1931–32 Scottish Division One

Transcription

>> --ALWAYS WANTED TO DO THIS, AND WHEN YOU GET READY TO DO SOMETHING LIKE THIS, AFTER YOU GET READY FOR A WHILE, YOU BEGIN TO WONDER IF IT WAS SUCH A GOOD IDEA. (audience laughing) BECAUSE, I MEAN, THE 50 MINUTES THING HAS GOT ME REALLY NERVOUS. THERE'S PROBABLY ABOUT 50 WEEKS' WORTH OF MATERIAL HERE, WHICH WILL NOW BE DISTILLED IN 50 MINUTES OR LESS. THE REASON FOR THE 50 MINUTES IS BECAUSE I'D LIKE TO HAVE A FEW MINUTES AT THE END FOR QUESTIONS. IF YOU HAVE A QUESTION THAT YOU JUST CAN'T-- YOU CAN'T SURVIVE UNLESS YOU ASK IT IMMEDIATELY, YOU MAY CERTAINLY DO SO, AND I MAY GIVE AN ANSWER, OR I MAY SAY, "THAT'S A GREAT QUESTION, I'LL ANSWER IT AT THE END," OR I MAY GIVE THE ANSWER THAT I GIVE MOST OF THE TIME, NOW THAT I AM DEPARTMENT HEAD, WHICH IS, "I HAVE NO IDEA." (audience laughing) UM, WHEN YOU'RE GONNA GIVE A TALK LIKE THIS, YOU GOTTA FIGURE OUT WHERE YOU'RE GONNA START. AND... OKAY, SO, MATHEMATICS-- BASIC MATHEMATICS-- COUNTING OR MEASURING THINGS, MEASURING AREAS, MEASURING LENGTHS. SOMEBODY HAD TO DO THAT FIRST, BUT WE DON'T KNOW WHO, WE DON'T KNOW WHERE, WE DON'T KNOW WHEN. AND YOU KNOW, THERE ARE SOME EXTREMELY DEEP, FUNDAMENTAL IDEAS HERE THAT WE'RE PROBABLY NEVER GONNA KNOW THE ANSWER TO. I MEAN, IT WAS GENIUS FOR SOMEONE TO REALIZE THAT IF YOU HAD THIS, AND IF YOU HAD... THOSE, AND IF YOU HAD... THESE... THAT SOMEHOW, THEY WERE ALL RELATED BY SOMETHING THAT WE NOW CALL "FIVE." AND TO ACTUALLY REALIZE THAT THERE WAS THAT UNITING THEME THAT NEEDED A NAME AND A NOTATION, THAT'S DEEP. AND WE HAVE NO IDEA WHO DID THAT. AND ANTHROPOLOGISTS MIGHT HAVE THEORIES, BUT THAT'S WAY OUT OF MY REALMS. SO, WE DON'T KNOW WHO DID THAT, SO WE'LL START SOMEPLACE WHERE WE HAVE SOME IDEA OF WHAT WENT ON. WHEN YOU TALK ABOUT EARLY CIVILIZATIONS-- EGYPT, UH, MESOPOTAMIA, THE BABYLONIANS, THE PRESENT-DAY IRAQ, IRAN, SYRIA, THAT AREA, THE TIGRIS AND EUPHRATES... THE CRADLE OF CIVILIZATION. UH, CHINA AND INDIA COME TO MIND. NOW, ALL THESE ALL HAD CERTAIN THINGS IN COMMON. THEY ALL DEALT WITH ARITHMETIC AND GEOMETRY-- IN SOME CASES, FAIRLY ADVANCED. BABYLONIAN GEOMETRY ACTUALLY GOT PRETTY INTRICATE. THEY KNEW THE PYTHAGOREAN THEOREM A LONG TIME BEFORE PYTHAGORAS WAS AROUND. CHINA WAS USING THE EQUIVALENT OF OUR DECIMAL FRACTIONS-- NOT WITH OUR NOTATION, OF COURSE-- UM, SEVERAL HUNDRED YEARS B.C., WHICH IS ABOUT 1,500 YEARS BEFORE THEY WERE USED IN EUROPE. AND INDIA-- THE NUMERALS THAT WE USE ARE CALLED "HINDU-ARABIC NUMERALS" BECAUSE THEY BEGAN IN INDIA PROBABLY AROUND THE 3rd CENTURY B.C. SO, THEY WORKED WITH ARITHMETIC, THEY DID SOME AMAZING THINGS WITH GEOMETRY, CONSIDERING THE ERA, BUT ONE THING THAT IS NOTABLE ABOUT ALL OF THE MATHEMATICS FROM THIS ERA IS THERE IS NOT ONE INSTANCE ANYWHERE OF ANYTHING THAT WOULD BE CONSIDERED A FORMAL PROOF, OR EVEN A DISCUSSION OF OR DEMONSTRATION. "HERE'S WHY SOMETHING WORKS"-- THEY NEVER DID THAT. NEVER. IT WAS "HERE'S A PROBLEM. "DO THIS, DO THIS, DO THIS, DO THIS, DO THIS," AND THEY OFTEN ENDED WITH SOMETHING LIKE, "AND YOU WILL SEE THAT IT IS RIGHT." THEY NEVER EXPLAINED WHY. IT WAS JUST SOMETHING YOU WERE SUPPOSED TO "SEE." >> THAT'S NOT MATHEMATICS, IS IT? I'M SORRY... (audience laughing) >> THERE'S A HECKLER IN EVERY CROWD. (audience laughing) UM... AND THE MATHEMATICS THAT THEY DID WAS OFTEN VERY PRACTICAL, I MEAN, COMMERCE AND FINDING AREAS FOR LAND AND WHAT NOT, BUT IT WAS OFTEN RECREATIONAL. YOU FIND PROBLEMS THAT WERE CLEARLY POSED MERELY TO BE PUZZLES OR TO MAYBE SEE IF YOU COULD STUMP SOMEONE ELSE-- YOU KNOW, LIKE THE WORD PROBLEMS WE GIVE OUR ALGEBRA STUDENTS. NOW, FOR THOSE OF US WHO ARE A LITTLE MORE REFINED IN OUR TASTES AND FEEL THAT YOU DON'T HAVE MATHEMATICS UNLESS YOU HAVE FORMAL PROOF, THEN WE HAVE PROOF BY DEDUCTIVE REASONING, WHICH WAS STARTED BY THE GREEKS... AND THALES, 600 B.C., THE FLYER-- THE PERSON IN THE UPPER LEFT-HAND CORNER OF THE FLYER-- IS SOME ARTIST'S RENDITION OF WHO THALES WAS. HE IS THE ONE WHO IS CREDITED WITH FIRST GIVING LOGICAL PROOFS. NOW, A LOGICAL PROOF IN MATHEMATICS IS AIRTIGHT IF DONE CORRECTLY. YOU START WITH AXIOMS, POSTULATES-- YOU START WITH STATEMENTS THAT EVERYBODY BELIEVES TO BE TRUE. GIVEN TWO POINTS, THERE IS EXACTLY ONE STRAIGHT LINE THAT GOES THROUGH THEM-- THAT IS AN AXIOM. YOU START WITH DEFINITIONS... "A CIRCLE IS THE SET OF ALL POINTS IN THE PLANE "A FIXED DISTANCE FROM ANY GIVEN POINT." AND THEN, YOU USE LOGIC TO DEDUCE OTHER PROPOSITIONS FROM THOSE DEFINITIONS AND AXIOMS. THAT APPROACH-- I MEAN, THE IDEA THAT YOU HAVE TO PROVE THINGS BASED ON PREVIOUSLY ESTABLISHED RESULTS OR PREVIOUS DEFINITIONS AND AXIOMS-- THAT IS STILL USED TODAY. THE GREEKS SET THE STANDARD FOR THAT. AND THE GEOMETRY THAT THEY DID-- MOST OF THEIR WORK WAS IN GEOMETRY. THEY DID VERY LITTLE WITH NUMBER THEORY AND EVEN LESS WITH ALGEBRA... UM, A LITTLE BIT WITH TRIG. I WANNA SAY SOMETHING ABOUT THAT IN A MOMENT, BUT THEIR GEOMETRY-- ALL THE GEOMETRY THAT YOU LEARNED IN HIGH SCHOOL AND IF YOU HAD ANY GEOMETRY PRIOR TO SOME NON-EUCLIDEAN STUFF-- ALL THAT GEOMETRY, THE GREEKS KNEW 2,500 YEARS AGO. THEY KNEW ALL OF THAT AND A LOT MORE. THE TRIGONOMETRY IS INTERESTING BECAUSE IT WAS ASTRONOMERS WHO STARTED THE STUDY OF TRIGONOMETRY AND WAS ESSENTIALLY AN OUTGROWTH OF-- AN EXTENSION OF GEOMETRY, AND THEY WERE TRYING TO FIGURE OUT HOW THE EARTH MOVED AROUND THE SUN. NOW, THIS IS 3rd CENTURY B.C. GREEKS KNEW-- OR AT LEAST SOME OF THEM KNEW-- THAT THE EARTH MOVED. THE EARTH MOVED AROUND THE SUN. IT WAS NOT SOMETHING THAT WAS BORN IN EUROPE IN THE 16th OR 15th CENTURY. IT'S ACTUALLY MUCH OLDER THAN THAT. IN TERMS OF WHO WAS DOING THESE THINGS-- WELL, PYTHAGORAS. WE ALL KNOW THAT NAME. IT'S THE PYTHAGOREAN THEOREM, NAMED AFTER HIM. HE MAY HAVE PROVED IT-- NO ONE KNOWS FOR SURE. IT'S FAIRLY CERTAIN THAT THE SCHOOL OF PHILOSOPHY THAT HE FOUNDED-- SOMEONE IN THAT SCHOOL PROVED IT. THAT SCHOOL WAS ALSO THE FIRST TO COME UP WITH THE IDEA THAT THERE IS SUCH A THING AS AN IRRATIONAL NUMBER-- A NUMBER THAT IS NOT SIMPLY THE RATIO OF TWO INTEGERS. AS FAR AS THE PYTHAGOREANS WERE CONCERNED, THAT WAS A HUGE FINDING. EUCLID IS NOT NECESSARILY AS COMMON A NAME, BUT HIS "ELEMENTS" IS THE MOST INFLUENTIAL MATHEMATICS TEXT EVER. IT'S BEEN IN PRINT FOR 2,400 YEARS. IT BASICALLY-- IF YOU STUDIED MATHEMATICS FROM THE TIME OF EUCLID UNTIL INTO THE 19th CENTURY, WHEN YOU STARTED STUDYING SERIOUS MATHEMATICS, YOU STUDIED EUCLID. LINCOLN STUDIED EUCLID. NEWTON STUDIED EUCLID. ALL THE BIG NAMES STUDIED EUCLID. AND EUCLID BASED HIS GEOMETRY ON THAT AXIOMATIC METHOD WITH DEDUCTIVE REASONING, AND IT WAS CONSIDERED TO BE SUCH AN EXCELLENT WORK THAT IT WAS VIEWED AS INFALLIBLE. IT WAS ALMOST AS THE SAME LEVEL-- IN THE CHRISTIAN ERA, IT WAS ALMOST AT THE SAME LEVEL AS THE BIBLE. YOU KNOW, PEOPLE WHO KNOW THE BIBLE AND WILL QUOTE IT BY CHAPTER AND VERSE? PEOPLE WHO KNEW EUCLID-- WHICH WAS ANYONE VERSED IN MATHEMATICS IN MEDIEVAL EUROPE AND LATER-- THEY WOULD QUOTE EUCLID, AS WELL. "IN BOOK I, PROPOSITION 47, WE FIND--" AND BY THE WAY, THAT'S THE PYTHAGOREAN THEOREM. (audience laughing) IT WAS REVERED. WELL, SOMEBODY HAS TO KNOW THIS STUFF. ARCHIMEDES-- WE'LL PROBABLY SAY A LITTLE BIT MORE ABOUT HIM LATER, BUT FOR RIGHT NOW, YOU HAVE TO SAY THAT ARCHIMEDES WAS 2,000 YEARS AHEAD OF HIS TIME, IN TERMS OF WHAT HE WAS ABLE TO DO WITH MATHEMATICS. NOT ONLY DID GEOMETRY AND NUMBER THEORY-- HE DID SOME VERY AMAZING THINGS WITH SOME CALCULUS IDEAS. ONE OF THE GREATEST MATHEMATICIANS OF ALL TIME. HIS IDEAS PREDATED THOSE OF EUROPE BY ABOUT 2,000 YEARS. AND IF YOU'VE EVER STUDIED CONIC SECTIONS-- THOSE GOT BROUGHT UP RECENTLY. YEAH, CONIC SECTIONS-- UH, ELLIPSE, CIRCLE, PARABOLA, HYPERBOLA. THE GREEKS STUDIED THOSE, AND APOLLONIUS DID SIGNIFICANT TREATISE ON THOSE IN THE 2nd CENTURY B.C. CONIC SECTIONS PLAYED A VERY IMPORTANT ROLE LATER WITH ASTRONOMY AND THE ORBITS OF PLANETS AROUND THE SUN. MEANWHILE, FROM 400 TO 1200 A.D. IN EUROPE, NOTHING WAS HAPPENING. (audience giggling) NOTHING. I MEAN, THEY CALL IT "THE DARK AGES" FOR A REASON. IF YOU'RE LOOKING FOR ANYTHING OF SIGNIFICANCE IN EUROPE BETWEEN 400 AND 1200 A.D., THERE'S NOTHING THERE. YOU'RE LOOKING IN THE WRONG PLACE. MAYBE BE LOOKING IN THE MIDDLE EAST. MESOPOTAMIA, THE BABYLONIANS-- THE PROPHET MUHAMMAD FOUNDED THE RELIGION OF ISLAM IN THE 7th CENTURY A.D. BY AROUND 700, THE FOLLOWERS OF ISLAM WERE BASICALLY HEEDING THE CALL OF THAT RELIGION WHICH, AT THAT TIME, WAS VERY STRONG ON "WE WANT TO SEEK KNOWLEDGE, "WE WANT TO SEEK TRUTH," AND SO, THERE WAS A... A TREMENDOUS CALLING AMONG SCHOLARS TO LEARN FOR THE SAKE OF LEARNING. AND SO, BECAUSE OF THAT, UM-- (cell phone ringing) MIDDLE EASTERN-- I WON'T DO WHAT I DO IN CLASS, BUT-- MIDDLE EASTERN MATHEMATICIANS ACQUIRED ANCIENT GEOMETRIC TEXTS FROM THE GREEKS, TRANSLATED THEM INTO ARABIC. THEY TOOK THE NUMERALS FROM INDIA-- THE HINDU NUMERALS-- AND NOW, WE CALL THEM "HINDU-ARABIC," AND CREATED WHAT WE WOULD NOW CALL "ARITHMETIC WITH HINDU-ARABIC NUMERALS." YOU KNOW, ADDING, SUBTRACTING, MULTIPLYING, DIVIDING, LIKE WE DO ON A NORMAL BASIS. THAT WAS DONE ON A REGULAR BASIS IN THE MIDDLE EAST BY 1000 A.D. THIS WORD RIGHT HERE... UH, "THE CONDENSED BOOK OF CALCULATION OF AL-JABR." AL-JABR. ALGEBRA. THAT'S WHERE "ALGEBRA" CAME FROM. THE WORD "ALGEBRA" COMES FROM THIS WORD BY AL-KHWARIZMI IN 825. A TREMENDOUS AMOUNT OF ALGEBRA WAS BEING DONE IN THE MIDDLE EAST AT THAT TIME. NOW, YOU WOULDN'T RECOGNIZE IT AS SUCH BECAUSE IT WAS VERY GEOMETRIC IN NATURE, IN THE SENSE THAT YOU BASE THINGS ON GEOMETRIC IDEAS, AND THEY DIDN'T HAVE THE NOTATION WE HAD. BUT IF YOU LOOK CAREFULLY, A LOT OF WHAT WE DO TODAY, A LOT OF THE FACTORING WE DO-- YOU KNOW, SQUARING BINOMIALS, CUBING BINOMIALS, ALL THAT STUFF THAT ARE STUDENTS SOMETIMES DON'T DO SO WELL WITH... THEY WERE DOING THAT. AND IF YOU ARE OF A LITERARY BENT AND YOU'RE FAMILIAR WITH "THE RUBAIYAT OF OMAR KHAYYAM," THAT IS THE SAME OMAR KHAYYAM-- HE WAS ALSO AN EXCELLENT MATHEMATICIAN AND DID SOME WORK WITH THIRD-DEGREE POLYNOMIAL EQUATIONS-- CUBIC EQUATIONS-- THAT WAS NOT SURPASSED IN EUROPE UNTIL ABOUT 16th CENTURY. INTERESTING THING ABOUT THIS IS THERE IS A HUGE AMOUNT ABOUT MIDDLE EASTERN MATHEMATICS FROM THIS TIME THAT IS VIRTUALLY UNTOUCHED. THERE ARE MOSQUES AND PALACES AND BASEMENTS FILLED WITH MANUSCRIPTS THAT HAVE NEVER BEEN LOOKED AT. AND SO, THE CONTRIBUTION FROM MIDDLE EAST AT PRESENT IS SOMEWHAT KNOWN, BUT NOT EVEN CLOSE TO KNOWN COMPLETELY. WELL, OKAY, SO, 1200 A.D.-- EVENTUALLY, EUROPE STARTED TO WAKE UP. THE HINDU-ARABIC NUMERALS THAT WE USE STARTED MAKING THEIR WAY INTO EUROPE BY AROUND THE 10th CENTURY, AND IF YOU'VE EVER HEARD OF THE FIBONACCI SEQUENCE... LET'S SEE, HOW DOES THAT GO AGAIN? 1, 1, 2-- WHAT'S THE NEXT ONE? >> THREE. >> AND THEN? >> FIVE. >> JUST KEEP ADDING... >> EIGHT. >> OKAY. FIBONACCI IS KNOWN FOR THAT, AND THAT'S WONDERFUL, BUT THAT'S REALLY, IN A SENSE, TRIVIAL. WHAT HE'S REALLY TO BE HONORED FOR IS THE FACT THAT HE WAS A VERY VOCAL ADVOCATE OF HINDU-ARABIC NUMERALS BEING USED IN EUROPE. UH, PRINTED THE "LIBRE ABACI" IN 1228 A.D. AND IT ONLY TOOK ABOUT 300 YEARS FOR HINDU-ARABIC NUMERALS TO BECOME WIDESPREAD IN EUROPE. I MEAN, IF YOU THINK ABOUT THAT, YOU LOOK AT THE NUMBERS WE USE ON A REGULAR BASIS, YOU'D THINK, "WHY WOULDN'T YOU USE SOMETHING LIKE THAT?" THEY WERE USING ROMAN NUMERALS PRIOR TO THAT. AND ROMAN NUMERALS ARE REALLY... CUMBERSOME, HARD TO READ, HARD TO COMPUTE WITH, OKAY? THEY BASICALLY USED ROMAN NUMERALS TO RECORD THINGS WITH, IN TERMS OF THE MERCANTILE CLASS. BUT 300 YEARS IT TOOK TO ACTUALLY GET UNIVERSAL ADOPTION OF HINDU-ARABIC NUMERALS ACROSS EUROPE. AND JUST TO GIVE YOU A SENSE OF HOW ADVANCED HINDU-ARABIC NUMERALS WERE CONSIDERED, I'M GONNA READ YOU SOMETHING HERE. THIS IS FROM ABOUT 1450. "A GERMAN MERCHANT HAD A SON WHOM HE DESIRED "TO GIVE AN ADVANCED COMMERCIAL EDUCATION. "HE APPEALED TO A PROMINENT PROFESSOR OF A UNIVERSITY "FOR ADVICE AS TO WHERE HE SHOULD SEND HIS SON. "THE REPLY WAS THAT IF THE MATHEMATICAL CURRICULUM "OF THE YOUNG WAS TO BE CONFINED TO ADDING AND SUBTRACTING, "HE COULD GO TO A GERMAN UNIVERSITY," BUT IF HE WANTED TO LEARN HOW TO MULTIPLY AND DIVIDE, THE ONLY COUNTRY WHERE HE COULD LEARN THAT WAS ITALY. IT WAS HIGH-POWERED STUFF BACK IN 1450. (audience giggling) SO, I DON'T KNOW IF SOME OF YOU WANNA SHARE THAT WITH YOUR 095 CLASSES OR NOT... (audience laughing) SO, EUROPE BEGINS TO AWAKEN. FRACTIONS HAD BEEN AROUND, ESSENTIALLY FOREVER. I MEAN, IN RECORDED HISTORY, FRACTIONS HAVE ALWAYS BEEN PRESENT. THE EGYPTIANS WERE USING FRACTIONS 5,000 YEARS AGO. UM, NOW, FRACTIONS GAVE WAY TO FRACTIONS THAT WERE A LITTLE MORE PREDICTABLE. BASE 60 FRACTIONS. BABYLONIANS USED BASE 60 NUMBERS-- SEXAGESIMAL NUMBER SYSTEM. NOT DECIMAL-- SEXAGESIMAL. THEY USED BASE 60, AND THEY USED BASE 60 FRACTIONS. AND WE STILL DO TODAY, RIGHT? THAT'S WHAT THIS IS. THIS IS 57-60ths OF A DEGREE, AND THEN THIS IS 48 SECONDS, BUT THERE ARE 3,600 SECONDS IN ONE DEGREE. AND SO, THIS IS 48-- THESE ARE SEXAGESIMAL FRACTIONS. SO, IT'S NOT LIKE YOU'VE NEVER SEEN 'EM BEFORE. BUT THOSE WERE USED PRETTY EXTENSIVELY IN EUROPE AND THE MIDDLE EAST WELL INTO THE 17th CENTURY. DECIMAL FRACTIONS AS WE KNOW THEM-- UM, SIMON STEVIN, BELGIUM, ADVOCATED BASE 10 FRACTIONS USING, UM... A NOTATION THAT WE MIGHT FIND A BIT ODD. WE WOULD USE THIS NOTATION. STEVIN'S "ART OF TENTHS" FROM 1585 USED THIS NOTATION... WHICH, IF YOU LOOK AT IT THE RIGHT WAY, YOU'RE REALLY TALKING ABOUT NEGATIVE EXPONENTS, AREN'T YA? 'CAUSE THIS IS ONE-TENTH, RIGHT? TEN TO THE NEGATIVE ONE. THIS IS 4/100ths, TEN TO THE NEGATIVE TWO. IT'S NOT A HORRIBLE NOTATION, BUT IT'S KINDA CUMBERSOME. BUT THE IMPORTANT THING WAS TWO THINGS-- HE PUBLISHED A BOOK SAYING, "WE SHOULD ALL BE USING "THESE DECIMAL FRACTIONS AND HERE'S HOW THEY WORK." AND AT LEAST HE HAD A NOTATION. SO, IT WAS A STEP IN THE RIGHT DIRECTION. UM, DID NOT GET MUCH ATTENTION FOR QUITE SOME TIME. IT WAS ONE OF THOSE BOOKS THAT GOT PUBLISHED AND HARDLY ANYBODY READ IT... BUT THEN, ALONG CAME LOGARITHMS. NOW, THIS WILL WARM THE HEARTS OF CERTAIN PEOPLE IN HERE WHO ARE OF A CERTAIN AGE-- I'M SEEING SOME SMILES FROM SOME AND OTHERS JUST BLANK LOOKS 'CAUSE YOU'RE TOO YOUNG. (audience chuckling) UM, JOHN NAPIER, SCOTTISH, HAD THE BRILLIANT IDEA THAT YOU COULD TRANSFORM MULTIPLICATION INTO ADDITION. AND THE THEORY IS HE GOT THAT IDEA FROM TRIGONOMETRIC IDENTITIES. THERE ARE TRIGONOMETRIC IDENTITIES THAT ALLOW YOU TO TRANSFORM PRODUCTS TO THE SUMS... AND HE BASICALLY FIGURED, "WELL, YOU OUGHTA BE ABLE TO DO THAT "FOR JUST PLAIN OLD NUMBERS," SO HE CAME UP WITH AN IDEA THAT ALLOWED YOU TO TRANSFORM PRODUCTS INTO SUMS, QUOTIENTS INTO DIFFERENCES, POWERS INTO MULTIPLICATION, AND ROOTS-- LIKE CUBE ROOT-- INTO DIVISION. AND FOR THOSE OF YOU OLD ENOUGH TO REMEMBER, IT WASN'T A LOT OF FUN BUT IT WAS A HECK OF A LOT EASIER THAN DOING IT BY HAND, OKAY? THE PROBLEM WITH NAPIER, THOUGH, WAS THAT HE USED A BASE THAT-- WELL, OKAY. THIS GETS COMPLICATED. HE ESSENTIALLY USED LOGARITHMS OF BASE 1-OVER-E. IT WASN'T EXACTLY THAT, AND "E" HADN'T EVEN BEEN DISCOVERED YET, OKAY? BUT THAT'S BASICALLY WHAT HE HAD. THEY'RE VERY DIFFICULT TO USE. WITH LOGARITHMS OF THAT BASE, AS THE NUMBERS GOT BIGGER, THE LOGARITHMS GOT SMALLER-- IT WAS JUST WEIRD. SO, NAPIER AND BRIGGS MET, AND BRIGGS HAD THE IDEA THAT WE'D USE BASE 10 LOGARITHMS-- THE COMMON LOGARITHMS THAT WERE IN THE TABLES THAT SOME OF US USED, AND YOU HAVE A "LOG" KEY ON YOUR CALCULATOR-- L-O-G WILL FIND LOG-BASE 10 LOGARITHMS. BRIGGS PROBABLY GOES DOWN IN HISTORY AS ONE OF THE MOST HEROIC FIGURES IN THE ERA OF COMPUTATIONAL MATHEMATICS. IN ORDER TO CREATE "LOG" TABLES THAT HE KNEW WOULD BE ACCURATE, HE BEGAN BY TAKING THE NUMBER TEN AND EXTRACTING 54 CONSECUTIVE SQUARE-ROOTS OF IT, BY HAND, TO 30 DECIMAL PLACES. IT TOOK HIM A COUPLE OF MONTHS TO DO THAT. AND THEN, HE BUILT HIS LOGARITHM TABLES METICULOUSLY TO 14 DECIMAL PLACES... AND I SAY "DECIMAL PLACES"-- IF YOU'RE DEALING WITH "LOG"S-- THEY'RE BASE 10 "LOG"S, YOU GOTTA USE BASE 10 FRACTIONS, YOU WANT TO USE DECIMAL NUMBERS. SIMON STEVIN'S DECIMAL NUMBERS REALLY DIDN'T CATCH ON THAT FAST BECAUSE THERE WASN'T A NEED FOR IT, BUT AS SOON AS YOU HAD LOGARITHMS IN PLAY, IT WAS LIKE, "OH, MY GOSH-- WE NEED DECIMAL POINTS," AND THAT'S WHEN DECIMAL NUMBERS REALLY TOOK OFF. NOW, SYMBOLIC ALGEBRA-- THAT'S THE ALGEBRA WE ALL KNOW, OKAY? ALGEBRA WAS NOT LIKE THAT UNTIL VERY RECENTLY. ALGEBRA STARTED OUT BEING RHETORICAL. RHETORICAL ALGEBRA WAS COMMON IN EUROPE AND THE MIDDLE EAST FOR A VERY LONG TIME. HERE IS A STATEMENT IN RHETORICAL ALGEBRA, OKAY? SOME OF YOU CAN TRY THIS ON IN 095, 6, 7, OR 8, ALL RIGHT? "IN THE RULE OF THREE, ARGUMENT, FRUIT, "AND REQUISITION ARE THE NAMES OF THE TERMS. "THE FIRST AND LAST TERMS MUST BE SIMILAR. "REQUISITION MULTIPLIED BY FRUIT "AND DIVIDED BY ARGUMENT IS PRODUCE." (audience giggling) OKAY? "THE FIRST AND LAST MUST BE SIMILAR." ARGUMENT... REQUISITION. REQUISITION MULTIPLIED BY FRUIT PRODUCES PRODUCE-- YOU'RE TALKING ABOUT A PROPORTION. THIS TIMES THIS DIVIDED BY THAT... WILL GIVE YOU "P." IT WAS CALLED "THE RULE OF THREE." IT WAS CALLED "THE GOLDEN RULE." MERCHANTS USE THAT ALL THE TIME TO FIGURE OUT HOW MUCH-- YOU KNOW, IF IT'S THIS MUCH FOR HOW MANY, HOW MUCH IT'LL BE FOR THIS MANY-- THEY USE THAT A LOT. BUT THAT WAS RHETORICAL ALGEBRA. IT WAS PROSAIC BUT IT WASN'T EASY TO DO ANYTHING WITH. SYNCOPATED ALGEBRA WAS A STEP UP. INSTEAD OF USING WORDS, YOU USED ABBREVIATIONS, OKAY, LIKE THIS. THIS IS FROM ITALY, 1494. UM, MY KNOWLEDGE OF ITALIAN IS VAST... IT'S NON-EXISTENT, OKAY? THIS STANDS FOR "CUBO"-- THIS IS "X" CUBED. "MENO," "LESS" MEANS "MINUS." THIS IS "X" CUBED MINUS-- "CENSO" IS FOR "X" SQUARED. YOU GOT "X" CUBED MINUS "X" SQUARED. THIS IS FOR "PLUS." UH, THIS IS THE MOST INTERESTING ONE. THIS IS FOR "COSA." "COSA" TRANSLATES INTO "THING." "THING." WHAT IS THE "THING" YOU ARE SOLVING FOR? THAT'S THE VARIABLE "X." SO, "X" CUBED MINUS "X" SQUARED PLUS "X" EQUALS-- THAT'S "(indistinct)," I THINK-- ZERO. THAT'S A CUBIC EQUATION. OKAY? UM, WHAT MAKES THIS INTERESTING IS THAT "COSA"-- ALGEBRA BECAME KNOWN IN EUROPE-- SEE, THIS IS WHAT, 1494-- YEAH, 1400s, 1500s, 1600s-- ALGEBRA WAS KNOWN AS THE "COSIC ART." "COSIC ART," BECAUSE YOU WORK WITH "COSA," YOU WORK WITH "THINGS," YOU TRY TO FIND "THINGS." YOU TRY TO SOLVE FOR THE UNKNOWN. BUT YOU KNOW, THIS IS NOT OUR ALGEBRA. OUR ALGEBRA-- THE SYMBOLIC ALGEBRA THAT WE'RE FAMILIAR WITH-- REALLY BEGAN TAKING OFF IN THE 16th CENTURY, AND BY THE MIDDLE OF THE 17th CENTURY, IT WAS PRETTY MUCH STANDARD. NOT COMPLETELY, BUT YOU WOULD RECOGNIZE IT. IF YOU PICKED UP SOMETHING FROM THE 1650s IN ALGEBRA AND READ IT, YOU'D RECOGNIZE IT. UM, IT'S HARD TO SAY WHO DID WHAT, WHEN, BECAUSE SO MUCH WAS GOING ON THEN, SO WE'LL JUST SAY, "MANY PEOPLE IN EUROPE DID THIS" BETWEEN 16th AND MIDDLE OF 17th CENTURY. NOW THAT YOU'VE GOT ALGEBRA-- OKAY, SO YOU GOT ALGEBRA, RIGHT? YOU GOT THINGS LIKE "X" CUBED MINUS "X" SQUARED PLUS "X" EQUALS ZERO. WELL, THEN, YOU'VE HAD GEOMETRY, RIGHT? REMEMBER-- GREEKS, GEOMETRY, EUCLID, REVERED, EVERYONE-- ANYONE WHO STUDIED MATHEMATICS KNEW GEOMETRY. NOW YOU'VE GOT ALGEBRA. SO, WHAT'S THE NEXT STEP? UNITE THEM. NOW, I HAVE TO MENTION NICOLE ORESME BECAUSE HE WAS SO FAR AHEAD OF HIS TIME THAT IT'S ALMOST SCARY. 1350-- THAT WAS THE ERA OF THE BLACK DEATH IN EUROPE, AND HERE IS ORESME STUDYING, AMONG OTHER THINGS, VELOCITY-TIME GRAPHS. ARE YOU FAMILIAR WITH THIS AT ALL FROM ANYTHING IN PHYSICS? >> SAY IT AGAIN. >> UH, VELOCITY-TIME GRAPHS. >> SURE. >> BUT FROM THIS ERA. >> NO. >> HE WAS DOING VELOCITY-TIME GRAPHS IN AROUND 1350, WHICH IS ABOUT 300 YEARS AHEAD OF ANYBODY ELSE. AND THEN, HE DIED YOUNG AND PEOPLE PRETTY MUCH FORGOT WHAT HE DID. THE TWO PEOPLE FOR WHOM WE GIVE CREDIT FOR ESTABLISHING MODERN-DAY COORDINATE SYSTEM ARE RIGHT HERE-- FERMAT AND DESCARTES. BOTH FRENCH. BOTH CAME UP WITH THEIR COORDINATE SYSTEMS AT VIRTUALLY THE SAME TIME. FERMAT WAS A LITTLE AHEAD OF DESCARTES... THERE'S PROBABLY A JOKE IN THERE SOMEPLACE. (audience chuckling) BUT I WON'T DO IT. I'M NOT PRUIS. (audience laughing) UM... IT'S AN "IN" JOKE. FERMAT WAS FIRST, BUT FERMAT WAS AN AMATEUR MATHEMATICIAN. HE WAS A LAWYER BY TRADE, AND FROM WHAT I'VE READ, HE WASN'T A VERY GOOD LAWYER. (audience laughing) AND YOU KNOW WHY? HE SPENT SO MUCH TIME DOING MATH. SO, OBVIOUSLY, HE'S ONE OF MY PERSONAL HEROES. UM, HE ACTUALLY CAME UP WITH HIS IDEA FIRST, BUT HE DIDN'T PUBLISH ANYTHING. HE JUST DIDN'T CARE. HE DIDN'T PUBLISH A THING. HE CIRCULATED A FEW MANUSCRIPTS. AND ACTUALLY, DESCARTES GOT A HOLD OF ONE OF THOSE MANUSCRIPTS, AND HE WAS JUST PUTTING THE FINISHING TOUCHES ON HIS METHOD AND PUBLISHED THIS RIGHT AWAY BECAUSE HE WANTED CREDIT. AND OKAY, FINE, I WON'T GET INTO THAT. I MEAN, HE PUBLISHED-- THEY BOTH CAME OVER ESSENTIALLY THE SAME TIME. THEY HAD VERY DIFFERENT TAKES ON THINGS. UM, FERMAT ESSENTIALLY WAS INTERESTED IN GETTING AN EQUATION, "LET'S SEE WHAT WE CAN FIND OUT ABOUT IT, "LET'S STUDY THE EQUATION GEOMETRICALLY." YOU KNOW, LIKE WE DO WITH A GRAPHING CALCULATOR. YOU WANNA FIGURE OUT WHERE THAT-- YOU KNOW THAT EQUATION UP THERE EARLIER-- "X" CUBED MINUS "X" SQUARED PLUS "X" EQUALS ZERO-- WHAT ARE ITS SOLUTIONS? GRAPH IT AND LOOK AT THE "X" INTERCEPTS. THAT'S ROUGHLY WHAT FERMAT WAS INTERESTED IN DOING. DESCARTES WAS MORE INTERESTED IN... "GIVE ME A CURVE-- I WANNA FIND ITS EQUATION, "THEN STUDY THE CURVE ALGEBRAICALLY." HIS INTENT WAS "I WANNA SEE IF WE CAN STUDY EUCLID'S GEOMETRY "FROM AN ALGEBRAIC POINT OF VIEW." OKAY. THAT WAS THE METHOD. A VERY IMPORTANT IDEA THAT UNDERLIES ALL THIS IS THAT VARIABLES, NOW ACTUALLY VARIED, THERE IS A HUGE DIFFERENCE BETWEEN THIS... AND THIS. EQUATIONS LIKE THIS HAD BEEN WORKED ON IN VARIOUS FORMS. BABYLONIANS DID THIS 2,000 YEARS B.C. FIND A NUMBER THAT YOU CAN SUBTRACT TWO FROM IN ORDER TO GET TEN. THIS ACTS AS A PLACEHOLDER AND NOTHING ELSE. THAT'S ALL IT IS. IT JUST TAKES THE SPOT OF SOMETHING YOU DON'T KNOW. THIS, WITHIN THIS CONTEXT-- THIS HAS A CONTINUOUS GRAPH. IT'S A LINE. YOU KNOW, IT LOOKS SOMETHING LIKE THIS... I HOPE IT LOOKS SOMETHING LIKE THAT. IT LOOKS SOMETHING-- IT'S CONTINUOUS. I MEAN, YOU CAN GO FROM POINT TO POINT TO POINT IN A CONTINUOUS MANNER, WHICH MEANS THESE VARIABLES CAN VARY CONTINUOUSLY, AND THAT WAS A BRAND NEW IDEA. THAT WAS HUGE BECAUSE THAT ACTUALLY GAVE RISE TO SOMETHING REALLY SIGNIFICANT IN JUST A FEW YEARS. THE STAGE IS SET. IN 1650 IN EUROPE, GREEK GEOMETRY WAS KNOWN, SYMBOLIC ALGEBRA WAS ESSENTIALLY AS WE HAVE IT TODAY, HINDU-ARABIC NUMERALS, PEOPLE WERE VERY COMFORTABLE WITH THEM, LOGARITHMS WERE BEING USED, AND WE HAD A COORDINATE SYSTEM. SO, NOW, WE ARE READY FOR... ENTER THE CALCULUS. NOW, THE CALCULUS-- THERE IS A DISTINCTION TO BE MADE BETWEEN THIS AND THIS, AND I THINK THE EASIEST WAY I CAN EXPLAIN IT IS SIMPLE. I CAN DO THIS. I'M JUST A GUY. HIGH SCHOOL STUDENTS AROUND THE COUNTRY DO THIS EVERY YEAR. YOU DON'T HAVE TO BE A GENIUS TO DO THIS. I'M LIVING PROOF OF THAT. BUT IF YOU'RE GOING TO DO THIS, YOU HAD TO BE EXTRAORDINARILY CLEVER, BECAUSE THIS IS A COLLECTION OF RULES, NOTATION, AND PROCEDURES THAT CREATES A SYSTEM BY WHICH YOU CAN SOLVE PROBLEMS IN AN ORGANIZED MANNER. THE CALCULUS IS A TOOL THAT APPLIES TO A WIDE VARIETY OF SITUATIONS, AND YOU DON'T HAVE TO BE A GENIUS TO USE IT. THESE-- THE PEOPLE WHO DID THIS HAD TO BE EXTRAORDINARILY CLEVER-- I MEAN, ARCHIMEDES WAS A GENIUS OF THE FIRST RANK, AND TO DO WHAT HE DID, HE HAD TO BE. NOBODY ELSE COULD DO WHAT HE DID IN 3rd CENTURY B.C. HE WAS PRETTY MUCH ALONE... BECAUSE HE WOULD TAKE SOME REALLY FUNDAMENTAL CALCULUS IDEAS AND APPLY THEM IN EXTRAORDINARILY CLEVER WAYS, MOSTLY GEOMETRIC FOR HIM, BUT FOR SOME OF THESE PEOPLE, IT WAS LARGELY GEOMETRIC AND ALGEBRAIC, AND COME UP WITH RESULTS. THESE PEOPLE DOWN HERE-- THESE ARE ALL-- THEY DID THEIR WORK FROM 1600 TO AROUND 1660... AND IMPRESSIVE RESULTS. KEPLER, WHO CAME UP WITH HIS "THREE LAWS OF PLANETARY MOTION"-- ONE OF WHICH IS ALL THE PLANETS ORBIT THE SUN IN THE SHAPE OF AN ELLIPSE, WITH THE SUN AT ONE FOCUS. AND IN ORDER TO DEAL WITH THAT, HE ALSO DEALT WITH SOME PROBLEMS INVOLVING AREA. HE USED CALCULUS TECHNIQUES TO WORK WITH THOSE AREAS. FERMAT, WHO WE MET EARLIER, WAS WORKING TANGENT LINE PROBLEMS. YOU KNOW YOUR CALC PROBLEMS WHERE YOU WANNA FIND MAXIMUMS AND MINIMUMS WHILE LOOKING FOR A TANGENT LINE THAT'S HORIZONTAL? FERMAT WAS DOING THAT IN 1629... BUT HE HAD TO TREAT EACH PROBLEM AS IF IT WAS A BRAND NEW PROBLEM. ALL THESE PEOPLE DID A LOT WITH CALCULUS IDEAS BUT THEY DIDN'T GET TO THE CALCULUS UNTIL SOMEONE WHO MAY HAVE BEEN SITTING IN ON ONE OF BARROW'S CLASSES. ISAAC BARROW WAS AT CAMBRIDGE AND, IN 1664 TO 1665, HE GAVE LECTURES IN WHICH HE DEMONSTRATED THE GEOMETRIC CONNECTION BETWEEN DERIVATIVES AND INTERVALS. AND FOR THOSE OF YOU THAT DON'T KNOW ANY CALCULUS, HE DEMONSTRATED GEOMETRICALLY THAT THE TANGENT LINE PROBLEM WAS THE INVERSE OF THE AREA PROBLEM. HE DEMONSTRATED THAT, AND HE MOVED ON BECAUSE HE DIDN'T SEE ANY SIGNIFICANCE. THAT WAS LEFT TO ONE OF HIS PUPILS. ISAAC NEWTON... WHAT CAN YOU SAY? I MEAN, THERE ARE GENIUSES AND THEN THERE ARE TRANSCENDENT GENIUSES. HIS WORK IN THE CALCULUS-- HIS DISCOVERY OF CALCULUS TOOK PLACE OVER A VERY BRIEF PERIOD. HE PUBLISHED, ESSENTIALLY, NONE OF THAT. HE PUBLISHED NONE OF THAT. SOME OF THAT WORK WAS NOT PUBLISHED TILL THE MID-1900s. IT'S TRUE. AND SOME OF YOU WILL GLARE AT ME WHEN I SAY THIS-- HE WAS A TOP-NOTCH MATHEMATICIAN. ONE OF THE GREATEST EVER. I WOULD NOT SAY HE IS THE GREATEST EVER. HE WAS THE GREATEST PHYSICIST EVER. HIS FAME, IN MY EYES, LIES RIGHT HERE. HE PUBLISHED THIS BOOK, WHICH IS THE MOST INFLUENTIAL BOOK IN THE HISTORY OF SCIENCE-- NO QUESTION. IT TRANSFORMED SCIENCE. SCIENCE BECAME MATHEMATICAL WITH THE PUBLICATION OF THIS BOOK. NEWTON DEMONSTRATED "IF YOU WANNA UNDERSTAND THE WORLD, "IF YOU WANNA UNDERSTAND THE UNIVERSE, "I HAVE SOME FUNDAMENTAL IDEAS. "I'M NOT GONNA TELL YOU WHY THEY WORK. "I FEIGN NO HYPOTHESIS. "I DON'T KNOW WHY GRAVITY WORKS, BUT HERE'S HOW IT WORKS." I MEAN, IF YOU USE THESE IDEAS TOGETHER WITH CALCULUS, YOU CAN DO A LOT. AND THAT TRANSFORMED SCIENCE. BUT HE DIDN'T PUBLISH MUCH... AND HE WASN'T A LOUSY WRITER BUT HE WASN'T THE GREATEST WRITER, AND HE USED LOUSY NOTATION. HIS NOTATION WAS POOR. HE DIDN'T CARE. IF HE WASN'T WRITING FOR THE MASSES, HE WAS WRITING FOR SCHOLARS AND FOR HIMSELF. SO, NEWTON DEVELOPED HIS CALCULUS, LEIBNIZ DEVELOPED HIS A FEW YEARS LATER. IT IS LEIBNIZ'S CALCULUS THAT WE USE TODAY. EVERY TIME YOU WRITE THIS... WHICH IS JUST BEAUTIFUL-- THAT'S LEIBNIZ'S. OCTOBER 1675-- I FORGET THE EXACT DATE. COPIOUS NOTES. I MEAN, YOU CAN LOOK AT HIS NOTES. YOU CAN FIND OUT EXACTLY WHEN HE CAME UP WITH THIS NOTATION, EXACTLY WHEN HE CAME UP WITH THIS NOTATION. THE COMMON NOTATIONS USED IN CALCULUS, THOSE ARE LEIBNIZ'S NOTATION. AND HE WAS VERY-- YOU KNOW, THE CALCULUS-- NEWTON REALLY WASN'T SO CONCERNED ABOUT CREATING THE CALCULUS THAT OTHER PEOPLE COULD USE. HE WANTED TO USE IT BUT HE DIDN'T REALLY CARE TOO MUCH IF OTHER PEOPLE COULD USE IT. LEIBNIZ WANTED OTHER PEOPLE TO KNOW HOW TO USE THE CALCULUS, AND SO, HE ACTUALLY CREATED A JOURNAL-- HE FOUNDED A JOURNAL IN 1684 FOR THE SOLE PURPOSE OF PUBLISHING HIS CALCULUS RESULTS. IT'S ONE WAY TO GET ATTENTION. IT WAS BEFORE THE INTERNET. YOU COULDN'T JUST POST THIS STUFF ONLINE YET-- HAD TO HAVE A JOURNAL, OKAY? IMAGINE SENDING A 140-CHARACTER TWEET, AND "GUESS WHAT-- TODAY, I FOUNDED CALCULUS." (audience laughing) UM, NOW, THERE IS-- IF YOU KNOW ANYTHING AT ALL ABOUT THE HISTORY OF MATHEMATICS, YOU'VE PROBABLY HEARD OF THE PRIORITY DISPUTE BETWEEN NEWTON FOLLOWERS AND LEIBNIZ'S FOLLOWERS, AND I WANNA EMPHASIZE THEIR FOLLOWERS. NEITHER NEWTON NOR LEIBNIZ STARTED THE PRIORITY DISPUTE. NEITHER ONE DID. BUT THEY WERE BOTH DRAWN INTO IT. IT GOT UGLY. IT'S JUST-- IT'S A SAD STATE OF AFFAIRS IN THE HISTORY OF MATHEMATICS. IT REALLY HURT ENGLISH MATHEMATICS BECAUSE THE ENGLISH MATHEMATICIANS-- I MEAN, THEY WERE ARDENT FOLLOWERS OF NEWTON, AND NEWTON WAS HARDER TO UNDERSTAND AND HAD LOUSY NOTATION, AND SO, MATHEMATICAL WORK IN ENGLAND SORT OF STAGNATED FOR ABOUT 100 YEARS. MEANWHILE, IN EUROPE, THINGS WERE REALLY HEATING UP. NOW, I HAD TO INCLUDE THIS. THIS IS SOMETHING I INSERTED A COUPLE OF DAYS AGO-- WE HAVE TO PAUSE FOR JUST A MOMENT. 17th CENTURY IN EUROPE IS KNOWN AS "THE HEROIC CENTURY" IN MATHEMATICS, AND THE BEST WAY TO DESCRIBE THAT IS IF YOU WERE TO COMPARE MATHEMATICS IN 1600 TO MATHEMATICS IN 1700... A VAST CHANGE HAD TAKEN PLACE. EUROPEAN MATHEMATICS IN 1600, ASIDE FROM THE LANGUAGE DIFFERENCES, LOOK PRETTY MUCH LIKE THE MATHEMATICS OF GREECE. THERE WAS A LITTLE MORE ALGEBRA, BUT REMEMBER, THERE WASN'T MUCH ALGEBRAIC NOTATION. SO, IT WAS BASICALLY NOT MUCH DIFFERENT THAN GREEK MATHEMATICS IN 200 B.C. BY 1700, EVERYTHING HAD CHANGED. YOU HAD THE ALGEBRAIC SYMBOLISM, YOU HAD LOGARITHMS, YOU HAD HINDU-ARABIC NUMERALS BEING USED EVERYWHERE, YOU HAD CALCULUS, YOU HAD GRAPHS. THE EMPHASIS HAD CHANGED. 1600 MATHEMATICS WAS ALMOST ALL GEOMETRICS. 1700-- GEOMETRY WAS STILL IMPORTANT, BUT IT WAS DEFINITELY IN THE BACKGROUND. SYMBOLIC ALGEBRA, SYMBOLIC CALCULUS HAD TAKEN THE FOREGROUND, AND IT WAS THE EXPONENTS OF LEIBNIZ WHO TOOK THE CHARGE ON THAT. THE BERNOULLI FAMILY... JACOB AND JOHANN BERNOULLI-- AND THERE WERE OTHERS-- LEARNED FROM LEIBNIZ. LEIBNIZ TAUGHT THEM, THE BERNOULLI'S TAUGHT L'HOSPITAL AND LEONHARD EULER, ANOTHER GIANT TITAN OF MATHEMATICS-- GREATEST MATHEMATICIAN OF THE 18th CENTURY, NO QUESTION ABOUT THAT. UM, BRILLIANT AND SCARILY INTUITIVE. HE JUST SORT OF SAID, "OH, THIS'LL WORK." AND THEN, HE'D WORK OUT A BUNCH OF STUFF AND HE WAS CONVINCED IT WAS RIGHT, AND HE MOVED ON. WE NEED TO COME BACK TO THAT IN A MOMENT. BUT HE WAS DEFINITELY THE CHIEF PROPONENT OF "WE'VE GOT THIS POWERFUL TOOL CALLED 'CALCULUS.' "WE'RE JUST GONNA KEEP USING IT "BECAUSE IT'S GIVING US THESE AMAZING RESULTS," AND THE AMAZING RESULTS WERE SOMETIMES MATHEMATICAL AND SOMETIMES THEY WERE IN THE APPLIED WORLD-- PHYSICS, ASTRONOMY, ENGINEERING. ONE OF MY FAVORITE EPISODES FROM THIS ERA IS THE DISCOVERY OF THE PLANET NEPTUNE, WHICH WAS DONE NOT BY TAKING A WHOLE BUNCH OF TELESCOPES AND PROBING THE NIGHT SKY FOR MONTHS AND YEARS. IT WAS DONE MATHEMATICALLY, USING THE PHYSICS OF NEWTON, AS IMPROVED BY LAPLACE... CELESTIAL MECHANICS. AND THE MATHEMATICS OF THE ERA, WHICH WAS CALCULUS-- THEY DID WHAT'S CALLED "INVERSE PERTURBATION THEORY," AND THEY ESSENTIALLY SAID, "THERE'S GOT TO BE A PLANET "OUT THERE THAT IS SCREWING UP THE ORBIT "OF THE PLANET URANUS... "AND WE HAVE DONE THE MATH, "AND THE PLANET WILL BE THERE AT 9 O'CLOCK "ON THIS CERTAIN NIGHT" IN 1846, AND THEY POINTED THEIR TELESCOPES AND THERE IT WAS. IT'S JUST-- IT'S ASTOUNDING. SO, THE MATHEMATICS WAS WORKING MARVELOUSLY WELL, AND EULER WAS THEIR LEADER, OKAY? NOW, THAT'S GREAT. IF YOU'RE GOING TO MAKE A LOT OF DISCOVERIES, YOU WANT TO JUST TRY STUFF. I TELL MY STUDENTS THAT ALL THE TIME. "JUST TRY STUFF!" OKAY? BUT EVERY ONCE IN A WHILE, IT'S IMPORTANT THAT YOU CAN PROVE THAT WHAT YOU'RE DOING IS CORRECT. AND AFTER A GIDDY 100 YEARS OF JUST FLYING WITH CALCULUS TO SEE WHAT IT WOULD DO, WE GET TO THE 19th CENTURY, WHICH WAS KIND OF A TUMULTUOUS CENTURY IN MATHEMATICS. UH, IT BEGAN BY-- I DON'T KNOW, "BEGAN" IS-- THIS IS NOT NECESSARILY PURELY CHRONOLOGICAL. CHALLENGING TRUTH. TRUTH. IN THIS SLIDE, "TRUTH" MEANS EUCLID. REMEMBER, EUCLID WAS INFALLIBLE. EUCLID'S GEOMETRY WAS THE GEOMETRY. IF YOU'RE GONNA DO GEOMETRY-- IT'S EUCLID. THAT'S IT, THERE'S NOTHING ELSE TO DISCUSS, MOVE ON. WELL... THE PARALLEL POSTULATE IS ONE OF THOSE THINGS-- REMEMBER, NOW IT'S A POSTULATE, IT'S AN AXIOM, IT'S ONE OF THOSE THINGS THAT'S SUPPOSED TO BE REALLY OBVIOUS. YOU KNOW, LIKE, "HERE'S A POINT, HERE'S A POINT-- "THERE'S EXACTLY ONE LINE." EUCLID'S PARALLEL POSTULATE IS VERY WORDY, AND YOU LOOK AT IT AND YOU THINK, "OH, THIS IS SOMETHING WE COULD PROVE." THIS ISN'T AN AXIOM. THIS IS SOMETHING YOU STATE WITHOUT PROOF. WE CAN PROVE THIS, AND MATHEMATICIANS FOR OVER 2,000 YEARS WERE CONVINCED THAT THEY COULD PROVE EUCLID'S POSTULATE FROM OTHER POSTULATES... THAT IT WAS ACTUALLY A THEOREM. AND THEY TRIED AND THEY FAILED, AND THEY TRIED AND THEY FAILED, AND AFTER ABOUT 2,000 YEARS OF TRYING AND FAILING, PEOPLE BEGAN TO GET THE IDEA THAT, "WELL, MAYBE WE CAN'T DO IT." AND SOME INTERESTING THINGS BEGAN TO HAPPEN. IF YOU HAVE EUCLID'S PARALLEL POSTULATE, THEN ONE OF THE THINGS YOU CAN PROVE IS THAT IN ANY TRIANGLE-- WHEN YOU ADD UP ALL THE ANGLES IN ANY TRIANGLE, WHAT DO YOU GET? >> 180 DEGREES. >> 180. NOW, THAT'S EUCLIDEAN GEOMETRY-- E.G.-- OKAY? GAUSS-- WHO MORE HAS TO BE SAID OF-- WHEN HE WAS 15 YEARS OLD-- HE'S 15 YEARS OLD AND HE BEGINS LOOKING AT THE PARALLEL POSTULATE AND SAYING, "THERE'S SOMETHING ABOUT THIS THAT BUGS ME." AND BY 1870, WHICH WAS QUITE A FEW YEARS LATER-- BUT WHEN HE WAS QUITE YOUNG, ACTUALLY, HE HAD CONVINCED HIMSELF THAT EUCLID'S PARALLEL POSTULATE COULDN'T BE PROVEN AND, IN FACT, WASN'T EVEN NECESSARY. THAT YOU COULD HAVE OTHER PARALLEL POSTULATES BESIDES EUCLID'S. HE DIDN'T PUBLISH ANY OF THIS, BECAUSE IT WASN'T GOOD ENOUGH FOR HIM. GAUSS HAD PROBABLY THE HIGHEST STANDARDS OF ANY MATHEMATICIAN WELL INTO THE 20th CENTURY. BUT OTHER PEOPLE WERE THINKING ABOUT THE SAME THINGS, AND HYPERBOLIC GEOMETRY, UM-- HERE, HYPERBOLIC GEOMETRY UNPUBLISHED WAS GAUSS. PUBLISHED-- LOBACHEVSKI AND BOLYAI. THEY CAME UP WITH A PARALLEL POSTULATE WHICH SAID THAT IT'S POSSIBLE TO HAVE A TRIANGLE WITH A SUM OF THE ANGLES INSTEAD OF EQUAL TO 180, LESS THAN 180. NOW, IF YOU'RE SAYING, "THAT'S NOT A TRIANGLE," I CAN UNDERSTAND THAT, BUT IT ALL DEPENDS ON WHAT SPACE YOU'RE IN. IF YOU'RE IN EUCLIDEAN SPACE, THIS ISN'T A TRIANGLE. BUT IF YOU'RE IN HYPERBOLIC SPACE, IT IS. AND YOU SAY, "THERE'S NO SUCH THING AS HYPERBOLIC SPACE..." EINSTEIN FOUND IT VERY USEFUL WHEN HE WAS COMING UP WITH THE MATHEMATICS OF GENERAL RELATIVITY. THE SPACE-TIME CONTINUUM-- CURVED SPACE-- IT'S HYPERBOLIC SPACE. AND THEN, RIEMANN CAME UP WITH... FAT TRIANGLES. SOME OF THE ANGLES ARE GREATER THAN 180 DEGREES. THAT'S RIEMANNIAN GEOMETRY OR ELLIPTIC GEOMETRY. NOW, THE POINT OF ALL THIS WAS... PRIOR TO THE 1860s AND EARLIER, THERE WAS ONE GEOMETRY-- IT WAS EUCLID'S. WHEN THESE CAME OUT, FIRST THEY WERE VIEWED AS NOTHING MORE THAN CURIOSITIES. IT'S LIKE, "OH, YOU HAVE A VERY INTERESTING GEOMETRY. "ISN'T THAT NICE?" AND THEN, PEOPLE MOVED ON. BUT BY AROUND 1870, IT FINALLY HIT HOME-- THIS GEOMETRY AND THIS GEOMETRY, AS WEIRD AS THEY MIGHT HAVE SEEMED, WERE ABSOLUTELY AS VALID, AS TRUE, AS THIS ONE. MATHEMATICALLY, THERE WAS-- IF THERE WAS SOMETHING WRONG WITH THIS, THERE WAS SOMETHING EQUALLY WRONG WITH THIS OR SOMETHING EQUALLY WRONG WITH THIS. OR TO PUT IT IN ANOTHER WAY, IF THIS WAS INVALID AND THIS WAS INVALID, THEN SO WAS THIS. SO, EITHER ACCEPT THEM ALL OR YOU DON'T ACCEPT ANY OF 'EM. AND THAT BECAME APPARENT BY AROUND 1870. SO, "EUCLID EQUALS TRUTH"-- GONE. GONE. GAUSS... ANOTHER CANDIDATE FOR GREATEST MATHEMATICIAN OF ALL TIME. DID NOT PUBLISH NEARLY AS MUCH AS MANY OTHER MATHEMATICIANS DID, BUT HIS MOTTO WAS, "FEW BUT RIPE." HE DIDN'T PUBLISH A LOT, BUT WHAT HE PUBLISHED WAS ESSENTIALLY PERFECT. RIGOROUS, CORRECT IN EVERY DETAIL, HIGH STANDARDS OF RIGOR. IF YOU'VE EVER DONE A FITTING OF CURVES TO A SET OF DATA-- LEAST-SQUARES ANALYSIS-- GAUSS CAME UP WITH THAT. DIFFERENTIAL GEOMETRY, WHICH FOLKS IN CALC 3 ARE GONNA START STUDYING SOON-- DIFFERENTIAL GEOMETRY-- THAT WAS GAUSS'S. HYPERBOLIC GEOMETRY-- GAUSS DIDN'T PUBLISH BUT HE KNEW ABOUT THAT. NUMBER THEORY-- GAUSS DID TREMENDOUS WORK IN NUMBER THEORY. DEFINITELY THE GREATEST MATHEMATICIAN IN THE 19th CENTURY-- NO QUESTION ABOUT THAT. AND BELIEVE ME, THE 19th CENTURY WAS QUITE A CENTURY FOR MATHEMATICIANS. ALGEBRA GOT MODIFIED QUITE A BIT. YOU KNOW THAT CUBIC EQUATION WE LOOKED AT EARLIER? ARE YOU AWARE OF THE FACT THAT IN EUROPE, IN ITALY, IN THE 15th-- ER, EXCUSE ME, 16th CENTURY ITALY, THERE WERE ACTUALLY PUBLICLY HELD EQUATION-SOLVING CONTESTS? I KID YOU NOT. THAT IS NOT HARD TO FIND ONLINE AND IN MANY OTHER SOURCES. BUT THEY ONLY WORKED WITH EQUATIONS UP TO THE FOURTH-DEGREE... BECAUSE AFTER THE FOURTH-DEGREE, THINGS JUST DIDN'T WORK VERY WELL. AND GALOIS ESTABLISHED WHAT IS NOW KNOWN AS "GALOIS THEORY"-- I MEAN, HE DIDN'T CALL IT THAT HIMSELF, OKAY? (audience chuckling) UM, GALOIS THEORY, WHICH ESSENTIALLY PROVED THAT IF IT'S FIFTH-DEGREE OR HIGHER, THERE'S NOT GONNA BE A FORMULA FOR SOLVING IT. YOU'RE JUST GONNA HAVE TO USE AD HOC TECHNIQUES. UM, SOMETHING A LITTLE CLOSER TO HOME, SOMETHING WE CAN ALL RELATE TO-- HAMILTON AND CAYLEY. OKAY-- OH, AND IF YOU'RE IN CALC 3 OR IF YOU ARE IN LINEAR ALGEBRA AND TAKING ANY MATRIX THEORY-- WITH REAL NUMBERS, WE ALL KNOW THIS IS TRUE, AND WHAT IS THAT CALLED? >> (all) COMMUTATIVE PROPERTY. >> COMMUTATIVE PROPERTY. THE ORDER IN WHICH YOU MULTIPLY DOESN'T MATTER, RIGHT? BOTH HAMILTON AND CAYLEY CAME UP WITH NON-COMMUTATIVE ALGEBRAS, BOTH AS THEY WERE LOOKING FOR WAYS TO DESCRIBE ROTATIONS IN THREE-DIMENSIONAL SPACE. THE IDEA THAT THIS WAS NOT TRUE FOR CERTAIN AREAS OF MATHEMATICS WAS... STUNNING. IT WAS LIKE, "YOU COULD DO THAT?" AND IT DIDN'T TAKE LONG FOR MATHEMATICIANS TO GO, "YEAH, WE CAN DO THAT!" AND THEN THEY JUST DISCOVERED ALL SORTS OF OTHER ALGEBRAS. YOU KNOW, YOU GIVE MATHEMATICIANS AN INCH, THEY'LL TAKE A KILOMETER. (audience chuckling) BECAUSE YOU HAVE TO CONVERT THE UNITS. (audience laughing) UM... EUCLID DIDN'T EQUAL TRUTH ANYMORE. THE PARALLEL POSTULATE WAS SHOWN TO BE UNESSENTIAL. YOU COULD USE OTHER POSTULATES WHICH-- NOW, LOOK, FOR 2,000 YEARS, MATHEMATICIANS THOUGHT THAT EUCLID WAS INFALLIBLE, AND IT TURNED OUT THAT THERE WERE OTHER GEOMETRIES. AND SO, THE WHOLE IDEA OF, "WELL, WHAT ABOUT OTHER MATHEMATICS?" WELL, WHAT ABOUT CALCULUS? IS THE FOUNDATION OF CALCULUS INFALLIBLE? WELL, JOSEPH FOURIER DID SOME WORK WITH HEAT TRANSFER THAT INVOLVED TRIGONOMETRIC SERIES, AND THE SHORT STORY ON THAT WAS WHAT HE CAME UP WITH DID REALLY WEIRD STUFF THAT NOBODY COULD EXPLAIN. IT WAS LIKE, "WHAT-- WHAT ARE YOU DOING?" AND HE SAID, "WELL, IT WORKS, SO I'M GONNA KEEP DOING IT." (audience chuckling) THERE WERE SOME REAL CRACKS IN THE FOUNDATION OF CALCULUS. IN THE MID-1800s, THE DEFINITE INTEGRAL HAD NEVER BEEN DEFINED, LIMITS HAD NEVER BEEN CAREFULLY DEFINED, THERE WAS CONFUSION ABOUT THE RELATIONSHIP BETWEEN CONTINUOUS AND DIFFERENTIAL FUNCTIONS. PEOPLE DIDN'T KNOW WHAT THEY WERE DOING. AND SO, THE ARITHMETIZATION OF ANALYSIS TOOK PLACE. VERY QUICKLY WHAT HAPPENED WAS IF YOU'RE GONNA DO LIMITS, YOU GOTTA BASE 'EM ON REAL NUMBERS. REAL NUMBERS CAN BE BASED ON THE RATIONALS, RATIONALS ON THE INTEGERS, INTEGERS ON NATURAL NUMBERS, AND SO... PEANO CAME UP WITH HIS AXIOMS FOR ESTABLISHING THE ENTIRE SET OF NATURAL NUMBERS. DEDEKIND CAME UP WITH HIS VERSION OF DEDEKIND CUTS, WHICH DEFINED REAL NUMBERS IN TERMS OF RATIONALS, WHICH THEN COULD BE BROUGHT BACK DOWN TO THE NATURAL NUMBERS. WEIERSTRASS GAVE THE FINAL PERFECT DEFINITION OF WHAT A "LIMIT" IS, AND IF YOU'VE EVER DONE EPSILON-DELTA PROOFS OF LIMITS, THIS IS THE GUY. RIEMANN DEFINED THE DEFINITE INTEGRAL. CAUCHY-- NOT AS CAREFULLY AS WEIERSTRASS, BUT CAUCHY WORKED WITH CONTINUITY AND DIFFERENTIABILITY. AND THEN, OF COURSE, GAUSS-- ALWAYS THE MOST CAREFUL OF ANYONE-- WAY BACK WHEN HAD WORKED WITH CONVERGENCE OF SERIES. BY 1900, THERE WAS, SUPPOSEDLY, A FIRM, UNSHAKABLE FOUNDATION FOR ALL OF ANALYSIS FOR CALCULUS AND EVERYTHING RELATED TO CALCULUS. THAT TAKES US TO 1900. SO, NOW-- OH, I ALMOST FORGOT. HOW COULD I FORGET THIS? TO-- OKAY, I WAS TEMPTED TO INFINITY AND BEYOND-- (audience laughing) THAT IS THE HEBREW LETTER "ALEPH," SUBSCRIPT IS ZERO, AND THAT IS PRONOUNCED "ALEPH-NULL"... WHICH IS A SYMBOL FOR A CERTAIN LEVEL OF INFINITY. SO, I COULD HAVE SAID, "TO INFINITY AND BEYOND," WHICH I BELIEVE WAS A VERY POPULAR PHRASE IN A MOVIE OF SOME TIME AGO, OKAY, BUT I TYPED THAT UP. UM, CANTOR-- CANTOR DEVELOPED HIS THEORY OF SETS TO HELP BRING SOLID STRUCTURE TO THIS EDIFICE THAT WE CALL "CALCULUS." HE OVERTHREW GREEK THOUGHT OF MORE THAN 2,000 YEARS, WHICH WAS "POTENTIAL INFINITY IS FINE, "BUT THERE'S NO SUCH THING AS ACTUAL INFINITY." THE GREEKS, WITH MAYBE A FEW EXCEPTIONS-- THERE IS SOME EVIDENCE THAT ARCHIMEDES MIGHT HAVE ACCEPTED ACTUAL INFINITY, BUT THAT'S TENUOUS EVEN NOW. FOR THE MOST PART, GREEKS FELT THAT INFINITY WAS POTENTIAL. YOU KNOW, ONE, TWO, THREE, FOUR, FIVE, SIX-- HOW FAR CAN YOU GO? "WELL, YOU CAN GO FOREVER," YOU KNOW? "I CAN NAME A NUMBER BIGGER THAN YOU." "OH, YEAH?" "YEAH-- A MILLION." "A MILLION AND ONE." "A BILLION." "A BILLION AND ONE." YOU CAN KEEP GOING... BUT THAT WAS POTENTIAL. THEY ACCEPTED THAT. WHAT THEY DID NOT ACCEPT IS, OKAY, ALL OF THE NATURAL NUMBERS, THERE IS THIS SET THAT CONTAINS ALL THE NATURAL NUMBERS AND HERE IT IS. THERE'S THE SET OF ALL NATURAL NUMBERS. IT CAN BE THOUGHT OF AS AN ACTUAL COLLECTION. YOU NEED TO GIVE THAT BACK WHEN YOU'RE DONE. (audience laughing) THERE'S AN ACTUAL COLLECTION THAT THEY'D, "NO, THERE'S NO SUCH THING." THEY DO NOT BELIEVE THAT WAS POSSIBLE. CANTOR NOT ONLY EMBRACED THIS BUT RAN WITH IT, AND IF YOU WANT TO LEARN MORE ABOUT THIS IDEA THAT SOME INFINITIES ARE BIGGER THAN OTHERS-- IN FACT, SOME ARE A LOT BIGGER THAN OTHERS-- OUR OCTOBER SEMINAR, KELLY ROZIN WILL BE SPEAKING TO US ON INFINITY. SO, I HAVE A FEELING CANTOR'S NAME MIGHT COME UP AGAIN. SO, NOW, OKAY, AS I WAS SAYING, IT WAS AROUND 1900 AND NOW WE ARE AT ESSENTIALLY THE MODERN, MODERN ERA-- UM, YOU KNOW, MATHEMATICS OF THE LAST 100 YEARS. THAT FIRST BULLET-POINT IS A GROSS UNDERSTATEMENT. MUCH, MUCH, MUCH MORE MATHEMATICS HAS BEEN DISCOVERED OR CREATED IN THE LAST 100 YEARS THAT IN THE PREVIOUS 5,000. IT IS JUST MIND-BOGGLING HOW MUCH HAS BEEN DONE. AND SO, YOU KNOW, SUMMARIZING THAT IN TWO MINUTES-- NO. BUT ALSO THE PROBLEM IS-- MOST OF IT IS PRETTY DEEP, DIFFICULT, ABSTRACT STUFF. AND IT'S HARD TO SAY WHICH OF THAT IS REALLY GONNA BE SEEN AS IMPORTANT 100 YEARS FROM NOW, BUT SINCE IT'S MY TALK... (audience laughing) I WILL NOW TELL YOU TWO THINGS THAT-- WHEN I'M THINKING ABOUT "HOW AM I GONNA END THIS?" I THOUGHT, "YOU KNOW WHAT? "I'M GONNA THINK OF A COUPLE THINGS "THAT I THINK WILL STILL BE SIGNIFICANT "MAYBE, YOU KNOW, 200 YEARS FROM NOW." AND I THOUGHT, "OH, THE FIRST TWO THINGS THAT CAME TO MIND"-- WELL, HERE'S THE FIRST ONE. THIS IS ACTUALLY ABSTRACT AND FAIRLY DEEP, BUT THE GIST OF IT, A LITTLE BIT SIMPLIFIED, IS NOT HARD TO GRASP, OKAY? KURT GODEL-- IN MY OPINION, GREATEST LOGICIAN OF ALL TIME. BRILLIANT ENOUGH TO BE ABLE TO WALK AROUND THE GROUNDS OF THE ADVANCED-- THE INSTITUTE FOR ADVANCED STUDY AT PRINCETON WITH EINSTEIN AND CARRY ON CONVERSATIONS AT EINSTEIN'S LEVEL. THEY WERE VERY GOOD FRIENDS. I WISH, I WISH, I WISH SOMEONE HAD WANDERED WITH THEM EVERY ONCE IN A WHILE AND TAKEN NOTES, 'CAUSE IT WOULD HAVE BEEN INTERESTING. YOU KNOW, MAYBE THEY DID TALK ABOUT THE WEATHER. MAYBE THEY TALKED ABOUT OTHER THINGS, BUT I GUESS WE'LL NEVER KNOW. GODEL, IN 1931, ESSENTIALLY SAID, "YOU KNOW THESE SOLID FOUNDATIONS THAT YOU THINK YOU'VE GOT? "IT'S AN ILLUSION." 'CAUSE HERE'S WHAT HE WAS ABLE TO PROVE-- IF YOU TAKE ANY FORMAL AXIOMATIC SYSTEM THAT'S RICH ENOUGH TO CONTAIN THE NATURAL NUMBERS-- SO, LET'S JUST USE THE NATURAL NUMBERS AS AN EXAMPLE-- YOU'VE GOT YOUR AXIOMS, YOU'VE GOT YOUR DEFINITIONS. SO, YOU CREATE THIS SYSTEM FOR THE NATURAL NUMBERS. IT IS IMPOSSIBLE TO PROVE THAT THAT-- IT IS IMPOSSIBLE TO PROVE WITHIN THAT SYSTEM THAT THAT SYSTEM IS COMPLETE. IN OTHER WORDS, GODEL SHOWED THAT THERE WILL ALWAYS BE STATEMENTS IN THAT SYSTEM THAT YOU CAN'T PROVE WITHIN THE SYSTEM. FURTHERMORE, HE SHOWED THAT ONE OF THOSE STATEMENTS THAT YOU CAN'T PROVE WITHIN THE SYSTEM IS THE STATEMENT THAT THIS SYSTEM IS CONSISTENT. IF YOU WANT TO ESTABLISH THAT THE SYSTEM OF ALL NATURAL NUMBERS DOESN'T CONTAIN CONTRADICTIONS, YOU CAN'T DO IT WITHIN THE SYSTEM. GODEL PROVED THAT. SO, THE DREAM OF CREATING ROCK-SOLID FOUNDATIONS THAT ARE INFALLIBLE-- IT MAY BE TRUE... WE CAN'T PROVE IT. SOME OF THE PEOPLE IN THIS ROOM WHO ARE NOT IN THE MATH DEPARTMENT LIKE TO REMIND ME OF THINGS LIKE THIS EVERY ONCE IN A WHILE, JUST TO TRY AND KEEP ME HONEST, AND I TELL THEM TO GO AWAY. (audience laughing) AND THIS IS ALMOST DIAMETRICALLY OPPOSITE. THIS IS NOT ABSTRACT. THIS IS PRACTICAL. YOU'VE GOT CELL PHONES, YOU'RE ON FACEBOOK, YOU USE THE INTERNET-- HECK, I USE THE INTERNET. I DON'T KNOW WHAT A CELL PHONE IS. I'VE SEEN THEM. (audience chuckling) YOUTUBE-- ALL THAT STUFF THAT INVOLVES COMMUNICATION FROM ONE PLACE TO ANOTHER-- THAT ALL DEPENDS ON TRANSFERRING INFORMATION USING ONES AND ZEROS. ONES AND ZEROS. IT'S ALL ABOUT ONES AND ZEROS. SOMEONE HAD TO COME UP WITH THE MATHEMATICAL FORMULATION WHICH SAID, "YOU CAN DO IT, AND HERE'S HOW YOU DO IT," AND THAT WAS CLAUDE SHANNON. NOW, CLAUDE SHANNON IS NATIVE-- HE WAS BORN IN GAYLORD, WENT TO THE UNIVERSITY OF MICHIGAN, ENGINEERING AND MATHEMATICS BACHELOR'S DEGREES FROM U. OF M., MASTER'S IN ENGINEERING FROM M.I.T., PhD IN MATHEMATICS FROM M.I.T. THE ENGINEERS CLAIM HIM AS THEIR OWN, BUT HE HAS A PhD IN MATHEMATICS, SO HE BELONGS TO US. (audience laughing) AND THIS PAPER-- THIS IS-- THIS IS, I MEAN, A GROUND-BREAKING PAPER. WELL, YOU'RE GONNA USE YOUR CELL PHONE LATER? A COUPLE OF YOU HAVE BEEN CHECKING IT IN HERE WHILE I'VE BEEN TALKING-- THAT'S OKAY. I'M NOT GLARING AT YOU. UM, YOU'RE DOING THAT BASED ON THIS MATHEMATICS. THAT MADE IT POSSIBLE. SO, WHETHER YOU LOVE THE DIGITAL ERA THAT WE'RE IN OR YOU LOATHE IT, HE'S RESPONSIBLE. WITHOUT THAT MATHEMATICS, YOU WOULDN'T HAVE ANY OF IT. UM... THAT'S ABOUT IT IN TERMS OF WHAT I NEED TO SAY. NOW, IF YOU WANT TO STUDY FURTHER, THESE ARE ALL GOOD-- UM, MY FAVORITE IS PROBABLY THIS ONE. THIS IS GOOD BUT IT'S HIGHER LEVEL. THIS IS GOOD BUT IT'S LOWER LEVEL. THIS IS FANTASTIC BUT IT'S A TOUGH READ SOMETIMES. IF YOU WANT TO KNOW ANYTHING ABOUT THE HISTORY OF MATHEMATICAL NOTATIONS, YOU CANNOT DO ANY BETTER THAN THIS. THIS IS THE SOURCE. YOU GO ONLINE AND YOU LOOK UP "HISTORY OF THE PLUS SIGN," THEY TOOK IT FROM CAJORI-- MAYBE WITH OR WITHOUT ATTRIBUTION. AND IF YOU ACTUALLY WANT A SITE ONLINE THAT YOU CAN GO TO FOR HISTORY OF MATHEMATICS, THAT'S OUT OF THE UNIVERSITY OF ST. ANDREWS IN SCOTLAND-- THAT'S A GOOD PLACE. UH, DO YOU HAVE ANY QUESTIONS ABOUT ANYTHING? WELL, THEN WE'RE DONE. (audience chuckling) IF YOU DO HAVE ANY QUESTIONS AND WANNA HANG AROUND, THAT'S FINE. THANKS FOR COMING. (applause)