GoKawiilIn 1931, by turning logic on itself, Kurt Gödel proved a pair of theorems that transformed the landscape of knowledge and truth. These “incompleteness theorems” established that no formal system of mathematics — no finite set of rules, or axioms, from which everything is supposed to follow — can ever be complete. There will always be true mathematical statements that don’t logically follow from those axioms.
I spent the early weeks of the Covid pandemic learning how the 25-year-old Austrian logician and mathematician did such a thing, and then writing a rundown of his proof in fewer than 2,000 words. (My wife, when I reminded her of this period: “Oh yeah, that time you almost went crazy?” A slight exaggeration.)
In philosophy, “qualia” refers to the subjective qualities of our experience: what it’s like for Alice to see blue or for Bob to feel delighted. Qualia are “the ways things seem to us,” as the late philosopher Daniel Dennett put it. In these essays, our columnists follow their curiosity, and explore important but not necessarily answerable scientific questions.
But even after grasping the steps of Gödel’s proof, I was unsure what to make of his theorems, which are commonly understood as ruling out the possibility of a mathematical “theory of everything.” It’s not just me. In Gödel’s Proof (a classic 1958 book that I heavily relied upon for my account), philosopher Ernest Nagel and mathematician James R. Newman wrote that the meaning of Gödel’s theorems “has not been fully fathomed.”
Maybe not, but six decades have passed since then. Where are we with these ideas today? Recently, I asked logicians, mathematicians, philosophers, and one physicist to discuss the meaning of incompleteness. They had plenty to say about the implications of Gödel’s strange intellectual achievement and how it changed the course of humanity’s unending search for truth.
PANU RAATIKAINEN, philosopher at Tampere University and author of the Stanford Encyclopedia of Philosophy entry on Gödel’s incompleteness theorems Ever since the ancient Greeks, the axiomatic method has been widely taken as the ideal way of organizing scientific knowledge. The aim is to have a small number of “self-evident” basic propositions — axioms, principles, or laws — such that all truths of the field in question can be logically derived from them. Gödel’s incompleteness theorems show with mathematical precision that this ideal necessarily fails for large parts of mathematics. The whole of mathematical truth concerning even just positive integers (1, 2, 3 …) is so perplexingly complex that it does not follow from any finite set of axioms. This means that some mathematical problems are not even in principle solvable by our current mathematical methods. Progress may require creative conceptual innovation. As a result, mathematical truths do not make up a unified whole of equally indubitable truths; instead, their status as knowledge varies gradually from doubtless facts to increasingly uncertain hypotheses.
Raatikainen makes a good point that Gödel’s theorems muddy the waters between where objective truth ends and invented math begins. One historical way people have tried to overcome the limitations of Gödel’s theorems has been to propose additional axioms beyond the commonly accepted ones. Say you want to prove a statement with the traditional axioms, but you find that you can’t — that it is undecidable. If you add a new axiom to your starting set, you may then be able to prove the statement true. Adding a different axiom, however, and you may be able to prove it false. So whether it’s true or false depends on the choice you’ve made. Suddenly, “truth” is more contingent on one’s preferences or assumptions.
REBECCA GOLDSTEIN, philosopher and author of Incompleteness: The Proof and Paradox of Kurt Gödel Intuitions have always played an important role in mathematics. After all, we can’t prove everything; we need to accept some truths (i.e., the axioms) without proof in order to get our proofs off the ground. But we’ve learned over the centuries that sometimes intuitions prove unreliable — so unreliable as to generate actual paradoxes — meaning we’re driven to assert out-and-out contradictions. In the early 20th century, Bertrand Russell and Alfred North Whitehead were working on The Principles of Mathematics, which attempted to reduce arithmetic to logic. [The view that math is nothing but logic is known as “logicism.”] The work led Russell to the discovery of what came to be called Russell’s Paradox. It concerns the set of all sets that aren’t members of themselves. The paradox reveals itself when you ask: Is this set a member of itself? The contradiction: If it is, then it isn’t. And if it isn’t, then it is. (Georg Cantor, considered the founder of set theory, had already realized the contradiction back in the 1890s.) How Gödel’s Proof Works explainers How Gödel’s Proof Works Save Article Read Later The response of mathematicians — most forcefully David Hilbert, the leading mathematician of that time — was to rid mathematics of iffy intuitions by way of formally axiomatizing mathematics into a consistent and complete set of algorithmic, recursive rules, essentially reducing math to a mechanical game of symbol manipulation. This goal of formalization was christened the Hilbert Program. What Gödel proved was that the Hilbert Program was unrealizable. His first incompleteness theorem states that in every formal system of mathematics that is rich enough to express arithmetic, there will be propositions that are both true and unprovable. So, although formal systems comprised of mechanical rules of symbol manipulation successfully eliminate all intuitions, they also fail to capture all that we know to be mathematically true — a knowledge enriched by intuitions concerning the infinite structures that we call numbers.
It’s fascinating that our intuitions about numbers might go beyond what we can prove.
Personally, my intuition is silent on the mathematical statement that, in the years after Gödel’s proof, made incompleteness real. It is called the continuum hypothesis, and it asserts that the set of all real numbers (the continuum) is the second-smallest infinite set after the set of natural numbers (1, 2, 3 …). It was found to be undecidable using the standard axioms of mathematics. Extra axioms can be engineered to establish it as true or false, but logicians are divided on which way to go.
... continue reading