We are well aware by now of the observer problem in quantum mechanics. Human subjectivity appears to play a key role in the results of quantum experiments. However, the observer problem reaches far beyond just quantum mechanics, argues Edward Frenkel.
In the episode The Path to the Black Lodge of David Lynch’s iconic series Twin Peaks, Annie (played by Heather Graham) recites to Dale Cooper (Kyle MacLachlan) a famous quote by the great German physicist Werner Heisenberg, “What we observe is not reality itself but reality exposed to our method of questioning.”
This quote elegantly encapsulates what is often referred to as “observer-dependence” in quantum mechanics: Depending on how we set up an experiment, quantum reality will expose itself in various ways, with different experimental setups revealing different, seemingly contradictory forms. For example, in the famous double-slit experiment, electrons will reveal themselves as waves if we don’t put detectors behind the slits but will appear to us as particles if we do put the detectors. Thus, our choice of experimental protocol influences what pattern of behavior we observe. This makes the first-person perspective an integral part of modern physics.
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Is there room for the first-person perspective in mathematics as well?
At first glance, the answer appears to be “no.” After all, math seems to be a “paragon of reliability and truth,” as the eminent mathematician David Hilbert put it. It is the most objective of all sciences, and mathematicians take pride in the certainty and the timeless nature of mathematical truths. Indeed, if Leo Tolstoy wasn't born or died before he wrote Anna Karenina, this book wouldn’t exist; no one else would have written the exact same novel. But if Pythagoras had not lived, someone else would have discovered the same Pythagoras theorem (and in fact, many did). Moreover, his theorem means the same thing to everyone today as it meant 2,500 years ago when he discovered it, and there’s every reason to believe that it will mean the same thing to everyone 2,500 years from now, regardless of their culture, upbringing, religion, gender, or skin color.
Now, Pythagoras theorem is a mathematical statement about the right triangles (also known as right-angled triangles, such that one of its angles is the right angle; that is, 90 degrees) that holds in a specific framework — namely, that of Euclidean geometry on the plane (think of an idealized flat tabletop extended to infinity in all directions; that’s what mathematicians call a “plane”). It states that x2+y2=z2 where x and y are the two sides of the right angle and z is the side opposite to it; it’s called the hypothenuse.
However, Pythagoras’ theorem is not true in the framework of non-Euclidean geometry.
The words “non-Euclidean geometry” may sound intimidating, but actually, it’s something very much down-to-earth. Indeed, an example of non-Euclidean geometry is the geometry of a sphere. Imagine a globe. In its spherical geometry, the role of lines is played by its largest possible circles, such as the meridians or the equator. Therefore, an analogue of the right triangle in spherical geometry is a figure bounded by three large circles, with the condition that two of them intersect at the right angle. Let’s carve one such triangle by connecting the north pole of the globe to two points on the equator via two meridians and then connecting those two points by a segment of the equator. The resulting triangle actually has two right angles, at the intersection points of each of the two meridians and the equator. Hence it has two “hypothenuses” of the same length, so that y=z. And since the third side (the one going along the equator) has a non-zero length x, the familiar equation x2+y2=z2 obviously doesn’t hold. Thus, Pythagoras’ theorem fails for the right triangles on the sphere!
What’s going on here? To answer this question, we need to look more closely at what it means to prove a mathematical theorem. A theorem does not exist in a vacuum; it exists in what mathematicians call a formal system. A formal system comes with its own formal language; that is, an alphabet and a collection of words as well as grammar, which enables one to construct sentences that are considered meaningful in this language. Euclidean geometry is an example of a formal system. Its language includes words such as “point” and “line” and sentences such as “Point p belongs to line L.”
Next, out of all sentences of our formal system we distinguish the ones that we stipulate to be valid, or true. These are the theorems. They are constructed in two steps: First, we have to choose the initial theorems, the ones we declare to be valid without proof. These are called axioms. They constitute the seed of the formal system, if you will. We have to start somewhere! For example, here’s one of the axioms of Euclidean geometry: “If p and q are two distinct points on the plane, then there is a unique line L to which both p and q belong.” Reasonable. But if you look for a proof of this sentence in the formal system of Euclidean geometry, you won’t find it. The point is that we accept it as valid without proof, along with the other axioms.
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This creates the impression that all of mathematics could be done by a computer. But that’s not the case.
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Second, we use the logical rules of inference to derive other valid sentences from the axioms we have chosen; these sentences are called the theorems of our formal system. For example, suppose that sentences “A” and “A implies B” are axioms. Then sentence B is a theorem, derived from those two. Next, if “B implies C” is also a theorem (it could be an axiom or it could be a theorem we have derived from axioms before), then so is “C” and so on. Producing theorems this way is a linear, step-by-step procedure of purely syntactic (or symbolic) manipulation that could in principle be delegated to a computer.
This creates the impression that all of mathematics could be done by a computer. But that’s not the case. It’s true that once the axioms have been chosen, there’s no ambiguity in what constitutes a theorem in our formal system. This is the objective part that could indeed be programmed on a computer. But one still has to choose the axioms, and this choice is crucial. For example, Euclidean geometry of the plane and non-Euclidean geometry of the sphere differ in just one of their five axioms; the other four are the same. But this one axiom (the famous “Euclid’s fifth postulate”) changes everything. Theorems of Euclidean geometry aren’t theorems in non-Euclidean geometry and vice versa. (Fun fact: for about 2,000 years mathematicians tried to prove the fifth axiom of Euclidean geometry from the first four — only to realize, in the 19th century, that it is impossible; on the contrary, one can replace it with a different axiom, and this leads to the non-Euclidean geometry discussed above.)
How do mathematicians choose their axioms? In the case of Euclidean and non-Euclidean geometry, the answer is clear: they correspond to what we wish to describe — if it’s the geometry of a plane, then the former; if it’s the geometry of a sphere, then the latter. But this simplicity is deceptive because the subject matter is just too narrow. Mathematics is vast and the question of how to choose the axioms becomes much more poignant when we go deeper, to the foundations of math.
For the past hundred years, mathematics has been based on set theory. The idea is that every mathematical object is what we call a set equipped with some additional structures. For example, we have the set of natural numbers: 1,2,3,4,… equipped with the operations of addition and multiplication. What exactly is a general set has never been properly defined in math. The creator of set theory, Georg Cantor, has given the following poetic description: “A set is a Many that allows itself to be thought of as a One.” It’s beautiful and gives one an intuitive understanding of what a set is but it’s hardly a rigorous definition. Nevertheless, like everything in math, set theory is described by a particular formal system. It’s called ZFC, in honor of its creators Ernst Zermelo and Abraham Fraenkel (no relation, as far as I know) as well as one of its axioms called the Axiom of Choice. (Set theory is not the only foundational system of math, but other systems also exhibit effects similar to what I describe below.)
This is where things get complicated.
Today, most mathematicians accept ZFC as the formal system of set theory, underpinning all of mathematics. But there’s a small minority of mathematicians (perhaps, less than 1%) who call themselves finitists. They refuse to include one of the axioms of ZFC, called the Axiom of Infinity. In other words, the formal system they work with is ZFC without the Axiom of Infinity. I’ll call it “ZFC light.”
Axiom of Infinity states that the set of natural numbers 1,2,3,4,… exists. This is a much stronger statement than the statement (called “potential infinity”) that for every natural number there’s a bigger number. Finitists agree that the list of natural numbers never ends, but at any given time they restrict themselves to considering only finite subsets of the set of natural numbers. They refuse to accept that the totality of all natural numbers, taken together at once, is real. Hence, they remove the Axiom of Infinity from ZFC.
Does this matter? Yes! Because they remove this axiom, there are fewer theorems finitists can prove. Significantly fewer. It turns out that they can’t even prove some theorems about the finite sets they like so much. In other words, it turns out that to prove some statements about finite sets, one needs to step out of the world of finite sets, which means accepting the reality of infinite sets.
Suppose you are a young mathematician just starting your career. Which formal system should you adhere to? ZFC or “ZFC light” (without infinity)? This is a big question! To be fair, most mathematicians use infinity in their research without thinking about it too much. They check the “Agree to Terms and Conditions” box of ZFC, so to speak, without reading the fine print. But that doesn’t make the question any less real. In your proofs, you either use infinite sets or you don’t. And the difference is enormous.
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You might think: surely, there must be some objective criteria by which we could judge formal systems and decide which one to choose. Spoiler alert: There isn’t.
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For me, it’s a no-brainer. I love infinity because it symbolizes something that transcends time and space. Come to think of it, I became a mathematician because I wanted to experience a world of pure Platonic forms that reside somewhere far beyond the minutiae of daily life. And infinity leads me to this kind of experience. But I also respect others who say, “You’d have to show me infinity for me to believe it’s real.” And I also know there’s no trick I can use to show you infinity the way I can show you a particular number, or a particular object. The reality (or non-reality) of infinity is something each of us has to figure out on our own. It is inherently subjective. And so, one does have the option to refuse using infinity. To me, this is akin to what the French writer George Perec from the Oulipo literary group in France did when he wrote his novel without using the letter “e.” I see it as a special form of art but also as a self-imposed limitation.
You might think: surely, there must be some objective criteria by which we could judge formal systems and decide which one to choose. Spoiler alert: There isn’t.
To explain this, let’s discuss what properties we’d like a formal system to have. The first and foremost is consistency, which means that the axioms of the formal system do not contradict each other. Otherwise, our formal system would be useless because then every sentence in the language of this formal system would be a theorem (including, for every sentence A, its negation “not A”). We certainly don’t want that. What we want is to be able to prove some — but not all — sentences. However, Kurt Gödel’s Second Incompleteness Theorem states that a sufficiently sophisticated formal system (such as ZFC or “ZFC light”) cannot prove its own consistency. In principle.
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Think about it: mathematicians actually don’t know whether ZFC, the foundation of all of math today, is on solid ground. And most likely, we’ll never know. Why? Because, by Gödel’s Second Incompleteness Theorem, we could only prove the consistency of ZFC in a “bigger” formal system, obtained from ZFC by adding more axioms. But how do we know that this bigger formal system is consistent? We don’t. And if it’s not consistent, then its proof that ZFC is consistent has no value because this bigger formal system would also be able to prove its negation. Since this bigger formal system can’t prove its own consistency (again, by Gödel’s theorem), the only way to prove its consistency would be to create an even bigger formal system, and thus ad infinitum.
Not only does this imply that we have to take consistency of mathematics, as we understand it today, on faith. It also suggests that there’s actually no objective criterion as to which axioms we should choose for doing mathematics.
But then, who has the authority to choose these axioms?
The answer might surprise you: We do — living, breathing mathematicians. No one else can do it for us. Not even Pythagoras or Euclid or Gödel. They have shown us the way but now it’s in our hands. It’s our choice, it’s our free will. And we exercise it every day — in particular, by accepting or rejecting the Axiom of Infinity.
My criterion for axioms is simple: I choose the ones that lead us to richer, more diverse, more fruitful mathematics (this is close to the position advocated by philosopher Penelope Maddy, which she calls naturalism). Why limit oneself? That’s why I do accept the Axiom of Infinity.
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The act (or shall we say, the art) of choosing a particular set of axioms is akin to the act of setting up a particular experiment in quantum physics. There’s a choice inherent in it, and it brings the observer into the picture. This is how the first-person perspective, and the freedom that comes with it, takes its rightful place in mathematics.
At the end of the 19th century, William James predicted that the belief of the science of his era that “in its own essential and innermost nature our world is a strictly impersonal world” would turn out to be “the very defect that our descendants will be most surprised at in our own boasted science, the omission that, to their eyes, will most tend to make it look perspectiveless and short.” Physics and mathematics of the 20th century have proven him right.
