My first poetry collection, Ode to Numbers, was published by Antrim House in September 2017 (http://www.antrimhousebooks.com/glaz.html). The book contains poems written over a quarter of a century and inspired by mathematics and my life as a mathematician. The poems in this folder are a small selection from the book—a series of seven poems focusing on events from the history of mathematics

Glaz, Sarah

English

Keck Graduate Institute

Journal of Humanistic Mathematics 
Volume 9 | Issue 1 January 2019 
A Selection of Poems from Ode to Numbers 
Sarah Glaz 
Department of Mathematics, University of Connecticut 
Follow this and additional works at: https://scholarship.claremont.edu/jhm 
 Part of the Arts and Humanities Commons, and the Mathematics Commons 
Recommended Citation 
Glaz, S. "A Selection of Poems from Ode to Numbers," Journal of Humanistic Mathematics, Volume 9 
Issue 1 (January 2019), pages 350-359. DOI: 10.5642/jhummath.201901.22 . Available at: 
https://scholarship.claremont.edu/jhm/vol9/iss1/22 
©2019 by the authors. This work is licensed under a Creative Commons License. 
JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and 
published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ 
The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds 
professional ethical guidelines. However the views and opinions expressed in each published manuscript belong 
exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for 
them. See https://scholarship.claremont.edu/jhm/policies.html for more information. 
Poetry Folder
♦
A Selection of Poems from Ode to Numbers
Sarah Glaz
Sarah.Glaz@uconn.edu
My first poetry collection, Ode to Numbers, was published by Antrim House in
September 2017 (http://www.antrimhousebooks.com/glaz.html). The book
contains poems written over a quarter of a century and inspired by mathemat-
ics and my life as a mathematician. The poems trace the story of my life from
early childhood in Europe to the forty years long research and teaching career in
mathematics in the United States. The poems respond, in turn, to my students
and the topics of the courses I have taught, the colleagues I have worked with
and the research projects in the area of Commutative Algebra that I have been
involved in, the joys of professional travel, the pleasures and the difficulties facing
a woman pursuing a career in mathematics, and most of all, my life-long love of
both mathematics and poetry.
Ode to Numbers is a poetry category finalist for both 2018 Next Generation Indie
Book Awards and 2018 Book Excellence Awards.
The poems in this folder are a small selection from the second chapter of the book
titled Pythagoras Plays His Lyre, a name derived from the poem by the same title,
which appeared in 2016 in this journal.1 As the title implies, the poems in this
chapter of the book were inspired by mathematical results and the history of
mathematics. The present selection, driven by history, time-travels from ancient
Mesopotamia, where numbers and letters were invented circa 4000 B.C., to post
World War II Britain, the home of the Number Theorist, G.H. Hardy. Along the
way we visit the 5th century B.C. Pythagoreans in Metapontum and Euclid in
3rd century B.C. Alexandria, we meet the 17th century cofounders of Calculus,
Newton and Leibniz, in London, Paris and Hanover; and during the 19th and early
20th centuries, we briefly encounter Cantor, Hilbert, Go¨del, and their followers
in Germany, Austria, and United States.
–Sarah Glaz (Sarah.Glaz@uconn.edu), Connecticut, USA.
1Glaz, S. “Pythagoras plays his lyre”, Journal of Humanistic Mathematics, Volume 6 Issue
1 (January 2016), page 291. Available at: https://scholarship.claremont.edu/jhm/vol6/
iss1/21, last accessed on January 25, 2019.
Journal of Humanistic Mathematics Volume 9 Number 1 (January 2019)
Numbers and Letters
The symbols of Mnemosyne
don’t get along.
Today I heard a quarrel between B and 2,
and even as we speak, this lazy eight ∞
shamelessly stretched without a limit on a limb,
pushes numbers and letters towards the outer edge.
From a to z or 0 to ∞ —
a single source does not suffice
for peaceful coexistence.
They once were born-together twins,
offspring of mother memory and father wealth,
both trained from infancy as record keepers
of the first thousand kegs of wine and beer
produced by the first thousand settlers around Ur
who, tired of forever shearing wool,
started a very lucrative and joyful business:
planting of hops, barley, and grapes,
fermenting juices,
and who, in their spare time,
sloshed and satisfied,
took to the flute and lute
and sang long hymns of thanks,
recited endless stories of nomadic pasts,
composed numerous tunes of praise:
Praised be Marduk,
Nisaba shall be praised,
revered the moon god Nana,
and powerful Inana.
And to ensure
they left tradition for the children,
in brewing, counting, and song alike,
imprinted symbols
with sharp-angled wedges
into the region’s malleable clay
and baked them in hot kilns.
In the beginning, letters and numbers
looked alike, happily shared a tablet. . .
351
The First Negative
In fact, there is no real reason why negative numbers
should be introduced at all. Nobody owned −2 books.
J.J. O’Connor & E.F. Robertson
An Overview of the History of Mathematics
Down the funnel of history
I search for you—the first man
to cup in his hands
minus one grain of fire:
“Ghost of departed quantity!
Mysterious absence!
I am afraid
I am cradling an illusion!”
Yet from my vantage point
I discern a sign
upon the still uncharted number line
where negatives
began to light the way
for abstract algebra to come into existence
several thousand years
in the future.
352
√
2 = 1.41421 . . .
We started our voyage on the gulf of Tarentum.
The sea was choppy
and the brothers were restless.
At dawn, we gathered on the deck
intent to solve the conflict like rational men.
Hippasus still refused to keep the secret.
He had discovered that
the diagonal of a square
is incommensurable
with its side.
Alas! Our world had collapsed
and so did our geometric proofs.
Too much to lose, we heaved him overboard.
Historical Note: In the 5th century B.C., the Pythagorean Hippasus of Metapontum dis-
covered the existence of irrational numbers. Particularly, he had shown that
√
2 (the length
of the diagonal of a square with a unit side) is an irrational number. For his sin, legend has
it, Hippasus was thrown overboard during a sea voyage. The poem plays with the imaginary
possibility that his murder occurred before he breached the Pythagorean code of secrecy and
made his discovery public.
The line count of the poem’s stanzas follows the decimal expansion of to five decimal places.
353
The Death of Euclid
So great was Euclid’s fame that he was known to the
Greeks as ὁ στοιχειωτής, “teacher of the Elements.”
D. E. Smith
History of Mathematics
And onto the barge decked with flowers
we lifted him our arms dark and
strong his body shrunk in its
shroud of warm roses The Nile’s
sluggish brown flow Myth floating
in front of us like the cloud of God
We lifted his eyes towards heaven
where he saw the next generation
inherit and we mourned how
we mourned his fine-tuned mind
his irreplaceable jewels of thought Not
one of us came close in our search
for truth to match his erudition
Not one of us came close And
the barge groaned under his almost
weightless body once as if in pain
then clutched his mutilated heart
to its own Crocodiles circled silently
plucking at wilted flowers And the
procession started slowly heavy
with sighs ghost deep oars into
water and heaving mud splashing
dipping and cutting a path upstream
we left on the shore rooted tangled
in tears our imaginations seeking
relief in words of reverence
Historical Note: Euclid (323 B.C.–285 B.C.) lived in Alexandria, a Greek city situated in
Egypt at the mouth of the Nile, and famous for its legendary library (housing 600,000 papyrus
rolls), the Museum (a center of leaning rivaling Plato’s Academy in Athens), and its lighthouse
(an engineering marvel considered one of the Seven Wonders of the Ancient World). Euclid
founded a school of mathematics at the Museum. Very little is known about his life beyond
the monumental achievement of writing The Elements, a book that had profound impact on
Western thought, and set the foundation for mathematical standards of reasoning for all time.
Only the Bible has been more widely reprinted and studied.
354
Calculus
I tell my students the story of Newton versus Leibniz,
the war of symbols, lasting five generations,
between The Continent and British Isles,
involving deeply hurt sensibilities
and grievous blows to national pride
on such weighty issues as publication priority
and working systems of logical notation:
whether the derivative must be denoted by a “prime,”
an apostrophe atop the right hand-corner of a function,
evaluated by Newton’s fluxions method ∆y/∆x;
or by a formal quotient of differentials dy/dx,
intimating future possibilities,
terminology that guides the mind.
The genius of both men lies in grasping simplicity
out of the swirl of ideas guarded by Chaos,
becoming channels through which her light poured clarity
on the relation binding slope of tangent line
to area of planar region lying below a curve.
The Fundamental Theorem of Calculus,
basis of modern mathematics, claims nothing more.
While Leibniz—suave, debonair, philosopher and politician,
published his proof to jubilant cheers of continental followers,
the Isles seethed, unnerved.
They knew of Newton’s secret files,
locked in deep secret drawers—
for fear of theft and from stranger paranoid delusions,
hiding an earlier version of the same result.
The battle escalated to public accusations,
charges of blatant plagiarism,
excommunication from The Royal Math. Society,
a few blackened eyes,
(no duels),
and raged long after both men were buried,
splitting Isles from Continent, barring unified progress,
till black bile drained and turbulent spirits becalmed.
Calculus—Latin for small stones,
primitive means of calculation; evolving to abaci;
later to principles of enumeration advanced by widespread use
355
of the Hindu-Arabic numeral system employed to this day,
as practiced by algebristas—barbers and bone setters in Medieval Spain;
before Calculus came the Σ (sigma) notion—
sums of infinite yet countable series;
and culminating in addition of uncountable many dimensionless line segments—
the integral
∫
—snake,
first to thirst for knowledge, at any price.
That abstract concepts, applicable in the beginning
merely to unseen, unsensed objects—orbits of distant stars—
could generate intense earthly passion
is inconceivable today,
when Mathematics is considered a dry discipline,
depleted of life sap, devoid of emotion,
alive only in convoluted brain cells of weird scientific minds.
Historical Note: The British mathematician and physicist Isaac Newton (1642–1727) and
the German mathematician and philosopher Gottfried Leibniz (1646–1716) are considered to
be the co-founders of Calculus. Their seminal achievement was the formulation and proof of
The Fundamental Theorem of Calculus—a theorem that relates two disparate concepts: the
concept of integration (used to calculate areas below a curve) and the concept of differentiation
(employed to find formulas for tangent lines to curves). Although no one had made a successful
connection between the two concepts before Newton and Leibniz, The Fundamental Theorem
of Calculus has its roots in the work of numerous other mathematicians who studied each of
these concepts separately for many years. In fact, the roots of this area of mathematics go back
several thousand years to the invention of numerals by Hindu mathematicians.
A version of this poem was first published in Humanistic Mathematics Network Journal, 2002.
356
What Can We Do If We Crave Certainty in Mathematics?
Thus, to salvage traditional mathematics, Hilbert proposed a
bold new program. It required first that the whole of existing
mathematics should be axiomatized, and second that this
axiomatic theory should then be proven consistent.
David M. Burton
The History of Mathematics
Hilbert said, “No one shall expel us from
the paradise that Cantor has created.” And we believed,
like Cantor himself, that God had opened the gates
to the forbidden garden,
had invited us to enter,
meet ℵ (aleph) face to face,
converse in the language of sets,
admire the ascent of transfinite cardinals
into an infinitude of infinities.
No one can possess such knowledge and remain unscathed.
We lost our footing; doubts assaulted us
from the very first step,
set roadblocks in our path—
unanswerable questions, inexplicable paradoxes
and baﬄing results.
Before Kurt Go¨del, we could still have hoped.
Attempting to resolve inconsistencies,
we could have spent our lives trying to grasp the tantalizing
cloud of certainty hanging above our heads—
just beyond reach.
The na¨ıvete´ of Frege, Russell, Hilbert,
and all of us, their followers,
divided into schools with bombastic titles—
Axiomatic, Logistic, Formalistic—
with methods and approaches,
plans for the future,
a list of problems to last to the end of time.
Little did we know of logic’s limitations:
that our system would backfire,
stating its own incompleteness,
in its own ink signing the QED—
357
that with cymbals and umlauts
it would prove its inability to prove
consistency of axioms within the system.
And what else was there but the system
we took for granted
as we did our ability to breathe?
After Go¨del proved the Incompleteness Theorems
a grey cloud descended upon us—
we could touch the fog.
Nothing was pure logic.
Pure logic was nothing.
We could not even count on knowing what truth
can be proved.
Uncertainty permeated everything.
We prayed that this was not the end of the road—
that there was more of it to travel.
Historical Note: Georg Cantor (1845–1918) is responsible for the rigorous mathematical
representation of infinity and the development of set theory, the “language” that allowed math-
ematicians to work with the concept of infinity. Almost immediately after its discovery, a num-
ber of paradoxes were found in set theory, among them the famous Barber Paradox, posed by
Bertrand Russell (1872–1970). Considerable efforts were made to fix these paradoxes, including
the program proposed by the prominent German mathematician, David Hilbert (1862–1943),
mentioned in the epigraph to this poem. These efforts were dealt a severe blow by the Incom-
pleteness Theorems proved by Kurt Go¨del (1906–1978). Fortunately, it was not the end of the
road. Realizing the exact extent of uncertainty inherent in the mathematical system had the
beneficial effect of removing vague anxieties and redirecting research focus.
358
Hardy
It is a melancholy experience for a professional mathematician
to find himself writing about mathematics.
G. H. Hardy
A Mathematician’s Apology
It used to be mathematics once
I visited to drink a cup of peace;
I climbed the ladder rung by rung,
reaching the highest platform.
It used to be mathematics once
I went to meet.
I left the world behind in search of peace of mind;
I flew out of myself on a trajectory of hope.
It used to be mathematics once,
clear unadulterated thought,
for which I yearned.
Among the stars it led.
I followed unafraid.
No fall, no bread, no tears,
not even cries of joy allowed—
untouched pure beauty
my mind alone perceived
and brought it out
of its invisibility into the light.
It used to be mathematics once—
both quest and goal—
the only place of rest my mind had known.
Historical Note: G. H. Hardy (1877—1947) was a British mathematician who made signif-
icant contributions to Number Theory. His mathematical powers declined when, at the age of
62, he suffered a heart attack. Shortly after, he wrote A Mathematician’s Apology, a book of
haunting beauty and sadness, which inspired many towards mathematics.
359


A Selection of Poems from Ode to Numbers

Scholarship@Claremont

Journal of Humanistic Mathematics Volume 9 | Issue 1 January 2019 A Selection of Poems from Ode to Numbers Sarah Glaz Department of Mathematics, University of Connecticut Follow this and additional works at: https://scholarship.claremont.edu/jhm  Part of the Arts and Humanities Commons, and the Mathematics Commons Recommended Citation Glaz, S. "A Selection of Poems from Ode to Numbers," Journal of Humanistic Mathematics, Volume 9 Issue 1 (January 2019), pages 350-359. DOI: 10.5642/jhummath.201901.22 . Available at: https://scholarship.claremont.edu/jhm/vol9/iss1/22 ©2019 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. Poetry Folder♦A Selection of Poems from Ode to NumbersSarah GlazSarah.Glaz@uconn.eduMy first poetry collection, Ode to Numbers, was published by Antrim House inSeptember 2017 (http://www.antrimhousebooks.com/glaz.html). The bookcontains poems written over a quarter of a century and inspired by mathemat-ics and my life as a mathematician. The poems trace the story of my life fromearly childhood in Europe to the forty years long research and teaching career inmathematics in the United States. The poems respond, in turn, to my studentsand the topics of the courses I have taught, the colleagues I have worked withand the research projects in the area of Commutative Algebra that I have beeninvolved in, the joys of professional travel, the pleasures and the difficulties facinga woman pursuing a career in mathematics, and most of all, my life-long love ofboth mathematics and poetry.Ode to Numbers is a poetry category finalist for both 2018 Next Generation IndieBook Awards and 2018 Book Excellence Awards.The poems in this folder are a small selection from the second chapter of the booktitled Pythagoras Plays His Lyre, a name derived from the poem by the same title,which appeared in 2016 in this journal.1 As the title implies, the poems in thischapter of the book were inspired by mathematical results and the history ofmathematics. The present selection, driven by history, time-travels from ancientMesopotamia, where numbers and letters were invented circa 4000 B.C., to postWorld War II Britain, the home of the Number Theorist, G.H. Hardy. Along theway we visit the 5th century B.C. Pythagoreans in Metapontum and Euclid in3rd century B.C. Alexandria, we meet the 17th century cofounders of Calculus,Newton and Leibniz, in London, Paris and Hanover; and during the 19th and early20th centuries, we briefly encounter Cantor, Hilbert, Go¨del, and their followersin Germany, Austria, and United States.–Sarah Glaz (Sarah.Glaz@uconn.edu), Connecticut, USA.1Glaz, S. “Pythagoras plays his lyre”, Journal of Humanistic Mathematics, Volume 6 Issue1 (January 2016), page 291. Available at: https://scholarship.claremont.edu/jhm/vol6/iss1/21, last accessed on January 25, 2019.Journal of Humanistic Mathematics Volume 9 Number 1 (January 2019)Numbers and LettersThe symbols of Mnemosynedon’t get along.Today I heard a quarrel between B and 2,and even as we speak, this lazy eight ∞shamelessly stretched without a limit on a limb,pushes numbers and letters towards the outer edge.From a to z or 0 to ∞ —a single source does not sufficefor peaceful coexistence.They once were born-together twins,offspring of mother memory and father wealth,both trained from infancy as record keepersof the first thousand kegs of wine and beerproduced by the first thousand settlers around Urwho, tired of forever shearing wool,started a very lucrative and joyful business:planting of hops, barley, and grapes,fermenting juices,and who, in their spare time,sloshed and satisfied,took to the flute and luteand sang long hymns of thanks,recited endless stories of nomadic pasts,composed numerous tunes of praise:Praised be Marduk,Nisaba shall be praised,revered the moon god Nana,and powerful Inana.And to ensurethey left tradition for the children,in brewing, counting, and song alike,imprinted symbolswith sharp-angled wedgesinto the region’s malleable clayand baked them in hot kilns.In the beginning, letters and numberslooked alike, happily shared a tablet. . .351The First NegativeIn fact, there is no real reason why negative numbersshould be introduced at all. Nobody owned −2 books.J.J. O’Connor & E.F. RobertsonAn Overview of the History of MathematicsDown the funnel of historyI search for you—the first manto cup in his handsminus one grain of fire:“Ghost of departed quantity!Mysterious absence!I am afraidI am cradling an illusion!”Yet from my vantage pointI discern a signupon the still uncharted number linewhere negativesbegan to light the wayfor abstract algebra to come into existenceseveral thousand yearsin the future.352√2 = 1.41421 . . .We started our voyage on the gulf of Tarentum.The sea was choppyand the brothers were restless.At dawn, we gathered on the deckintent to solve the conflict like rational men.Hippasus still refused to keep the secret.He had discovered thatthe diagonal of a squareis incommensurablewith its side.Alas! Our world had collapsedand so did our geometric proofs.Too much to lose, we heaved him overboard.Historical Note: In the 5th century B.C., the Pythagorean Hippasus of Metapontum dis-covered the existence of irrational numbers. Particularly, he had shown that√2 (the lengthof the diagonal of a square with a unit side) is an irrational number. For his sin, legend hasit, Hippasus was thrown overboard during a sea voyage. The poem plays with the imaginarypossibility that his murder occurred before he breached the Pythagorean code of secrecy andmade his discovery public.The line count of the poem’s stanzas follows the decimal expansion of to five decimal places.353The Death of EuclidSo great was Euclid’s fame that he was known to theGreeks as ὁ στοιχειωτής, “teacher of the Elements.”D. E. SmithHistory of MathematicsAnd onto the barge decked with flowerswe lifted him our arms dark andstrong his body shrunk in itsshroud of warm roses The Nile’ssluggish brown flow Myth floatingin front of us like the cloud of GodWe lifted his eyes towards heavenwhere he saw the next generationinherit and we mourned howwe mourned his fine-tuned mindhis irreplaceable jewels of thought Notone of us came close in our searchfor truth to match his eruditionNot one of us came close Andthe barge groaned under his almostweightless body once as if in painthen clutched his mutilated heartto its own Crocodiles circled silentlyplucking at wilted flowers And theprocession started slowly heavywith sighs ghost deep oars intowater and heaving mud splashingdipping and cutting a path upstreamwe left on the shore rooted tangledin tears our imaginations seekingrelief in words of reverenceHistorical Note: Euclid (323 B.C.–285 B.C.) lived in Alexandria, a Greek city situated inEgypt at the mouth of the Nile, and famous for its legendary library (housing 600,000 papyrusrolls), the Museum (a center of leaning rivaling Plato’s Academy in Athens), and its lighthouse(an engineering marvel considered one of the Seven Wonders of the Ancient World). Euclidfounded a school of mathematics at the Museum. Very little is known about his life beyondthe monumental achievement of writing The Elements, a book that had profound impact onWestern thought, and set the foundation for mathematical standards of reasoning for all time.Only the Bible has been more widely reprinted and studied.354CalculusI tell my students the story of Newton versus Leibniz,the war of symbols, lasting five generations,between The Continent and British Isles,involving deeply hurt sensibilitiesand grievous blows to national prideon such weighty issues as publication priorityand working systems of logical notation:whether the derivative must be denoted by a “prime,”an apostrophe atop the right hand-corner of a function,evaluated by Newton’s fluxions method ∆y/∆x;or by a formal quotient of differentials dy/dx,intimating future possibilities,terminology that guides the mind.The genius of both men lies in grasping simplicityout of the swirl of ideas guarded by Chaos,becoming channels through which her light poured clarityon the relation binding slope of tangent lineto area of planar region lying below a curve.The Fundamental Theorem of Calculus,basis of modern mathematics, claims nothing more.While Leibniz—suave, debonair, philosopher and politician,published his proof to jubilant cheers of continental followers,the Isles seethed, unnerved.They knew of Newton’s secret files,locked in deep secret drawers—for fear of theft and from stranger paranoid delusions,hiding an earlier version of the same result.The battle escalated to public accusations,charges of blatant plagiarism,excommunication from The Royal Math. Society,a few blackened eyes,(no duels),and raged long after both men were buried,splitting Isles from Continent, barring unified progress,till black bile drained and turbulent spirits becalmed.Calculus—Latin for small stones,primitive means of calculation; evolving to abaci;later to principles of enumeration advanced by widespread use355of the Hindu-Arabic numeral system employed to this day,as practiced by algebristas—barbers and bone setters in Medieval Spain;before Calculus came the Σ (sigma) notion—sums of infinite yet countable series;and culminating in addition of uncountable many dimensionless line segments—the integral∫—snake,first to thirst for knowledge, at any price.That abstract concepts, applicable in the beginningmerely to unseen, unsensed objects—orbits of distant stars—could generate intense earthly passionis inconceivable today,when Mathematics is considered a dry discipline,depleted of life sap, devoid of emotion,alive only in convoluted brain cells of weird scientific minds.Historical Note: The British mathematician and physicist Isaac Newton (1642–1727) andthe German mathematician and philosopher Gottfried Leibniz (1646–1716) are considered tobe the co-founders of Calculus. Their seminal achievement was the formulation and proof ofThe Fundamental Theorem of Calculus—a theorem that relates two disparate concepts: theconcept of integration (used to calculate areas below a curve) and the concept of differentiation(employed to find formulas for tangent lines to curves). Although no one had made a successfulconnection between the two concepts before Newton and Leibniz, The Fundamental Theoremof Calculus has its roots in the work of numerous other mathematicians who studied each ofthese concepts separately for many years. In fact, the roots of this area of mathematics go backseveral thousand years to the invention of numerals by Hindu mathematicians.A version of this poem was first published in Humanistic Mathematics Network Journal, 2002.356What Can We Do If We Crave Certainty in Mathematics?Thus, to salvage traditional mathematics, Hilbert proposed abold new program. It required first that the whole of existingmathematics should be axiomatized, and second that thisaxiomatic theory should then be proven consistent.David M. BurtonThe History of MathematicsHilbert said, “No one shall expel us fromthe paradise that Cantor has created.” And we believed,like Cantor himself, that God had opened the gatesto the forbidden garden,had invited us to enter,meet ℵ (aleph) face to face,converse in the language of sets,admire the ascent of transfinite cardinalsinto an infinitude of infinities.No one can possess such knowledge and remain unscathed.We lost our footing; doubts assaulted usfrom the very first step,set roadblocks in our path—unanswerable questions, inexplicable paradoxesand baﬄing results.Before Kurt Go¨del, we could still have hoped.Attempting to resolve inconsistencies,we could have spent our lives trying to grasp the tantalizingcloud of certainty hanging above our heads—just beyond reach.The na¨ıvete´ of Frege, Russell, Hilbert,and all of us, their followers,divided into schools with bombastic titles—Axiomatic, Logistic, Formalistic—with methods and approaches,plans for the future,a list of problems to last to the end of time.Little did we know of logic’s limitations:that our system would backfire,stating its own incompleteness,in its own ink signing the QED—357that with cymbals and umlautsit would prove its inability to proveconsistency of axioms within the system.And what else was there but the systemwe took for grantedas we did our ability to breathe?After Go¨del proved the Incompleteness Theoremsa grey cloud descended upon us—we could touch the fog.Nothing was pure logic.Pure logic was nothing.We could not even count on knowing what truthcan be proved.Uncertainty permeated everything.We prayed that this was not the end of the road—that there was more of it to travel.Historical Note: Georg Cantor (1845–1918) is responsible for the rigorous mathematicalrepresentation of infinity and the development of set theory, the “language” that allowed math-ematicians to work with the concept of infinity. Almost immediately after its discovery, a num-ber of paradoxes were found in set theory, among them the famous Barber Paradox, posed byBertrand Russell (1872–1970). Considerable efforts were made to fix these paradoxes, includingthe program proposed by the prominent German mathematician, David Hilbert (1862–1943),mentioned in the epigraph to this poem. These efforts were dealt a severe blow by the Incom-pleteness Theorems proved by Kurt Go¨del (1906–1978). Fortunately, it was not the end of theroad. Realizing the exact extent of uncertainty inherent in the mathematical system had thebeneficial effect of removing vague anxieties and redirecting research focus.358HardyIt is a melancholy experience for a professional mathematicianto find himself writing about mathematics.G. H. HardyA Mathematician’s ApologyIt used to be mathematics onceI visited to drink a cup of peace;I climbed the ladder rung by rung,reaching the highest platform.It used to be mathematics onceI went to meet.I left the world behind in search of peace of mind;I flew out of myself on a trajectory of hope.It used to be mathematics once,clear unadulterated thought,for which I yearned.Among the stars it led.I followed unafraid.No fall, no bread, no tears,not even cries of joy allowed—untouched pure beautymy mind alone perceivedand brought it outof its invisibility into the light.It used to be mathematics once—both quest and goal—the only place of rest my mind had known.Historical Note: G. H. Hardy (1877—1947) was a British mathematician who made signif-icant contributions to Number Theory. His mathematical powers declined when, at the age of62, he suffered a heart attack. Shortly after, he wrote A Mathematician’s Apology, a book ofhaunting beauty and sadness, which inspired many towards mathematics.359

https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1427&amp;context=jhm

A Selection of Poems from Ode to Numbers

Abstract

Similar works

Full text

Available Versions

Keck Graduate Institute

Scholarship@Claremont