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Single Idea 8716
[filed under theme 6. Mathematics / C. Sources of Mathematics / 7. Formalism
]
Full Idea
There are not enough constraints in the Formalist view of mathematics, so there is no way to select a direction for trying to develop mathematics. There is no part of mathematics that is more important than another.
Gist of Idea
Formalism is unconstrained, so cannot indicate importance, or directions for research
Source
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 6.6)
Book Ref
Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.149
A Reaction
One might reply that an area of maths could be 'important' if lots of other areas depended on it, and big developments would ripple big changes through the interior of the subject. Formalism does, though, seem to reduce maths to a game.
The
41 ideas
from Michèle Friend
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8713
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In classical/realist logic the connectives are defined by truth-tables
[Friend]
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8663
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Raising omega to successive powers of omega reveal an infinity of infinities
[Friend]
|
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8662
|
The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega
[Friend]
|
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8661
|
The natural numbers are primitive, and the ordinals are up one level of abstraction
[Friend]
|
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8664
|
Cardinal numbers answer 'how many?', with the order being irrelevant
[Friend]
|
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8671
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The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps
[Friend]
|
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8669
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Between any two rational numbers there is an infinite number of rational numbers
[Friend]
|
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8665
|
A 'proper subset' of A contains only members of A, but not all of them
[Friend]
|
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8672
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A 'powerset' is all the subsets of a set
[Friend]
|
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8666
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Infinite sets correspond one-to-one with a subset
[Friend]
|
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8667
|
The 'integers' are the positive and negative natural numbers, plus zero
[Friend]
|
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8668
|
The 'rational' numbers are those representable as fractions
[Friend]
|
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8670
|
A number is 'irrational' if it cannot be represented as a fraction
[Friend]
|
|
8675
|
Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets'
[Friend]
|
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8674
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The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal
[Friend]
|
|
8677
|
Set theory makes a minimum ontological claim, that the empty set exists
[Friend]
|
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3678
|
Reductio ad absurdum proves an idea by showing that its denial produces contradiction
[Friend]
|
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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]
|
|
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]
|
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8695
|
Structuralism focuses on relations, predicates and functions, with objects being inessential
[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]
|
|
8700
|
'In re' structuralism says that the process of abstraction is pattern-spotting
[Friend]
|
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8704
|
Structuralists call a mathematical 'object' simply a 'place in a structure'
[Friend]
|
|
8706
|
Constructivism rejects too much mathematics
[Friend]
|
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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]
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