The Existence and Reality of Numbers: Beyond Natural and Real Numbers

Numbers are a fundamental concept in mathematics, but what if there were never numbers and counting didn't exist? Many of us take for granted the numbers we know, but what if they are not what they appear to be? In this article, we will explore the nature of numbers, particularly natural numbers, integers, rational numbers, and real numbers, and argue that the real numbers may not be as 'real' as we think.

Abstract and the Definition of a Number

I believe that many of the things we call numbers don’t truly exist, or at least, they aren’t what I would like to call numbers. A number, as I see it, is something we can write on a page, and for which we have a well-defined arithmetic, at least addition and multiplication. We start with natural numbers, which we can write and which allow us to perform addition and multiplication through concrete, finite processes.

The Construction of Numbers

Natural numbers form the basic building blocks of all other number systems. Once we have natural numbers, we can construct integers and rational numbers. These systems can be extended to complex rational numbers, each with its own arithmetic rules that are based on the previous steps. This process is straightforward and enables us to write two numbers on a page and perform operations to find their sum and product.

The Fall of Real Numbers

The construction of real numbers, however, is where things begin to falter. The real numbers represent the passage from the discrete to the continuous, and this transition is one of the greatest mysteries of mathematics. There are several existing constructions, including infinite decimals, Dedekind cuts, and equivalence classes of Cauchy sequences, but each requires an infinite number of operations to establish their arithmetic. This is not a practical method for me as an 'arithmetic.'

For me, real numbers are more of a marketing term than a description. The term 'real' was coined by Descartes, who introduced the idea of a number line where each point corresponds to a real number. In other words, we can assign a numerical length to every segment. However, the ancient Greeks avoided this idea, focusing on comparing two segments. They developed a theory based on Eudoxus to deal with incommensurate segments and areas.

The real numbers, it turns out, have a deep philosophical and theoretical underpinning. For instance, there is no rational number whose square is 2. Instead, we use the symbol (sqrt{2}) to represent this irrational number. We can define (mathbb{Q}[sqrt{2}]), a system of numbers with a finite arithmetic, but the real numbers, (mathbb{R}), present more complex issues. When we see statements like (sqrt{2} 1.414ldots), I begin to question their validity.

The Irrationality of Real Numbers

Of all the irrational real numbers, (sqrt{2}) is probably the most 'real.' It is an algebraic symbol, and when we square it, we get 2. However, we can further divide the transcendentals into those that can be expressed with a finite formula or algorithm and those that cannot. The transcendentals with no finite representation are significant because there are only countably infinite finite representations, but the reals are uncountable. According to Cantor, almost all reals have no possible way to write them down, making the concept of 'real' numbers even more elusive.

The Arithmetic of Infinite Representations

Even when we can write down the formulas for numbers like (pi), (e), and (gamma), the arithmetic of these numbers is often vacuous. Consider the expression (pi cdot e cdot sqrt{2}). The result is (pi cdot e cdot sqrt{2}), and even when we have formulas, the arithmetic is merely symbolic. This leads to an interesting paradox: if we write (pi cdot e cdot sqrt{2} supset), we can claim to solve every math expression while it equals itself!

The Axiomatic Approach to Real Numbers

Instead of constructing the real numbers and proving their properties through careful, finite steps, we often rely on axioms. The real numbers are often asserted to satisfy the field axioms, including closure, associativity, and commutativity of addition and multiplication, without proving them. This is in contrast to the construction of rational numbers or complex rational numbers, which we prove satisfy the definition of a field. The axiomatic approach is more of a wishful thinking than a concrete method, which explains why I don't consider the reals as 'numbers' but rather as a 'marketing term.'

Numbers, then, are more like deli meat that we can locate between two slices of bread. As we work with numbers, we can get narrower those slices, reducing the range, but we will always have a range, never a number.