>>MATHEMATICS DESCRIBES THE REAL WORLD OF ATOMS AND ACORNS, STARS AND STAIRS. SIMPLE ABSTRACT EQUATIONS DEFINE COMPLEX PHYSICAL THINGS BEAUTIFULLY, ELEGANTLY. WHY SHOULD THIS BE? THE MORE I THINK ABOUT THIS THE MORE ASTONISHED I GET. ALBERT EINSTEIN SAID, "THE MOST INCOMPREHENSIBLE THING ABOUT THE UNIVERSE IS THAT IT IS COMPREHENSIBLE." PHYSICIST EUGENE WIGNER NOTED THE UNREASONABLE EFFECTIVENESS OF MATHEMATICS IN SCIENCE. SO IS MATHEMATICS INVENTED BY HUMANS LIKE CHISELS AND HAMMERS AND PIECES OF MUSIC? OR IS MATHEMATICS DISCOVERED, ALWAYS OUT THERE SOMEWHERE, LIKE MYSTERIOUS ISLANDS WAITING TO BE FOUND? THE QUESTION PROBES THE DEEPEST SECRETS OF EXISTENCE. IS MATHEMATICS INVENTED OR DISCOVERED? I'M ROBERT LAWRENCE KUHN AND CLOSER TO TRUTH IS MY JOURNEY TO FIND OUT. I SHOULD BEGIN AT OXFORD WITH ONE OF THE WORLD'S MOST DISTINGUISHED MATHEMATICIAN ROGER PENROSE. IN HIS VISION ROGER SETS THE STANDARD. ROGER, HOW ACCURATELY DOES MATH DESCRIBE THE PHYSICAL WORLD? >THAT IS EXTRAORDINARILY PRECISE, BUT I THINK PEOPLE OFTEN FIND IT PUZZLING THAT SOMETHING ABSTRACT LIKE MATHEMATICS COULD REALLY DESCRIBE REALITY AS WE UNDERSTAND IT. I MEAN REALITY YOU THINK OF SOMETHING LIKE THIS CHAIR OR SOMETHING, OR SOMETHING MADE OF SOLID STUFF. AND THEN YOU SAY, WELL, WHAT'S OUR BEST SCIENTIFIC UNDERSTANDING OF WHAT THAT IS? WELL, YOU SAY IT'S MADE OF FIBERS AND CELLS AND SO ON, AND THESE ARE MADE OF MOLECULES AND THOSE MOLECULES ARE MADE OF ATOMS, AND THOSE ATOMS ARE MADE OUT OF NUCLEI AND ELECTRONS GOING AROUND. THEN YOU SAY WELL, WHAT'S A NUCLEUS? THEN YOU SAY, WELL IT'S PROTONS AND NEUTRONS AND THEY'RE HELD TOGETHER BY THINGS CALLED GLUONS, AND THEIR NEUTRONS AND PROTONS ARE MADE OF THINGS CALLED QUARKS AND SO ON. AND THEN YOU SAY, WELL, WHAT IS AN ELECTRON AND WHAT'S A QUARK? AND AT THAT STAGE THE BEST YOU CAN DO IS TO DESCRIBE SOME MATHEMATICAL STRUCTURE. YOU SAY THEY'RE THINGS THAT SATISFY THE DIRAC EQUATION, OR SOMETHING LIKE THAT, WHICH YOU CAN'T UNDERSTAND WHAT THAT MEANS WITHOUT MATHEMATICS. I MEAN, THE MATHEMATICAL DESCRIPTION OF REALITY IS WHERE WE'RE ALWAYS LED AND THESE EQUATIONS ARE FANTASTICALLY ACCURATELY. FEYNMAN HAD A VERY GOOD DESCRIPTION. HE SAID, YOU CAN, IT DESCRIBES THE DISTANCE BETWEEN NEW YORK AND LOS ANGELES TO AN ACCURACY OF LESS THAN THE THICKNESS OF A HUMAN HAIR. SO, THAT'S PRETTY PRECISE. I MEAN, NEWTON'S THEORY ALREADY HAD A PRECISION OF SOMETHING LIKE ONE PART IN 10 TO THE 7TH, SO THAT'S 10 MILLION. AND THEN EINSTEIN COMES ALONG AND PRODUCES A THEORY WHICH IS NOW KNOWN TO HAVE A PRECISION SOMETHING LIKE 10 TO THE POWER OF 14. SO IN A SENSE THIS IS TELLING US THAT OUR PICTURE OF PHYSICAL REALITY DEPENDS ON SOMETHING WHICH IS MORE PRECISE, AT LEAST IN OUR UNDERSTANDING OF IT, THAN HOW WE THINK ABOUT THE WORLD. AND THIS PRECISION REALLY DATES BACK TO THE ANCIENT GREEKS, THE TIME OF PYTHAGORAS, AND LATER WHERE THEY DEVELOPED THE MATHEMATICAL IDEAS AS A FIELD OF STUDY STIMULATED TO SOME DEGREE BY PHYSICAL REALITY. BUT THEY DEVELOPED THIS MATHEMATICAL SCHEME PURELY AS A STUDY ON ITS OWN, AND EVER SINCE THEN MATHEMATICS HAS BEEN A SUBJECT WHICH YOU CAN STUDY FOR ITS OWN SAKE. IT HAS ITS OWN LIFE IN A SENSE, AND CERTAINLY MATHEMATICIANS VIEW IT THIS WAY, AS SOMETHING OUT THERE THAT SEEMS TO HAVE A REALITY INDEPENDENT OF THE ORDINARY KIND OF REALITY LIKE THINGS LIKE CHAIRS AND SO ON, WHICH ARE WHAT WE NORMALLY THINK OF AS REAL. IT'S SOMETIMES REFERRED TO AS A PLATONIC WORLD, A PLATONIC REALITY, AND SOMETIMES PEOPLE HAVE A LOT OF TROUBLE THINKING OF THAT AS REAL. I MEAN, PHILOSOPHERS WORRY ABOUT THAT AND SO ON. >>WHAT WOULD THAT MEAN, A PLATONIC REALITY? >IT'S A DIFFERENT KIND OF REALITY FROM THE REALITY OF THE PHYSICAL WORLD. I MEAN I TEND TO THINK OF THERE BEING DIFFERENT WAYS OF LOOKING AT REALITY. THERE'S THE REALITY OF OUR MENTAL EXPERIENCE WHICH, OKAY, INTERRELATES WITH THE PHYSICAL REALITY, BUT SO THEN DOES THE MATHEMATICAL REALITY OF THIS PLATONIC WORLD WHICH GIVES REALITY TO THESE MOTIONS. SO IF YOU LIKE MATHEMATICAL FACTS, LIKE THERE IS NO LARGEST PRIME NUMBER, IT'S SOMETHING INDEPENDENT OF OURSELVES. IT'S ALWAYS BEEN TRUE THEN SOMEHOW BECOME TRUE AS SOON AS SOMEBODY SAW HOW TO PROVE IT. IT'S ALWAYS BEEN TRUE. >>AND IT WOULD HAVE BEEN TRUE IF NOBODY EVER PROVED IT. >IF NOBODY ... EXACTLY. YES. IN A SENSE THAT HAD TO BE SO BECAUSE IF THE PHYSICAL WORLD DEPENDED SO PRECISELY ON THESE MATHEMATICAL LAWS I COULDN'T HAVE KNOWN WHAT TO DO IN A CERTAIN SENSE IF THE MATHEMATICS HADN'T ALREADY BEEN THERE. I MEAN, IT'S NOT US THAT IMPOSES THIS ON THE WORLD. IT'S OUT THERE. SOMETIMES PEOPLE THINK THAT, YOU KNOW, MAYBE THE REASON WE HAVE GOOD MATHEMATICAL LAWS OF PHYSICS IS THAT'S THE BEST WAY WE CAN COME TO UNDERSTAND THE WORLD, BUT IT'S SOMETHING MORE THAN THAT. IT REALLY IS OUT THERE IN THE WORLD. I LIKE TO THINK OF MATHEMATICS AS A BIT LIKE GEOLOGY OR ARCHAEOLOGY WHERE YOU'RE REALLY EXPLORING OUT THERE IN THE WORLD AND YOU'RE FINDING BEAUTIFUL THINGS OR THINGS WHICH HAVE BEEN THERE, IN FACT, FOR AGES AND AGES AND AGES, AND YOU'RE REVEALING THEM FOR THE FIRST TIME. >>SOME OF WHICH YOU NEVER DREAMED OF. >I'VE NEVER DREAMED OF SOME OF THEM. ABSOLUTELY RIGHT. >>ROGER FAMOUSLY BELIEVES THAT MATHEMATICS HAS AN INDEPENDENT EXISTENCE. A PLATONIC EXISTENCE RADICALLY DISTINCT FROM PHYSICAL SPACE AND TIME. I LIKE THAT. EXISTENCE SOMEHOW BEYOND THE MATERIAL. BUT I WORRY. IT'S AN ANSWER TOO PAT, TOO EASY? AND NOT EVERYONE AGREES. SO I GO TO LOS ANGELES TO SEE MARK BALAGUER, WHO SPECIALIZES IN THE PHILOSOPHY OF MATHEMATICS. SO WHAT'S SO SPECIAL ABOUT MATHEMATICS, MARK? >MOST PEOPLE THINK THAT MATHEMATICS IS ABOUT SOLVING PROBLEMS AND, YOU KNOW, THEY TALK ABOUT BEING GOOD AT MATH, SO IT'S LIKE A SKILL. BUT REAL MATHEMATICS THAT MATHEMATICIANS DO IS A THEORY ABOUT THE WORLD IN THE SAME WAY THAT PHYSICS AND BIOLOGY ARE. SO, FOR EXAMPLE, IT LOOKS LIKE MATHEMATICS IS THE STUDY OF STRUCTURES, SO THE MOST BASIC ONE THAT PEOPLE KNOW ABOUT IS THE STRUCTURE OF THE NATURAL NUMBER SO THERE'S THIS STRUCTURE THAT WE KNOW ABOUT. IT STARTS WITH ZERO THEN IT'S GOT ONE, TWO, THREE, AND IT GOES FOREVER. SO FROM THESE BASIC LAWS MATHEMATICIANS START THINKING ABOUT MORE ADVANCED QUESTIONS ABOUT WHAT THE STRUCTURE IS LIKE AND THEY START PROVING THEOREMS AND THE THEOREMS ARE JUST CLAIMS ABOUT THE NATURE OF THIS STRUCTURE. THE MYSTERY IS, WHY ARE THESE STRUCTURES THAT SEEM TO BE COMPLETELY ABSTRACT AND INTELLECTUAL, WHY ARE THESE RELEVANT TO DESCRIBING THE WAY THE PHYSICAL WORLD WORKS? >>OKAY, SO NOW HOW DO WE SOLVE THE MYSTERY, HOW DOES THAT MAKE SENSE? >WELL, ONE EXPLANATION IS THERE ARE TONS AND TONS OF MATHEMATICAL STRUCTURES THAT WE COULD HAVE STUDIED THAT WOULD BE NO USE AT ALL IN STUDYING THE PHYSICAL WORLD, AND WHY DID MATHEMATICIANS START STUDYING THESE STRUCTURES THAT TURNED OUT TO BE USEFUL, AND THE ANSWER IS BECAUSE THEY LIVE IN THE PHYSICAL WORLD AND THEIR THOUGHTS ABOUT STRUCTURES ARE GENERATED BY LIVING IN THE PHYSICAL WORLD. >>SO WHAT WE HAVE WITH THESE ABSTRACT MATHEMATICAL STRUCTURES, IDEAS, NUMBERS, RELATIONSHIPS IS SOMETHING THAT DOESN'T EXIST IN THE PHYSICAL WORLD BUT DOES IT EXIST? >IT DEPENDS ON WHO YOU TALK TO. THERE ARE FOUR VIEWS: A MENTALISTIC VIEW THAT IT'S IN THE HEAD; A PHYSICALISTIC VIEW THAT IT'S IN THE PHYSICAL WORLD; A PLATONISTIC VIEW THAT IT'S NON-PHYSICAL AND NON-MENTAL ABSTRACT OBJECTS; AND THEN AN ANTI-REALIST VIEW THAT THERE JUST ARE NO MATHEMATICAL OBJECTS AND THERE'S NOTHING. >>SO THE THIRD VIEW IS A PLATONISTIC VIEW, NOW WHY IS IT CALLED THAT? >SO AN ABSTRACT OBJECT IS AN OBJECT THAT DOESN'T EXIST IN SPACE AND TIME SO IT'S NOT PHYSICAL, IT'S NOT MENTAL, IT DOESN'T ENTER INTO CAUSAL RELATIONS, SO IT'S NOT LIKE ANY OBJECT WE EVER ENCOUNTER IN OUR ORDINARY LIVES. AND THE BELIEF IN THOSE KINDS OF OBJECTS, ABSTRACT OBJECTS, THAT'S CALLED PLATONISM BECAUSE IT WAS FAMOUSLY PLATO'S VIEW THAT THERE WERE SUCH THINGS. THE RIGHT KIND OF PLATONISM IS THE STRONGEST KIND OF REALISM YOU CAN HAVE AND FICTIONALISM IS THE STRONGEST KIND OF ANTI-REALISM YOU CAN HAVE, THAT MATHEMATICS IS LITERALLY FALSE. >>SO YOU'VE DEFENDED BOTH OF THESE VIEWS. >YES. SO, THE ONLY THING THEY DISAGREE ON IS DO THE OBJECTS EXIST OR NOT, AND IT DOESN'T LOOK LIKE WE HAVE ANY WAY OF KNOWING WHETHER THEY EXIST SO WE CAN'T DISCOVER WHETHER PLATONISM OR FICTIONALISM IS THE RIGHT VIEW. >>BUT YOU THINK IT'S ONE OR THE OTHER. >NO. I DON'T. I THINK THAT THERE'S NO RIGHT ANSWER. >>SO IT'S NOT JUST THAT WE CAN'T KNOW BUT THERE IS NO RIGHT ANSWER. >THERE IS NO RIGHT ANSWER ABOUT, THERE'S NO FACT OF THE MATTER ABOUT WHETHER THE OBJECTS EXIST. >>IT'S NOT THAT WE JUST CAN'T DISCOVER IT? >RIGHT. AND YOU MIGHT GO, WELL HOW COULD THAT BE? WELL, I'LL TELL YOU. WHEN YOU FIRST HEAR ABOUT PLATONISM - I TEACH THIS TO MY STUDENTS AND THEY SCRATCH THEIR HEADS, THEY'RE GOING WHAT? WHAT'S THE VIEW? IT'S NON-PHYSICAL, NON - WHAT? AND I REMEMBER HAVING THIS EXPERIENCE WHEN I FIRST HEARD ABOUT IT, WHEN I WAS AN UNDERGRAD, AND THEN I WORKED AND WORKED AND WORKED ON THIS AND I GOT MORE COMFORTABLE WITH THE IDEA OF ABSTRACT OBJECTS, THAT'S WHAT PHILOSOPHERS DO. THEY GET REALLY COMFORTABLE WITH THIS AND THEN THEY TALK AND THEY ACT LIKE OH, IT'S JUST AN EASY QUESTION. BUT I ACTUALLY THINK THE BEGINNING STUDENT WAS RIGHT. IT'S COMPLETELY UNCLEAR WHAT IT COULD MEAN FOR THERE TO BE AN ABSTRACT OBJECT, THIS THING THAT'S COMPLETELY NON-PHYSICAL, NON-MENTAL, NON-SPATIAL, NON-TEMPORAL, NON-CAUSAL. WHAT IS IT? YOU'RE TELLING ME WHAT IT ISN'T, I DON'T KNOW WHAT IT IS. THE NOTION OF AN ABSTRACT OBJECT IS SO UNCLEAR THAT WE DON'T EVEN KNOW WHAT WOULD COUNT AS IT BEING THE CASE THAT ABSTRACT OBJECTS EXIST. IT'S NOT LIKE THERE'S A VILLAGE IN NEPAL SOMEWHERE THAT WE COULD ACTUALLY GO AND DISCOVER. I JUST DON'T THINK THERE'S A RIGHT ANSWER TO IT. >>THE EFFECTIVENESS OF MATHEMATICS IS CLEAR. WHY THEN IS THE ESSENCE OF MATHEMATICS SO FOGGY? IS MATH MENTAL, PHYSICAL, PLATONIC, OR JUST NOT REAL? I NEED A MATHEMATICIAN. IN ICELAND FOR A GATHERING OF COSMOLOGISTS I MEET GREGORY CHAITIN, A MATHEMATICIAN, NOT A PHILOSOPHER. GREG SPECIALIZES IN COMPLEXITY THEORY AND COMPUTATION, WHICH DRIVE HIS STRONG OPINIONS OF WHAT MATH IS ALL ABOUT. GREG, IS THERE ANYTHING IN MATHEMATICS THAT'S SO FUNDAMENTAL THAT IT'S DISCOVERED? >WHEN YOU'RE A MATHEMATICIAN AND YOU FIND SOMETHING THAT FEELS REALLY FUNDAMENTAL YOU MAY THINK THAT IF YOU HADN'T FOUND IT SOMEBODY ELSE WOULD BECAUSE IN SOME SENSE IT'S GOT TO BE THERE, BUT SOME MATHEMATICS FEELS MUCH MORE CONTRIVED, LIKE AN EXERCISE THAT YOU DO TO BE ABLE TO PUBLISH A PAPER. I SORT OF SWING BOTH WAYS ON THIS. I THINK IF YOU LOOK INTO THE INNER RECESSES OF THE UNCONSCIOUS OF A LOT OF MATHEMATICS, AND I INCLUDE MYSELF, I THINK IN A WAY WE HAVE THIS THEOLOGICAL MEDIEVAL BELIEF. WE BELIEVE IN THIS WORLD OF IDEAS, IN THIS PLATONIC WORLD OF PERFECT IDEAS OF MATHEMATICAL CONCEPTS OTHERWISE WE'RE WASTING OUR LIVES. >>WELL, THERE'S NOTHING WRONG WITH - >IS IT ALL A GAME? IS IT ALL A GAME THAT WE JUST INVENT AS WE GO ALONG? >>THERE'S NOTHING WRONG WITH THAT IF THAT'S TRUE. >WELL, SOME PEOPLE FOR SOME - >>I WANT TO KNOW IT'S TRUE. I DON'T WANT TO MAKE YOU HAPPY. >WELL, THERE ARE SOME FIELDS IN MATHEMATICS THAT I DON'T LIKE - I WON'T MENTION ANY NAMES - I THINK ARE PROBABLY INVENTED NOT DISCOVERED, BUT I'D LIKE TO HAVE THE FANTASY THAT I HAVEN'T THROWN MY LIFE AWAY COMPLETELY AND I DIDN'T JUST INVENT IT AND IT EXPRESSES SOME KIND OF FUNDAMENTAL REALITY OUT THERE. IT'S INDEPENDENT OF THE PHYSICAL WORLD. IT'S A SEPARATE REALITY, AND I DON'T KNOW WHERE IT IS. DON'T ASK ME WHERE THE POSITIVE INTEGERS LIVE, BUT THEY DON'T LIVE HERE I DON'T THINK. YOU KNOW, IT'S LIKE A RELIGION IN A WAY. IT'S LIKE A SUBJECT THAT IS STUCK IN THE MIDDLE AGES. I MEAN IN A WAY I WOULD SAY, TO BE PROVOCATIVE, THEY'RE THOUGHTS IN THE MIND OF GOD. I MEAN, WHERE ARE THESE - WHERE IS THE SUBJECT? IT'S NOT HERE. SO YOU BELIEVE IN SOME INVISIBLE WORLD, BETTER THAN OURS, PURER. THIS IS BEGINNING TO SOUND A LITTLE RELIGIOUS, ISN'T IT, IN SOME, IT'S A STRANGE FORM OF RELIGION, BUT IT'S - >>DOES THAT UPSET YOU, THAT IT SOUNDS RELIGIOUS? >NO. >>YOU SOUND DEFENSIVE. >WELL, GIVEN THE CURRENT SITUATION ONE HAS TO SOUND DEFENSIVE. LET ME EXPLAIN THE STRANGE TURN THAT MY OWN THINKING HAD. I STARTED OFF LIKE A TRADITIONAL MATH STUDENT, WHICH IS I BELIEVE IN MATHEMATICAL REALITY, SO IN A WAY THAT'S SAYING MATHEMATICS IS IN THE MIND OF GOD IT'S PERFECT. OKAY? SO I START OFF THAT WAY. AND WHAT IS MY FINAL CONCLUSION 40 YEARS LATER? MY FINAL CONCLUSION AFTER A LIFETIME OBSESSED, TOTALLY OBSESSED WITH ALL OF THIS, IS THAT MATHEMATICIANS SHOULD BEHAVE A LITTLE BIT MORE LIKE EXPERIMENTAL SCIENTISTS DO, AND IF THEY DO COMPUTER EXPERIMENTS AND THEY SEE THAT SOMETHING SEEMS TO BE THE CASE AND THEY CAN'T PROVE IT, AND THIS IS A VERY USEFUL TRUTH, IF IT WERE TRUE, THEN MAYBE THEY SHOULD ADD THAT AS A NEW AXIOM. >>AND ADMIT THAT YOU'RE NOT GOING TO BE ABLE TO PROVE IT IN A TRADITIONAL WAY. >GIVE UP ON IT. NOW, A MATHEMATICIAN WILL REEL BACK IN HORROR AT THIS, BUT I THINK MY WORK PUSHES IN THIS DIRECTION. THE THING IS, IN MY CASE, I WAS SORT OF FORCED AGAINST MY WILL IN THE DIRECTION OF SAYING THAT MATHEMATICS IS EMPIRICAL, OR TO PUT IT IN OTHER WORDS, WE INVENTED IT AS WE GO, SO IT'S SORT OF CONTRADICTORY PSYCHOLOGICALLY TO START WITH THIS POSITION AND THEN END UP WITH THE OPPOSITE POSITION. SO I DON'T KNOW WHERE WE STAND. IN A WAY IT'S A RIDICULE ABSURDUM OF THE TRADITIONAL VIEW OF MATHEMATICS BECAUSE IF YOU BELIEVE IN IT THEN YOU HAVE TO DISAGREE WITH IT. SO WHERE ARE WE? I'M NOT QUITE SURE WHERE WE ARE. >>INVENTED OR DISCOVERED? GREG'S IDEAS ABOUT MATH HAVE CHANGED FROM CERTAINTY TO UNCERTAINTY ABOUT WHETHER MATH HAS ALWAYS EXISTED. AFTER A LIFETIME IN MATH GREG IS NOT QUITE SURE? SO WHERE DOES THAT LEAVE ME? AND YES, IT DOES MATTER. MATH IS FUNDAMENTAL TO EXISTENCE. ARE THERE NEW WAYS OF THINKING? STEPHEN WOLFRAM, A PHYSICIST AND ORIGINAL THINKER, HAS DEVELOPED WHAT HE CALLS A NEW KIND OF SCIENCE, WHICH IS FOUNDED ON THE SHOCKING IDEA THAT SIMPLE RULES, NOT COMPLEX MATHEMATICS, CONSTRUCT REALITY. IS THIS POSSIBLE? COULD ALL MATH BE AN ARTIFACT OF SIMPLE RULES? >I'VE BEEN INTERESTED FOR A LONG TIME IN QUESTIONS ABOUT SORT OF WHAT IS THE ESSENCE OF MATHEMATICS? I MAKE MY LIVING BUILDING THIS THING CALLED MATHEMATICA, WHICH ATTEMPTS TO COVER IN THE BROADEST POSSIBLE SENSE THE KINDS OF THINGS THAT MATHEMATICS MIGHT ENCOMPASS. BUT SO, THE QUESTION THAT I'VE BEEN INTERESTED IN, AND ALSO FROM THE POINT OF VIEW OF BASIC SCIENCE IS, IS THE MATHEMATICS THAT WE SORT OF PRACTICE TODAY THE ONLY POSSIBLE MATHEMATICS, OR IS IT A MATHEMATICS THAT IS SORT OF A GREAT ARTIFACT OF OUR CIVILIZATION BUT SORT OF AN HISTORICAL ACCIDENT ARTIFACT? THE CONCLUSION THAT I'VE SORT OF RESOUNDINGLY COME TO IS THAT THE MATHEMATICS THAT WE HAVE TODAY IS, IN FACT, REALLY AN HISTORICAL ARTIFACT. NOW THAT'S NOT HISTORICALLY IN THE TRADITION OF MATHEMATICS ITSELF, THAT'S NOT WHAT PEOPLE HAVE TENDED TO CONCLUDE. THEY'VE TENDED TO THINK THAT MATHEMATICS IS SORT OF THE MOST GENERAL POSSIBLE FORMAL ABSTRACT SYSTEM. IF YOU LOOK AT THE HISTORY OF MATHEMATICS THAT'S CERTAINLY NOT HOW IT ORIGINALLY STARTED OUT. I MEAN, IN ANCIENT BABYLON YOU KNOW, THERE WAS ARITHMETIC FOR COMMERCE AND OTHER THINGS, AND THERE WAS GEOMETRY FOR LAND SURVEYING, AND WHAT I THINK HAS REALLY BEEN THE HISTORY OF MATHEMATICS IS THE PROGRESSIVE GENERALIZATION OF ARITHMETIC AND GEOMETRY PLUS ONE KEY METHODOLOGICAL IDEA THAT ONE CAN MAKE THEOREMS AND ABSTRACT PROOFS OF THOSE THEOREMS. ONE CAN ASK THE QUESTION IF ONE JUST SORT OF ARBITRARILY LOOKS AT FORMAL SYSTEMS WILL THEY TEND TO HAVE THE CHARACTER OF MATHEMATICS AS WE KNOW IT TODAY? WILL THEY TEND TO HAVE THE FEATURE OF THAT? MOST OF THE THINGS ONE ASKS ABOUT ONE CAN SUCCESSFULLY PROVE THEOREMS ABOUT. I THINK IN BOTH CASES THE ANSWER IS NO, NOT REALLY. SO, FOR EXAMPLE, ONE THING ONE CAN DO IS TO KIND OF ULTIMATELY DECONSTRUCT MATHEMATICS. IF ONE LOOKS AT, YOU KNOW, THERE ARE MAYBE THREE MILLION PAPERS THAT HAVE BEEN PUBLISHED ABOUT MATHEMATICS, OKAY, AND THESE ARE ALL BASED ON A CERTAIN SET OF AXIOMS. THE AXIOMS ARE WHAT YOU GROW MATHEMATICS FROM. THE AXIOMS ARE QUITE SIMPLE. OUR PARTICULAR MATHEMATICS IS THE PARTICULAR SET OF AXIOMS YOU CAN WRITE DOWN ON THESE COUPLE OF PAGES, BUT THERE'S A WHOLE UNIVERSE OF POSSIBLE MATHEMATICSES OUT THERE, WHAT ARE THEY LIKE? FIRST QUESTION MIGHT BE, WHERE DOES OUR PARTICULAR MATHEMATICS LIE IN THIS UNIVERSE OF POSSIBLE MATHEMATICSES? IS IT POSSIBLE MATHEMATICS NUMBER 1, IS IT POSSIBLE MATHEMATICS NUMBER 10, IS IT POSSIBLE MATHEMATICS NUMBER QUINTILLION? WHERE DOES IT LIE? THE ANSWER IS, DEPENDS ON EXACTLY HOW YOU ENUMERATE THE SPACE, BUT ROUGHLY IT'S ABOUT THE 50,000TH POSSIBLE AXIOM SYSTEM, SO RIGHT THERE, SORT OF IN THE UNIVERSE OF POSSIBLE AXIOM SYSTEMS, THE UNIVERSE OF POSSIBLE MATHEMATICSES, THERE'S LOGIC. IF THE ALIENS DELIVERED, YOU KNOW, A DIFFERENT POSSIBLE MATHEMATICS YOU KNOW I DON'T THINK WE WOULD BE ABLE TO IMMEDIATELY SAY THAT'S NOT A REASONABLE VALID MATHEMATICS. >>BECAUSE IT WOULD BE SELF CONSISTENT EVEN THOUGH IT WOULD BE RADICALLY DIFFERENT. >YES. NOW, WHAT'S HAPPENED IN THE HISTORY OF MATHEMATICS IS THAT MOST THINGS PEOPLE HAVE BEEN INTERESTED IN HAVE ENDED UP BEING EVENTUALLY SOLVABLE, ALTHOUGH SOMETIMES WITH EFFORT AND CENTURIES OF WORK AND SO ON, BUT ONE OF THE THINGS THAT I SUSPECT IS THAT THAT'S ACTUALLY NOT REALLY THE WAY THAT IS THE TRUE, THE TRUE REALITY TO MATHEMATICS. THAT REALLY, IF WE WERE TO JUST SORT OF ASK MATHEMATICAL QUESTIONS ARBITRARILY THAT THE VAST MAJORITY OF THEM WOULD END UP TURNING OUT TO BE UNSOLVABLE, AND IN FACT THAT UNSOLVABILITY IS ACTUALLY CLOSE AT HAND IN MATHEMATICS. WE JUST DON'T SEE IT BECAUSE THE PARTICULAR WAY THAT MATHEMATICS HAS PROGRESSED HISTORICALLY HAS TENDED TO AVOID IT. NOW YOU MIGHT SAY, BUT MATHEMATICS IS A GOOD MODEL OF THE NATURAL WORLD AND MATHEMATICS HAS BEEN SORT OF DRIVEN BY MODELING THE NATURAL WORLD, I THINK THERE'S KIND OF A CIRCULAR ARGUMENT BECAUSE WHAT'S HAPPENED IS THAT THOSE THINGS WHICH HAVE BEEN SUCCESSFULLY ADDRESSED IN SCIENCE IN STUDYING THE NATURAL WORLD ARE JUST THOSE THINGS THAT METHODS LIKE MATHEMATICS HAVE SUCCESSFULLY ALLOWED US TO ADDRESS. SO I THINK ONE OF THE EXCITING THINGS THAT ONE REALIZES IS THAT HUMAN MATHEMATICS, IT'S ONE OF THE GREAT ARTIFACTS OF OUR CIVILIZATION. IT'S ONE OF THE SORT OF PERFECT, WONDERFUL THINGS THAT'S BEEN PRODUCED BY A HUGE AMOUNT OF HUMAN EFFORT. >>BUT IT'S AN ARTIFACT. >BUT IT'S AN ARTIFACT AND THERE IS MUCH MORE OUT THERE IN THE SORT OF SPACE OF ALL POSSIBLE MATHEMATICSES AND I THINK IN THE FUTURE WE WILL SEE AN INCREASING KIND OF REALIZATION AND AN INCREASING ABILITY TO EXPLORE ALL THAT OTHER UNIVERSE OF MATHEMATICSES AND IT WILL BE PROFOUNDLY IMPORTANT, NOT ONLY FOR MATHEMATICS BUT FOR OUR SCIENCE AND FOR OUR TECHNOLOGY. >>STEPHEN REJECTS THE IDEA THAT OUR MATHEMATICS HAS DEEP SIGNIFICANCE, RATHER HE LOOKS TO THE VASTLY LARGE SPACE OF ALL POSSIBLE MATHEMATICS. BUT MATH AS MERE ARTIFACT STILL TROUBLES ME. THAT'S WHY I GO TO MIT TO MEET FRANK WILCZEK, A NOBEL LAUREATE IN PHYSICS WHO SEEKS COMMON SENSE IN GETTING AT THE ESSENCE OF MATHEMATICS. FRANK, WHAT DO YOU DO WITH THAT FUNDAMENTAL QUESTION ABOUT MATHEMATICS? IS IT INVENTED BY MINDS, HUMAN MINDS, OR IS IT DISCOVERED? IT WAS ALWAYS THERE IN SOME PLATONIC FORM OR PLATONIC HEAVEN? >THE ANSWER IS BOTH. IT'S BOTH INVENTED AND DISCOVERED BUT I THINK IT'S MOSTLY DISCOVERED. I THINK MATHEMATICS IS THE PROCESS OF TAKING AXIOMS, TAKING A DEFINITE SET OF ASSUMPTIONS AND DRAWING OUT THEIR CONSEQUENCES. SO MAKING UP AXIOMS IS INVENTION AND DRAWING OUT THEIR CONSEQUENCE IS DISCOVERY. IF YOU LOOK AT WHAT MATHEMATICIANS ACTUALLY DO, MOST MATHEMATICIANS DON'T SPEND A LOT OF TIME MAKING UP NEW AXIOMS. THEY SPEND THEIR TIME DRAWING OUT THE CONSEQUENCES OF AXIOMS THAT HAVE PROVEN TO BE RICH AND INTERESTING AND CONSEQUENCES. OCCASIONALLY YOU HAVE TO INTRODUCE NEW SETS OF AXIOMS LIKE THE PASSAGE FROM EUCLIDEAN GEOMETRY TO NON-EUCLIDEAN GEOMETRY. THESE ARE EPICAL EVENTS IN MATHEMATICS AND THOSE ARE, IN A SENSE, INVENTION. >>THAT'S A GOOD ONE. IS THAT AN INVENTION, BECAUSE IF INDEED THE UNIVERSE MAY NEED NON-EUCLIDEAN GEOMETRY FOR EINSTEIN'S THEORY OF RELATIVITY, YOU KNOW, IT WAS THERE ALL THE TIME. >WELL, INVENTIONS HAVE TO COME FROM SOMEWHERE SO THEY COULD BE INSPIRED BY NATURAL PHENOMENA. IN THE CASE OF NON-EUCLIDEAN GEOMETRY THEY WERE SOMEWHAT. GAUST DEVELOPED THOSE CONCEPTS IN THE CONTEXT OF SURVEYING THE EARTH, THE EARTH IS ROUND AND, BUT THE UNDERLYING DYNAMIC IS THAT YEAH, YOU CAN INVENT AXIOMS AD LIB TO DO ANYTHING, BUT MOST OF THEM WON'T BE INTERESTING, AND THE ONES THAT ARE INTERESTING ARE DISCOVERIES. SO EVEN THE INVENTIONS HAVE SOME ELEMENT OF DISCOVERY. YOU DISCOVER WHAT ARE INTERESTING AXIOMS. SO, I SAID ORIGINALLY THAT MATHEMATICS IS MORE DISCOVERED THAN INVENTED, THAT ONLY MAKES IT MORE SO. >>SO IS MATHEMATICS INVENTED OR DISCOVERED? HERE'S WHAT WE KNOW. MATHEMATICS DESCRIBES THE PHYSICAL WORLD WITH REMARKABLE PRECISION. WHY? THERE ARE TWO POSSIBILITIES. FIRST, MATH SOMEHOW UNDERLIES THE PHYSICAL WORLD, GENERATES IT. OR SECOND, MATH IS A HUMAN DESCRIPTION OF HOW WE DESCRIBE CERTAIN REGULARITIES IN NATURE. AND BECAUSE THERE IS SO MUCH POSSIBLE MATHEMATICS, SOME EQUATIONS ARE BOUND TO FIT. AS FOR THE ESSENCE OF MATHEMATICS THERE ARE FOUR POSSIBILITIES: MATH COULD BE PHYSICAL IN THE REAL WORLD, ACTUALLY EXISTING; OR, MENTAL, IN THE MIND, ONLY A HUMAN CONSTRUCT; OR, PLATONIC, NON-PHYSICAL, NON-MENTAL, ABSTRACT OBJECTS; OR, FICTIONAL, ANTI-REALIST, UTTERLY MADE UP. MATH IS EITHER PHYSICAL, MENTAL, OR PLATONIC, OR FICTIONAL. CHOOSE ONLY ONE. IN PEERING DOWN THE DARK WELL OF DEEP REALITY, MATHEMATICS BRINGS US CLOSER TO TRUTH.
