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Single Idea 15935

[filed under theme 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism ]

Full Idea

In modern mathematics virtually all work is only up to isomorphism and no one cares what the numbers or points and lines 'really are'.

Clarification

'Isomorphism' is where all the models coincide in structure

Gist of Idea

Modern mathematics works up to isomorphism, and doesn't care what things 'really are'

Source

Shaughan Lavine (Understanding the Infinite [1994], VI.1)

Book Ref

Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.157

A Reaction

At least that leaves the field open for philosophers, because we do care what things really are. So should everybody else, but there is no persuading some people.

The 21 ideas with the same theme [reasons for doubting the existence of maths entities]:

12339 Aristotle removes ontology from mathematics, and replaces the true with the beautiful [Aristotle, by Badiou]
12556 Mathematics is just about ideas, so whether circles exist is irrelevant [Locke]
14162 Mathematics doesn't care whether its entities exist [Russell]
17881 Mathematician want performable operations, not propositions about objects [Skolem]
3663 How can you contemplate Platonic entities without causal transactions with them? [Putnam]
17927 Realists have semantics without epistemology, anti-realists epistemology but bad semantics [Benacerraf, by Colyvan]
9936 The platonist view of mathematics doesn't fit our epistemology very well [Benacerraf]
9910 Number-as-objects works wholesale, but fails utterly object by object [Benacerraf]
2521 'Real' maths objects have no causal role, no determinate reference, and no abstract/concrete distinction [Katz]
9963 If we all intuited mathematical objects, platonism would be agreed [Jubien]
9962 How can pure abstract entities give models to serve as interpretations? [Jubien]
9964 Since mathematical objects are essentially relational, they can't be picked out on their own [Jubien]
12211 It is plausible that x^2 = -1 had no solutions before complex numbers were 'introduced' [Fine,K]
17823 If mathematical objects exist, how can we know them, and which objects are they? [Maddy]
9934 Number words became nouns around the time of Plato [Burgess/Rosen]
8310 Does the existence of numbers matter, in the way space, time and persons do? [Lowe]
17454 Children can use numbers, without a concept of them as countable objects [Heck]
10089 Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman]
10008 Arithmetic is not about a domain of entities, as the quantifiers are purely inferential [Hofweber]
15935 Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine]
8681 The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend]