more on this theme
|
more from this thinker
Single Idea 8668
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
]
Full Idea
The 'rational' numbers are all those that can be represented in the form m/n (i.e. as fractions), where m and n are natural numbers different from zero.
Gist of Idea
The 'rational' numbers are those representable as fractions
Source
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5)
Book Ref
Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.17
A Reaction
Pythagoreans needed numbers to stop there, in order to represent the whole of reality numerically. See irrational numbers for the ensuing disaster. How can a universe with a finite number of particles contain numbers that are not 'rational'?
The
41 ideas
from 'Introducing the Philosophy of Mathematics'
8713
|
In classical/realist logic the connectives are defined by truth-tables
[Friend]
|
8663
|
Raising omega to successive powers of omega reveal an infinity of infinities
[Friend]
|
8662
|
The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega
[Friend]
|
8661
|
The natural numbers are primitive, and the ordinals are up one level of abstraction
[Friend]
|
8671
|
The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps
[Friend]
|
8669
|
Between any two rational numbers there is an infinite number of rational numbers
[Friend]
|
8665
|
A 'proper subset' of A contains only members of A, but not all of them
[Friend]
|
8672
|
A 'powerset' is all the subsets of a set
[Friend]
|
8666
|
Infinite sets correspond one-to-one with a subset
[Friend]
|
8664
|
Cardinal numbers answer 'how many?', with the order being irrelevant
[Friend]
|
8667
|
The 'integers' are the positive and negative natural numbers, plus zero
[Friend]
|
8668
|
The 'rational' numbers are those representable as fractions
[Friend]
|
8670
|
A number is 'irrational' if it cannot be represented as a fraction
[Friend]
|
8677
|
Set theory makes a minimum ontological claim, that the empty set exists
[Friend]
|
3678
|
Reductio ad absurdum proves an idea by showing that its denial produces contradiction
[Friend]
|
8676
|
Is mathematics based on sets, types, categories, models or topology?
[Friend]
|
8678
|
Most mathematical theories can be translated into the language of set theory
[Friend]
|
8675
|
Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets'
[Friend]
|
8674
|
The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal
[Friend]
|
8680
|
Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects
[Friend]
|
8681
|
The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts?
[Friend]
|
8682
|
Major set theories differ in their axioms, and also over the additional axioms of choice and infinity
[Friend]
|
8685
|
Studying biology presumes the laws of chemistry, and it could never contradict them
[Friend]
|
8688
|
Concepts can be presented extensionally (as objects) or intensionally (as a characterization)
[Friend]
|
8694
|
Free logic was developed for fictional or non-existent objects
[Friend]
|
8696
|
Structuralist says maths concerns concepts about base objects, not base objects themselves
[Friend]
|
8695
|
Structuralism focuses on relations, predicates and functions, with objects being inessential
[Friend]
|
8700
|
'In re' structuralism says that the process of abstraction is pattern-spotting
[Friend]
|
8701
|
The number 8 in isolation from the other numbers is of no interest
[Friend]
|
8702
|
In structuralism the number 8 is not quite the same in different structures, only equivalent
[Friend]
|
8699
|
Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)?
[Friend]
|
8704
|
Structuralists call a mathematical 'object' simply a 'place in a structure'
[Friend]
|
8706
|
Constructivism rejects too much mathematics
[Friend]
|
8705
|
Anti-realists see truth as our servant, and epistemically contrained
[Friend]
|
8708
|
Double negation elimination is not valid in intuitionist logic
[Friend]
|
8709
|
The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false
[Friend]
|
8707
|
Intuitionists typically retain bivalence but reject the law of excluded middle
[Friend]
|
8711
|
Intuitionists read the universal quantifier as "we have a procedure for checking every..."
[Friend]
|
8712
|
Mathematics should be treated as true whenever it is indispensable to our best physical theory
[Friend]
|
8716
|
Formalism is unconstrained, so cannot indicate importance, or directions for research
[Friend]
|
8721
|
An 'impredicative' definition seems circular, because it uses the term being defined
[Friend]
|