18 ideas
| 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. |
| 6368 | If my ticket won't win the lottery (and it won't), no other tickets will either [Kyburg, by Pollock/Cruz] |
| Full Idea: The Lottery Paradox says you should rationally conclude that your ticket will not win the lottery, and then apply the same reasoning to all the other tickets, and conclude that no ticket will win the lottery. | |
| From: report of Henry E. Kyburg Jr (Probability and Logic of Rational Belief [1961]) by J Pollock / J Cruz - Contemporary theories of Knowledge (2nd) §7.2.8 | |
| A reaction: (Very compressed by me). I doubt whether this is a very deep paradox; the conclusion that I will not win is a rational assessment of likelihood, but it is not the result of strict logic. |
| 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. |
| 7647 | The imagination alone perceives all objects; it is the soul, playing all its roles [La Mettrie] |
| Full Idea: The imagination alone perceives; it forms an idea of all objects, with the words and figures that characterise them; thus the imagination is the soul, because it plays all its roles. | |
| From: Julien Offray de La Mettrie (Machine Man [1747], p.15) | |
| A reaction: This is not just a big claim for the importance of imagination, in strong opposition to Descartes's rather dismissive view (Idea 1399), but also appears to be the germ of an interesting theory about the nature of personal identity. |
| 7645 | When falling asleep, the soul becomes paralysed and weak, just like the body [La Mettrie] |
| Full Idea: The soul and body fall asleep together. The soul slowly becomes paralysed, together with all the body's muscles. They can no longer hold up the weight of the head, while the soul can no longer bear the burden of thought. | |
| From: Julien Offray de La Mettrie (Machine Man [1747], p.6) | |
| A reaction: A very nice observation, to place alongside other evidence such as drunkenness and blushing. Personally I find it hard to see why anyone ever believed dualism. You don't need modern brain scans and brain lesion research to see the problem. |
| 23225 | The soul's faculties depend on the brain, and are simply the brain's organisation [La Mettrie] |
| Full Idea: All the soul's faculties depend so much on the specific organisation of the brain and of the whole body that they are clearly nothing but that organisation. | |
| From: Julien Offray de La Mettrie (Machine Man [1747], p.26) | |
| A reaction: An interesting idea because it suggests that La Mettrie is a functionalist, rather than simply a reductive physicalist. |
| 7652 | Man is a machine, and there exists only one substance, diversely modified [La Mettrie] |
| Full Idea: Let us conclude boldly that man is a machine and that there is in the whole universe only one diversely modified substance. | |
| From: Julien Offray de La Mettrie (Machine Man [1747], p.39) | |
| A reaction: What courage it must have taken to write what now seems a perfectly acceptable and normal view. One day there should be a collective monument to Hobbes, Gassendi, Spinoza, La Mettrie and Hume, who thought so boldly. |
| 7650 | All thought is feeling, and rationality is the sensitive soul contemplating reasoning [La Mettrie] |
| Full Idea: Thought is only a capacity to feel, and the rational soul is only the sensitive soul applied to the contemplation of ideas and to reasoning. | |
| From: Julien Offray de La Mettrie (Machine Man [1747], p.33) | |
| A reaction: What a very nice idea. La Mettrie wants to bring us closer to animals. Because we can pursue a train of rational thought, it does not follow that we have a faculty called 'rationality'. A dog can follow a clever series of clues that lead to food. |
| 7651 | With wonderful new machines being made, a speaking machine no longer seems impossible [La Mettrie] |
| Full Idea: If wonderful machines like Huygens's planetary clock can be made, it would take even more cogs and springs to make a speaking machine, which can no longer be considered impossible, particularly at the hands of a new Prometheus. | |
| From: Julien Offray de La Mettrie (Machine Man [1747], p.34) | |
| A reaction: Compare Descartes in Idea 3614. The idea of artificial intelligence does not arise with the advent of computers; it follows naturally from the materialist view of the mind, along with a bit of ambition to build complex machines. |
| 7648 | The sun and rain weren't made for us; they sometimes burn us, or spoil our seeds [La Mettrie] |
| Full Idea: The sun was not made in order to heat the earth and all its inhabitants - whom it sometimes burns - any more than the rain was created in order to grow seeds - which it often spoils. | |
| From: Julien Offray de La Mettrie (Machine Man [1747]) | |
| A reaction: This denial of Aristotelian (and divine) teleology is as much part of the movement against religion, as are concerns about natural evil, and about the weakness of arguments for God's existence. These facts were obvious long before La Mettrie. |
| 7646 | There is no abrupt transition from man to animal; only language has opened a gap [La Mettrie] |
| Full Idea: From animals to man there is no abrupt transition. What was man before he invented words and learnt languages? An animal of a particular species, with much less natural instinct than the others. | |
| From: Julien Offray de La Mettrie (Machine Man [1747], p.13) | |
| A reaction: This shows how strongly the evolutionary idea was in the air, a century before Darwin proposed a mechanism for it. This thought is the beginning of a very new view of man, and also of a very new view of animals. |
| 7649 | There is no clear idea of the soul, which should only refer to our thinking part [La Mettrie] |
| Full Idea: The soul is merely a vain term of which we have no idea and which a good mind should use only to refer to that part of us which thinks. | |
| From: Julien Offray de La Mettrie (Machine Man [1747]) | |
| A reaction: I have always found the concept of the soul particularly baffling. It seems that it is only believed in to make immortality possible, with no other purpose to the belief, let alone evidence. I suspect that Descartes agreed with La Mettrie on this. |