Would mathematics exist if people didn't? Since ancient times, mankind has hotly debated whether mathematics was discovered or invented. Did we create mathematical concepts to help us understand the universe around us, or is math the native language of the universe itself, existing whether we find its truths or not? Are numbers, polygons and equations truly real, or merely ethereal representations of some theoretical ideal? The independent reality of math has some ancient advocates. The Pythagoreans of 5th Century Greece believed numbers were both living entities and universal principles. They called the number one, "the monad," the generator of all other numbers and source of all creation. Numbers were active agents in nature. Plato argued mathematical concepts were concrete and as real as the universe itself, regardless of our knowledge of them. Euclid, the father of geometry, believed nature itself was the physical manifestation of mathematical laws. Others argue that while numbers may or may not exist physically, mathematical statements definitely don't. Their truth values are based on rules that humans created. Mathematics is thus an invented logic exercise, with no existence outside mankind's conscious thought, a language of abstract relationships based on patterns discerned by brains, built to use those patterns to invent useful but artificial order from chaos. One proponent of this sort of idea was Leopold Kronecker, a professor of mathematics in 19th century Germany. His belief is summed up in his famous statement: "God created the natural numbers, all else is the work of man." During mathematician David Hilbert's lifetime, there was a push to establish mathematics as a logical construct. Hilbert attempted to axiomatize all of mathematics, as Euclid had done with geometry. He and others who attempted this saw mathematics as a deeply philosophical game but a game nonetheless. Henri Poincaré, one of the father's of non-Euclidean geometry, believed that the existence of non-Euclidean geometry, dealing with the non-flat surfaces of hyperbolic and elliptical curvatures, proved that Euclidean geometry, the long standing geometry of flat surfaces, was not a universal truth, but rather one outcome of using one particular set of game rules. But in 1960, Nobel Physics laureate Eugene Wigner coined the phrase, "the unreasonable effectiveness of mathematics," pushing strongly for the idea that mathematics is real and discovered by people. Wigner pointed out that many purely mathematical theories developed in a vacuum, often with no view towards describing any physical phenomena, have proven decades or even centuries later, to be the framework necessary to explain how the universe has been working all along. For instance, the number theory of British mathematician Gottfried Hardy, who had boasted that none of his work would ever be found useful in describing any phenomena in the real world, helped establish cryptography. Another piece of his purely theoretical work became known as the Hardy-Weinberg law in genetics, and won a Nobel prize. And Fibonacci stumbled upon his famous sequence while looking at the growth of an idealized rabbit population. Mankind later found the sequence everywhere in nature, from sunflower seeds and flower petal arrangements, to the structure of a pineapple, even the branching of bronchi in the lungs. Or there's the non-Euclidean work of Bernhard Riemann in the 1850s, which Einstein used in the model for general relativity a century later. Here's an even bigger jump: mathematical knot theory, first developed around 1771 to describe the geometry of position, was used in the late 20th century to explain how DNA unravels itself during the replication process. It may even provide key explanations for string theory. Some of the most influential mathematicians and scientists of all of human history have chimed in on the issue as well, often in surprising ways. So, is mathematics an invention or a discovery? Artificial construct or universal truth? Human product or natural, possibly divine, creation? These questions are so deep the debate often becomes spiritual in nature. The answer might depend on the specific concept being looked at, but it can all feel like a distorted zen koan. If there's a number of trees in a forest, but no one's there to count them, does that number exist?
Da li bi bilo matematike da nema ljudi? Od davnina, čovečanstvo je raspravljalo o tome da li je matematika otkrivena ili izmišljena. Da li smo stvorili matematičke koncepte kako bismo razumeli univerzum oko nas, ili je matematika prirodni jezik samog univerzuma, koji postoji spoznali mi istinu ili ne? Da li su brojevi, poligoni i jednačine istinski stvarne, ili su samo uzvišena reprezentacija nekog teorijskog ideala? Postoje drevni zastupnici nezavisnog postojanja matematike. Pitagorejci Grčke iz 5-tog veka su verovali da su brojevi istovremeno živi entiteti i univerzalni principi. Broj jedan su zvali "monada", generator svih ostalih brojeva i izvor celog stvaranja. Brojevi su bili aktivni agensi u prirodi. Platon je tvrdio da su matematičke ideje konkretne i stvarne koliko i sam univerzum, bez obzira na naše znanje o njima. Euklid, otac geometrije, je verovao da je priroda sama po sebi fizička manifestacija matematičkih zakona. Drugi zagovaraju da dok brojevi mogu ili ne postojati fizički, matematičke izjave definitivno ne postoje. Njihove istinitosne vrednosti su osnovane na pravilima koje su ljudi stvorili. Matematika je dakle, stvorena logička vežba, koja ne postoji izvan ljudske svesti. Jezik apstraktnih odnosa zasnovan na osnovu šablona koje nazire mozak, napravljen da pomoću tih šablona stvori koristan ali veštački red od haosa. Jedan pobornik ove ideje bio je Leopold Kroneker, profesor matematike u 19. veku u Nemačkoj. Njegovo verovanje je sadržano u njegovoj slavnoj izjavi: "Bog je stvorio prirodne brojeve, sve ostalo je delo čoveka." Tokom života matematičara Dejvida Hilberta, postojao je pritisak da se utemelje logičke osnove matematike. Hilbert je pokušao da aksiomatizuje celu matematiku, kao što je Euklid uradio sa geometrijom. Zajedno sa ostalima uvideo je da je matematika duboka filozofska igra, ali iznad svega, ipak igra. Henri Poenkare, jedan od osnivača neeuklidske geometrije, verovao je da postojanje neeuklidske geometrije, koja se bavi krivim površima sa hiperboličkim i eliptičkim krivinama, dokazuje da euklidska geometrija, stara geometrija ravnih površi, nije univerzalna istina, već posledica korišćenja tačno određenih pravila igre. 1960. dobitnik Nobelove nagrade za fiziku Eugen Vigner skovao je frazu: "nerazumna efikasnost matematike", snažno podržavajući ideju da je matematika stvarna i da su je ljudi otkrili. Vigner je istakao da mnoge teorije čiste matematike koje su nastale ni iz čega, često ne opisujući fizičke pojave, dokazane decenijama i vekovima kasnije, zapravo su neophodan okvir za objašnjenje kako univerzum funkcioniše sve vreme. Npr, teorija brojeva britanskog matematičara Gotfrida Hardija, koji se hvalio da nijedan njegov rad neće biti koristan za opisivanje bilo koje pojave u stvarnom svetu, pomogao je da se osnuje kriptografija. Još jedan njegov čisto teorijski rad postao je poznat kao Hardi-Vajnbergov zakon u genetici, i dobio je Nobelovu nagradu. I Fibonači je nabasao na svoj čuveni niz dok je posmatrao rast idealizovane populacije zečeva. Zatim je čovečanstvo opažalo ovaj niz svuda u prirodi, od rasporeda zrna suncokreta i krunica cveća, do strukture ananasa, čak i do grananja bronhija u plućima. Takođe, tu je i neeuklidski rad Bernarda Rimana iz 1850-tih, koji je Ajnštajn koristio pri modeliranju teorije relativnosti vek kasnije. Evo još većeg skoka: matematička teorija čvorova, zasnovana oko 1771. kako bi se opisala geometrija položaja, primenjena je krajem 20-tog veka pri objašnjavanju kako se raspliće DNK tokom procesa replikacije. Možda pruži i ključno objašnjenje za teoriju struna. Neki od najuticajnijih matematičara i naučnika svih vremena takođe su dotakli problem, često na iznenađujuće načine. Dakle, da li je matematika izum ili otkriće? Veštačka konstrukcija ili univerzalna istina? Ljudska proizvod ili prirodna, verovatno božanska, kreacija? Ova pitanja su toliko duboka da debate često po prirodi postaju spiritualne. Odgovor može da zavisi od određenog koncepta koji se posmatra, ali isto tako može da izgleda kao uvrnuta zen koana. Ako je u šumi određen broj drveća, i nema nikoga da ih prebroji, da li taj broj postoji?