Philosophy

On Numbers: A Complex Existence

We all know what numbers are. In fact, we are so sure of this knowledge that we can easily point our fingers at anything that contains them and say, “This is a number.” But is it really?

If we think about it a bit, when visualising a number, what we observe is actually numerals. That is, the symbols that we humanly chose to represent numbers physically. But we are not able to say that a number has a specific figure. They only acquire this quality when they are attributed to tangible things.

So, is the existence of numbers subject to their physical representation? Of course not. Numbers present themselves independently of the things to which they are attributed, since they are both adjectives (a word that grants an attribute) and nouns (a word that identifies something). We must also remember that the numbers we use to express tangible quantities (those called natural) only represent a small fraction of the types of numbers that we know so far. Or is it that negative, irrational, complex and even imaginary digits are not numbers? Obviously they are, as they have demonstrated the need for their applicability.

Thus, the following question ensues: do numbers exist or not?

Photograph of a person who writes numerals.

From a philosophical standpoint, we could argue that, a number is one of two things:

  1. an unreal construct or
  2. an abstract object.

If we accept the former, we would enter the field of nominalism, or the doctrine where general or universal ideas are mere names with no corresponding reality. Ergo, things exist because they get manifested physically. However, this kind of thinking is not entirely correct once we consider that which exists but can’t be manipulated.

Take the novel The Great Gatsby by F. Scott Fitzgerald for example. We could comfortably get hold of a physical copy of the novel and say, “This is The Great Gatsby.” However, what if all copies ceased to exist and only the original manuscript remained? Would the novel still exist? Surely, despite not being 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? Such a calamity has happened before to many writings that nowadays lack a physical iteration but whose collective memory keeps them within the margin of exiting. As such, following this notion, The Great Gatsby and, therefore, all other undefined things deny the need to be tangible objects; becoming, instead, things that exist beyond the physical realm.

Enter then the second point.

Under this Platonic perspective, we admit the objective existence of abstract objects outside of our minds. Things like mathematical concepts, musical compositions and even movements in a chess game fall into this category. But how is it that these things are abstract?

Taking a chess move as an example, it is notorious how we assume that said move exists concretely as long as a chess player uses it. But this seemingly concrete movement could be the replica of another person’s game or it could be done in the head before even being put to use on the board. Thus, these concrete possibilities give rise to a metaphysical problem:

If there is only one concrete movement, then what is that movement? And what does that make the other movements? Copies of the concrete movement or instances of the same movement?

If we think about it in a nominal way, we would certainly get in trouble. But if we consider the Platonic notion, our luck changes a bit. We would say, as such, that a chess movement is an abstract object and that each physical version of that movement is a concrete iteration of itself.

In other words, none of the concrete physical movements are actually the movement; there is only one movement and it is an abstract object, while every physical movement is just an imitation. This is directly tied with Plato’s theory of forms, where it is argued that there are ideal forms, or perfect archetypes outside of space-time, of which objects in the real world are imperfect copies.

Artistic impression of the ideal form of beauty and its subsequent imitations.

But there is a “small” problem regarding this conception and that is its failure to deal with the way we learn about such abstract objects.

Allow me to explain.

Generally, we acquire knowledge as follows:

First, we perceive an object in the physical world through physical means.

Next, we process these perceptions in our brains, eventually working with mental representations of the object in question.

And, finally, we get the information or “knowledge”.

But an abstract object cannot be processed like this. It is not physical, so our usual way of obtaining knowledge fails. Especially for things like numbers, whose perception is not as intuitive as it seems.

So, if they really exist, how do we get knowledge 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 this is clearly an unacceptable answer to the problem with abstract objects. How do they impose their truths on us? To say that “they just do it” is not an explanation; leaving us, instead, in the dark as to how they do it.

The only way to clarify this would be to resort to magic, imitating Descartes a bit. After all, it was he who postulated that minds are substances different from brains and that they are not located in temporal space. But this forces us to think about how the mind appears in our brains in order to affect them. According to Descartes, the mind slides through the pineal gland to reach the brain. Although I’m afraid that this is not an answer; it is merely a delay at obtaining a true response. Plus, how is it that the mind, an abstract object outside of temporal space, crawls through the pineal gland and then into the brain? Descartes had no answer for this, because explaining how the non-physical interacts with the physical turns out to be widely tedious.

And this is where the nominalists again take the reins of the debate. Although their party quickly gets frustrated by the argument of indispensability.

Unconditional nominalists are often philosophers of scientific motivation and mentality. And it is said love for science what makes them have conflicts whenever they deny the existence of numbers. Especially when arguments such as the following are postulated:

Science is the best arbiter of what exists.

Therefore, if science says something exists, we must accept it.

Science is (strongly) based on mathematics.

Therefore, science admits the existence of numbers.

Therefore, numbers exist.

In other words, the indispensability of numbers to science makes them existing objects. And as nominalism tells us otherwise, we run into a paradox.

So what does all this tell us about the existence of numbers?

Well, if you follow Platonic doctrines, you will say that “yes, numbers exist”. Likewise, you will say that they posses some sort of abstract existence, distinct from that enjoyed by physical entities. However, this means that you are in the unenviable position of explaining the coherence of this type of existence and how we come to know anything about this abstract realm that lacks tangible sensibility.

On the other hand, if you are a nominalist, you will say that “no, numbers don’t exist”. However, this means that you now have the undesirable job of explaining why mathematics seems so indispensable to science. An enigma whose current responses are unsustainable [i].

What to believe then?

Well, as is often the case in philosophy, maintaining a dogmatic stance in a bilateral debate is inappropriate. It is best to admit that we really don’t know if numbers exist or not. Thereby merely rejoicing in the beauty of their applicability.

Graphic representation of Euler’s identity; an equality that is considered an example of mathematical beauty, since it shows a deep connection between the most fundamental mathematical numbers.


[i] For more information, consult the following books:

https://www.amazon.com/Science-without-Numbers-Hartry-Field/dp/0198777922

Controversial work where it is affirmed that science only seems to depend on numbers and that these are not necessary to understand scientific concepts.

https://www.amazon.com/Realism-Mathematics-Clarendon-Paperbacks-Penelope/dp/019824035X

Work that states that what is generally considered as abstract objects actually exist in some way in the physical world.

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