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RobbBB comments on What do professional philosophers believe, and why? - Less Wrong Discussion
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Wikipedia's fine, but I'd rely more on SEP for quick stuff like this. The question of what makes something 'mathematical' is a difficult one, but it's not important for evaluating abstract-object realism. What makes something abstract is just that it's causally inert and non-spatiotemporal. Tegmark's MUH asserts things like that. Sparser mathematical platonisms also assert things like that. For present purposes, their salient difference is how they motivate realism about abstract objects, not how they conceive of the nature of our own world.
If I understand this correctly, I disagree. Modern philosophical platonism means different things by 'abstract' than Tegmark's platonism. In philosophical platonism, I accept your definition that something is abstract if it is causally inert and non-spatiotemporal. For Tegmark, this doesn't really make sense though, since the universe is causal in the same sense that a mathematical model of a dynamical system is causal, and it is spatiotemporal in the same sense that the mathematical concept of Minkowski spacetime is spatiotemporal, since the universe is just (approximately) a dynamical system on (approximately) Minkowski spacetime. The usual definition of an abstract object implies that physical, spatiotemporal objects are not abstract, which contradicts the MUH. I don't think we really have a precise definition of abstract object that makes sense in Tegmark's platonism, since something like 'mathematical structure' is obviously imprecise.
I don't think that means that abstract objects in the ordinary sense don't make sense. It just means that he counts a lot of things as concrete that most people might think of as abstract. We don't need a definition of 'mathematical structure' for present purposes, just mathematically precise definitions of 'causal' and 'spatiotemporal'.
The abstract/concrete distinction is actually a separate ontic axis from the mathematical/physical one. You can have abstract (platonic) physical objects, and concrete mathematical objects.
Example of abstract physical objects: Fields
Example of concrete mathematical objects: Software
My definitions:
Abstract: universal , timeless and acausal (always everywhere true and outside time and space, and not causally connected to concrete things).
Concrete: can be located in space and time, is causal, has moving parts
Mathematical: concerned with categories, logics and models
Physical: concerned with space, time, and matter
My take on modern Platonism is that abstract objects are considered the only real (fundamental) objects. Abstract objects can’t interact with concrete objects, because concrete objects don’t actually exist! Rather, concrete things should be thought of as particular parts (cross-sections, aspects of) abstract things. Abstract objects encompass concrete objects. But the so-called concrete objects are really just categories in our own minds (a feature of the way we have chosen to ‘carve reality at the joints’).
This isn't modern Platonism.
A program is an abstract object. Particular copies of a program stored in your hard drive, are concrete.
Ok, then its Geddesian Platonism ;) The easiest solution is to do away with the concrete dynamic objects as anything fundamental and just regard reality as a timeless Platonia . I thought thats more or less what Julian Barbour suggests.
http://en.wikipedia.org/wiki/Platonia_(philosophy)
The actual timeless (abstract) math objects are the mathematical relations making up the algorithm in question. But the particular model or representation of a program stored on a computer can be regarded as a concrete math object. And an instantiated (running) program can be viewed as a concrete math object also ( a dynamical system with input, processing and output).
These analogies are exact:
Space is to physics as categories are to math
Time is to physics as dynamical systems (running programs) are to math
Matter is to physics as data models are to math