Will a point come where society moves past science? After all, there should come a point in dialogue (or monologue) where one questions the longevity of a pursuit. If that pursuit is a job, the questioning most likely comes from analyzing one’s self (acknowledgment of potential, dissatisfaction with a current situation, etc.). With an idea, its lasting power comes from its perception among that particular society; ideas that don’t have a following or are disagreed upon are often countered by other, more agreeable ideas.

If we define science as one of these pursuits, then we can break down the question of society moving past science into a more basic question: Will there come a point when the pursuit of science is pointless?

Many — myself included — would say “no”. But regardless, I think there’s an interesting discussion to be had here.

The Humble Axiom

An axiom is a statement which we know to be true without any further proof. In math, we know that, if a = b and b = c, then a = c; no proof needed. In science, empirical observations are axioms. For example, the fact that the universe has order and follows predictable rules which we could make conclusions about. Often, these axioms are building blocks that form the basis for questions that scientists and mathematicians then dig into to create mathematical systems or scientific systems, which are composed of theorems.

If these theorems can prove every unknown question, then a system exhibits completeness. In other words, the system has all the necessary theorems to tackle any question. That already sounds like a tall order.

Lastly, any supposedly complete system should have an answer that proves the truth of only one thing, not its opposite. For example, if a lawyer were to prove their client guilty of a specific crime, they should also not be able to prove them not guilty of that crime. If they could, then one of those proofs – guilty or not guilty – is false, yet proven, which would throw the entirety of a system into question, including the theorems and, by extension, the axioms that make up those theorems. Systems that don’t contradict themselves are known as consistent.

Gödel’s Incompleteness Theorem

In the early 1900s, Kurt Gödel, a logician and mathematician, created two theorems to demonstrate the limits of any axiomatic system, or any system built on axioms.

But the consensus for Gödel’s theorems is that they only work for mathematics. Why is that, when science also contains axioms? To find out, we have to dig deeper.

The first theorem states that a system can only be complete or consistent, but never both. This is true. There are no systems that exist that can answer every single question and have provable answers without some of those proofs contradicting each other. For example, if my system was inconsistent, then I would be able to prove that, for example, humans are both warm-blooded and cold-blooded. If I can prove that “humans are warm-blooded” and “humans are cold-blooded”, then my system is complete, even if one or both of those statements are wrong. This is, obviously, catastrophic. We don’t want that inconsistency; it’s disturbing and unwelcome for our curious minds.

But what about the opposite? Would a system that’s consistent, but incomplete be just as catastrophic? Actually, no. In fact, we would prefer if it was a system that couldn’t answer every question, so long as it doesn’t contradict itself…Wait, that sounds familiar…I can’t put my finger on it…what does that sound like?

Oh! Right! The answer is “reality”. No system can answer every truth; hence “incompleteness”.

But, there’s one problem. How do you prove consistency? Say that you only had one system to work with. How does that system prove itself to be consistent? Can you use theorems within a system to prove that system? The answer is “no”. We don’t say that gravity exists because of the theorem of gravity. We say it exists because of the force that it creates, which dips into a second system: physics, or, more specifically, the Second Law of Motion. Simply put, you can’t prove that your system is consistent because, within that system, you can’t prove that there are no contradictions. No consistent system can prove its own consistency; this is Gödel’s second incompleteness theorem.