Do Numbers Really Exist Out There?

Do numbers really exist out there or are they just figments of our imagination? This is a question that philosophers of mathematics have pondered upon for ages. When one associates oneself with mathematics long enough or is curious enough, one eventually stumbles upon this question.

When I stumbled upon this question myself, I developed my own model of reasoning (more on that later). At the same time, I was naturally curious about what other mathematicians thought about the topic.

What I learnt was a fascinating philosophical discussion involving three different schools of thought:

1. Platonism

2. Nominalism

3. Fictionalism

In this essay, I will present perspectives from each of these styles of reasoning. Finally, I will present my own mental model of this topic as well.

This essay is supported by Generatebg

Platonism on the Existence of Numbers

Platonism stems from the ancient Greek philosopher Plato. The Platonists (people who subscribe to Platonism) believe that numbers do indeed really exist.

According to them, the notion of numbers is as real as any other object, such as a book or a ball. The only catch is that they define numbers are abstract objects.

The Platonist numbers exist outside of time and space. Therefore, they do not have any direct causal effect on our materialistic world as such. But at the same time, when a mathematician uses a mathematically true concept, Platonists argue that the concept cannot be true if it did not exist.

For example, if I told you that you have 3 glasses, I cannot use ‘3’ as a description if the number did not exist in our world in some way; in this case, it supposedly exists in an abstract manner.

So, in essence, the Platonists believe that numbers do really exist. But they exist beyond the realms of time and space. Although this line of reasoning is helpful on numerous fronts, it comes with its own set of disadvantages.

Challenges with Mathematical Platonism

Firstly, what does outside of time and space mean? Depending on how one chooses to interpret that statement, it could mean a whole bunch of different things.

If numbers really are abstract objects outside of time and space, how do mathematically true concepts or mathematicians, for that matter, access these abstract objects ever so precisely and reliably? What is the common dimension that links our world and this abstract world of numbers?

The Platonists typically have challenges answering that question. This brings us directly to a school of that overcomes this challenge.

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Nominalism — Do Numbers Really Exist Out there?

Like the Platonists, the Nominalists believe that numbers really do exist. The point where they diverge from the Platonists is at the definition of mathematical numbers.

You see, the Nominalists strictly define mathematical numbers as claims about objects in the real world. For example, the statement “You have 3 glasses” uses the number ‘3’ to refer to glasses. Without the real-world link, 3 loses its meaning and purpose, according to Nominalists.

If you think about it, this is the line of reasoning that most of use employ when we learn numbers for the first time as children. Personally, I was interested in employing numbers to count the number of candies that I could buy with money.

I did not intuitively think about or care about numbers as abstract objects as a child. So, the biggest selling point of Nominalism is that it manages to establish an intuitive link between the concept of numbers and real-world objects; something Platonism seemingly failed at.

Although very practical, this school of thought is not immune to challenges either.

Challenges with Mathematical Nominalism

If numbers exist only if they can be intuitively linked to real-world objects, how do we make sense of higher order mathematical abstractions such as imaginary numbers. Is √(-1) not a number? The Nominalists struggle to answer this question, whereas the Platonists are quite happy to embrace complex numbers as abstract objects outside of space and time.

This issue extends beyond just numbers though. Think about the act of multiplying two negative numbers. The result is a positive number. Any practitioner of mathematics would call you crazy if you tell her that such an operation cannot exist.

But a Nominalist would struggle to justify the existence of such an operation purely because of its difficult/impossible link with the real world. Think about it. When we say (3*2 = 6), we could link it to the act of counting three sets of two candies.

However, how would you link (-5 * -3 = 15) to a real-world application? If you look outside the world of man-made abstractions such as banking, a nominalist link seems very challenging.

Fictionalism on the Existence of Numbers

The Fictionalists are a unique bunch. They believe that numbers do not exist. In fact, they claim that the entire business of mathematical discourse is an illusion and is false.

But mathematicians and scientists employ mathematical discourse to construct real-world applications such as computers and buildings all the time. How can mathematical discourse be false and an illusion if we are able to build such applications?

Well, the Fictionalist argues that the applications are only proof of the usefulness of mathematical discourse, and do not prove its truth. Of all the three schools of thought that we’ve seen, this is the most extreme one.

The Fictionalists term the sentence “7 is prime” as false just as they term “the tooth fairy is prime” as false. In their view, both ‘7’ and the tooth fairy do not exist. In essence, the Fictionalists believe that there is no such thing as an abstract object.

Needless to say, an extreme view such as this one leads to numerous controversies. For the sake of brevity, I’ll spare that discussion in this essay. For now, let us recollect what we have covered so far.

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Do Numbers Really Exist Out there? — Summary

Here is a quick summary of the three schools of thought we have covered so far:

1. The Platonists believe that numbers do exist, but are abstract objects that exist outside the realms of time and space.

2. The Nominalists also believe that numbers exist, but for them, numbers are claims about real-world objects.

3. The Fictionalists believe that numbers do not exist. They also believe that there is no such thing as an abstract object and think that mathematical discourse is false.

Author’s Comments

As far as I could research, philosophers often use statements made by famous and influential mathematicians as arguments for or against a particular school of thought. But these mathematicians never claimed (as far as I could tell) that they belong squarely to one camp or the other.

I find myself torn between Nominalism and Platonism. I love applied mathematics. So, the Nominalist style of reasoning comes naturally to me.

However, I am also an abstract thinker by nature, which gives me access to the Platonist point of view as well. This also enables me to tackle the disadvantages of the Nominalist school of thought.

Either way, I firmly believe that numbers do really exist both in the real as well as the abstract world. Do you think that numbers really exist? And what is your line of reasoning around this topic?

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Reference and credit: Jonathan Tallant.

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Further reading that might interest you:

• How To Casually Guess Numbers After Dice Throws?
• The New Industrial Revolution Is Here.
• The Hindsight Bias: Cause And Effect.

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