Mathematical Platonism: Have Some Delicious Pi
Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices.
Øystein Linnebo, Platonism in the Philosophy of Mathematics
In my previous blog post, I wrote at length about what numbers are (and consequently, whether my math homework exists or not). Unsurprisingly, I was not the first philosopher to ponder the properties of mathematical objects. The inspiration for the theory of mathematical Platonism dates back to Plato and his Theory of Forms, as one can infer from the name of the theory.
However, mathematical Platonism is not directly derived from Plato’s Theory of Forms; instead, many of its principles are based upon the work of the 19th-20th century mathematician and philosopher Gottlob Frege. Frege’s works have been adapted and intertwined with similar ideas, and an modern expert in the field of mathematical Platonism is Øystein Linnebo, whose ideas I will be quoting at length in this post.
Mathematical platonism can be defined as the conjunction of the following three theses:
There are mathematical objects.
Mathematical objects are abstract.
Mathematical objects are independent of intelligent agents and their language, thought, and practices.
A decent grasp of these three ideas is essential for an understanding of mathematical Platonism, so I’ll go through them one by one in more detail.
Linnebo starts by referencing some of Frege’s ideas:
The Fregean argument is based on two premises, the first of which concerns the semantics of the language of mathematics:
The singular terms of the language of mathematics purport to refer to mathematical objects, and its first-order quantifiers purport to range over such objects.
Most sentences accepted as mathematical theorems are true (regardless of their syntactic and semantic structure).
These premises are worded in complicated ways, but they boil down to simple logic:
1) Mathematical theorems are true.
2) Mathematical theorems refer to mathematical objects.
3) Therefore, mathematical objects exist.
The article goes into a little more detail (you can read more here), but this is the gist of it.
(Please note that the above logic is not intended to be sound. Instead, it is intended to facilitate the understanding of the idea of existence.)
Abstractness says that every mathematical object is abstract, where an object is said to be abstract just in case it is non-spatiotemporal and (therefore) causally inefficacious . . .
. . . For if these objects had spatiotemporal locations, then actual mathematical practice would be misguided and inadequate, since pure mathematicians ought then to take an interest in the locations of their objects, just as physicists take an interest in the locations of theirs.
The second of the three ideas, abstraction, is much less complicated than existence. If an object is abstract, it does not exist in space-time (also known as the material world). Other entities that are non-spatiotemporal may include ideas, thoughts, and concepts, depending on what philosophical outlook you have.
Abstract objects exist in an abstract world, sometimes thought of as a mirror to our own. This is one area of Platonism and mathematical Platonism differ. While Platonism states that the abstract world in the more fundamental/superior world to our own, mathematical Platonism does not assert this superiority.
Independence says that mathematical objects, if there are any, are independent of intelligent agents and their language, thought, and practices . . .
. . . had there not been any intelligent agents, or had their language, thought, or practices been different, there would still have been mathematical objects.
The last of the three ideas, independence, is perhaps the simplest idea of the three. Independence states that mathematical entities are more than a human construct, and that they exist independently of us. This means that they were discovered by humans instead of created by humans, which is an important distinction. What independence implies is that (if they exist), other conscious entities would also discover mathematics in a similar way to us, or at least the basic concepts would be the same.
To summarize, mathematical Platonism states that mathematical objects (such as 3 and π) exist, are non-spatiotemporal, and were discovered as opposed to created by humans. This theory for the explanation of the existence of mathematical objects makes the most sense to me – for as I discussed in my earlier blog post, numbers don’t really exist in the physical world (space-time). Five fingers are material objects and so are five sheep, but does five itself exist materially in the same manner? This theory offers what I see to be sound explanations for the properties of mathematical entities, helping to lay the foundations for the entire field of mathematics.