Combining Texts
Ideas for
'Structures and Structuralism in Phil of Maths', 'Psychosemantics' and 'Frege's Concept of Numbers as Objects'
expand these ideas
|
start again
|
choose
another area for these texts
display all the ideas for this combination of texts
35 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
|
13861
|
Number theory aims at the essence of natural numbers, giving their nature, and the epistemology [Wright,C]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
|
13892
|
One could grasp numbers, and name sizes with them, without grasping ordering [Wright,C]
|
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 / 4. Using Numbers / d. Counting via concepts
|
13867
|
Instances of a non-sortal concept can only be counted relative to a sortal concept [Wright,C]
|
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 / d. Peano arithmetic
|
17441
|
Wright thinks Hume's Principle is more fundamental to cardinals than the Peano Axioms are [Wright,C, by Heck]
|
|
13862
|
There are five Peano axioms, which can be expressed informally [Wright,C]
|
|
17853
|
Number truths are said to be the consequence of PA - but it needs semantic consequence [Wright,C]
|
|
17854
|
What facts underpin the truths of the Peano axioms? [Wright,C]
|
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
|
13894
|
Sameness of number is fundamental, not counting, despite children learning that first [Wright,C]
|
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
|
10140
|
We derive Hume's Law from Law V, then discard the latter in deriving arithmetic [Wright,C, by Fine,K]
|
|
8692
|
Frege has a good system if his 'number principle' replaces his basic law V [Wright,C, by Friend]
|
|
17440
|
Wright says Hume's Principle is analytic of cardinal numbers, like a definition [Wright,C, by Heck]
|
|
13893
|
It is 1-1 correlation of concepts, and not progression, which distinguishes natural number [Wright,C]
|
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
|
13888
|
If numbers are extensions, Frege must first solve the Caesar problem for extensions [Wright,C]
|
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
|
10172
|
Set-theory gives a unified and an explicit basis for mathematics [Reck/Price]
|
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 / a. For mathematical platonism
|
13869
|
Number platonism says that natural number is a sortal concept [Wright,C]
|
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
|
13870
|
We can't use empiricism to dismiss numbers, if numbers are our main evidence against empiricism [Wright,C]
|
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
|
13873
|
Treating numbers adjectivally is treating them as quantifiers [Wright,C]
|
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
|
13899
|
The Peano Axioms, and infinity of cardinal numbers, are logical consequences of how we explain cardinals [Wright,C]
|
|
13896
|
The aim is to follow Frege's strategy to derive the Peano Axioms, but without invoking classes [Wright,C]
|
|
7804
|
Wright has revived Frege's discredited logicism [Wright,C, by Benardete,JA]
|
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
|
13863
|
Logicism seemed to fail by Russell's paradox, Gödel's theorems, and non-logical axioms [Wright,C]
|
|
13895
|
The standard objections are Russell's Paradox, non-logical axioms, and Gödel's theorems [Wright,C]
|