17 ideas
13489 | Von Neumann treated cardinals as a special sort of ordinal [Neumann, by Hart,WD] |
Full Idea: Von Neumann's decision was to start with the ordinals and to treat cardinals as a special sort of ordinal. | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by William D. Hart - The Evolution of Logic 3 | |
A reaction: [see Hart 73-74 for an explication of this] |
12336 | A von Neumann ordinal is a transitive set with transitive elements [Neumann, by Badiou] |
Full Idea: In Von Neumann's definition an ordinal is a transitive set in which all of the elements are transitive. | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Alain Badiou - Briefings on Existence 11 |
18180 | Von Neumann numbers are preferred, because they continue into the transfinite [Maddy on Neumann] |
Full Idea: Von Neumann's version of the natural numbers is in fact preferred because it carries over directly to the transfinite ordinals. | |
From: comment on John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n9 |
18179 | For Von Neumann the successor of n is n U {n} (rather than {n}) [Neumann, by Maddy] |
Full Idea: For Von Neumann the successor of n is n U {n} (rather than Zermelo's successor, which is {n}). | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n8 |
15925 | Each Von Neumann ordinal number is the set of its predecessors [Neumann, by Lavine] |
Full Idea: Each Von Neumann ordinal number is the set of its predecessors. ...He had shown how to introduce ordinal numbers as sets, making it possible to use them without leaving the domain of sets. | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Shaughan Lavine - Understanding the Infinite V.3 |
22180 | Multiple realisability is said to make reduction impossible [Okasha] |
Full Idea: Philosophers have often invoked multiple realisability to explain why psychology cannot be reduced to physics or chemistry, but in principle the explanation works for any higher-level science. | |
From: Samir Okasha (Philosophy of Science: Very Short Intro (2nd ed) [2016], 3) | |
A reaction: He gives the example of a 'cell' in biology, which can be implemented in all sorts of ways. Presumably that can be reduced to many sorts of physics, but not just to one sort. The high level contains patterns that vanish at the low level. |
22172 | Not all sciences are experimental; astronomy relies on careful observation [Okasha] |
Full Idea: Not all sciences are experimental - astronomers obviously cannot do experiments on the heavens, but have to content themselves with careful observation instead. | |
From: Samir Okasha (Philosophy of Science: Very Short Intro (2nd ed) [2016], 1) | |
A reaction: Biology too. Psychology tries hard to be experimental, but I doubt whether the main theories emerge from experiments. |
22177 | Randomised Control Trials have a treatment and a control group, chosen at random [Okasha] |
Full Idea: In the Randomised Controlled Trial for a new drug, patients are divided at random into a treatment group who receive the drug, and a control group who do not. Randomisation is important to eliminate confounding factors. | |
From: Samir Okasha (Philosophy of Science: Very Short Intro (2nd ed) [2016], 2) | |
A reaction: [compressed] Devised in the 1930s, and a major breakthrough in methodology for that kind of trial. Psychologists use the method all the time. Some theorists say it is the only reliable method. |
22174 | The discoverers of Neptune didn't change their theory because of an anomaly [Okasha] |
Full Idea: Adams and Leverrier began with Newton's theory of gravity, which made an incorrect prediction about the orbit of Uranus. They explained away the conflicting observations by postulating a new planet, Neptune. | |
From: Samir Okasha (Philosophy of Science: Very Short Intro (2nd ed) [2016], 1) | |
A reaction: The falsificationists can say that the anomalous observation did not falsify the theory, because they didn't know quite what they were observing. It was not in fact an anomaly for Newtonian theory at all. |
22175 | Science mostly aims at confirming theories, rather than falsifying them [Okasha] |
Full Idea: The goal of science is not solely to refute theories, but also to determine which theories are true (or probably true). When a scientist collects data …they are trying to show that their own theory is true. | |
From: Samir Okasha (Philosophy of Science: Very Short Intro (2nd ed) [2016], 2) | |
A reaction: This is the aim of 'accommodation' to a wide set of data, rather than prediction or refutation. |
22182 | Theories with unobservables are underdetermined by the evidence [Okasha] |
Full Idea: According to anti-realists, scientific theories which posit unobservable entities are underdetermined by the empirical data - there will always be a number of competing theories which can account for the data equally well. | |
From: Samir Okasha (Philosophy of Science: Very Short Intro (2nd ed) [2016], 4) | |
A reaction: The fancy version is Putnam's model theoretic argument, explored by Tim Button. The reply, apparently, is that there are other criteria for theory choice, apart from the data. And we don't have to actually observe everything in a theory. |
22185 | Two things can't be incompatible if they are incommensurable [Okasha] |
Full Idea: If two things are incommensurable they cannot be incompatible. | |
From: Samir Okasha (Philosophy of Science: Very Short Intro (2nd ed) [2016], 5) | |
A reaction: Kuhn had claimed that two rival theories are incompatible, which forces the paradigm shift. He can't stop the slide off into total relativism. The point is there cannot be a conflict if there cannot even be a comparison. |
22176 | Induction is inferences from examined to unexamined instances of a given kind [Okasha] |
Full Idea: Some philosophers use 'inductive' to just mean not deductive, …but we reserve it for inferences from examined to unexamined instances of a given kind. | |
From: Samir Okasha (Philosophy of Science: Very Short Intro (2nd ed) [2016], 2) | |
A reaction: The instances must at least be comparable. Must you know the kind before you start? Surely you can examine a sequence of things, trying to decide whether or not they are of one kind? Is checking the uniformity of a kind induction? |
22178 | If the rules only concern changes of belief, and not the starting point, absurd views can look ratiional [Okasha] |
Full Idea: If the only objective constraints concern how we should change our credences, but what our initial credences should be is entirely subjective, then individuals with very bizarre opinions about the world will count as perfectly rational. | |
From: Samir Okasha (Philosophy of Science: Very Short Intro (2nd ed) [2016], 2) | |
A reaction: The important rationality has to be the assessement of a diverse batch of evidence, for which there can never be any rules or mathematics. |
1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
Full Idea: Archelaus was the first person to say that the universe is boundless. | |
From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
22173 | Galileo refuted the Aristotelian theory that heavier objects fall faster [Okasha] |
Full Idea: Galileo's most enduring contribution lay in mechanics, where he refuted the Aristotelian theory that heavier bodies fall faster than lighter. | |
From: Samir Okasha (Philosophy of Science: Very Short Intro (2nd ed) [2016], 2) | |
A reaction: This must the first idea in the theory of mechanics, allowing mathematical treatment and accurate comparisons. |
5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |