Axioms of Arithmetic 1

Before we study the properties & relations of arithmetic operations, we need to understand about the origin of the “Axioms of Arithmetic”.

The 2 fundamental operations that can be performed on numbers are addition & multiplication.

Subtraction & division were in a way “inverse” operations of addition & multiplication.

The “fundamental laws of arithmetic” lay down the properties & rules of operation of each of these 4 operations individually and together with the other operations.

The “fundamental laws of operations” are also called “axioms of arithmetic”.

How they came to be adopted by mathematicians, and how they lost their importance, is an interesting story in the history of mathematics.

Logic & Geometry of Greeks

We can say the formal study of geometry started among Greeks around 600 BCE. When Aristotle developed his theory of Logic, mathematicians found it a useful guide in developing mathematical ideas, both in Arithmetic and in Geometry.

As noted elsewhere in these articles, Greeks focused mostly on Geometry. Their Arithmetic did not develop very much.

Euclid compiled all the geometrical ideas of the past 5 centuries into his Elements.

His development of Plane Geometry starting with a few simple assumptions and by deducing all of them sequentially through arguments using deduction, was a model of developing mathematics though logic.

It also established the paradigm that truth in mathematics can be deduced by use of logic & the method of proof.

So mathematics developed by an internal logic of its own, without any need for confirmation from the physical world.

It was unchallenged for almost 2000 years.

Fundamental Laws of Arithmetic

By early 19th century, the field of arithmetic had become wide and varied. Different kinds of numbers & different kinds of operations were invented giving rise to many doubts in the minds of mathematicians.

Some of the simpler questions were

1. What is a number? Under what circumstances can a new entity can be accepted as a number?

2. What operations were permitted on numbers?

3. When multiple operations were performed, what should be the order of operations?

There was a movement among mathematicians to reconstruct the whole of arithmetic on “solid” foundations, starting with a few assumptions, the same way Euclid had developed Plane Geometry.

Though several mathematicians were working on it, Gotlieb Frege announced in the last decade of the 20th century that he has completed the work. He called his 2-volume work as “Fundamental Laws of Arithmetic”.

Frege had used principles of Set Theory in developing his structure of Arithmetic.

Around 1901, Frege was about to publish his books.

Around the same time the Barber’s Paradox was used in paper by English mathematician Bertrand Russell to prove a point about the inconsistencies with naive set theory. It brought into question the validity of Frege's work which was based on the validity of Set Theory.

Frege's frustrated response is a famous one. “Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished."

Axioms of Arithmetic

This led to the reevaluation of set theory, eventually being replaced by consistent axiomatic systems that introduced some limitations on how sets could be formed.

But the “Fundamental Laws of Arithmetic” are still useful at the level of arithmetic we learn in school.

What we call today as “Fundamental Laws of Arithmetic” are a slightly modified/ abridged version of Frege's work.

The next document gives a brief description of the Axioms of Arithmetic