16 ideas
10807 | Mathematics reduces to set theory, which reduces, with some mereology, to the singleton function [Lewis] |
Full Idea: It is generally accepted that mathematics reduces to set theory, and I argue that set theory in turn reduces, with some aid of mereology, to the theory of the singleton function. | |
From: David Lewis (Mathematics is Megethology [1993], p.03) |
10809 | We can accept the null set, but not a null class, a class lacking members [Lewis] |
Full Idea: In my usage of 'class', there is no such things as the null class. I don't mind calling some memberless thing - some individual - the null set. But that doesn't make it a memberless class. Rather, that makes it a 'set' that is not a class. | |
From: David Lewis (Mathematics is Megethology [1993], p.05) | |
A reaction: Lewis calls this usage 'idiosyncratic', but it strikes me as excellent. Set theorists can have their vital null class, and sensible people can be left to say, with Lewis, that classes of things must have members. |
10811 | The null set plays the role of last resort, for class abstracts and for existence [Lewis] |
Full Idea: The null set serves two useful purposes. It is a denotation of last resort for class abstracts that denote no nonempty class. And it is an individual of last resort: we can count on its existence, and fearlessly build the hierarchy of sets from it. | |
From: David Lewis (Mathematics is Megethology [1993], p.09) | |
A reaction: This passage assuages my major reservation about the existence of the null set, but at the expense of confirming that it must be taken as an entirely fictional entity. |
10812 | The null set is not a little speck of sheer nothingness, a black hole in Reality [Lewis] |
Full Idea: Should we accept the null set as a most extraordinary individual, a little speck of sheer nothingness, a sort of black hole in the fabric of Reality itself? Not that either, I think. | |
From: David Lewis (Mathematics is Megethology [1993], p.09) | |
A reaction: Correct! |
10813 | What on earth is the relationship between a singleton and an element? [Lewis] |
Full Idea: A new student of set theory has just one thing, the element, and he has another single thing, the singleton, and not the slightest guidance about what one thing has to do with the other. | |
From: David Lewis (Mathematics is Megethology [1993], p.12) |
10814 | Are all singletons exact intrinsic duplicates? [Lewis] |
Full Idea: Are all singletons exact intrinsic duplicates? | |
From: David Lewis (Mathematics is Megethology [1993], p.13) |
10806 | Megethology is the result of adding plural quantification to mereology [Lewis] |
Full Idea: Megethology is the result of adding plural quantification, as advocated by George Boolos, to the language of mereology. | |
From: David Lewis (Mathematics is Megethology [1993], p.03) |
10816 | We can use mereology to simulate quantification over relations [Lewis] |
Full Idea: We can simulate quantification over relations using megethology. Roughly, a quantifier over relations is a plural quantifier over things that encode ordered pairs by mereological means. | |
From: David Lewis (Mathematics is Megethology [1993], p.18) | |
A reaction: [He credits this idea to Burgess and Haven] The point is to avoid second-order logic, which quantifies over relations as ordered n-tuple sets. |
10808 | Mathematics is generalisations about singleton functions [Lewis] |
Full Idea: We can take the theory of singleton functions, and hence set theory, and hence mathematics, to consist of generalisations about all singleton functions. | |
From: David Lewis (Mathematics is Megethology [1993], p.03) | |
A reaction: At first glance this sounds like a fancy version of the somewhat discredited Greek idea that mathematics is built on the concept of a 'unit'. |
10815 | We don't need 'abstract structures' to have structural truths about successor functions [Lewis] |
Full Idea: We needn't believe in 'abstract structures' to have general structural truths about all successor functions. | |
From: David Lewis (Mathematics is Megethology [1993], p.16) |
16639 | Only individual bodies exist [Bacon] |
Full Idea: Nothing truly exists in nature beyond individual bodies. | |
From: Francis Bacon (The New Organon [1620]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 182 | |
A reaction: [Unusually, Pasnau gives no reference in the text; possibly II:1-2] What this leaves out, from even an auster nominalist ontology, is undifferentiated stuff like water. Even electrons don't seem quite distinct from one another. |
16033 | There are only individual bodies containing law-based powers, and the Forms are these laws [Bacon] |
Full Idea: Though nothing exists in nature except individual bodies which exhibit pure individual acts [powers] in accordance with law…It is this law and its clauses which we understand by the term Forms. | |
From: Francis Bacon (The New Organon [1620], p.103), quoted by Jan-Erik Jones - Real Essence §3 | |
A reaction: This isn't far off what Aristotle had in mind, when he talks of forms as being 'principles', though there is more emphasis on mechanisms in the original idea. Note that Bacon takes laws so literally that he refers to their 'clauses'. |
10810 | I say that absolutely any things can have a mereological fusion [Lewis] |
Full Idea: I accept the principle of Unrestricted Composition: whenever there are some things, no matter how many or how unrelated or how disparate in character they may be, they have a mereological fusion. ...The trout-turkey is part fish and part fowl. | |
From: David Lewis (Mathematics is Megethology [1993], p.07) | |
A reaction: This nicely ducks the question of when things form natural wholes and when they don't, but I would have thought that that might be one of the central issues of metaphysicals, so I think I'll give Lewis's principle a miss. |
21950 | Science must clear away the idols of the mind if they are ever going to find the truth [Bacon] |
Full Idea: We must clear away the idols and false notions which are now in possession of the human understanding, and have taken deep root therein, and so beset men's minds that truth can hardly find an entrance. | |
From: Francis Bacon (The New Organon [1620], 38), quoted by Mark Wrathall - Heidegger: how to read 2 | |
A reaction: [He goes on to list the types of idol] |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |