17 ideas
| 18274 | Analysis complicates a statement, but only as far as the complexity of its meaning [Wittgenstein] |
| Full Idea: Analysis makes the statement more complicated than it was; but it cannot and ought not to make it more complicated than its meaning (Bedeutung) was to begin with. When the statement is as complex as its meaning, then it is completely analysed. | |
| From: Ludwig Wittgenstein (Notebooks 1914-1916 [1915], 46e) | |
| A reaction: But how do you assess how complex the 'Bedeutung' was before you started? |
| 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. |
| 16908 | We can dispense with self-evidence, if language itself prevents logical mistakes [Jeshion on Wittgenstein] |
| Full Idea: The 'self-evidence' of which Russell talks so much can only be dispensed with in logic if language itself prevents any logical mistake. | |
| From: comment on Ludwig Wittgenstein (Notebooks 1914-1916 [1915], 4) by Robin Jeshion - Frege's Notion of Self-Evidence 4 | |
| A reaction: Jeshion presents this as a key idea, turning against Frege, and is the real source of the 'linguistic turn' in philosophy. If self-evidence is abandoned, then language itself is the guide to truth, so study language. I think I prefer Frege. See Quine? |
| 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. |
| 18276 | A statement's logical form derives entirely from its constituents [Wittgenstein] |
| Full Idea: The logical form of the statement must already be given in the forms of its constituents. | |
| From: Ludwig Wittgenstein (Notebooks 1914-1916 [1915], 23e) | |
| A reaction: This would evidently require each constituent to have a 'logical form'. It is hard to see what that could beyond its part of speech. Do two common nouns have the same logical form? |
| 6563 | 'And' and 'not' are non-referring terms, which do not represent anything [Wittgenstein, by Fogelin] |
| Full Idea: Wittgenstein's 'fundamental idea' is that the 'and' and 'not' which guarantee the truth of "not p and not-p" are meaningful, but do not get their meaning by representing or standing for or referring to some kind of entity; they are non-referring terms. | |
| From: report of Ludwig Wittgenstein (Notebooks 1914-1916 [1915], §37) by Robert Fogelin - Walking the Tightrope of Reason Ch.1 | |
| A reaction: Wittgenstein then defines the terms using truth tables, to show what they do, rather than what they stand for. This seems to me to be a candidate for the single most important idea in the history of the philosophy of logic. |
| 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. |
| 23472 | The sense of propositions relies on the world's basic logical structure [Wittgenstein] |
| Full Idea: In order for a proposition to be CAPABLE of making sense, the world must already have the logical structure it has. The logic of the world is prior to all truth and falsehood. | |
| From: Ludwig Wittgenstein (Notebooks 1914-1916 [1915], p.14c) | |
| A reaction: It seems that in Tractatus it is propositions about facts which are true or false, but prior to the facts are substance and the objects, and it is there that we find the logical structure of the world. I see this view as modern stoicism. |
| 14296 | Dispositions are physical states of mechanism; when known, these replace the old disposition term [Quine] |
| Full Idea: Each disposition, in my view, is a physical state or mechanism. ...In some cases nowadays we understand the physical details and set them forth explicitly in terms of the arrangement and interaction of small bodies. This replaces the old disposition. | |
| From: Willard Quine (The Roots of Reference [1990], p.11), quoted by Stephen Mumford - Dispositions 01.3 | |
| A reaction: A challenge to the dispositions and powers view of nature, one which rests on the 'categorical' structural properties, rather than the 'hypothetical' dispositions. But can we define a mechanism without mentioning its powers? |
| 23500 | My main problem is the order of the world, and whether it is knowable a priori [Wittgenstein] |
| Full Idea: The great problem around which everything turns that I write is: is there an order in the world a priori, and if so what does it consist in? | |
| From: Ludwig Wittgenstein (Notebooks 1914-1916 [1915], 15.06.01) | |
| A reaction: Morris identifies this as a 'Kantian question'. I trace it back to stoicism. This question has never bothered me. It just seems weird to think that you can infer reality from the examination of your own thinking. Perhaps I should take it more seriously? |
| 22323 | The philosophical I is the metaphysical subject, the limit - not a part of the world [Wittgenstein] |
| Full Idea: The philosophical I is not the man, not the human body, or the human soul of wh9ch psychology treats, but the metaphysical subject, the limit - not a part of the world. | |
| From: Ludwig Wittgenstein (Notebooks 1914-1916 [1915], 1916. 2 Sep), quoted by Michael Potter - The Rise of Analytic Philosophy 1879-1930 58 Intro | |
| A reaction: This is to treat the self as a phenomenon of thought, rather than of a human being. So if a machine could think, would it hence necessarily have a metaphysical self? |
| 23481 | Propositions assemble a world experimentally, like the model of a road accident [Wittgenstein] |
| Full Idea: In the proposition a world is as it were put together experimentally. (As when in the law court in Paris a motor-car accident is represented by means of dolls, etc). | |
| From: Ludwig Wittgenstein (Notebooks 1914-1916 [1915], 14.09.29) | |
| A reaction: [see Tractatus 4.031] This is the first appearance of LW's picture (or model) theory of meaning. It may well be the best theory of meaning anyone has come up with, since meaning being out in the world strikes me as absurd. |
| 4678 | Absolute prohibitions are the essence of ethics, and suicide is the most obvious example [Wittgenstein] |
| Full Idea: If suicide is allowed, then everything is allowed. If anything is not allowed, then suicide is not allowed. This throws a light on the nature of ethics, for suicide is, so to speak, the elementary sin. | |
| From: Ludwig Wittgenstein (Notebooks 1914-1916 [1915], end), quoted by Jonathan Glover - Causing Death and Saving Lives §13 | |
| A reaction: This reveals the religious streak in Wittgenstein. I am reluctant to judge suicide, but this seems wrong. Should a 'jumper' worry if they land on someone else and kill them? Of course they should. |