Combining Philosophers
Ideas for Archimedes, Hartry Field and Brian Clegg
expand these ideas
|
start again
|
choose
another area for these philosophers
display all the ideas for this combination of philosophers
23 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
9226
|
If mathematical theories conflict, it may just be that they have different subject matter [Field,H]
|
10880
|
Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
8958
|
In Field's version of science, space-time points replace real numbers [Field,H, by Szabó]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
10861
|
Beyond infinity cardinals and ordinals can come apart [Clegg]
|
10860
|
An ordinal number is defined by the set that comes before it [Clegg]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10854
|
Transcendental numbers can't be fitted to finite equations [Clegg]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / k. Imaginary numbers
10858
|
By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg]
|
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero
10853
|
Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg]
|
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
10866
|
Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg]
|
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
10869
|
The Continuum Hypothesis is independent of the axioms of set theory [Clegg]
|
10862
|
The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg]
|
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
13007
|
Archimedes defined a straight line as the shortest distance between two points [Archimedes, by Leibniz]
|
18221
|
'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space [Field,H]
|
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
8757
|
The Indispensability Argument is the only serious ground for the existence of mathematical entities [Field,H]
|
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
18212
|
Nominalists try to only refer to physical objects, or language, or mental constructions [Field,H]
|
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
10261
|
The application of mathematics only needs its possibility, not its truth [Field,H, by Shapiro]
|
18218
|
Hilbert explains geometry, by non-numerical facts about space [Field,H]
|
9623
|
Field needs a semantical notion of second-order consequence, and that needs sets [Brown,JR on Field,H]
|
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
18215
|
It seems impossible to explain the idea that the conclusion is contained in the premises [Field,H]
|
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
18216
|
Abstractions can form useful counterparts to concrete statements [Field,H]
|
18214
|
Mathematics is only empirical as regards which theory is useful [Field,H]
|
8714
|
Fictionalists say 2+2=4 is true in the way that 'Oliver Twist lived in London' is true [Field,H]
|
18210
|
Why regard standard mathematics as truths, rather than as interesting fictions? [Field,H]
|