Editing Reality (section)

=== Abstract objects and mathematics ===
The status of [[Abstraction (mathematics)|abstract]] entities, particularly numbers, is a topic of discussion in mathematics.

In the philosophy of mathematics, the best known form of realism about numbers is [[Platonic realism]], which grants them abstract, immaterial existence. Other forms of realism identify mathematics with the concrete physical universe.

Anti-realist stances include [[formalism (mathematics)|formalism]] and [[mathematical fictionalism|fictionalism]].

Some approaches are selectively realistic about some mathematical objects but not others. [[Finitism]] rejects [[Infinity|infinite]] quantities. [[Ultra-finitism]] accepts finite quantities up to a certain amount. [[Constructivism (mathematics)|Constructivism]] and [[intuitionism]] are realistic about objects that can be explicitly constructed, but reject the use of the [[principle of the excluded middle]] to prove existence by [[reductio ad absurdum]].

The traditional debate has focused on whether an abstract (immaterial, intelligible) realm of numbers has existed ''in addition to'' the physical (sensible, concrete) world. A recent development is the [[mathematical universe hypothesis]], the theory that ''only'' a mathematical world exists, with the finite, physical world being an illusion within it.

An extreme form of realism about mathematics is the [[mathematical multiverse hypothesis]] advanced by [[Max Tegmark]]. Tegmark's sole postulate is: ''All structures that exist mathematically also exist physically''. That is, in the sense that "in those [worlds] complex enough to contain self-aware substructures [they] will subjectively perceive themselves as existing in a physically 'real' world".<ref name="Tegmark2008">{{cite journal|last=Tegmark |first=Max |date=February 2008 |title=The Mathematical Universe |journal=Foundations of Physics |volume=38 |issue=2 |pages=101–150 |doi=10.1007/s10701-007-9186-9 |arxiv=0704.0646|bibcode = 2008FoPh...38..101T |s2cid=9890455 }}</ref><ref>Tegmark (1998), p. 1.</ref> The hypothesis suggests that worlds corresponding to different sets of initial conditions, physical constants, or altogether different equations should be considered real. The theory can be considered a form of [[Platonism]] in that it posits the existence of mathematical entities, but can also be considered a [[philosophy of mathematics|mathematical monism]] in that it denies that anything exists except mathematical objects.