16 ideas
| 7454 | Gassendi is the first great empiricist philosopher [Hacking] |
| Full Idea: Gassendi is the first in the great line of empiricist philosophers that gradually came to dominate European thought. | |
| From: Ian Hacking (The Emergence of Probability [1975], Ch.5) | |
| A reaction: Epicurus, of course, was clearly an empiricist. British readers should note that Gassendi was not British. |
| 9193 | ZF set theory has variables which range over sets, 'equals' and 'member', and extensionality [Dummett] |
| Full Idea: ZF set theory is a first-order axiomatization. Variables range over sets, there are no second-order variables, and primitive predicates are just 'equals' and 'member of'. The axiom of extensionality says sets with the same members are identical. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 7) | |
| A reaction: If the eleven members of the cricket team are the same as the eleven members of the hockey team, is the cricket team the same as the hockey team? Our cricket team is better than our hockey team, so different predicates apply to them. |
| 9194 | The main alternative to ZF is one which includes looser classes as well as sets [Dummett] |
| Full Idea: The main alternative to ZF is two-sorted theories, with some variables ranging over classes. Classes have more generous existence assumptions: there is a universal class, containing all sets, and a class containing all ordinals. Classes are not members. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 7.1.1) | |
| A reaction: My intuition is to prefer strict systems when it comes to logical theories. The whole point is precision. Otherwise we could just think about things, and skip all this difficult symbolic stuff. |
| 9195 | Intuitionists reject excluded middle, not for a third value, but for possibility of proof [Dummett] |
| Full Idea: It must not be concluded from the rejection of excluded middle that intuitionistic logic operates with three values: true, false, and neither true nor false. It does not make use of true and false, but only with a construction being a proof. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 8.1) | |
| A reaction: This just sounds like verificationism to me, with all its problems. It seems to make speculative statements meaningless, which can't be right. Realism has lots of propositions which are assumed to be true or false, but also unknowable. |
| 10121 | Contradiction is not a sign of falsity, nor lack of contradiction a sign of truth [Pascal] |
| Full Idea: Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth. | |
| From: Blaise Pascal (works [1660]), quoted by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
| A reaction: [Quoted in Auden and Kronenberger's Book of Aphorisms] Presumably we would now say that contradiction is a purely formal, syntactic notion, and not a semantic one. If you hit a contradiction, something has certainly gone wrong. |
| 9186 | First-order logic concerns objects; second-order adds properties, kinds, relations and functions [Dummett] |
| Full Idea: First-order logic is distinguished by generalizations (quantification) only over objects: second-order logic admits generalizations or quantification over properties or kinds of objects, and over relations between them, and functions defined over them. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 3.1) | |
| A reaction: Second-order logic was introduced by Frege, but is (interestingly) rejected by Quine, because of the ontological commitments involved. I remain unconvinced that quantification entails ontological commitment, so I'm happy. |
| 9187 | Logical truths and inference are characterized either syntactically or semantically [Dummett] |
| Full Idea: There are two ways of characterizing logical truths and correct inference. Proof-theoretic or syntactic characterizations, if the formalization admits of proof or derivation; and model-theoretic or semantic versions, being true in all interpretations. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 3.1) | |
| A reaction: Dummett calls this distinction 'fundamental'. The second one involves truth, and hence meaning, where the first one just responds to rules. ..But how can you have a notion of correctly following a rule, without a notion of truth? |
| 9191 | Ordinals seem more basic than cardinals, since we count objects in sequence [Dummett] |
| Full Idea: It can be argued that the notion of ordinal numbers is more fundamental than that of cardinals. To count objects, we must count them in sequence. ..The theory of ordinals forms the substratum of Cantor's theory of cardinals. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 5) | |
| A reaction: Depends what you mean by 'fundamental'. I would take cardinality to be psychologically prior ('that is a lot of sheep'). You can't order people by height without first acquiring some people with differing heights. I vote for cardinals. |
| 9192 | The number 4 has different positions in the naturals and the wholes, with the same structure [Dummett] |
| Full Idea: The number 4 cannot be characterized solely by its position in a system, because it has different positions in the system of natural numbers and that of the positive whole numbers, whereas these systems have the very same structure. | |
| From: Michael Dummett (The Philosophy of Mathematics [1998], 6.1) | |
| A reaction: Dummett seems to think this is fairly decisive against structuralism. There is also the structure of the real numbers. We will solve this by saying that the wholes are abstracted from the naturals, which are abstracted from the reals. Job done. |
| 7447 | Probability was fully explained between 1654 and 1812 [Hacking] |
| Full Idea: There is hardly any history of probability to record before Pascal (1654), and the whole subject is very well understood after Laplace (1812). | |
| From: Ian Hacking (The Emergence of Probability [1975], Ch.1) | |
| A reaction: An interesting little pointer on the question of whether the human race is close to exhausting all the available intellectual problems. What then? |
| 7448 | Probability is statistical (behaviour of chance devices) or epistemological (belief based on evidence) [Hacking] |
| Full Idea: Probability has two aspects: the degree of belief warranted by evidence, and the tendency displayed by some chance device to produce stable relative frequencies. These are the epistemological and statistical aspects of the subject. | |
| From: Ian Hacking (The Emergence of Probability [1975], Ch.1) | |
| A reaction: The most basic distinction in the subject. Later (p.124) he suggests that the statistical form (known as 'aleatory' probability) is de re, and the other is de dicto. |
| 7449 | Epistemological probability based either on logical implications or coherent judgments [Hacking] |
| Full Idea: Epistemological probability is torn between Keynes etc saying it depends on the strength of logical implication, and Ramsey etc saying it is personal judgement which is subject to strong rules of internal coherence. | |
| From: Ian Hacking (The Emergence of Probability [1975], Ch.2) | |
| A reaction: See Idea 7449 for epistemological probability. My immediate intuition is that the Ramsey approach sounds much more plausible. In real life there are too many fine-grained particulars involved for straight implication to settle a probability. |
| 7450 | In the medieval view, only deduction counted as true evidence [Hacking] |
| Full Idea: In the medieval view, evidence short of deduction was not really evidence at all. | |
| From: Ian Hacking (The Emergence of Probability [1975], Ch.3) | |
| A reaction: Hacking says the modern concept of evidence comes with probability in the 17th century. That might make it one of the most important ideas ever thought of, allowing us to abandon certainties and live our lives in a more questioning way. |
| 7451 | Formerly evidence came from people; the new idea was that things provided evidence [Hacking] |
| Full Idea: In the medieval view, people provided the evidence of testimony and of authority. What was lacking was the seventeenth century idea of the evidence provided by things. | |
| From: Ian Hacking (The Emergence of Probability [1975], Ch.4) | |
| A reaction: A most intriguing distinction, which seems to imply a huge shift in world-view. The culmination of this is Peirce's pragmatism, in Idea 6948, of which I strongly approve. |
| 7452 | An experiment is a test, or an adventure, or a diagnosis, or a dissection [Hacking, by PG] |
| Full Idea: An experiment is a test (if T, then E implies R, so try E, and if R follows, T seems right), an adventure (no theory, but try things), a diagnosis (reading the signs), or a dissection (taking apart). | |
| From: report of Ian Hacking (The Emergence of Probability [1975], Ch.4) by PG - Db (ideas) | |
| A reaction: A nice analysis. The Greeks did diagnosis, then the alchemists tried adventures, then Vesalius began dissections, then the followers of Bacon concentrated on the test, setting up controlled conditions. 'If you don't believe it, try it yourself'. |
| 7459 | Follow maths for necessary truths, and jurisprudence for contingent truths [Hacking] |
| Full Idea: Mathematics is the model for reasoning about necessary truths, but jurisprudence must be our model when we deliberate about contingencies. | |
| From: Ian Hacking (The Emergence of Probability [1975], Ch.10) | |
| A reaction: Interesting. Certainly huge thinking, especially since the Romans, has gone into the law, and creating rules of evidence. Maybe all philosophers should study law and mathematics? |