27 ideas
| 23657 | The existence of tensed verbs shows that not all truths are necessary truths [Reid] |
| Full Idea: If all truths were necessary truths, there would be no occasion for different tenses in the verbs by which they are expressed. | |
| From: Thomas Reid (Essays on Intellectual Powers 6: Judgement [1785], 5) | |
| A reaction: This really is like modern linguistic analysis. Of course the tensed verbs might only indicate times when the universal necessities have been noticed by speakers. …But then the noticing would be contingent! |
| 23655 | An ad hominem argument is good, if it is shown that the man's principles are inconsistent [Reid] |
| Full Idea: It is a good argument ad hominem, if it can be shewn that a first principle which a man rejects, stands upon the same footing with others which he admits, …for he must then be guilty of an inconsistency. | |
| From: Thomas Reid (Essays on Intellectual Powers 6: Judgement [1785], 4) | |
| A reaction: Good point. You can't divorce 'pure' reason from the reasoners, because the inconsistency of two propositions only matters when they are both asserted together. …But attacking the ideas isn't quite the same as attacking the person. |
| 18189 | ZFC could contain a contradiction, and it can never prove its own consistency [MacLane] |
| Full Idea: We have at hand no proof that the axioms of ZFC for set theory will never yield a contradiction, while Gödel's second theorem tells us that such a consistency proof cannot be conducted within ZFC. | |
| From: Saunders MacLane (Mathematics: Form and Function [1986], p.406), quoted by Penelope Maddy - Naturalism in Mathematics | |
| A reaction: Maddy quotes this, while defending set theory as the foundation of mathematics, but it clearly isn't the most secure foundation that could be devised. She says the benefits of set theory do not need guaranteed consistency (p.30). |
| 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. |
| 23659 | If someone denies that he is thinking when he is conscious of it, we can only laugh [Reid] |
| Full Idea: If any man could be found so frantic as to deny that he thinks, while he is conscious of it, I may wonder, I may laugh, or I may pity him, but I cannot reason the matter with him. | |
| From: Thomas Reid (Essays on Intellectual Powers 6: Judgement [1785], 5) | |
| A reaction: An example of the influence of Descartes' Cogito running through all subsequent European philosophy. There remain the usual questions about personal identity which then arise, but Reid addresses those. |
| 23662 | The existence of ideas is no more obvious than the existence of external objects [Reid] |
| Full Idea: If external objects be perceived immediately, we have the same reason to believe their existence as philosophers have to believe the existence of ideas. | |
| From: Thomas Reid (Essays on Intellectual Powers 6: Judgement [1785], 5) | |
| A reaction: He doesn't pay much attention to mirages and delusions, but in difficult conditions of perception we are confident of our experiences but doubtful about the objects they represent. |
| 23661 | We are only aware of other beings through our senses; without that, we are alone in the universe [Reid] |
| Full Idea: We can have no communication, no correspondence or society with any created being, but by means of our senses. And, until we rely on their testimony, we must consider ourselves as being alone in the universe. | |
| From: Thomas Reid (Essays on Intellectual Powers 6: Judgement [1785], 5) | |
| A reaction: I'm not aware of any thinker before this so directly addressing solipsism. Even the champion of direct and common sense realism has to recognise the intermediary of our senses when accepting other minds. |
| 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. |
| 23654 | In obscure matters the few must lead the many, but the many usually lead in common sense [Reid] |
| Full Idea: In matters beyond the reach of common understanding, the many are led by the few, and willingly yield to their authority. But, in matters of common sense, the few must yield to the many, when local and temporary prejudices are removed. | |
| From: Thomas Reid (Essays on Intellectual Powers 6: Judgement [1785], 4) | |
| A reaction: Wishful thinking in the 21st century, when the many routinely deny the authority of the expert few, and the expert few occasionally prove that the collective common sense of the many is delusional. I still sort of agree with Reid. |
| 23660 | The theory of ideas, popular with philosophers, means past existence has to be proved [Reid] |
| Full Idea: The theory concerning ideas, so generally received by philosophers, destroys all the authority of memory. …This theory made it necessary for them to find out arguments to prove the existence of external objects …and of things past. | |
| From: Thomas Reid (Essays on Intellectual Powers 6: Judgement [1785], 5) | |
| A reaction: Reid was a very articulate direct realist. He seems less aware than the rest of us of the problem of delusions and false memories. Our strong sense that immediate memories are reliable is certainly inexplicable. |
| 23658 | Consciousness is an indefinable and unique operation [Reid] |
| Full Idea: Consciousness is an operation of the understanding of its own kind, and cannot be logically defined. | |
| From: Thomas Reid (Essays on Intellectual Powers 6: Judgement [1785], 5) | |
| A reaction: It is interesting that has tried to define consciousness, rather than just assuming it. I note that he calls consciousness an 'operation', rather than an entity. Good. |
| 23656 | The structure of languages reveals a uniformity in basic human opinions [Reid] |
| Full Idea: What is common in the structure of languages, indicates an uniformity of opinion in those things upon which that structure is grounded. | |
| From: Thomas Reid (Essays on Intellectual Powers 6: Judgement [1785], 4) | |
| A reaction: Reid was more interested than his contemporaries in the role of language in philosophy. The first idea sounds like Chomsky. I would add to this that the uniformity of common opinion reflects uniformities in the world they are talking about. |
| 23653 | If you can't distinguish the features of a complex object, your notion of it would be a muddle [Reid] |
| Full Idea: If you perceive an object, white, round, and a foot in diameter, if you had not been able to distinguish the colour from the figure, and both from the magnitude, your senses would only give you one complex and confused notion of all these mingled together | |
| From: Thomas Reid (Essays on Intellectual Powers 6: Judgement [1785], 1) | |
| A reaction: His point is that if you reject the 'abstraction' of these qualities, you still cannot deny that distinguishing them is an essential aspect of perceiving complex things. Does this mean that animals distinguish such things? |
| 23663 | There are axioms of taste - such as a general consensus about a beautiful face [Reid] |
| Full Idea: I think there are axioms, even in matters of taste. …I never heard of any man who thought it a beauty in a human face to want a nose, or an eye, or to have the mouth on one side. | |
| From: Thomas Reid (Essays on Intellectual Powers 6: Judgement [1785], 6) | |
| A reaction: It is hard to disagree, but the human face may be a special case, since it is so deeply embedded in the minds of even the youngest infants. More recent artists seem able to discover beauty in very unlikely places. |