more from this thinker
|
more from this text
Single Idea 17776
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
]
Full Idea
The ultimate principles upon which mathematics rests are those to which mathematicians appeal without proof; and the primitive concepts of mathematics ...themselves are grasped directly, if grasped at all, without the mediation of definition.
Gist of Idea
The ultimate principles and concepts of mathematics are presumed, or grasped directly
Source
John Mayberry (What Required for Foundation for Maths? [1994], p.405-1)
Book Ref
'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.405
A Reaction
This begs the question of whether the 'grasping' is purely a priori, or whether it derives from experience. I defend the latter, and Jenkins puts the case well.
The
15 ideas
with the same theme
[existence of fundamentals as a basis for mathematics]:
|
18271
|
We can't prove everything, but we can spell out the unproved, so that foundations are clear
[Frege]
|
|
21224
|
Pure mathematics is the relations between all possible objects, and is thus formal ontology
[Husserl, by Velarde-Mayol]
|
|
17880
|
Integers and induction are clear as foundations, but set-theory axioms certainly aren't
[Skolem]
|
|
17810
|
The study of mathematical foundations needs new non-mathematical concepts
[Kreisel]
|
|
9937
|
I do not believe mathematics either has or needs 'foundations'
[Putnam]
|
|
12688
|
Mathematics is the formal study of the categorical dimensions of things
[Ellis]
|
|
17776
|
The ultimate principles and concepts of mathematics are presumed, or grasped directly
[Mayberry]
|
|
17775
|
If proof and definition are central, then mathematics needs and possesses foundations
[Mayberry]
|
|
17777
|
Foundations need concepts, definition rules, premises, and proof rules
[Mayberry]
|
|
17804
|
Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms
[Mayberry]
|
|
10236
|
There is no grounding for mathematics that is more secure than mathematics
[Shapiro]
|
|
8764
|
Categories are the best foundation for mathematics
[Shapiro]
|
|
8676
|
Is mathematics based on sets, types, categories, models or topology?
[Friend]
|
|
17922
|
Reducing real numbers to rationals suggested arithmetic as the foundation of maths
[Colyvan]
|
|
18846
|
Cantor and Dedekind aimed to give analysis a foundation in set theory (rather than geometry)
[Rumfitt]
|
email your comments