Combining Philosophers
Ideas for Roderick Firth, E Reck / M Price and Penelope Maddy
expand these ideas
|
start again
|
choose
another area for these philosophers
display all the ideas for this combination of philosophers
45 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10165
|
'Analysis' is the theory of the real numbers [Reck/Price]
|
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
18171
|
Cantor and Dedekind brought completed infinities into mathematics [Maddy]
|
18190
|
Completed infinities resulted from giving foundations to calculus [Maddy]
|
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
17615
|
Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy]
|
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
18172
|
Infinity has degrees, and large cardinals are the heart of set theory [Maddy]
|
18175
|
For any cardinal there is always a larger one (so there is no set of all sets) [Maddy]
|
18196
|
An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy]
|
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
18187
|
Theorems about limits could only be proved once the real numbers were understood [Maddy]
|
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
10174
|
Mereological arithmetic needs infinite objects, and function definitions [Reck/Price]
|
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
10164
|
Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price]
|
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
18182
|
The extension of concepts is not important to me [Maddy]
|
18177
|
In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy]
|
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
18164
|
Frege solves the Caesar problem by explicitly defining each number [Maddy]
|
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
17825
|
Set theory (unlike the Peano postulates) can explain why multiplication is commutative [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]
|
10718
|
A natural number is a property of sets [Maddy, by Oliver]
|
18184
|
Making set theory foundational to mathematics leads to very fruitful axioms [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]
|
18188
|
The line of rationals has gaps, but set theory provided an ordered continuum [Maddy]
|
17618
|
Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy]
|
10172
|
Set-theory gives a unified and an explicit basis for mathematics [Reck/Price]
|
18163
|
Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy]
|
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
17830
|
Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy]
|
17827
|
Sets exist where their elements are, but numbers are more like universals [Maddy]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
10167
|
Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
10169
|
Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price]
|
10179
|
There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price]
|
10181
|
Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price]
|
10182
|
There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
10168
|
Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price]
|
10178
|
Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
10176
|
Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price]
|
10177
|
Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
10171
|
The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price]
|
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
17823
|
If mathematical objects exist, how can we know them, and which objects are they? [Maddy]
|
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
8756
|
Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro]
|
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
17733
|
We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins]
|
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
18204
|
Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy]
|
18207
|
Maybe applications of continuum mathematics are all idealisations [Maddy]
|
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
17614
|
The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy]
|
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
17829
|
Number words are unusual as adjectives; we don't say 'is five', and numbers always come first [Maddy]
|
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
18167
|
We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy]
|