Posted in Science & Nature

## Grandi’s Series

In 1703, Italian mathematician and monk Guido Grandi posed a deceptively simple-sounding question:

What is the sum of the following infinite series?
1 – 1 + 1 – 1 + 1 – 1 + 1 – 1…

With simple arithmetic, we can easily divide the series using parentheses (brackets):

(1 – 1) + (1 – 1) + (1 – 1) + (1 – 1)… = 0 + 0 + 0 + 0 +… = 0

But what if we changed the way we used the parentheses?

1 + (-1 + 1) + (-1 + 1) + (-1 + 1)… = 1 + 0 + 0 + 0 +… = 1

Because of the way negative numbers work, this solution is equally feasible. Ergo, both 0 and 1 are acceptable answers.

How can one series possibly have two different answers? Grandi used the fact that both 0 and 1 are possible from his series as proof that God exists, as something (1) can be made from nothing (0).

Grandi’s series becomes even stranger when a more advanced technique is applied.

Let us say that Grandi’s series is denoted by S (S = 1 – 1 + 1 – 1…).
We can then break down the series as 1 – (1 + 1 -1 + 1…), because the plus and minus signs can be inverted together.
Ergo, S = 1 – S → 2S = 1 → S = ½

Now we have three answers to Grandi’s question: 0, 1 and ½.
For over 150 years, mathematicians fiercely debated the answer to Grandi’s question. By the 19th century, mathematics had evolved and mathematicians had figured out better ways to solve infinite series.

The classic example is the solution to the series: 1 + ½ + ¼ + ⅛…
To solve this, you can add the partial sums, where you add each number to the sum of the previous numbers to see what number you are approaching (the limit).

1 → 1.5 → 1.75 → 1.875 → 1.9375… until we infinitely approach 2 (or 1.9999999…)

If we apply this method to Grandi’s series, we do not approach a single number because we keep swinging between 0 and 1. (1 → 0 → 1 → 0 → 1…)

So we can apply another method, where we average the partial sums as we go instead of adding.

e.g. 1 → ½(1 + 1.5) = 1.25 → ⅓(1 + 1.5 + 1.75) = 1.416 → ¼(1 + 1.5 + 1.75 + 1.875) = 1.531… until we approach 2.

Using this method on Grandi’s series:

1 → ½(1 + 0) = ½ → ⅓(1 + 0 + 1) = ⅔ → ¼(1 + 0 + 1 + 0) = ½…

Eventually, the series appears to converge on ½, showing that the answer to Grandi’s series seems to be ½.

The problem with this method is that Grandi’s series does not actually have a limit, but we are applying a solution as if it has a limit. This is similar to using a divide by 0 trick to prove that 1 + 1 = 3. In mathematics, when rules are bent, we end up with weird, paradoxical results.

To show this empirically, consider the thought experiment of Thomson’s Lamp:

Imagine a lamp that is turned on after 1 minute, turned off after ½ minute, turned on again after ¼ minute ad infinitum.
This incorporates both infinite series discussed above.
Ergo, we know that the sum of time is 2 minutes.
So, at the end of 2 minutes, is the lamp on or off?
If Grandi’s series solves to 0, the light is off; if it is 1, the light is on.
Then what does it mean if Grandi’s series solves to ½?
Is the light on or off?

Posted in Philosophy

## Quantum Immortality

The famous Schrödinger’s cat thought experiment illustrates the Copenhagen interpretation of quantum physics. Quantum physics is an extremely complicated field of study, but the gist of the Copenhagen interpretation is that a probability remains in a superposition – that is a state where many possibilities exist at the same time – until it is observed, when it collapses into a certain state.

For example, imagine a cat that is locked in a box sealed with a vial of poison, that is set to break open only 50% of the time. Until the box is opened, we do not know if the cat has been killed by poison or not. Therefore, the cat can be said to be both alive and dead at the same time (Erwin Schrödinger initially devised the experiment to mock the Copenhagen interpretation).

There is a fascinating theory that takes this strange thought experiment one step further. Another interpretation of quantum physics is the Everett many-worlds interpretation. This explains that instead of the wavefunction collapsing (i.e. producing a single result such as alive or dead) on observation, two parallel universes are created instead: one universe where the cat died and another universe where the cat is still alive. Essentially, it states there are infinite universes containing every permutation of possibilities that can exist and that whenever a probability is observed, we enter a specific universe.

This is a very confusing concept to grasp, so let us return to the cat in the box. According to the Copenhagen interpretation, the cat has a 50% chance of surviving the experiment the first time. From then on, the chance of the cat being dead grows exponentially with every experiment. However, according to the many-worlds interpretation, no matter how many experiments we perform, there always will be a universe where the cat miraculously survived each one. From the cat’s perspective, it would not know of the universe if it had died. Therefore, the only universe where the cat is able to tell this story to its friends at the end of the day is one where it survives every single experiment

Now let us apply that to our own lives. Imagine that you are crossing the road and a bus is about to hit you. If there is even a 1% chance you might survive this event, your quantum self will move to a universe where it is possible (otherwise you would be dead and your consciousness ceases to exist). By extrapolation, you can never really die as a version of you will forever live on, beating improbable odds until a point where there are literally no possible universes you could be alive.

Quantum immortality is a thought experiment that relies on the many-worlds interpretation. However, it is also extremely difficult to prove wrong. The only way you could confirm this is if you attempted to kill yourself over and over (quantum suicide) and failed each time. But if you were wrong, you would die and not be able to tell anyone. Ergo, you cannot rule out the possibility that you will live forever.

The scariest part of the theory is not that you are potentially immortal. It is that quantum immortality does not account for your well-being – just your consciousness. If an accident were to leave you horribly disfigured but alert, it would still satisfy quantum immortality. You could be trapped in a motionless body for the rest of eternity, unable to communicate to anyone. Yet quantum immortality will keep you alive, forever and ever.

(Infinity Mirror Room by Yayoi Kusama)

Posted in Philosophy

## The First Move

In the game of chess, every move counts. Each action you take can drastically change the way the game will play out from there on. By the second move, there are 72,084 possible games. By the third move, 9 million possible games exist. By the fourth, there are 318 trillion possible games. Essentially, after the first move, the game becomes nearly impossible to predict. There are more possible games on a chessboard than there are atoms in the universe. What spawns all of these possibilities is the first move.

This makes the first move all that terrifying. One mistake and you have destroyed countless possibilities where you are victorious. On the last few moves of the game, the results are much more predictable as the possibilities have been weeded out. Therefore, you can have more confidence in your moves. But the first move is as far as you can get from the end move, with an infinite sea of possibilities between you and the other side.

The corollary to this is that if you do make a mistake on your first move, then you have infinite ways to fix that mistake. So don’t be afraid of taking the first move – simply relax and play the game.

(inspiration/paraphrased from Harold Finch, Person of Interest)

Posted in Philosophy

## Eternity

“High in the North in a land called Svithjod there is a mountain. It is a hundred miles long and a hundred miles high and once every thousand years a little bird comes to this mountain to sharpen its beak. When the mountain has thus been worn away a single day of eternity will have passed.”

~ Hendrik Willem van Loon

Posted in Philosophy

## Ouroboros

The Ouroboros is a symbol that depicts a serpent or a dragon biting its own tail, forming a ring. It is the symbol of cyclicality – something that is in a constant cycle of rebirth through the three steps of creation, maintenance and destruction.

The concept of a serpent devouring itself likely stems from the ancient belief that a snake shedding its skin is the act of leaving an old, inferior body to be reborn into a better, new body. The ancient Greeks explained that the Ouroboros connects its beginning (mouth) and end (tail) to form a metaphor for the link between life and death. By forming a circle, the Ouroboros has no beginning and no end; it is an infinite, linear path that cycles endlessly. Because of this, the Ouroboros is also the symbol of infinity, immortality and the cycle of time. An alternate ancient explanation for the Ouroboros is that because it eats itself, it will ultimately end up as nothing.

The Ouroboros was an important symbol in medieval alchemy. Alchemists used the symbol “O” to represent the Ouroboros. To the alchemists, the Ouroboros was an entity that did not place importance in the two natural processes of creation and destruction, but the often-neglected third force – maintenance. This neutral process is the connection between the start and end of anything. Alchemists knew that in any chemical reaction, the process is just as important as the starting ingredients and the final product. The Ouroboros also represented “everything” and “perfection” to alchemists as it connected its own beginning and end. Because of this, the Ouroboros came to represent the Philosopher’s stone.

Perhaps the most relevant application of the Ouroboros to us is the concept of rebirth and cycling. Nothing in nature is permanent. Matter changes states, chemicals react and species evolve. We too are never permanent. There is always room for change – to destroy what you do not like about yourself, create something better and then maintain that state until the next cycle comes. As much as it is important to know to love who you are, it is vital that you continuously recycle, refine and develop yourself to become the person that you are truly happy to call “me”.

Posted in Science & Nature

## Pi

Pi (π) a mathematical constant that is defined as the ratio of a circle’s circumference to its diameter. It is approximately equal to 3.14159, but since it is an irrational number (cannot be expressed as a ratio), the decimal places go on and on with no repeating segments. The history of pi extends back to almost 5000 years ago, as it plays such a crucial role in geometry, such as finding the area of a circle (A = π ²). It is not an understatement to say that pi is among the top five most important numbers discovered in history (0, 1, i and e being the others).

The interesting thing about pi is that it is an irrational number. As mentioned above, this means that pi has an infinite number of non-repeating decimal places, with numbers appearing in random sequence. For example, pi to a 30 decimal places is 3.141592653589793238462643383279… Because of this feature, pi contains all possible sequences and combinations of numbers at a certain point. The corollary to this fact is, if pi is converted into binary code (a number system of only 0 and 1, used by computers to encode information), somewhere in that infinite string of digits is every combination of digits, letters and symbols imaginable. The name of every person you will ever love. The date, time and manner of your death. Answers to all the great questions of the universe. All of this is encoded in one letter: π.

That, is the power of infinity.

Posted in Philosophy

## Achilles And The Tortoise

In 450 BC, a Greek philosopher named Zeno thought of the following paradox. Let us imagine that Achilles and a tortoise were to have a footrace. Achilles, obvious being faster than the tortoise, allows the tortoise to have a head start of 100 metres. Once the race starts, Achilles will quickly catch up to the tortoise. However, within the time he took to cover the distance, the tortoise would have travelled some distance as well (say 10 metres). When Achilles runs the 10m to catch up again, the tortoise has once again toddled on another metre. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Because there are an infinite number of points Achilles must reach where the tortoise has already been, theoretically the tortoise will be ahead of Achilles for eternity.

According to this thought experiment, motion is paradoxical and theoretically impossible. However, we know for a fact that motion happens. So how can we break Zeno’s paradox?

The main flaw of Zeno’s paradox is that he uses the concept of “eternity”. If we record the story mathematically, the time taken for Achilles to run the footrace is (if it took him 10 seconds to run 100m): 10 + 1 + 0.1 + 0.01 + 0.001… = 11.111… Ergo, the tortoise is only ahead of Achilles for less than 11.2 seconds (rounded). After 11.2 seconds pass, the time passed exceeds the sum of the infinite series and the paradox no longer applies.

Although it is a flawed paradox, the story of Achilles and the tortoise teaches the concept of geometric series – that something finite can be divided an infinite amount of times. For example, 1 = ½ + ¼ + 1/8 + 1/16… ad infinitum. This principle is a crucial part of mathematics and has significant implications in the field of economics. For example, it can be used to calculate the value of money in the future, which is necessary for working out mortgage payments and investment returns. Perhaps it is because of this mathematical principle that it seemingly takes an infinite amount of time to pay off a mortgage.

Zeno’s paradox teaches us that one should not take the concept of infinity for granted.

Posted in Science & Nature

## Dimensions: To Infinity And Beyond

So far we have covered 6 dimensions: point, lengthwidth, depth, time, alternate universe and travelling between possible futures being the key feature of each. Hopefully, you as the reader have caught on to the pattern of dimensions so far: point (0D) -> line (1D) -> branch (2D) -> fold/point (3D) -> line (4D) -> branch (5D) -> fold (6D)… This pattern of point/line/branch/fold continues past the 6th dimension. Therefore, the 6th dimension acts quite similarly to our 3D world in that we can assume it to be a point.
The 3rd dimension was a point in time. So what could the 6th dimension be a point of? The answer is infinity.

It was mentioned that the 5th dimension carries all of the possible alternate timelines that are created from a certain point. In the 6th dimension, these branches fold up to meet so that we can travel freely between every point. Therefore, the 6th dimension is a point that contains every possible timeline – where anything that can happen in the universe exists. This is infinity.

But by definition, infinity encompasses everything as there is no “end”. Then has our journey ended? In a fascinating turn of events, it turns out that we can not only ascend to the 7th dimension, but there are still 3 more dimensions to travel through.
By now we know that as the 6th dimension was a “fold/point” dimension, the 7th dimension must be a line connecting different 6D points (infinity). How can there be more than one infinity? Actually, infinity is only as great as the initial conditions from whence it was born – the Big Bang. In terms of universes, these initial conditions are the laws of physics such as gravity, the speed of light and hundreds of other constants. For example, gravity is 9.81ms-² in our universe. But if this value was off by even 0.0001, our entire universe would be completely different. Ergo, our universe and all the timelines that have and will form depend on the Big Bang. This also means that there can be many other “infinities” with different laws of physics. The line that these infinities lie on is the 7th dimension.

A good analogy for this is genetics. People’s lives have different outcomes depending on their choice, actions and random chance, but they cannot escape their pre-programmed genes. For example, it is not expected that a boy will (naturally) grow into a woman or sprout wings and fly. But if they were born with two X chromosomes or born with the DNA of a bird, this life would be possible.

Now let us follow the basic pattern to move to the 8th dimension. Here, the 7D line branches to meet yet another point of infinity (6D). And yet again, we can bend these branches through the 9th dimension to jump from one universe to another.

Lastly, we can take all of these branches and folds that encompass all possible timelines and all possible universes and draw it as a single point in the 10th dimension. This one point is the relative and absolute “everything”.

But what now? It is impossible to reach past the point of “everything possible”. This means that we cannot jump up another dimension as no other 10D point exists to be connected to. Ergo, the highest possible dimension is 10D and this is the basis of string theory. The 10th dimension is where the so-called superstring vibrates to form the subatomic particles that are building blocks of every matter in our universe.

As mind-boggling a journey it was, if you were able to follow through from the start, we have travelled from a single point that occupies no space to another point that encompasses all things possible in our universe in all possible timelines. We have zoomed out to the point that there is no longer a box to “think outside of”.

Can one ever reach that point where one knows everything that was, is and will be? To know every piece of knowledge that is the absolute yet relative truth? Although we cannot physically jump through dimensions, our minds can keep rising up to raise our level of understanding and enlightenment higher and higher. As we only live in the 3rd dimension, we have no less than 7 more dimensions to explore and understand. Only when we have reached the 10th dimension can we say that everything possible has been discovered.

Until then, anything is possible.

(This post is part of a series exploring the concepts of dimensions. Read all of them here: https://jineralknowledge.com/tag/dimensions/?order=asc)

Posted in Science & Nature

## Dimensions: Time Warp

The first three dimensions covered the three variables that determine space: length, width and depth. Then what could the fourth dimension possibly add? The answer is duration. The third dimension effectively becomes a point and the 4th dimension connects different 3D points to form a line that we refer to as time. For example, the “you” at this exact time is different to the “you” in five years time. These two “you”s are different (3D) points that lie on a 4D line (more specifically, your life). This is exactly the same principle as the 0th dimension being points in a 1D line, except three dimensions higher.

As a “moment” in 3D space is just a point on a 4D line, travelling from one point in time to another (i.e. time-travelling) would be as easy as walking along a straight line for a four-dimensional being. This concept is mind-blowing for us as we cannot fathom the concept of manipulating time. We are unable to see time as a dimension as we exist in a lower dimension. Every single moment in our lives is a brick that paves the road called time, meaning that we can only see each individual brick and not the overall picture. This is exactly the same as how a Flatlander could not understand the concept of depth and how we can be above them.

Although we see time as linear and straight, we are under the same illusion as the ant walking down the Mobius strip. Time is actually twisting and turning in the fifth dimension, creating multiple timelines that branch out like a tree of possibilities. These branches are influenced by our own choice, chance and the actions of others.
In other words, if a man proposes to girl A then he will go down the branch where he marries girl A. However, if he chose not to propose, he would end up marrying a different girl (or not at all). Therefore, he has entered a different branch than the girl A branch.
Now, if the man wanted to go from a timeline where he married girl A to another timeline where he married girl B, what could he do?

One method would be if he bent the 4th dimension (time) on itself through the 5th dimension to travel back in time to when he met the girl and not ask her number. This is exactly like folding a 1D line into a 2D circle to make the end point meet the starting point. However, to marry girl B he would have to make the right sequence of choices that lead to him marrying girl B (much like a role-playing game). This is the long-way round that would be too time-consuming and complex.

A simpler approach would be if he folded the 5th dimension through the sixth dimension and jump from the ending of timeline A to timeline B. This would be like the finger-lifting analogy we have been using time after time. If we pretend that the 5th dimension was a piece of paper, then we could fold it into a 6D cylinder so that the two edges meet. Now we are able to jump from one ending to another effortlessly, just as we did in the 2nd dimension.

It is easy to confuse the 5th and 6th dimension as they both deal with “alternate realitites”. Here is one way to differentiate the two: 5D space is like a 2D space for time – a flat plane where different timelines cross each other. Therefore, it can contain all of the possible outcomes from an initial condition – that depends on an action, choice or chance – such as your conception (visualise a dot on a piece of paper with many lines radiating out from it).
If we were to put a dot on a piece of paper above the first piece of paper (in the 3D space for time), we have entered a universe where there is a completely different starting point (before your conception), such as dinosaurs not existing. This means that the line called “you” may not even exist, and the two pieces of paper would never meet. The only way to jump from one piece of paper (5D) to the other would be through the 6th dimension.
Therefore, by jumping up a dimension, we gain a degree of freedom where we can move in yet another direction. This is seen between every dimension, such as 2D versus 3D. The 6th dimension merely lets us travel between different sheets of 5D paper.

So somehow we have reached the 6th dimension where one can not only time-travel, but jump from one alternate reality to another. Shall we venture further into the seventh dimension – infinity – and beyond?

(This post is part of a series exploring the concepts of dimensions. Read all of them here: https://jineralknowledge.com/tag/dimensions/?order=asc)

(Stick figures from xkcd)

Posted in Philosophy

## Lines

There are two lines.
If they are parallel, they have many things in common but will never meet.
Sadly, all other pairs would still meet just once and then go their own ways for infinity.

One solution to this problem is making one line bend at an angle to make it continue in the same trajectory as the other line.
In life, there are moments when you must bend your line. Those who do not know how to bend will tread alone, ad infinitum.