17 ideas
| 13750 | Analysis aims at the structure of facts, which are needed to give a rationale to analysis [Urmson, by Schaffer,J] |
| Full Idea: Urmson explains the direction of analysis as 'towards a structure...more nearly similar to the structure of the fact', adding that this metaphysical picture is needed as a 'rationale of the practice of analysis'. | |
| From: report of J.O. Urmson (Philosophical Analysis [1956], p.24-5) by Jonathan Schaffer - On What Grounds What n30 | |
| A reaction: In other words, only realists can be truly motivated to keep going with analysis. Merely analysing language-games is doable, but hardly exciting. |
| 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. |
| 4020 | The modern self has disengaged reason, self-exploration, and personal commitment [Taylor,C] |
| Full Idea: The modern notion of the self is defined by disengaged reason (with its associated freedom and dignity), by self-exploration, and by personal commitment. | |
| From: Charles Taylor (Sources of the Self [1989], §13.1) | |
| A reaction: Taylor makes a good case that this broader view of how the self is seen is as important as narrow debates about personal identity. |
| 4002 | My aim is to map the connections between our sense of self and our moral understanding [Taylor,C] |
| Full Idea: My entire way of proceeding involves mapping connections between the sense of the self and moral visions, between identity and the good. | |
| From: Charles Taylor (Sources of the Self [1989], Pref) | |
| A reaction: An interesting project. Modern brain research supports the idea that emotions and values are tightly integrated into al thought. |
| 4006 | I can only be aware of myself as a person who changes by means of my personal history [Taylor,C] |
| Full Idea: As a being who grows and becomes I can only know myself through the history of my maturations and regressions, overcomings and defeats. | |
| From: Charles Taylor (Sources of the Self [1989], §2.3) | |
| A reaction: An important insight. My immediate sense of self makes my personal history central, not an extra. But a history must be a history OF something. |
| 4003 | Selfhood and moral values are inextricably intertwined [Taylor,C] |
| Full Idea: Selfhood and the good, or in another way selfhood and morality, turn out to be inextricably intertwined. | |
| From: Charles Taylor (Sources of the Self [1989], §1.1) | |
| A reaction: This seems an inevitable convergence of three centuries of thought about personal identity and morality. |
| 4021 | Willingness to risk life was the constitutive quality of the man of honour [Taylor,C] |
| Full Idea: Willingness to risk life was the constitutive quality of the man of honour. | |
| From: Charles Taylor (Sources of the Self [1989], §13.1) | |
| A reaction: Which is why war is required. The growth of civil society meant the inevitable rise of other virtues. |
| 4005 | To have respect for people, you must feel their claims, or their injustices, or hold them in awe [Taylor,C] |
| Full Idea: If you want to discriminate more finely what makes humans worthy of respect, you must call to mind the claim of human suffering, or what is repugnant about justice, or the awe you feel about human life. | |
| From: Charles Taylor (Sources of the Self [1989], §1.1) | |
| A reaction: A persuasive part of the claim that such feelings are inseparable from thinking about people in any way at all. |
| 4004 | Consistency presupposes intrinsic description [Taylor,C] |
| Full Idea: The issue of consistency presupposes intrinsic description. | |
| From: Charles Taylor (Sources of the Self [1989], §1.1) | |
| A reaction: This may be the key criticism of Kant. The so-called 'maxim' of an action can be almost infinitely re-expressed to suit the agent. |
| 4010 | In later utilitarianism the modern stress on freedom leads to the rejection of paternalism [Taylor,C] |
| Full Idea: In mature utilitarianism , the stress on modern freedom emerges in the rejection of paternalism. | |
| From: Charles Taylor (Sources of the Self [1989], §3.3) | |
| A reaction: This seems good; it is the beginnings of a rejection of paternalism. What is better, happiness or freedom? What is the value of freedom? |
| 4009 | Nominalists defended the sovereignty of God against the idea of natural existing good and evil [Taylor,C] |
| Full Idea: Late medieval nominalism defended the sovereignty of God as incompatible with there being an order in nature which by itself defined good and bad. | |
| From: Charles Taylor (Sources of the Self [1989], §3.3) | |
| A reaction: Part of their attack on Platonism. But what made them place such a high value on the sovereignty of God? |