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72 ideas

1. Philosophy / B. History of Ideas / 5. Later European Thought
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
1. Philosophy / C. History of Philosophy / 4. Later European Philosophy / d. Nineteenth century philosophy
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.
1. Philosophy / E. Nature of Metaphysics / 1. Nature of Metaphysics
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]
1. Philosophy / E. Nature of Metaphysics / 3. Metaphysical Systems
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.
1. Philosophy / E. Nature of Metaphysics / 7. Against Metaphysics
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
2. Reason / B. Laws of Thought / 2. Sufficient Reason
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!]
2. Reason / C. Styles of Reason / 1. Dialectic
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.
3. Truth / F. Semantic Truth / 1. Tarski's Truth / b. Satisfaction and truth
13634 Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro]
     Full Idea: In a sense, satisfaction is the notion of 'truth in a model', and (as Hodes 1984 elegantly puts it) 'truth in a model' is a model of 'truth'.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1)
     A reaction: So we can say that Tarski doesn't offer a definition of truth itself, but replaces it with a 'model' of truth.
4. Formal Logic / A. Syllogistic Logic / 1. Aristotelian Logic
13643 Aristotelian logic is complete [Shapiro]
     Full Idea: Aristotelian logic is complete.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.5)
     A reaction: [He cites Corcoran 1972]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
13651 A set is 'transitive' if contains every member of each of its members [Shapiro]
     Full Idea: If, for every b∈d, a∈b entails that a∈d, the d is said to be 'transitive'. In other words, d is transitive if it contains every member of each of its members.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.2)
     A reaction: The alternative would be that the members of the set are subsets, but the members of those subsets are not themselves members of the higher-level set.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
13647 Choice is essential for proving downward Löwenheim-Skolem [Shapiro]
     Full Idea: The axiom of choice is essential for proving the downward Löwenheim-Skolem Theorem.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1)
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
13631 Are sets part of logic, or part of mathematics? [Shapiro]
     Full Idea: Is there a notion of set in the jurisdiction of logic, or does it belong to mathematics proper?
     From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
     A reaction: It immediately strikes me that they might be neither. I don't see that relations between well-defined groups of things must involve number, and I don't see that mapping the relations must intrinsically involve logical consequence or inference.
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
13654 It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro]
     Full Idea: In set theory it is central to the iterative conception that the membership relation is well-founded, ...which means there are no infinite descending chains from any relation.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.4)
13640 Russell's paradox shows that there are classes which are not iterative sets [Shapiro]
     Full Idea: The argument behind Russell's paradox shows that in set theory there are logical sets (i.e. classes) that are not iterative sets.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.3)
     A reaction: In his preface, Shapiro expresses doubts about the idea of a 'logical set'. Hence the theorists like the iterative hierarchy because it is well-founded and under control, not because it is comprehensive in scope. See all of pp.19-20.
13666 Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro]
     Full Idea: Iterative sets do not exhibit a Boolean structure, because the complement of an iterative set is not itself an iterative set.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1)
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
13653 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro]
     Full Idea: A 'well-ordering' of a set X is an irreflexive, transitive, and binary relation on X in which every non-empty subset of X has a least element.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.3)
     A reaction: So there is a beginning, an ongoing sequence, and no retracing of steps.
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
13627 There is no 'correct' logic for natural languages [Shapiro]
     Full Idea: There is no question of finding the 'correct' or 'true' logic underlying a part of natural language.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
     A reaction: One needs the context of Shapiro's defence of second-order logic to see his reasons for this. Call me romantic, but I retain faith that there is one true logic. The Kennedy Assassination problem - can't see the truth because drowning in evidence.
13642 Logic is the ideal for learning new propositions on the basis of others [Shapiro]
     Full Idea: A logic can be seen as the ideal of what may be called 'relative justification', the process of coming to know some propositions on the basis of others.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.3.1)
     A reaction: This seems to be the modern idea of logic, as opposed to identification of a set of 'logical truths' from which eternal necessities (such as mathematics) can be derived. 'Know' implies that they are true - which conclusions may not be.
5. Theory of Logic / A. Overview of Logic / 2. History of Logic
13668 Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro]
     Full Idea: Bernays (1918) formulated and proved the completeness of propositional logic, the first precise solution as part of the Hilbert programme.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2.1)
13669 Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro]
     Full Idea: In 1910 Weyl observed that set theory seemed to presuppose natural numbers, and he regarded numbers as more fundamental than sets, as did Fraenkel. Dedekind had developed set theory independently, and used it to formulate numbers.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2.2)
13667 Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro]
     Full Idea: Skolem and Gödel were the main proponents of first-order languages. The higher-order language 'opposition' was championed by Zermelo, Hilbert, and Bernays.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.2)
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
13662 First-order logic was an afterthought in the development of modern logic [Shapiro]
     Full Idea: Almost all the systems developed in the first part of the twentieth century are higher-order; first-order logic was an afterthought.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1)
13624 The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro]
     Full Idea: The 'triumph' of first-order logic may be related to the remnants of failed foundationalist programmes early this century - logicism and the Hilbert programme.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
     A reaction: Being complete must also be one of its attractions, and Quine seems to like it because of its minimal ontological commitment.
13660 Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro]
     Full Idea: Tharp (1975) suggested that compactness, semantic effectiveness, and the Löwenheim-Skolem properties are consequences of features one would want a logic to have.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5)
     A reaction: I like this proposal, though Shapiro is strongly against. We keep extending our logic so that we can prove new things, but why should we assume that we can prove everything? That's just what Gödel suggests that we should give up on.
13673 The notion of finitude is actually built into first-order languages [Shapiro]
     Full Idea: The notion of finitude is explicitly 'built in' to the systems of first-order languages in one way or another.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1)
     A reaction: Personally I am inclined to think that they are none the worse for that. No one had even thought of all these lovely infinities before 1870, and now we are supposed to change our logic (our actual logic!) to accommodate them. Cf quantum logic.
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
15944 Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine]
     Full Idea: Shapiro preferred second-order logic to set theory because second-order logic refers only to the relations and operations in a domain, and not to the other things that set-theory brings with it - other domains, higher-order relations, and so forth.
     From: report of Stewart Shapiro (Foundations without Foundationalism [1991]) by Shaughan Lavine - Understanding the Infinite VII.4
13629 Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro]
     Full Idea: Three systems of semantics for second-order languages: 'standard semantics' (variables cover all relations and functions), 'Henkin semantics' (relations and functions are a subclass) and 'first-order semantics' (many-sorted domains for variable-types).
     From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
     A reaction: [my summary]
13650 Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro]
     Full Idea: In 'Henkin' semantics, in a given model the relation variables range over a fixed collection of relations D on the domain, and the function variables range over a collection of functions F on the domain.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 3.3)
13645 In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro]
     Full Idea: In the standard semantics of second-order logic, by fixing a domain one thereby fixes the range of both the first-order variables and the second-order variables. There is no further 'interpreting' to be done.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 3.3)
     A reaction: This contrasts with 'Henkin' semantics (Idea 13650), or first-order semantics, which involve more than one domain of quantification.
13649 Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro]
     Full Idea: The counterparts of Completeness, Compactness and the Löwenheim-Skolem theorems all fail for second-order languages with standard semantics, but hold for Henkin or first-order semantics. Hence such logics are much like first-order logic.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1)
     A reaction: Shapiro votes for the standard semantics, because he wants the greater expressive power, especially for the characterization of infinite structures.
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
13626 Semantic consequence is ineffective in second-order logic [Shapiro]
     Full Idea: It follows from Gödel's incompleteness theorem that the semantic consequence relation of second-order logic is not effective. For example, the set of logical truths of any second-order logic is not recursively enumerable. It is not even arithmetic.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
     A reaction: I don't fully understand this, but it sounds rather major, and a good reason to avoid second-order logic (despite Shapiro's proselytising). See Peter Smith on 'effectively enumerable'.
13637 If a logic is incomplete, its semantic consequence relation is not effective [Shapiro]
     Full Idea: Second-order logic is inherently incomplete, so its semantic consequence relation is not effective.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.2.1)
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
13632 Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro]
     Full Idea: It is sometimes difficult to find a formula that is a suitable counterpart of a particular sentence of natural language, and there is no acclaimed criterion for what counts as a good, or even acceptable, 'translation'.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1)
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / c. not
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.
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
13674 We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro]
     Full Idea: The main role of substitutional semantics is to reduce ontology. As an alternative to model-theoretic semantics for formal languages, the idea is to replace the 'satisfaction' relation of formulas (by objects) with the 'truth' of sentences (using terms).
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4)
     A reaction: I find this very appealing, and Ruth Barcan Marcus is the person to look at. My intuition is that logic should have no ontology at all, as it is just about how inference works, not about how things are. Shapiro offers a compromise.
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
13633 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro]
     Full Idea: The 'satisfaction' relation may be thought of as a function from models, assignments, and formulas to the truth values {true,false}.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1)
     A reaction: This at least makes clear that satisfaction is not the same as truth. Now you have to understand how Tarski can define truth in terms of satisfaction.
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
13644 Semantics for models uses set-theory [Shapiro]
     Full Idea: Typically, model-theoretic semantics is formulated in set theory.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.5.1)
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
13636 An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro]
     Full Idea: An axiomatization is 'categorical' if all its models are isomorphic to one another; ..hence it has 'essentially only one' interpretation [Veblen 1904].
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.2.1)
13670 Categoricity can't be reached in a first-order language [Shapiro]
     Full Idea: Categoricity cannot be attained in a first-order language.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.3)
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
13648 The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro]
     Full Idea: The Löwenheim-Skolem theorems mean that no first-order theory with an infinite model is categorical. If Γ has an infinite model, then it has a model of every infinite cardinality. So first-order languages cannot characterize infinite structures.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1)
     A reaction: So much of the debate about different logics hinges on characterizing 'infinite structures' - whatever they are! Shapiro is a leading structuralist in mathematics, so he wants second-order logic to help with his project.
13675 Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro]
     Full Idea: The Upward Löwenheim-Skolem theorem fails (trivially) with substitutional semantics. If there are only countably many terms of the language, then there are no uncountable substitution models.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4)
     A reaction: Better and better. See Idea 13674. Why postulate more objects than you can possibly name? I'm even suspicious of all real numbers, because you can't properly define them in finite terms. Shapiro objects that the uncountable can't be characterized.
13658 Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro]
     Full Idea: A language has the Downward Löwenheim-Skolem property if each satisfiable countable set of sentences has a model whose domain is at most countable.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5)
     A reaction: This means you can't employ an infinite model to represent a fact about a countable set.
13659 Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro]
     Full Idea: A language has the Upward Löwenheim-Skolem property if for each set of sentences whose model has an infinite domain, then it has a model at least as big as each infinite cardinal.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5)
     A reaction: This means you can't have a countable model to represent a fact about infinite sets.
5. Theory of Logic / K. Features of Logics / 3. Soundness
13635 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro]
     Full Idea: A logic is 'weakly sound' if every theorem is a logical truth, and 'strongly sound', or simply 'sound', if every deduction from Γ is a semantic consequence of Γ. Soundness indicates that the deductive system is faithful to the semantics.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.1)
     A reaction: Similarly, 'weakly complete' is when every logical truth is a theorem.
5. Theory of Logic / K. Features of Logics / 4. Completeness
13628 We can live well without completeness in logic [Shapiro]
     Full Idea: We can live without completeness in logic, and live well.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
     A reaction: This is the kind of heady suggestion that American philosophers love to make. Sounds OK to me, though. Our ability to draw good inferences should be expected to outrun our ability to actually prove them. Completeness is for wimps.
5. Theory of Logic / K. Features of Logics / 6. Compactness
13630 Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro]
     Full Idea: It is sometimes said that non-compactness is a defect of second-order logic, but it is a consequence of a crucial strength - its ability to give categorical characterisations of infinite structures.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
     A reaction: The dispute between fans of first- and second-order may hinge on their attitude to the infinite. I note that Skolem, who was not keen on the infinite, stuck to first-order. Should we launch a new Skolemite Crusade?
13646 Compactness is derived from soundness and completeness [Shapiro]
     Full Idea: Compactness is a corollary of soundness and completeness. If Γ is not satisfiable, then, by completeness, Γ is not consistent. But the deductions contain only finite premises. So a finite subset shows the inconsistency.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 4.1)
     A reaction: [this is abbreviated, but a proof of compactness] Since all worthwhile logics are sound, this effectively means that completeness entails compactness.
5. Theory of Logic / K. Features of Logics / 9. Expressibility
13661 A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro]
     Full Idea: A logical language is 'semantically effective' if the collection of logically true sentences is a recursively enumerable set of strings.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 6.5)
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
13641 Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro]
     Full Idea: 'Definitions' of integers as pairs of naturals, rationals as pairs of integers, reals as Cauchy sequences of rationals, and complex numbers as pairs of reals are reductive foundations of various fields.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 2.1)
     A reaction: On p.30 (bottom) Shapiro objects that in the process of reduction the numbers acquire properties they didn't have before.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
13676 Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro]
     Full Idea: The main problem of characterizing the natural numbers is to state, somehow, that 0,1,2,.... are all the numbers that there are. We have seen that this can be accomplished with a higher-order language, but not in a first-order language.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4)
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13677 Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro]
     Full Idea: By convention, the natural numbers are the finite ordinals, the integers are certain equivalence classes of pairs of finite ordinals, etc.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 9.3)
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
13652 The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro]
     Full Idea: The 'continuum' is the cardinality of the powerset of a denumerably infinite set.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.2)
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
13657 First-order arithmetic can't even represent basic number theory [Shapiro]
     Full Idea: Few theorists consider first-order arithmetic to be an adequate representation of even basic number theory.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 5 n28)
     A reaction: This will be because of Idea 13656. Even 'basic' number theory will include all sorts of vast infinities, and that seems to be where the trouble is.
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
13656 Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro]
     Full Idea: There are sets of natural numbers definable in set-theory but not in arithmetic.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.3.3)
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
13664 Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro]
     Full Idea: It is claimed that aiming at a universal language for all contexts, and the thesis that logic does not involve a process of abstraction, separates the logicists from algebraists and mathematicians, and also from modern model theory.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1)
     A reaction: I am intuitively drawn to the idea that logic is essentially the result of a series of abstractions, so this gives me a further reason not to be a logicist. Shapiro cites Goldfarb 1979 and van Heijenoort 1967. Logicists reduce abstraction to logic.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
13625 Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro]
     Full Idea: I extend Quinean holism to logic itself; there is no sharp border between mathematics and logic, especially the logic of mathematics. One cannot expect to do logic without incorporating some mathematics and accepting at least some of its ontology.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
     A reaction: I have strong sales resistance to this proposal. Mathematics may have hijacked logic and warped it for its own evil purposes, but if logic is just the study of inferences then it must be more general than to apply specifically to mathematics.
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
13663 Some reject formal properties if they are not defined, or defined impredicatively [Shapiro]
     Full Idea: Some authors (Poincaré and Russell, for example) were disposed to reject properties that are not definable, or are definable only impredicatively.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.1)
     A reaction: I take Quine to be the culmination of this line of thought, with his general rejection of 'attributes' in logic and in metaphysics.
7. Existence / A. Nature of Existence / 3. Being / c. Becoming
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'.
7. Existence / A. Nature of Existence / 5. Reason for Existence
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?
7. Existence / E. Categories / 4. Category Realism
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.
7. Existence / E. Categories / 5. Category Anti-Realism
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.
8. Modes of Existence / B. Properties / 10. Properties as Predicates
13638 Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro]
     Full Idea: Properties are often taken to be intensional; equiangular and equilateral are thought to be different properties of triangles, even though any triangle is equilateral if and only if it is equiangular.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.3)
     A reaction: Many logicians seem to want to treat properties as sets of objects (red being just the set of red things), but this looks like a desperate desire to say everything in first-order logic, where only objects are available to quantify over.
12. Knowledge Sources / A. A Priori Knowledge / 5. A Priori Synthetic
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.
23. Ethics / F. Existentialism / 1. Existentialism
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.
24. Political Theory / A. Basis of a State / 1. A People / c. A unified people
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.
24. Political Theory / B. Nature of a State / 2. State Legitimacy / c. Social contract
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?).
26. Natural Theory / A. Speculations on Nature / 1. Nature
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.
28. God / B. Proving God / 2. Proofs of Reason / a. Ontological Proof
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.
29. Religion / B. Monotheistic Religion / 4. Christianity / a. Christianity
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?
29. Religion / B. Monotheistic Religion / 4. Christianity / d. Heresy
16713 Philosophers are the forefathers of heretics [Tertullian]
     Full Idea: Philosophers are the forefathers of heretics.
     From: Tertullian (works [c.200]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.2
29. Religion / D. Religious Issues / 1. Religious Commitment / e. Fideism
6610 I believe because it is absurd [Tertullian]
     Full Idea: I believe because it is absurd ('Credo quia absurdum est').
     From: Tertullian (works [c.200]), quoted by Robert Fogelin - Walking the Tightrope of Reason n4.2
     A reaction: This seems to be a rather desperate remark, in response to what must have been rather good hostile arguments. No one would abandon the support of reason if it was easy to acquire. You can't deny its engaging romantic defiance, though.