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Single Idea 14420
[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
]
Full Idea
The nature of infinity and continuity belonged in former days to philosophy, but belongs now to mathematics.
Gist of Idea
Infinity and continuity used to be philosophy, but are now mathematics
Source
Bertrand Russell (Introduction to Mathematical Philosophy [1919], Pref)
Book Ref
Russell,Bertrand: 'Introduction to Mathematical Philosophy' [George Allen and Unwin 1975], p.-6
A Reaction
It is hard to disagree, since mathematicians since Cantor have revealed so much about infinite numbers (through set theory), but I think it remains an open question whether philosophers have anything distinctive to contribute.
The
26 ideas
with the same theme
[the status and nature of infinity as a number]:
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18090
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Without infinity time has limits, magnitudes are indivisible, and numbers come to an end
[Aristotle]
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7555
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Zeno achieved the statement of the problems of infinitesimals, infinity and continuity
[Russell on Zeno of Citium]
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8738
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Postulate 2 says a line can be extended continuously
[Euclid, by Shapiro]
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13151
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Not all infinites are equal
[Newton]
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10856
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A truly infinite quantity does not need to be a variable
[Bolzano]
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15902
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Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties
[Cantor, by Lavine]
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15908
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It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers
[Cantor, by Lavine]
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9633
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No one shall drive us out of the paradise the Cantor has created for us
[Hilbert]
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12460
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We extend finite statements with ideal ones, in order to preserve our logic
[Hilbert]
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12462
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Only the finite can bring certainty to the infinite
[Hilbert]
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14420
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Infinity and continuity used to be philosophy, but are now mathematics
[Russell]
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14119
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We do not currently know whether, of two infinite numbers, one must be greater than the other
[Russell]
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14133
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There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal)
[Russell]
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18708
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Infinity is not a number, so doesn't say how many; it is the property of a law
[Wittgenstein]
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17809
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Gödel showed that the syntactic approach to the infinite is of limited value
[Kreisel]
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10491
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Infinite natural numbers is as obvious as infinite sentences in English
[Boolos]
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9813
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Mathematics shows that thinking is not confined to the finite
[Badiou]
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13509
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We can establish truths about infinite numbers by means of induction
[Hart,WD]
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17799
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Cantor's infinite is an absolute, of all the sets or all the ordinal numbers
[Mayberry]
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17797
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Cantor extended the finite (rather than 'taming the infinite')
[Mayberry]
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18518
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Infinite numbers are qualitatively different - they are not just very large numbers
[Heil]
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10866
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Cantor's account of infinities has the shaky foundation of irrational numbers
[Clegg]
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15947
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The infinite is extrapolation from the experience of indefinitely large size
[Lavine]
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15949
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The theory of infinity must rest on our inability to distinguish between very large sizes
[Lavine]
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17730
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Combining the concepts of negation and finiteness gives the concept of infinity
[Jenkins]
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17923
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Intuitionists only accept a few safe infinities
[Colyvan]
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