102 ideas
| 22733 | Epicurus accepted God in his popular works, but not in his writings on nature [Epicurus, by Sext.Empiricus] |
| Full Idea: Epicurus in his popular exposition allows the existence of God, but in expounding the real nature of things he does not allow it. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Sextus Empiricus - Against the Physicists (two books) I.58 | |
| A reaction: Plato and Aristotle also distinguished their esoteric from their exoteric writings, but this is an indication that thei popular works may always have presented safer doctrines. |
| 13291 | Slavery to philosophy brings true freedom [Epicurus] |
| Full Idea: To win true freedom you must be a slave to philosophy. | |
| From: Epicurus (fragments/reports [c.289 BCE]), quoted by Seneca the Younger - Letters from a Stoic 008 | |
| A reaction: A lovely idea. It is one thing to free the body, or to free one's social situation, but the challenge to 'free your mind' is either romantic nonsense or totally baffling, apart from the suggestion offered here. Reason is freedom. Very Kantian. |
| 22758 | Philosophy aims at a happy life, through argument and discussion [Epicurus] |
| Full Idea: Philosophy is an activity which secures the happy life by arguments and discussions. | |
| From: Epicurus (fragments/reports [c.289 BCE]), quoted by Sextus Empiricus - Against the Ethicists (one book) VI.169 | |
| A reaction: Presumably this aims at the happiness of the participant. Universal happiness would need to be much more political. If this is your aim then you can't just follow the winds of the argument, but must channel it towards happiness. No nasty truths? |
| 14523 | We should come to philosophy free from any taint of culture [Epicurus] |
| Full Idea: I congratulate you, sir, because you have come to philosophy free of any taint of culture. | |
| From: Epicurus (fragments/reports [c.289 BCE]) | |
| A reaction: [source: Athenaeus, 'Deipnosophists' 13 588b] No one nowadays thinks such an aspiration remotely possible, not least because the culture is embedded in your native language, but I find the idea very appealing. |
| 22240 | The aim of medicine is removal of sickness, and philosophy similarly removes our affections [Epicurus] |
| Full Idea: Just as there is no benefit to medicine if it does not heal the sicknesses [nosos] of bodies, so too there is none to philosophy unless it expels that affections of the soul. | |
| From: Epicurus (fragments/reports [c.289 BCE], fr 221), quoted by James Allen - Soul's Virtue and the Health of the Body p.78 | |
| A reaction: This sounds rather Buddhist, if the only route to happiness is to suppress the emotions. Epicurus probably refers to the more extreme desires, which only lead to harm. Galen quotes Chrysippus as endorsing this idea (see footnote 5). |
| 1484 | We should say nothing of the whole if our contact is with the parts [Epicurus, by Plutarch] |
| Full Idea: We should make no assertion about the whole when our contact is with the parts. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Plutarch - 74: Reply to Colotes 1109e |
| 2670 | Epicurus despises and laughs at the whole of dialectic [Epicurus, by Cicero] |
| Full Idea: Epicurus despises and laughs at the whole of dialectic. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by M. Tullius Cicero - Academica II.30.97 |
| 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. |
| 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] |
| 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. |
| 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) |
| 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. |
| 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) |
| 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. |
| 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. |
| 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) |
| 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. |
| 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 |
|
| 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. |
| 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) |
| 21668 | Epicurus rejected excluded middle, because accepting it for events is fatalistic [Epicurus, by Cicero] |
| Full Idea: Epicurus said that not every proposition is either true or false. ...Epicurus was afraid that if he admits that every proposition is true or false he will also have to admit that all events are caused by fate (if they are so from all eternity). | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by M. Tullius Cicero - On Fate ('De fato') 10.21 | |
| A reaction: Epicurus proposed his 'swerve' in the movements of atoms to avoid this fatalism. Epicurus is agreeing with Aristotle, who did not accept excluded middle for a future contingent sea-fight. |
| 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) |
| 21676 | Epicureans say disjunctions can be true whiile the disjuncts are not true [Epicurus, by Cicero] |
| Full Idea: Epicureans make the impudent assertion that disjunctions consisting of contrary propositions are true, but that the statements contained in the propositions are neither of them true. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by M. Tullius Cicero - On Fate ('De fato') 16.36 | |
| A reaction: Is that 'it is definitely one or the other, but we haven't a clue which one'? Seems to fit speculations about Goldbach's Conjecture. It doesn't sound terribly impudent to me. Or is it the crazy 'It's definitely one of them, but it's neither of them'? |
| 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. |
| 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. |
| 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) |
| 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) |
| 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. |
| 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. |
| 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. |
| 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. |
| 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. |
| 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) |
| 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. |
| 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) |
| 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) |
| 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) |
| 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. |
| 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) |
| 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. |
| 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. |
| 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. |
| 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. |
| 7301 | The phenomenalist says that to be is to be perceivable [Cardinal/Hayward/Jones] |
| Full Idea: Where the idealist says that to be (i.e. to exist) is to be perceived, the phenomenalist says that to be is to be perceivable. | |
| From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.4) | |
| A reaction: This is a nice phenomenalist slogan to add to Mill's well known one (Idea 3583). Expressed in this form, it looks false to me. What about neutrinoes? They weren't at all perceivable until recently. Maybe some physical stuff can never be perceived. |
| 7302 | Linguistic phenomenalism says we can eliminate talk of physical objects [Cardinal/Hayward/Jones] |
| Full Idea: Linguistic phenomenalism argues that it is possible to remove all talk of physical objects from our speech with no loss of meaning. | |
| From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.4) | |
| A reaction: I find this proposal unappealing. My basic objection is that I cannot understand why anyone would refuse to even contemplate the question of WHY I am having a given group of consistent experiences, of (say) a table kind. |
| 7303 | If we lack enough sense-data, are we to say that parts of reality are 'indeterminate'? [Cardinal/Hayward/Jones] |
| Full Idea: The problem with taking sense-data as basic is that some data can appear indeterminate. If we can't discern the colour of someone's eyes, or the number of sides of a complex figure, are we to say that there is no fact about those things? | |
| From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.4) | |
| A reaction: I like that. How many electrons are there in the sun? Such things cannot just be reduced to talk of sense-data, as there is obviously a vast gap between the data and the facts. As usual, ontology and epistemology must be kept separate. |
| 1823 | We can't seek for things if we have no idea of them [Epicurus, by Diog. Laertius] |
| Full Idea: We could not seek for anything if we had not some notion of it. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 10.21 |
| 1824 | To name something, you must already have an idea of what it is [Epicurus, by Diog. Laertius] |
| Full Idea: We could not give names to things, if we had not a preliminary notion of what the things were. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 10.21 |
| 7299 | Primary qualities can be described mathematically, unlike secondary qualities [Cardinal/Hayward/Jones] |
| Full Idea: All the primary qualities lend themselves readily to mathematical or geometric description. ...but it seems that secondary qualities are less amenable to being represented mathematically. | |
| From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.4) | |
| A reaction: As a believer in the primary/secondary distinction, I welcome this point. This is either evidence for the external reality of primary qualities, or an interesting observation about maths. Do we make the primary/secondary distinction because we do maths? |
| 7300 | An object cannot remain an object without its primary qualities [Cardinal/Hayward/Jones] |
| Full Idea: An object cannot lack shape, size, position or motion and remain an object. | |
| From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.4) | |
| A reaction: This points towards the essentialist view (see Idea 5453). This does raise the question of whether an object could lose its colour with impugnity, or the quality of sound that it makes when struck. |
| 5949 | Epicurus says colours are relative to the eye, not intrinsic to bodies [Epicurus, by Plutarch] |
| Full Idea: Epicurus says that colours are not intrinsic to bodies but a result of certain arrangements and positions relative to the eye, which implies that body is no more colourless than coloured. | |
| From: report of Epicurus (fragments/reports [c.289 BCE], Fr 30) by Plutarch - 74: Reply to Colotes §1110 | |
| A reaction: This seems to me such a self-evident truth that I am puzzled as to why anyone would claim that colours are real features of bodies. Epicurus points out that entering a dark room we see no colour, but then colour appears after a while. |
| 1821 | Sensations cannot be judged, because similar sensations have equal value, and different ones have nothing in common [Epicurus, by Diog. Laertius] |
| Full Idea: Sensation is out of reach of control, because one sensation cannot judge another which resembles itself, as they have equal value, and different sensations have different objects. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 10.20 | |
| A reaction: Scepticism about the possibility of purely empirical knowledge; an interesting comment on the question of whether perceptions contain any intrinsic knowledge. |
| 1820 | The criteria of truth are senses, preconceptions and passions [Epicurus, by Diog. Laertius] |
| Full Idea: The criteria of truth are the senses, the preconceptions, and the passions. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 10.20 |
| 1822 | Reason can't judge senses, as it is based on them [Epicurus, by Diog. Laertius] |
| Full Idea: Reason cannot judge the senses, because it is based on them. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 10.20 |
| 7297 | My justifications might be very coherent, but totally unconnected to the world [Cardinal/Hayward/Jones] |
| Full Idea: My beliefs could be well justified in coherentist terms, while not accurately representing the world, and my system of beliefs could be completely free-floating. | |
| From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.3) | |
| A reaction: This nicely encapsulates to correspondence objection to coherence theory. One thing missing from the coherence account is that beliefs aren't chosen for their coherence, but are mostly unthinkingly triggered by experiences. |
| 4549 | Epicurus denied knowledge in order to retain morality or hedonism as the highest values [Nietzsche on Epicurus] |
| Full Idea: Epicurus denied the possibility of knowledge in order to retain moral (or hedonistic) values as the highest values. | |
| From: comment on Epicurus (fragments/reports [c.289 BCE]) by Friedrich Nietzsche - The Will to Power (notebooks) §578 | |
| A reaction: The history of philosophy suggests that this dichotomy is unnecessary. Dogmatist place a high value on multitudes of things. |
| 2668 | Epicurus says if one of a man's senses ever lies, none of his senses should ever be believed [Epicurus, by Cicero] |
| Full Idea: Epicurus says that if one sense has told a lie once in a man's life, no sense must ever be believed. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by M. Tullius Cicero - Academica II.25.79 |
| 1482 | If two people disagree over taste, who is right? [Epicurus, by Plutarch] |
| Full Idea: If one person says the wine is dry and the other that it is sweet, and neither errs in his sensation, how is the wine any more dry than sweet? | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Plutarch - 74: Reply to Colotes 1109b |
| 1483 | Bath water is too hot for some, too cold for others [Epicurus, by Plutarch] |
| Full Idea: In the very same bath some treat the water as too hot, others as too cold. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Plutarch - 74: Reply to Colotes 1109b |
| 1487 | When entering a dark room it is colourless, but colour gradually appears [Epicurus] |
| Full Idea: On entering a dark room we see no colour, but do so after waiting a short time. | |
| From: Epicurus (fragments/reports [c.289 BCE]), quoted by Plutarch - 74: Reply to Colotes 1110d |
| 14526 | The rational soul is in the chest, and the non-rational soul is spread through the body [Epicurus] |
| Full Idea: Democritus and Epicurus say the soul has two parts, one which is rational and is situated in the chest area, and the other which is non-rational and is spread throughout the entire compound of the body | |
| From: Epicurus (fragments/reports [c.289 BCE]) | |
| A reaction: [source Aetius 4.4.6] |
| 6035 | Soul is made of four stuffs, giving warmth, rest, motion and perception [Epicurus, by Aetius] |
| Full Idea: Epicurus says the soul is a blend of fiery stuff (for bodily warmth), airy stuff (rest), breath (motion), and a nameless stuff (sense-perception). | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Aetius - fragments/reports 4.3.11 | |
| A reaction: Obviously Epicurus thought the four stuffs were different combinations of atoms, rather than being elements. Is there no stuff which gives reason? Reason must reduce to motion, presumably. |
| 6018 | Epicurus was the first to see the free will problem, and he was a libertarian [Epicurus, by Long/Sedley] |
| Full Idea: By posing the problem of determinism, Epicurus became arguably the first philosopher to recognise the philosophical centrality of what we call the Free Will Question. His strongly libertarian approach is strongly contrasted with Stoic determinism. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by AA Long / DN Sedley - Hellenic Philosophers commentary | |
| A reaction: Epicurus introduced the rather dubious 'swerve' of the atoms to make room for free will. It seems to me more consistent to stick with the determinism of Democritus. Zeno became a determinist in reaction to Epicurus. |
| 20922 | Epicurus showed that the swerve can give free motion in the atoms [Epicurus, by Diogenes of Oen.] |
| Full Idea: There is a free motion in the atoms, which Democritus did not discover, but which Epicurus brought to light, and which consists in a swerve, as he demonstrated on the basis of what is seen to be the case? | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Diogenes (Oen) - Wall inscription 54.II-III | |
| A reaction: I presume the last bit means that we see that we have freedom of choice, and infer the swerve in the atoms as the only possible explanation. The worry for libertarians is, of course, who is in charge of the swerve. |
| 14516 | There is no necessity to live with necessity [Epicurus] |
| Full Idea: Necessity is a bad thing, but there is no necessity to live with necessity. | |
| From: Epicurus (fragments/reports [c.289 BCE], 9) |
| 1909 | How can pleasure or judgement occur in a heap of atoms? [Sext.Empiricus on Epicurus] |
| Full Idea: If Epicurus makes the end consist in pleasure and asserts that the soul, like all else, is composed of atoms, it is impossible to explain how in a heap of atoms there can come about pleasure, or judgement of the good. | |
| From: comment on Epicurus (fragments/reports [c.289 BCE]) by Sextus Empiricus - Outlines of Pyrrhonism III.187 | |
| A reaction: This is a nice statement of the mind-body problem. Ontologically, physics still seems to present reality as a 'heap of particles', which gives no basis for the emergence of anything as strange as consciousness. But then magnetism is pretty strange. |
| 7814 | It was Epicurus who made the question of the will's freedom central to ethics [Epicurus, by Grayling] |
| Full Idea: Epicurus was responsible for the innovatory recognition that the question of the will's freedom is central to ethics. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by A.C. Grayling - What is Good? Ch.3 | |
| A reaction: Compare Ideas 7672 and 6018. Obviously ethical action needs freedom, but the idea of a 'free will' is quite different. It is a fiction, created to give some sort of arrogant ultimate responsibility to our actions, like God. |
| 3562 | Fine things are worthless if they give no pleasure [Epicurus] |
| Full Idea: I spit on the fine and those who emptily admire it, when it doesn't make any pleasure. | |
| From: Epicurus (fragments/reports [c.289 BCE]), quoted by Julia Annas - The Morality of Happiness Ch.16 | |
| A reaction: in Athenaeus |
| 1840 | Pleasure is the chief good because it is the most natural, especially for animals [Epicurus, by Diog. Laertius] |
| Full Idea: Pleasure is the chief good, because all animals from the moment of their birth are delighted with pleasure and offended by pain by their natural instinct, without the employment of reason. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 10.29 | |
| A reaction: The highest pleasure of predators is likely to be the killing of weaker animals. What all animals do isn't much of a criterion for the natural chief good. They also breathe. |
| 1839 | Pains of the soul are worse than pains of the body, because it feels the past and future [Epicurus, by Diog. Laertius] |
| Full Idea: The pains of the soul are worst, for the flesh is only sensible of present affliction, but the soul feels the past, present and future. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 10.29 | |
| A reaction: I don't think feeling extended across time is very relevant. What matters is that pains of the soul usually endure far longer than physical suffering. |
| 1842 | Pleasures only differ in their duration and the part of the body affected [Epicurus] |
| Full Idea: If every pleasure lasted long, and affected the whole body, then there would be no difference between one pleasure and another | |
| From: Epicurus (fragments/reports [c.289 BCE]), quoted by Diogenes Laertius - Lives of Eminent Philosophers 10.31.08 | |
| A reaction: This seems to miss out on intensity, which is of great importance to most pleasure seekers. Also it is a pleasure to be alive, which is lifelong, but we barely notice it. |
| 3557 | The end for Epicurus is static pleasure [Epicurus, by Annas] |
| Full Idea: Epicurus identifies our final end with what he calls tranquillity or 'ataraxia', which is static pleasure. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Julia Annas - The Morality of Happiness Ch.7 | |
| A reaction: I don't recall any Greek ever spotting that boredom is a problem. But then they didn't have privacy, so other people always hold their attention. Maybe this is a dream of privacy. |
| 1845 | Justice has no independent existence, but arises entirely from keeping contracts [Epicurus] |
| Full Idea: Justice has no independent existence; it results from mutual contracts, and establishes itself wherever there is a mutual engagement to guard against doing or sustaining mutual injury. | |
| From: Epicurus (fragments/reports [c.289 BCE]), quoted by Diogenes Laertius - Lives of Eminent Philosophers 10.31.35 |
| 1841 | We choose virtue because of pleasure, not for its own sake [Epicurus, by Diog. Laertius] |
| Full Idea: We choose the virtues for the sake of pleasure, and not on their own account. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 10.30 |
| 1829 | A wise man would be happy even under torture [Epicurus, by Diog. Laertius] |
| Full Idea: Even if the wise man were put to the torture, he would still be happy. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 10.26 |
| 1843 | Friendship is by far the most important ingredient of a complete and happy life [Epicurus] |
| Full Idea: Of all the things which wisdom provides for the happiness of the whole life, by far the most important is the acquisition of friendship. | |
| From: Epicurus (fragments/reports [c.289 BCE]), quoted by Diogenes Laertius - Lives of Eminent Philosophers 10.31.28 |
| 1831 | Wise men should partake of life even if they go blind [Epicurus, by Diog. Laertius] |
| Full Idea: Even though he lose his eyes, a wise man should still partake of life. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 10.26 |
| 12044 | Only Epicurus denied purpose in nature, for the whole world, or for its parts [Epicurus, by Annas] |
| Full Idea: Epicurus alone among the ancient schools denies that in nature we find any teleological explanations. Nothing in nature is for anything, neither the world as a whole nor anything in it. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Julia Annas - Ancient Philosophy: very short introduction | |
| A reaction: This may explain the controversial position that epicureanism held in the seventeenth century, as well as its incipient atheism. |
| 20907 | Democritus says atoms have size and shape, and Epicurus added weight [Epicurus, by Ps-Plutarch] |
| Full Idea: Democritus said that the properties of the atoms are in number two, magnitude and shape, but Epicurus added to these a third one, weight. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by Pseudo-Plutarch - On the Doctrine of the Philosophers 1.3.18 | |
| A reaction: The addition of Epicurus seems very sensible, and an odd omission by Democritus. He seems to think that atoms have a uniform density, so that volume indicates weight. |
| 21669 | Atoms don't swerve by being struck, because they move in parallel, so the swerve is uncaused [Cicero on Epicurus] |
| Full Idea: The swerve of Epicurus takes place without a cause; it does not take place in consequence of being struck by another atom, since how can that take place if they are indivisible bodies travelling perpendicularly in straight lines by the force of gravity? | |
| From: comment on Epicurus (fragments/reports [c.289 BCE]) by M. Tullius Cicero - On Fate ('De fato') 10.22 | |
| A reaction: The swerve is the most ad hoc proposal in the history of theoretical physics. This is interesting for spelling out that the travel in vertical parallels. What's that all about, then? |
| 21680 | What causes atomic swerves? Do they draw lots? What decides the size or number of swerves? [Cicero on Epicurus] |
| Full Idea: What fresh cause exists in nature to make the atom swerve (or do the atoms cast lots among them which is to swerve and which not?), or to serve as the reason for making a very small swerve and not a large one, or one swerve, and not two or three swerves? | |
| From: comment on Epicurus (fragments/reports [c.289 BCE]) by M. Tullius Cicero - On Fate ('De fato') 20.46 | |
| A reaction: This is an appeal to the Principle of Sufficient Reason, which seems to be the main ground for rejecting the swerve. The only reason to accept the swerve is reluctance to accept determinism or fatalism. |
| 14525 | Stoics say time is incorporeal and self-sufficient; Epicurus says it is a property of properties of things [Epicurus] |
| Full Idea: Stoics posited that time is an incorporeal which is conceived of all by itself, while Epicurus thinks that it is an accident of certain things, ...and he called in a property of properties. | |
| From: Epicurus (fragments/reports [c.289 BCE]) | |
| A reaction: [Source Sextus 'Adversus Mathematicos' 10.219-227] |
| 2637 | For Epicureans gods are made of atoms, and are not eternal [Epicurus, by Cicero] |
| Full Idea: For Epicureans the gods are made of atoms, so in that case they are not eternal. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by M. Tullius Cicero - On the Nature of the Gods ('De natura deorum') I.68 |
| 2633 | Epicurus saw that gods must exist, because nature has imprinted them on human minds [Epicurus, by Cicero] |
| Full Idea: Epicurus alone saw that gods must exist because nature herself has imprinted an idea of them in the minds of all mankind. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by M. Tullius Cicero - On the Nature of the Gods ('De natura deorum') I.43 |
| 2639 | Some say Epicurus only pretended to believe in the gods, so as not to offend Athenians [Epicurus, by Cicero] |
| Full Idea: Some believe that Epicurus gave lip-service only to the gods, so as not to offend the Athenians, but in fact did not believe in them. | |
| From: report of Epicurus (fragments/reports [c.289 BCE]) by M. Tullius Cicero - On the Nature of the Gods ('De natura deorum') I.84 |
| 14527 | If god answered prayers we would be destroyed, because we pray for others to suffer [Epicurus] |
| Full Idea: If god acted in accordance with the prayers of men, all men would rather quickly be destroyed, since they constantly pray for many sufferings to befall each other. | |
| From: Epicurus (fragments/reports [c.289 BCE]) | |
| A reaction: [source Maximus the Abbott 'Gnom.' 14] |