19 ideas
| 1597 | Thales was the first western thinker to believe the arché was intelligible [Roochnik on Thales] |
| Full Idea: Thales was the first thinker in the west to believe that the arché (the basis of things) was intelligible. | |
| From: comment on Thales (fragments/reports [c.585 BCE]) by David Roochnik - The Tragedy of Reason p.138 |
| 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) |
| 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. |
| 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. |
| 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! |
| 10938 | The extremes of essentialism are that all properties are essential, or only very trivial ones [Rami] |
| Full Idea: It would be natural to label one extreme view 'maximal essentialism' - that all of an object's properties are essential - and the other extreme 'minimal' - that only trivial properties such as self-identity of being either F or not-F are essential. | |
| From: Adolph Rami (Essential vs Accidental Properties [2008]) | |
| A reaction: Personally I don't accept the trivial ones as being in any way describable as 'properties'. The maximal view destroys any useful notion of essence. Leibniz is a minority holder of the maximal view. I would defend a middle way. |
| 10940 | An 'individual essence' is possessed uniquely by a particular object [Rami] |
| Full Idea: An 'individual essence' is a property that in addition to being essential is also unique to the object, in the sense that it is not possible that something distinct from that object possesses that property. | |
| From: Adolph Rami (Essential vs Accidental Properties [2008], §5) | |
| A reaction: She cites a 'haecceity' (or mere bare identity) as a trivial example of an individual essence. |
| 10939 | 'Sortal essentialism' says being a particular kind is what is essential [Rami] |
| Full Idea: According to 'sortal essentialism', an object could not have been of a radically different kind than it in fact is. | |
| From: Adolph Rami (Essential vs Accidental Properties [2008], §4) | |
| A reaction: This strikes me as thoroughly wrong. Things belong in kinds because of their properties. Could you remove all the contingent features of a tiger, leaving it as merely 'a tiger', despite being totally unrecognisable? |
| 10934 | Unlosable properties are not the same as essential properties [Rami] |
| Full Idea: It is easy to confuse the notion of an essential property that a thing could not lack, with a property it could not lose. My having spent Christmas 2007 in Tennessee is a non-essential property I could not lose. | |
| From: Adolph Rami (Essential vs Accidental Properties [2008], §1) | |
| A reaction: The idea that having spent Christmas in Tennessee is a property I find quite bewildering. Is my not having spent my Christmas in Tennessee one of my properties? I suspect that real unlosable properties are essential ones. |
| 10933 | Physical possibility is part of metaphysical possibility which is part of logical possibility [Rami] |
| Full Idea: The usual view is that 'physical possibilities' are a natural subset of the 'metaphysical possibilities', which in turn are a subset of the 'logical possibilities'. | |
| From: Adolph Rami (Essential vs Accidental Properties [2008], §1) | |
| A reaction: [She cites Fine 2002 for an opposing view] I prefer 'natural' to 'physical', leaving it open where the borders of the natural lie. I take 'metaphysical' possibility to be 'in all naturally possible worlds'. So is a round square a logical possibility? |
| 10932 | If it is possible 'for all I know' then it is 'epistemically possible' [Rami] |
| Full Idea: There is 'epistemic possibility' when it is 'for all I know'. That is, P is epistemically possible for agent A just in case P is consistent with what A knows. | |
| From: Adolph Rami (Essential vs Accidental Properties [2008], §1) | |
| A reaction: Two problems: maybe 'we' know, and A knows we know, but A doesn't know. And maybe someone knows, but we are not sure about that, which seems to introduce a modal element into the knowing. If someone knows it's impossible, it's impossible. |
| 3013 | Nothing is stronger than necessity, which rules everything [Thales, by Diog. Laertius] |
| Full Idea: Necessity is the strongest of things, for it rules everything. | |
| From: report of Thales (fragments/reports [c.585 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 01.2.9 |
| 1494 | Thales said water is the first principle, perhaps from observing that food is moist [Thales, by Aristotle] |
| Full Idea: Thales says water is the first principle (which is why he declared the earth is on water); perhaps he concluded this from seeing that all food is moist. | |
| From: report of Thales (fragments/reports [c.585 BCE], A12) by Aristotle - Metaphysics 983b12 |
| 1713 | Thales must have thought soul causes movement, since he thought magnets have soul [Thales, by Aristotle] |
| Full Idea: Thales seems, from what is recorded of him, to have supposed that the soul is something productive of movement, if he really said that the magnet has soul because it produces movement in iron. | |
| From: report of Thales (fragments/reports [c.585 BCE]) by Aristotle - De Anima 405a20 |
| 1742 | Thales said the gods know our wrong thoughts as well as our evil actions [Thales, by Diog. Laertius] |
| Full Idea: When asked whether a man who did wrong could escape the notice of the gods, Thales is said to have replied: 'No, not even if he thinks wrong.' | |
| From: report of Thales (fragments/reports [c.585 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 01.Th.9 |