30 ideas
| 7740 | There exists a realm, beyond objects and ideas, of non-spatio-temporal thoughts [Frege, by Weiner] |
| Full Idea: There is, in addition to the external world of physical objects and the internal world of ideas, a third realm of non-spatio-temporal objective objects, among which are thoughts. | |
| From: report of Gottlob Frege (The Thought: a Logical Enquiry [1918]) by Joan Weiner - Frege Ch.7 | |
| A reaction: This seems to be Platonism, and, in particular, to give a Platonic existent status to propositions. Personally I believe in propositions, but as glimpses of how our brains actually work, not as mystical objects. |
| 19466 | The word 'true' seems to be unique and indefinable [Frege] |
| Full Idea: It seems likely that the content of the word 'true' is sui generis and indefinable | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918], p.327 (60)) | |
| A reaction: This is the view I associate with Davidson, though fans of Axiomatic Truth give up defining it, and just describe how it behaves. Defining it is very elusive, but I don't accept that nothing can be said about the contents of the concept of truth. |
| 19465 | There cannot be complete correspondence, because ideas and reality are quite different [Frege] |
| Full Idea: It is essential that the reality shall be distinct from the idea. But then there can be no complete correspondence, no complete truth. | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918], p.327 (60)) | |
| A reaction: He thinks that logic can give a perfect account of truth, or at least the extension of truth, where ordinary language will always fail. I wonder what he would have thought of Tarski's theory? |
| 19468 | The property of truth in 'It is true that I smell violets' adds nothing to 'I smell violets' [Frege] |
| Full Idea: The sentence 'I smell the scent of violets' has just the same content as 'It is true that I smell the scent of violets'. So it seems that nothing is added to the thought by my ascribing to it the property of truth. | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918], p.328 (61)) | |
| A reaction: This idea predates Ramsey's similar proposal, for which, oddly, Ramsey always seems to get the credit. To a logician they may have identical content, but pragmatically they are likely to differ in context. 'True' certainly doesn't add to the thought. |
| 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. |
| 19470 | Thoughts in the 'third realm' cannot be sensed, and do not need an owner to exist [Frege] |
| Full Idea: Thoughts are neither things in the external world nor ideas. A third realm must be recognised. Anything in this realm has it in common with ideas that it cannot be perceived by the senses, and does not need an owner to belong with his consciousness. | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918], p.337(69)) | |
| A reaction: This important idea is the creed for modern platonists. We don't have to accept Forms, or any particular content, but there is a mode of existence which is distinct from both mental and physical, and is the residence of 'abstracta'. I deny it! |
| 19471 | A fact is a thought that is true [Frege] |
| Full Idea: A fact is a thought that is true. | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918], p.342(74)) | |
| A reaction: It strikes me as pretty obvious that facts are not thoughts, because they concern the contents of thoughts. You can't discuss facts without the notion of what a thought is 'about'. If I think about my garden, the relevant fact is aspects of my garden. |
| 9877 | Late Frege saw his non-actual objective objects as exclusively thoughts and senses [Frege, by Dummett] |
| Full Idea: Earlier, Frege divided objects into subjective, actual objective, and non-actual objective; in the 'Grundgesetze' he emphasised logical objects; but in 'The Thought' the non-actual objects become exclusively thoughts and their constituent senses. | |
| From: report of Gottlob Frege (The Thought: a Logical Enquiry [1918]) by Michael Dummett - Frege philosophy of mathematics Ch.18 | |
| A reaction: Sounds to me like Frege was finally waking up and taking a dose of common sense. The Equator is the standard example of a non-actual objective object. |
| 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. |
| 3364 | All mental phenomena contain an object [Brentano] |
| Full Idea: Every mental phenomenon contains something as object within itself. | |
| From: Franz Brentano (Psychology from an empirical standpoint [1874], p. 88), quoted by Jaegwon Kim - Philosophy of Mind p.21 | |
| A reaction: This gives rise to the slogan that 'intentionality is the mark of the mental', which notoriously seems to miss out the phenomenal aspect of mental life. We note now, though, that even emotions have objects. |
| 3225 | Mental unity suggests that qualia and intentionality must connect [Brentano, by Rey] |
| Full Idea: Brentano's thesis is that all mental phenomena are intentional i.e. representational. Support for this view is that assimilating phenomenal experience to attitudes we explain the essential unity of the mind. | |
| From: report of Franz Brentano (Psychology from an empirical standpoint [1874]) by Georges Rey - Contemporary Philosophy of Mind 11.5 | |
| A reaction: Unifying intentionality and qualia in a single theory looks like a good move, but which one has priority? Evolutionary theory says priority goes to whatever produces behaviour. My intuition is that qualia are more basic - in tiny insects, say. |
| 19469 | We grasp thoughts (thinking), decide they are true (judgement), and manifest the judgement (assertion) [Frege] |
| Full Idea: We distinguish the grasp of a thought, which is 'thinking', from the acknowledgement of the truth of a thought, which is the act of 'judgement', from the manifestation of this judgement, which is an 'assertion'. | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918], p.329 (62)) |
| 8162 | Thoughts have their own realm of reality - 'sense' (as opposed to the realm of 'reference') [Frege, by Dummett] |
| Full Idea: For Frege, thoughts belong to a special realm of reality, which he called the 'realm of sense' and distinguished from the 'realm of reference'. | |
| From: report of Gottlob Frege (The Thought: a Logical Enquiry [1918]) by Michael Dummett - Thought and Reality 1 | |
| A reaction: A thought is, for Frege, a proposition. There is a halfway Platonism possible here, where the 'realm' for such things exists, but within that realm the objects might be conventional, or some such. Real possible worlds containing fictions! |
| 9818 | A thought is distinguished from other things by a capacity to be true or false [Frege, by Dummett] |
| Full Idea: On Frege's view, what distinguishes thoughts from everything else is that they may meaningfully be called 'true' and 'false'. | |
| From: report of Gottlob Frege (The Thought: a Logical Enquiry [1918]) by Michael Dummett - Frege philosophy of mathematics Ch.2 | |
| A reaction: A lot of thinking is imagistic, and while the image may or may not truly picture the world, we tend to think that the truth or otherwise of daydreaming is simply irrelevant. Does Frege take all thought to be propositional? |
| 16379 | Thoughts about myself are understood one way to me, and another when communicated [Frege] |
| Full Idea: When Dr Lauben thinks he has been wounded, ..only Dr Lauben can grasp thoughts determined in this way. But he cannot communicate a thought which only he can grasp. To say 'I have been wounded' he must use 'I' in a sense graspable by others. | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918]), quoted by François Recanati - Mental Files 16.1 | |
| A reaction: [compressed] This seems to be the first, and very influential, attempt to explain the unusual and revealing semantics of indexicals. It seems to be the ultimate source of 2-D semantics, by introducing two modes of meaning for one term. |
| 19467 | A 'thought' is something for which the question of truth can arise; thoughts are senses of sentences [Frege] |
| Full Idea: I call a 'thought' something for which the question of truth can arise at all. ...So I can say: thoughts are senses of sentences, without wishing to assert that the sense of every sentence is a thought. | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918], p.327-8 (61)) | |
| A reaction: This builds on his distinction between sense and reference. The reference of every truth sentence is just 'the true', and the sense is the proposition. The concept of a proposition seems indispensable to logic, I would say. |
| 19472 | A sentence is only a thought if it is complete, and has a time-specification [Frege] |
| Full Idea: Only a sentence with the time-specification filled out, a sentence complete in every respect, expresses a thought. | |
| From: Gottlob Frege (The Thought: a Logical Enquiry [1918], p.343(76)) | |
| A reaction: I take the 'every respect' to include the avoidance of ambiguity, and some sort of perspicacious reference for the terms. I wish philosophers would focus on the thoughts in their subject, and not nit-pick about the sentences. Does he mean 'utterances'? |