2+3=4

Harys Dalvi

February 2022

I once saw a math joke something along the lines of “if Joe has ten apples in his right hand, and twelve apples in his right hand, what does Joe have in total?” Answer: “really big hands.” While this seems ridiculous, it's definitely not wrong, and I would argue it's even more clever and insightful than it seems at first.

In fact, you could construct an entire self-consistent mathematical system to describe the number of apples Joe can have in his hands. Say Joe can only hold a total of four apples at a time. Then we could write \(2+3=4\), because if there were 5 apples, one would just fall. This is in fact valid mathematics within this mathematical framework specifically designed to model the apples in Joe's hands.

What does this mean? First of all, it means I have just shown an example of a mathematical framework where \(2+3=4\) is a completely valid statement. This isn't so strange: modular arithmetic is accepted and widely-used, and there you can have \(2+3 \equiv 1 \mod 4\).

Second, this tells us something about the relationship between math and the real world. Sometimes people wonder how math describes the physical world so accurately. I am going to argue that this is because math was designed to do this, at least to some extent.

The Universe Can't Do Math

Plants can't talk, but we can talk about plants. The universe can't do math, but we can use math to describe the universe. I'm going to show some examples of this idea.

Bacterial Populations

At least during the log phase, bacterial populations grow exponentially [1]. As an equation, that looks like $$P = Ce^{kt}.$$ How is this possible? Do the bacteria conduct regular censuses and conduct centralized business meetings, making sure to divide at such a rate that the population grows exponentially? Of course that's ridiculous. The underlying cause is that each bacterial cell produces about the same number of offspring in a given amount of time. And the underlying cause of that, in turn, has to do with how long it takes for the bacteria to collect all the things they need to divide in the cell cycle.

If each bacterial cell produces the same number of offspring in a given amount of time, what does that look like as an equation? It's $$\frac{dP}{dt} = kP.$$ And if you solve that equation, you get the exponential growth equation above. So it's not that the bacteria do math: instead, the physical world places certain constraints on how the bacteria will behave. Based on those constraints, we can logically deduce what will happen to the bacteria. The way we deduce that is called mathematics.

Coulomb's Law

Newton's well-known law of gravitation states $$F=\frac{Gm_1m_2}{r^2}$$ to describe the force of gravity between two masses. However, since gravity is weird and I do not have a theory of everything, I'll use the similar Coulomb's law from electromagnetism: $$F=\frac{kq_1q_2}{r^2}.$$ This law is quite surprising. Why should the electric force between two charges be directly proportional to \(1/r^2\)? Do electrons have little calculators and rulers that we can't see, making complex measurements every instant in order to obey this law? Do they take into account all the charges in the entire universe when deciding where to go in an instant?

Obviously these little electron masterminds are unrealistic. Another theory is to have electric field vectors coming out of charges and spreading out. That way instead of measuring distances to every charge in the entire universe every instant, charges only have to look at the electric field where they are. With Gauss's law, this actually gives the same 1/\(r^2\). Why? Because the surface area of a sphere is proportional to \(r^2\).

Derivation of Coulomb's law from Gauss's law

Gauss's law states $$\oint \mathbf{E} \cdot d \mathbf{A} = \frac{Q_{enc}}{\epsilon_0}$$ Consider a spherical surface around a charge \(Q_{enc}\). If there are no other charges, the electric field should have the same magnitude all around the surface, and should always be perpendicular to the surface. Therefore \(\mathbf{E} \cdot d \mathbf{A} = E \, dA\), where \(E\) is the magnitude of the electric field. $$\oint \mathbf{E} \cdot d \mathbf{A} = \int E \, dA = EA = \frac{Q_{enc}}{\epsilon_0}$$ Since the surface is a sphere, \(A = 4 \pi r^2\). Also note that the \(k\) in Coulomb's law equals \(1/4\pi \epsilon_0\). $$E = \frac{Q_{enc}}{4 \pi r^2 \epsilon_0} = \frac{kQ_{enc}}{r^2}$$ Finally, the magnitude of electric force on a charge \(q\) equals the magnitude of the electric field times \(q\). If we say \(Q_{enc}=q_1\), that gives $$\boxed{F=\frac{kq_1q_2}{r^2}}$$

Let's go back to the big picture. It seemed at first that the universe somehow knew about inverse square laws and mathematics. In reality, this mysterious law comes down to the surface area of a sphere, and electrons have no mathematical abilities.

Conservation of Momentum

Conservation of momentum is a really interesting physical law. It was first used for classical physics, but it turns out to work for quantum physics too. So does the universe measure the momentum of each particle and add it up to make sure the total momentum is the same?

In fact, conservation of momentum takes place on a smaller scale. You can show that interactions between objects always conserve momentum, and therefore, the momentum of the universe is conserved. One way to do that is with Newton's second and third laws: \(\mathbf{F}=d\mathbf{p}/dt\) and \(\mathbf{F}_{AB}=-\mathbf{F}_{BA}\)

But then how can you derive Newton's second and third laws? You can't: they are laws, in the language of math, that describe the physical universe. It's a lot like the differential equation for bacterial populations: mathematics is a tool made to describe the universe, which explains why the universe seems to run on mathematics.

Axioms

If mathematics is just a tool we use to describe the universe, does that make it invented? I think the idea of axioms is helpful here. Euclid proposed five axioms that are at the base of his geometry [2], but other forms of geometry use different axioms and come to different conclusions [3].

If these geometries have different conclusions, how do we know which geometry is correct? Here comes the cop-out answer: it depends. If you are doing geometry on a plane, Euclidean geometry is correct. If you are doing geometry on a sphere, spherical geometry is correct.

But there's another way to find out which geometry is correct. Just check the axioms, and see if they apply to whatever thing you're looking at. This technique should work for anything, not just geometry, as long as the math is valid.

Let's go back to Joe and his apples. Conventional math tells us 2+3=5, but if Joe has 2 apples and we add 3, we find that Joe has only 4 apples since one fell out of his hands. What's the problem? Is math wrong? No, it's just that a fact of conventional mathematics does not apply to the case of Joe and his apples: specifically, the idea that the set of counting numbers is infinite. Because of this, we need to define addition in a different way than in conventional mathematics.

Mathematicians define addition in different ways all the time. Just look at linear algebra: you start with a set of axioms and a definition for addition. If whatever real-world thing you are studying follows all the axioms of linear algebra, then you can use all the methods of linear algebra which have been developed by mathematicians over the years. Some of this conflicts with normal mathematics: for example, \(AB-BA\) is not necessarily 0 when you work with matrices.

Richard Feynman noted that in physics, unlike in mathematics, starting with fundamental axioms isn't always the best way to do things [4]. Why is that? I think it's because we need to start with the physical world, then see what axioms it follows, and finally do math assuming those axioms are true.

I think we need to reframe our idea of axioms. Instead of a fundamental truth, axioms are a starting place from which you do further reasoning. If you ever find yourself at the same starting place as some branch of mathematics, you can use the reasoning from that part of mathematics.

So is mathematics discovered or invented? It's hard to tell. I would say that the axioms, the starting points of mathematics, are observed in nature or just invented. From there, the reasoning we do is a process of discovery, not invention.

Finally, I want to emphasize the interplay between reasoning and observation. Mathematics is the process of first picking axioms that appear to be true based on the physical world, and then doing numerical reasoning from there. The first part explains why the physical world seems to follow mathematics: it's the other way around. But that doesn't make mathematics lesser in any way, because the second part is what lets us make accurate predictions about the real world. Reasoning is what makes mathematics so powerful.

References

1. Growth of bacterial populations (Britannica) ^
2. Axioms and Postulates of Euclid (The Elements of Euclid translated by Sir Thomas Heath) ^
3. The Three Geometries (EscherMath) ^
4. The Relation of Mathematics to Physics (Richard Feynman, Youtube) ^