10 ideas
10882 | Predicative definitions only refer to entities outside the defined collection [Horsten] |
Full Idea: Definitions are called 'predicative', and are considered sound, if they only refer to entities which exist independently from the defined collection. | |
From: Leon Horsten (Philosophy of Mathematics [2007], §2.4) |
10884 | A theory is 'categorical' if it has just one model up to isomorphism [Horsten] |
Full Idea: If a theory has, up to isomorphism, exactly one model, then it is said to be 'categorical'. | |
From: Leon Horsten (Philosophy of Mathematics [2007], §5.2) |
10885 | Computer proofs don't provide explanations [Horsten] |
Full Idea: Mathematicians are uncomfortable with computerised proofs because a 'good' proof should do more than convince us that a certain statement is true. It should also explain why the statement in question holds. | |
From: Leon Horsten (Philosophy of Mathematics [2007], §5.3) |
10881 | The concept of 'ordinal number' is set-theoretic, not arithmetical [Horsten] |
Full Idea: The notion of an ordinal number is a set-theoretic, and hence non-arithmetical, concept. | |
From: Leon Horsten (Philosophy of Mathematics [2007], §2.3) |
5062 | First: there must be reasons; Second: why anything at all?; Third: why this? [Leibniz] |
Full Idea: We rise to metaphysics by saying 'nothing takes place without a reason', then asking 'why is there something rather than nothing?, and then 'why do things exist as they do?' | |
From: Gottfried Leibniz (Principles of Nature and Grace based on Reason [1714], §7) | |
A reaction: Wonderful. This is what we pay philosophers for - to attempt to go to the heart of the mystery, and then start formulating the appropriate questions. The question of 'why this?' is the sweetest question. The first one seems a little intractable. |
19377 | A monad and its body are living, so life is everywhere, and comes in infinite degrees [Leibniz] |
Full Idea: Each monad, together with a particular body, makes up a living substance. Thus, there is not only life everywhere, joined to limbs or organs, but there are also infinite degrees of life in the monads, some dominating more or less over others. | |
From: Gottfried Leibniz (Principles of Nature and Grace based on Reason [1714], 4) | |
A reaction: Two key ideas: that each monad is linked to a body (which is presumably passive), and the infinite degrees of life in monads. Thus rocks consist of monads, but at an exceedingly low degree of life. They are stubborn and responsive. |
19353 | 'Perception' is basic internal representation, and 'apperception' is reflective knowledge of perception [Leibniz] |
Full Idea: We distinguish between 'perception', the internal state of the monad representing external things, and 'apperception', which is consciousness, or the reflective knowledge of this internal state, not given to all souls, nor at all times to a given soul. | |
From: Gottfried Leibniz (Principles of Nature and Grace based on Reason [1714], §4) | |
A reaction: The word 'apperception' is standard in Kant. I find it surprising that modern analytic philosophers don't seem to use it when they write about perception. It strikes me as useful, but maybe specialists have a reason for avoiding it. |
5061 | Animals are semi-rational because they connect facts, but they don't see causes [Leibniz] |
Full Idea: There is a connexion between the perceptions of animals, which bears some resemblance to reason: but it is based only on the memory of facts or effects, and not at all on the knowledge of causes. | |
From: Gottfried Leibniz (Principles of Nature and Grace based on Reason [1714], §5) | |
A reaction: This amounts to the view that animals can do Humean induction (where you see regularities), but not Leibnizian induction (where you see necessities). I say all minds perceive patterns, but only humans can think about the patterns they have perceived. |
5063 | Music charms, although its beauty is the harmony of numbers [Leibniz] |
Full Idea: Music charms us although its beauty only consists in the harmony of numbers. | |
From: Gottfried Leibniz (Principles of Nature and Grace based on Reason [1714], §17) | |
A reaction: 'Only'! This is a super-pythagorean view of music, as you might expect from a great mathematician. Did he understand the horrible compromises that had just been made to achieve even-tempered tuning? Patterns are the key, as always. |
2594 | A true cause must involve a necessary connection between cause and effect [Malebranche] |
Full Idea: A true cause as I understand it is one such that the mind perceives a necessary connection between it and its effects. | |
From: Nicolas Malebranche (The Union of Body and Soul [1675], p.116) |