First, we will look at what it means to be a contemporary platonist towards some domain. Being a contemporary platonist means to affirm that such things called Abstract Objects exist, that they are neither physical (sensible) nor mental (mere ideas), and that they therefore exist independently of us. The term abstract object is meant to umbrella various objects of philosophical interest and not only mathematics but properties, relations, propositions, and possible worlds. This series will only be looking at Mathematical Objects (a fancy word for numbers, sets, and basically anything mathematicians study).
Historically, platonism with regard to mathematics has been affirmed by the great German philosopher Gottlob Frege, well known British Atheist Bertrand Russell, the highly influential Willard Van Orman Quine and the famous mathematician Kurt Gödel.
Again to be a platonist with respect to mathematics is, as Standford Encyclopedia (from here on SEP) notes, to be committed to three claims.
1. There are mathematical objects. (Existence)
2. Mathematical objects are abstract. (Abstractness)
3. Mathematical Objects exist independently of our ideas, beliefs, feelings, practices etc. (Indepdenence)
This post will be looking at the reasons why we should believe 1 (that these objects, however they exist, exist nonetheless).
Let’s review how we generally talk about things and truth in the world before I present the Singular Term Argument developed by Frege.
Example: When we say “That Sally is a girl is literally true”,
We seem to be saying…
"There is some actually existing girl, whose name is actually Sally."
From the above we derive a “principle of ontological commitment” which is explained by our first premise.
Premise 1. If we say of some A, that it is F, and that this statement is literally true, A must exist.
Not convinced? Suppose I were to say:
“Here I am speaking literally about a person, her name is Sally, and it is true that she is a girl, but she does not exist.”
We would likely wait for a qualification. Perhaps Sally is a fictional character? Or perhaps Sally is a metaphor or again, a symbol? But of course, if Sally was fictional or symbolic, one would not be speaking literally.
In the example, Sally stands for a Singular Term which is defined by Frege as a term that is inherently about the object to which it applies, and I should add that it is uniquely about that object. “Sally” (the term) is about the unique object Sally (the actually existing girl).
So if we accept as plausible this “principle of ontological commitment” we move onto the second premise, again I will quote SEP.
Premise 2. There are literally true simple sentences containing singular terms that refer to things that could only be (mathematical) objects.
SEP’s example of a sentence that fits premise 2’s description:
"3 is prime."
Therefore, from premise 1 and 2, it necessarily follows that mathematical objects exist.
Again, this only proves our first claim, that these mathematical objects exist in some, as of yet, unspecified sense. This isn’t extraordinary, there are many non-platonic philosophical accounts of mathematics that accept that numbers exist at least in some sense, and accept that there are mathematical truths, so this first claim isn’t so controversial as the platonists’ other two claims.
But before we move on to our second claim, can’t it be reasonably asked
"Why should we at all accept that we are speaking literally about mathematics in the sense of actually existing objects, aren’t there other ways of accounting for math talk?"
To entertain perhaps that math talk is just a game of language and metaphor, we ought to ask what exactly such a game would be about, and if a metaphor a metaphor about what? The answers to this will usually be that math is just a language game or a way of speaking about our ideas, or our experience. I will address this in the next post.
More controversial but possible, would be to question why we should even assume that we are speaking truth when we talk about mathematics.
Such an account known as Fictionalism can be summarized by saying that mathematical talk is nothing more than ‘useful fictions’ and ‘irrefutable errors’ (as Nietzsche would say). Fictionalists deny mathematical truth in general, to them all mathematical statements are simply false. While they agree that our way of speaking seems to commit us to the existence of mathematical objects, and so if such sentences were true these objects would exist, such objects do not exist, so any statement that seems to purport a mathematical truth must by default be false.
I find it hard to conceive of someone denying that “2+2=4 is true”. I would assume that they must not have understood what I said, or lacked the intellectual capacity to carry out the operation. Nonetheless the Fictionalist will say I’m no better than a myth-teller. Well, to such a position how can a platonist respond?
The trick is to meet them where they are, Fictionalists do not deny that mathematics is useful. For instance, maths is heavily relied upon and has been extremely successful for discovering truth about the physical world through the sciences. Now, the platonist will argue that if there is at least one truth in science that relies upon mathematics, then it is possible for mathematics to lead us to truth.
Fictionalists that deny that mathematics can lead to truth would seem, to my mind at least, to commit them to a three views which are inconsistent if held at the same time:
- Our best scientific theories represent the objects it studies through Mathematical concepts and derives successful predictions about these objects and thus the physical world through mathematics.
- We should believe that the success of our best scientific theories are a result of accurately representing the truth about the physical world.
- There are no truths that can be derived from mathematics.
It seems obvious that the Fictionalist cannot commit to the first bullet point and also affirm both the second and third. The first is undisputed, this is simply how science is done, so this cannot be denied. To deny the second is to believe that the success of science is somehow not due to our models accurately representing reality, but rather extremely lucky guesses. The Fictionalist is committed to affirming three, so she must deny two and thus conclude that Science’s accurately representing the world is not evidence of the truth of it’s claims. I argue, they must give up belief in the truths of science, or accept that truths can be derived from mathematics. And if we can derive truth from mathematics, then the Singular Term Argument holds.
My argument for the paradox Fictionalists find themselves in, is only as strong as science and mathematics’ relationship.
To further strengthen this relationship, and support why affirming the third bullet point commits to a denial of two I will cite the influential Indispensability Argument from the Internet Encyclopedia of Philosophy developed by Quine and Putnam.
i. We should acknowledge the existence of—or, as Quine and Putnam would prefer to put it, be ontologically committed to—all those entities that are indispensable to our best scientific theories.
ii. Mathematical objects or structures are indispensable to our best scientific theories.
Therefore,iii. We should acknowledge the existence of—be ontologically committed to—mathematical objects or structures.
Whether Science cannot get rid of mathematics in principle is a matter of contested debate in academic philosophy. Perhaps some will say that it is conceivable that someday it will be possible to do Science without mathematics, but I do not know how such a thing would be done. For now however, Science cannot dispense with mathematics, and that is all that is needed for the argument.
Next time, I’ll be looking at the different accounts of just how exactly Numbers are supposed to exist.
Perhaps numbers exist independently because (ignoring Idealism) there is an external world, and numbers are instantiated or tied up with the concrete physical objects of this world.
Or maybe they are abstract but only because they’re ideas, and therefore, being only concepts, exist because of us.
The committed Platonist must deny both.