Combining Philosophers
Ideas for Donald Davidson, John M. Cooper and Stewart Shapiro
expand these ideas
|
start again
|
choose
another area for these philosophers
display all the ideas for this combination of philosophers
54 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 / b. Types of number
13641
|
Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro]
|
8763
|
The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
13676
|
Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13677
|
Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [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 / h. Reals from Cauchy
18249
|
Cauchy gave a formal definition of a converging sequence. [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 / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
13652
|
The 'continuum' is the cardinality of the powerset of a denumerably infinite set [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]
|
8764
|
Categories are the best foundation for 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
13657
|
First-order arithmetic can't even represent basic number theory [Shapiro]
|
10202
|
Natural numbers just need an initial object, successors, and an induction principle [Shapiro]
|
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
10294
|
Second-order logic has the expressive power for mathematics, but an unworkable model theory [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 / 5. Definitions of Number / f. Zermelo numbers
8762
|
Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
|
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
13656
|
Some sets of natural numbers are definable in set-theory but not in 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]
|
8760
|
Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
|
8761
|
A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [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 / 6. Logicism / c. Neo-logicism
13664
|
Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro]
|
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
13625
|
Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro]
|
8744
|
Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
|
6. Mathematics / C. Sources of Mathematics / 7. Formalism
8749
|
Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
|
8750
|
Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
|
8752
|
Deductivism says mathematics is logical consequences of uninterpreted axioms [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]
|
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
8753
|
Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
|
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
8731
|
Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
|
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
13663
|
Some reject formal properties if they are not defined, or defined impredicatively [Shapiro]
|
8730
|
'Impredicative' definitions refer to the thing being described [Shapiro]
|