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Single Idea 9553
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
]
Full Idea
With the invention of analytic geometry (by Fermat and then Descartes) physical space could be represented as having a mathematical structure, which could eventually lead to its axiomatization (by Hilbert).
Gist of Idea
Analytic geometry gave space a mathematical structure, which could then have axioms
Source
Charles Chihara (A Structural Account of Mathematics [2004], 02.3)
Book Ref
Chihara,Charles: 'A Structural Account of Mathematics' [OUP 2004], p.43
A Reaction
The idea that space might have axioms seems to be pythagoreanism run riot. I wonder if there is some flaw at the heart of Einstein's General Theory because of this?
The
17 ideas
from 'A Structural Account of Mathematics'
10192
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We can replace existence of sets with possibility of constructing token sentences
[Chihara, by MacBride]
|
9547
|
Mathematical entities are causally inert, so the causal theory of reference won't work for them
[Chihara]
|
9550
|
We only know relational facts about the empty set, but nothing intrinsic
[Chihara]
|
9551
|
What is special about Bill Clinton's unit set, in comparison with all the others?
[Chihara]
|
9549
|
The set theorist cannot tell us what 'membership' is
[Chihara]
|
9552
|
Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced
[Chihara]
|
9553
|
Analytic geometry gave space a mathematical structure, which could then have axioms
[Chihara]
|
9559
|
If a successful theory confirms mathematics, presumably a failed theory disconfirms it?
[Chihara]
|
9561
|
The mathematics of relations is entirely covered by ordered pairs
[Chihara]
|
9562
|
In simple type theory there is a hierarchy of null sets
[Chihara]
|
9563
|
A pack of wolves doesn't cease when one member dies
[Chihara]
|
9566
|
No scientific explanation would collapse if mathematical objects were shown not to exist
[Chihara]
|
9568
|
I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes'
[Chihara]
|
9571
|
ZFU refers to the physical world, when it talks of 'urelements'
[Chihara]
|
9572
|
Realists about sets say there exists a null set in the real world, with no members
[Chihara]
|
9573
|
The null set is a structural position which has no other position in membership relation
[Chihara]
|
9574
|
'Gunk' is an individual possessing no parts that are atoms
[Chihara]
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