Prologue~
“The heart asks pleasure first,
And then, excuse from pain”
~Emily Dickinson
In life some things are always close to being perfect. An almost lingers there- half a syllable away from being the perfect thing that can be. A heartbreak then seeps in...
Similarly, in mathematics we have these structures and objects which suffer a(n almost) heartbreak! Overqualified for certain things but underqualified for the others- lingering between a here and a there... an almost there!
Verse I~ A king with no field.
We have a Ring and a king who speaks
But the subjects are civilians with unique voices, indeed.
The king demands to bow before him,
Once under 1, these civilians will have kins
but the adamant civilians refuse to bow~
And hence remain kin-less in a field of doubt.
Not a field but more than a ring;
With no inverses in sight
The king then rules a Ring so kin-less...
What an autumnal plight!
The ring here is $\mathbb{Z}$ which does not qualify for a field, simply because the multiplicative inverses do not exist. When one comes to ponder, one sympathises with the set of integers due to the lack of their kins. No hand in sight, and none to be called a friend.
There exists an identity, and two operations indeed, where one stands complete but the other- suffers defeat.
Verse II~ Exempla
We can also consider $2 \mathbb{Z}$ which is certainly a ring but lacks identity. Another almost there, another heartbreak...
For instance, the symmetric group $S_n$ which consists of all the possible permutations of $n$ elements of the underlying set. Essentially, this group is a set of maps. For $n \geq 3$, the group is non-abelian, but given any element in $S_n$, the element commutes with itself!
A set of introverts, communicating within themselves, while the world around doesn't reciprocate?
Also, consider the General Linear Group $GL_n$ which consists of all the invertible matrices of order $n \times n$ but fails to fulfil another essential condition and hence the group (again), suffers an unfortunate fate, but worse- it does not even form a ring! Since the sum of $n \times n$ invertible matrices might not be invertible.
Or perhaps, Cantor Set which is uncountable but has a zero (Lebesgue) measure! Or a Möbius strip which is locally orientable but globally not!
Verse III~ Wabi-Sabi (Imperfect, Impermanent and Incomplete)
Mathematics, much like life, reveals to us the imperfections- a mirror in the face of adversity and self retrospection. A mirror that tells you of your imperfections, which might even mock you but to look at these in the light of optimisim, where the deeper truths reveal, is an art, and an essential one–very much so.
The set of integers is not a field but that flaw reveals to us the elegnace of irreducible wholes! The rationals- incomplete but adamant, force us to look into the expansive continuum of the reals.
A very fun example of the Möbius strip- an extremely fascinating surface, in refusal of being orientable, becomes something stranger and richer- alluring people towards its charms.
Epilogue~
Perhaps the cracks and the imperfections in the structures are simply a way to let the light in.
"There is a crack in everything, that’s how the light gets in."
~ Leonard Cohen
Mathematics speaks to us of the ‘almost’s while teaching one to hover at the edge of revelation. The integers are not broken—they are unfinished. The Möbius strip is not flawed—it is unfathomable. And we, like Cauchy sequences in $\mathbb{Q}$, are always converging towards something, just beyond our grasp (for now).