Combining Texts
Ideas for
'Metaphysics', 'Substances without Substrata' and 'Thinking About Mathematics'
expand these ideas
|
start again
|
choose
another area for these texts
display all the ideas for this combination of texts
29 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
560
|
Mathematical precision is only possible in immaterial things [Aristotle]
|
9076
|
Mathematics studies the domain of perceptible entities, but its subject-matter is not perceptible [Aristotle]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
10958
|
Perhaps numbers are substances? [Aristotle]
|
13273
|
Pluralities divide into discontinous countables; magnitudes divide into continuous things [Aristotle]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
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 / 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 / m. One
12074
|
The one in number just is the particular [Aristotle]
|
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
17845
|
If only rectilinear figures existed, then unity would be the triangle [Aristotle]
|
17844
|
The unit is stipulated to be indivisible [Aristotle]
|
17859
|
Units came about when the unequals were equalised [Aristotle]
|
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
646
|
When we count, are we adding, or naming numbers? [Aristotle]
|
17861
|
Two men do not make one thing, as well as themselves [Aristotle]
|
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
8764
|
Categories are the best foundation for mathematics [Shapiro]
|
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
17851
|
Number is plurality measured by unity [Aristotle]
|
17843
|
The idea of 'one' is the foundation of number [Aristotle]
|
17850
|
Each many is just ones, and is measured by the one [Aristotle]
|
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 / 7. Mathematical Structuralism / a. Structuralism
9793
|
Mathematics studies abstracted relations, commensurability and proportion [Aristotle]
|
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 / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
13738
|
It is a simple truth that the objects of mathematics have being, of some sort [Aristotle]
|
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
12339
|
Aristotle removes ontology from mathematics, and replaces the true with the beautiful [Aristotle, by Badiou]
|
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
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 / 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
8730
|
'Impredicative' definitions refer to the thing being described [Shapiro]
|