Numbers
I have to write about my thoughts recently on numbers. As a disclaimer, I have very little understanding of what I’m about to discuss, and am writing this as an outlet for my thoughts and interests.
I had come across the perspective that numbers are the only thing, known to humans, that declare what they are . This is to say, We give names to various living and nonliving things, but numbers in and of themselves explain what they are. For example, the number 1 does not exist, it simply is an abstraction that helps us understand the world around us. Even certain Mathematical equations are not real in the sense that it has a self-declarative form.
British Mathematician G.H. Hardy said this about prime numbers.“317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is, because mathematical reality is built that way”
Prime numbers and cryptography
In a popularly used form of cryptography known as RSA encryption, prime numbers are what’s used to encrypt data. A given explanation for RSA is the equation n = p * q such that n is an RSA number where p and q are prime numbers. This method is referred to as prime number factorization
Finding a prime is especially difficult because there are no numbers prior to it that are factors of it. For example 9 = 3 * 3. We can find 9 be multiplying 2 numbers lower than it. 11 is a prime number and we can’t conclude it is prime unless we try factoring it out. Now imagine having two large prime numbers, p and q, and multiplying them together. Using this method, it becomes hard to find the values of p and q for n due to the nature of finding these prime factors by testing all numbers below it. Finding the prime number factorization of n becomes challenging, even for computers when counting up to higher numbers. RSA cryptography is essentially this.
Universal language
Numbers are, in this way, the only universal method of communication.
One scenario I find interesting would be if an Alien species could communicate with us, I don’t think they would want to know aspects of our cultures, religions, politics, or maybe even math and sciences. Something I could see them asking is, “What is the largest prime number humanity has found”. This means something to them because it is universally proven across the barriers present with an alien species. It also implies humanity has found an effective way of finding primes, which require extensive work, and therefore advanced technological ability.
Another thought exercise I have done is the following. If an individual was born outside of earthly social constructs such as language, math, science, culture, religion, there would be no practical way of communicating with them. I struggle to think of an effective way to communicate with this individual outside of observing the natural world and drawing connections to show them.
One example I can think of would be to hold a branch in my hand. This person will not understand the words I use to describe it, what I refer to it as, or any of its scientifically makeup. However I can demonstrate that there is a singular branch by laying it down by itself on the ground, grabbing 2 more branches and lying them next to the singular branch. I can do this up to any hypothetical number. Let’s say 5. Now I have 5 piles of branches on the ground each with 1, 2, 3, 4, 5 branches respectfully. The term “Monad”, taken from ancient greek, refers to the philosophical concept of “oneness” or “unity”. It can represent the most basic building block of existence, or what we think of as 1.
Given that the individual has the same rational mind as me, they would most likely start to comprehend what is being said here. Roman numerals, interestingly, are easier to demonstrate this concept. The Arabic numbers (0-9), and the base 10 system, are more common to us nowadays. Roman numerals are represented as I, II, III, IV, V. The concept of monads and adding 1 to the previous number is evident here with the numbers 1-3. Much more can be said, but through this process, we can hope to relay more information that doesn’t require any prior knowledge that this individual is dependent on having.
Zero
The number 0 was discovered/invented. 0 is more of a concept than a number however, due to the way it behaves with other numbers. Adding and subtracting other numbers by 0 does nothing. Multiplying any number by it results in 0. Dividing by 0 is mathematically impossible and dividing 0 by any number is still 0. Regardless of these problems, using 0 helps mathematicians, especially when it comes to calculus and the concept of limits.
People in Ancient Greece had complex understanding of math and figures like Pythagoras, Euclid, and others, made notable contributions. Nonetheless, Ancient Greece refused to accept that 0 could be real, due to various cultural norms and the ways they viewed the natural world. My instinctual reaction and understanding is that I think I agree with this perspective. I understand that many modern tools and measurements would not be possible without a concept of 0, such as calculus and computers. I still think we should be careful when assuming it is “real”.
Zeno’s paradox, when attempted to be solved through purely mathematics is impossible. The concept of 0 and limits allows for an explanation of the problem. I recommend reading more in-depth explanations of this topic. I, for one, am not satisfied with this answer. If we can’t solve this with basic math, then fabricating formulas or rules to explain what doesn’t work. Like I said before, I understand the practical application of these concepts, and how they actually help improve our understanding of the physical world. It just doesn’t sit right with me and I think it might be harming our understanding of math and numbers by assuming that because it works that it is correct.
Our Future Understanding of Numbers
My personal theory is that, if we are to develop into a more complex society and hopefully understand the universe, we might need to rely on numbers. Numbers, seem to be fundamentally tied to the universe itself, or at least the way we, as humans, understand it. Are there possibly secrets hidden in numbers that can explain things we already know about reality, but in a more elegant and simple manner?
I believe Mathematical concepts should be beautiful, and through my experiences, I have had mixed feelings. The mathematical constant, Pi, has always fascinated me, along with Euclidean geometry. Other concepts introduced in Calculus, that are incredibly practical for our understanding of continuous change, seem to me to be overly complex albeit useful. What do I know though? This could be a naive take by me, and I would love to be proven wrong and shown material to help make it more understandable.
To end, I also want to comment on my curiosity around prime numbers, the limits that they go to, and whether or not there is more to them than we are currently aware.