We all know what numbers are. In fact, we are so sure of this knowledge that we can easily point our fingers at their written depiction and exclaim, “These are numbers!”. But are they really?
If we think about it, when visualizing a number, what we see are numerals, or the symbols that we humanly chose to represent the physical form of numbers. For this reason, we are unable to claim that a number has a specific embodiment. Numbers only acquire such a quality when they are attributed to tangible things.
So, is the existence of numbers subject to their physical representation? Not quite. Numbers show up regardless of the things to which they are assigned, being that they are both concrete values expressed through words and uncountable amounts. Besides, we must remember that the numbers we employ to convey tangible quantities (those known as natural) only represent a small fraction of the types of numbers that we know so far. Or could it be that negative integers, irrational, complex and even imaginary numbers are not numbers? It’s obvious that they are since these have demonstrated the need for their applicability.
As a result, we must ask ourselves the following question: do numbers exist or not?
From a philosophical standpoint, we could say that a number is one of two things: a) an unreal construct, or b) an abstract object.
If we accept the former, we enter the field of nominalism. This doctrine postulates that general or universal ideas are mere labels without any corresponding reality. Thereby, things exist because they manifest themselves physically. Unfortunately, this view falls apart when considering that which exists but cannot be manipulated.
Take the novel The Great Gatsby by F. Scott Fitzgerald, for instance. We could comfortably pick up a physical copy of the novel and say, “This is The Great Gatsby”. However, what if all the copies ceased to exist and only the original manuscript remained? Would the novel still exist? Certainly, though it would not be available to the masses. But what if we go a little further and say that the original manuscript was burnt? Would the novel still exist? This has happened before with other writings, after all. Only public memory keeps them at the margin of “reality” despite lacking a physical iteration. Thus, if we’re to follow nominalism, indefinite things need not be tangible objects for their existence is beyond the physical. Somehow(?)
Enter the second point.
Under this Platonic perspective, we allow for the objective existence of abstract objects outside our minds. Things like mathematical concepts, music compositions, and even moves in a game of chess fall into this category. But how come these things are abstract?
Taking a chess move as an example, we assume that it exists in a concrete way so long as a chess player uses it. Yet, such a seemingly concrete move could be a replica of someone else’s game, or it could be thought of in the head before it’s even used on the board. That being the case, these concrete possibilities give rise to a metaphysical problem: If there is only one move, what is the original, untainted move? And what are the other moves? Copies of the concrete move or instances of the same move?
If we thought about it nominally, we would certainly get into a thicket. Luckily, we are considering the Platonic notion, which put the odds in our favour.
We could assert that a chess move is an abstract object, and that each physical version of that move is a concrete iteration of itself. In other words, none of the concrete physical moves are actually “the move”. There is only one move, and it is an abstract object that every physical replica imitates. This idea is directly related to Plato’s theory of forms, which suggests that ideal forms, or perfect archetypes, exist outside of space-time, and that objects in the real world are imperfect copies of said ideal forms.
Sadly, there is a “teeny, little” problem with this concept: its failure to deal with the way we learn about abstract objects.
Let me explain.
Generally speaking, we acquire knowledge in the following way:
Step Number 1. An object is perceived in the physical world through physical means.
Step Number 2. Our brains process what we observed, leading us to work with mental images of the object in question.
Step Number 3. Information is obtained.
An abstract object cannot be processed like this. It is not physical, so our usual way of gaining knowledge fails. This is especially applicable to things like numbers, whose perception is not as intuitive as it seems. As such, how can we attain any insight from abstract objects? Some of the followers of the Platonic school resort to the idea that certain abstract things are imposed on us as true. But how is it that they impose their truths on us? To say that they “simply do it” is not an acceptable explanation.
I guess the only way to clarify this would be to resort to magic, mimicking Descartes’ speculation that minds are substances outside of the brain, and that they are not located in space-time. Such a presumption forces us to think about how the mind shows up in our brains. According to Descartes, the mind slips through the pineal gland to reach the brain. Though I’m afraid this is more or less wishful thinking and a massive delay in the search for the right response. Also, how is it that the mind, an abstract object outside of temporal space, crawls through the pineal gland and then to the brain? Descartes had no answer to this, given that explaining how the non-physical interacts with the physical can be largely illogical.
Hence, it is here that the nominalists take the reins of the debate once again. Although their party is quickly watered down by the indispensability argument.
Hardcore nominalists are often scientifically minded and motivated philosophers. However, their love of science can be put into question whenever they deny the existence of numbers. Consider the following argument:
Science is the best arbiter of what exists.
If science says that something exists, we must accept it.
Science is (strongly) based on mathematics.
Science admits the existence of numbers.
Therefore, numbers exist.
To put it simply, the indispensability of numbers to science means they exist. Nominalism tells us otherwise. Ergo, paradox time!
So, what does all this tell us about the existence of numbers?
Well, if you choose to go by the platonic doctrine, you agree that “yes, numbers exist”. You also declare that numbers have some sort of abstract existence, different from the one enjoyed by physical entities. Yet, this means that you are in the unenviable position of explaining the coherence behind such an existence and how it is that we come to know anything about this abstract realm, since it’s devoid of tangible sentience.
On the other hand, if you are a nominalist, you affirm that “no, numbers do not exist”. Sadly, you now have the daunting task of explaining why mathematics seems so indispensable to science, and that’s an enigma whose current answers are untenable.
What to believe then?
Often, in philosophy, maintaining a dogmatic stance in a bilateral debate is inappropriate. In this case, we cannot assert whether numbers exist or not; therefore, we concede that we don’t know.
But want to know what we know? That numbers are useful, important, and, at times, even beautiful. So rejoice, my dear reader, for a life unaccounted for is not a life worth living.
Thank you for reading. (:
[*] For further information on the subject, see the following:
Science without Numbers, by Hartry Field
Realism in Mathematics, by Penelope Maddy