Is Mathematics Invented or Discovered? | Episode 409 | Closer To Truth

MATHEMATICS HAS CHANGED FROM CERTAINTY TO UNCERTAINTY ABOUT WHETHER MATHEMATICS HAS ALWAYS EXISTED. AFTER A LIFETIME IN MATHEMATICS, GREG IS NOT VERY SURE? SO WHERE DOES THAT LEAVE ME? AND YES, IT MATTERS. MATHEMATICS ARE FUNDAMENTAL FOR EXISTENCE. ARE THERE NEW WAYS OF THINKING? STEPHEN WOLFRAM, A PHYSICIST AND ORIGINAL THINKER, HAS DEVELOPED WHAT HE CALLS A NEW TYPE OF SCIENCE, WHICH IS BASED ON THE SHOCKING IDEA THAT SIMPLE RULES, NOT COMPLEX MATHEMATICS, CONSTRUCT REALITY. IT'S POSSIBLE? COULD ALL MATHEMATICS BE AN ARTIFACT OF SIMPLE RULES? I HAVE BEEN INTERESTED FOR A LONG TIME IN QUESTIONS ABOUT WHAT IS THE ESSENCE OF MATHEMATICS?

I MAKE A LIVING BUILDING THIS THING CALLED MATHEMATICS, WHICH TRYES TO COVER IN THE BROADEST SENSE POSSIBLE THE KIND OF THINGS THAT MATHEMATICS CAN COVER. BUT THEN THE QUESTION THAT HAS INTERESTED ME, AND ALSO FROM THE POINT OF VIEW OF BASIC SCIENCE, IS, IS THE MATHEMATICS THAT WE PRACTICE TODAY THE ONLY POSSIBLE MATHEMATICS, OR IS IT A MATHEMATICS THAT IS GREAT? ARTIFACT OF OUR CIVILIZATION BUT A KIND OF ARTIFACT FROM A HISTORICAL ACCIDENT? The conclusion I have come to resoundingly is that the

mathematics

we have today is, in fact, truly a historical artifact. THAT IS NOT HISTORICALLY IN THE TRADITION OF MATHEMATICS ITSELF, THAT IS NOT WHAT PEOPLE HAVE TENDED TO CONCLUDE.

THEY HAVE TENDED TO THINK THAT MATHEMATICS IS THE MOST GENERAL ABSTRACT FORMAL SYSTEM POSSIBLE. IF WE LOOK AT THE HISTORY OF MATHEMATICS, THIS CERTAINLY IS NOT HOW IT ORIGINALLY STARTED. I mean, in ancient Babylon, you know, there was arithmetic for commerce and other things, and there was geometry for surveying, and what I think has really been the history of

mathematics

is the progressive generalization of arithmetic and geometry. plus a key methodological idea. THAT THEOREMS AND ABSTRACT DEMONSTRATIONS OF THOSE THEOREMS CAN BE MADE. THE QUESTION CAN BE ASKED IF WE ARBITRARILY LOOK AT FORMAL SYSTEMS, DO THEY TEND TO HAVE THE CHARACTER OF MATHEMATICS AS WE KNOW IT TODAY?

DO THEY TEND TO HAVE THE CHARACTERISTICS OF THAT? MOST THINGS ONE ASKS ABOUT ONE CAN SUCCESSFULLY PROVE THEOREMS. I THINK IN BOTH CASES THE ANSWER IS NO, NOT REALLY. So, for example, one thing one can do is ultimately deconstruct mathematics. IF YOU LOOK, YOU KNOW, THERE ARE MAYBE THREE MILLION ARTICLES THAT HAVE BEEN PUBLISHED ON MATHEMATICS, OK, AND THEY ARE ALL BASED ON A CERTAIN SET OF AXIOMS. AXIOMS ​​ARE WHAT MATHEMATICS GROWS FROM. THE AXIOMS ​​ARE VERY SIMPLE. OUR PARTICULAR MATHEMATICS ARE THE PARTICULAR SET OF AXIOMS ​​THAT YOU CAN WRITE ON THIS COUPLE OF PAGES, BUT THERE IS A WHOLE UNIVERSE OF POSSIBLE MATHEMATICS, WHAT ARE THEY LIKE?

THE FIRST QUESTION COULD BE: WHERE ARE OUR PARTICULAR MATHEMATICS FOUND IN THIS UNIVERSE OF POSSIBLE MATHEMATICS? IS MATH NUMBER 1 POSSIBLE, IS MATH NUMBER 10 POSSIBLE, IS MATH NUMBER QUINTILLION POSSIBLE? WHERE DOES IT LIE? THE ANSWER IS, IT DEPENDS EXACTLY ON HOW THE SPACE IS NUMBERED, BUT MORE APPROXIMATELY IT IS ABOUT THE SYSTEM OF POSSIBLE AXIOMS ​​NUMBER 50,000, SO RIGHT THERE, IN THE UNIVERSE OF THE SYSTEMS OF POSSIBLE AXIOMS, THE UNIVERSE OF POSSIBLE MATHEMATICS, THERE IS LOGIC . IF THE ALIENS GIVEN, YOU KNOW, A DIFFERENT POSSIBLE MATHEMATICS I DON'T THINK WE COULD IMMEDIATELY SAY THAT IT'S NOT REASONABLE VALID MATHEMATICS.

BECAUSE IT WOULD BE CONSISTENT EVEN THOUGH IT WOULD BE RADICALLY DIFFERENT. YES. NOW, WHAT HAS HAPPENED IN THE HISTORY OF MATHEMATICS IS THAT MOST OF THE THINGS PEOPLE HAVE BEEN INTERESTED IN END UP BEING SOLVABLE, ALTHOUGH SOMETIMES WITH EFFORT AND CENTURIES OF WORK AND SO ON, BUT ONE OF THE THINGS I SUSPECT THAT IS REALLY NOT REALLY THE WAY THAT IS THE TRUTH, THE TRUE REALITY FOR MATHEMATICS. REALLY, IF we were to ask mathematical questions arbitrarily, the vast majority of them would end up being unsolvable, and in fact, that unsolvability is really within reach in mathematics.

We just don't see it because the particular way mathematics has progressed historically has tended to avoid it. NOW YOU CAN SAY, BUT MATHEMATICS IS A GOOD MODEL OF THE NATURAL WORLD AND MATHEMATICS HAS BEEN DRIVEN BY MODELING THE NATURAL WORLD, I THINK THERE'S A KIND OF CIRCULAR ARGUMENT BECAUSE WHAT HAPPENED IS THAT THOSE THINGS THAT HAVE BEEN SUCCESSFULLY ADDRESSED IN THE SCIENCE IN THE STUDY OF THE NATURAL WORLD IS ONLY THOSE THINGS THAT METHODS LIKE MATHEMATICS HAVE ALLOWED US TO SUCCESSFULLY ADDRESS. So I think one of the exciting things that you realize is that human mathematics is one of the great artifacts of our civilization.

IT IS ONE OF THE KIND OF PERFECT AND WONDERFUL THINGS THAT HAS BEEN PRODUCED BY AN ENORMOUS AMOUNT OF HUMAN EFFORT. BUT IT'S AN ARTIFACT. BUT IT IS AN ARTIFACT AND THERE IS MUCH MORE IN THE TYPE OF SPACE OF ALL THE POSSIBLE MATHEMATICS AND I THINK THAT IN THE FUTURE WE WILL SEE A GROWING TYPE OF REALIZATION AND A GROWING ABILITY TO EXPLORE THAT OTHER UNIVERSE OF MATHEMATICS AND THUS BE DEEPLY IMPORTANT, NO ONLY FOR MATHEMATICS BUT FOR OUR SCIENCE AND OUR TECHNOLOGY. STEPHEN REJECTS THE IDEA THAT OUR MATHEMATICS HAS A DEEP MEANING, RATHER LOOKS INTO THE VERY BIG SPACE OF ALL POSSIBLE MATHEMATICS.

BUT MATHEMATICS AS A MERE ARTIFACT STILL CONCERNS ME. THAT'S WHY I'M GOING TO MIT TO MEET FRANK WILCZEK, NOBEL WINNER IN PHYSICS WHO SEEKS COMMON SENSE TO GET TO THE ESSENCE OF MATHEMATICS. FRANK, WHAT DO YOU DO WITH THAT FUNDAMENTAL MATH QUESTION? IS IT INVENTED BY MINDS, HUMAN MINDS OR IS IT DISCOVERED? WAS IT ALWAYS THERE IN SOME PLATONIC FORM OR PLATONIC HEAVEN? THE ANSWER IS BOTH. IT IS BOTH INVENTED AND DISCOVERED BUT I THINK IT IS MOSTLY DISCOVERED. I THINK MATHEMATICS IS THE PROCESS OF TAKING AXIOMS, TAKING A DEFINED SET OF ASSUMPTIONS AND DRAWING THEIR CONSEQUENCES.

THEREFORE, INVENTING AXIOMS ​​IS INVENTION AND DRAWING ITS CONSEQUENCES IS DISCOVERY. IF YOU LOOK AT WHAT MATHEMATICIANS ACTUALLY DO, MOST MATHEMATICIANS DO NOT SPEND A LOT OF TIME INVENTING NEW AXIOMS. THEY DEDICATE THEIR TIME TO DRAWING THE CONSEQUENCES OF AXIOMS ​​THAT HAVE PROVEN TO BE RICH AND INTERESTING AND CONSEQUENCES. OCCASIONALLY, NEW SETS OF AXIOMS ​​HAVE TO BE INTRODUCED, SUCH AS THE PASSAGE FROM EUCLIDEAN GEOMETRY TO NON-EUCLIDEAN GEOMETRY. THESE ARE EPIC EVENTS IN MATHEMATICS AND THESE, IN A SENSE, INVENTION. THAT'S A GOOD ONE. THIS IS AN INVENTION, BECAUSE IF THE UNIVERSE REALLY 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 IT WAS SOMETHING. GAUST DEVELOPED THOSE CONCEPTS IN THE CONTEXT OF THE STUDY OF THE EARTH, THE EARTH IS ROUND AND, BUT THE UNDERLYING DYNAMIC IS THAT YES, YOU CAN INVENT AXIOMS ​​AD FREELY TO DO ANYTHING, BUT MOST OF THEM WON'T BE INTERESTING, AND THE ONE'S THAT ARE THERE ARE INTERESTING DISCOVERIES. SO EVEN INVENTIONS HAVE SOME ELEMENT OF DISCOVERY. DISCOVER WHICH ARE INTERESTING AXIOMS. THEN I ORIGINALLY SAID THAT MATHEMATICS IS DISCOVERED MORE THAN IT IS INVENTED, AND THAT ONLY MAKES IT MORE SO.

SO, IS MATHEMATICS INVENTED OR DISCOVERED? THIS IS WHAT WE KNOW. MATHEMATICS DESCRIBE THE PHYSICAL WORLD WITH REMARKABLE PRECISION. BECAUSE? THERE ARE TWO POSSIBILITIES. FIRST, MATHEMATICS somehow underlies the physical world, generates it. OR SECOND, MATHEMATICS IS A HUMAN DESCRIPTION OF HOW WE DESCRIBE CERTAIN REGULARITIES OF NATURE. AND BECAUSE THERE IS SO MUCH POSSIBLE MATH, SOME EQUATIONS MUST FIT. REGARDING THE ESSENCE OF MATHEMATICS THERE ARE FOUR POSSIBILITIES: MATHEMATICS COULD BE PHYSICAL IN THE REAL WORLD, REALLY EXISTING; OR, MENTALLY, IN THE MIND, JUST A HUMAN CONSTRUCTION; OR, PLATONIC, NON-PHYSICAL, NON-MENTAL, ABSTRACT OBJECTS; OR FICTIONAL, ANTI-REALIST, COMPLETELY INVENTED. MATHEMATICS ARE PHYSICAL, MENTAL, PLATONIC OR FICTIONAL.

CHOOSE ONLY ONE. BY LOOKING INTO THE DARK PIT OF DEEP REALITY, MATHEMATICS BRINGS US CLOSER TO THE TRUTH.