Skip to main content

The man who broke Maths -

In the early 20th century, it led Austrian logician Kurt Gödel to a discovery that would change mathematics forever. Gödel’s discovery had to do with the limitations of mathematical proofs. A proof is a logical argument that demonstrates why a statement about numbers is true. 

Stack of books with pencil holder and glasses against a chalkboard

The building blocks of these arguments are called axioms— undeniable statements about the numbers involved. Every system built on mathematics, from the most complex proof to basic arithmetic, is constructed from axioms. And if a statement about numbers is true, mathematicians should be able to confirm it with an axiomatic proof. Since ancient Greece, mathematicians used this system to prove or disprove mathematical claims with total certainty. But when Gödel entered the field, some newly uncovered logical paradoxes were threatening that certainty. Prominent mathematicians were eager to prove that mathematics had no contradictions. 

Gödel himself wasn’t so sure. And he was even less confident that mathematics was the right tool to investigate this problem. While it’s relatively easy to create a self-referential paradox with words, numbers don't typically talk about themselves. A mathematical statement is simply true or false. But Gödel had an idea. First, he translated mathematical statements and equations into code numbers so that a complex mathematical idea could be expressed in a single number. This meant that mathematical statements written with those numbers were also expressing something about the encoded statements of mathematics. In this way, the coding allowed mathematics to talk about itself. Through this method, he was able to write:

 “This statement cannot be proved” as an equation, creating the first self-referential mathematical statement. However, unlike the ambiguous sentence that inspired him, mathematical statements must be true or false. So which is it? If it’s false, that means the statement does have a proof. But if a mathematical statement has a proof, then it must be true. This contradiction means that Gödel’s statement can’t be false, and therefore it must be true that “this statement cannot be proved.” Yet this result is even more surprising, because it means we now have a true equation of mathematics that asserts it cannot be proved. This revelation is at the heart of Gödel’s Incompleteness Theorem, which introduces an entirely new class of mathematical statement.

 In Gödel’s paradigm, statements still are either true or false, but true statements can either be provable or unprovable within a given set of axioms. Furthermore, Gödel argues these unprovable true statements exist in every axiomatic system. This makes it impossible to create a perfectly complete system using mathematics, because there will always be true statements we cannot prove. Even if you account for these unprovable statements by adding them as new axioms to an enlarged mathematical system, that very process introduces new unprovably true statements.

 No matter how many axioms you add, there will always be unprovably true statements in your system. It’s Gödels all the way down! This revelation rocked the foundations of the field, crushing those who dreamed that every mathematical claim would one day be proven or disproven. While most mathematicians accepted this new reality, some fervently debated it. Others still tried to ignore the newly uncovered a hole in the heart of their field. But as more classical problems were proven to be un provably true, some began to worry their life's work would be impossible to complete. Still, Gödel’s theorem opened as many doors as a closed. 

Knowledge of unprovably true statements inspired key innovations in early computers. And today, some mathematicians dedicate their careers to identifying provably unprovable statements. So while mathematicians may have lost some certainty, thanks to Gödel they can embrace the unknown at the heart of any quest for truth.

Thank You 

Comments

Post a Comment

Popular posts from this blog

Santizer VS Soap / Which is best for Corona ?

Your hands, up close, are anything but smooth. With peaks and valleys, folds and rifts, there are plenty of hiding places for a virus to stick. If you then touch your face, the virus can infect you. But there are two extraordinarily simple ways you can keep that from happening: soap and water, and hand sanitizer. So which is better?  The coronavirus that causes COVID-19 is one of many viruses whose protective outer surface is made of a lipid bilayer. These lipids are pin shaped molecules whose heads are attracted to water, and tails are repulsed by it. So in water-rich environments, lipids naturally form a shell like this, with the heads outside and the tails inside. Their shared reaction to water makes the lipids stick loosely together— this is called the hydrophobic effect. This outer structure helps the molecular machinery of the virus break through cellular membranes and hijack our cells. But it has thousands upon thousands of weak points where the right molecules could pr...
Post a Comment
Read more

The incredible history of China's terracotta warriors -

What happens after death? Is there a restful paradise? An eternal torment? A rebirth? Or maybe just nothingness? Well, one Chinese emperor thought that whatever the hereafter was, he better bring an army.  We know that because in 1974, farmers digging a well near their small village stumbled upon one of the most important finds in archeological history: vast underground chambers surrounding that emperor's tomb, and containing more than 8,000life-size clay soldiers ready for battle.  The story of the subterranean army begins with Ying Zheng, who came to power as the king of the Qin state at the age of 13 in 246 BCE. Ambitious and ruthless, he would go on to become Qin Shi Huangdi, the first emperor of China after uniting its seven warring kingdoms. His 36 year reign saw many historic accomplishments, including a universal system of weights and measures, a single standardized writing script for all of China, and a defensive barrier that would later come to be known as the Great ...
Post a Comment
Read more

Are the illuminati real ?

The year was 1776. In Bavaria, new ideals of rationalism , religious freedom, and universal human rights competed with the Catholic church’s heavy influence over public affairs.  Across the Atlantic, a new nation staked its claim for independence on the basis of these ideas. But back in Bavaria, law professor Adam Weishaupt’s attempts to teach secular philosophy continued to be frustrated. Weishaupt decided to spread his ideas through a secret society that would shine a light on the shortcomings of the Church’s ideology. He called his secret society the Illuminati. Weishaupt modelled aspects of his secret society off a group called the Freemasons. Originally an elite stoneworkers’ guild in the late Middle Ages, the Freemasons had gone from passing down the craft of masonry to more generally promoting ideals of knowledge and reason. Over time, they had grown into a semi-secret, exclusive order that included many wealthy and influential individuals, with elaborate, secret initia...
Post a Comment
Read more