51 ideas
18702 | Names, descriptions and predicates refer to things; without that, language and thought are baffling [Davidson] |
Full Idea: The simple thesis that names and descriptions often refer to things, and that predicates often have an extension in the world of things, is obvious, and essential to the most elementary appreciation of both language and the thoughts we express. | |
From: Donald Davidson (Replies to Critics [1998], p.323) | |
A reaction: In 1983 Davidson had been a rare modern champion of the coherence theory of truth, but this is his clearest later renunciation of that view (and quite right too). |
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
A reaction: The sequence is 'Cauchy' if N exists. |
8764 | Categories are the best foundation for mathematics [Shapiro] |
Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
3942 | I do not believe in the existence of anything, if I see no reason to believe it [Berkeley] |
Full Idea: It is to me a sufficient reason not to believe the existence of anything, if I see no reason for believing it. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], II p.205) | |
A reaction: This may just be a reasonable application of Ockham's Razor, but I fear that Berkeley painted himself into corner by demanding too many 'reasons' for everything. |
3952 | I know that nothing inconsistent can exist [Berkeley] |
Full Idea: I know that nothing inconsistent can exist. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], III p.224) | |
A reaction: Fine, but the problem is to assess with confidence what is inconsistent. Human imagination seems to be the test for existence. But what else can we do? |
3959 | There is no other substance, in a strict sense, than spirit [Berkeley] |
Full Idea: There is no other substance, in a strict sense, than spirit. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], III p.257) | |
A reaction: A nice clear statement of idealism. Why is he so confident of making this assertion. Note the addition, though, of 'in a strict' sense. He is presenting an epistemological claim as if it was an ontological one. |
3946 | A thing is shown to be impossible if a contradiction is demonstrated within its definition [Berkeley] |
Full Idea: A thing is shown to be impossible when a repugnancy is demonstrated between the ideas comprehended in its definition. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], II p.214) | |
A reaction: The problem is always that imagination is needed to see the 'repugnancy', and that is relative and limited. |
3958 | Since our ideas vary when the real things are said to be unchanged, they cannot be true copies [Berkeley] |
Full Idea: As our ideas are perpetually varied, without any change in the supposed real things, it necessarily follows that they cannot all be true copies of them. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], III p.239) | |
A reaction: This seems a good objection to any direct or naïve realist view. Colours get darker as the sun goes down, and objects become blurred as they recede into the distance. |
3943 | If existence is perceived directly, by which sense; if indirectly, how is it inferred from direct perception? [Berkeley] |
Full Idea: Either you perceive the being of matter immediately, or mediately; if immediately, pray inform me by which of the senses you perceive it; if mediately, let me know by what reasonings it is inferred from those things which you perceive immediately. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], II p.208) | |
A reaction: A problem for strong empiricists, and he is right that existence can't be directly perceived, but it seems a good explanation (for which some reason can be shown), and supports a more rationalist view. |
3931 | Sensible objects are just sets of sensible qualities [Berkeley] |
Full Idea: Sensible things are nothing else but so many sensible qualities. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], I p.154) | |
A reaction: As it stands this is phenomenalism, but Berkeley eventually votes for idealism. He should acknowledge possible sensations which aren't actually experienced. |
5192 | Berkeley did not deny material things; he merely said they must be defined through sensations [Berkeley, by Ayer] |
Full Idea: Berkeley did not (as we are commonly told) deny the reality of material things. ..What Berkeley discovered was that material things must be defined in terms of sense-contents. | |
From: report of George Berkeley (Three Dialogues of Hylas and Philonous [1713]) by A.J. Ayer - Language,Truth and Logic Ch.2 | |
A reaction: This seems to be a rather debatable attempt to claim that Berkeley was a phenomenalist (like Ayer), rather than an idealist. Try ideas 3942, 3944, 3945, 3957, 3959 in this database. |
5174 | Berkeley needed a phenomenalist account of the self, as well as of material things [Ayer on Berkeley] |
Full Idea: The considerations which make it necessary, as Berkeley saw, to give a phenomenalist account of material things, make it necessary also, as Berkeley did not see, to give a phenomenalist account of the self. | |
From: comment on George Berkeley (Three Dialogues of Hylas and Philonous [1713]) by A.J. Ayer - Language,Truth and Logic Ch.7 | |
A reaction: Phenomenalism involves 'possible' experiences as well as actual ones. That could add up to quite a rich and stable account of the self, as opposed to Hume's notorious introspection, which only saw an actual shifting 'bundle' of experience. |
1103 | 'To be is to be perceived' is a simple confusion of experience with its objects [Russell on Berkeley] |
Full Idea: Berkeley thinks 'to be is to be perceived', and only God provides continuity. He has simply confused the experience of perception with the thing being perceived. Ideas have content. | |
From: comment on George Berkeley (Three Dialogues of Hylas and Philonous [1713]) by Bertrand Russell - Problems of Philosophy |
6403 | For Berkelely, reality is ideas and a community of minds, including God's [Berkeley, by Grayling] |
Full Idea: Berkeley's thesis is that reality ultimately consists of a community of minds and their ideas; one of the minds (God) is infinite, and causes most of the ideas. | |
From: report of George Berkeley (Three Dialogues of Hylas and Philonous [1713]) by A.C. Grayling - Russell Ch.2 | |
A reaction: I think Russell nicely pinpoints what is wrong with Berekely, which is that he confuses ideas with their contents. If I think about my garden, the garden is real (probably), which is the content, and they idea is just a way of thinking. |
3936 | Time is measured by the succession of ideas in our minds [Berkeley] |
Full Idea: Time is measured by the succession of ideas in our minds. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], I p.172) | |
A reaction: But we distinguish between subjective time (which flies when you are having fun), and objective time, judged from observation of clocks and nature. |
3930 | There is no such thing as 'material substance' [Berkeley] |
Full Idea: That there is no such thing as what philosophers call 'material substance', I am seriously persuaded. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], I p.150) | |
A reaction: I'm sorry, but I can't do with this. It confuses epistemology with ontology. Ontology is a matter of judgement; epistemology is the evidence on which we base it. We know sensations; personally I judge that there are material substances. What about you? |
3939 | I conceive a tree in my mind, but I cannot prove that its existence can be conceived outside a mind [Berkeley] |
Full Idea: I may conceive in my own thoughts the idea of a tree, but that is all. And this is far from proving that I can conceive it existing out of the minds of all spirits. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], I p.184) | |
A reaction: If Berkeley has based a world view on this point, then his mistake is to require a 'proof'. Aristotle explained why you can't prove everything (not to mention Gödel). |
3945 | There is nothing in nature which needs the concept of matter to explain it [Berkeley] |
Full Idea: I challenge you to show me that thing in nature which needs matter to explain or account for it. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], II p.212) | |
A reaction: I disagree. Physics is a good theory for explaining why we have perceptions. Failing that there is not even a glimmer of an explanation of our experiences. |
3947 | Perceptions are ideas, and ideas exist in the mind, so objects only exist in the mind [Berkeley] |
Full Idea: Wood, fire, water, flesh, iron, are things that I know, and only known because I perceive them by my senses; these are immediately perceived, and so are ideas; ideas cannot exist without the mind; their existence consists therefore in being perceived. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], III p.220) | |
A reaction: This makes no distinction between an idea and its content. Berkeley fails to grasp the weird concept of intentionality. Trees aren't in my head, just because I think about them! |
3933 | Primary qualities (such as shape, solidity, mass) are held to really exist, unlike secondary qualities [Berkeley] |
Full Idea: Sensible qualities are by philosophers divided into primary and secondary; the former are extension, figure, solidity, gravity, motion and rest, which exist really in bodies. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], I p.169) | |
A reaction: A crucial distinction, which anti-realists such as Berkeley end up denying. I think it is a good distinction, and philosophers should fight to preserve it. |
3934 | A mite would see its own foot as large, though we would see it as tiny [Berkeley] |
Full Idea: A mite must be supposed to see his own foot as a body of some considerable dimension, though they appear to you scarcely discernible. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], I p.170) | |
A reaction: Berkeley is confused. Hot is secondary, but temperature is primary. Bigness is secondary, size primay. Midgets and tall people don't disagree over the size of a table. |
3935 | The apparent size of an object varies with its distance away, so that can't be a property of the object [Berkeley] |
Full Idea: As we approach to or recede from an object, the visible extension varies, being at one distance ten or a hundred times greater than at another; doth it not follow that it is not really inherent in the object? | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], I p.171) | |
A reaction: Berkeley is confused, because he is too literally empirical. Qualities are not self-evidently primary or secondary, but are judged so after comparisons (e.g. with testimony, or with the other senses). |
3937 | 'Solidity' is either not a sensible quality at all, or it is clearly relative to our senses [Berkeley] |
Full Idea: By 'solidity' either you do not mean any sensible quality, and so it is beside our enquiry; or if you do, it must be hardness or resistance, which are plainly relative to our senses. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], I p.173) | |
A reaction: Berkeley fails to recognise that a quality can have primary and secondary aspects (hot/high temperature). He is right that primary qualities are not directly perceived. They are judgements. |
3940 | Distance is not directly perceived by sight [Berkeley] |
Full Idea: Distance is not properly and immediately perceived by sight. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], I p.186) | |
A reaction: Interestingly, if secondary qualities are not strictly perceptions of the object, and primary qualities are not directly perceived, then we don't seem to perceive anything at all. Perhaps we should drop the concept of 'perception'? |
3957 | Immediate objects of perception, which some treat as appearances, I treat as the real things themselves [Berkeley] |
Full Idea: Those immediate objects of perception, which, according to you, are only appearances of things, I take to be the real things themselves. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], III p.237) | |
A reaction: If that is a judgement, which it seems to be, it is a strange one. Realists offer a much better explanation of perceptions. |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
3953 | Real things and imaginary or dreamed things differ because the latter are much fainter [Berkeley] |
Full Idea: The difference between real things, and chimeras formed by the imagination, or the visions of a dream, is that the latter are faint and indistinct. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], III p.225) | |
A reaction: In Hume this becomes 'impressions' and 'ideas'. It does raise the question of WHY some ideas are not as faint as others. |
3938 | Geometry is originally perceived by senses, and so is not purely intellectual [Berkeley] |
Full Idea: Figures and extension, being originally perceived by sense, do not belong to pure intellect. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], I p.176) | |
A reaction: Is the square root of 169 less 'pure' in my mind if I learn it from laying out bricks instead of by thinking about numbers? Confusion of how you learn with what you learn? |
3944 | It is possible that we could perceive everything as we do now, but nothing actually existed. [Berkeley] |
Full Idea: We might perceive all things just as we do now, though there was no matter in the world. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], II p.209) | |
A reaction: An old Greek argument. Now we have an explanation of experience, but we wouldn't if nothing existed. Which doesn't prove that anything exists. Is some explanation always preferable to none? Cf. religion. |
3932 | A hot hand and a cold hand will have different experiences in the same tepid water [Berkeley] |
Full Idea: Suppose now one of your hands hot, and the other cold, and that they are both at once put into the same vessel of water, in an intermediate state; will not the water seem cold to one hand, and warm to the other? | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], I p.158) | |
A reaction: A nice clear example of how some relativism must be acknowledged. It feels hot, but what is its temperature in degrees C? |
3948 | Experience tells me that other minds exist independently from my own [Berkeley] |
Full Idea: It is plain that other minds have an existence exterior to my mind, since I find them by experience to be independent of it. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], III p.220) | |
A reaction: This is a surprising claim from Berkeley. If trees only exist through their experience in my mind, why don't other minds exist in the same way? |
3941 | How can that which is unthinking be a cause of thought? [Berkeley] |
Full Idea: How can that which is unthinking be a cause of thought? | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], II p.203) | |
A reaction: Presumably, though, he thinks that thought can cause 'that which is unthinking' to move'. He likes one half of the interaction problem (which supports dualism), but avoids the other half. |
5374 | Berkeley probably used 'idea' to mean both the act of apprehension and the thing apprehended [Russell on Berkeley] |
Full Idea: Berkeley seems to have confused the colour of the thing apprehended with the act of apprehension; probably either of these would have been called an 'idea' be Berkeley. | |
From: comment on George Berkeley (Three Dialogues of Hylas and Philonous [1713]) by Bertrand Russell - Problems of Philosophy | |
A reaction: If we are saying that Berkeley's error was entirely verbal, there is a chicken-and-egg problem. He was an idealist, so he wouldn't have thought that there were two separate concepts behind the word 'idea'. Russell merely asserts that there are. |
3954 | Immorality is not in the action, but in the deviation of the will from moral law [Berkeley] |
Full Idea: Sin or moral turpitude doth not consist in the outward physical action or motion, but in the internal deviation of the will from the laws of reason and religion. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], III p.227) | |
A reaction: A Kantian view (that the only good thing is a good will). It is a very empiricist (and anti-Greek) view to deny that actions have any intrinsic value. |
3950 | There must be a God, because all sensible things must be perceived by him [Berkeley] |
Full Idea: I immediately and necessarily conclude the being of a God, because all sensible things must be perceived by him. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], II p.198) | |
A reaction: Daft. This contradicts Berkeley's whole empiricist position, that existence depends on known experience. Who knows whether God is thinking about trees? |
3951 | There must be a God, because I and my ideas are not independent [Berkeley] |
Full Idea: From the dependency I find in myself and my ideas, I do by an act of reason necessarily infer the existence of a God. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], III p.222) | |
A reaction: No. Hume answered this, by showing how big abstract ideas are built up from experience. This is a future bishop's wish-fulfilment. |
3949 | It has been proved that creation is the workmanship of God, from its beauty and usefulness [Berkeley] |
Full Idea: Divines and philosophers have proved beyond all controversy, from the beauty and usefulness of the several parts of creation, that it was the workmanship of God. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], II p.198) | |
A reaction: Not convincing. Beauty is probably a sublimation of sexual desire (or an echo of the human mind in the external world, in music), and utility is relative to homo sapiens, I presume. |
3956 | People are responsible because they have limited power, though this ultimately derives from God [Berkeley] |
Full Idea: Thinking rational beings, in the production of motions, have the use of limited powers, ultimately derived from God, but immediately under the direction of their own wills, which is sufficient to entitle them to all the guilt of their own actions. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], III p.228) | |
A reaction: An episcopal evasion. A classic attempt to have cake and eat it. Either God is in charge or he isn't. |
3955 | If sin is not just physical, we don't consider God the origin of sin because he causes physical events [Berkeley] |
Full Idea: If sin doth not consist of purely physical actions, the making God a cause of all such actions, is not making him the author of sin. | |
From: George Berkeley (Three Dialogues of Hylas and Philonous [1713], III p.227) | |
A reaction: An equivocation. If responsibility resides in consciousness, God is presumably conscious, and we can judge the events he causes. |