This article intends
to illustrate how mathematics can become a gateway of understanding. For
simplicity, I would concentrate on the
idea of distance and the effects of adding or removing the relevant physical
dimensions on the understanding of it.
Let’s consider the Distanceformula from 1 to 4 dimensions as follows:
Since human
beings are basically living in a 3D world, the physical distance (d) between 2
points (x₁, y₁, z₁) and (x₂, y₂, z₂) is usually given by
If we try to
extend the foregoing formula by adding 1 more dimension to make it 4D, then we
may arrive at a similar distance formula as follows:
4D formula
The difficulty
here is how can we understand the
additional dimension? As 3D persons, it’s difficult to think of any
realistic 4th dimension physically as we could NOT think of a
dimensional axis (W) in addition to the existing X, Y and Z axes. As an
illustration, it is relatively easier to downgrade a 3D to a 2D situation. We
simply drop the Z axis to form the X-Y plane. Accordingly, the distance simply
becomes

Similarly, we can
downgrade a 2D to 1D and got
Most important, we don’t have a problem of visualizing the said distances
as straight lines in their respective dimensions (i.e. either 2D or 1D).
However, we could only somehow presume
that in a 4D situation the distance is still a STRAIGHT LINE. Yet, how such a 4D straight line exists would be
an enigma as it CANNOT be another 3D straight line in our world!
We often heard
that we are living in a 4D world
comprising space and time where time is usually referred to as the 4th
dimension. Yet, can we simply replace w
by t in the above 4D formula? The answer is “NO” because it doesn’t
make any realistic sense in the physical world.
Indeed, time is
an element of speed which is the rate of change of distance over time. As such,
we don’t usually think of time as another “solid” dimension. As this article is
aimed at illustrating how mathematics can become a gateway of understanding, it
suffices to note that one can move forward, backward, etc… freely in space. In
contrast, time flows in only one
direction (the arrow of time) always pointing towards the future. Besides,
relativity theories predict that gravity and speed cause time dilation and
length contraction. Besides, time cannot become a standard spatial axis. Hence,
time could be considered as imaginary or subjective.
Points to ponder
Can we formulate a
distance formula without considering its dimensions? The answer is NO as illustrated
above. Strictly speaking, it’s a complicated matter when one try to understand
distance in the various dimensions. For example, a 3D space is different from a
2D space. 2D physical objects do not exist in our universe because all real matter
requires a minimum of 3 spatial dimensions and volume to occupy space. 1D or 2D concepts are in fact mathematical concepts to help build mental
models or simplifications to understand and calculate complex 3D reality by
ignoring the depth dimension (say).
When we are
watching a shadow play, the figures projected on the screen could move up and
down or left or right on the screen. The movements are controlled by the
puppeteers behind the screen. Hence, the
defining movements happen behind the screen in a 3D space.
Even though we
may imagine that some kind of 2D persons did exist in the screen’s 2D plane, we
have to admit that they cannot move as freely and intentionally as watched in
shadow play. For instance, if these 2D persons
are presumed to be “solid” in their own right, person A on the screen have to
move around person B on the screen without detaching itself from the screen
(as the puppeteers do otherwise) in
order to overcome B, say. Similarly, if there is a solid ground on the
screen, a group of persons thereon could only move in a lineup order. Any one in the said line could only “jump”
or “fly” over the others in order to change position supposing that there
is an atmosphere above the ground.
I hope you can grasp the
complexity of the movement as discussed in the previous paragraph in the various
dimensions when we try to add or remove the dimensions of space in question.
Let me give one more example here. Let’s imagine
that we are 1D entities with a similar rationality as who we truly are. How
could we make an extrapolation to a 2D world? As 1D entities, we must assume that we have no experience or perception
of a physical 2D world. We can also suppose that our 1D world is solid and
sufficient for our activities and there exist NO transcendental capability for
us to transcend our bodies to a different world (or universe). Thus, we could only imagine mentally that we can
add another dimension to our existing dimension. The question is how? One way to do it is by supposing that there
are an infinite numbers of 1D (linear) worlds (universes) besides our particular
one. The figure is produced by AI Gemini

Also, these 1D worlds could never touch or connect with each other (otherwise we can easily move into
a 2D world consequently). However, they
can be packed so closely together to form a 2D plane in our imagination which can
only be a vague and loosely defined concept by us as 1D entities.
Coming back to
our 3D reality, we could imagine more or less similarly when we try to expand
our 3D to a 4D world while time is unchanged. Then, one can either postulate
some kind of mathematical concept to formulate the 4D ideas (distance, say) or
make up funny but unreal ideas based on our 3D experience and perception.
While we can
understand a physical object or an idea (especially a scientific one) through
the gateway of mathematics, we can hardly exhaust all the possibilities and implications
in the mathematical process of adding or removing dimensions in our
investigation and pondering. Obviously, one can understand our world from various
disciplines, faculties or reasonable spectacles available to us. I believe that
such avenues for understanding are valuable and meaningful in their respective
field and vision. Perhaps, each one of us would gain a beneficial understanding
thereby. However, it would be difficult if not impossible for us to have a complete
and all-compassing understanding of a subject in our individual capacity.
Takeaway
1. Math is a useful
tool aiding our understanding of physical phenomena and concepts.
2. Math does has its
own limitation when applied typologically to the formation of ideas or notions.
3. Our understanding
would become more realistic and reliable when we know the limits of our tool, e.g.
math.
4. Our understanding
can be enhanced when we can delineate the context of our investigation into the
physical, philosophical and spiritual dimensions.
5. The uses of our
understanding should also be properly delineated.