Contemporary Platonism is the view that there exist such things as abstract objects. An abstract object is an object that does not exist in space or time and is entirely non-physical and non-mental. This form of Platonism is a contemporary one and it is unclear whether or not the historical Plato would have held it, although it is still connected in important ways to him. 

As non-physical entities, abstract objects are not made of matter or anything physical; as non-mental entities, abstract objects are not ideas, minds, disembodied souls, or gods. The Platonist views numbers as abstract objects: the number ‘3’, for example, is a real, objective object. It exists in the same way that a table or a car exists independently of the human mind; it is not just an idea existing in one’s head. The number 3 would still exist even if the Earth was devoid of human beings. But unlike tables and cars, Platonism maintains that the number 3 is immaterial; it is non-physical and it does not exist in space or time. It is believed to exist in a platonic heaven, which is not a physical space.

Abstract objects are acausal as they do not have any causal influence over other objects or items, and other objects or items do not have a causal influence over them. This is because causal relations require a spatio-temporal realm; for example, kicking a ball causes it to move in space and time, but one cannot kick the number 3. Abstract objects are eternal in that they exist at all times and/or exist outside of temporal relations. Abstract objects are also metaphysically necessary or exist necessarily. In other words, the item could not have failed to exist.

Many Platonists also view properties as abstract objects. The color red is a property (“redness”) that is believed to exist independently of any red object or thing. There are red cars and shirts that exist in the physical world. But Platonists posit the property of redness over and above red cars, shirts, and objects. Such objects exemplify or instantiate redness.

In the mathematical sense, many embrace Platonism because of everyday mathematical activities. We accept, for example, that “2 + 2 = 4” and that this is an objective truth in the universe. Platonists take this conviction to support the view that mathematical theories have a subject matter that is independent of human minds and rational activities. Some refer to this as the prima facie plausibility of Platonism. Some philosophers have attempted to provide arguments in favor of Platonism. An important argument is provided by Friedrich Ludwig Gottlob Frege (1848-1925) and is stated as follows:

P1: If a simple sentence (i.e., a sentence of the form ‘a is F’, or ‘a is R-related to b’, or…) is literally true, then the objects that its singular terms denote exist. (Likewise, if an existential sentence is literally true, then there exist objects of the relevant kinds; e.g., if ‘There is an F’ is true, then there exist some Fs.).

P2: There are literally true simple sentences containing singular terms that refer to things that could only be abstract objects. (Likewise, there are literally true existential statements whose existential quantifiers range over things that could only be abstract objects.) Therefore,

C: Abstract objects exist.

Most philosophers accept P1 as true, so most of the debate will hinge on whether or not we can accept P2, namely that there are true sentences that refer to things that could only be abstract objects. Platonists have attempted to provide examples in support of P2, such as possible worlds, relations, propositions, properties, sentence types, and logical objects.

References

Balaguer, Mark. 2016. Platonism in Metaphysics. Available.

Garvey, James., and Stangroom, Jeremy. 2012. The Story of Philosophy: A History of Western Thought. London: Hachette UK.