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Single Idea 13413

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers ]

Full Idea

Both ordinalists and cardinalists, to account for our number words, have to account for the fact that we know so many of them, and that we can 'recognize' numbers which we've neither seen nor heard.

Gist of Idea

We must explain how we know so many numbers, and recognise ones we haven't met before

Source

Paul Benacerraf (Logicism, Some Considerations (PhD) [1960], p.166)

A Reaction

This seems an important contraint on any attempt to explain numbers. Benacerraf is an incipient structuralist, and here presses the importance of rules in our grasp of number. Faced with 42,578,645, we perform an act of deconstruction to grasp it.

The 26 ideas with the same theme [general ideas concerning numbers]:

1729 We perceive number by the denial of continuity [Aristotle]
13273 Pluralities divide into discontinous countables; magnitudes divide into continuous things [Aristotle]
10958 Perhaps numbers are substances? [Aristotle]
13152 We can talk of 'innumerable number', about the infinite points on a line [Newton]
16920 Numbers are formed by addition of units in time [Kant]
9823 Numbers are free creations of the human mind, to understand differences [Dedekind]
23186 Numbers enable us to manage the world - to the limits of counting [Nietzsche]
13412 Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order [Benacerraf]
13413 We must explain how we know so many numbers, and recognise ones we haven't met before [Benacerraf]
9901 Numbers can't be sets if there is no agreement on which sets they are [Benacerraf]
9912 There are no such things as numbers [Benacerraf]
12334 There is no single unified definition of number [Badiou]
12335 Numbers are for measuring and for calculating (and the two must be consistent) [Badiou]
10599 For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P]
13861 Number theory aims at the essence of natural numbers, giving their nature, and the epistemology [Wright,C]
10027 Mathematics is higher-order modal logic [Hodes]
8958 In Field's version of science, space-time points replace real numbers [Field,H, by Szabó]
8865 If 'the number of Democrats is on the rise', does that mean that 50 million is on the rise? [Yablo]
9923 We should talk about possible existence, rather than actual existence, of numbers [Burgess/Rosen]
9622 'There are two apples' can be expressed logically, with no mention of numbers [Brown,JR]
17453 The meaning of a number isn't just the numerals leading up to it [Heck]
12238 The Aristotelian view is that numbers depend on (and are abstracted from) other things [Oderberg]
9998 What is the relation of number words as singular-terms, adjectives/determiners, and symbols? [Hofweber]
10002 '2 + 2 = 4' can be read as either singular or plural [Hofweber]
21644 Numbers are used as singular terms, as adjectives, and as symbols [Hofweber]
21646 The Amazonian Piraha language is said to have no number words [Hofweber]