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The two main alternatives to Platonism, then, are realistic anti-Platonism and nominalism. These alternatives are described more fully in the following two sections.

About Rationally Speaking

There are two different versions of realistic anti-Platonism, namely, psychologism and physicalism. Psychologism is the view that mathematical theorems are about concrete mental objects of some sort. As will become clearer below in the section Mathematical Platonism: for and against , psychologism has serious problems and is no longer endorsed by many philosophers; nonetheless, it was popular during the late 19th and early 20th centuries, the most notable proponents being the German philosopher Edmund Husserl and the Dutch mathematicians L.

Brouwer and Arend Heyting.

A platonist epistemology

Physicalism , on the other hand, is the view that mathematics is about concrete physical objects of some sort. Advocates of this view agree with Platonists that there exist such things as numbers and sets, and, unlike adherents of psychologism, they also agree that these things exist independently of people and their thoughts.

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Physicalists differ from Platonists, however, in holding that mathematics is about ordinary physical objects. There are a few different versions of this view. For example, one might hold that geometric objects, such as circles, are regions of actual physical space. Similarly, sets might be claimed to be piles of actual physical objects—thus, a set of eggs would be nothing more than the aggregate of physical matter that makes up the eggs.

Moving on to numbers, one strategy is to take them to be physical properties of some sort—for example, properties of piles of physical objects, so that, for instance, the number 3 might be a property of a pile of three eggs. It should be noted here that many people have endorsed a Platonistic view of properties. In particular, Plato thought that, in addition to all the red things he observed in the world, there exists an independent property of redness and that this property was an abstract object.

Aristotle , on the other hand, thought that properties exist in the physical world; thus, in his view, redness exists in particular objects, such as red houses and red apples, rather than as an abstract object outside of space and time. So in order to motivate a physicalistic view of mathematics by claiming that numbers are properties, one would also have to argue for an Aristotelian, or physicalistic, view of properties.

Platonism and Anti-Platonism in Mathematics - Mark Balaguer - Google книги

One person who has developed a view of this sort since Aristotle is the Australian philosopher David Armstrong. Another strategy for interpreting talk of numbers to be about the physical world is to interpret it as talk about actual piles of physical objects rather than properties of such piles. In Part III, Balaguer extends his conclusions, arguing that it is not simply that we do not currently have any good argument for or against platonism, but that we could never have such an argument, and indeed, that there is no fact of the matter as to whether platonism is correct ie.

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Philosophy of mathematics

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