Summary: Zero: The Biography of a Dangerous Idea by Charles Seife

Zero (2000) is the com­pelling tale of a fig­ure dis­al­lowed by the ancient Greeks and revered by ancient Indi­ans. Zero – along with its com­pan­ion, infin­i­ty – is a dig­it that has played a sig­nif­i­cant role in both math­e­mat­ics and phi­los­o­phy through­out the ages.

Commencement: Uncover the chronicle of a heretical digit

Zero is a pecu­liar type of dig­it. It’s not quite like 4, or 32, or 83.

When you incor­po­rate zero with oth­er dig­its, noth­ing changes. When you mul­ti­ply oth­er dig­its by zero, you invari­ably obtain zero in return. And when you attempt to divide by zero, chaos ensues.

It’s such an uncon­ven­tion­al dig­it, in truth, that many esteemed math­e­mati­cians in the ancient era refused to acknowl­edge its exis­tence entire­ly. And, in con­tem­po­rary times, even the philoso­pher René Descartes main­tained it was not genuine.

Nonethe­less, fol­low­ing its even­tu­al recog­ni­tion, it has been dis­cov­ered at the core of vir­tu­al­ly every advance­ment in math or physics. How so? You’re on the brink of discovering.

In these digests, you’ll understand

• why the Baby­lo­ni­ans intro­duced zero;
• why Aris­to­tle pro­hib­it­ed it;
• why infin­i­ty is the coun­ter­part of zero.

Zero did not exist in the earliest days of mathematics

Can you envi­sion a world devoid of digits?

In the Stone Age, that’s how things were – until some inno­v­a­tive cave dwellers com­menced notch­ing onto a wolf bone.

What were they tal­ly­ing? That remains a mys­tery. Yet, it like­ly per­tained to some­thing prag­mat­ic, such as ani­mals or spear­heads. Since pre­his­toric math­e­mat­ics was strict­ly prag­mat­ic, there was no neces­si­ty for the notion of zero. A unique term for “zero” deer was unnec­es­sary; there sim­ply … were no deer.

How­ev­er, as time pro­gressed, math­e­mat­ics advanced, and indi­vid­u­als devel­oped intri­cate count­ing sys­tems. The ancient Baby­lo­ni­ans even­tu­al­ly real­ized that some­thing – or rather, noth­ing – was absent.

This is the prin­ci­pal point: Zero did not exist in the ear­li­est days of math­e­mat­ics; it orig­i­nat­ed in ancient Babylonia.

To grasp why Zero made its debut, you will need to com­pre­hend how the ancient Baby­lon­ian count­ing sys­tem oper­at­ed. Here’s a brief breakdown.

You are most like­ly aware that our con­tem­po­rary count­ing sys­tem is dec­i­mal, or in base 10: we group ele­ments into 1s, 10s, and 100s. Con­verse­ly, in ancient Baby­lo­nia, the sys­tem was sex­a­ges­i­mal – it was in base 60. Aston­ish­ing­ly, it sole­ly fea­tured two symbols.

These two sym­bols rep­re­sent­ed “1” and “10.” The Baby­lo­ni­ans mere­ly repeat­ed those sym­bols as need­ed – anal­o­gous to the sub­se­quent­ly renowned Roman sys­tem. For instance, fifty would be five times the “10” sym­bol; fifty-one would be iden­ti­cal plus a “1” sym­bol – and so forth, until it reached 60.

Here’s the per­plex­ing part: at 60, they would recom­mence with the “1” sym­bol. Six­ty and 1 were indi­cat­ed by the same sym­bol. Like­wise, 60 times 60, or 3,600.

If you dis­cern that this appears ambigu­ous, you are cor­rect. Par­tic­u­lar­ly con­cern­ing num­bers like 61 and 3,601. Both were rep­re­sent­ed mere­ly by two “1” sym­bols, adja­cent­ly. How would you dif­fer­en­ti­ate between them?

Ulti­mate­ly, the Baby­lo­ni­ans devised a solu­tion: zero. To inscribe 3,601, they intro­duced a com­plete­ly new sym­bol between the two “1” sym­bols; this clar­i­fied that the ini­tial dig­it was not 60 but a degree high­er. This marked the gen­e­sis of zero.

Notwith­stand­ing, this was not exact­ly our cur­rent-day zero. Essen­tial­ly, it func­tioned sole­ly as a place­hold­er indi­cat­ing an absence. It was only after­ward that the pecu­liar, mys­ti­cal char­ac­ter­is­tics of zero would be ful­ly unveiled – much to the aston­ish­ment, and dread, of the ancient Greeks.

Philosophically inclined ancient Greeks spurned zero despite its utility

For many ancient soci­eties, num­bers served sole­ly as tools for com­pu­ta­tion and divid­ing land. Con­verse­ly, for the ancient Greeks, num­bers con­sti­tut­ed an entire phi­los­o­phy. Math­e­mati­cian-philoso­phers like Pythago­ras observed a sym­pho­ny of num­bers with­in every shape.

Nev­er­the­less, the ancient Greeks did not embrace zero – not in the least. In truth, Aris­to­tle pro­claimed that it sim­ply did not exist; it was mere­ly a fig­ment of human imag­i­na­tion. His stances on this top­ic, as on innu­mer­able oth­ers, rever­ber­at­ed through the eons – to the detri­ment of math­e­mat­ics in the West­ern sphere.

The cen­tral idea here is: Philo­soph­i­cal­ly inclined ancient Greeks spurned zero despite its utility.

The con­sen­sus in ancient Greece was that zero was fic­tion­al. How­ev­er, one philoso­pher, Zeno, com­posed a para­dox that chal­lenged this wide­ly accept­ed belief.

Pon­der a sce­nario where the renowned ath­lete Achilles is in a race against a tor­toise. The tor­toise gar­ners a head start of one foot. Can Achilles over­take the tor­toise and emerge victorious?

Achilles man­ages to neu­tral­ize the tor­toise’s one-foot lead in, let’s say, a sec­ond. Yet, by that instance, the tor­toise has slight­ly advanced. Sub­se­quent­ly, Achilles arrives at the tor­toise’s updat­ed posi­tion in a frac­tion of a sec­ond – but, unsur­pris­ing­ly, the tor­toise has inched for­ward again. This sequence repeats indefinitely.

Upon each instance Achilles catch­es up to the tor­toise’s past loca­tion, it has already inched for­ward. The gap between them dimin­ish­es bit by bit … but Achilles nev­er whol­ly reach­es it. Correct?

In real­i­ty, we know Achilles would effort­less­ly over­take the tor­toise. The rea­son is, that the gap between Achilles and the tor­toise has a lim­it: zero. Undoubt­ed­ly, an infi­nite num­ber of pro­gres­sive­ly minute stages are nec­es­sary to close the gap, but it inevitably occurs.

Yet, the Greek math­e­mat­i­cal sys­tem strug­gled to account for Zeno’s para­dox – because it ostra­cized zero.

Accord­ing to Aris­to­tle, zero and infin­i­ty were non-exis­tent; every­thing was finite. The cos­mos had an exter­nal sphere and then abrupt­ly ceased. Time was like­wise finite; at a cer­tain junc­ture in the dis­tant past, it sim­ply com­menced. This con­sti­tut­ed the core of their belief system.

How­ev­er, what exist­ed before time com­menced? The response is either sheer empti­ness – zero – or there was no incep­tion point – infin­i­ty. Dis­avow­ing the exis­tence of zero and infin­i­ty is illog­i­cal. Nonethe­less, this tran­spired in ancient Greece and through­out the Mid­dle Ages in the West­ern domain, due to Aris­totle’s immense influence.

Else­where in the East, how­ev­er, Aris­totle’s influ­ence waned.

Ancient Indian and Arabic mathematicians embraced zero and achieved monumental mathematical progress

In ancient India, there was no trep­i­da­tion of infin­i­ty or zero. While Aris­to­tle whol­ly rebuffed these con­cepts, they con­sti­tut­ed an inte­gral facet of Indi­an phi­los­o­phy. Ancient Indi­ans per­ceived that the cos­mos emerged from a void of empti­ness and that it was infi­nite – yet since the world emanat­ed from noth­ing­ness, it would even­tu­al­ly revert to nothingness.

Ancient Indi­an math­e­mati­cians acknowl­edged that zero mer­it­ed a stand­ing among the dig­its. This real­iza­tion cat­alyzed an array of innovations.

Here’s the pri­ma­ry point: Ancient Indi­an and Ara­bic math­e­mati­cians embraced zero and accom­plished mon­u­men­tal math­e­mat­i­cal feats.

Anoth­er crit­i­cal con­trast between ancient Greece and India per­tained to geom­e­try. To the Greeks, Math­e­mat­ics resided at the core of math; numer­i­cal val­ues essen­tial­ly indi­cat­ed ratios and forms. How­ev­er, in India, math­e­mati­cians viewed num­bers in con­cep­tu­al terms.

Here’s an instance exem­pli­fy­ing the dis­crep­an­cy this view­point results in. What is the result of sub­tract­ing 3 from 2?

To an indi­vid­ual in ancient Greece, this inquiry holds no coher­ence. If you have a field that mea­sures two acres, you can­not deduct three acres from it. Nonethe­less, when the num­bers do not sym­bol­ize any spe­cif­ic enti­ties, you can effort­less­ly solve the equa­tion. Con­se­quent­ly, you arrive at ‑1.

Along­side neg­a­tive num­bers, the ancient Indi­ans glad­ly incor­po­rat­ed zero into their numer­al sys­tem; it seam­less­ly fit in between the pos­i­tive and neg­a­tive fig­ures. Yet, they still found it some­what peculiar.

Mul­ti­ply­ing any val­ue by zero yields zero again. As for divi­sion, well, that stirs up dis­or­der. How many 0s equal 1? The Indi­an math­e­mati­cian Bhaskara from the twelfth cen­tu­ry com­pre­hend­ed the solu­tion as infin­i­ty. Inci­den­tal­ly, infin­i­ty also bore pecu­liar attrib­ut­es; you could add or sub­tract any fig­ure, and it would remain unaltered.

When these math­e­mat­i­cal advance­ments dif­fused to Mus­lim, Jew­ish, and Chris­t­ian schol­ars, it gen­er­at­ed sig­nif­i­cant impli­ca­tions — not sole­ly due to the pecu­liar traits of zero and infin­i­ty. The trio of reli­gions had been pro­found­ly influ­enced by Aris­to­tle, mak­ing these nov­el con­cepts dis­rup­tive to their world­view. Although, even­tu­al­ly, all three affil­i­a­tions acknowl­edged and assim­i­lat­ed them.

Chris­tians were the last to embrace zero — ulti­mate­ly, it was the com­mer­cial pres­sures that impelled them to adopt it. Ital­ian mer­chants real­ized that the con­tem­po­rary method of count­ing using ten dig­its, rec­og­nized as the Ara­bic sys­tem, proved far sim­pler com­pared to the Roman sys­tem man­dat­ed by the church. How­ev­er, at that junc­ture, the Ara­bic sys­tem inte­grat­ed a dig­it rep­re­sent­ing zero.

Con­se­quent­ly, dur­ing the Mid­dle Ages, zero sub­tly crept into the West­ern numer­i­cal sys­tem. Yet, it remained dis­trust­ed — even by some of the most emi­nent math­e­mat­i­cal intellects.

Integrating zero in the Occident posed theological challenges, nevertheless, it facilitated a mathematical revolution: calculus

René Descartes was born in 1596. Anal­o­gous to numer­ous great thinkers before him, such as Pythago­ras, he func­tioned both as a math­e­mati­cian and a philoso­pher. How­ev­er, he nev­er whole­heart­ed­ly embraced zero.

Descartes left his imprint on the Carte­sian coor­di­nate sys­tem — the x and y axes instruct­ed in sec­ondary school. These two axes neces­si­tate hav­ing a zero in the low­er left cor­ner. Embark­ing from 1 would swift­ly lead to errors.

This potent new coor­di­nate frame­work ush­ered in an array of math­e­mat­i­cal advance­ments. Nev­er­the­less, Descartes con­sis­tent­ly con­tend­ed that zero did not have actu­al exis­tence. Hav­ing been nur­tured by Aris­totle’s doc­trines, zero marked the bound­ary for him.

Sub­se­quent math­e­mati­cians exhib­it­ed less reluc­tance — ulti­mate­ly yield­ing spec­tac­u­lar outcomes.

This is the cru­cial mes­sage: Assim­i­lat­ing zero in the Occi­dent proved the­o­log­i­cal­ly com­pli­cat­ed, how­ev­er, it yield­ed a math­e­mat­i­cal rev­o­lu­tion: calculus.

You pos­si­bly recall some cal­cu­lus from your aca­d­e­m­ic days, but you may not entire­ly grasp its intrin­sic con­nec­tion with zero and infin­i­ty. Hence, let’s review some fun­da­men­tal aspects.

Imag­ine sketch­ing a curve on a Carte­sian grid. How would you com­pute the area beneath it?

You might ini­ti­ate by delin­eat­ing a rec­tan­gle below the curve — one that cov­ers the max­i­mum area fea­si­ble. While this is a rea­son­able start­ing point, its pre­ci­sion remains mediocre.

To enhance pre­ci­sion, you could por­tray two small­er rec­tan­gles instead; by doing so, a larg­er por­tion of the area could be cov­ered. Pro­gress­ing with three rec­tan­gles brings about even fur­ther accu­ra­cy — and so forth. How­ev­er, to com­pute the actu­al area beneath the curve, you would neces­si­tate an infi­nite num­ber of rec­tan­gles, each pos­sess­ing an area infin­i­tes­i­mal­ly small — essen­tial­ly, an area of zero.

This con­cept may appear illog­i­cal, yet the bizarre aspect is that it func­tions. Math­e­mati­cians Isaac New­ton and Got­tfried Leib­niz essen­tial­ly acknowl­edged this simul­ta­ne­ous­ly and devised cal­cu­lus sys­tems. This enabled them to com­pute the area math­e­mat­i­cal­ly, despite involv­ing pecu­liar math­e­mat­ics revolv­ing around zero and infinity.

This notion was the­o­ret­i­cal­ly per­turb­ing. Irish bish­op George Berke­ley observed that dis­tinct from oth­er domains of math­e­mat­ics, which had been entire­ly sub­stan­ti­at­ed, cal­cu­lus relied on faith — no one gen­uine­ly com­pre­hend­ed the oper­a­tions involv­ing all those zeros.

The math­e­mati­cian who resolved this predica­ment was Jean Le Rond d’Alem­bert. Anal­o­gous to Zeno’s para­dox, d’Alem­bert expli­cat­ed the res­o­lu­tion uti­liz­ing lim­i­ta­tions. A sequence might length­en towards infin­i­ty — yet it may still con­verge on a finite limit.

Mathematicians promptly unearthed the intricate but captivating relationship between zero and its antithesis, infinity

Qua­drat­ic equa­tions con­sti­tute the foun­da­tion of sec­ondary school math­e­mat­ics. How­ev­er, this does not sig­ni­fy they are uncom­pli­cat­ed. Quite the contrary.

Take a seem­ing­ly sim­ple sce­nario: x2 + 1 = 0. What does x equal?

<p,>As you might rec­ol­lect, qua­drat­ic equa­tions nor­mal­ly present two solu­tions, one pos­i­tive and one neg­a­tive. The solu­tions to that equa­tion seem some­what pecu­liar: the square root of ‑1 and the neg­a­tive square root of ‑1.

Con­sid­er­ing these num­bers do not exist, math­e­mati­cians assert that they are imag­i­nary. Con­se­quent­ly, they name them i and ‑i.

How does this per­tain to zero? It might appear unre­lat­ed — yet it is intri­cate­ly linked.

The focal point here is: Math­e­mati­cians prompt­ly learned that zero and its coun­ter­part, infin­i­ty, boast­ed an elab­o­rate yet cap­ti­vat­ing relationship.

Once the exis­tence of imag­i­nary num­bers is fac­tored in, you can amal­ga­mate them and obtain val­ues like i + 2, or 2i — 4. To visu­al­ize num­bers of this nature — known as com­plex num­bers, it proves expe­di­ent to chart them on a Carte­sian grid, with the x‑axis sym­bol­iz­ing the real por­tion of the num­ber (e.g., ‑4) and the y‑axis indi­cat­ing the imag­i­nary part (e.g., 2i).

How­ev­er, this grid does not func­tion akin to a tra­di­tion­al set­up. Sup­pose you posi­tion the point i one unit upwards on the y‑axis and at zero on the x‑axis. What tran­spires when you square that val­ue? By def­i­n­i­tion, i2 equals ‑1. Con­se­quent­ly, that point rotates 90 degrees to the left. This is applic­a­ble for any com­plex num­ber: its square mul­ti­pli­ca­tion prompts rota­tion around the grid.

<p,>Should we con­tin­ue on a two-dimen­sion­al grid, the sce­nario prompt­ly grows intri­cate — hence, the math­e­mati­cian Bern­hard Rie­mann dis­cerned that it made more sense to con­cep­tu­al­ize things on a sphere. Envi­sion a sphere with i placed at one point and ‑i posi­tioned direct­ly oppo­site. Ver­ti­cal to these points lie 1 and ‑1. And who occu­pies the upper and low­er­most points of the sphere? Zero and infinity.

The pecu­liar log­ic under­pin­ning math­e­mat­ics involv­ing com­plex num­bers dis­clos­es that zero and infin­i­ty are rec­i­p­ro­cal ends, akin to 1 and ‑1.

The Rie­mann sphere alle­vi­ates com­pre­hen­sion of cer­tain equa­tions that posed dif­fi­cul­ties pre­vi­ous­ly. For instance, con­sid­er y = 1/x. In two-dimen­sion­al space, it presents some­what hap­haz­ard­ly: the curve extends into infin­i­ty as x approach­es zero. Yet, on the sphere, the causal­i­ty becomes trans­par­ent: the curve mere­ly reach­es the apex.

This might appear some­what the­o­ret­i­cal. Be that as it may, if you are pon­der­ing over the rel­e­vance of all these dis­cov­er­ies to the tan­gi­ble, real world, remain engaged — for zero and infin­i­ty are not sole­ly con­fined to math­e­mat­ics but also con­sti­tute inte­gral com­po­nents of physics!

Zero and infinity are not merely mathematical concepts — they are abundant in physics as well

We recent­ly touched upon imag­i­nary num­bers — nonethe­less, zero and infin­i­ty are so real that they man­i­fest them­selves in the actu­al world through man­i­fold impor­tant avenues. They, in fact, under­lie many of the break­throughs achieved in physics over the past cen­tu­ry or so.

An illus­tra­tion dates back even pri­or to that. In the 1850s, physi­cist Lord Kelvin uncov­ered that itIt was prac­ti­cal­ly unat­tain­able to cool an object below rough­ly ‑273 degrees Cel­sius. In essence, he unveiled absolute zero.

 

Anoth­er zero in physics was revealed through the research of Albert Ein­stein: the black hole. Ein­stein’s hypothe­ses aid­ed in clar­i­fy­ing a fas­ci­nat­ing and trou­ble­some phe­nom­e­non occur­ring in deep space. When a mas­sive star per­ish­es, its grav­i­ta­tion­al pull becomes so force­ful that it caves in on itself, con­tract­ing con­tin­u­ous­ly, until it even­tu­al­ly occu­pies zero space. Despite occu­py­ing zero space, it retains mass. This con­tra­dic­to­ry com­bi­na­tion results in a dis­tor­tion in space-time itself, draw­ing in any­thing that comes close to it.

Fur­ther break­throughs in physics have alter­na­tive con­nec­tions with zero entire­ly. For instance, string the­o­ry takes the curi­ous mea­sure of essen­tial­ly out­law­ing it — but not entire­ly in the same man­ner as Aris­to­tle all those cen­turies ago.

Accord­ing to string the­o­ry, the uni­verse exists in ten or per­haps eleven dimen­sions, so what appears as zero to us may not gen­uine­ly be zero when all the oth­er dimen­sions are account­ed for. This the­o­ry helps in elu­ci­dat­ing spe­cif­ic enig­mat­ic aspects of the uni­verse — although some con­tend that this is more a phi­los­o­phy than true sci­ence because it is not demon­stra­ble through experimentation.

Then again, zero and phi­los­o­phy have always been inter­twined. Right from the ori­gin of time — the Big Bang itself, anoth­er zero, nat­u­ral­ly — all the way to the even­tu­al con­clu­sion of the uni­verse, zero has always pos­sessed an enig­mat­ic, emp­ty influence.

Noth­ing can arise out of noth­ing, the poet and philoso­pher Lucretius once assert­ed. How­ev­er, noth­ing pos­sess­es pecu­liar, mys­ti­cal char­ac­ter­is­tics. And these traits are still being uncov­ered today.

Summary

The pri­ma­ry les­son in these digests is that:

Zero did not have a pres­ence in the ear­li­est days of math­e­mat­ics — and in Baby­lo­nia, where it was ini­tial­ly cre­at­ed, it was sim­ply a place­hold­er. Although the ancient Greeks excelled in math, zero was barred by Aris­to­tle. This entailed that it was not ful­ly rec­og­nized in the West­ern world for numer­ous cen­turies. Con­verse­ly, in locales like India, it was embraced — and arith­metic made sig­nif­i­cant strides. Zero has since right­ful­ly secured its posi­tion in the numer­i­cal sys­tem along­side its coun­ter­part, infin­i­ty. It has proven to be a cru­cial yet mys­te­ri­ous ele­ment of every new con­cept in math­e­mat­ics or physics, from cal­cu­lus to relativity.

About the Author

Charles Seife is the writer of five pre­ced­ing books, includ­ing Proofi­ness and Vir­tu­al Unre­al­i­ty. He has con­tributed to a vast array of peri­od­i­cals, includ­ing The New York Times, Wired, New Sci­en­tist, Sci­ence, Sci­en­tif­ic Amer­i­can, and The Econ­o­mist. He is a Jour­nal­ism pro­fes­sor at New York Uni­ver­si­ty and resides in New York City.