The infinite is infinite.

In a behavioral experiment, infants were presented with a stage that was partially hidden by a wall. An experimenter then placed one puppet on a stage behind the wall. Another puppet was added to the stage while the wall was still present. Even though the events taking place behind the wall are not visible, we are able to mentally represent the two puppets. The wall was then removed, and infants’ reactions were recorded. Infants looked longer when there was only one puppet revealed compared to when two puppets were revealed. This longer looking time implies that they were expecting one outcome, and their expectation was violated. This result indicates that infants understand the concept of "1 + 1 = 2."

In the traditional Japanese method of counting, each place is labeled with a corresponding word. Kei and Gai are defined as 10^16 and 10^20, respectively. The largest number in this counting system is one Muryotaisu, which represents 10^68. To give some scale of how vast one Muryotaisu is, the number of fixed stars in the whole universe is 6×10^23 and the number of H2O molecules on our planet is 4.5 x 10^46. One Muryotaisu is considerably larger than either of these numbers. Modern mathematics describes places by using a concept called power, which is a shorthand describing how many times 10 is multiplied. While this does make it easier to express large numbers, this system still has limitations in fully describing numbers.

Sometimes we need to represent extremely large numbers. Consider the function "y = 1/x." As x gets larger, the value of y gets closer and closer to zero. The value of y approaches zero infinitely, but it never actually reaches zero. The concept of infinity is useful for describing the y = 1/x function.

 

What does the infinite tell us?

To date, mathematics has not yet found an algorithm for detecting prime numbers. The only method to identify prime numbers is the "Sieve of Eratosthenes". You start by preparing a finite list of sequential numbers (for example, 1-1000). Then, you determine the smallest prime number (in this case 2) and eliminate all multiples of two. You can then identify the next smallest prime number, which is three, and eliminate all multiples of three. By repeating this process, you will eventually find all the prime numbers up the largest number in your list. In the Peano axioms, there is an unlimited number of integers, and thus we would never be able to prove whether the full set of prime numbers is finite or infinite. Modern science has not proven whether there is any regularity in the distribution of prime numbers.  However, with mathematical induction, we can easily prove that there is an unlimited number of prime numbers. For full details, please see the following URL for the mathematical proof. (https://manabitimes.jp/math/677)

Time is an invisible entity, which is why we often forget that "time" does indeed exist. Time simply passes from the past to the future (According to leaders in the field of physics, the dimension of time meanders and the passage of time is not a one-way straight road (Hawking, 2019; Takeuchi, 2005)). Time has a beginning and end, and this can be easily proven by the concept of infinity. If the converse was true and time had no beginning and the past was infinite, a question arises: "Why does time still go on, even though an infinite amount of time has already passed?" Consider the second law of thermodynamics; entropy must increase, and the opposite never happens. Burning coal raises its temperature and increases its entropy. If all materials in the universe were brought to a high state of entropy through heat transfer (as we expect will happen), a state of equilibrium should be reached, and the temperature would be uniform throughout the universe. If time began at the infinite past, everything would have reached equilibrium a long time ago. This contradiction would arise if time had no beginning and no end.

Adults are not the only ones capable of performing mathematical calculations. Infants possess this ability as well.  However, scientists have not yet determined a way to test if non-human animals also understand infinity. Whether the concept of infinity is a privilege exclusive to humans is a question that remains.