Have you ever tried to create a paradox, or think about one? Maybe you've come up with the infamous "this sentence is a lie" paradox, where it is true if and only if it is false, or vice versa. This type of paradox is called self-reference, which is caused by having some statement to describe itself in some way. These types of word paradoxes hold little to no meaning since we cannot expect all words to hold some sort of truth value. But in mathematics, this is different. When we see some statement, like 3+1 = 4, it can only either be true or false, in this case, it is true. But then here's the question: can we prove it? Can we prove that 3+1 is 4? Well of course we can, thanks to the axioms, or the statements that are regarded as true without proof, in which all theorems and such in math are built off of. But what if we can't? This is the idea at the heart of the Incompleteness Theorem, but, before we talk about that, we have to talk about David Hilbert.
David Hilbert was one of the most influential mathematicians in the late 18th and early 19th centuries. Making significant contributions to mathematics and physics, he led a group of mathematicians known as the formalists. The formalists aimed to represent all mathematics down to just a few symbols, including "~" for not and "V" for or, to name a few.
Thanks to this extremely rigorous and exact way of representing math, Hilbert and the other formalists could prove different kinds of theorems, still based on the axioms, using this method without any incorrect or unclear logic phasing through. Now with this method, there were 3 questions that Hilbert wanted to answer.
Is Math Complete (Is there a way to prove every true statement)
Is Math Consistent (Is Math free from contradictions)
Is Math Decidable (Is there an algorithm that can always determine if a given statement is based on the axioms)
These weren't the only big problems that Hilbert was wondering about, but these three were perhaps the most significant, as if any one were proven to be true or untrue the nature of the system could be thought of completely different. Now, Hilbert himself was confident that all 3 of those properties applied to math, and when he addressed a meeting of mathematicians in 1930, he said this: "We must not believe those, who today with philosophical bearing and a tone of superiority prophesy the downfall of culture and accept the ignorabimus. For us there is no ignorabimus, and in my opinion even none whatever in natural science. In place of the foolish ignorabimus let stand our slogan: We must know, We will know." Unfortunately for him, this dream was about to get crushed.
Kurt Gödel was born in 1906 in what is now the Czech Republic, as a child he was plagued by poor health, and despite fully recovering from rheumatic fever at the age of 6, for the rest of his life he was convinced he had lasting health issues. Gödel was a gifted student, and received a doctorate in mathematics at the University of Vienna in 1929. 2 years later, just a year after Hilbert gave his 1930 address, Gödel published his first Incompleteness Theorem, disproving Hilbert's first idea, that math was complete. Here's how he did it.
Using Hilbert's own system against him, he devised a clever system that would allow math to talk about itself. First, he gave each of the mathematical symbols a number for example:
Symbol | Gödel Number |
~ | 1 |
V | 2 |
⊃ | 3 |
∃ | 4 |
= | 5 |
0 | 6 |
s | 7 |
( | 8 |
) | 9 |
' | 10 |
+ | 11 |
* | 12 |
x | 13 |
y | 17 |
z | 19 |
(Chart Credit: Veritasium)
Then using this chart, you could write out different statements. For example, 0 = 0 would correspond to 6 5 6. Then, you use each number as an exponent for increasing primes. So the first prime number is 2, then 3, then 5, so 0 = 0 would be 2^6 x 3^5 x 5^6 = 243,000,000 in Gödel numbers. Using this system, it is possible to make mathematics talk about itself, because each statement is encoded in a number. Using some clever logical tricks, he manages to create a statement which translates into: There is no proof for the statement with Gödel number N, where N is a huge number where it would take until the end of the universe to finish writing. The issue is that statement, when translated, has Gödel number N. Using the ideas of self-reference, Gödel essentially created a paradox. How can something that's true be unprovable? Unless, it is possible. What Gödel had discovered is that truthfulness and provableness are two completely different ideas, debunking Hilbert's first question. Gödel later went on to expand it in his second Theorem of Incompleteness, showing that any consistent system of mathematics cannot prove its own consistency, making Hilbert's second assertion questionable at best.
Gödel made some other important contributions, like how in 1949 he showed that Einstein's Theory of Relativity could possibly allow Time Travel. But during the last few decades of his life, his mental health began to deteriorate, until 1978 he starved himself to death for fear of being poisoned. But his legacy lives on. 15 years before his death in 1963, Paul Cohen finished the prove of incompleteness for the Continuum Hypothesis, which was a hypothesis about the nature of infinite sets (a huge simplification), proving that it could not be proven true when Gödel himself had proven that it could not be proven false earlier. And many other problems that have remained unsolved for centuries, like the Twin Prime Conjecture or even the Riemann Hypothesis, which if solved could unlock secrets to the true nature of mathematics, could be true but unsolvable. When Hilbert died, engraved in his tombstone wrote: "We must know, we will know." But in light of Gödel's discovery, unfortunately, a more accurate answer may be: "We will either know, or we will never know."
Works Cited
Kennedy, Juliette. “Kurt Gödel (Stanford Encyclopedia of Philosophy).” Stanford.edu, 2015, plato.stanford.edu/entries/goedel/.
Muller, Derek. “Math’s Fundamental Flaw.” Www.youtube.com, 22 May 2021, www.youtube.com/watch?v=HeQX2HjkcNo&t=966s. Accessed 30 Apr. 2024.
O’Connor, J, and E Robertson. “David Hilbert - Biography.” Maths History, mathshistory.st-andrews.ac.uk/Biographies/Hilbert/.
Smith, James. “David Hilbert’s Radio Address - English Translation | Mathematical Association of America.” Maa.org, maa.org/press/periodicals/convergence/david-hilberts-radio-address-english-translation. Accessed 30 Apr. 2024.
The Editors of Encyclopaedia Britannica. “Formalism | Philosophy of Mathematics | Britannica.” Www.britannica.com, www.britannica.com/topic/formalism-philosophy-of-mathematics.