13 Facts About Mathematical Platonism

Mathematical Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging.

The term Mathematical Platonism is used because such a view is seen to parallel Plato's Theory of Forms and a "World of Ideas" described in Plato's allegory of the cave: the everyday world can only imperfectly approximate an unchanging, ultimate reality.

Full-blooded Platonism is a modern variation of Platonism, which is in reaction to the fact that different sets of mathematical entities can be proven to exist depending on the axioms and inference rules employed.

Max Tegmark's mathematical universe hypothesis goes further than Platonism in asserting that not only do all mathematical objects exist, but nothing else does.

Mathematical Platonism held that axioms in geometry should be chosen for the results they produce, not for their apparent coherence with human intuitions about the physical world.

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Mathematical Platonism empiricism is a form of realism that denies that mathematics can be known a priori at all.

Mathematical Platonism advocated a form of "pure realism" that rejected mystical notions of truth and accepted much quasi-empiricism in mathematics.

Mathematical Platonism proposed that an alien species doing mathematics might well rely on quasi-empirical methods primarily, being willing often to forgo rigorous and axiomatic proofs, and still be doing mathematics—at perhaps a somewhat greater risk of failure of their calculations.

Mathematical Platonism gave a detailed argument for this in New Directions.

Mathematical Platonism fictionalism was brought to fame in 1980 when Hartry Field published Science Without Numbers, which rejected and in fact reversed Quine's indispensability argument.

Mathematical Platonism did this by giving a complete axiomatization of Newtonian mechanics with no reference to numbers or functions at all.

Mathematical Platonism started with the "betweenness" of Hilbert's axioms to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields.

Mathematical Platonism showed that mathematical physics is a conservative extension of his non-mathematical physics, so that mathematics is a reliable process whose physical applications are all true, even though its own statements are false.