If Numbers Didn't Exist: A Severe Consequence of Abstraction

Numbers, as we understand them, might not exist in the way we conceive. Here, we explore how numbers are constructed, the challenges in defining them, and the potential implications of a world without numbers.

The Nature of Numbers

Defining a number is a fascinating challenge. At its core, a number is something we can write down, add, and multiply. Natural numbers, for instance, are the numbers we can easily write and perform arithmetic operations on:

Natural Numbers: These are the simplest form of numbers, like 1, 2, 3, etc. We know how to write them down and perform addition and multiplication. Integers and Rational Numbers: These extend the natural numbers to include negative numbers and fractions. Arithmetic operations are well-defined and finite. Real Numbers: This is where the complexity arises. Real numbers are difficult to define because they require us to perform an infinite number of operations. This is problematic for our definition of arithmetic.

The Real Numbers

The construction of real numbers is a monumental achievement, but it comes with its own set of challenges. The existing constructions, such as infinite decimals, Dedekind cuts, and equivalence classes of Cauchy sequences, require us to perform an infinite number of operations. This is not something we can easily do in practice.

Their arithmetic requires us to do an infinity of operations and then go on with our business. That doesn't work for me as arithmetic which is why I have trouble conceiving of the reals as numbers.

Descartes' idea of the real number line, where each point corresponds to a real number, is a marketing term rather than a precise description. The ancient Greeks, on the other hand, had a more practical approach:

Incommensurable Segments: Segments that cannot be expressed as rational numbers. For example, the side and diagonal of a square are incommensurate. GCD and Euclidean Algorithm: The Greeks used the GCD algorithm to find the greatest common divisor, which is a precursor to the modern Euclidean algorithm.

The Uncountable and the Unrepresentable

The set of real numbers is uncountable, meaning there are more real numbers than can be represented by any finite sequence. This implies that almost all real numbers have no finite representation. Formally, the number of finite representations is countable, while the set of real numbers is uncountable.

There are only countably infinite finite representations, but the reals are uncountable and per Cantor, those cannot be put into one-to-one correspondence. So almost all reals have no possible way to write them down.

Even with the finite representations of numbers like (pi), (e), and (gamma), the arithmetic involved is often vacuous. For instance, the multiplication of (pi), (e), and (sqrt{2}) is:

(pi times e times sqrt{2} pi times e times sqrt{2})

This is not a useful or practical way to perform arithmetic. The way we handle this is by using axioms, but even this approach is abstract and theoretical:

It's too hard to do the constructions and then prove the properties we want so we have the Field Axioms of the real numbers where we just assert the properties we want – closure, associativity, commutativity, etc. As if wishing made it so.

This abstraction means that the real numbers are more a construct than a concrete entity, much like deli meat that can be placed between two slices of bread. The more we work with real numbers, the narrower the range becomes, but we will never have a precise number.

Conclusion

The concept of numbers, especially real numbers, is deeply rooted in mathematical abstraction. While we can define and work with them, the practical limitations and theoretical challenges imply that the idea of numbers as we know them might not hold up in a world without such abstractions.

So, next time you think about numbers, consider the depth and complexity of their existence, or non-existence, and the challenges they pose in our understanding of the mathematical world.