Combining Texts
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'Structures and Structuralism in Phil of Maths', 'Letter to Mersenne' and 'Elements of Geometry'
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19 ideas
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
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10302
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Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Euclid, by Bernays]
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22278
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Euclid relied on obvious properties in diagrams, as well as on his axioms [Potter on Euclid]
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8673
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Euclid's parallel postulate defines unique non-intersecting parallel lines [Euclid, by Friend]
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10250
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Euclid needs a principle of continuity, saying some lines must intersect [Shapiro on Euclid]
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14157
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Modern geometries only accept various parts of the Euclid propositions [Russell on Euclid]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
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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 / e. Peano arithmetic 2nd-order
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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 / 5. Definitions of Number / b. Greek arithmetic
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1600
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Euclid's common notions or axioms are what we must have if we are to learn anything at all [Euclid, by Roochnik]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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10172
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Set-theory gives a unified and an explicit basis for mathematics [Reck/Price]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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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
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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
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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
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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
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10171
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The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price]
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