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Single Idea 9606
[filed under theme 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
]
Full Idea
We could not discover irrational numbers by physical measurement. The discovery of the irrationality of the square root of two was an intellectual achievement, not at all connected to sense experience.
Gist of Idea
The irrationality of root-2 was achieved by intellect, not experience
Source
James Robert Brown (Philosophy of Mathematics [1999], Ch. 1)
Book Ref
Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.5
A Reaction
Brown declares himself a platonist, and this is clearly a key argument for him, and rather a good one. Hm. I'll get back to you on this one...
The
33 ideas
from 'Philosophy of Mathematics'
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9605
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If a proposition is false, then its negation is true
[Brown,JR]
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9604
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Mathematics is the only place where we are sure we are right
[Brown,JR]
|
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9606
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The irrationality of root-2 was achieved by intellect, not experience
[Brown,JR]
|
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9610
|
Numbers are not abstracted from particulars, because each number is a particular
[Brown,JR]
|
|
9612
|
There is an infinity of mathematical objects, so they can't be physical
[Brown,JR]
|
|
9608
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There are no constructions for many highly desirable results in mathematics
[Brown,JR]
|
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9611
|
'Abstract' nowadays means outside space and time, not concrete, not physical
[Brown,JR]
|
|
9609
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The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars
[Brown,JR]
|
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9615
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Nowadays conditions are only defined on existing sets
[Brown,JR]
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9613
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Naïve set theory assumed that there is a set for every condition
[Brown,JR]
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9617
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The 'iterative' view says sets start with the empty set and build up
[Brown,JR]
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|
9619
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David's 'Napoleon' is about something concrete and something abstract
[Brown,JR]
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9620
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Empiricists base numbers on objects, Platonists base them on properties
[Brown,JR]
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9622
|
'There are two apples' can be expressed logically, with no mention of numbers
[Brown,JR]
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9621
|
Mathematics represents the world through structurally similar models.
[Brown,JR]
|
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9625
|
To see a structure in something, we must already have the idea of the structure
[Brown,JR]
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|
9628
|
Sets seem basic to mathematics, but they don't suit structuralism
[Brown,JR]
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9634
|
Set theory says that natural numbers are an actual infinity (to accommodate their powerset)
[Brown,JR]
|
|
9638
|
Berry's Paradox finds a contradiction in the naming of huge numbers
[Brown,JR]
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9630
|
The most brilliant formalist was Hilbert
[Brown,JR]
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9629
|
For nomalists there are no numbers, only numerals
[Brown,JR]
|
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9635
|
Given atomism at one end, and a finite universe at the other, there are no physical infinities
[Brown,JR]
|
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9640
|
A term can have not only a sense and a reference, but also a 'computational role'
[Brown,JR]
|
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9639
|
Does some mathematics depend entirely on notation?
[Brown,JR]
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9643
|
Set theory may represent all of mathematics, without actually being mathematics
[Brown,JR]
|
|
9644
|
When graphs are defined set-theoretically, that won't cover unlabelled graphs
[Brown,JR]
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|
9641
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Definitions should be replaceable by primitives, and should not be creative
[Brown,JR]
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9642
|
A flock of birds is not a set, because a set cannot go anywhere
[Brown,JR]
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9645
|
Constructivists say p has no value, if the value depends on Goldbach's Conjecture
[Brown,JR]
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9646
|
There is no limit to how many ways something can be proved in mathematics
[Brown,JR]
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9647
|
Computers played an essential role in proving the four-colour theorem of maps
[Brown,JR]
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9648
|
π is a 'transcendental' number, because it is not the solution of an equation
[Brown,JR]
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|
9649
|
Axioms are either self-evident, or stipulations, or fallible attempts
[Brown,JR]
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