Whether it’s some sort of brain teaser, a personal observation,
a visual or sound, or a seemingly unresolvable problem within math, science, and philosophy,
every time we try to make sense of something through accurate, consistent premises and logic,
but no matter how hard we try, fail to do so, we encounter what is known as a paradox.
These strange intersections of knowledge and ignorance, absurdity and simplicity, intuitively
true yet apparently false, can not only bend the mind in harmless brain teasers and puzzles but
can also reveal hugely significant insights into the nature of human thinking and its limitations.
The word paradox originates from the Greek word, paradoxon, which translates into, “distinct from
our opinion.” Nowadays the term is often applied to a variety of things in a variety of contexts,
be it literature, math, science, philosophy, logic, economics, perception, as well as
many other looser pseudo applications. In terms of a more complete contemporary
philosophical definition, a paradox occurs when you start out with a set of premises that are
understood to be true, or are at least believed to be, you evaluate and follow the premises through
accurate, consistent logic, and then determine their conclusion. However, despite this—despite
everything else being true and in the right order—the conclusion appears false,
impossible, inconsistent, or absurd. However, despite all appearing under the same umbrella,
not all paradoxes are equal, and not all paradoxes are actually paradoxical.
In his book, The Ways of Paradox and Other Essays, philosopher and logician,
Willard Van Orman Quine, explores into the nature of paradoxes and standardizes three separate,
distinct categories—what he would classify as falsidical, veridical, and antinomy.
To help illustrate the first category, falsidical, consider one of the classic motion paradoxes
from 400 BC Greek philosopher, Zeno of Elea, known as The Arrow Paradox. The Arrow Paradox
argues that an arrow flying through the air is at rest. Zeno argues for this by claiming that
in each single, isolated instant of time that the arrow is flying, it is in one specific,
isolated position in space, unmoving. In this moment, the arrow cannot be moving to where it is,
because it is already there, nor can it be moving to where it is not, because no time has elapsed in
this isolated moment. Thus, Zeno concludes, in every one of these individual moments,
no motion is occurring; and since time is composed of a string of all these individual moments,
then no motion is ever occurring at all. Thus, the flying arrow is at rest.
At first, and at the time of Zeno, this argument might somehow sound correct. But of course,
in reality, we know that the arrows is in motion. This is what makes it initially paradoxical.
However, this particular paradox is considered falsidical, which are, according to Quine,
paradoxes that appear to be true based on a certain logical assessment within a certain state
or condition of knowledge and understanding, but said state is fallacious in some sense,
lacking the necessary insight to resolve the paradox. The paradox, however,
is capable of being resolved and made non-paradoxical.
In this particular case, Zeno’s argument is fallacious because it assumes at least one or more
false premises to be true. Firstly, it assumes that an instant of time can be divisible into a
finitely small moment or chunk that can last zero units of itself. But time cannot be made into
none-time. If time is made of units equal to zero and you multiply or divide any measure of time by
this, you would simply always have zero time. This would essentially render time inexistent,
and if we are to agree that time, in at least some sense, does exist, then this premise would be
false. Rather, time must be infinitely divisible into instants that still contain some vanishingly
minute degree of elapsing duration. Secondly, motion is improperly defined by Zeno as well.
Motion is not measured in abstract isolation in singular dimensionless moments. Rather,
motion necessitates that an object is in different places at different times. Zeno’s arrow perhaps
brings up some metaphysical questions about time, but in terms of physics, with the right sufficient
information considered from the correct angle, the paradox is understood to be false, or falsidical.
Quine’s next category, veridical, are paradoxes that also appear false at first,
but in this case, not because they are made by false premises or faulty logic, but rather,
the premises are in fact true, the logic used is in fact correct, and the conclusion is in fact
also true. It only appears false, however, because its truth runs counter to our initial intuitions.
This can be exemplified most simply by one of Bertrand Russell’s paradoxes, The Barber Paradox.
In which, imagine a small town with only one barber. This particular barber shaves all and only
men who do not shave themselves. Initially, this seems fairly straight forward, but the paradox is,
such a barber can never exist. This might like an extreme jump or false claim at first,
but it’s true, which is what makes it veridical. When considering the following question,
it becomes clearer. Does the barber shave himself? If he does not, then he is part of the group which
he does shave, since he shaves those who do not shave themselves. But if he’s in this group, then
he does shave himself. But if does shave himself, then he cannot be the barber since the barber only
shaves those who do not shave themselves. Thus, we have an inescapable contradiction,
and no barber can ever exist in such a condition. The important part of this paradox being veridical
is not merely the inescapable contradiction, but that, taken from the position that that no
such barber can exist, the paradox simply demonstrates that an initially obvious or
simple seeming statement can be logically impossible upon further consideration.
The third and final category from Quine’s analysis is what he classified as antinomies. This category
is what we most likely think of when we think of a paradox. And arguably, this is by far the
most interesting and relevant kind. According to Quine, antinomies are paradoxes that achieve
a contradictory or absurd conclusion even by correctly applying true premises and consistent
logic. They defy all accepted laws of logic and information, and for some, don’t even appear to
be resolvable by any conceivable information at all. They create, in Quine’s words, a “crisis
in thought.” And ironically, they can come in as little as a three-to-four-word statements.
Consider the following. “This statement is false.” This four-word statement implodes all logic.
If the statement is true, then the statement is false. But if the statement is false,
then the statement is true. For the statement’s claim that it is false to be true, necessitates
it being false and vice versa, ad Infinitum. There is no way out, no rope of logic to escape with.
Another similar example includes the statement, "There is no truth.” Likewise, this statement
is inescapably self-contradictory. If there is no truth, then this statement itself cannot be true,
and therefore, must be false. But if the statement is false,
it must be true that there is no truth, and the statement is true. But if the statement is true,
there must be no truth, and again, it must be false. And then repeat, over and over.
Other antimony paradoxes that are a bit more involved and seem to imply different sorts of
gaps between their conclusion and reality include problems like The Fermi Paradox, which reveals the
contradiction between the strong logic that there are exceptionally high odds for there being other,
advanced extraterrestrial life scattered throughout the galaxy, and there being no signs
of any. Also, The Faint Young Sun Paradox, which reveals the contradiction between astrophysical
understandings that the output of Earth’s sun would have only been about 70 percent as intense
during Earth’s early history, while simultaneously evidence shows that Earth had the presence of
liquid water and life during this same time with relatively consistent temperatures, which would be
impossible if the sun’s output was that low. One could also go so far to include problems like the
entire origins of the cosmos—often summarize by the question, “Why is there something
rather than nothing?” As in, how did something apparently come from nothing, or how has something
always been? This appears absolutely inconceivable to answer in any way that makes any sense. Also,
the very nature of our personal, subjective experience as self-conscious beings, at the very
least, has strong traces of an antinomy paradox. Summarized by the hard problem of consciousness,
how does physical activity in the brain equate to a subjective, felt state of being? Arguably, there
appears to be no concept or evidence that could ever even really answer how this is possible.
Antinomous paradoxes of this broader form seem to exist at the very foundation of our being—from
the origins of the cosmos all the way down into the very core of our consciousness.
Existence, as a whole, is perhaps the greatest unresolvable paradox.
Throughout different periods of history and for different people, like Zeno and his cohorts in
400 BC, paradoxes that were once believed with conviction to be antimonies have since become
falsidical with the advent of new knowledge and understanding. We now know how and why Zeno’s math
and arguments were wrong in his motion paradoxes. And so, it is fair to assume that this will
also occur for some portion of what appear to be antinomies today, sometime in the future. However,
arguably, it is overzealously romantic to assume that all or even most will. And more yet, will
not other, new paradoxes emerge in time alongside old ones being resolved, creating an endless loop,
never really allowing for the resolution of man’s intrinsic relationship with paradox?
Ultimately, no paradox is ever resolved without some cost. Something must be broken. Our current
knowledge must reveal faulty. Or our ability to ever really think consistently and accurately
about reality must reveal incompetent. In cases like, “This statement is false”,
what information could possibly be missing that would resolve this paradox?
Or in the cases of perceptual paradoxes and so on? Perhaps some paradoxes are more like shorts
in the wiring of our brain and our language rather than some missing information. “What
we cannot speak about we must pass over in silence,” wrote Ludwig Wittgenstein.
Although the human mind is the most powerful and impressive thing that it itself is aware of,
it can also only visually perceive about 0.0035 percent of the electromagnetic spectrum,
can only hear between 20 Hz to 20 kHz of all possible audio frequencies, and can only know
what it thinks it knows. Paradoxes, at least of a certain kind, perhaps situate themselves at the
outer perimeter of our mind’s abilities, revealing to us what we cannot comprehend within the limits
of the comprehensible. On these paradoxes, we seemingly stand on the cliff of our understanding,
knowing where we are while looking out at a vast, looming valley covered in fog.
Paradoxes challenge us to continually explore this valley, to fill it in with new checkpoints
and create new maps of understanding. But perhaps there are simply unresolvable paradoxes within
this valley—areas we can likely just never see or discover. Or perhaps antimonies don’t even
really exist at all, and all paradoxes are merely created by human fallacy
that can be resolved. Or perhaps the opposite, all things are fundamentally paradoxical,
and all resolutions are contrived by false human meaning. Whatever the case may be,
the valley is likely far too big, and we are likely far too small to ever know for sure.
Ultimately, the fact that any amount of our existence can make any sense, and that any
amount of paradoxes, past or present, have and can be resolved by us, is a paradox in of itself.