Topic: number representation

topics > philosophy > Group: philosophy of mathematics

data value

kinds of numbers
mathematics as a formal system
names independent of objects
number as a named set of numbers
referential transparency
symbolic representation
what is a number
words in natural languages


Numerals represent numbers. They are arbitrary and yet decimal notation is universal. The notation selected clearly matters, otherwise we'd still be using Roman numerals.

Numbers of course are important to computers. Or more accurately to the users and builders of computers. A computer doesn't understand the number; it is only people who do that. And they tend to be restrictive about what means what. Fixed length bit strings are usually used.

Kripke argues that decimal notation is intrinsic because of the need for a vivid representation. And somehow the notation is the number. Consider if it wasn't. Then any notation could be used and all multiplication becomes trivial. (cbb 4/92)

Subtopic: numerals vs. numbers up

Quote: a numeral, an R-value expression, is the written representation of a number, an abstract object; only the single digits are also names [»straC8_1967]
Quote: a value, e.g., 'six feet', is an entity with many representations [»kentW_1978]
Quote: numerals do not stand for number entities; look at the conditions and utility for using numerals [»kripSA_1982]
Quote: Roman numerals demonstrate that notation matters; they were an important advance over the Greek system of one number per letter [»greeTR_1980]
Quote: a denotation is a self-defining name for a value; e.g., '2' [»maclBJ12_1983]

Subtopic: tally up

Quote: use tally sticks or parts of your body to keep track of numbers; no need to count as we do [»ifraG_2000]

Subtopic: numbers and computers up

Quote: computers only operate on the marks of numbers; humans set up a correspondence between any set of things and numbers [»jakiSL_1969]
Quote: while mathematical values are independent of an encoding, values in an occurrence system or a programming language have constrained codings [»holtAW9_1968]
Note: a number is a bit string while memory is at a location and has a size [»cbb_1990, OK]
Quote: C's INT_MAX limit effects the design of a program but rarely leads to errors [»ritcDM_1996]

Subtopic: positional notation up

Quote: positional notation independently discovered by Babylonians, Chinese, Mayan, and Indians [»ifraG_2000]

Subtopic: decimal notation up

Quote: numerals 1..9 are the first letters for number words in Sanskrit; universally used [»blisCK_1965]
Quote: decimal notation is vivid because it mirrors the structure it is meant to denote [»cbb_1990, OK]
Quote: a number is an ordered sequence of objects selected from 10 set-theoretic ones [»cbb_1990, OK]

Subtopic: number base up

Quote: if we know a number in any base, multiplication and exponentiation is trivial [»cbb_1990, OK]

Subtopic: word numbers up

Quote: India has a place-value system of word numerals; 1 is anything markedly unique, e.g., the moon, the earth; 0 is denoted by words meaning void, sky, complete; still used today

Related Topics up

Group: naming   (32 topics, 789 quotes)
Group: data value   (19 topics, 433 quotes)

Topic: kinds of numbers (24 items)
Topic: mathematics as a formal system (30 items)
Topic: names independent of objects (34 items)
Topic: number as a named set of numbers (15 items)
Topic: referential transparency (26 items)
Topic: symbolic representation (26 items)
Topic: what is a number (55 items)
Topic: words in natural languages
(40 items)

Updated barberCB 9/04
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