more from this thinker
|
more from this text
Single Idea 9886
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
]
Full Idea
The cardinals and the reals are completely disjoint domains. The cardinal numbers answer the question 'How many objects of a given kind are there?', but the real numbers are for measurement, saying how large a quantity is compared to a unit quantity.
Gist of Idea
Cardinals say how many, and reals give measurements compared to a unit quantity
Source
Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903], §157), quoted by Michael Dummett - Frege philosophy of mathematics Ch.19
Book Ref
Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.246
A Reaction
We might say that cardinals are digital and reals are analogue. Frege is unusual in totally separating them. They map onto one another, after all. Cardinals look like special cases of reals. Reals are dreams about the gaps between cardinals.
The
25 ideas
with the same theme
[the various families of numbers]:
|
9603
|
An assumption that there is a largest prime leads to a contradiction
[Euclid, by Brown,JR]
|
|
15905
|
Cantor proved the points on a plane are in one-to-one correspondence to the points on a line
[Cantor, by Lavine]
|
|
9886
|
Cardinals say how many, and reals give measurements compared to a unit quantity
[Frege]
|
|
14438
|
New numbers solve problems: negatives for subtraction, fractions for division, complex for equations
[Russell]
|
|
18254
|
Russell's approach had to treat real 5/8 as different from rational 5/8
[Russell, by Dummett]
|
|
14144
|
Ordinals result from likeness among relations, as cardinals from similarity among classes
[Russell]
|
|
9896
|
A prime number is one which is measured by a unit alone
[Dummett]
|
|
15273
|
Points can be 'dense' by unending division, but must meet a tougher criterion to be 'continuous'
[Harré/Madden]
|
|
12333
|
Each type of number has its own characteristic procedure of introduction
[Badiou]
|
|
12322
|
Must we accept numbers as existing when they no longer consist of units?
[Badiou]
|
|
18100
|
ω + 1 is a new ordinal, but its cardinality is unchanged
[Bostock]
|
|
18101
|
Each addition changes the ordinality but not the cardinality, prior to aleph-1
[Bostock]
|
|
3330
|
Negatives, rationals, irrationals and imaginaries are all postulated to solve baffling equations
[Benardete,JA]
|
|
3337
|
Natural numbers are seen in terms of either their ordinality (Peano), or cardinality (set theory)
[Benardete,JA]
|
|
10573
|
Dedekind cuts lead to the bizarre idea that there are many different number 1's
[Fine,K]
|
|
13641
|
Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals
[Shapiro]
|
|
8763
|
The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex
[Shapiro]
|
|
10106
|
Rational numbers give answers to division problems with integers
[George/Velleman]
|
|
10102
|
The integers are answers to subtraction problems involving natural numbers
[George/Velleman]
|
|
8454
|
The whole numbers are 'natural'; 'rational' numbers include fractions; the 'reals' include root-2 etc.
[Orenstein]
|
|
15712
|
1 and 0, then add for naturals, subtract for negatives, divide for rationals, take roots for irrationals
[Kaplan/Kaplan]
|
|
15942
|
Every rational number, unlike every natural number, is divisible by some other number
[Lavine]
|
|
8667
|
The 'integers' are the positive and negative natural numbers, plus zero
[Friend]
|
|
8668
|
The 'rational' numbers are those representable as fractions
[Friend]
|
|
8670
|
A number is 'irrational' if it cannot be represented as a fraction
[Friend]
|