18 ideas
23295 | Truth cannot be reduced to anything simpler [Davidson] |
Full Idea: We cannot hope to underpin the concept of truth with something more transparent or easier to grasp. | |
From: Donald Davidson (The Folly of Trying to Define Truth [1999], p.21) | |
A reaction: I suppose precise accounts of correspondence or coherence are offered as replacements for truth, but neither of those ever seem to be possible. I agree with accepting truth as a primitive. |
23298 | Neither Aristotle nor Tarski introduce the facts needed for a correspondence theory [Davidson] |
Full Idea: Neither Aristotle's formula nor Tarski's truth definitions are sympathetic to the correspondence theory, because they don't introduce entities like facts or states of affairs for sentences to correspond. | |
From: Donald Davidson (The Folly of Trying to Define Truth [1999], p.25) | |
A reaction: This seems convincing, although it is often claimed that both theories offer a sort of correspondence. |
23297 | The language to define truth needs a finite vocabulary, to make the definition finite [Davidson] |
Full Idea: If the definition of the truth predicate is to be finite (Tarski insisted on this), the definition must take advantage of the fact that sentences, though potentially infinite in number, are constructed from a finite vocabulary. | |
From: Donald Davidson (The Folly of Trying to Define Truth [1999], p.23) | |
A reaction: Not sure whether this is in the object language or the meta-language, though I guess the former. |
23296 | We can elucidate indefinable truth, but showing its relation to other concepts [Davidson] |
Full Idea: We can still say revealing things about truth, by relating it to other concepts like belief, desire, cause and action. | |
From: Donald Davidson (The Folly of Trying to Define Truth [1999], p.21) | |
A reaction: The trickiest concept to link it to is meaning. I think Davidson's view points to the Axiomatic account of truth, which flourished soon after Davidson wrote this. We can give rules for the correct use of 'true'. |
17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
Full Idea: One feature of the Axiom of Choice that troubled many mathematicians was the so-called Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres the same size as the original. | |
From: Penelope Maddy (Defending the Axioms [2011], 1.3) | |
A reaction: (The key is that the parts are non-measurable). To an outsider it is puzzling that the Axiom has been universally accepted, even though it produces such a result. Someone can explain that, I'm sure. |
17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
Full Idea: If-thenism denies that mathematics is in the business of discovering truths about abstracta. ...[their opponents] obviously don't regard any starting point, even a consistent one, as equally worthy of investigation. | |
From: Penelope Maddy (Defending the Axioms [2011], 3.3) | |
A reaction: I have some sympathy with if-thenism, in that you can obviously study the implications of any 'if' you like, but deep down I agree with the critics. |
17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
Full Idea: At the end of the nineteenth century there was a renewed emphasis on rigor, the central tool of which was axiomatization, along the lines of Hilbert's axioms for geometry and Dedekind's axioms for real numbers. | |
From: Penelope Maddy (Defending the Axioms [2011], 1.3) |
17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
Full Idea: The fact that two apparently fruitful mathematical themes turn out to coincide makes it all the more likely that they're tracking a genuine strain of mathematical depth. | |
From: Penelope Maddy (Defending the Axioms [2011], 5.3ii) |
17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
Full Idea: One form of the Continuum Hypothesis is the claim that every infinite set of reals is either countable or of the same size as the full set of reals. | |
From: Penelope Maddy (Defending the Axioms [2011], 2.4 n40) |
17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
Full Idea: Our set-theoretic methods track the underlying contours of mathematical depth. ...What sets are, most fundamentally, is markers for these contours ...they are maximally effective trackers of certain trains of mathematical fruitfulness. | |
From: Penelope Maddy (Defending the Axioms [2011], 3.4) | |
A reaction: This seems to make it more like a map of mathematics than the actual essence of mathematics. |
17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
Full Idea: Ordinary perceptual cognition is most likely involved in our grasp of elementary arithmetic, but ...this connection to the physical world has long since been idealized away in the infinitary structures of contemporary pure mathematics. | |
From: Penelope Maddy (Defending the Axioms [2011], 2.3) | |
A reaction: Despite this, Maddy's quest is for a 'naturalistic' account of mathematics. She ends up defending 'objectivity' (and invoking Tyler Burge), rather than even modest realism. You can't 'idealise away' the counting of objects. I blame Cantor. |
10502 | We can rise by degrees through abstraction, with higher levels representing more things [Arnauld,A/Nicole,P] |
Full Idea: I can start with a triangle, and rise by degrees to all straight-lined figures and to extension itself. The lower degree will include the higher degree. Since the higher degree is less determinate, it can represent more things. | |
From: Arnauld / Nicole (Logic (Port-Royal Art of Thinking) [1662], I.5) | |
A reaction: [compressed] This attempts to explain the generalising ability of abstraction cited in Idea 10501. If you take a complex object and eliminate features one by one, it can only 'represent' more particulars; it could hardly represent fewer. |
18258 | We can only know the exterior world via our ideas [Arnauld,A/Nicole,P] |
Full Idea: We can have knowledge of what is outside us only through the mediation of ideas in us. | |
From: Arnauld / Nicole (Logic (Port-Royal Art of Thinking) [1662], p.63), quoted by J. Alberto Coffa - The Semantic Tradition from Kant to Carnap 1 'Conc' |
23294 | It is common to doubt truth when discussing it, but totally accept it when discussing knowledge [Davidson] |
Full Idea: You are following Plato's lead if you worry about the concept of truth when it is the focus of your attention, but you pretend you understand it when trying to cope with knowledge (or belief, memory, perception etc.). | |
From: Donald Davidson (The Folly of Trying to Define Truth [1999], p.20) | |
A reaction: Nice to find someone pointing out this absurdity. He says Hume does the same with doubts about the external world, which he ignores when discussing other minds. Belief is holding true; only truths are actually remembered…. |
16784 | Forms make things distinct and explain the properties, by pure form, or arrangement of parts [Arnauld,A/Nicole,P] |
Full Idea: The form is what renders a thing such and distinguishes it from others, whether it is a being really distinct from the matter, according to the Schools, or whether it is only the arrangement of the parts. By this form one must explain its properties. | |
From: Arnauld / Nicole (Logic (Port-Royal Art of Thinking) [1662], III.18 p240), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 27.6 | |
A reaction: If we ask 'what explains the properties of this thing' it is hard to avoid coming up with something that might be called the 'form'. Note that they allow either substantial or corpuscularian forms. It is hard to disagree with the idea. |
10499 | We know by abstraction because we only understand composite things a part at a time [Arnauld,A/Nicole,P] |
Full Idea: The mind cannot perfectly understand things that are even slightly composite unless it considers them a part at a time. ...This is generally called knowing by abstraction. (..the human body, for example). | |
From: Arnauld / Nicole (Logic (Port-Royal Art of Thinking) [1662], I.5) | |
A reaction: This adds the interesting thought that the mind is forced to abstract, rather than abstraction being a luxury extra feature. Knowledge through analysis is knowledge by abstraction. Also a nice linking of abstraction to epistemology. |
10501 | A triangle diagram is about all triangles, if some features are ignored [Arnauld,A/Nicole,P] |
Full Idea: If I draw an equilateral triangle on a piece of paper, ..I shall have an idea of only a single triangle. But if I ignore all the particular circumstances and focus on the three equal lines, I will be able to represent all equilateral triangles. | |
From: Arnauld / Nicole (Logic (Port-Royal Art of Thinking) [1662], I.5) | |
A reaction: [compressed] They observed that we grasp composites through their parts, and now that we can grasp generalisations through particulars, both achieved by the psychological act of abstraction, thus showing its epistemological power. |
10500 | No one denies that a line has width, but we can just attend to its length [Arnauld,A/Nicole,P] |
Full Idea: Geometers by no means assume that there are lines without width or surfaces without depth. They only think it is possible to consider the length without paying attention to the width. We can measure the length of a path without its width. | |
From: Arnauld / Nicole (Logic (Port-Royal Art of Thinking) [1662], I.5) | |
A reaction: A nice example which makes the point indubitable. The modern 'rigorous' account of abstraction that starts with Frege seems to require more than one object, in order to derive abstractions like direction or number. Path widths are not comparatives. |