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 Life & Happiness

## Best Friend

Whether we’d admit it or not, we all have someone in our lives that we consider a best friend. A best friend is someone who you enjoy spending time with, trust with your deepest secrets and talk openly and honestly with when something is troubling you.

For some, this may be a childhood friend with whom they had endured the hardships of life together. For others, it may be their parents, sibling or significant other. In some cases, a person who was a stranger to you less than a year ago may quickly develop in to your most valuable friend. Many of us will even have multiple “best friends” who we can call upon in times of need, or if we just need to rant over a drink.

These friendships do not happen without effort. Sure, it requires basic chemistry and connection. But to build a great friendship, it requires both parties to invest time, care and empathy. Loyalty is built on acts of kindness. You need to actively listen to delve deeper into the emotions and thoughts that drive your friend’s worries. We improve each other over time by calling out bad behaviours, while offering endless support and love when the other person feels worthless or unattractive. We take for granted the sheer amount of emotional energy invested in cultivating a true friendship.

When we forget this fact, we become terrible friends. We can be selfish, becoming angry with our friend that they aren’t giving us the support that we need. If this ever happens, consider the fact that your friend is also human and that they might be in exactly the same position as you. To parody John F. Kennedy:

There is also one other friendship we must discuss – the friendship between you and yourself. This sounds strange, but you should be your own best friend. You are the person that has truly lived your life with you. You know of all the dramas, thoughts and feelings you have experienced. Yet when we are in a time of need, we neglect to support ourselves as a friend. Instead of support and love, we criticise ourselves, neglect ourselves and drive ourselves to stress and fatigue.

Be generous with your kindness to yourself and don’t forget to treat yourself. If you are having a bad day, take a break so that you can be there for yourself. Watch a movie, go for a walk, introspect and have a deep and meaningful chat with yourself. If you feel like a failure, remind yourself that you are being stupid and remind yourself of how amazing you are.

No matter how many great friends we have, we cannot truly be happy if we treat ourselves like an enemy.

Posted in Science & Nature

## Intelligent Life

One of the great questions in science is “could intelligent life develop on planets other than Earth?”. Even the general populace has heard of programmes such as SETI (Search for Extraterrestrial Intelligent Life) and mathematical models such as the Drake equation that attempt to predict the possibility of extraterrestrial intelligent civilisations. But an equally intriguing question we seem to neglect is: “could intelligent life develop on Earth?”.

The definition of “intelligent life” is hugely varying, but nonetheless attempts have been made to compare our intelligence level to other animals. From the pool of research throughout the decades, the most “intelligent” non-human animals appear to be chimpanzees, bonobos, great apes, dolphins, elephants, certain parrots, ravens and rats. There is much research on the intelligence of cephalopods (e.g. the octopus) that has shown promise. If we were to shift the focus from individual intelligence, we could also consider “civilised” animals such as ants, as they are capable of building vast cities with intricate societies. All of this shows that intelligence is not exclusive to our species. We have simply walked down the path of evolution where the trait of ever-increasing intelligence, knowledge and wisdom have allowed us to adapt to and survive our environment. Ergo, it is fair to consider the possibility that other animals are walking a similar path that may lead to the making of a species with intelligence comparable to us.

However, this only raises the theoretical possibility of intelligent life. What is the realistic, practical possibility of intelligent life developing on Earth in the near future? Put another way, could intelligent life develop in the presence of a higher intelligent life (e.g. humans)? The road that brought us to throne of “the most intelligent species on Earth” was not an easy one. We are but one of many other hominid (human-like) species that evolution produced while tinkering with the concept. For example, there was a time when we (Homo sapiens) shared the Earth with other intelligent hominids such as the Neanderthals. The Neanderthals are commonly pictured as simple, knuckle-dragging apes but in reality they were just as intelligent as Homo sapiens during that time. They had a culture similar to our own, developed stone tools just as complex and even made cave paintings in a display of art. The reason why we are not breaking bad with Neanderthal neighbours now is that (according to one theory) we successfully outcompeted them, driving them to extinction (there is debate whether genocide and cannibalism was involved).

Evolutionarily speaking, it makes sense for an intelligent species to wipe out another species trying to compete with the ecological niche of intelligence. This has been discussed in many works of science fiction, such as Planet of the Apes where the emergence of intelligent apes leads to the destruction of human civilisation. Arthur C. Clarke discussed this as a side plot in his novel The Songs of Distant Earth. Upon discovering a species of sea scorpions that show signs of intelligence such as social hierarchy and metal collecting, the scientists suggest that they should allow it to develop, but ultimately the government decides to eradicate them as soon as they attempt to migrate to land.

Suffice to say, given our track record in history involving the countless times colonists wiped out other civilisations to serve their purpose, there is a good chance that any new intelligent life would immediately be removed by us if they had the misfortune of arising during our time.

Posted in Simple Pleasures of Life

## Simple Pleasures of Life #6

Re-reading comics you loved as a child, loving and appreciating them even more now.

So I recently acquired The Complete Calvin and Hobbes set!!! It’s made of three beautiful hardcover books that contain ALL the Calvin and Hobbes strips that have been drawn~ 😀 Extremely ecstatic at the moment because Calvin and Hobbes is one of my favourite comics ever (seconded only by Peanuts) hehe. It has content ranging from hilariously innocent to deeply philosophical and thought-provoking. There’s so much stuff that I wouldn’t have gotten or appreciated as a kid, but make for very good life advice now that I’m more grown up. Here’s an example:

(click for bigger image)

I’m gonna marathon the whole thing from start to finish, hopefully not taking up too much study time haha. But a few pages in between study sessions won’t be harmful 🙂

Pure bliss~

Posted in Simple Pleasures of Life

## Simple Pleasures of Life #14

Taking time off to do something fun when you can’t afford to.

This is most relevant to my life right now with my O&G OSCE (clinical examinations) on Friday, short cases (biggest clinical exam of med school) next Tuesday and two gigantic exams (testing us on literally everything we learnt in med school) the week after that all crashing down at once. The sheer pressure of all of this is really pushing all of us (5th year med students) to a point of anxiety, depression and insanity, and without some way to relieve that stress somewhere, we’d probably end up jumping off somewhere tall or go on a shooting spree.

I personally believe human beings feel the happiest when they are enjoying a luxury they cannot afford. For example, having a nice dinner out once in a while even though your budget is straining. The same applies to time. No matter how busy life gets, just stop what you’re doing and take some time for yourself. Whether it be spending a couple hours having lunch with a friend, playing card games for half an hour, or singing a song on your guitar for ten minutes. Hell, go to the park and lay on the grass for even five minutes, doing nothing but staring into the blue, blue sky. Trust me, it will make a huge difference to your mental health and work productivity.

Here’s an analogy from an old ARK post:

There are two clocks: a still clock and a clock that is always a minute late.
Which clock is more accurate?

The answer is the still clock – it tells the correct time twice every day, but the other clock is always wrong despite the fact that it constantly ticks.
Sometimes when life gets you down, it is better to stop and rest instead of trudging on forever, always out of sync.

Posted in Simple Pleasures of Life

## Simple Pleasures of Life #26

Laughing over even the silliest of things.

There are many people who frown upon cheesy humour and say “that’s just immature” or something along the lines of “that’s not sophisticated enough for someone like me”. Essentially, everyone has a standard that they measure all things humorous against.

Mine happens to be rather low, so I’ll easily laugh over the littlest things or find a TV show episode very funny even when others just LOOK at me funny haha. But my rationale is that the end result is that I’m laughing and releasing tons of endorphins into my bloodstream.

So who cares if other people think the things you like are lame? As long as you’re happy. 🙂

Relevant ARK: https://jineralknowledge.com/laughter

Posted in Simple Pleasures of Life

## Simple Pleasures of Life #30

Meeting people, hearing their stories, sharing a moment and leaving a positive mark on their lives.

There’s a branch of philosophy that believes that the world only exists within our minds. That the people around us are merely figments of our imagination. As crazy as this thought is, to some degrees, we human beings all live at least a little selfishly, absorbed in our own worlds. For example, thanks to a psychological phenomenon known as special pleading, we are prone to judging others by how their actions affect us while we only judge our own intentions.

Because of this human nature, we often forget one simple fact: each and every person that you see around you has a life of their own. That person that you accidentally bumped into has a name, a job, a family (maybe). He or she will have hobbies and interests, goals and dreams, experiences and memories.

Of course, it’s understandable that your brain tends to ignore this simple fact on a routine basis. The poor bugger has to process an overload of sensory and cognitive information every nanosecond, and if it was to consider every little details of everything, your head might just explode. But that’s no excuse to not think about it every now and then. You know, just ponder the implications.

If there’s one thing I’ve learnt about the world, it’s that each person has a very unique story to tell. They are walking books that are in the process of being written into a grander, more epic tale by the day. You’d be surprised what kind of stories you hear from complete strangers. One of the reasons I wanted to go into medicine was to hear all the stories every patient had to tell. I am reminded of this fact every day in the hospital. I’ve met old men who survived wars, kids who’ve suffered more diseases than you would in three lifetimes and even an old lady who gave the best tips on how to cook fish (which I’ve sadly forgotten…). Point is, if you take the time to stop and have a chat, you might hear some amazing stories that put movies and books to shame.

I think one of the greatest joys we can experience as human beings is really opening up to someone, getting to know who they are and showing them who you are to them. Memories, stories, values, dreams, opinions, humour, all that jazz. It might be crazy to think that the world only exists in our heads, but it’s not at all crazy to think that everyone’s perspective of the world is very different. When you share a bond with another person, it’s like connecting those two different worlds. Yup, getting to know a person is like opening a wormhole between dimensions. What that’s pretty cool right. The best part is that this could happen between two lovers who’ve known each other for ten years, or even between two people that happened to meet by chance an hour ago. That’s human interaction I guess. It’s not about the amount of time you’ve spent with them. It’s about the quality of the time you shared with them. Opening wormholes and stuff.

My last point is to do with leaving a mark. It’s one thing to be aware that the person that’s in front of you is a unique life of their own, it’s another to be conscious of the role you play in their story. For example, a compliment you paid in passing might completely change the person’s day. A simple act of kindness you thought nothing of could be recorded in someone’s life book as a life-changing event.
I said at the start that we judge others by their actions and ourselves by our intentions. Well then, by that logic, once you pass away the only thing that will be left to affirm your existence will be recordings of you in other people’s life books. So consider the mark you leave. Even a smile can make a difference.

I’ll end my 30 day challenge with a quote I particularly like regarding human interactions, by a certain Mr. Rogers:

“If you could only sense how important you are to the lives of those you meet; how important you can be to the people you may never even dream of. There is something of yourself that you leave at every meeting with another person.” ~ Mr. Rogers

May your life be full of little, simple pleasures 🙂

Posted in Life & Happiness

## Life Review

They say that when you face your mortality, your entire life flashes before your eyes like a sped-up autobiographical film. This tends to happen in situation where a person feels they are in danger of imminent death, such as moments before a car crash. Reports say that the event typically lasts anywhere between less than a second to few seconds, and what they perceive as major life events flash before their eyes, usually in chronological order. However, reports are very subjective and variable.

This phenomenon sounds very clichéd, but it has been widely reported throughout time and space. Over 8 million people in the United States of America stated that they experienced this “life review” in a near-death experience, with countless records in historical texts, reaching far back as at least 1795 in a letter by Rear-Admiral Sir Francis Beaufort. It is fascinating to see that there is even a set name or phrase for this phenomenon deeply ingrained in various languages, such as English, German, French, Dutch, Russian, Persian, Arabian and Korean, suggesting that the phenomenon is widespread and common.

There is no strong evidence for why this phenomenon occurs, but there is one theory that is persuading. The brain is always subconsciously referring to past experiences and knowledge to apply to the present to help solve a problem. It has been suggested that when you are at the brink of death, the brain frantically searches through everything in an attempt to save you from demise. This is a rather messy process as the brain does not routinely encounter such near-death experiences and does not have much information to refer to immediately. In this process, it brings up every memory that you thought you had forgotten, which you see as a montage flashing before your eyes. For example, a man who was attacked by a great white shark reported that out of nowhere, he recalled his son watching a documentary on sharks and remembered that putting your hands down a shark’s gills will incapacitate it. Thanks to this, he survived.

The brain does indeed have an amazing ability to alter your speed of thought and delay time perception when you are in danger, or the so-called “fight-or-flight” mode. There is much anecdotal evidence of firefighters instinctively knowing that a building will collapse very soon, or emergency physicians making complex clinical decisions in the blink of an eye by drawing from a well of past experiences.