EPILOGUE-SCIENCE, SCIENCE!Chiho HarutaTechnical Development Center. She is in-volved in the development of hearing aid software and new functions. She also does graduate research on acoustic sig-nal processing.Don’t understand at allNo counterargumentBrain understandsTimeWholeheartedly understandFun and excitingNeeds serious concentration for contemplationFun to ponder while taking a breakAlways keeping in mindImprovement of the algorithm Application of a new algorithm Confronted with a new fact and proposition, our thinking may follow a curve like this. Both our brain and heart may ultimately reach an understanding.(Drafted by Chiho Haruta)In nity Comes in Different Sizes?!?My fascination with math started when I came across the problem of infinity. Of course, infinity isn’t a specific number. Rather, it’s a term referring to the state where something can’t be counted. The time I spend considering infinity isn’t just an enjoyable pastime. It has positive effects on my work, too. That’s how interesting infinity is.* *What’s so interesting about infinity? I think the topic of the different sizes of infinity will hook many others besides me. Let’s take the example of natural numbers and even numbers. There’s an infinite number of natural numbers. The same applies to even numbers. So is the infinite number of natural numbers the same as the infinite number of even numbers? The sequence of natural numbers begins 1, 2, 3, 4, … and goes to infinity. The sequence of even numbers begins 2, 4, 6, 8, … and likewise goes to infinity. If you had to guess, I think most of you would say while both are infinite, the infinity of natural numbers is bigger. That turns out not to be the case: The infinite size of natural numbers and the infinite size of even numbers are the same. How can that be?* *There are several ways to determine whether the number and amount of elements in two different sets are the same. The clearest example is one analogous to how the number of balls in the basket are counted at the end of a tamaire (ball tossing) game. The balls are taken out one by one from the baskets of two teams and thrown into the air as the numbers “one, two, three, …” are counted out loud. The winning team has more balls, or a larger number of balls, in the basket. In other words, putting the two teams’ balls into 1:1 correspondence with each other, you can see the difference in size when there are balls that cannot be placed in a 1:1 relationship and you see the equality in size when all the balls can be placed in the 1:1 relationship. If you know that you can establish the 1:1 relationship between the balls, there is no need to count to the end. Keep this thought in mind, and let’s get back to the topic of natural numbers and even numbers. You can form a 1:1 relationship between a sequence of 1,2,3,4,… and a sequence of 2,4,6,8,… by doubling and halving. For example, natural number 13 can be paired with even number 26 and even number 512 can be paired with natural number 256. If sets have a finite number of elements like 1 to 10 natural numbers and 1 to 10 even numbers, you cannot make the 1:1 relationship rule. However, in the case of infinite sets, there are certainly even numbers or natural numbers which correspond to no matter how small the even number or dauntingly large natural number. This is because the sets are endless. I was flabbergasted the first time I learned about this.* *If you cap the numbers of natural and even numbers to below 100, you’d clearly have more natural numbers. But things are different in infinite worlds. Normally, we live and think in a finite world. That way of thinking does not work in an infinite world. There are just as many even numbers as there are natural numbers. One day I think about it, I see that as a fact, and I let it go. But it comes back to me—later, at some random moment in my life or on my way somewhere, I’ll find myself thinking about the problem again. After reconsidering the problem, I might be reconciled with the fact—but the doubt returns once again, followed once again by reaffirmation and reconciliation. This cycle of doubt and conviction goes round and round until it finally converges to the point of a deeper understanding. The idea of infinity has taught me that this cycle of thinking applies whenever we try to come to terms with new facts.* *When we come across a new fact and proposition, we don’t quite understand it, but we don’t actively argue against it. That’s the first stage. In the second stage, we apprehend the fact in our brain, but we’re not wholeheartedly convinced. In the third stage, we’re sort of convinced. Finally, we reach the phase of deeper understanding, the fourth stage. There’s a lag between the time our brain apprehends the fact and the time wholehearted conviction kicks in. Doesn’t looking at things this way somewhat deepen your appreciation of all the time you spend wavering between questioning and conviction?* *One of my work responsibilities is developing the software needed for fitting hearing aids. It takes time for my brain and heart to completely grasp a highly advanced algorithm when developing new functions. Now I find myself rather enjoying the time going back and forth between questioning and understanding before I reach a full understanding. And I’m grateful to infinity for that. I’ve come to realize that the joy I find in mathematics and my work comes from the same place inside me.002Endless Thoughts on the Concept of Infinity!Rion is supported by many science-loving staff members.In this series, our science and math-loving staff members will write about their enthusiasm for science and math.Part 2 discusses the concept of infinity.Because Weʻre Science LoversColumn by Rion’s staff on their obsession with science20
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