Combining Texts
Ideas for
'Monadology', 'Structures and Structuralism in Phil of Maths' and 'Foundations without Foundationalism'
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23 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
13641
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Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
13676
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Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13677
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Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10165
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'Analysis' is the theory of the real numbers [Reck/Price]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
13652
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The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
10174
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Mereological arithmetic needs infinite objects, and function definitions [Reck/Price]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
13657
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First-order arithmetic can't even represent basic number theory [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
10164
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Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
10172
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Set-theory gives a unified and an explicit basis for mathematics [Reck/Price]
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13656
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Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
10167
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Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
10169
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Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price]
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10179
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There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price]
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10181
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Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price]
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10182
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There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
10168
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Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price]
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10178
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Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
10176
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Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price]
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10177
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Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
10171
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The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
13664
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Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
13625
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Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
13663
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Some reject formal properties if they are not defined, or defined impredicatively [Shapiro]
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