What is metaphysical infinity?
Infinity in Physical Science. From a metaphysical perspective, the theories of mathematical physics seem to be ontologically committed to objects and their properties. If any of those objects or properties are infinite, then physics is committed to there being infinity within the physical world.
Infinity is bigger than you think
The video Infinity is bigger than you think [1] mainly introduces the concept that there are different types of infinities than we thought, symbolizing that some infinities are bigger than others. The content of the video is based on the idea of Georg Cantor who demonstrated that the set of real numbers is bigger than the set of natural numbers. As we usually know, Infinity is not a number but an idea of being endless or going on forever and is not countable or listable as well.
However, if we pretend that we could list all the decimals between 0 and 1, we can randomly assign some numbers between 0 and 1 such as 0.1100554. . . and the next one 0.2132434.. and so on. Then, we could write a new number between 0 and 1 that disagrees with the first digit in the first place and the second digit in the second place and so on.
So no matter how many numbers we write on that list, we can always find a completely different number that won’t be on the list as a result of which it is uncountable and that means it is an entirely new type of infinity, a bigger type of infinity.
What I have learned is that we cannot list all the decimals between 0 and 1 which recognize infinity cannot be listed one by one. The concept of a bigger type of infinity inspires me to learn more about the complexity and beauty of mathematics because the video introduces a familiar concept with different angles.
The Golden Ratio (why it is so irrational)
The video Golden Ratio (why it is so irrational) [3] basically concludes the idea of Golden Ratio by illustrating how the flower chooses the direction to put their seeds efficiently in order to make itself grow more nicely, reducing the unused space.
The video first lists some examples of fractions. The numerator of the fraction decides how many spikes (growing direction of flower) will be and the irrational number will show a spiral shape instead of exact spike and somehow the flower uses “irrational number” to utilize more space. The author is wondering whether there is one that is the most irrational?
The golden ratio (0.61803398874989) surprisingly turns out to be the most efficient number for the flower.
Because the spikes are trying to grow randomly in every direction. The way that sunflower expands has an obscure connection with the utilization of golden ratio. The correlation between the seeds of sunflower and the placement on a computer shows surprisingly few differences.
The author tells his audience directly that the golden ratio makes the placement most efficiently simply through looking at them.
But I was wondering if there might be other numbers that will do better than it. However, the later proof is pretty captivating. x = 1 + 1 1+ a bit, a bit means to do this fraction infinitely, can acquire the most irrational number which will be a continued fraction 1 because it contains no large numbers such as 1. By solving the equation, we have the result of x = 0.61803398874989 as exactly the same as the golden ratio.
The most attractive part is that there can be some connections between nature and mathematics.
The pattern of natural growth eventually evolves to the most efficient type that stays close to mathematics. How amazing it is.
ASTOUNDING: 1 + 2 + 3 + 4 + 5 + … = -1/12 The video[2] proves that the sum of all natural numbers (from 1 to infinity) produces an ”astounding” result.
X∞ n=1
n = −112
First of all, we will have three equations: S1 = 1 − 1 + 1 − 1… and S2 = 1 − 2 + 3 − 4… and S = 1 + 2 + 3 + 4… For S1, if it goes to an odd position, we will have 1. If it goes to an even position, we will have 0. So taking the average answer of two, we will have 12.
By looking at S2, let’s adding S2 with it self: 2S2 = 1 − 2 + 3 − 4… + 1 − 2 + 3 − 4… then 2S2 = 1 − 1 + 1 − 1… then 2S2 = S1 so S2 =14.
Last, if we made the equation S −S2 = 1 + 2 + 3 + 4…−(1−2 + 3−4…) and minus the number one by one, we will have S − S2 = 4 + 8 + 12 + 16… Then, S − S2 = 4S. As we solved above, S2 =14. Therefore, S = −112 .
This amazing proof that the sum of all natural numbers has been demonstrated by Professor Tony Padilla at the University of Nottingham.
The conclusion is magnificent as an application in many science fields. We admitted that math has to follow the established foundations of what is already laid out. From my perspective, although no mathematicians in their mind believe that 1 + 2 + 3 + 4… = −112 , they make the distinction that it is the part of the analytic continuation of the series that does not diverge.
What is metaphysical infinity? Written by Hong Cao
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References For What is metaphysical infinity?
[1] Dr James Grime — Georg Cantor. “Infinity is bigger than you think”. In: (July 6, 2012). doi: https://www.youtube.com/watch?v=elvOZm0d4H0.
[2] Numberphile — Tony Padilla. “ASTOUNDING: 1 + 2 + 3 + 4 + 5 + … = −1/12”. In: (Jan 9, 2014). doi: https://www.youtube.com/watch?v=w-I6XTVZXww.
[3] Numberphile — Ben Sparks. “The Golden Ratio (why it is so irrational)”. In: (May 9, 2018). doi: https://www.youtube.com/watch?v=sj8Sg8qnjOg.
