display all the ideas for this combination of texts
16 ideas
15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
Full Idea: Mathematics is today thought of as the study of abstract structure, not the study of quantity. That point of view arose directly out of the development of the set-theoretic notion of abstract structure. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.2) | |
A reaction: It sounds as if Structuralism, which is a controversial view in philosophy, is a fait accompli among mathematicians. |
15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
Full Idea: One reason to introduce the rational numbers is that it simplifes the theory of division, since every rational number is divisible by every nonzero rational number, while the analogous statement is false for the natural numbers. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VI.3) | |
A reaction: That is, with rations every division operation has an answer. |
15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
Full Idea: The chief importance of the Continuum Hypothesis for Cantor (I believe) was that it would show that the real numbers form a set, and hence that they were encompassed by his theory. | |
From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
Full Idea: The Cauchy convergence criterion for a sequence: the sequence S0,S1,... has a limit if |S(n+r) - S(n)| is less than any given quantity for every value of r and sufficiently large values of n. He proved this necessary, but not sufficient. | |
From: Shaughan Lavine (Understanding the Infinite [1994], 2.5) |
15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
Full Idea: Roughly speaking, the upper and lower parts of the Dedekind cut correspond to the commensurable ratios greater than and less than a given incommensurable ratio. | |
From: Shaughan Lavine (Understanding the Infinite [1994], II.6) | |
A reaction: Thus there is the problem of whether the contents of the gap are one unique thing, or many. |
15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
Full Idea: Counting a set produces a well-ordering of it. Conversely, if one has a well-ordering of a set, one can count it by following the well-ordering. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
A reaction: Cantor didn't mean that you could literally count the set, only in principle. |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
Full Idea: My proposal is that the concept of the infinite began with an extrapolation from the experience of indefinitely large size. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2) | |
A reaction: I think it might be better to talk of an 'abstraction' than an 'extrapolition', since the latter is just more of the same, which doesn't get you to concept. Lavine spends 100 pages working out his proposal. |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
Full Idea: The indiscernibility of indefinitely large sizes will be a critical part of the theory of indefinitely large sizes. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2) |
15940 | The intuitionist endorses only the potential infinite [Lavine] |
Full Idea: The intuitionist endorse the actual finite, but only the potential infinite. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VI.2) |
15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
Full Idea: The symbol 'aleph-nought' denotes the cardinal number of the set of natural numbers. The symbol 'aleph-one' denotes the next larger cardinal number. 'Aleph-omega' denotes the omega-th cardinal number. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.3) |
15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
Full Idea: The ordinals are basic because the transfinite sets are those that can be counted, or (equivalently for Cantor), those that can be numbered by an ordinal or are well-ordered. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
A reaction: Lavine observes (p.55) that for Cantor 'countable' meant 'countable by God'! |
15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
Full Idea: The paradox of the largest ordinal (the 'Burali-Forti') is that the class of all ordinal numbers is apparently well-ordered, and so it has an ordinal number as order type, which must be the largest ordinal - but all ordinals can be increased by one. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.5) |
15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
Full Idea: The paradox of the largest cardinal ('Cantor's Paradox') says the diagonal argument shows there is no largest cardinal, but the class of all individuals (including the classes) must be the largest cardinal number. | |
From: Shaughan Lavine (Understanding the Infinite [1994], III.5) |
15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
Full Idea: Every theorem of mathematics has a counterpart with set theory - ...but that theory cannot serve as a basis for the notion of proof. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.3) |
15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
Full Idea: In modern mathematics virtually all work is only up to isomorphism and no one cares what the numbers or points and lines 'really are'. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) | |
A reaction: At least that leaves the field open for philosophers, because we do care what things really are. So should everybody else, but there is no persuading some people. |
15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
Full Idea: Intuitionism in philosophy of mathematics rejects set-theoretic foundations. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.3 n33) |