23 ideas
| 18781 | Inconsistency doesn't prevent us reasoning about some system [Mares] |
| Full Idea: We are able to reason about inconsistent beliefs, stories, and theories in useful and important ways | |
| From: Edwin D. Mares (Negation [2014], 1) |
| 6299 | Axioms are often affirmed simply because they produce results which have been accepted [Resnik] |
| Full Idea: Many axioms have been proposed, not on the grounds that they can be directly known, but rather because they produce a desired body of previously recognised results. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.5.1) | |
| A reaction: This is the perennial problem with axioms - whether we start from them, or whether we deduce them after the event. There is nothing wrong with that, just as we might infer the existence of quarks because of their results. |
| 18790 | Intuitionism as natural deduction has no rule for negation [Mares] |
| Full Idea: In intuitionist logic each connective has one introduction and one elimination rule attached to it, but in the classical system we have to add an extra rule for negation. | |
| From: Edwin D. Mares (Negation [2014], 5.5) | |
| A reaction: How very intriguing. Mares says there are other ways to achieve classical logic, but they all seem rather cumbersome. |
| 18789 | Intuitionist logic looks best as natural deduction [Mares] |
| Full Idea: Intuitionist logic appears most attractive in the form of a natural deduction system. | |
| From: Edwin D. Mares (Negation [2014], 5.5) |
| 18787 | Three-valued logic is useful for a theory of presupposition [Mares] |
| Full Idea: One reason for wanting a three-valued logic is to act as a basis of a theory of presupposition. | |
| From: Edwin D. Mares (Negation [2014], 3.1) | |
| A reaction: [He cites Strawson 1950] The point is that you can get a result when the presupposition does not apply, as in talk of the 'present King of France'. |
| 18793 | Material implication (and classical logic) considers nothing but truth values for implications [Mares] |
| Full Idea: The problem with material implication, and classical logic more generally, is that it considers only the truth value of formulas in deciding whether to make an implication stand between them. It ignores everything else. | |
| From: Edwin D. Mares (Negation [2014], 7.1) | |
| A reaction: The obvious problem case is conditionals, and relevance is an obvious extra principle that comes to mind. |
| 18784 | In classical logic the connectives can be related elegantly, as in De Morgan's laws [Mares] |
| Full Idea: Among the virtues of classical logic is the fact that the connectives are related to one another in elegant ways that often involved negation. For example, De Morgan's Laws, which involve negation, disjunction and conjunction. | |
| From: Edwin D. Mares (Negation [2014], 2.2) | |
| A reaction: Mares says these enable us to take disjunction or conjunction as primitive, and then define one in terms of the other, using negation as the tool. |
| 18786 | Excluded middle standardly implies bivalence; attacks use non-contradiction, De M 3, or double negation [Mares] |
| Full Idea: On its standard reading, excluded middle tells us that bivalence holds. To reject excluded middle, we must reject either non-contradiction, or ¬(A∧B) ↔ (¬A∨¬B) [De Morgan 3], or the principle of double negation. All have been tried. | |
| From: Edwin D. Mares (Negation [2014], 2.2) |
| 18780 | Standard disjunction and negation force us to accept the principle of bivalence [Mares] |
| Full Idea: If we treat disjunction in the standard way and take the negation of a statement A to mean that A is false, accepting excluded middle forces us also to accept the principle of bivalence, which is the dictum that every statement is either true or false. | |
| From: Edwin D. Mares (Negation [2014], 1) | |
| A reaction: Mates's point is to show that passively taking the normal account of negation for granted has important implications. |
| 18782 | The connectives are studied either through model theory or through proof theory [Mares] |
| Full Idea: In studying the logical connectives, philosophers of logic typically adopt the perspective of either model theory (givng truth conditions of various parts of the language), or of proof theory (where use in a proof system gives the connective's meaning). | |
| From: Edwin D. Mares (Negation [2014], 1) | |
| A reaction: [compressed] The commonest proof theory is natural deduction, giving rules for introduction and elimination. Mates suggests moving between the two views is illuminating. |
| 18783 | Many-valued logics lack a natural deduction system [Mares] |
| Full Idea: Many-valued logics do not have reasonable natural deduction systems. | |
| From: Edwin D. Mares (Negation [2014], 1) |
| 18792 | Situation semantics for logics: not possible worlds, but information in situations [Mares] |
| Full Idea: Situation semantics for logics consider not what is true in worlds, but what information is contained in situations. | |
| From: Edwin D. Mares (Negation [2014], 6.2) | |
| A reaction: Since many theoretical physicists seem to think that 'information' might be the most basic concept of a natural ontology, this proposal is obviously rather appealing. Barwise and Perry are the authors of the theory. |
| 18785 | Consistency is semantic, but non-contradiction is syntactic [Mares] |
| Full Idea: The difference between the principle of consistency and the principle of non-contradiction is that the former must be stated in a semantic metalanguage, whereas the latter is a thesis of logical systems. | |
| From: Edwin D. Mares (Negation [2014], 2.2) |
| 6304 | Mathematical realism says that maths exists, is largely true, and is independent of proofs [Resnik] |
| Full Idea: Mathematical realism is the doctrine that mathematical objects exist, that much contemporary mathematics is true, and that the existence and truth in question is independent of our constructions, beliefs and proofs. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.12.9) | |
| A reaction: As thus defined, I would call myself a mathematical realist, but everyone must hesitate a little at the word 'exist' and ask, how does it exist? What is it 'made of'? To say that it exists in the way that patterns exist strikes me as very helpful. |
| 6300 | Mathematical constants and quantifiers only exist as locations within structures or patterns [Resnik] |
| Full Idea: In maths the primary subject-matter is not mathematical objects but structures in which they are arranged; our constants and quantifiers denote atoms, structureless points, or positions in structures; they have no identity outside a structure or pattern. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.1) | |
| A reaction: This seems to me a very promising idea for the understanding of mathematics. All mathematicians acknowledge that the recognition of patterns is basic to the subject. Even animals recognise patterns. It is natural to invent a language of patterns. |
| 6303 | Sets are positions in patterns [Resnik] |
| Full Idea: On my view, sets are positions in certain patterns. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.5) | |
| A reaction: I have always found the ontology of a 'set' puzzling, because they seem to depend on prior reasons why something is a member of a given set, which cannot always be random. It is hard to explain sets without mentioning properties. |
| 6295 | There are too many mathematical objects for them all to be mental or physical [Resnik] |
| Full Idea: If we take mathematics at its word, there are too many mathematical objects for it to be plausible that they are all mental or physical objects. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.1) | |
| A reaction: No one, of course, has ever claimed that they are, but this is a good starting point for assessing the ontology of mathematics. We are going to need 'rules', which can deduce the multitudinous mathematical objects from a small ontology. |
| 6296 | Maths is pattern recognition and representation, and its truth and proofs are based on these [Resnik] |
| Full Idea: I argue that mathematical knowledge has its roots in pattern recognition and representation, and that manipulating representations of patterns provides the connection between the mathematical proof and mathematical truth. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], One.1) | |
| A reaction: The suggestion that patterns are at the basis of the ontology of mathematics is the most illuminating thought I have encountered in the area. It immediately opens up the possibility of maths being an entirely empirical subject. |
| 6301 | Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest [Resnik] |
| Full Idea: Of the equivalence relationships which occur between patterns, congruence is the strongest, equivalence the next, and mutual occurrence the weakest. None of these is identity, which would require the same position. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.3) | |
| A reaction: This gives some indication of how an account of mathematics as a science of patterns might be built up. Presumably the recognition of these 'degrees of strength' cannot be straightforward observation, but will need an a priori component? |
| 6302 | Structuralism must explain why a triangle is a whole, and not a random set of points [Resnik] |
| Full Idea: An objection is that structuralism fails to explain why certain mathematical patterns are unified wholes while others are not; for instance, some think that an ontological account of mathematics must explain why a triangle is not a 'random' set of points. | |
| From: Michael D. Resnik (Maths as a Science of Patterns [1997], Three.10.4) | |
| A reaction: This is an indication that we are not just saying that we recognise patterns in nature, but that we also 'see' various underlying characteristics of the patterns. The obvious suggestion is that we see meta-patterns. |
| 18788 | For intuitionists there are not numbers and sets, but processes of counting and collecting [Mares] |
| Full Idea: For the intuitionist, talk of mathematical objects is rather misleading. For them, there really isn't anything that we should call the natural numbers, but instead there is counting. What intuitionists study are processes, such as counting and collecting. | |
| From: Edwin D. Mares (Negation [2014], 5.1) | |
| A reaction: That is the first time I have seen mathematical intuitionism described in a way that made it seem attractive. One might compare it to a metaphysics based on processes. Apparently intuitionists struggle with infinite sets and real numbers. |
| 18791 | In 'situation semantics' our main concepts are abstracted from situations [Mares] |
| Full Idea: In 'situation semantics' individuals, properties, facts, and events are treated as abstractions from situations. | |
| From: Edwin D. Mares (Negation [2014], 6.1) | |
| A reaction: [Barwise and Perry 1983 are cited] Since I take the process of abstraction to be basic to thought, I am delighted to learn that someone has developed a formal theory based on it. I am immediately sympathetic to situation semantics. |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |