Mathematical Sonnets (Immersive)

SONNET I

Reversibility

        When in the throes of vigorous spring cleaning
        I find myself permuting furniture,
        the conservation laws acquire meaning
        other mathematicians would abjure.
      

        My wife and I have spent ten years together
        and, entropy be damned, our energy
        persists in ways not even Emmy Noether
        saw when characterizing symmetry.
      

        Touching my chin, I come to understand
        the purpose in the angle of her frown—
        a silent plea for me to lend a hand
        and turn the hanging mirror upside down.
      

I tell myself, while humoring her whim: Mirror rim, sides reversed, is mirror rim.

SONNET II

Lost and Found

        I run a service for abandoned hearts
        within the basement of a shuttered house
        to which rich would-be clients come to browse
        in hopes of finding perfect counterparts.
      

        When one points out a sculpted beau or belle,
        I step into the back room with my sword;
        then, five cuts later, when the clavichord
        delivers its refrain, I cast the spell.
      

        The models speak and bow and fall in line
        looking exactly as the clients want:
        they greet; they yearn; they tease; they even taunt;
        they don't betray the flaws in their design.
      

For every earning, manic, lonesome body, forever yearning man, I clone somebody.

SONNET III

Heirloom

I turn with terror and horror from this
lamentable plague of functions which
have no derivatives. — Charles Hermite

        However hard you try, you must admit
        there isn't any moment of the day
        at which you could predict what I would say
        or do the very moment after it.
      

        You may be reading this, for instance, now,
        and come to trust you know where it will go,
        but you don't have the slightest clue, no, no,
        because a turn is coming, anyhow.
      

        So you proclaim me pathological,
        my failure to be smooth at any time
        engendering a Dantean, slopeless climb,
        a wager that would terrify Pascal.
      

However, I am legion, to be fair, while you are meager, in the sense of Baire.

SONNET IV

The Observer Effect

For every function \(f\) assigning to each real number
a countable set of real numbers, there exist \(x\) and \(y\)
such that \(x \notin f(y)\) and \(y \notin f(x)\).

        No matter how the logos preordains
        the set of nuisances each person weathers;
        no matter how affection slips its tethers
        and rusts within a tide that seldom wanes;
      

        no matter that my father and my mother
        have met each other's glances with a sigh,
        there must exist a couple, \(x\) and \(y\),
        who'd never scowl or barb at one another.
      

        Imagine \(x\) and \(y\) one day agree
        to go out on a date devoid of grief,
        and \(x\) expounds at length on his belief
        in Freiling's Axiom of Symmetry;
      

\(y\) can't quite take the measure of the claim and starts to think that \(x\) might be to blame.

SONNET V

Self-Portrait as a Cauchy Sequence

For Zhong-Jin Ruan

        However tiny epsilon may be,
        there is a moment after which I'll stay
        within an epsilon-vicinity
        of who I am that moment of the day.
      

        I'll waver just as though I were a string
        someone had plucked to hear the note it made,
        a wanderer in sunlight wondering:
        Who is this shadow-self to whom I fade?
      

        And when the slow unwinding dims the skies,
        my failure to converge—so bittersweet—
        will lead me to lie down and recognize
        the logos didn't make the world complete.
      

So what? I'll raise my glass and make a toast to life in search of my departed ghost.

SONNET VI

In Praise of Smaller Models

For Ward Henson

        When I was twenty-five and in the flower
        of mathematical ingenuousness,
        I read Fréchet and took my ultrapower
        to have more witnesses to my excess.
      

        I satisfied all types: each dot-dot-dot,
        no longer brush strokes of some mystery,
        resolved itself into a finished thought
        that I could find embodied within me.
      

        When my maturity arrived, I used
        Löwenheim-Skolem to slim down a bit:
        all those nonstandard objects I refused
        to keep had swollen up the soul of wit.
      

Yet \(\hat{I}\) and I are, in our discontent, elementarily equivalent.

SONNET VII

Counter-Exemplary Tranquility

Conway's base-13 function:
Write a real number \(x\) in base 13 using \(0,1,\ldots,9,A,B,C\).
If from some point onward the expansion reads
\(\ldots Ax_1x_2\ldots x_nCy_1y_2\ldots\) then \(f(x) = +x_1x_2\ldots x_n.y_1y_2\ldots\)
If instead \(\ldots Bx_1x_2\ldots x_nCy_1y_2\ldots\) then \(f(x) = -x_1x_2\ldots x_n.y_1y_2\ldots\)
Otherwise, \(f(x) = 0\).

        These fire-and-brimstone, MAGA days I go
        from numb estrangement to indignant rage,
        a Weierstrassian zig-zag-vertigo
        and dread at the news cycle of the age.
      

        Between those two extremes, one would surmise,
        there should be times when wisdom reigns supreme:
        no Felon, Pee Wee German, DOGE, or ICE;
        no damn doomscrolling and no dopey meme.
      

        The Intermediate Value Theorem pledges,
        so long as continuity is met,
        that here and there, along the blur of edges,
        there's fleeting equanimity, and yet
      

I sense a Conway-demiurge has been enciphering the days in base thirteen.

SONNET VIII

Arithmetic Ozymandias

The termination of Goodstein sequences is
unprovable in first-order Peano arithmetic.

        I met a traveler from an antique land
        who said: “I saw a sequence grow—it grew,
        its number dwarfing all the desert sand,
        but in what base it did so no one knew.
      

        Each term was vastly larger than the last,
        as if to boast of boasting even more;
        no sooner one observed its growth was fast,
        the growth became much faster than before.
      

        A puny minus one—memento mori—
        nipped at its heels while listening to it say:
        ‘My name is Goodstein, watch my golden glory!’
        Nothing of it remains. Round the decay
      

of that colossal wreck, the number zero has claimed the bones of the mysterious hero.”

SONNET IX

Recurrence

Pólya's Theorem: a simple random walk on a \(d\)-dimensional
lattice is recurrent for \(d = 1, 2\) and transient for \(d > 2\).

        Xenakis in my earbuds—Rebonds B—,
        today I grabbed a coffee on the go
        and took a random walk across the snow,
        ignoring whether ICE would question me.
      

        I traced the neighborhood's familiar ground.
        Each corner I would flip a quarter twice:
        I figured that the outcomes would suffice
        to code for left or right, straight, turn around.
      

        And as I walked, I thought about migration:
        Am I no one, who Polyphemus tried
        to trap within the cave—so says my pride—
        or just a brown dot in this whitening nation?
      

I cursed the impulse that had made me roam, but somehow Pólya's Theorem got me home.

SONNET X

Dichotomy

An arbitrary infinite-dimensional Banach space either
contains a subspace with an unconditional basis or a
hereditarily indecomposable subspace.

        Once in a while I scrutinize the sky
        and sense the logos is a Banach space,
        my little section represented by
        some basis giving structure to the place.
      

        The branches bending in the wind are there,
        as is the river, cold against my toes,
        where I skipped countless pebbles to compare
        the many forms a single motion knows.
      

        But lacking any natural sense of angle,
        this territory that I call my lot
        may be no idyll but a boundless tangle
        ensnaring even this discordant thought.
      

If so, this sonnet has no way to flee its own indecomposability.

SONNET XI

Divertimentum Ornithologicum

After Jorge Luis Borges's
Argumentum Ornithologicum

        A synchrony of wings across the sky
        is quavering in its feathered beats of flight.
        Their number is too high to count— I try
        to estimate it but I can't: the night
      

        is dark, the birds are black, my eyes are weak.
        Certainly less than \(N\) but more than \(k\),
        I tell myself, but then, in an oblique
        arrow of thought, I argue with dismay
      

        that if \(k\) is too small, then \(k + 1\)
        can't be enough, and so, inductively,
        all of God's natural numbers fail—there's none
        determining how many birds I see.
      

I start to sketch a proof He can't exist, but \(N\) being hyperfinite, I desist.

SONNET XII

Upon Inspecting the Mandelbrot Set

        I find myself absorbed by what I find
        within a hole within a hole within
        a surface I don't know how to begin
        to trace without perceiving that my mind
      

        is spiraling toward the undefined,
        its curlicues with curlicues that spin
        like pegs I turn to tune a violin
        I nonetheless can't play. I stand resigned
      

        to sink into the void, to simply sink,
        to sigh because I cannot help but sigh,
        to mean to see what's meant for me to see,
        to think the thoughts I think, or thought I'd think,
      

to know the die's been cast before I die, and be the wonder wondering to be.

Mathematical Sonnets

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