On Understanding the Void through Dividing by Zero
A lot of the following ideas are built on notes from a great podcast episode called Zeroworld from Radiolab. Listen here.
As you may have remembered from math class, we're taught that you can do almost anything with the medium of numbers, except divide by zero.
That’s because in math, division by zero is undefined. Here’s two simple points as to why:
Division is fundamentally defined in mathematics as the inverse of multiplication. For example, 6/3=2 because 2*3=6. However, when you try to apply this to division by zero, things fall apart. A key concept of math is that things need to be "undone", but there is no number that you can multiply by 0 to get a non-zero number, and therefore division by zero cannot be "undone" through multiplication. Therefore by allowing division by zero, it would create contradictions within mathematics. For example, if you were to define 1/0 = some number, let's say 'Y', then according to multiplication, 0*Y should be 1. But we know that anything multiplied by 0 is 0, so this breaks the basic rules of arithmetic.
When you divide a number by another number that gets closer and closer to zero, the result becomes larger and larger, approaching infinity. The problem with this is infinity is actually not a real number; it's a concept. Dividing by zero doesn't produce a definable number but rather leads towards an infinite limit, which is not a specific, finite value.
For example:
1/1 = 1
1/0.1 = 10
1/0.01 = 100
1/0.001 = 1000
and so forth
Therefore, as we divide something with numbers that approach zero, the result approaches infinity.
This relationship between zero and infinity can give us insights into another reason why division by zero is “not allowed” - when we divide something by zero, we’re actually getting an answer that’s not a number. Remember, infinity is a spatial concept. Maybe, then, division by zero is actually a portal into understanding their inextricably linked qualities.
Zero & Infinity
As Charles Seife states, "Zero and Infinity are two sides of the same coin--equal and opposite, yin and yang, equally powerful adversaries at either end of the realm of numbers". The essence of zero cannot be fully appreciated without its counterpart, infinity, and vice versa. Their interdependence is such that the presence of one suggests the existence of the other.
Zero represents a numerical foundation, a point of origin. In mathematics, it's the central point in the number line, demarcating the positive from the negative. As a number, zero signifies nullity or the absence of quantity, which is essential in arithmetic and algebra for maintaining the integrity of the number system.
Infinity on the other hand, is often perceived as a spatial concept. It represents boundlessness or unending extension. In geometry and calculus, infinity is used to describe endless lines, curves, or surfaces. It's not a number in the conventional sense but rather a state or quality of being limitless. In math, infinity is used to describe the behaviour of functions as they stretch out towards endlessness. In physics, infinity is used to describe concepts like infinitely small scales (quantum mechanics) or infinitely large expanses (cosmology). Infinity's relationship with zero in this sense is clear, where operations involving limits explore how functions behave as they approach zero (infinitesimally small) or stretch towards infinity.
A math concept that discovered an interesting relationship between zero and infinity is the Riemann Sphere. Imagine a sphere sitting in a 3D plane, where the south pole represents the origin of 0 and the north pole symbolizes infinity. This sphere is studied to visually perceive changes in the sphere’s shape, such as rotations and deformations, which takes place when you apply mathematical operations to it like multiplication and division.
This spherical model opens up avenues for transforming and manipulating mathematical concepts. For example, by changing the way we rotate the sphere – say, by using the formula (x-1)/(x+1) for multiplication – we can shift the poles to the equator, offering a fresh perspective on zero and infinity’s relationship. One of the most intriguing transformations occurs when multiplying by the reciprocal, which inverts the sphere, transforming zero into infinity and vice versa. This beautifully illustrates the equations 1/0=∞ and 1/∞=0.
However, the effects of multiplying by infinity or zero are distinct from this reciprocal relationship. Multiplying by infinity is like inflating a balloon to its breaking point, where it bursts and scatters - a metaphor for the omnipresence of infinity in such multiplication. Conversely, multiplying by zero is akin to bursting the balloon and obliterating it. Thus, in their unique ways, both zero and infinity become agents of transformation (of destruction) to the Riemann Sphere, demonstrating their symmetrical states in the mathematical universe.
Zero to Infinity & Beyond
Let’s conceptualize zero and infinity in a new way: through the Fibonacci sequence. The Fibonacci sequence can serve as a bridge, illustrating how a series can grow from zero (a definite number) towards increasingly larger numbers, extending towards infinity in a conceptual sense. This reflects the progression from a finite state to an infinite one.
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. That is, the sequence goes
0, 1, 1, 2, 3, 5, 8, 13, 21, 34… (and so forth)
Philosophically, starting the Fibonacci sequence with zero can be seen as a microcosm of the universe itself; everything starting from the Void. This is interesting considering the Fibonacci sequence is associated with visual patterns of the natural world and growth processes, like the arrangement of leaves on a stem or the spiral patterns of shells (more on this later). Starting with zero, a symbol of emptiness, and progressing to larger and larger numbers, infinity mirrors the concept of natural growth and complexity emerging from the Void.
We’ll bring this full circle, but first let’s discuss…
Space & Time… Spacetime
In traditional physics, space and time were viewed as separate entities. Space was considered as a three-dimensional arena in which objects could be positioned and moved, and time was seen as a completely separate, one-dimensional continuum in which events occur sequentially.
Einstein's theory of relativity changed everything; he introduced that space and time are not separate, but rather are intricately linked in a four-dimensional continuum. This four-dimensional continuum is what we call "Spacetime."
In Spacetime, the three dimensions of space (length, width, and height) are combined with the one dimension (a single point) of time to create a single, unified entity. This means that when we talk about the position of an object, we must consider not just its location in space but also its position in time.
What if this four dimensional fabric of Spacetime was composed of the symmetrical states between zero and infinity?
Space: endless, unending extension
Space = ∞
Time, a single dimension all happening at once
Time = 0
In this case, ∞ represents the concept of material space, and 0 represents the concept of time. They’re two perspectives on the same thing; two sides of the same Spacetime coin.
As we know, zero is nothingness. It’s not really anything; it’s value null. In this case, 0 is time: the undefined present moment. It is the fleeting “here and now”, which from our perspective seems to only be able to continuously slip away toward the future. Time (as we experience it) is like the sequence of pages in the book. We feel it moving from one page to another, creating a story or a flow. But, just like zero, each moment in time by itself is kind of empty or still. Only the present moment exists between what feels like a perpetual space of what just came before it and what is about to come. In this analogy, each moment of time is like the number zero. By itself, it's just a point or a pause — it doesn't move or change. But when we string these moments together, like counting numbers in sequence, we get the illusion of linear time.
Infinity, on the other hand, is not really a number but a spatial concept. Space naturally requires division in order to be perceived. Without division, there would be no magnitude of anything. You could think of the literal definition of infinity as “without limits”: we require limitations in order to measure space. For example, “units” is a set of defined measurements that we use to measure other defined things. If you were to think about your height, it would have no meaning unless there was another thing relative to compare it to.
For example, let’s consider a typical statement of
I am 6 feet tall.
The unit of measurement in this statement is the ‘foot’. The foot, or any unit of measurement, is required to measure yourself against or else the statement is meaningless. For example:
I am tall.
This statement immediately makes us think: ‘tall relative to what?’. Are you tall relative to your mom, your friend, a pencil, a basketball player?
Another way to think about this concept is through the Void itself. If you were simply existing in a black, infinite void, you would have no way to perceive your size. You would understand the proportions of your individual components (for example, your hand size relative to your foot size), but without another thing to compare yourself to, you could be the size of an ant or the size of a mountain. Without measurements or units, or something else that is not you, scale does not exist.
Through this analogy, we can comprehend that relationships between things in space require division in order for it to be perceived as something other than infinitely vast and eternal. Therefore, if we exist in a universe that is fundamentally the intertwining of nothingness and endlessness, the only way it could be perceived would be to create limitations.
Let’s think about division as a medium itself. If we want to understand more about the implications of why we can’t divide by zero, we should understand what we are trying to do when we divide things in the first place. Defining division in a philosophical sense requires us to step back from the strict arithmetic interpretation and consider the underlying concepts and intentions that division embodies. Division represents a fundamental aspect of human cognition and our interaction with the world: it is a method of understanding proportion and relationship between entities. When we divide one number by another, we are essentially exploring how these entities relate to each other in terms of scale, quantity, or magnitude. This process is akin to seeking an understanding of the balance or imbalance, equality or inequality, in the distribution or allocation of resources, qualities, or properties. At a basic level, division represents the idea of sharing or distributing something into parts. This concept is deeply rooted in human social and economic interactions, reflecting the principles of fairness, equity, and the division of labor or resources. When we divide, we are essentially
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