14 ideas
23445 | Naïve set theory says any formula defines a set, and coextensive sets are identical [Linnebo] |
Full Idea: Naïve set theory is based on the principles that any formula defines a set, and that coextensive sets are identical. | |
From: Øystein Linnebo (Philosophy of Mathematics [2017], 4.2) | |
A reaction: The second principle is a standard axiom of ZFC. The first principle causes the trouble. |
23447 | In classical semantics singular terms refer, and quantifiers range over domains [Linnebo] |
Full Idea: In classical semantics the function of singular terms is to refer, and that of quantifiers, to range over appropriate domains of entities. | |
From: Øystein Linnebo (Philosophy of Mathematics [2017], 7.1) |
23444 | To investigate axiomatic theories, mathematics needs its own foundational axioms [Linnebo] |
Full Idea: Mathematics investigates the deductive consequences of axiomatic theories, but it also needs its own foundational axioms in order to provide models for its various axiomatic theories. | |
From: Øystein Linnebo (Philosophy of Mathematics [2017], 4.1) | |
A reaction: This is a problem which faces the deductivist (if-then) approach. The deductive process needs its own grounds. |
23443 | The axioms of group theory are not assertions, but a definition of a structure [Linnebo] |
Full Idea: Considered in isolation, the axioms of group theory are not assertions but comprise an implicit definition of some abstract structure, | |
From: Øystein Linnebo (Philosophy of Mathematics [2017], 3.5) | |
A reaction: The traditional Euclidean approach is that axioms are plausible assertions with which to start. The present idea sums up the modern approach. In the modern version you can work backwards from a structure to a set of axioms. |
23446 | You can't prove consistency using a weaker theory, but you can use a consistent theory [Linnebo] |
Full Idea: If the 2nd Incompleteness Theorem undermines Hilbert's attempt to use a weak theory to prove the consistency of a strong one, it is still possible to prove the consistency of one theory, assuming the consistency of another theory. | |
From: Øystein Linnebo (Philosophy of Mathematics [2017], 4.6) | |
A reaction: Note that this concerns consistency, not completeness. |
23448 | Mathematics is the study of all possible patterns, and is thus bound to describe the world [Linnebo] |
Full Idea: Philosophical structuralism holds that mathematics is the study of abstract structures, or 'patterns'. If mathematics is the study of all possible patterns, then it is inevitable that the world is described by mathematics. | |
From: Øystein Linnebo (Philosophy of Mathematics [2017], 11.1) | |
A reaction: [He cites the physicist John Barrow (2010) for this] For me this is a major idea, because the concept of a pattern gives a link between the natural physical world and the abstract world of mathematics. No platonism is needed. |
23441 | Logical truth is true in all models, so mathematical objects can't be purely logical [Linnebo] |
Full Idea: Modern logic requires that logical truths be true in all models, including ones devoid of any mathematical objects. It follows immediately that the existence of mathematical objects can never be a matter of logic alone. | |
From: Øystein Linnebo (Philosophy of Mathematics [2017], 2) | |
A reaction: Hm. Could there not be a complete set of models for a theory which all included mathematical objects? (I can't answer that). |
23442 | Game Formalism has no semantics, and Term Formalism reduces the semantics [Linnebo] |
Full Idea: Game Formalism seeks to banish all semantics from mathematics, and Term Formalism seeks to reduce any such notions to purely syntactic ones. | |
From: Øystein Linnebo (Philosophy of Mathematics [2017], 3.3) | |
A reaction: This approach was stimulated by the need to justify the existence of the imaginary number i. Just say it is a letter! |
11993 | Jones may cease to exist without some simple property, but that doesn't make it essential [Kung] |
Full Idea: If Jones ceases to be a father, or ceases to be over eight years old, he will cease to exist, yet these properties surely do not belong essentially to him. | |
From: Joan Kung (Aristotle on Essence and Explanation [1977], II) | |
A reaction: This seems to correct, though I would doubt whether either of these count as true properties, in the causal sense I prefer. If being 'over 8' is a property, how many 'over n' or 'under m' properties does he have? One for each quantum moment? |
11997 | A property may belong essentially to one thing and contingently to another [Kung] |
Full Idea: It is possible that a property may belong essentially to one thing and contingently to another. | |
From: Joan Kung (Aristotle on Essence and Explanation [1977], III) | |
A reaction: Thus a love of blues music may be part of your essence, but only a minor part of me. Sounds right. Spin or charge are part of the essence of an electron, but only contingently part of a child's top. |
11992 | Aristotelian essences underlie a thing's existence, explain it, and must belong to it [Kung] |
Full Idea: Three essentialist claims are labelled 'Aristotelian': the thing would cease to exist without the property; an essential property is explanatory; and it is such that it must belong to everything to which it belongs. | |
From: Joan Kung (Aristotle on Essence and Explanation [1977], Intro) | |
A reaction: She says the second one is indispensable, and that it rules out the third one. My working assumption, like hers, is that the second one is the key part of the game, because Aristotle wanted to explain things. |
11996 | Some non-essential properties may explain more than essential-but-peripheral ones do [Kung] |
Full Idea: It seems highly likely that some non-essential properties may explain more about the individual or about things of his kind than the peripheral properties. | |
From: Joan Kung (Aristotle on Essence and Explanation [1977], II) | |
A reaction: Another important issue, if one is defending the explanatory role of essences. It is not only essences which explain. A key question is whether we endorse individual essences as well as generic ones. I think we should. They explain the details. |
11995 | Some peripheral properties are explained by essential ones, but don't themselves explain properties [Kung] |
Full Idea: There will be demonstrated properties at the edge of the system, so to speak. They will be explained in terms of the essential properties of the basic entities and principles of the science, but will themselves not be explanatory of further properties. | |
From: Joan Kung (Aristotle on Essence and Explanation [1977], II) | |
A reaction: This is an important line of thought which needs clarification. We can't glibly say that essences are what explain the other properties. Some properties do more than others to explain subsequent dependent properties. |
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'). |