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All the ideas for 'fragments/reports', 'Reality without Reference' and 'Philosophy of Mathematics'

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129 ideas

1. Philosophy / A. Wisdom / 1. Nature of Wisdom
Wisdom for one instant is as good as wisdom for eternity [Chrysippus]
     Full Idea: If a person has wisdom for one instant, he is no less happy than he who possesses it for eternity.
     From: Chrysippus (fragments/reports [c.240 BCE]), quoted by Pierre Hadot - Philosophy as a way of life 8
     A reaction: [Hadot quotes Plutarch 'On Common Conceptions' 8,1062a] This makes it sound awfully like some sort of Buddhist 'enlightenment', which strikes like lightning. He does wisdom recognise itself - by a warm glow, or by the cautious thought that got you there?
1. Philosophy / A. Wisdom / 2. Wise People
Wise men should try to participate in politics, since they are a good influence [Chrysippus, by Diog. Laertius]
     Full Idea: The wise man will participate in politics unless something prevents him, for he will restrain vice and promote virtue.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.121
     A reaction: [from lost On Ways of Life Bk 1] We have made modern politics so hostile for its participants, thanks to cruel media pressure, that the best people now run a mile from it. Disastrous.
1. Philosophy / D. Nature of Philosophy / 4. Divisions of Philosophy
Three branches of philosophy: first logic, second ethics, third physics (which ends with theology) [Chrysippus]
     Full Idea: There are three kinds of philosophical theorems, logical, ethical, and physical; of these the logic should be placed first, ethics second, and physics third (and theology is the final topic in physics).
     From: Chrysippus (fragments/reports [c.240 BCE]), quoted by Plutarch - 70: Stoic Self-contradictions 1035a
     A reaction: [in his lost 'On Lives' Bk 4] 'Theology is the final topic in physics'! That should create a stir in theology departments. Is this an order of study, or of importance? You come to theology right at the end of your studies.
2. Reason / A. Nature of Reason / 6. Coherence
Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro]
     Full Idea: I take 'coherence' to be a primitive, intuitive notion, not reduced to something formal, and so I do not venture a rigorous definition of it.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8)
     A reaction: I agree strongly with this. Best to talk of 'the space of reasons', or some such. Rationality extends far beyond what can be formally defined. Coherence is the last court of appeal in rational thought.
2. Reason / B. Laws of Thought / 2. Sufficient Reason
Chrysippus said the uncaused is non-existent [Chrysippus, by Plutarch]
     Full Idea: Chrysippus said that the uncaused is altogether non-existent.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Plutarch - 70: Stoic Self-contradictions 1045c
     A reaction: The difficulty is to see what empirical basis there can be for such a claim, or what argument of any kind other than an intuition. Induction is the obvious answer, but Hume teaches us scepticism about any claim that 'there can be no exceptions'.
2. Reason / D. Definition / 7. Contextual Definition
An 'implicit definition' gives a direct description of the relations of an entity [Shapiro]
     Full Idea: An 'implicit definition' characterizes a structure or class of structures by giving a direct description of the relations that hold among the places of the structure.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
     A reaction: This might also be thought of as a 'functional definition', since it seems to say what the structure or entity does, rather than give the intrinsic characteristics that make its relations and actions possible.
3. Truth / B. Truthmakers / 10. Making Future Truths
The causes of future true events must exist now, so they will happen because of destiny [Chrysippus, by Cicero]
     Full Idea: True future events cannot be such as do not possess causes on account of which they will happen; therefore that which is true must possess causes: and so, when the [true future events] happen they will have happened as a result of destiny.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by M. Tullius Cicero - On Fate ('De fato') 9.23-8
     A reaction: [exact ref unclear] Presumably the current causes are the truthmakers for the future events, and so the past is the truthmaker of the future, if you are a determinist.
3. Truth / C. Correspondence Truth / 2. Correspondence to Facts
Graspable presentations are criteria of facts, and are molded according to their objects [Chrysippus, by Diog. Laertius]
     Full Idea: Of presentations, some are graspable, some non-graspable. The graspable presentation, which they say is the criterion of facts [pragmata], is that which comes from an existing object and is stamped and molded in accordance wth the existing object itself.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.46
     A reaction: [in lost Physics Bk 2] The big modern anguish over truth-as-correspondence is how you are supposed to verify the 'accordance'. This idea seems to blur the ideas of truth and justification (the 'criterion'), and you can't have both as accordance.
3. Truth / C. Correspondence Truth / 3. Correspondence Truth critique
How could you ever know that the presentation is similar to the object? [Sext.Empiricus on Chrysippus]
     Full Idea: One cannot say that the soul grasps the externally existing objects by means of the states of the senses on the basis of the similarity of these states to the externally existing objects. For on what basis will it know the similarity?
     From: comment on Chrysippus (fragments/reports [c.240 BCE]) by Sextus Empiricus - Outlines of Pyrrhonism 2.74
     A reaction: This exactly the main modern reason for rejecting the correspondence theory of truth. You are welcome to affirm a robust view of truth, but supporting it by claiming a correspondence or resemblance is dubious.
4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
Stoic propositional logic is like chemistry - how atoms make molecules, not the innards of atoms [Chrysippus, by Devlin]
     Full Idea: In Stoic logic propositions are treated the way atoms are treated in present-day chemistry, where the focus is on the way atoms fit together to form molecules, rather than on the internal structure of the atoms.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Keith Devlin - Goodbye Descartes Ch.2
     A reaction: A nice analogy to explain the nature of Propositional Logic, which was invented by the Stoics (N.B. after Aristotle had invented predicate logic).
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
Chrysippus has five obvious 'indemonstrables' of reasoning [Chrysippus, by Diog. Laertius]
     Full Idea: Chrysippus has five indemonstrables that do not need demonstration:1) If 1st the 2nd, but 1st, so 2nd; 2) If 1st the 2nd, but not 2nd, so not 1st; 3) Not 1st and 2nd, the 1st, so not 2nd; 4) 1st or 2nd, the 1st, so not 2nd; 5) 1st or 2nd, not 2nd, so 1st.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.80-81
     A reaction: [from his lost text 'Dialectics'; squashed to fit into one quote] 1) is Modus Ponens, 2) is Modus Tollens. 4) and 5) are Disjunctive Syllogisms. 3) seems a bit complex to be an indemonstrable.
4. Formal Logic / D. Modal Logic ML / 1. Modal Logic
Modal operators are usually treated as quantifiers [Shapiro]
     Full Idea: It is common now, and throughout the history of philosophy, to interpret modal operators as quantifiers. This is an analysis of modality in terms of ontology.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
Axiom of Choice: some function has a value for every set in a given set [Shapiro]
     Full Idea: One version of the Axiom of Choice says that for every set A of nonempty sets, there is a function whose domain is A and whose value, for every a ∈ A, is a member of a.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 1)
The Axiom of Choice seems to license an infinite amount of choosing [Shapiro]
     Full Idea: If the Axiom of Choice says we can choose one member from each of a set of non-empty sets and put the chosen elements together in a set, this licenses the constructor to do an infinite amount of choosing.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.3)
     A reaction: This is one reason why the Axiom was originally controversial, and still is for many philosophers.
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
Anti-realists reject set theory [Shapiro]
     Full Idea: Anti-realists reject set theory.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
     A reaction: That is, anti-realists about mathematical objects. I would have thought that one could accept an account of sets as (say) fictions, which provided interesting models of mathematics etc.
5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence
The two standard explanations of consequence are semantic (in models) and deductive [Shapiro]
     Full Idea: The two best historical explanations of consequence are the semantic (model-theoretic), and the deductive versions.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.2)
     A reaction: Shapiro points out the fictionalists are in trouble here, because the first involves commitment to sets, and the second to the existence of deductions.
5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens
Modus ponens is one of five inference rules identified by the Stoics [Chrysippus, by Devlin]
     Full Idea: Modus ponens is just one of the five different inference rules identified by the Stoics.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Keith Devlin - Goodbye Descartes Ch.2
     A reaction: Modus ponens strikes me as being more like a definition of implication than a 'rule'. Implication is what gets you from one truth to another. All the implications of a truth must also be true.
Intuitionism only sanctions modus ponens if all three components are proved [Shapiro]
     Full Idea: In some intuitionist semantics modus ponens is not sanctioned. At any given time there is likely to be a conditional such that it and its antecedent have been proved, but nobody has bothered to prove the consequent.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.7)
     A reaction: [He cites Heyting] This is a bit baffling. In what sense can 'it' (i.e. the conditional implication) have been 'proved' if the consequent doesn't immediately follow? Proving both propositions seems to make the conditional redundant.
5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
Either logic determines objects, or objects determine logic, or they are separate [Shapiro]
     Full Idea: Ontology does not depend on language and logic if either one has the objects determining the logic, or the objects are independent of the logic.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.4)
     A reaction: I favour the first option. I think we should seek an account of how logic grows from our understanding of the physical world. If this cannot be established, I shall invent a new Mad Logic, and use it for all my future reasoning, with (I trust) impunity.
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
The law of excluded middle might be seen as a principle of omniscience [Shapiro]
     Full Idea: The law of excluded middle might be seen as a principle of omniscience.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.3)
     A reaction: [E.Bishop 1967 is cited] Put that way, you can see why a lot of people (such as intuitionists in mathematics) might begin to doubt it.
Every proposition is either true or false [Chrysippus, by Cicero]
     Full Idea: We hold fast to the position, defended by Chrysippus, that every proposition is either true or false.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by M. Tullius Cicero - On Fate ('De fato') 38
     A reaction: I am intrigued to know exactly how you defend this claim. It may depend what you mean by a proposition. A badly expressed proposition may have indeterminate truth, quite apart from the vague, the undecidable etc.
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro]
     Full Idea: To some extent, every truth-functional connective differs from its counterpart in ordinary language. Classical conjunction, for example, is timeless, whereas the word 'and' often is not. 'Socrates runs and Socrates stops' cannot be reversed.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 3)
     A reaction: Shapiro suggests two interpretations: either the classical connectives are revealing the deeper structure of ordinary language, or else they are a simplification of it.
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
A function is just an arbitrary correspondence between collections [Shapiro]
     Full Idea: The modern extensional notion of function is just an arbitrary correspondence between collections.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 1)
     A reaction: Shapiro links this with the idea that a set is just an arbitrary collection. These minimalist concepts seem like a reaction to a general failure to come up with a more useful and common sense definition.
5. Theory of Logic / G. Quantification / 6. Plural Quantification
Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro]
     Full Idea: Maybe plural quantifiers should themselves be understood in terms of classes (or sets).
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.4)
     A reaction: [Shapiro credits Resnik for this criticism]
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
A sentence is 'satisfiable' if it has a model [Shapiro]
     Full Idea: Normally, to say that a sentence Φ is 'satisfiable' is to say that there exists a model of Φ.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8)
     A reaction: Nothing is said about whether the model is impressive, or founded on good axioms. Tarski builds his account of truth from this initial notion of satisfaction.
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
Model theory deals with relations, reference and extensions [Shapiro]
     Full Idea: Model theory determines only the relations between truth conditions, the reference of singular terms, the extensions of predicates, and the extensions of the logical terminology.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.9)
The central notion of model theory is the relation of 'satisfaction' [Shapiro]
     Full Idea: The central notion of model theory is the relation of 'satisfaction', sometimes called 'truth in a model'.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.9)
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro]
     Full Idea: No object-language theory determines its ontology by itself. The best possible is that all models are isomorphic, in which case the ontology is determined 'up to isomorphism', but only if the domain is finite, or it is stronger than first-order.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 2.5)
     A reaction: This seems highly significant when ontological claims are being made, and is good support for Shapiro's claim that the structures matter, not the objects. There is a parallel in Tarksi's notion of truth-in-all-models. [The Skolem Paradox is the problem]
The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro]
     Full Idea: Set theorists often point out that the set-theoretical hierarchy contains as many isomorphism types as possible; that is the point of the theory.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8)
     A reaction: Hence there are a huge number of models for any theory, which are then reduced to the one we want at the level of isomorphism.
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
Any theory with an infinite model has a model of every infinite cardinality [Shapiro]
     Full Idea: The Löwenheim-Skolem theorems (which apply to first-order formal theories) show that any theory with an infinite model has a model of every infinite cardinality.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8)
     A reaction: This aspect of the theorems is the Skolem Paradox. Shapiro argues that in first-order this infinity of models for arithmetic must be accepted, but he defends second-order model theory, where 'standard' models can be selected.
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Virtually all of mathematics can be modeled in set theory [Shapiro]
     Full Idea: It is well known that virtually every field of mathematics can be reduced to, or modelled in, set theory.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
     A reaction: The word 'virtually' is tantalising. The fact that something can be 'modeled' in set theory doesn't mean it IS set theory. Most weather can be modeled in a computer.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro]
     Full Idea: Real numbers are either Cauchy sequences of rational numbers (interpreted as pairs of integers), or else real numbers can be thought of as Dedekind cuts, certain sets of rational numbers. So π is a Dedekind cut, or an equivalence class of sequences.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 2.5)
     A reaction: This question is parallel to the question of whether natural numbers are Zermelo sets or Von Neumann sets. The famous problem is that there seems no way of deciding. Hence, for Shapiro, we are looking at models, not actual objects.
Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro]
     Full Idea: There is no more to understanding the real-number structure than knowing how to use the language of analysis. .. One learns the axioms of the implicit definition. ...These determine the realtionships between real numbers.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.9)
     A reaction: This, of course, is the structuralist view of such things, which isn't really interested in the intrinsic nature of anything, but only in its relations. The slogan that 'meaning is use' seems to be in the background.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro]
     Full Idea: A Dedekind Cut is a division of rationals into two set (A1,A2) where every member of A1 is less than every member of A2. If n is the largest A1 or the smallest A2, the cut is produced by n. Some cuts aren't produced by rationals.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.4)
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
There is no grounding for mathematics that is more secure than mathematics [Shapiro]
     Full Idea: We cannot ground mathematics in any domain or theory that is more secure than mathematics itself.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8)
     A reaction: This pronouncement comes after a hundred years of hard work, notably by Gödel, so we'd better believe it. It might explain why Putnam rejects the idea that mathematics needs 'foundations'. Personally I'm prepare to found it in countable objects.
6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
For intuitionists, proof is inherently informal [Shapiro]
     Full Idea: For intuitionists, proof is inherently informal.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.7)
     A reaction: This thought is quite appealing, so I may have to take intuitionism more seriously. It connects with my view of coherence, which I take to be a notion far too complex for precise definition. However, we don't want 'proof' to just mean 'persuasive'.
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
Natural numbers just need an initial object, successors, and an induction principle [Shapiro]
     Full Idea: The natural-number structure is a pattern common to any system of objects that has a distinguished initial object and a successor relation that satisfies the induction principle
     From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
     A reaction: If you started your number system with 5, and successors were only odd numbers, something would have gone wrong, so a bit more seems to be needed. How do we decided whether the initial object is 0, 1 or 2?
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro]
     Full Idea: Originally, the focus of geometry was space - matter and extension - and the subject matter of arithmetic was quantity. Geometry concerned the continuous, whereas arithmetic concerned the discrete. Mathematics left these roots in the nineteenth century.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
     A reaction: Mathematicians can do what they like, but I don't think philosophers of mathematics should lose sight of these two roots. It would be odd if the true nature of mathematics had nothing whatever to do with its origin.
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
Mathematical foundations may not be sets; categories are a popular rival [Shapiro]
     Full Idea: Foundationalists (e.g. Quine and Lewis) have shown that mathematics can be rendered in theories other than the iterative hierarchy of sets. A dedicated contingent hold that the category of categories is the proper foundation (e.g. Lawvere).
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.3)
     A reaction: I like the sound of that. The categories are presumably concepts that generate sets. Tricky territory, with Frege's disaster as a horrible warning to be careful.
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
Baseball positions and chess pieces depend entirely on context [Shapiro]
     Full Idea: We cannot imagine a shortstop independent of a baseball infield, or a piece that plays the role of black's queen bishop independent of a chess game.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.1)
     A reaction: This is the basic thought that leads to the structuralist view of things. I must be careful because I like structuralism, but I have attacked the functionalist view in many areas, because it neglects the essences of the functioning entities.
The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro]
     Full Idea: The even numbers and the natural numbers greater than 4 both exemplify the natural-number structure. In the former, 6 plays the 3 role, and in the latter 8 plays the 3 role.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.5)
     A reaction: This begins to sound a bit odd. If you count the even numbers, 6 is the third one. I could count pebbles using only evens, but then presumably '6' would just mean '3'; it wouldn't be the actual number 6 acting in a different role, like Laurence Olivier.
Could infinite structures be apprehended by pattern recognition? [Shapiro]
     Full Idea: It is contentious, to say the least, to claim that infinite structures are apprehended by pattern recognition.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.1)
     A reaction: It only seems contentious for completed infinities. The idea that the pattern continues in same way seems (pace Wittgenstein) fairly self-evident, just like an arithmetical series.
The 4-pattern is the structure common to all collections of four objects [Shapiro]
     Full Idea: The 4-pattern is the structure common to all collections of four objects.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.2)
     A reaction: This seems open to Frege's objection, that you can have four disparate abstract concepts, or four spatially scattered items of unknown pattern. It certainly isn't a visual pattern, but then if the only detectable pattern is the fourness, it is circular.
The main mathematical structures are algebraic, ordered, and topological [Shapiro]
     Full Idea: According to Bourbaki, there are three main types of structure: algebraic structures, such as group, ring, field; order structures, such as partial order, linear order, well-order; topological structures, involving limit, neighbour, continuity, and space.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.5)
     A reaction: Bourbaki is mentioned as the main champion of structuralism within mathematics.
Some structures are exemplified by both abstract and concrete [Shapiro]
     Full Idea: Some structures are exemplified by both systems of abstracta and systems of concreta.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.2)
     A reaction: It at least seems plausible that one might try to build a physical structure that modelled arithmetic (an abacus might be an instance), so the parallel is feasible. Then to say that the abstract arose from modelling the physical seems equally plausible.
Mathematical structures are defined by axioms, or in set theory [Shapiro]
     Full Idea: Mathematical structures are characterised axiomatically (as implicit definitions), or they are defined in set theory.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.3)
     A reaction: Presumably earlier mathematicians had neither axiomatised their theories, nor expressed them in set theory, but they still had a good working knowledge of the relationships.
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
The main versions of structuralism are all definitionally equivalent [Shapiro]
     Full Idea: Ante rem structuralism, eliminative structuralism formulated over a sufficiently large domain of abstract objects, and modal eliminative structuralism are all definitionally equivalent. Neither is to be ontologically preferred, but the first is clearer.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.5)
     A reaction: Since Shapiro's ontology is platonist, I would have thought there were pretty obvious grounds for making a choice between that and eliminativm, even if the grounds are intuitive rather than formal.
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
Is there is no more to structures than the systems that exemplify them? [Shapiro]
     Full Idea: The 'in re' view of structures is that there is no more to structures than the systems that exemplify them.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.3)
     A reaction: I say there is more than just the systems, because we can abstract from them to a common structure, but that doesn't commit us to the existence of such a common structure.
Number statements are generalizations about number sequences, and are bound variables [Shapiro]
     Full Idea: According to 'in re' structuralism, a statement that appears to be about numbers is a disguised generalization about all natural-number sequences; the numbers are bound variables, not singular terms.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.3.4)
     A reaction: Any theory of anything which comes out with the thought that 'really it is a variable, not a ...' has my immediate attention and sympathy.
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro]
     Full Idea: Because the same structure can be exemplified by more than one system, a structure is a one-over-many.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.3)
     A reaction: The phrase 'one-over-many' is a classic Greek hallmark of a universal. Cf. Idea 10217, where Shapiro talks of arriving at structures by abstraction, through focusing and ignoring. This sounds more like a creation than a platonic universal.
There is no 'structure of all structures', just as there is no set of all sets [Shapiro]
     Full Idea: There is no 'structure of all structures', just as there is no set of all sets.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.4)
     A reaction: If one cannot abstract from all the structures to a higher level, why should Shapiro have abstracted from the systems/models to get the over-arching structures?
Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend]
     Full Idea: Shapiro's structuralism champions model theory as the branch of mathematics that best describes mathematics. The essence of mathematical activity is seen as an exercise in comparing mathematical structures to each other.
     From: report of Stewart Shapiro (Philosophy of Mathematics [1997], 4.4) by Michèle Friend - Introducing the Philosophy of Mathematics
     A reaction: Note it 'best describes' it, rather than being foundational. Assessing whether propositional logic is complete is given as an example of model theory. That makes model theory a very high-level activity. Does it capture simple arithmetic?
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro]
     Full Idea: According to structuralism, someone who uses small natural numbers in everyday life presupposes an infinite structure. It seems absurd that a child who learns to count his toes applies an infinite structure to reality, and thus presupposes the structure.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.2)
     A reaction: Shapiro says we can meet this objection by thinking of smaller structures embedded in larger ones, with the child knowing the smaller ones.
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro]
     Full Idea: We must distinguish between 'realism in ontology' - that mathematical objects exist - and 'realism in truth-value', which is suggested by the model-theoretic framework - that each well-formed meaningful sentence is non-vacuously either true or false.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
     A reaction: My inclination is fairly strongly towards realism of the second kind, but not of the first. A view about the notion of a 'truth-maker' might therefore be required. What do the truths refer to? Answer: not objects, but abstractions from objects.
If mathematical objects are accepted, then a number of standard principles will follow [Shapiro]
     Full Idea: One who believes in the independent existence of mathematical objects is likely to accept the law of excluded middle, impredicative definitions, the axiom of choice, extensionality, and arbitrary sets and functions.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 1)
     A reaction: The underlying thought is that since the objects pre-exist, all of the above simply describe the relations between them, rather than having to actually bring the objects into existence. Personally I would seek a middle ground.
Platonists claim we can state the essence of a number without reference to the others [Shapiro]
     Full Idea: The Platonist view may be that one can state the essence of each number, without referring to the other numbers.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.1)
     A reaction: Frege certainly talks this way (in his 'borehole' analogy). Fine, we are asked to spell out the essence of some number, without making reference either to any 'units' composing it, or to any other number adjacent to it or composing it. Reals?
Platonism must accept that the Peano Axioms could all be false [Shapiro]
     Full Idea: A traditional Platonist has to face the possibility that all of the Peano Axioms are false.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.7)
     A reaction: This would be because the objects exist independently, and so the Axioms are a mere human attempt at pinning them down. For the Formalist the axioms create the numbers, and so couldn't be false. This makes me, alas, warm to platonism!
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
Intuition is an outright hindrance to five-dimensional geometry [Shapiro]
     Full Idea: Even if spatial intuition provides a little help in the heuristics of four-dimensional geometry, intuition is an outright hindrance for five-dimensional geometry and beyond.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 5.2)
     A reaction: One might respond by saying 'so much the worse for five-dimensional geometry'. One could hardly abolish the subject, though, so the point must be taken.
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro]
     Full Idea: For each stone, there is at least one pattern such that the stone is a position in that pattern. The stone can be treated in terms of places-are-objects, or places-are-offices, to be filled with objects drawn from another ontology.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.4)
     A reaction: I believe this is the story J.S. Mill had in mind. His view was that the structures move off into abstraction, but it is only at the empirical and physical level that we can possibly learn the structures.
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
Can the ideal constructor also destroy objects? [Shapiro]
     Full Idea: Can we assume that the ideal constructor cannot destroy objects? Presumably the ideal constructor does not have an eraser, and the collection of objects is non-reducing over time.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.5)
     A reaction: A very nice question, which platonists should enjoy.
Presumably nothing can block a possible dynamic operation? [Shapiro]
     Full Idea: Presumably within a dynamic system, once the constructor has an operation available, then no activity can preclude the performance of the operation?
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.5)
     A reaction: There seems to be an interesting assumption in static accounts of mathematics, that all the possible outputs of (say) a function actually exist with a theory. In an actual dynamic account, the constructor may be smitten with lethargy.
7. Existence / A. Nature of Existence / 1. Nature of Existence
Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro]
     Full Idea: Can we 'discover' whether a deck is really identical with its fifty-two cards, or whether a person is identical with her corresponding time-slices, molecules, or space-time points? This is like Benacerraf's problem about numbers.
     From: Stewart Shapiro (Philosophy of Mathematics [1997])
     A reaction: Shapiro is defending the structuralist view, that each of these is a model of an agreed reality, so we cannot choose a right model if they all satisfy the necessary criteria.
7. Existence / A. Nature of Existence / 6. Criterion for Existence
Chrysippus says action is the criterion for existence, which must be physical [Chrysippus, by Tieleman]
     Full Idea: Chrysippus regarded power to act and be acted upon as the criterion for existence or being - a test satisfied by bodies alone.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Teun L. Tieleman - Chrysippus
     A reaction: This defines existence in terms of causation. Is he ruling out a priori a particle (say) which exists, but never interacts with anything? If so, he is inclining towards anti-realism.
7. Existence / C. Structure of Existence / 7. Abstract/Concrete / a. Abstract/concrete
The abstract/concrete boundary now seems blurred, and would need a defence [Shapiro]
     Full Idea: The epistemic proposals of ontological realists in mathematics (such as Maddy and Resnik) has resulted in the blurring of the abstract/concrete boundary. ...Perhaps the burden is now on defenders of the boundary.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.1)
     A reaction: As Shapiro says, 'a vague boundary is still a boundary', so we need not be mesmerised by borderline cases. I would defend the boundary, with the concrete just being physical. A chair is physical, but our concept of a chair may already be abstract.
Mathematicians regard arithmetic as concrete, and group theory as abstract [Shapiro]
     Full Idea: Mathematicians use the 'abstract/concrete' label differently, with arithmetic being 'concrete' because it is a single structure (up to isomorphism), while group theory is considered more 'abstract'.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.1 n1)
     A reaction: I would say that it is the normal distinction, but they have moved the significant boundary up several levels in the hierarchy of abstraction.
7. Existence / D. Theories of Reality / 7. Fictionalism
Fictionalism eschews the abstract, but it still needs the possible (without model theory) [Shapiro]
     Full Idea: Fictionalism takes an epistemology of the concrete to be more promising than concrete-and-abstract, but fictionalism requires an epistemology of the actual and possible, secured without the benefits of model theory.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.2)
     A reaction: The idea that possibilities (logical, natural and metaphysical) should be understood as features of the concrete world has always struck me as appealing, so I have (unlike Shapiro) no intuitive problems with this proposal.
Structuralism blurs the distinction between mathematical and ordinary objects [Shapiro]
     Full Idea: One result of the structuralist perspective is a healthy blurring of the distinction between mathematical and ordinary objects. ..'According to the structuralist, physical configurations often instantiate mathematical patterns'.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.4)
     A reaction: [The quotation is from Penelope Maddy 1988 p.28] This is probably the main reason why I found structuralism interesting, and began to investigate it.
7. Existence / D. Theories of Reality / 8. Facts / b. Types of fact
There are simple and complex facts; the latter depend on further facts [Chrysippus, by Cicero]
     Full Idea: Chrysippus says there are two classes of facts, simple and complex. An instance of a simple fact is 'Socrates will die at a given date', ...but 'Milo will wrestle at Olympia' is a complex statement, because there can be no wrestling without an opponent.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by M. Tullius Cicero - On Fate ('De fato') 13.30
     A reaction: We might say that there are atomic and complex facts, but our atomic facts tend to be much simpler, usually just saying some object has some property.
7. Existence / E. Categories / 3. Proposed Categories
Stoics categories are Substrate, Quality, Disposition, and Relation [Chrysippus, by Pasnau]
     Full Idea: The Stoics proposed a rather modest categorisation of Substrate, Quality, Disposition, and Relation.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Robert Pasnau - Metaphysical Themes 1274-1671 12.1
9. Objects / A. Existence of Objects / 1. Physical Objects
The notion of 'object' is at least partially structural and mathematical [Shapiro]
     Full Idea: The very notion of 'object' is at least partially structural and mathematical.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.1)
     A reaction: [In the context, Shapiro clearly has physical objects in mind] This view seems to fit with Russell's 'relational' view of the physical world, though Russell rejected structuralism in mathematics. I take abstraction to be part of perception.
9. Objects / B. Unity of Objects / 3. Unity Problems / b. Cat and its tail
Dion and Theon coexist, but Theon lacks a foot. If Dion loses a foot, he ousts Theon? [Chrysippus, by Philo of Alexandria]
     Full Idea: If two individuals occupied one substance …let one individual (Dion) be thought of as whole-limbed, the other (Theon) as minus one foot. Then let one of Dion's feet be amputated. Theon is the stronger candidate to have perished.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Philo (Alex) - On the Eternity of the World 48
     A reaction: [SVF 2.397 - from Chrysippus's lost 'On the Growing Argument'] This is the original of Tibbles the Cat. Dion must persist to change, and then ousts Theon (it seems). Philo protests at Theon ceasing to exist when nothing has happened to him.
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
A blurry border is still a border [Shapiro]
     Full Idea: A blurry border is still a border.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 8.3)
     A reaction: This remark deserves to be quoted in almost every area of philosophy, against those who attack a concept by focusing on its vague edges. Philosophers should focus on central cases, not borderline cases (though the latter may be of interest).
9. Objects / E. Objects over Time / 2. Objects that Change
Change of matter doesn't destroy identity - in Dion and Theon change is a condition of identity [Chrysippus, by Long/Sedley]
     Full Idea: The Growing Argument said any change of matter is a change of identity. Chrysippus presents it with a case (Dion and Theon) where material diminution is the necessary condition of enduring identity, since the diminished footless Dion survives.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by AA Long / DN Sedley - Hellenic Philosophers commentary 28:175
     A reaction: [The example, in Idea 16058, is the original of Tibbles the Cat] This is a lovely bold idea which I haven't met in the modern discussions - that identity actually requires change. The concept of identity is meaningless without change?
10. Modality / A. Necessity / 6. Logical Necessity
Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro]
     Full Idea: For many philosophers the logical notions of possibility and necessity are exceptions to a general scepticism, perhaps because they have been reduced to model theory, via set theory. Thus Φ is logically possible if there is a model that satisfies it.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.1)
     A reaction: Initially this looks a bit feeble, like an empiricist only believing what they actually see right now, but the modern analytical philosophy project seems to be the extension of logical accounts further and further into what we intuit about modality.
10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro]
     Full Idea: The fact that the 'myth' of possible worlds happens to produce the correct modal logic is itself a phenomenon in need of explanation.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 7.4)
     A reaction: The claim that it produces 'the' correct modal logic seems to beg a lot of questions, given the profusion of modal systems. This is a problem with any sort of metaphysics which invokes fictionalism - what were those particular fictions responding to?
15. Nature of Minds / A. Nature of Mind / 7. Animal Minds
Dogs show reason in decisions made by elimination [Chrysippus, by Sext.Empiricus]
     Full Idea: A dog makes use of the fifth complex indemonstrable syllogism when, arriving at a spot where three ways meet, after smelling at two roads by which the quarry did not pass, he rushes off at once by the third without pausing to smell.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Sextus Empiricus - Outlines of Pyrrhonism I.69
     A reaction: As we might say: either A or B or C; not A; not B; therefore C. I wouldn't want to trust this observation without a lot of analysis of slow-motion photography of dogs as crossroads. Even so, it is a nice challenge to Descartes' view of animals.
15. Nature of Minds / C. Capacities of Minds / 3. Abstraction by mind
We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro]
     Full Idea: The epistemological account of mathematical structures depends on the size and complexity of the structure, but small, finite structures are apprehended through abstraction via simple pattern recognition.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
     A reaction: Yes! This I take to be the reason why John Stuart Mill was not a fool in his discussion of the pebbles. Successive abstractions (and fictions) will then get you to more complex structures.
16. Persons / F. Free Will / 4. For Free Will
Chrysippus allows evil to say it is fated, or even that it is rational and natural [Plutarch on Chrysippus]
     Full Idea: Chrysippus gives vice blatant freedom to say not only that it is necessary and according to fate, but even that it occurs according to god's reason and the best nature.
     From: comment on Chrysippus (fragments/reports [c.240 BCE]) by Plutarch - 70: Stoic Self-contradictions 1050c
     A reaction: This is Plutarch's criticism of stoic determinism or fatalism. Zeno replied that the punishment for vice may also be fated. It seems that Chysippus did believe that punishments were too harsh, given that vices are fated [p.109].
16. Persons / F. Free Will / 5. Against Free Will
A swerve in the atoms would be unnatural, like scales settling differently for no reason [Chrysippus, by Plutarch]
     Full Idea: Chrysippus argues against the 'swerve' of the Epicureans, on the grounds that they are doing violence to nature by positing something which is uncaused, and cites dice or scales, which can't settle differently without some cause or difference.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Plutarch - 70: Stoic Self-contradictions 1045c
     A reaction: That is, the principle of sufficient reason (or of everything having a cause) is derived from observation, not a priori understanding. Pace Leibniz. As in modern discussion, free will or the swerve only occur in our minds, and not elsewhere.
16. Persons / F. Free Will / 6. Determinism / a. Determinism
Chrysippus is wrong to believe in non-occurring future possibilities if he is a fatalist [Plutarch on Chrysippus]
     Full Idea: Chrysippus's accounts of possibility and fate are in conflict. If he is right that 'everything that permits of occurring even if it is not going to occur is possible', then many things are possible which are not according to fate.
     From: comment on Chrysippus (fragments/reports [c.240 BCE]) by Plutarch - 70: Stoic Self-contradictions 1055e
     A reaction: A palpable hit, I think. Plutarch refers to Chrysippus's rejection of Diodorus Cronus's Master Argument. Fatalism seems to entail that the only future possibilities are the ones that actually occur.
Everything is fated, either by continuous causes or by a supreme rational principle [Chrysippus, by Diog. Laertius]
     Full Idea: Chrysippus says (in his 'On Fate') that everything happens by fate. Fate is a continuous string of causes of things which exist or a rational principle according to which the cosmos is managed.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.148
16. Persons / F. Free Will / 6. Determinism / b. Fate
The Lazy Argument responds to fate with 'why bother?', but the bothering is also fated [Chrysippus, by Cicero]
     Full Idea: Chrysippus responded to the Lazy Argument (that the outcome of an illness is fated, so there is no point in calling the doctor) by saying 'calling the doctor is fated just as much as recovering', which he calls 'co-fated'.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by M. Tullius Cicero - On Fate ('De fato') 28-30
     A reaction: From a pragmatic point of view, this idea also nullifies fatalism, since you can plausibly fight against your fate to your last breath. No evidence could ever be offered in support of fatalism, not even the most unlikely events.
When we say events are fated by antecedent causes, do we mean principal or auxiliary causes? [Chrysippus]
     Full Idea: Some causes are perfect and principal, others auxiliary and proximate. Hence when we say that everything takes place by fate owing to antecedent causes, what we wish to be understood is not perfect and principal causes but auxiliary and proximate causes.
     From: Chrysippus (fragments/reports [c.240 BCE]), quoted by M. Tullius Cicero - On Fate ('De fato') 18.41
     A reaction: This move is described by Cicero as enabling Chrysippus to 'escape necessity and to retain fate'.
Fate is an eternal and fixed chain of causal events [Chrysippus]
     Full Idea: Fate is a sempiternal and unchangeable series and chain of things, rolling and unravelling itself through eternal sequences of cause and effect, of which it is composed and compounded.
     From: Chrysippus (fragments/reports [c.240 BCE]), quoted by Aulus Gellius - Noctes Atticae 7.2.01
     A reaction: It seems that Chrysippus (called by Aulus Gellius 'the chief Stoic philosopher') had a rather grandly rhetorical prose style.
16. Persons / F. Free Will / 7. Compatibilism
Destiny is only a predisposing cause, not a sufficient cause [Chrysippus, by Plutarch]
     Full Idea: Chrysippus considered destiny to be not a cause sufficient of itself but only a predisposing cause.
     From: report of Chrysippus (fragments/reports [c.240 BCE], fr 997) by Plutarch - 70: Stoic Self-contradictions 1056b
     A reaction: This appears to be a rejection of determinism, and is the equivalent of Epicurus' introduction of the 'swerve' in atoms. They had suddenly become bothered about the free will problem in about 305 BCE. There must be other non-destiny causes?
18. Thought / E. Abstraction / 2. Abstracta by Selection
Simple types can be apprehended through their tokens, via abstraction [Shapiro]
     Full Idea: Some realists argue that simple types can be apprehended through their tokens, via abstraction.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.2)
     A reaction: One might rephrase that to say that types are created by abstraction from tokens (and then preserved in language).
18. Thought / E. Abstraction / 3. Abstracta by Ignoring
We can apprehend structures by focusing on or ignoring features of patterns [Shapiro]
     Full Idea: One way to apprehend a particular structure is through a process of pattern recognition, or abstraction. One observes systems in a structure, and focuses attention on the relations among the objects - ignoring features irrelevant to their relations.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 3.1)
     A reaction: A lovely statement of the classic Aristotelian abstractionist approach of focusing-and-ignoring. But this is made in 1997, long after Frege and Geach ridiculed it. It just won't go away - not if you want a full and unified account of what is going on.
We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro]
     Full Idea: One can observe a system and focus attention on the relations among the objects - ignoring those features of the objects not relevant to the system. For example, we can understand a baseball defense system by going to several games.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], p.74), quoted by Charles Chihara - A Structural Account of Mathematics
     A reaction: This is Shapiro perpetrating precisely the wicked abstractionism which Frege and Geach claim is ridiculous. Frege objects that abstract concepts then become private, but baseball defences are discussed in national newspapers.
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro]
     Full Idea: Perhaps we can introduce abstract objects by abstraction over an equivalence relation on a base class of entities, just as Frege suggested that 'direction' be obtained from parallel lines. ..Properties must be equinumerous, but need not be individuated.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.5)
     A reaction: [He cites Hale and Wright as the originators of this} It is not entirely clear why this is 'abstraction', rather than just drawing attention to possible groupings of entities.
19. Language / A. Nature of Meaning / 1. Meaning
A minimum requirement for a theory of meaning is that it include an account of truth [Davidson]
     Full Idea: Whatever else it embraces, a theory of meaning must include an account of truth - a statement of the conditions under which an arbitrary sentence of the language is true.
     From: Donald Davidson (Reality without Reference [1977], p.132)
     A reaction: It is a moot point whether we can define meaning if we assume truth, or if we can define truth by assuming meaning. Tarski seems to presuppose meaning when he defines truth (Idea 2345). I like Davidson's taking of truth as basic.
19. Language / A. Nature of Meaning / 4. Meaning as Truth-Conditions
A theory of truth tells us how communication by language is possible [Davidson]
     Full Idea: A theory of truth lets us answer the underlying question how communication by language is possible.
     From: Donald Davidson (Reality without Reference [1977], p.137)
     A reaction: If, instead, you explain communication by understood intentions (á la Grice), you have to say more about what sort of intentions are meant. If you use reference, you still have more to say about the meaning of sentences. Davidson looks good.
19. Language / B. Reference / 1. Reference theories
Is reference the key place where language and the world meet? [Davidson]
     Full Idea: The essential question is whether reference is the, or at least one, place where there is direct contact between linguistic theory and events, actions, or objects described in nonlinguistic terms.
     From: Donald Davidson (Reality without Reference [1977], p.134)
     A reaction: How do you 'describe objects in nonlinguistic terms'? The causal theory of reference (e.g. Idea 4957) is designed to plug language straight into the world via reference. It simplifies things nicely, but I don't quite believe it.
With a holistic approach, we can give up reference in empirical theories of language [Davidson]
     Full Idea: I defend a version of the holistic approach, and urge that we must give up the concept of reference as basic to an empirical theory of language.
     From: Donald Davidson (Reality without Reference [1977], p.136)
     A reaction: He proposes to connect language to the world via the concept of truth, rather than of reference. It is a brilliant idea, and is the key issue in philosophy of language. I go back to animals, which seem to care about situations rather than things.
19. Language / B. Reference / 4. Descriptive Reference / b. Reference by description
To explain the reference of a name, you must explain its sentence-role, so reference can't be defined nonlinguistically [Davidson]
     Full Idea: It is inconceivable that one should be able to explain the relationship between 'Kilimanjiro' and Kilimanjiro without first explaining the role of the word in sentences; hence there is no chance of explaining reference directly in nonlinguistic terms.
     From: Donald Davidson (Reality without Reference [1977], p.135)
     A reaction: I point at the mountain, and a local says 'Kilimanjiro'? There is a 'gavagai'-type problem with that. The prior question might be 'what is it about this word that enables it to have a role in sentences?' Unlike whimpering or belching.
19. Language / D. Propositions / 1. Propositions
A proposition is what can be asserted or denied on its own [Chrysippus]
     Full Idea: A proposition is what can be asserted or denied on its own, for example, 'It is day' or 'Dion is walking'.
     From: Chrysippus (fragments/reports [c.240 BCE]), quoted by Diogenes Laertius - Lives of Eminent Philosophers 07.65
     A reaction: Note the phrase 'on its own'. If you say 'it is day and Dion is walking', that can't be denied on its own, because first the two halves must each be evaluated, so presumably that doesn't count as a stoic proposition.
20. Action / B. Preliminaries of Action / 2. Willed Action / d. Weakness of will
Passions are judgements; greed thinks money is honorable, and likewise drinking and lust [Chrysippus, by Diog. Laertius]
     Full Idea: Chrysippus says (in his On Passions) that the passions are judgements; for greed is a supposition that money is honorable, and similarly for drunkennes and wantonness and others.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.111
     A reaction: This is an endorsement of Socrates's intellectualist reading of weakness of will, as against Aristotle's assigning it to overpowering passions.
20. Action / C. Motives for Action / 5. Action Dilemmas / c. Omissions
The highest degree of morality performs all that is appropriate, omitting nothing [Chrysippus]
     Full Idea: He who makes moral progress to the highest degree performs all the appropriate actions in all circumstances, and omits none.
     From: Chrysippus (fragments/reports [c.240 BCE]), quoted by Sophocles - Sophocles' Electra 4.39.22
     A reaction: Hence concerns about omission as well as commission in the practice of ethics can be seen in the light of character and virtue. The world is fully of nice people who act well, but don't do so well on omissions. Car drivers, for example.
21. Aesthetics / C. Artistic Issues / 7. Art and Morality
Stoics say that beauty and goodness are equivalent and linked [Chrysippus, by Diog. Laertius]
     Full Idea: Stoics say the beautiful is the only good. Good is an equivalent term to the beautiful; since a thing is good, it is beautiful; and it is beautiful, therefore it is good.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.1.59
22. Metaethics / A. Ethics Foundations / 1. Nature of Ethics / g. Moral responsibility
Fate initiates general causes, but individual wills and characters dictate what we do [Chrysippus]
     Full Idea: The order and reason of fate set in motion the general types and starting points of the causes, but each person's own will [or decisions] and the character of his mind govern the impulses of our thoughts and minds and our very actions.
     From: Chrysippus (fragments/reports [c.240 BCE]), quoted by Aulus Gellius - Noctes Atticae 7.2.11
     A reaction: So if you try and fail it was fate, but if you try and succeed it was you?
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / e. Human nature
Human purpose is to contemplate and imitate the cosmos [Chrysippus]
     Full Idea: The human being was born for the sake of contemplating and imitating the cosmos.
     From: Chrysippus (fragments/reports [c.240 BCE]), quoted by M. Tullius Cicero - On the Nature of the Gods ('De natura deorum') 2.37
     A reaction: [This seems to be an idea of Chrysippus] Remind me how to imitate the cosmos. Presumably this is living according to nature, but that becomes more obscure when express like this.
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / j. Ethics by convention
Stoics say justice is a part of nature, not just an invented principle [Chrysippus, by Diog. Laertius]
     Full Idea: Stoics say that justice exists by nature, and not because of any definition or principle.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.1.66
     A reaction: cf Idea 3024. Stoics thought that nature is intrinsically rational, and therein lies its justice. 'King Lear' enacts this drama about whether nature is just.
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / k. Ethics from nature
Only nature is available to guide action and virtue [Chrysippus]
     Full Idea: What am I to take as the principle of appropriate action and raw material for virtue if I give up nature and what is according to nature?
     From: Chrysippus (fragments/reports [c.240 BCE]), quoted by Plutarch - On Common Conceptions 1069e
     A reaction: 'Nature' is awfully vague as a guideline, even when we are told nature is rational. I can only make sense of it as 'human nature', which is more Aristotelian than stoic. 'Go with the flow' and 'lay the cards you are dealt' might capture it.
22. Metaethics / B. Value / 1. Nature of Value / f. Ultimate value
Live in agreement, according to experience of natural events [Chrysippus]
     Full Idea: The goal of life is to live in agreement, which is according to experience of the things which happen by nature.
     From: Chrysippus (fragments/reports [c.240 BCE]), quoted by John Stobaeus - Anthology 2.06a
     A reaction: Cleanthes added 'with nature' to Zeno's slogan, and Chyrisppus added this variation. At least it gives you some idea of what the consistent rational principle should be. You still have to assess which aspects of nature should influence us.
22. Metaethics / C. The Good / 1. Goodness / d. Good as virtue
Living happily is nothing but living virtuously [Chrysippus, by Plutarch]
     Full Idea: According to Chrysippus, living happily consists solely in living virtuously.
     From: report of Chrysippus (fragments/reports [c.240 BCE], fr139) by Plutarch - 72: Against Stoics on common Conceptions 1060d
     A reaction: This, along with 'live according to nature', is the essential doctrine of stoicism. This is 'eudaimonia', not the modern idea of feeling nice. Is it possible to admire another person for anything other than virtue? (Yes! Looks, brains, strength, wealth).
22. Metaethics / C. The Good / 1. Goodness / f. Good as pleasure
Pleasure is not the good, because there are disgraceful pleasures [Chrysippus, by Diog. Laertius]
     Full Idea: Pleasure is not the good, because there are disgraceful pleasures, and nothing disgraceful is good.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.Ze.60
     A reaction: I certainly approve of the idea that not all pleasure is intrinsically good. Indeed, I think good has probably got nothing to do with pleasure. 'Disgraceful' is hardly objective though.
Justice can be preserved if pleasure is a good, but not if it is the goal [Chrysippus, by Plutarch]
     Full Idea: Chrysippus thinks that, while justice could not be preserved if one should set up pleasure as the goal, it could be if one should take pleasure to be not a goal but simply a good.
     From: report of Chrysippus (fragments/reports [c.240 BCE], fr 23) by Plutarch - 72: Against Stoics on common Conceptions 1070d
     A reaction: This is an interesting and original contribution to the ancient debate about pleasure. It shows Aristotle's moderate criticism of pleasure (e.g. Idea 84), but attempts to pinpoint where the danger is. Aristotle says it thwarts achievement of the mean.
22. Metaethics / C. The Good / 3. Pleasure / c. Value of pleasure
There are shameful pleasures, and nothing shameful is good, so pleasure is not a good [Chrysippus, by Diog. Laertius]
     Full Idea: Chrysippus (in his On Pleasure) denies even of pleasure that it is a good; for there are also shameful pleasures, and nothing shameful is good.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.103
     A reaction: Socrates seems to have started this line of the thought, to argue that pleasure is not The Good. Stoics are more puritanical. Nothing counts as good if it is capable of being bad. Thus good pleasures are not good, which sounds odd.
23. Ethics / A. Egoism / 2. Hedonism
People need nothing except corn and water [Chrysippus, by Plutarch]
     Full Idea: Chrysippus praises ad nauseam the lines "For what need mortals save two things alone,/ Demeter's grain and draughts of water clear".
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Plutarch - 70: Stoic Self-contradictions 1043e
     A reaction: "Oh, reason not the need!" says King Lear. The remark shows the close affinity of stoicism and cynicism, as the famous story of Diogenes is that he threw away his drinking cup when he realised you could drink with your hands.
23. Ethics / C. Virtue Theory / 1. Virtue Theory / a. Nature of virtue
All virtue is good, but not always praised (as in not lusting after someone ugly) [Chrysippus]
     Full Idea: Although deeds done in accordance with virtue are congenial, not all are cited as examples, such as courageously extending one's finger, or continently abstaining from a half-dead old woman, or not immediately agreeing that three is four.
     From: Chrysippus (fragments/reports [c.240 BCE], fr 211), quoted by Plutarch - 70: Stoic Self-contradictions 1038f
     A reaction: Presumably the point (so elegantly expressed - what a shame we have lost most of Chrysippus) is that virtue comes in degrees, even though its value is an absolute. The same has been said (by Russell and Bonjour) about self-evidence.
23. Ethics / C. Virtue Theory / 1. Virtue Theory / b. Basis of virtue
Chrysippus says virtue can be lost (though Cleanthes says it is too secure for that) [Chrysippus, by Diog. Laertius]
     Full Idea: Chrysippus says that virtue can be lost, owing to drunkenness and excess of black bile, whereas Cleanthes says it cannot, because it consists in secure intellectual grasps, and it is worth choosing for its own sake.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.127
     A reaction: Succumbing to drunkenness looks like evidence that you were not truly virtuous. Mental illness is something else. On the whole I agree the Cleanthes.
Chrysippus says nothing is blameworthy, as everything conforms with the best nature [Chrysippus, by Plutarch]
     Full Idea: Chrysippus has often written on the theme that there is nothing reprehensible or blameworthy in the universe since all things are accomplished in conformity with the best nature.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Plutarch - 70: Stoic Self-contradictions 1051b
     A reaction: This is Leibniz's "best of all possible worlds", but deriving the idea from the rightness of nature rather than the perfection of God. Chrysippus has a more plausible ground than Leibniz, as for him nasty things follow from conscious choice.
24. Political Theory / A. Basis of a State / 1. A People / b. The natural life
Rational animals begin uncorrupted, but externals and companions are bad influences [Chrysippus, by Diog. Laertius]
     Full Idea: The rational animal is corrupted, sometimes because of the persuasiveness of external activities and sometimes because of the influence of companions. For the starting points provided by nature are uncorrupted.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.89
     A reaction: If companions corrupt us, what corrupted the companions? Aren't we all in this together? And where do the 'external activities' originate?
25. Social Practice / D. Justice / 2. The Law / c. Natural law
Justice, the law, and right reason are natural and not conventional [Chrysippus, by Diog. Laertius]
     Full Idea: Chrysippus says (in On the Honourable) that justice is natural and not conventional, as are the law and right reason.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.128
     A reaction: How does he explain variations in the law between different states? Presumably some of them have got it wrong. What is the criterion for deciding which laws are natural?
25. Social Practice / F. Life Issues / 6. Animal Rights
We don't have obligations to animals as they aren't like us [Chrysippus, by Diog. Laertius]
     Full Idea: We have no obligations of justice to other animals, because they are dissimilar to us.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.Ze.66
     A reaction: "Dissimilar" begs questions. Some human beings don't seem much like me. How are we going to treat visiting aliens?
Justice is irrelevant to animals, because they are too unlike us [Chrysippus, by Diog. Laertius]
     Full Idea: There is no justice between us and other animals because of the dissimilarity between us and them.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.129
     A reaction: [from lost On Justice Bk 1] What would he make of modern revelations about bonobos and chimpanzees? If there is great dissimilarity between some peoples, does that invalidate justice between them? He also said animals exist for our use.
26. Natural Theory / A. Speculations on Nature / 2. Natural Purpose / a. Final purpose
Covers are for shields, and sheaths for swords; likewise, all in the cosmos is for some other thing [Chrysippus]
     Full Idea: Just as the cover was made for the sake of the shield, and the sheath for the sword, in the same way everything else except the cosmos was made for the sake of other things.
     From: Chrysippus (fragments/reports [c.240 BCE]), quoted by M. Tullius Cicero - On the Nature of the Gods ('De natura deorum') 2.37
     A reaction: Chrysippus was wise to stop at the cosmos. Similarly, religious teleology had better not ask about the purpose of God. What does he think pebbles are for? Nature is the source of stoic value, so it needs to be purposeful.
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / f. Ancient elements
The later Stoics identified the logos with an air-fire compound, called 'pneuma' [Chrysippus, by Long]
     Full Idea: From Chrysippus onwards, the Stoics identified the logos throughout each world-cycle not with pure fire, but with a compound of fire and air, 'pneuma'.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by A.A. Long - Hellenistic Philosophy 4.4.2
     A reaction: I suspect this was because breath is so vital to the human body.
Fire is a separate element, not formed with others (as was previously believed) [Chrysippus, by Stobaeus]
     Full Idea: In his theory fire is said independently to be an element, since it is not formed together with another one, whereas according to the earlier theory fire is formed with other elements.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by John Stobaeus - Anthology 1.10.16c
     A reaction: The point is that fire precedes the other elements, and is superior to them.
Stoics say earth, air, fire and water are the primary elements [Chrysippus, by Plutarch]
     Full Idea: The Stoics call the four bodies - earth and water and air and fire - primary elements.
     From: report of Chrysippus (fragments/reports [c.240 BCE], fr 444) by Plutarch - 72: Against Stoics on common Conceptions 1085c
     A reaction: Elsewhere (fr 413) Chrysippus denies that they are all 'primary'. Essentially, though, he seems to be adopting the doctrine of Empedocles and Aristotle, in specific opposition to Epicurus' atomism.
27. Natural Reality / D. Time / 1. Nature of Time / h. Presentism
The past and the future subsist, but only the present exists [Chrysippus, by Plutarch]
     Full Idea: When he wished to be subtle, Chrysippus wrote that the past part of time and the future part do not exist but subsist, and only the present exists.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Plutarch - On Common Conceptions 1081f
     A reaction: [from lost On Void] I think I prefer the ontology of Idea 20818. Idea 20819 does not offer an epistemology. Is the present substantial enough to be known? The word 'subsist' is an ontological evasion (even though Russell briefly relied on it).
27. Natural Reality / D. Time / 3. Parts of Time / e. Present moment
The present does not exist, so our immediate experience is actually part past and part future [Chrysippus, by Plutarch]
     Full Idea: Stoics do not allow a minimal time to exist, and do not want to have a partless 'now'; so what one thinks one has grasped as present is in part future and in part past.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Plutarch - On Common Conceptions 1081c
     A reaction: [from lost On Parts Bk3-5] I agree with the ontology here, but I take our grasp of the present to be very short-term memory of the past. I ignore special relativity. Chrysippus expressed two views about this; in the other one he was a Presentist.
Time is continous and infinitely divisible, so there cannot be a wholly present time [Chrysippus, by Stobaeus]
     Full Idea: Chrysippus says most clearly that no time is wholly present; for since the divisibility of continuous things is infinite, time as a whole is also subject to infinite divisibility by this method of division.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by John Stobaeus - Anthology 1.08.42
     A reaction: But what is his reason for thinking that time is a continuous thing? There is a minimum time in quantum mechanics (the Planck Time), but do these quantum intervals overlap? Compare Idea 20819.
28. God / A. Divine Nature / 3. Divine Perfections
Stoics say that God the creator is the perfection of all animals [Chrysippus, by Diog. Laertius]
     Full Idea: Stoics say that God is an animal immortal, rational, perfect, and intellectual in his happiness, unsusceptible of any kind of evil, having a foreknowledge of the world; however, he is not the figure of a man, and is the creator of the universe.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.1.72
28. God / A. Divine Nature / 6. Divine Morality / a. Divine morality
The origin of justice can only be in Zeus, and in nature [Chrysippus]
     Full Idea: One can find no other starting point or origin for justice except the one derived from Zeus and that derived from the common nature; for everything like this must have that starting point, if we are going to say anything at all about good and bad things.
     From: Chrysippus (fragments/reports [c.240 BCE]), quoted by Plutarch - 70: Stoic Self-contradictions 1035c
     A reaction: [in lost 'On Gods' bk 3] This appears to offer two starting points, in the mind of Zeus, and in nature, though since nature is presumed to be rational the two may run together. Is Zeus the embodiment, or the unconscious source, or the maker of decrees?
28. God / A. Divine Nature / 6. Divine Morality / d. God decrees morality
The source of all justice is Zeus and the universal nature [Chrysippus]
     Full Idea: It is not possible to discover any other beginning of justice or any source for it other than that from Zeus and from the universal nature.
     From: Chrysippus (fragments/reports [c.240 BCE], fr 326), quoted by Plutarch - 70: Stoic Self-contradictions 1035c
     A reaction: If the source is 'universal nature', that could agree with Plato, but if the source is Zeus, then stoicism is a religion rather than a philosophy.
Stoics teach that law is identical with right reason, which is the will of Zeus [Chrysippus, by Diog. Laertius]
     Full Idea: Stoics teach that common law is identical with that right reason which pervades everything, being the same with Zeus, who is the regulator and chief manager of all existing things.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.1.53
29. Religion / B. Monotheistic Religion / 1. Monotheistic Religion
Stoics teach that God is a unity, variously known as Mind, or Fate, or Jupiter [Chrysippus, by Diog. Laertius]
     Full Idea: Stoics teach that God is unity, and that he is called Mind, and Fate, and Jupiter, and by many names besides.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.Ze.68
29. Religion / D. Religious Issues / 2. Immortality / b. Soul
Death can't separate soul from body, because incorporeal soul can't unite with body [Chrysippus]
     Full Idea: Death is a separation of soul from body. But nothing incorporeal can be separated from a body. For neither does anything incorporeal touch a body, and the soul touches and is separated from the body. Therefore the soul is not incorporeal.
     From: Chrysippus (fragments/reports [c.240 BCE]), quoted by Tertullian - The Soul as an 'Astral Body' 5.3
     A reaction: This is the classic interaction difficulty for substance dualist theories of mind.
29. Religion / D. Religious Issues / 3. Problem of Evil / d. Natural Evil
There is a rationale in terrible disasters; they are useful to the whole, and make good possible [Chrysippus]
     Full Idea: The evil which occurs in terrible disasters has a rationale [logos] peculiar to itself: for in a sense it occurs in accordance with universal reason, and is not without usefulness in relation to the whole. For without it there could be no good.
     From: Chrysippus (fragments/reports [c.240 BCE]), quoted by A.A. Long - Hellenistic Philosophy 4.4.5
     A reaction: [a quotation from Chrysippus. Plutarch, Comm Not 1065b] A nice question about any terrible disaster is whether it is in some way 'useful', if we take a broader view of things. Almost everything has a good aspect, from that perspective.