In the season one finale of the TV show ‘The Good Place’, one of the main characters, Eleanor, realizes that she and her friends, contrary to what they had previously been told, are in fact trapped in the bad place. After being told at the beginning that they are in the good place, they struggled really hard to shape everything in their reality to fit that narrative.
A preconceived assumption that had been left unquestioned for a really long time that leads to less-than-accurate conclusions. This can be observed in various fields of thought, including mathematics.
Mathematics is famously known for its absoluteness/ objectiveness. However, in this article, I’m going to prove that it is not so. I’m going to prove mathematics is subjective to the observer – that it has no absolute right or wrong answers – by taking the simplest mathematical equation there is, which is 1+1=2, and proving it wrong. And just like in the NBC TV series, I’m going to do that by questioning a preconceived assumption that has existed since the origin of mathematics.
By the way, I openly invite anyone reading this article to prove me wrong. I have tried to do it myself many times, but failed.
Does 1 = 1 ?
We will approach 1 + 1 = 2 from the left side of the equation.
1 + 1
1+1 denotes two of something (the same thing) being ‘added’ together. We will get into what it means to ‘add’ something to another in a separate article.
What I want to focus on is the implication that the first 1 is equal to the second 1, because it is here that our problem lies.
1 = 1
For starters, in the above statement, the two 1’s have different locations on your screen. And if they were written on paper by hand, then they would have different shapes as well. However, we agree to ignore those characteristics and take them for the symbols that stand for the meanings behind them.
1 here means one of a thing — something, anything. The next 1 too means one of the same thing. And we assume those two things are ‘equal’. We assume 1 = 1.
And here lies our problem.
There have never existed two things in reality which are (absolutely) ‘equal’, nor will such ever exist. Let me explain.
The most common example we take to teach 1 + 1 = 2 to children is oranges. You take one orange, another orange, put them together, and you have two oranges. When we perform this function, we assume both oranges to be equal/ identical.
However, I would argue that no two oranges have ever been identical, nor will they ever be.
Take these two oranges for an example. They are clearly different. For starters, one is on a grey background, the other is on a white one. And you can see that all the other factors too, like shape, color, size, orientation etc are different.
Now, someone could say, it doesn’t matter the minute differences – you can ignore them – both of these things are basically the same – they are inherently oranges. And I would agree. That is how I have added oranges when I buy them at the supermarket all of my life. However, the definitions of the word ‘basically’ is arbitrary/ subjective. It depends on how much you are willing to ignore/disregard to validate the existence of an inherent form which is common to both objects which you consider to define them.
In other words, the things I consider as insignificant enough to neglect, another person could consider as too significant to do so, and vice versa. It’s a slippery slope, because once you allow one thing to be ignored, then you have to allow another. How do you decide where to draw the line?
If someone claims these two oranges are equal, on what basis do you disagree with them? And if you agree with them, on what basis do you disagree with the next person turns up with an orange and an apple claiming they are basically the same, hence equal? You really cannot draw an objective line.
Naturally, at this point, the urge is to leave our everyday ‘practical’ reality and jump into the extremes that exist theoretically. I too have done that.
Let’s go straight from complex oranges to the most basic objects known to us. Quantum objects/ particles.
Consider two electrons. Now immediately, all those concerns regarding differences of shape, size, color, texture etc fly out the window. Except for one.
Location.
There exist no two particles in our reality which have identical location property. The only particle which holds identical location property (to another) is that particle itself (essentially making 1 + 1 = 1; imagine your kindergarten teacher pointing to the same orange twice to explain 1 + 1, because hey, the two 1’s have to be equal right?). Therefore, there can exist no two things/ objects in our reality which are absolutely/ perfectly equal to each other.
Just like for the two oranges at the beginning, the moment you and me agree to ignore one difference (albeit extremely small) between the two particles to consider them as basically equal, we have to respect the next person who considers a slightly bigger difference as neglectable, until we agree to the person who compares a grain of sand to the Eiffel Tower and considers them basically the same, because there is no objective point where we can draw the line.
The assumption of the existence of an absolute equality between two things is at the core of mathematics, and is flawed.
Equal Sign
The implied meaning of the equal sign is an objective equality independent of any observer between the two sides of an equation. However, as I have shown above, such cannot exist.
Identically Equal (≡)
The existence of a sign to denote an equality that is identical dictates that the equal sign (=) is used to denote an equality between two entities where only some aspects are considered (some are ignored).
The acknowledgment of the existence of an equality beyond what is denoted by the equal sign automatically commands that the sign stands for an equality that is less than perfect. Almost equal, but never exactly.
Further, since there can never exist two things which are objectively/ identically equal, this sign too can only mean a less than perfect equality at most. Hence, the distinction between the two signs (=) and (≡) is arbitrary. No objective line can be drawn between where the equality indicated by (=) stops and where that denoted by (≡) starts. Therefore, arguably the two signs can be used interchangeably.
Approximately equal (≃)
This sign denotes a lesser equality than what the equal sign stands for. However, as we learned that objective equality doesn’t exist, and therefore the equal sign too denotes an approximate equality, what degree of difference warrants the use of one sign over the other is arbitrary. No objective line exists to make the distinction between the two.
Therefore, the sign (≃) too can be be used interchangeably with (=) and (≡).
Not Equal (≠)
For the same 1 and 1, both the statements 1 + 1 = 2 and 1 + 1 ≠ 2 are valid. Neither one of the statements are more (objectively) correct than the other. How correct any one of them is depends on the opinion/ perception of the observer — how much they are willing to neglect as insignificant to consider the two entities represented by 1 and 1 as equal.
Since no objective distinction can be made between the signs (≠) and (=), they can technically be used interchangeably, based on the observer’s preference. Accordingly, since (=) can be be used interchangeably with (≃) and (≡), we arrive at the following.
All signs (≠), (=), (≃) and (≡) can be treated as basically the same, and therefore can be used interchangeably.
Out of them, the sign (≠) holds the meaning (of inequality) which will always be technically/ indisputably true.
According to Buddhist teachings, it is only the changing nature of reality that will never change.
Sets
In set theory, a set is defined as a ‘gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set’.
The problem we faced with the oranges can be described using sets.
As soon as we define the set that contains all the particles (considering location in space as well) which can be named as ‘an orange’, it makes it impossible to call any other set without exactly the same elements as ‘an orange’ as well.
The only set that is absolutely equal to another set is that set itself.
The Problem With Analogies
An analogy is a comparison between two things. We claim one thing to be like another. However, every analogy is guilty of purposefully ignoring at least a single aspect of a thing that does not fit its narrative.
Let me explain using an analogy in the form of a somewhat offensive joke I found on the internet as an example. It’s a spin on the famous quote from the movie Forrest Gump. It starts like this.
Life is like a box of chocolates…
This is the first part of the joke. Now, let’s try to guess the rest of the line.
Is life like a box of chocolates because it…
- Is delicious?
- Has compartments inside?
- Is a result of capitalism?
- Cheap?
- Has different flavors?
These are all valid properties of a box of chocolates that may or may not work as a clever and humorous analogy to life.
Here’s how the original joke ends.
Life is like a box of chocolates, it doesn’t last as long for the obese.
As you can see, the creator of the joke makes a single statement to represent a shared aspect between both things. The ability of the end phrase to simultaneously convey the two contrasting meanings (we don’t treat the finishing of a box of chocolates and the end of a life the same) is what creates surprise and therefore humor.
In an analogy, only a chosen common aspect of two things are taken into account. Others are considered as invalid. Therefore, no analogy is absolutely perfect — no two things compared are objectively equal in all aspects.
So, just like our deduction regarding particles, the only perfect analogy that exists to a certain box of chocolates is that box of chocolates itself. However, an analogy by definition should compare two different things, therefore a perfect analogy does not exist (here, the term different itself proves that the two things cannot be absolutely similar, if there are to be two things).
Just like in the joke, when we compare two oranges and claim one to be like the other, what we are making is an analogy. Although the differences between two oranges are less apparent than those between life and a box of chocolates, they do exist.
Same But Different
A defense against the argument of no two oranges being equal would be that, in mathematical addition, only the chosen common property is considered. This is known as value — what matters. The differences are not considered — they don’t matter.
Let’s take a look at this using an example I found on the internet written by a certain math professor. I think his comment beautifully summarizes the most common explanations of 1 + 1 by academics.
“…For example, whether we are holding a pencil, a fork, a basketball, or a cat, there is a philosophical concept of counting that lets us know that we have one of each. ‘One’ does not mean basketball or cat; the number is independent of the objects themselves.
If we allow for simple definitions of one, two, three, and so on, then at that point it’s easy to see that 1 + 1 = 2. If we hold a fork and a pencil, then we have two objects. Regardless of the identity of the objects, we have two of them when we put one and one together.”
At first glance, this seems to be the perfect defense against my argument. If we don’t consider any of the differences between the two oranges, then we can take them for what they both fundamentally are — objects. It doesn’t even have to be two oranges, it can be an orange and a tennis ball… or a shoe.
So yes, it does seem like we have arrived at an objective answer. However, there lies a contradiction in it that hides in plain sight.
Let me explain.
An object can be defined as a thing that occupies space. We can consider one object, then consider another object to add to it; on one condition.
It has to be a different object.
But different how? We agreed not to take any differences into consideration — that no two objects are different; all objects are equal for the purpose of our calculation. Therefore, we cannot discuss a ‘different’ object, because we got rid of any property/ characteristic that can be used to differentiate between objects.
In other words, what we agreed upon earlier is that there is no difference between an object that occupies a certain space and another that occupies a different space in reality. The space a basketball takes up in New York is considered to be no different to the space an identical-in-size basketball takes up in Tokyo.
Therefore, the second object can be substituted with the first — the first object can act as both the objects. After all, the most similar object to the first object is the first object itself.
However then, the meaning of addition gets lost. Addition by definition requires two different (yet similar) things to work. You cannot discuss addition with just one object. Yet, when you demand a difference between the objects, aka two different objects, you acknowledge properties that are not similar between them, which you decided to disregard in the first place.
Can two objects be both similar and different at the same time? At which point do they become too different to become similar?
Time
“No man ever steps in the same river twice, for it’s not the same river and he’s not the same man.” – Heraclitus
So far in our discussion we have ignored the factor of time. However, just like location, time is a factor that indisputably debunks the absence of objective equality in our reality. Whereas the location property concerns the inequality between two objects visually distinguishable as separate, the time property concerns the ‘same’ object at different times.
Take a look at any ID card of yourself that you possess. Is the photo in it absolutely similar to your appearance? First of all, it’s two dimensional. But you are not so — your face does not look like a sheet of paper. Also, you are not just a head hovering in the air. There’s more to your body (unless you’re the guy in the movie Source Code). But the photo in your ID doesn’t show any of that. It’s a mere analogy to your appearance.
Your hair, skin and shape has changed. However, for most instances of usage of an ID card, the differences can be considered as to be insignificant enough to be neglected. But when is it too much? When does it cross the line?
Technically, every ID photo expires the moment after it’s taken. The question is, how expired is too expired? Where is the right place to draw the line? And who draws it? God?
The elements in the set of self keep changing constantly. Therefore it is not consistent. Any set defined at a certain time cannot be objectively equal to ‘itself’ at a later time.
Axioms and God
Both the concepts of axioms in mathematics and god in religion share the quality of being unable to be proven, yet being required to exist as the representation of objective truth. This should come hardly as a surprise because both are essentially approaches to understanding the same reality.
However subjective reality may be, conscious beings cannot avoid a perceived objectiveness (aka truth). The most common appearance of this is the seemingly objective line that separates the (moral) right from wrong, and of course the inescapable conviction of the continuation of oneself from one’s oldest memory to now.
Gödel’s Incompleteness Theorems
Published by Kurt Gödel in 1931, the two incompleteness theorems are concerned with the limits of provability in formal axiomatic theories.
In brief, the first theorem argues that there can be statements that are true yet cannot be proven as such. This is because there is no one objective truth. The axioms which are treated as objective truths and used to ‘prove’ other statements are only subjectively true. Therefore it is not only some statements that cannot be proven, it is all of them.
The second theorem states that the system (of mathematics) cannot demonstrate its own consistency. This is because it is not so. It is inconsistent.
Can a belief system based on the consideration of two entities as simultaneously equal and unequal to each other really be consistent? It contradicts itself, just like all truth claims in reality. It requires the simultaneous acknowledgment and negligence of properties in reality. This is of course not a flaw in mathematics, but the nature of the subjective reality we experience itself. Reality itself is hypocritical and constantly contradicting itself.
Mathematics as an Invention
There has long been a debate whether mathematics is discovered or invented.
The argument I have put forward proves that the mathematical qualities in reality are not inherent to it but rather attributed to it as per the perception of the observer. They do not exist independently/ externally.
Therefore mathematics, like reality itself, is an invention of the consciousness.
What Matters
Here is a linguistic approach to explain the argument at hand.
Consider two quantum particles which are identical in every way but physical location. Now, if we claim the miniscule space in between them as something that doesn’t matter, we are left with two identically objectively equal particles.
However, what doesn’t matter can be gotten rid of. But when we delete the space in between, we are left with only one particle.
Conclusion
The equation 1 + 1 = 2 (and 1 = 1) implies that one of something and another of the same thing can be added together to arrive at two of it. However, no two things can be considered as two of the same, without neglecting at least a single characteristic of them.
Since there is no objective level of accepted negligence, it is up to the judgment of the observer to decide if two things can be considered as equal or not. Therefore, the validity of the conception of equality which is at the core of mathematical functions, is subjective. It cannot be proven objectively.
The concept of equality by definition demands the consideration of two different objects, which automatically disqualifies them from being treated as absolutely equal.
I believe mathematics and science are extremely useful frameworks to understand our reality, including the subjective nature of it. In fact, I believe the knowledge of absence of objective truth would be the next step in understanding the nature of the reality we are living in.
Please do share with people who you think should see this. I would love to hear all kinds of feedback and counter arguments.
Take care in the matrix. Goodbye!