Hegel, by Houlgate] |
| Full Idea: For Hegel, the 'negation of negation' is negation that, as it were, doubles back on itself and 'relates itself to itself'. | |
| From: report of Georg W.F.Hegel (works [1812]) by Stephen Houlgate - An Introduction to Hegel 6 'Space' | |
| A reaction: [ref VNP 1823 p.108] Glad we've cleared that one up. |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
| A reaction: The sequence is 'Cauchy' if N exists. |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
| A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
| A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
| A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
| A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
| A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
| A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
| A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
| 5645 | The dialectical opposition of being and nothing is resolved in passing to the concept of becoming [Hegel, by Scruton] |
| Full Idea: The concept of being contains within itself it own negation - nothing - and the dialectical opposition between these two concepts is resolved only in the passage to a new concept, becoming, which contains the truth of the passage from nothing to being. | |
| From: report of Georg W.F.Hegel (works [1812]) by Roger Scruton - Short History of Modern Philosophy Ch.12 | |
| A reaction: The idea that one concept 'contains' another, or that an opposition could be 'resolved' by a new concept, sounds doubtful to me. For most analytical philosophers, and for Aristotle, oppositions are contradictions, and cannot and should not be 'resolved'. |
| 5646 | Hegel gives an ontological proof of the existence of everything [Hegel, by Scruton] |
| Full Idea: It would not be unfair to say that Hegel's metaphysics consists of an ontological proof of the existence of everything. | |
| From: report of Georg W.F.Hegel (works [1812]) by Roger Scruton - Short History of Modern Philosophy Ch.12 | |
| A reaction: This is so gloriously far from David Hume that we must all find some appeal in it. The next question would be whether necessary existence has been proved. If so, given death, decay and entropy, what is it that has to exist? 2nd Law of Thermodynamics? |
| 21755 | For Hegel, categories shift their form in the course of history [Hegel, by Houlgate] |
| Full Idea: For Hegel, the categories of thought are not fixed, eternal forms that remain unchanged throughout history, but are concepts that alter their meaning in history. | |
| From: report of Georg W.F.Hegel (works [1812]) by Stephen Houlgate - An Introduction to Hegel 01 | |
| A reaction: This results from a critique of Kant's rather rigid view of categories. This idea is very influential, and certainly counts among Hegel's better ideas. |
| 21754 | Our concepts and categories disclose the world, because we are part of the world [Hegel, by Houlgate] |
| Full Idea: For Hegel, the structure of our concepts and categories is identical with, and thus discloses, the structure of the world itself, because we ourselves are born into and so share the character of the world we encounter. | |
| From: report of Georg W.F.Hegel (works [1812]) by Stephen Houlgate - An Introduction to Hegel 01 | |
| A reaction: This is a reasonable speculation, but it makes more sense in the context of natural selection, and an empiricist theory of concepts. |
| 22079 | Hegel said Kant's fixed categories actually vary with culture and era [Hegel, by Houlgate] |
| Full Idea: Hegel's disagreement with Kant is that categories are not unambiguously universal forms of human understanding, but are conceived in subtly different ways in different cultures and in different historical epochs. | |
| From: report of Georg W.F.Hegel (works [1812]) by Stephen Houlgate - Hegel p.95 | |
| A reaction: This may be Hegel's most influential idea. Though he hoped that categories would contain truth, by arising untrammelled from reason, and thereby matching reality. His successors seem to have given up on that hope, and settled for relativism. |
| 594 | Speusippus suggested underlying principles for every substance, and ended with a huge list [Speussipus, by Aristotle] |
| Full Idea: Speusippus suggested principles for each substance, including principles for numbers, magnitude and the soul. He thus arrived at no mean list of substances. | |
| From: report of Speussipus (thirty titles (lost) [c.367 BCE]) by Aristotle - Metaphysics 1028b |
| 9225 | Hegel reputedly claimed to know a priori that there are five planets [Hegel, by Field,H] |
| Full Idea: Hegel is reputed to have claimed to have deduced on a priori grounds that the number of planets is exactly five. | |
| From: report of Georg W.F.Hegel (works [1812]) by Hartry Field - Recent Debates on the A Priori 1 | |
| A reaction: Even if this is a wicked travesty of Hegel, it will do nicely to represent the extremes of claims to a priori synthetic knowledge. Field doesn't offer any evidence. I would love it to be true. |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
| A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
| 21758 | Humans have no fixed identity, but produce and reveal their shifting identity in history [Hegel, by Houlgate] |
| Full Idea: For Hegel, the absolute truth of humanity is that human beings have no fixed, given identity, but rather determine and produce their own identity and their world in history, and that they gradually come to the recognition of this fact in history. | |
| From: report of Georg W.F.Hegel (works [1812]) by Stephen Houlgate - An Introduction to Hegel 01 | |
| A reaction: This quintessentially existentialist idea, most obvious in Sartre, seems to have originated with this view of Hegel's. |
| 20414 | Hegel's Absolute Spirit is the union of human rational activity at a moment, and whatever that sustains [Hegel, by Eldridge] |
| Full Idea: We may take Hegel's Absolute Spirit to be the union of collective, human rational activity at a historical moment with its proper object, the forms of social and individual life that the rational activity is devoted to understanding and sustaining. | |
| From: report of Georg W.F.Hegel (works [1812]) by Richard Eldridge - G.W.F. Hegel (aesthetics) 1 | |
| A reaction: From this formulation it sounds as if the whole human race might have momentary union, but presumably it is more local 'peoples' that can exhibit this. |
| 3909 | Society isn’t founded on a contract, since contracts presuppose a society [Hegel, by Scruton] |
| Full Idea: For Hegel, society cannot be founded on a contract, since contracts have no reality until society is in place. | |
| From: report of Georg W.F.Hegel (works [1812]) by Roger Scruton - Modern Philosophy:introduction and survey 28.2 | |
| A reaction: Interesting, and reminiscent of the private language argument, but contracts surely start as deals between individuals (on a desert island?). |
| 4347 | When man wills the natural, it is no longer natural [Hegel] |
| Full Idea: When man wills the natural, it is no longer natural. | |
| From: Georg W.F.Hegel (works [1812]), quoted by Rosalind Hursthouse - On Virtue Ethics Ch.4 | |
| A reaction: Sounds good, though I'm not sure what it means. The application of the word 'natural' seems a bit arbitrary to me. No objective joint exists between the natural and unnatural. The default position has to be that everything is natural. |
| 4188 | Hegel's entire philosophy is nothing but a monstrous amplification of the ontological proof [Schopenhauer on Hegel] |
| Full Idea: Hegel's entire philosophy is nothing but a monstrous amplification of the ontological proof. | |
| From: comment on Georg W.F.Hegel (works [1812]) by Arthur Schopenhauer - Abstract of 'The Fourfold Root' Ch.II | |
| A reaction: All massive a priori metaphysics is summed up in this argument, which is right at the core of philosophy. |
| 2632 | Speusippus said things were governed by some animal force rather than the gods [Speussipus, by Cicero] |
| Full Idea: Speusippus, following his uncle Plato, held that all things were governed by some kind of animal force, and tried to eradicate from our minds any notion of the gods. | |
| From: report of Speussipus (thirty titles (lost) [c.367 BCE]) by M. Tullius Cicero - On the Nature of the Gods ('De natura deorum') I.33 |
| 6686 | Hegel said he was offering an encyclopaedic rationalisation of Christianity [Hegel, by Graham] |
| Full Idea: Hegel claimed that his philosophy was nothing less than an encyclopaedic rationalisation of the Christian religion. | |
| From: report of Georg W.F.Hegel (works [1812]) by Gordon Graham - Eight Theories of Ethics Ch.5 | |
| A reaction: Why did he pick Christianity to rationalise? How can you reason properly if you start with a dogma? |