14 ideas
| 13472 | Hilbert aimed to eliminate number from geometry [Hilbert, by Hart,WD] |
| Full Idea: One of Hilbert's aims in 'The Foundations of Geometry' was to eliminate number [as measure of lengths and angles] from geometry. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by William D. Hart - The Evolution of Logic 2 | |
| A reaction: Presumably this would particularly have to include the elimination of ratios (rather than actual specific lengths). |
| 14775 | Numbers are just names devised for counting [Peirce] |
| Full Idea: Numbers are merely a system of names devised by men for the purpose of counting. | |
| From: Charles Sanders Peirce (Scientific Attitude and Fallibilism [1899], II) | |
| A reaction: This seems a perfectly plausible view prior to the advent of Cantor, set theory and modern mathematical logic. I suppose the modern reply to this is that Peirce may be right about origin, but that men thereby stumbled on an Aladdin's Cave of riches. |
| 9546 | Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Hilbert, by Chihara] |
| Full Idea: Hilbert's geometrical axioms were existential in character, asserting the existence of certain geometrical objects (points and lines). Euclid's postulates do not assert the existence of anything; they assert the possibility of certain constructions. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by Charles Chihara - A Structural Account of Mathematics 01.1 | |
| A reaction: Chihara says geometry was originally understood modally, but came to be understood existentially. It seems extraordinary to me that philosophers of mathematics can have become more platonist over the centuries. |
| 18742 | Hilbert's formalisation revealed implicit congruence axioms in Euclid [Hilbert, by Horsten/Pettigrew] |
| Full Idea: In his formal investigation of Euclidean geometry, Hilbert uncovered congruence axioms that implicitly played a role in Euclid's proofs but were not explicitly recognised. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by Horsten,L/Pettigrew,R - Mathematical Methods in Philosophy 2 | |
| A reaction: The writers are offering this as a good example of the benefits of a precise and formal approach to foundational questions. It's hard to disagree, but dispiriting if you need a PhD in maths before you can start doing philosophy. |
| 18217 | Hilbert's geometry is interesting because it captures Euclid without using real numbers [Hilbert, by Field,H] |
| Full Idea: Hilbert's formulation of the Euclidean theory is of special interest because (besides being rigorously axiomatised) it does not employ the real numbers in the axioms. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by Hartry Field - Science without Numbers 3 | |
| A reaction: Notice that this job was done by Hilbert, and not by the fictionalist Hartry Field. |
| 14776 | That two two-eyed people must have four eyes is a statement about numbers, not a fact [Peirce] |
| Full Idea: To say that 'if' there are two persons and each person has two eyes there 'will be' four eyes is not a statement of fact, but a statement about the system of numbers which is our own creation. | |
| From: Charles Sanders Peirce (Scientific Attitude and Fallibilism [1899], II) | |
| A reaction: One eye for each arm of the people is certainly a fact. Frege uses this equivalence to build numbers. I think Peirce is wrong. If it is not a fact that these people have four eyes, I don't know what 'four' means. It's being two pairs is also a fact. |
| 14770 | Reasoning is based on statistical induction, so it can't achieve certainty or precision [Peirce] |
| Full Idea: All positive reasoning is judging the proportion of something in a whole collection by the proportion found in a sample. Hence we can never hope to attain absolute certainty, absolute exactitude, absolute universality. | |
| From: Charles Sanders Peirce (Scientific Attitude and Fallibilism [1899], II) | |
| A reaction: This is the basis of Peirce's fallibilism - that all 'positive' reasoning (whatever that it?) is based on statistical induction. I'm all in favour of fallibilism, but find Peirce's claim to be a bit too narrow. He was too mesmerised by physical science. |
| 14774 | Innate truths are very uncertain and full of error, so they certainly have exceptions [Peirce] |
| Full Idea: It seems to me there is the most historic proof that innate truths are particularly uncertain and mixed up with error, and therefore a fortiori not without exception. | |
| From: Charles Sanders Peirce (Scientific Attitude and Fallibilism [1899], II) |
| 14773 | A truth is hard for us to understand if it rests on nothing but inspiration [Peirce] |
| Full Idea: A truth which rests on the authority of inspiration only is of a somewhat incomprehensible nature; and we can never be sure that we rightly comprehend it. | |
| From: Charles Sanders Peirce (Scientific Attitude and Fallibilism [1899], II) |
| 14772 | If we decide an idea is inspired, we still can't be sure we have got the idea right [Peirce] |
| Full Idea: Even if we decide that an idea really is inspired, we cannot be sure, or nearly sure, that the statement is true. We know one of the commandments of the Bible was printed without a 'not' in it. | |
| From: Charles Sanders Peirce (Scientific Attitude and Fallibilism [1899], II) |
| 14771 | Only reason can establish whether some deliverance of revelation really is inspired [Peirce] |
| Full Idea: We never can be absolutely certain that any given deliverance [of revelation] really is inspired; for that can only be established by reasoning. | |
| From: Charles Sanders Peirce (Scientific Attitude and Fallibilism [1899], II) |
| 14769 | Only imagination can connect phenomena together in a rational way [Peirce] |
| Full Idea: We can stare stupidly at phenomena; but in the absence of imagination they will not connect themselves together in any rational way. | |
| From: Charles Sanders Peirce (Scientific Attitude and Fallibilism [1899], I) | |
| A reaction: The importance of this is its connection between imagination and 'rational' understanding. This is an important corrective to a crude traditional picture of the role of imagination. I would connect imagination with counterfactuals and best explanation. |
| 7861 | Libet says the processes initiated in the cortex can still be consciously changed [Libet, by Papineau] |
| Full Idea: Libet himself points out that the conscious decisions still have the power to 'endorse' or 'cancel', so to speak, the processes initiated by the earlier cortical activity: no action will result if the action's execution is consciously countermanded. | |
| From: report of Benjamin Libet (Unconscious Cerebral Initiative [1985]) by David Papineau - Thinking about Consciousness 1.4 | |
| A reaction: This is why Libet's findings do not imply 'epiphenomenalism'. It seems that part of a decisive action is non-conscious, undermining the all-or-nothing view of consciousness. Searle tries to smuggle in free will at this point (Idea 3817). |
| 6660 | Libet found conscious choice 0.2 secs before movement, well after unconscious 'readiness potential' [Libet, by Lowe] |
| Full Idea: Libet found that a subject's conscious choice to move was about a fifth of a second before movement, and thus later than the onset of the brain's so-called 'readiness potential', which seems to imply that unconscious processes initiates action. | |
| From: report of Benjamin Libet (Unconscious Cerebral Initiative [1985]) by E.J. Lowe - Introduction to the Philosophy of Mind Ch.9 | |
| A reaction: Of great interest to philosophers! It seems to make conscious choices epiphenomenal. The key move, I think, is to give up the idea of consciousness as being all-or-nothing. My actions are still initiated by 'me', but 'me' shades off into unconsciousness. |