The whole is greater than the part, isn’t it?

Euclid flourished some 50 years after Aristotle and was certainly familiar with Aristotle’s Logic. Euclid’s organization of the work of the early geometers was truly innovative. The results of it depended on basic assumptions, called axioms and “common notions”. There are a total of 23 definitions, five axioms, and five common notions in The Elements. Axioms are specific assumptions that can be taken as self-evident, eg “the whole is greater than the part”.

Euclid imposed order on mathematics, creating an axiomatic system that endures to this day. His goal was to deduce a large number of theorems or propositions from a small number of axioms.

Your approach can be summarized as follows: (a) define your terms; (b) state your assumptions; (c) derive the logical consequences. Using this approach, Euclid proved nearly 500 theorems from a handful of axioms. The deductions had to be logically flawless, with each step clearly following the previous one.

But are Euclid’s axioms really beyond doubt? Is it essential to assume that “the whole is greater than the part”? Is this even justified? Galileo was one of the first to question it. He reasoned that the set of natural or counting numbers, 1, 2, 3, … and the set of even numbers, 2, 4, 6, … are of equal magnitude. It is very easy to make a perfect match between each number and its double: we match 1 with 2, 2 with 4, 3 with 6, and so on. Since both sets of numbers are infinite, the process goes on forever.

Galileo concluded that the number of even numbers is equal to the number of natural numbers. But this is counterintuitive. The even numbers form only a part of the set of natural numbers. We can argue that since odd and even numbers can easily be paired, they are equally abundant. But, with the logic of Galileo, they are also equinumerous with the natural numbers. However, Galileo did not dare deny Euclid’s axiom that the whole is greater than the part.

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Galileo reached a similar conclusion about continuous sets. For two concentric circles, a radial line from center O cuts them both. Thus, each point p1 on the inner circle is paired with a point p2 on the outer circle. Galileo concluded that there must be the same number of points on each circle. But this is surprising: the outer circle is clearly longer than the inner one. Galileo concluded that “the attributes of equality, majority and minority have no place in infinities”. He was on the verge of a great discovery, but he did not develop his ideas. That had to wait for the arrival of Georg Cantor, the founder of modern set theory, some three centuries later.

Cantor recognized that the hallmark of an infinite set is that it can be matched one-to-one with a part of itself. He used this feature to define infinite sets. He then defined two infinite sets that can be paired to be the same size. Thus, Cantor abandoned Euclid’s axiom and reasoned that, when dealing with infinite sets, “the whole is not necessarily greater than the part”.

So, we can ask, who was right? Mathematicians are free to assume whatever system of axioms they find most useful or interesting. A system of axioms is consistent if it is impossible to deduce contradictory results from it. The choice depends on the purpose; to model the physical world, axioms are required that produce physically realistic results. For purely mathematical explorations, any consistent system can be used as a starting point. The search continues today for the most fruitful axiom systems.

Logic Press has just released a new collection, That’s Maths III, on https://logicpress.ie/

Peter Lynch is an emeritus professor at the UCD School of Mathematics and Statistics; he writes a blog on esomaths.com

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