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Single Idea 9889
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
]
Full Idea
Frege fixed on construing real numbers as ratios of quantities (in agreement with Newton).
Gist of Idea
Real numbers are ratios of quantities
Source
report of Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Michael Dummett - Frege philosophy of mathematics Ch.20
Book Ref
Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.261
A Reaction
If 3/4 is the same real number as 6/8, which is the correct ratio? Why doesn't the square root of 9/16 also express it? Why should irrationals be so utterly different from rationals? In what sense are they both 'numbers'?
The
29 ideas
with the same theme
[all numbers, including those inexpressible as fractions]:
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11042
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Parts of a line join at a point, so it is continuous
[Aristotle]
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22962
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Two is the least number, but there is no least magnitude, because it is always divisible
[Aristotle]
|
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13445
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Descartes showed a one-one order-preserving match between points on a line and the real numbers
[Descartes, by Hart,WD]
|
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17611
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We want the essence of continuity, by showing its origin in arithmetic
[Dedekind]
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15906
|
Cantor tried to prove points on a line matched naturals or reals - but nothing in between
[Cantor, by Lavine]
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11015
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Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1
[Cantor, by Read]
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18252
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Real numbers are ratios of quantities, such as lengths or masses
[Frege]
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9889
|
Real numbers are ratios of quantities
[Frege, by Dummett]
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18253
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I wish to go straight from cardinals to reals (as ratios), leaving out the rationals
[Frege]
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14135
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Real numbers are a class of rational numbers (and so not really numbers at all)
[Russell]
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18738
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We don't get 'nearer' to something by adding decimals to 1.1412... (root-2)
[Wittgenstein]
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14648
|
Could I name all of the real numbers in one fell swoop? Call them all 'Charley'?
[Plantinga]
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18095
|
Instead of by cuts or series convergence, real numbers could be defined by axioms
[Bostock]
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18099
|
The number of reals is the number of subsets of the natural numbers
[Bostock]
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10610
|
The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals
[Smith,P]
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12395
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Real numbers stand to measurement as natural numbers stand to counting
[Kitcher]
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13446
|
19th century arithmetization of analysis isolated the real numbers from geometry
[Hart,WD]
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17784
|
Real numbers can be eliminated, by axiom systems for complete ordered fields
[Mayberry]
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10213
|
Real numbers are thought of as either Cauchy sequences or Dedekind cuts
[Shapiro]
|
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18243
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Understanding the real-number structure is knowing usage of the axiomatic language of analysis
[Shapiro]
|
|
10165
|
'Analysis' is the theory of the real numbers
[Reck/Price]
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10632
|
The real numbers may be introduced by abstraction as ratios of quantities
[Hale, by Hale/Wright]
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10107
|
Real numbers provide answers to square root problems
[George/Velleman]
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15711
|
The rationals are everywhere - the irrationals are everywhere else
[Kaplan/Kaplan]
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10854
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Transcendental numbers can't be fitted to finite equations
[Clegg]
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15922
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For the real numbers to form a set, we need the Continuum Hypothesis to be true
[Lavine]
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8671
|
The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps
[Friend]
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10684
|
I take the real numbers to be just lengths
[Hossack]
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15364
|
English expressions are denumerably infinite, but reals are nondenumerable, so many are unnameable
[Horsten]
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