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Single Idea 18187

[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits ]

Full Idea

Even the fundamental theorems about limits could not [at first] be proved because the reals themselves were not well understood.

Gist of Idea

Theorems about limits could only be proved once the real numbers were understood

Source

Penelope Maddy (Naturalism in Mathematics [1997], I.2)

Book Ref

Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.27

A Reaction

This refers to the period of about 1850 (Weierstrass) to 1880 (Dedekind and Cantor).

The 57 ideas from Penelope Maddy

13011 New axioms are being sought, to determine the size of the continuum [Maddy]
13014 Extensional sets are clearer, simpler, unique and expressive [Maddy]
13013 The Axiom of Extensionality seems to be analytic [Maddy]
13019 The Iterative Conception says everything appears at a stage, derived from the preceding appearances [Maddy]
13018 Limitation of Size is a vague intuition that over-large sets may generate paradoxes [Maddy]
13021 The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics [Maddy]
13022 Infinite sets are essential for giving an account of the real numbers [Maddy]
13023 The Power Set Axiom is needed for, and supported by, accounts of the continuum [Maddy]
13024 Efforts to prove the Axiom of Choice have failed [Maddy]
13025 Modern views say the Choice set exists, even if it can't be constructed [Maddy]
13026 A large array of theorems depend on the Axiom of Choice [Maddy]
17610 The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy]
17605 Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy]
17614 The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy]
17615 Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy]
17620 Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy]
17618 Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy]
17625 If two mathematical themes coincide, that suggest a single deep truth [Maddy]
18182 The extension of concepts is not important to me [Maddy]
18163 Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy]
18167 We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy]
18164 Frege solves the Caesar problem by explicitly defining each number [Maddy]
18175 For any cardinal there is always a larger one (so there is no set of all sets) [Maddy]
18172 Infinity has degrees, and large cardinals are the heart of set theory [Maddy]
18168 'Propositional functions' are propositions with a variable as subject or predicate [Maddy]
18169 Axiom of Reducibility: propositional functions are extensionally predicative [Maddy]
18171 Cantor and Dedekind brought completed infinities into mathematics [Maddy]
18187 Theorems about limits could only be proved once the real numbers were understood [Maddy]
18184 Making set theory foundational to mathematics leads to very fruitful axioms [Maddy]
18188 The line of rationals has gaps, but set theory provided an ordered continuum [Maddy]
18177 In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy]
18185 Unified set theory gives a final court of appeal for mathematics [Maddy]
18183 Set theory brings mathematics into one arena, where interrelations become clearer [Maddy]
18186 Identifying geometric points with real numbers revealed the power of set theory [Maddy]
18190 Completed infinities resulted from giving foundations to calculus [Maddy]
18193 The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy]
18191 Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy]
18194 'Forcing' can produce new models of ZFC from old models [Maddy]
18195 A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy [Maddy]
18196 An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy]
18204 Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy]
18205 The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy]
18206 Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy]
18207 Maybe applications of continuum mathematics are all idealisations [Maddy]
8755 Maddy replaces pure sets with just objects and perceived sets of objects [Maddy, by Shapiro]
8756 Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro]
17733 We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins]
10718 A natural number is a property of sets [Maddy, by Oliver]
10594 Henkin semantics is more plausible for plural logic than for second-order logic [Maddy]
17823 If mathematical objects exist, how can we know them, and which objects are they? [Maddy]
17825 Set theory (unlike the Peano postulates) can explain why multiplication is commutative [Maddy]
17824 The master science is physical objects divided into sets [Maddy]
17826 Standardly, numbers are said to be sets, which is neat ontology and epistemology [Maddy]
17828 Numbers are properties of sets, just as lengths are properties of physical objects [Maddy]
17827 Sets exist where their elements are, but numbers are more like universals [Maddy]
17829 Number words are unusual as adjectives; we don't say 'is five', and numbers always come first [Maddy]
17830 Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy]