13 ideas
23449 | Interpreting a text is representing it as making sense [Morris,M] |
Full Idea: Interpreting a text is a matter of making sense of it. And to make sense of a text is to represent it as making sense. | |
From: Michael Morris (Guidebook to Wittgenstein's Tractatus [2008], Intro.2) | |
A reaction: 'Making sense' is obviously not a very precise or determinate concept. It is probably better to say that the process is 'trying' to make sense of the text, because most texts don't totally make sense. |
23484 | Bipolarity adds to Bivalence the capacity for both truth values [Morris,M] |
Full Idea: According to the Principle of Bipolarity, every meaningful sentence must be capable both of being true and of being false. It is not enough merely that every sentence must be either true or false (which is Bivalence). | |
From: Michael Morris (Guidebook to Wittgenstein's Tractatus [2008], 3D) | |
A reaction: It is said that early Wittgenstein endorses this. That is, in addition to being true, the sentence must be capable of falsehood (and vice versa). This seems to be flirting with the verification principle. I presume it is 'affirmative' sentences. |
23494 | Conjunctive and disjunctive quantifiers are too specific, and are confined to the finite [Morris,M] |
Full Idea: There are two problems with defining the quantifiers in terms of conjunction and disjunction. The general statements are unspecific, and do not say which things have the properties, and also they can't range over infinite objects. | |
From: Michael Morris (Guidebook to Wittgenstein's Tractatus [2008], 5C) | |
A reaction: That is, the universal quantifier is lots of ands, and the existential is lots of ors. If there only existed finite objects, then naming them all would be universal, and the infinite wouldn't be needed. |
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). |
23460 | To count, we must distinguish things, and have a series with successors in it [Morris,M] |
Full Idea: Distinguishing between things is not enough for counting. …We need the crucial extra notion of a successor in a series of a certain kind. | |
From: Michael Morris (Guidebook to Wittgenstein's Tractatus [2008], Intro) | |
A reaction: This is the thinking that led to the Dedekind-Peano axioms for arithmetic. E.g. each series member can only have one successor. There is an unformalisable assumption that the series can then be applied to the things. |
23451 | Counting needs to distinguish things, and also needs the concept of a successor in a series [Morris,M] |
Full Idea: Just distinguishing things is not enough for counting (and hence arithmetic). We need the crucial extra notion of the successor in a series of some kind. | |
From: Michael Morris (Guidebook to Wittgenstein's Tractatus [2008], Intro.5) | |
A reaction: This is a step towards the Peano Axioms of arithmetic. The successors could be fingers and toes, taken in a conventional order, and matched one-to-one to the objects. 'My right big toe of cows' means 16 cows (but non-verbally). |
23452 | Discriminating things for counting implies concepts of identity and distinctness [Morris,M] |
Full Idea: The discrimination of things for counting needs to bring with it the notion of identity (and, correlatively, distinctness). | |
From: Michael Morris (Guidebook to Wittgenstein's Tractatus [2008], Intro.5) | |
A reaction: Morris is exploring how practices like counting might reveal necessary truths about the world. |
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. |
23491 | There must exist a general form of propositions, which are predictabe. It is: such and such is the case [Morris,M] |
Full Idea: The existence of a general propositional form is proved by the fact that there cannot be a proposition whose form could not have been foreseen (i.e. constructed). The general form of the proposition is: Such and such is the case. | |
From: Michael Morris (Guidebook to Wittgenstein's Tractatus [2008], 4.5) | |
A reaction: [last bit in Ogden translation] LW eventually expresses this symbolically. We could just say a proposition is an assertion. This strikes as either a rather empty claim, or an unfounded one. |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |