MathPhys

Undoubtedly, real numbers are the most fundamental things to describe literally everything we know about our physical world, so questioning if real numbers are real may appear to be as silly as questioning if our world is real. But seriously, no one has ever actually seen irrational numbers, for example $\sqrt{2}$. All we have seen or we could ever possibly see is a floating point approximation of $\sqrt{2}$ like 1.414213562373095, which is a rational number, but not the irrational number $\sqrt{2}$ itself as a whole.  So how do we know $\sqrt{2}$ exists? One may point to the equation $$x^2=2$$ for that. However, writing the solution symbolically as $x=\pm\sqrt{2}$ does not prove anything about the existence of the irrational number $\sqrt{2}$, though if the existence of $\sqrt{2}$ is assumed, it can be shown that $\sqrt{2}$ cannot be a rational number. (See here for the proof.)

Traditionally, there are two ways to construct real (irrational) numbers in mathematics. One is by Dedekind cuts (named after Richard Dedekind) and the other is by Cauchy sequences. The construction by Cauchy sequences was introduced by Georg Cantor.

Let me explain the construction by Dedekind cuts first. Let $A=\{x\in\mathbb{Q}: x^2<2\}$, i.e. the set of all rational numbers whose square is less than 2, and $B=\{x\in\mathbb{Q}:x^2>2\}$, i.e. the set of all rational numbers whose square is greater than 2. Then there is a gap or a hole between the two sets $A$ and $B$, that cannot be filled by a rational number. If $A$ were to have the least upper bound (also called the supremum) or $B$ were to have the greatest lower bound (also called the infimum), the gap could be filled, and the filler is none other than what we call $\sqrt{2}$. So, how do we know the set $A$ has the least upper bound or $B$ has the greatest lower bound? Actually we don’t, but we want it to exist. So here comes an axiom called the L.U.B. (Least Upper Bound) Property (or the Supremum Property): Every infinite set which is bounded above has the least upper bound. Its dual property is the G.L.B. (Greatest Lower Bound) Property (or the Infimum Property): Every infinite set which is bounded below has the greatest lower bound. This is one of the most basic properties regarding real numbers. The axiom is so fundamental that if we were not to accept this axiom, we would’t have had mathematics as we know it.

A sequence $\{x_n\}$ is called a Cauchy sequence if given a positive number $\epsilon>0$ (no matter how small it is), one can find a natural number $N$ such that for all $m,n\geq N$, the distance between $x_m$ and $x_n$ is smaller than $\epsilon$. Suppose that a sequence $\{x_n\}$ converges to $x$. Then one can easily show that $\{x_n\}$ is a Cauchy sequence using the triangle inequality $$|x_m-x_n|\leq |x_m-x|+|x-x_n|$$ How about the converse? Is a Cauchy sequence necessarily a convergent sequence? Often, we introduce the statement “every Cauchy sequence (of real numbers or of complex numbers) is a convergent sequence” in class as if it were a proven theorem. But the truth is that it is not a theorem because we can’t prove that it is true (without assuming an axiom such as the l.u.b. property), though we want it to be true for a good reason. Georg Cantor introduced the statement that “every Cauchy sequence is a convergent sequence” as an axiom and used it to construct real numbers. For instance, define a sequence $\{x_n\}$ of rational numbers recursively as follows: $$x_1=1,\ x_{n+1}=\frac{x_n}{2}+\frac{1}{x_n}$$ Then $\{x_n\}$ is a Cauchy sequence. By the Cantor’s axiom, it must converge to a number $x$ and it satisfies the equation $x^2=2$, i.e. the solution $\sqrt{2}$ exists.

The rational numbers have density property, meaning between any two rational numbers one can find another rational number (in fact infinitely many rational numbers) in between them, nonetheless  they still have gaps (irrational numbers) and those gaps can be filled by Dedekind cuts or by Cauchy sequences, hence the (real) number system becomes complete or a continuum. The existence of real numbers appears to be inconceivable without having the natural order relation $\leq$. But then another baffling mystery comes in. In set theory, the so-called Well-Ordering Theorem states that “every set can be well-ordered”. A totally ordered set or a chain (a set with an order relation in which any two elements are related by the order) is said to be well-ordered if its every non-empty subset has a least element. Although it is called a theorem, it is actually an axiom. It can be proved by assuming another axiom, for example, Axiom of Choice. According to Well-Ordering Theorem, the set of real numbers can be well-ordered also, though no one has ever found one yet. It’s quite puzzling as to how the set of real number can possibly be well-ordered. By the way, a consequence of Well-Ordering Theorem or Axiom of Choice is Banach-Tarski paradox. What it states is that a solid ball in 3-dimensional space can be decomposed into a finite number of disjoint pieces in such a way that those pieces can be put back together differently via translations and rotations (without changing their shapes) to yield two identical copies of the original ball. Banach-Tarski paradox is pretty counter-intuitive and for that reason it is called a paradox.

As we have seen, the existence of real numbers (irrational numbers) hangs on the axioms like the l.u.b. property or the convergence of Cauchy sequences. So, can we still say real numbers are definitely real? You may say yes if your notion of reality is that of Platonism or of Max Tegmark’s M.U.H. (Mathematical Universe Hypothesis): Platonism in hyperdrive. You may be less convinced, however, if your notion of reality is physical one in usual sense, that can be observed or tested. In quantum field theory, particles are treated as mathematical points and that is believed to be the source of divergence issues arising from quantum field theory. Superstring theory is an attempt to reformulate quantum field theory by replacing point particles by vibrating strings or membranes (in M-theory, M stands for Membranes or Mother meaning the Mother of all string theories). Physicists also realized that to extend the notion of particles from mathematical points in a physically consistent manner, the spacetime itself needs to be quantized (i.e. discrete) as well. Such an attempt to reformulate quantum field theory by quantizing the spacetime goes back to Yukawa Hideki’s elementary domain theory and a notable modern attempt is loop quantum gravity. Someday, we may find out that the spacetime is actually quantum rather than continuum. If and when that happens, the reality of real numbers and continuum could very well be in question.

$1+1=2$ is often quoted by laypeople (in mathematics) as an epitome of the absolute truth. Those who know a bit of mathematics know that that is not the case. For example, there is a number system where $1+1=0$. It is denoted by $\mathbb{Z}_2$. $\mathbb{Z}_2$ has only two numbers $0$ and $1$ and it is the smallest field. Here, we don’t mean a field by a vector field in physics but a number system where two numbers can be added, a number can be subtracted from another, a number can be multiplied by another, or a number can be divided by another nonzero number. $\mathbb{Z}_2=\{0,1\}$ is also important in computer science as $0$ and $1$ can be identified with two possible values of a bit and to the boolean values true and false.

The mind of young Thomas Edison, the one who would become one of the greatest American inventors in the future, was full of curiosities and the traditional school education unfortunately could not satisfy the young kid’s endless desire to know. Often, what’s being taught in school didn’t make much sense to his creative mind. One day, his teacher was telling students that $1+1=2$ and this happened to be one of the things that did not make sense to him. Edison told his teacher that he did not understand why it is always true and gave a counter example: “If you put a tiger and a rabbit together in a closed room, soon there will be only tiger left, so $1+1$ can be $1$ also.” Such a creative kid in class can be a nightmare for a teacher but a pleasant nightmare, I might add.

We have seen that $1+1$ is not necessarily $2$, so can there be a number system where $1+1=1$ like young Thomas Edison contended? It turns out the answer is not that simple and it is even confusing as the short answer is yes and no. In order for something to be called a number system, it needs to be a field. Let me first state the formal definition of a field. A field $\mathbb{F}$ is a nonempty set of numbers with two binary operations $+$ and $\cdot$ such that

1. for $a,b\in\mathbb{F}$, $a+b=b+a$.
2. for $a,b,c\in\mathbb{F}$, $(a+b)+c=a+(b+c)$.
3. there exists $0\in\mathbb{F}$ such that $a+0=a$ for all $a\in\mathbb{F}$.
4. for each $a\in\mathbb{F}$ there exists $-a\in\mathbb{F}$ such that $a+(-a)=0$.
5. for $a,b\in\mathbb{F}$, $ab=ba$.
6. for $a,b,c\in\mathbb{F}$, $(ab)c=a(bc)$.
7. there exists $1\in\mathbb{F}$ such that $a1=a$ for all $a\in\mathbb{F}$.
8. for each $a\in\mathbb{F}\setminus\{0\}$, there exists $\frac{1}{a}\in\mathbb{F}$ such that $a\frac{1}{a}=1$.
9. for $a,b,c\in\mathbb{F}$, $a(b+c)=ab+ac$.

The properties 1-4 says $\mathbb{F}$ with $+$ is an abelian group. The properties 5-8 says $\mathbb{F}\setminus\{0\}$ with $\cdot$ is an abelian group. In universal algebra, the identities $0$ and $1$ and inverses $-$ and $/$ are also considered as operations, more specifically the identities are nullary operations and the inverses are unary operations. Since the inverse $/$ (division) is not defined for $0$, a field is not considered as an algebra or an algebraic structure, i.e. a set with operations in universal algebra. Examples of fields include $\mathbb{Q}$, the set of all rational numbers, $\mathbb{R}$, the set of all real numbers, and $\mathbb{C}$, the set of all complex numbers. There are also finite fields $\mathbb{Z}_p=\{0,1,2,\cdots,p-1\}$ where $p$ is a prime number. Fields are not necessarily sets of numbers. There are also fields called function fields, an example being the set of rational functions of one variable such as $\frac{x^3+2x+1}{x^2-1}$. The definition tells that a field must have at least two numbers $0$ and $1$. So, what happens if a field satisfies $1+1=1$? It implies that the field must have a single element and that $0=1$. This pretty much rules out the possibility of $1+1$ being $1$, right? But it is not. First, allowing $0=1$ does not violate the definition of a field, although in conventional mathematics, it is implicitly assumed that $0\not=1$. The definition says any nonzero element has a multiplicative inverse but it doesn’t say $0$ cannot have a multiplicative inverse (although again it is implicitly assumed that $0$ does not have a multiplicative inverse in conventional mathematics). For a moment, let us allow $0=1$. There is a dire consequence of allowing this: any vector space over this field with one element is $0$-dimensional as ${\mathbf v}=1{\mathbf v}=0{\mathbf v}={\mathbf 0}$! Perhaps, we shouldn’t be allowing $0=1$ which seems to be good-for-nothing and should end discussion on the possibility of the field with one element, right? Not so fast, this is not over yet. While we don’t even know whether it can exist, mathematicians are taking the field with one element pretty serious. They want it so badly that they are willing to bend the definition of a field. It is usually denoted by $\mathbb{F}_1$ or $\mathbb{F}_{\mathrm{un}}$. The subscript “un” is from the French word meaning “one”. The field with one element was first introduced by a Belgian-French mathematician Jacques Tits in 1957 in his paper titled “Sur les analogues algébriques des groupes semi-simples complexes.” One of the axioms of projective geometry states that a line must have at least three points. If one replaces this axiom, while keeping others, by “a line admits only two points”, we obtain a degenerate geometry. Tits conjectured that such degenerate geometry can have equal footing with projective geometry if one were to consider such geometry on a field of characteristic one, i.e. a field satisfying $1=0$. Since the late 1980s, $\mathbb{F}_{\mathrm{un}}$ has gained a lot of attention from major league mathematicians after they realized profound implications of $\mathbb{F}_{\mathrm{un}}$ in number theory (including the Riemann hypothesis), algebraic geometry, and noncommutative geometry. Despite many attempts and proposals, as far as I know, there is no widely accepted notion of $\mathbb{F}_{\mathrm{un}}$, yet. Here is my own take on it. I propose to replace the above conventional definition of a field by the following: A field $\mathbb{F}$ is a nonempty set of numbers with two binary operations $+$ and $\cdot$ such that

1. $\mathbb{F}$ with $+$ is a abelian group.
2. $\mathbb{F}$, possibly excluding an element, with $\cdot$ is an abelian group.
3. for $a,b,c\in\mathbb{F}$, $a(b+c)=ab+ac$.

In this definition, there is no specific mentioning of particular additive and multiplicative identities, only that they exist. This definition then may allow the existence of a field with one element $\{1\}$ without requiring that $1=0$. I have not examined this idea much further yet, so whether this proposal can be something useful remains to be seen.

Thus far, $\mathbb{F}_{\mathrm{un}}$ is a phantom object in mathematics and mathematicians may be, after all, chasing a phantom. But throughout the history of mathematics, we have had phantom objects which turned out to be extremely important entities not only in mathematics but also in physics. Examples include 0, negative integers, irrational numbers, the imaginary number $i$ (complex numbers), quaternions, non-Euclidean geometry, and the summability of divergent series such as $1+2+3+\cdots=-\frac{1}{12}$ (read here for details if you are curious). One day, we may eventually understand what $\mathbb{F}_{\mathrm{un}}$ is and if and when that happens, it will take up its rightful place in mathematics.

Returning to the original question, can $1+1$ be $1$? Je ne sais pas. (I don’t know.) But those major league mathematicians who are way smarter than I am desperately want that to be true. Perhaps time will tell.