The concept of infinity has not always been as widely or easily accepted as it is today.
The Pythagoreans believed strongly in Number, and thought of ‘finite’ quantities as good and ‘infinite’ as evil.Other Greek mathematicians considered infinity as a possibility, as it described the endlessness and unboundedness of time and space. Since you could add one to a really big number and get a bigger number, they considered ‘potential’ infinity, which was allowed only in dealing with finite numbers. They hesitated to consider a ‘continuous’ infinity because they could only really quantify known. So at this time, the modern concept of infinity was not accepted because it describes the unknown.
A few centuries later, early European mathematicians reflected on the concept of infinity, only in the context of religion, to describe the influence and ‘unlimitedness’ of God.
The modern concept of infinity was not established until the emergence of Calculus. Georg Cantor, an esteemed mathematician who contributed to the fields of Number Theory and Calculus, is deemed as the first mathematician to really understand what infinity means and gave it a more precise definition. Cantor sad that if you can add finite numbers like 1 and 1, and 50 and 50, then surely you could add infinity and infinity. Instead of thinking of infinity as a really, really big number, he realized that there are infinities even larger, more infinities even larger, and so on. This idea forms the basis of the argument for why infinity is a number. We know that for every number there exists a number that is even bigger and even smaller. Cantor had a similar idea about the concept of infinity- there can be bigger infinities and smaller infinities. Cantor’s concept of infinity pushed the boundaries of mathematics, and made it possible for future mathematicians to study more abstract fields of mathematics involving the concept of infinity.
What does 'Infinity' Mean to Us Today?
Today, we think of infinity as a way to describe the abstract notion of “something without any bound or larger than any number.” Infinity is viewed as something ‘numberlike’ in the sense that it can be used to count or measure things “in infinite terms”, however it is not a number like the real, imaginary, complex, or irrational numbers.
My impression of infinity is that I understand the argument for why it is not a number. I think of ‘infinity’ as a word we use to describe something, like time and space, which go beyond the concept of Number. When we think of really, really big numbers that we cannot possibly quantify or plot on a number line, we call it ‘infinity.’ Infinity is a boundless quantity without a beginning or end. I also think that the way we interpret infinity changes based on the given mathematical situation.
After further reflection, I also understand the argument for my ‘infinity’ is a number. My understanding of a number is that it is a measure of quantity, or a way of describing “how many” or “how big” a given set of mathematical objects is. To gain a true understanding of number, we must consider the three aspects of a number: quantitative, verbal, and symbolic. In other words, a number is “a mathematical object used to count, measure, and label.”
Considering this definition of a number, I can see why infinity must be a number. In theory, we can count the number of objects in an infinite set of objects. We know that the size of that set is ‘infinity”. We also know that the measure of a line of infinite length is ‘infinity.’ Furthermore, since we can measure and count an infinite value, we can also label mathematical objects with ∞. Therefore, because infinity has all the qualities of a number, it must be a number.