Combining Texts
Ideas for
'Through the Looking Glass', 'Philosophy of Mathematics' and 'Aristotle on Essence and Explanation'
expand these ideas
|
start again
|
choose
another area for these texts
display all the ideas for this combination of texts
31 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10201
|
Virtually all of mathematics can be modeled in set theory [Shapiro]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10213
|
Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro]
|
18243
|
Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
18245
|
Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro]
|
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
10236
|
There is no grounding for mathematics that is more secure than mathematics [Shapiro]
|
6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
10256
|
For intuitionists, proof is inherently informal [Shapiro]
|
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
10202
|
Natural numbers just need an initial object, successors, and an induction principle [Shapiro]
|
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
10205
|
Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro]
|
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
10222
|
Mathematical foundations may not be sets; categories are a popular rival [Shapiro]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
10218
|
Baseball positions and chess pieces depend entirely on context [Shapiro]
|
10224
|
The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro]
|
10228
|
Could infinite structures be apprehended by pattern recognition? [Shapiro]
|
10230
|
The 4-pattern is the structure common to all collections of four objects [Shapiro]
|
10249
|
The main mathematical structures are algebraic, ordered, and topological [Shapiro]
|
10273
|
Some structures are exemplified by both abstract and concrete [Shapiro]
|
10276
|
Mathematical structures are defined by axioms, or in set theory [Shapiro]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
10270
|
The main versions of structuralism are all definitionally equivalent [Shapiro]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
10221
|
Is there is no more to structures than the systems that exemplify them? [Shapiro]
|
10248
|
Number statements are generalizations about number sequences, and are bound variables [Shapiro]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
10220
|
Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro]
|
10223
|
There is no 'structure of all structures', just as there is no set of all sets [Shapiro]
|
8703
|
Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend]
|
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
10274
|
Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro]
|
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
10200
|
We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro]
|
10210
|
If mathematical objects are accepted, then a number of standard principles will follow [Shapiro]
|
10215
|
Platonists claim we can state the essence of a number without reference to the others [Shapiro]
|
10233
|
Platonism must accept that the Peano Axioms could all be false [Shapiro]
|
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
10244
|
Intuition is an outright hindrance to five-dimensional geometry [Shapiro]
|
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
10280
|
A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro]
|
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
10254
|
Can the ideal constructor also destroy objects? [Shapiro]
|
10255
|
Presumably nothing can block a possible dynamic operation? [Shapiro]
|