MATHEMATICS
CONTENTS CHAPTER 1: INTRODUCTION: THE MATHEMATICAL IMAGE FURTHER READING CHAPTER 2: PLATONISM THE ORIGINAL PLATONIST SOME RECENT VIEWS WHAT IS PLATONISM? THE PROBLEM OF ACCESS THE PROBLEM OF CERTAINTY PLATONISM AND ITS RIVALS CHAPTER 3: PICTURE-PROOFS AND PLATONISM BOLZANO’S ‘PURELY ANALYTIC PROOF’ WHAT DID BOLZANO DO? DIFFERENT THEOREMS, DIFFERENT CONCEPTS? INDUCTIVE MATHEMATICS SPECIAL AND GENERAL CASES INSTRUCTIVE EXAMPLES REPRESENTATION SEEING INDUCTION? THREE ANALOGIES ARE PICTURES EXPLANATORY? SO WHY WORRY? APPENDIX CHAPTER 4: WHAT IS APPLIED MATHEMATICS? REPRESENTATIONS ARTIFACTS BOGUS APPLICATIONS DOES SCIENCE NEED MATHEMATICS? REPRESENTATION VS DESCRIPTION STRUCTURALISM CHAPTER 5: HILBERT AND GÖDEL THE NOMINALISTIC INSTINCT EARLY FORMALISM HILBERT’S FORMALISM HILBERT’S PROGRAMME SMALL PROBLEMS GÖDEL’S THEOREM GÖDEL’S SECOND THEOREM THE UPSHOT FOR HILBERT’S PROGRAMME THE AFTERMATH CHAPTER 6: KNOTS AND NOTATION KNOTS THE DOWKER NOTATION THE CONWAY NOTATION POLYNOMIALS CREATION OR REVELATION? SENSE, REFERENCE AND SOMETHING ELSE CHAPTER 7: WHAT IS A DE?NITION? THE OF?CIAL VIEW THE FREGE–HILBERT DEBATE REDUCTIONISM GRAPH THEORY LAKATOS CONCLUDING REMARKS CHAPTER 8: CONSTRUCTIVE APPROACHES FROM KANT TO BROUWER BROUWER’S INTUITIONISM BISHOP’S CONSTRUCTIVISM DUMMETT’S ANTI-REALISM LOGIC PROBLEMS CHAPTER 9: PROOFS, PICTURES AND PROCEDURES IN WITTGENSTEIN A PICTURE AND A PROBLEM FOLLOWING A RULE PLATONISM ALGORITHMS DISPOSITIONS KNOWING OUR OWN INTENTIONS BROUWER’S BEETLE RADICAL CONVENTIONALISM BIZARRE EXAMPLES NATURALISM THE SCEPTICAL SOLUTION MODUS PONENS OR MODUS TOLLENS? WHAT IS A RULE? GRASPING A SENSE PLATONISM VERSUS REALISM SURVEYABILITY THE SENSE OF A PICTURE CHAPTER 10: COMPUTATION, PROOF AND CONJECTURE THE FOUR COLOUR THEOREM FALLIBILITY PERFECT NUMBERS COMPUTATION IS ? NORMAL? FERMAT’S LAST THEOREM THE RIEMANN HYPOTHESIS CLUSTERS OF CONJECTURES POLYA AND PUTNAM CONJECTURES AND AXIOMS CHAPTER 11: HOW TO REFUTE THE CONTINUUM HYPOTHESIS WHAT IS THE CONTINUUM HYPOTHESIS? HOW COULD WE DETERMINE THE TRUTH OF CH? KREISEL’S ANALOGY FREILING’S REFUTATION OF CH WHAT MIGHT THE CONTINUUM BE? TWO OBJECTIONS WHAT DID THE THOUGHT EXPERIMENT CONTRIBUTE? TWO MORALS APPENDIX: FREILING’S VERSION CHAPTER 12: CALLING THE BLUFF CALLING THE BLUFF MATH WARS: A REPORT FROM THE FRONT ONCE MORE: THE MATHEMATICAL IMAGE NOTES CHAPTER INTRODUCTION: THE MATHEMATICAL IMAGE L ET’S BEGIN WITH A NICE EXAMPLE, THE PROOF THAT THERE ARE IN?NITELY MANY PRIME NUMBERS IF ASKED FOR A TYPICAL BIT OF REAL MATHEMATICS, YOUR FRIENDLY NEIGHBOURHOOD MATHEMATICIAN IS AS LIKELY TO GIVE THIS EXAMPLE AS ANY FIRST, WE NEED TO KNOW THAT SOME NUMBERS, CALLED ‘COMPOSITE’, CAN BE DIVIDED WITHOUT REMAINDER OR BROKEN INTO FACTORS (E G 6 2 3, 561 3 17), WHILE OTHER NUMBERS, CALLED ‘PRIME’, CANNOT (E G 2, 3, 5, 7, 11, 13, 17, ) NOW WE CAN ASK: HOW MANY PRIMES ARE THERE? THE ANSWER IS AT LEAST AS OLD AS EUCLID AND IS CONTAINED IN THE FOLLOWING THEOREM: THERE ARE IN?NITELY MANY PRIME NUMBERS PROOF: SUPPOSE, CONTRARY TO THE THEOREM, THAT THERE IS ONLY A ?NITE NUM- BER OF PRIMES THUS, THERE WILL BE A LARGEST WHICH WE CAN CALL P NOW DE?NE A NUMBER N AS 1 PLUS THE PRODUCT OF ALL THE PRIMES: IS N ITSELF PRIME OR COMPOSITE? IF IT IS PRIME THEN OUR ORIGINAL SUPPOSI- TION IS FALSE, SINCE N IS LARGER THAN THE SUPPOSED LARGEST PRIME P SO NOW LET’S CONSIDER IT COMPOSITE THIS MEANS THAT IT MUST BE DIVISIBLE (WITH- OUT REMAINDER) BY PRIME NUMBERS HOWEVER, NONE OF THE PRIMES UP TO P WILL DIVIDE N (SINCE WE WOULD ALWAYS HAVE REMAINDER 1), SO ANY NUM- BER WHICH DOES DIVIDE N MUST BE GREATER THAN P THIS MEANS THAT THERE IS A PRIME NUMBER GREATER THAN P AFTER ALL THUS, WHETHER N IS PRIME OR COMPOSITE, OUR SUPPOSITION THAT THERE IS A LARGEST PRIME NUMBER IS FALSE THEREFORE, THE SET OF PRIME NUMBERS IS IN?NITE N P ( ) THE PROOF IS ELEGANT AND THE RESULT PROFOUND STILL, IT IS TYPICAL MATHEMATICS; SO, IT’S A GOOD EXAMPLE TO RE?ECT UPON IN DOING SO, WE WILL BEGIN TO SEE THE ELEMENTS OF THE MATHEMATICAL IMAGE, THE STANDARD CONCEPTION OF WHAT MATHEMATICS IS LET’S BEGIN A LIST OF SOME COMMONLY ACCEPTED ASPECTS BY ‘COMMONLY ACCEPTED’I MEAN THAT THEY WOULD BE ACCEPTED BY MOST WORKING MATHEMATICIANS, BY MOST EDUCATED PEOPLE, AND PROBABLY BY MOST PHILOSOPHERS OF MATHEMATICS, AS WELL IN LISTING THEM AS PART OF THE COMMON MATHEMATICAL IMAGE WE NEED NOT ENDORSE THEM LATER WE MAY EVEN COME TO REJECT SOME OF THEM – I CERTAINLY WILL WITH THIS CAUTION IN MIND, LET’S BEGIN TO OUTLINE THE STANDARD CONCEPTION OF MATHEMATICS CERTAINTY THE THEOREM PROVING THE IN?NITUDE OF PRIMES SEEMS ESTABLISHED BEYOND A DOUBT THE NATURAL SCIENCES CAN’T GIVE US ANYTHING LIKE THIS IN SPITE OF ITS WONDERFUL ACCOMPLISHMENTS, NEWTONIAN PHYSICS HAS BEEN OVERTURNED IN FAVOUR OF QUANTUM MECHANICS AND RELATIVITY AND NO ONE TODAY WOULD BET TOO HEAVILY ON THE LONGEVITY OF CURRENT THEORIES MATHEMATICS, BY CONTRAST, SEEMS THE ONE AND ONLY PLACE WHERE WE HUMANS CAN BE ABSOLUTELY SURE WE GOT IT RIGHT OBJECTIVITY WHOEVER ?RST THOUGHT OF THIS THEOREM AND ITS PROOF MADE A GREAT DISCOVERY THERE ARE OTHER THINGS WE MIGHT BE CERTAIN OF, BUT THEY AREN’T DISCOV- ERIES: ‘BISHOPS MOVE DIAGONALLY ’ THIS IS A CHESS RULE; IT WASN’T DISCOVERED; IT WAS INVENTED IT IS CERTAIN, BUT ITS CERTAINTY STEMS FROM OUR RESOLUTION TO PLAY THE GAME OF CHESS THAT WAY ANOTHER WAY OF DESCRIBING THE SITUATION IS BY SAYING THAT OUR THEOREM IS AN OBJECTIVE TRUTH, NOT A CONVENTION YET A THIRD WAY OF MAK- ING THE SAME POINT IS BY SAYING THAT MARTIAN MATHEMATICS IS LIKE OURS, WHILE THEIR GAMES MIGHT BE QUITE DIFFERENT PROOF IS ESSENTIAL WITH A PROOF, THE RESULT IS CERTAIN; WITHOUT IT, BELIEF SHOULD BE SUSPENDED THAT MIGHT BE PUTTING IT A BIT TOO STRONGLY SOMETIMES MATH- EMATICIANS BELIEVE MATHEMATICAL PROPOSITIONS EVEN THOUGH THEY LACK A PROOF PERHAPS WE SHOULD SAY THAT WITHOUT A PROOF A MATHEMATICAL PROPOSITION IS NOT JUSTI?ED AND SHOULD NOT BE USED TO DERIVE OTHER MATHEMATICAL PROPOSITIONS GOLDBACH’S CONJECTURE IS AN EXAMPLE IT SAYS THAT EVERY EVEN NUMBER IS THE SUM OF TWO PRIMES AND THERE IS LOTS OF EVIDENCE FOR IT, E G 4 2 2, 6 3 3, 3 5, 10 5 5, 12 7 5, AND SO ON IT’S BEEN CHECKED INTO THE BILLIONS WITHOUT A COUNTER-EXAMPLE BIOLOGISTS DON’T HESITATE TO CONCLUDE THAT ALL RAVENS ARE BLACK BASED ON THIS SORT OF EVIDENCE; BUT MATHEMATICIANS (WHILE THEY MIGHT BELIEVE THAT GOLDBACH’S CONJECTURE IS TRUE) WON’T CALL IT A THEOREM AND WON’T USE IT TO ESTABLISH OTHER THEOREMS – NOT WITHOUT A PROOF LET’S LOOK AT A SECOND EXAMPLE, ANOTHER CLASSIC, THE PYTHAGOREAN THEOREM THE PROOF BELOW IS MODERN, NOT EUCLID’S THEOREM: IN ANY RIGHT-ANGLED TRIANGLE, THE SQUARE OF THE HYPOTENUSE IS EQUAL TO THE SUM OF THE SQUARES ON THE OTHER TWO SIDES P H I L O S O P H Y O F M AT H E M AT I C S PROOF: CONSIDER TWO SQUARE ?GURES, THE SMALLER PLACED IN THE LARGER, MAKING FOUR COPIES OF A RIGHT-ANGLED TRIANGLE ABC (FIGURE 1 1) WE WANT TO PROVE THAT C 2 A 2 B 2 THE AREA OF THE OUTER SQUARE (A B) 2 C 2 4 (AREA OF ABC) C 2 2AB, SINCE THE AREA OF EACH COPY OF ABC IS AB FROM ALGEBRA WE HAVE (A B) 2 A 2 2AB B 2 SUBTRACTING 2AB FROM EACH, WE CONCLUDE C 2 A 2 B 2 THIS BRINGS OUT ANOTHER FEATURE OF THE RECEIVED VIEW OF MATHEMATICS DIAGRAMS THERE ARE NO ILLUSTRATIONS OR PICTURES IN THE PROOFS OF MOST THEOREMS IN SOME THERE ARE, BUT THESE ARE MERELY A PSYCHOLOGICAL AIDE THE DIA- GRAM HELPS US TO UNDERSTAND THE THEOREM AND TO FOLLOW THE PROOF – NOTHING MORE THE PROOF OF THE PYTHAGOREAN THEOREM OR ANY OTHER IS THE VERBAL/SYMBOLIC ARGUMENT PICTURES CAN NEVER PLAY THE ROLE OF A REAL PROOF REMEMBER, IN SAYING THIS I’M NOT ENDORSING THESE ELEMENTS OF THE MATHEMAT- ICAL IMAGE, BUT MERELY EXHIBITING THEM SOME OF THESE I THINK RIGHT, OTHERS, INCLUDING THIS ONE ABOUT PICTURES, QUITE WRONG READERS MIGHT LIKE TO FORM THEIR OWN TENTATIVE OPINIONS AS WE LOOK AT THESE EXAMPLES MISLEADING DIAGRAMS PICTURES, AT BEST, ARE MERE PSYCHOLOGICAL AIDS; AT WORST THEY MISLEAD US – OFTEN BADLY CONSIDER THE IN?NITE SERIES WHICH WE CAN ILLUSTRATE WITH A PICTURE (FIGURE 1 2) I N T R O D U C T I O N – T H E M AT H E M AT I C A L I M A G E THE SUM OF THIS SERIES IS ? 2/6 1 6449 IN THE PICTURE, THE SUM IS EQUAL TO THE SHADED AREA LET’S SUPPOSE WE PAINT THE AREA AND THAT THIS TAKES ONE CAN OF PAINT NEXT CONSIDER THE SO-CALLED HARMONIC SERIES HERE’S THE CORRESPONDING PICTURE (FIGURE 1 3): THE STEPS KEEP GETTING SMALLER AND SMALLER, JUST AS IN THE EARLIER CASE, THOUGH NOT QUITE SO FAST HOW BIG IS THE SHADED AREA? OR RATHER, HOW MUCH PAINT WILL BE REQUIRED TO COVER THE SHADED AREA? COMPARING THE TWO PICTURES, ONE WOULD BE TEMPTED TO SAY THAT IT SHOULD REQUIRE ONLY SLIGHTLY MORE – PERHAPS TWO OR THREE CANS OF PAINT AT MOST ALAS, SUCH A GUESS COULDN’T BE FURTHER OFF THE MARK IN FACT, THERE ISN’T ENOUGH PAINT IN THE ENTIRE UNIVERSE TO COVER THE SHADED AREA – IT’S IN?- NITE THE PROOF GOES AS FOLLOWS AS WE WRITE OUT THE SERIES, WE CAN GROUP THE TERMS 9--- { ?? ? ?? ? ? ? ? ? ?? ? ? ? ? ? N --- N FIGURE 1 2 SHADED BLOCKS CORRESPOND TO TERMS IN THE SERIES THE SIZE OF THE ?RST GROUP IS OBVIOUSLY 1 IN THE SECOND GROUP THE TERMS ARE BETWEEN AND , SO THE SIZE IS BETWEEN 2 AND 2 , THAT IS, BETWEEN AND 1 IN THE NEXT GROUPING OF FOUR, ALL TERMS ARE BIGGER THAN 1/8, SO THE SUM IS AGAIN BETWEEN AND 1 THE SAME HOLDS FOR THE NEXT GROUP OF 8 TERMS; IT, TOO, HAS A SUM BETWEEN AND 1 CLEARLY, THERE ARE IN?NITELY MANY SUCH GROUPINGS, EACH WITH A SUM GREATER THAN WHEN WE ADD THEM ALL TOGETHER, THE TOTAL SIZE IS IN?NITE IT WOULD TAKE MORE PAINT THAN THE UNIVERSE CONTAINS TO COVER IT ALL YET, THE PICTURE DOESN’T GIVE US AN INKLING OF THIS STARTLING RESULT ONE OF THE MOST FAMOUS RESULTS OF ANTIQUITY STILL AMAZES; IT IS THE PROOF OF THE IRRATIONALITY OF THE SQUARE ROOT OF 2 A RATIONAL NUMBER IS A RATIO, A FRACTION, SUCH AS 3/4 OR 6937/528, WHICH IS COMPOSED OF WHOLE NUMBERS ?9 3 IS RATIONAL AND SO IS ?(9/16) 3/4; BUT ?2 IS NOT RATIONAL AS THE FOLLOWING THEOREM SHOWS THEOREM: THE SQUARE ROOT OF 2 IS NOT A RATIONAL NUMBER PROOF: SUPPOSE, CONTRARY TO THE THEOREM, THAT ?2 IS RATIONAL, I E SUPPOSE THAT THERE ARE INTEGERS P AND Q SUCH THAT ?2 P/Q OR EQUIVALENTLY, (P/Q) 2 P 2/Q 2 LET US FURTHER ASSUME THAT P/Q IS IN LOWEST TERMS (NOTE THAT 3/4 9/12 21/28, BUT ONLY THE ?RST EXPRESSION IS IN LOWEST TERMS ) REARRANGING THE ABOVE EXPRESSION, WE HAVE P 2 2Q 2 THUS, P 2 IS EVEN (BECAUSE 2 IS A FACTOR OF THE RIGHT SIDE) HENCE, P IS EVEN (SINCE THE SQUARE OF AN ODD NUMBER IS ODD) SO IT FOLLOWS THAT P 2R, FOR SOME NUMBER R FROM THIS WE GET 2Q 2 P 2 (2R) 2 4R 2 THUS, Q 2 2R 2, WHICH IMPLIES THAT Q 2 IS EVEN, AND HENCE THAT Q IS EVEN NOW WE HAVE THE RESULT THAT BOTH P AND Q ARE EVEN, HENCE BOTH DIVIS- IBLE BY 2, AND SO, NOT IN LOWEST TERMS AS WAS EARLIER SUPPOSED THUS, WE HAVE ARRIVED AT THE ABSURDITY THAT P/Q BOTH IS AND IS NOT IN LOWEST TERMS THEREFORE, OUR INITIAL ASSUMPTION THAT ?2 IS A RATIONAL NUMBER IS FALSE CLASSICAL LOGIC NOTICE THE STRUCTURE OF THE PROOF OF THE IRRATIONALITY OF ?2 WE MADE A SUPPOSITION WE DERIVED A CONTRADICTION FROM THIS, SHOWING THE SUPPOSI- TION IS FALSE THEN WE CONCLUDED THAT THE NEGATION OF THE SUPPOSITION IS TRUE THE LOGICAL PRINCIPLES BEHIND THIS ARE: ?RST, NO PROPOSITION IS BOTH TRUE AND FALSE (NON-CONTRADICTION) AND SECOND, IF A PROPOSITION IS FALSE, THEN ITS NEGATION IS TRUE (EXCLUDED MIDDLE) CLASSICAL LOGIC IS A WORKING TOOL OF MATHEMATICS WITHOUT THIS TOOL, MUCH OF TRADITIONAL MATHEMATICS WOULD CRUMBLE STRICTLY SPEAKING, THE PROOF OF THE IRRATIONALITY OF ?2 IS ACCEPTABLE TO CON- STRUCTIVE MATHEMATICIANS, EVEN THOUGH THEY DENY THE GENERAL LEGITIMACY OF CLAS- SICAL LOGIC THE ISSUE WILL COME UP IN MORE DETAIL IN A LATER CHAPTER THE PROOF JUST GIVEN NICELY ILLUSTRATES REDUCTION AD ABSURDUM REASONING IT IS ALSO ONE OF THE ALL TIME GREAT RESULTS, WHICH EVERYONE SHOULD KNOW AS A MATTER OF GENERAL CULTURE, JUST AS EVERYONE SHOULD KNOW HAMLET THIS IS MY EXCUSE FOR USING AN IMPERFECT EXAMPLE TO MAKE THE POINT ABOUT CLASSICAL LOGIC I N T R O D U C T I O N – T H E M AT H E M AT I C A L I M A G E