Mathematicsandknowledge
TOKmathguestlecturebyMr.Chase
IsMathematicsInventedorDiscovered?
Classicquestioninthephilosophyofmathematics,thatmaybeyou'vealreadythought
aboutortalkedaboutinthisclass:
Ismathematicsinventedordiscovered?
Poll:Hereareyouroptions:
1. Invented
2. Discovered
3. Unresolvable
4. Idon’tknow
[Giveaminuteforstudentstothink,thenhavethemraisehandsandtallythevote.]Keepin
mindtheoptionsI’veallowedforinthispollbecausetheymightforeshadowsomethinglater
inthistalk.[thenotionofundecidablestatements]
Herearethetypicalargumentsonbothsidesoftheissue.
Invented: Aristotle would have been in this camp. People say things like “Newton and
Leibniz invented Calculus.” Someone invents a new algorithm for something. Surely long
divisionwasinvented,notdiscovered,youwouldsay.Surelywecameupwithournumber
systemandweinventedalltheconventionsandsymbolsweuse.And,tocounterthosewho
mightclaimthatmathematicsisdiscovered:ifamathematicaltheorygoesundiscovered,does
ittrulyexist?(Ifatreefallsinthewoods…)
1aprimeorcomposite?(In1903,
Discovered:Platowouldhavebeeninthiscamp.Is2
Frank Nelson Cole proved it was composite in a famous lecture without words, simply by
writingthefactorsontheboardandmultiplyingitout.Itwasactuallyknowntobecomposite
since1876...that’salmost30yearsofnooneknowingthefactors!)Manypeopledidn'tknow
foryears,eventhoughclearlythere’sananswer.Onceitwasknown,Ithinkwemightsayit
wasdiscovered.Isthetwinprimeconjecturetrue?[haveastudentdefinewhattwinprimes
are, if a student is able.] Well, someone will invent a proof, but its truth is itself
transcendent.Thisisanexampleofsomethingstillyettobediscovered.Actually,progresson
thisparticularquestionhasbeenmadeinthelastyear!
[Handout–outlineofthetalk/quotes/jokes]
Thebeautyofmathematics—or“whydiscoveredistherightanswertothefirstquestion”:‐)
[ShowthisGeogebraappletwithquadrilateralparallelogramtheoreminteractively...thenask
kids if this is a coincidence. Can they prove it? It's either true or false. But you have
1always
aninsatiabledesiretotrytoproveit—aholeinyourmathematicalheart!][Is9
divisibleby8?][HowcanyouprovethattwopeopleinDChavethesamenumberofhairson
theirheads?][talkaboutexistenceproofsvs.constructiveproofs,iftimeallows]
Make nomistake, I aminPlato’scamp onthisone.Ithink mathematicsisdiscovered.The
mathematics that we think of as having been ‘invented’ is merely a shadow of the true
mathematics, projected into our temporal reality (to borrow Platonist language).
Mathematicsispure.
<takenfromxkcd.com>
“Mathematicsisaqueenofscience.”‐CarlFriedrichGauss
Yousee,mathisaboutbeautyandpurity.Thatrealizationiswhatturnsyoung,unsuspecting
studentsintomathematicians.“Someofyoumayhavemetmathematiciansandwonderedhow
theygotthatway.”‐TomLehrer:‐)Howdoyouknowwhenamathematicianishardatwork?
Ifyouwalkintothehallsofamathdepartmentandseeonemathematicianbusilyworkingon
hiscomputertypingupapaperorworkingonalecture,heisnotactuallydoingwork.Ifyou
seeanothermathematicianstaringoutthewindow,heistheonehardatwork!
No, really, it’s true. Mathematicians do mathematics because it is beautiful, regardless of
whetheritmightalsobeuseful:
o
o
o
“Whereverthereisnumber,thereisbeauty.”‐Proclus(addtothatlogic,variable,and
proof…notjustnumber)
“It is impossible to be a mathematician without being a poet in soul.” ‐ Sofia
Kovalevskaya
“Themathematiciandoesnotstudypuremathematicsbecauseitisuseful;hestudies
itbecausehedelightsinitandhedelightsinitbecauseitisbeautiful.”‐JulesHenri
Poincaré
Insomeways,itprovidesasafeharborfromthedisorderandchaosoftherealworld.Math
issopureandcertainthatwecanavoidthedebatesanddeconstructionismthatplaguesso
manyotheracademicareas:
<takenfromxkcd.com>
Yes,I'minthediscoveredcampwhenitcomestotheopeningquestion.Mathhasalwaysbeen
thesame.Ifyouweretogotoadifferentplanet,mathwouldstillbethesame(maybedifferent
axiomsystems,maybetheywoulduseadifferentbase,andtheywouldofcourseusedifferent
symbols,buttheywouldarriveatthesamemathematics,andthesametheorems).Peoplesay
“math hasn't changed in 30 years since when I was in school.” True. It's worse than that,
actually.Ithasn'tchangedin4.6billionyears!That'sonereasonwhymathissobeautiful,so
austere.
Whatpartsofmathematicsarewefreetoinvent?
“Ourbasetennumbersystemwasinvented.”Icompletelyagree.Butthebeautyofmathis
thatthebaseinwhichwedoitdoesn’tactuallymatter.Allofourtheoremsstillworkinother
bases.Thechoiceofbaseisarbitrary.Oursymbolsarearbitrary.
Thisbridgesustothenotionofformalism:
“Mathematicsisagameplayedaccordingtocertainsimpleruleswithmeaninglessmarks
onpaper.”‐DavidHilbert
Hilbert’squotemightseemfunnytoyou,butit’saperfectdescriptionofmathematics.Atone
time, it seemed that mathematics was immutable. You might say “1+1=2” is founded in
reality—it’s an immutable, unchangeable reality over which you have no say. But
mathematicianshaveaslightlybroaderviewthanthat.Wesetupaxiomsystemsandthen
usingsimplesetsofaxioms,weproveawholehostofresults.Generally,wewanttheserules
toconfirmourintuitionsandreflecttruthintheoutsideworldinameaningfulway,butthis
isn’tnecessary.Forexample,alloftherulesofalgebrathatyouknowandlovearebasedon
justasmallsetofaxioms,calledtheFieldAxioms(takenfromWikipedia):
[Handout–fieldaxiomsandRudinproofs]
ClosureofFunderadditionandmultiplication
For all a, b in F, both a + b and a · b are in F (or more formally, + and · are binary
operationsonF).
Associativityofadditionandmultiplication
Foralla,b,andcinF,thefollowingequalitieshold:a+(b+c)=(a+b)+canda·(b·c)=
(a·b)·c.
Commutativityofadditionandmultiplication
ForallaandbinF,thefollowingequalitieshold:a+b=b+aanda·b=b·a.
Existenceofadditiveandmultiplicativeidentityelements
ThereexistsanelementofF,calledtheadditiveidentityelementanddenotedby0,suchthat
forallainF,a+0=a.Likewise,thereisanelement,calledthemultiplicativeidentityelement
anddenotedby1,suchthatforallainF,a·1=a.Toexcludethetrivialring,theadditive
identityandthemultiplicativeidentityarerequiredtobedistinct.
Existenceofadditiveinversesandmultiplicativeinverses
For every a in F, there exists an element −a in F, such that a + (−a) = 0. Similarly, for
any a in F other than 0, there exists an element a−1 in F, such that a · a−1 = 1. (The
elements a + (−b) and a · b−1 are also denoted a − b and a/b, respectively.) In other
words,subtractionanddivisionoperationsexist.
Distributivityofmultiplicationoveraddition
Foralla,bandcinF,thefollowingequalityholds:a·(b+c)=(a·b)+(a·c).
These are the “simple rules” with which we “play the game” of mathematics. From these
simpleaxioms,thenotionofsubtractionanddivisioncanbederived,thenotionofexponents
&logarithms,androots.Thisistheentiresubjectofinterestinananalysiscourse,ifyouhave
the privilege of taking one someday. For example, people wonder why a negative times a
negativeisapositive.Here’stheanswer:Wecanproveitfromthefieldaxioms!Ifwedon’t
defineitthatway,wegetcontradictions.This,tome,isamuchmorepowerfulandappealing
argumentthanthosewhotrytojustifyitwithrealworldsituations.Thisbecomestrueforso
manyothersimilarquestionstoo—whycan’twedividebyzero?whyisittruethat0times
anynumberis0?or1timesanynumberisitself?whyisanumberraisedtothezeropower
equalto1?Inalloftheanswerstothesequestions,ourhandisforcedbyoursetofaxioms
(which,Iremindyou,wefreelychose).
Proposition1.14fromRudin.Theaxiomsforadditionimplythefollowingstatements.
(a) If
then
(cancellation)
(b) If
then
0(uniquenessoftheadditiveidentity)
(c) If
0then
(uniquenessoftheadditiveinverse)
(d)
(doublenegation)
Proof.If
,theaxiomsforadditiongive
0
0
.
Thisproves(a).Take
0in(a)toobtain(b).Take
in(a)toobtain(c).Since
0,(c)(with– inplaceof )gives(d).∎
Proposition1.15fromRudin.Theaxiomsformultiplicationimplythefollowingstatements.
(a) If
0and
then
(cancellation)
(b) If
0and
then
1(uniquenessofthemultiplicativeidentity)
(c) If
0and
1then
1/ (uniquenessofthemultiplicativeinverse)
(d) If
0then1/ 1/
.
Proposition1.16fromRudin.Thefieldaxiomsimplythefollowingstatements,forany , ,
and ∈ .
(a) 0
0.
(b) If
0and
0then
0.
(c)
.
(d)
Proof. 0
assume
0
0,
0 0
0,but
0 . Hence 1.14(b) implies that 0
0.Then(a)gives
1
1
1
1
1
0
0, and (a) holds. Next,
0
acontradiction.Thus(b)holds.Thefirstequalityin(c)comesfrom
0
0,
combinedwith1.14(c);theotherhalfof(c)isprovedinthesameway.Finally,
by(c)and1.14(d).∎
[Passoutanalysistextbooksforstudentstoskim]
Ifyouwanttobreakanyofthefieldaxioms,youmay.Youjustcan’tcallthealgebraicstructure
thatyou’reworkingwithafield.Forexample,theabsenceofmultiplicativeinversesgivesrise
tothenotionofaring,whichisalsoaveryimportantideainmathematics(albeitoneyou
won’tencounterinhighschool).
[Passoutalgebratextbooksforstudentstoskim]
Well‐formedformulas.“Thetidyloveperforatesmachines.”“Thrhemmeajajaj”“Threeand
fivearetwenty.”areallwrong,butforverydifferentreasons(semantics,syntax,...)[struck
forthesakeoftime]
KurtGödelprovedsomepath‐breakingresultsinmathematicallogic.Heshowedthatinany
axiomsystem,therewillalwaysremainstatementsthatyoucannotprovetrueorfalse.For
example:
Axioms:
“itisrainingoutside.”
“ifitisraining,Iwilltakeanumbrella”
Statements: “Iwilltakeanumbrella.”–provablytrueinthisaxiomsystem
“Itisnotrainingoutside.”–provablyfalseinthisaxiomsystem
“Ifitisraining,Iwillbringmypethamsteraswell.”–undecidableinthis
axiomsystem
Okay, then, you might say, “let’s just expand the axiom system until it can speak about
hamsters.”Youcantrytodothat,butwhatGödelshowedisthatthisisafool’serrand.You
can try to expand your axiom system so that your system is complete—that is, every
statementcanbeassignedavalueoftrueorfalse—butthenyoursystemwillnecessarilybe
inconsistent.
At one time, mathematicians hadn’t really encountered such ‘undecidable’ statements in
mathematics.Butnow,thankstoGödel’swork,weknowofmany:TheContinuumHypothesis,
TheAxiomofChoice,Hilbert’sTenthProblem,andTheHaltingProblem.Somathematicians,
now,areinterestedinthreetypesofresultswhenconsideringaconjecture:Provablytrue,
provablyfalse,orundecidable.
Ifastatementisundecidable,thisalsomeansthatwecanchoosethestatementtobetrueor
falseandineithercaseitwillbeconsistentwithalltheotherstatementsinthesystem.
Furthermore,heprovedthatifanaxiomsystemiscomplete(thatis,weavoidallundecidable
statements),thenitnecessarilyhasinternalinconsistencies.
[PassoutRebeccaGoldstein’sbookIncompleteness]
Theusefulnessofmathematics
“It’slikeagorgeouspaintingthatalsofunctionsasadishwasher!”–BenOrlin
Mathematicsjusthappenstoalsobeuseful.Butitscorrelationtotherealworldisactually
quiteunreasonable.Whyshoulditbetruethatwhenweplaywithmathinasterileroomand
discoverinterestingmathematics,thatitapplytoordescribetherealworldatall??It’sreally
mind‐blowing.
The fact that mathematics so accurately describes the real world is more support for the
Platonic view. We invent notation, true. But the actual mathematics describes real world
phenomenon—like the interaction of particles or the growth of populations. We would
certainly agree that these other aspects of reality—biology, chemistry, and physics—are
discovered.Peoplediscoveratoms,thentheyinventanameforthem.
Ajustificationofmatheducation
Many educational reformers would want you to learn math because it prepares you for a
futureworkingforSpaceXordecryptingstatesecretsattheNSA.Thisistrue.Mathematics
provides an excellent foundation for a cadre of high‐powered careers that reward
mathematicalintelligence.
Butinmyopinion,thereasonmathisimportantforyoutolearnisbecauseitisbeautifuland
interesting,notnecessarilybecauseitisuseful.Ourgoalasmathteachersistoprovideyou
withaliberaleducation(meaning,onethatfreesyou)sothatyoucanappreciatethebeautiful
thingsandengageinintelligentconversationwithallkindsofpeople.Somepeoplethinkthe
goalofeducationistomakeyouagoodcitizen,tomakeyousuccessfulorrich,ortolandyou
ajob.ButIhaveaslightlymorereligiousmotivation...it’ssothatyou’llglimpsethemindof
God!
Inreality,99%ofyouwon’tdirectlyuseanyofthemathyoulearn.Depressing?Itshouldn’t
be. The same is true for all your other high school subjects. We’re giving you a liberal
education.
Conclusion
Mathispure,beautiful,certain.Andby“math”wealsomeanlogic,formalsystems,games,
rules, reasoning, argument. It is discovered, not invented, though our notation and many
otherstructuresareinventedarbitrarily.Unlikeotherarenas,wecantrulybecertainwith
mathinwaysthatnoothersubjectcanenjoy.