Learning about numbers as a kid, early on, presented problems that my numerical naivete wouldn’t let me solve. The first big problem that I came across, if I recall, was infinity. Most can recall learning numbers and hearing that they can increase “forever” or arbitrarily by repeatedly adding one to the previous number. Considering infinity this way, my early view of the concept was unilateral; much like a ray- if one can imagine it that way. Numbers started at some finite reference point, typically zero, and increased arbitrarily in one direction. I hadn’t yet considered the reverse spectrum of numbers, the opposite direction. This is likely so, because the concepts of negative numbers and fractions were still foreign to my knowledge so I hadn’t a need to consider such things. As they were introduced, however, this conceptual grey area could no longer be ignored.
Upon understanding the concepts of fractions and negative numbers, I had no choice but to revisit my earlier hypothesis on infinity. And I came across two more problems: a negative number problem and a fraction problem- a particularly jarring issue. The first problem was the linear issue. Upon learning about negative numbers, zero was no longer a satisfactory reference point. If there are a ton of numbers below zero, at which, then, should I begin to count? If numbers cease to stop, where should I stop counting them? My reckoning with the infinity issue made me change my “ray” view of infinity into a linear view. Numbers now started at some arbitrary reference point (not necessarily zero since sequences don’t always start there) and increased arbitrarily forward and back. Numbers now not only had no end, but also no beginning. Unlike a circle, however, numbers are not contained. That is to say, a circle may also have no observable end nor beginning but it does have a limit- a perceivable and measurable perimeter of two-dimensional space that it occupies. Numbers do not, and nothing in nature behaves that way. So do numbers even exist outside of mental property? Does it matter?
Neither of these issues were, or are rather, as particularly jarring as the next. My final problem with infinity came about with my understanding of fractions. Numbers could now get arbitrarily smaller, without end, simply by increasing their denominators.
Wouldn’t you agree?
Well, consider it this way then. Take a look at the picture at the beginning of this (finite, I promise) post. It’s a shot of measuring tape. Nothing particularly interesting about it at a glance; the units in observation are inches and the distance between the whole inches is denoted units of 1/10 an inch.
But what if those tenths of an inch were changed to elevenths (1/11)? What about twentieths (1/20)? Hundredths (1/100)? Hundred-thousandths (1/100000)? If fractions can get arbitrarily smaller, then are there not an infinite amount of (fractional) numbers in between any two finite numbers? If there are an infinite amount of numbers in between any two finite numbers, is there also not an infinite amount of distances between any two points? If there is an infinite amount of distances between any two points, is there then an infinite distance between any two points by adding them all? (What hypocrisy!) If there is an infinite distance between any two points, does anything ever really move from one point to another? If there is an infinite distance between any two points, is infinity finite?
Still not bizarre?
I realize I am charting into Zeno territory here. Anyone who’s dipped into Greek philosophy has probably at some point visited The Arrow Paradox:
If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.
Zeno asks the same question, through his statement. Do we, arrows, ever really move? In my defense, I had asked this question long before I’d studied Greek philosophy and learned of his reckoning with the same concept. However, the answer is yes- we do move. So no, the flying arrow is not motionless. And just before spring break, we learned one way to solve this problem.
Mister/Minister/Pastor/Doctor/Jonas Reitz taught us about convergent sequences; cases in which the terms of a sequence all add up to a finite number. This could be said for the case of any two points. Revisiting the question:
If there is an infinite amount of distances between any two points, is there then an infinite distance between any two points by adding them all?
No. One would not get infinity in between two points by adding every distance, derived from adding up an infinite amount of infinitely small fractional distances, between two points. One would end up with a finite number. And, if one ends up with a finite number, then we can revisit the proceeding question:
If there is an infinite distance between any two points, does anything ever really move from one point to another?
The premise of an infinite distance between any two points is now void, and there is no question of whether or not objects move from one point to another. So we can revisit the final question:
If there is an infinite distance between any two points, is infinity finite?
No. There is not an infinite distance between any two points. Infinity is not finite.
Infinity is infinite.
MAT1575 M/W 4PM