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Motivation and School

I haven’t posted in a while, and the reason is schoolwork has snatched all the time out of my hands. It just feels like I’ve constantly had something or the other on my plate at any given moment in time.

The odd thing is though, when I take a step back, I don’t really have all that many tasks, but they just take a strangely long amount of time for me to do. Why is it taking me weeks to work on one (1) physics assignment, that feels like it could be completed within a couple days of work? The reason is in my complete and utter lack of motivation.

That revelation just brought up another question for me, however. This is the final year of school for me. All the assignments and tests and essays matter so much more than they did before, they can decide the fate of my future education. There is more on the line now than ever before, so WHY is that not motivating me to do any work? The answer lies in the idea of extrinsic and intrinsic motivation

In a nutshell, intrinsic motivation comes from the joy of doing something. You may be intrinsically motivated to paint simply because you love painting, for example. As the name implies, it’s an internal drive; we push ourselves to do something because we enjoy it and derive satisfaction from it.

Extrinsic motivation then, is anything other than that. Any reason to do a task or an action other than the pure joy of the task itself. These include monetary incentives, recognition, awards, trophies and praise. While those seem like positive things, there’s also negative extrinsic motivators: fear of a deadline, of reprimand, financial liabilities like mortgages and debts. The image below sums it up pretty nicely.

I think my main issue here is that schools rely far too heavily on the pitchfork depicted on the picture, that is, negative extrinsic motivators. Most of the time us students just work on memorizing textbooks to pass tests because we don’t want to fail or have low grades. It’s rare for us to be intrinsically motivated to study or learn, and the few instances that are there are often only for select subjects. But even positive extrinsic motivators aren’t ideal when it comes to learning.

There was a paper by Princeton describing how positive extrinsic motivation is beneficial only in the short term, and in fact can actually be detrimental to intrinsic motivation.

I won’t delve into the mathematical complexity of it here (though it is a fascinating read), but essentially the gist of the idea is that higher rewards are associated with more “unattractive” tasks,” and hence makes us more apprehensive of doing or enjoying them. It cites the interesting example of Tom Sawyer, who asked other kids for bribes to paint the fence for him.

The boys associated the payment they had to give for the task with a fabricated expectation of enjoyment, thus intrinsically motivating themselves. The main takeaway here is that extrinsic motivators tend to instill no long-term passions or drives within people. Hence why I struggle to finish my school tasks. Making them the primary motivator for learning then, which should be a continual and lifelong endeavor, is not the best of ideas.

I don’t think the fact that intrinsic motivation is internal means that it can’t be ignited from the outside. If placed in an environment that emphasizes engaging discussion and holistic learning, I genuinely think students can grow to become intrinsically motivated to learn. And then maybe I won’t procrastinate every single task I’m given.

None of this is to say that external motivators are all bad, of course. They are almost a necessity when it comes to obligatory tasks. However, I think relying on them as a prime motivator for children’s education is highly detrimental and something that definitely needs to be turned around.

Does 2+2=4?

While scrolling through my twitter feed I came across a series of threads all essentially asking the same thing: what is the meaning of the phrase 2+2=5?

If you’ve read Orwell’s 1984, you’ll know this as the dogmatic statement used by the Party to brainwash Smith into mental submission via doublethink. If you haven’t read 1984… I don’t know what you’re doing with your life go read it.
The main take-away, though, is that this phrase has very strong political undertones of anti-intellectualism and governments trying to politicize truth and deny reality to “control the masses”.

Nazi theory, indeed, specifically denies that such a thing as “the truth” exists. There is, for instance, no such thing as “Science”. There is only “German Science”, “Jewish Science”, etc. The implied objective of this line of thought is a nightmare world in which the Leader, or some ruling clique, controls not only the future, but the past. If the Leader says of such and such an event, “It never happened” — well, it never happened. If he says that “two and two are five” — well, two and two are five.

George Orwell, “Looking Back on the Spanish War”

So this is going to be a mess to tackle, as you may already be able to tell. Let’s dive into it.

These are the types of tweets and posts that had people concerned. It started around summer but even now there are conversations and threads continuing.

The main premise of these kinds of posts is that “2+2=4” is a truism only in “white” culture, since those symbols and mathematical concepts were standardized by white societies. For example, we use base 10 but Babylonian societies used base 60 for their number system.

I think there’s a miscommunication taking place here on both sides. On the one hand, the guy who said “math is not universal” probably didn’t intend to say that the way he did. What I THINK he meant was “the mathematical symbols and constructs we use aren’t necessary universal”. And to that, I agree. Of course for someone working modulo 3, the expression “2+2 = 1” might as well be true.

The key distinction here is that there are different ways to represent the same abstract concept. Whether you represent 4 as “10” in base 3 or as “-1” in modulo 5, we are all talking about the same thing here: the 4th smallest positive integer.

But what of the argument that “2+2 = 5”?

I think the argument here enters a realm of philosophy more than pure mathematics at this point, since we have to consider how we construct mathematical knowledge in the first place. There are essentially two schools of thought: mathematical platonism and mathematical formalism.

Mathematical Platonism is the idea that the concepts of “2” and “4” and “addition” exist beyond the realm of human though. That is, if there were no humans or intelligent life there to write it down or imagine it, the concept of “2” and “4” would still exist. This goes not just for basic numbers and operations but even for the more advanced things, like perfect circles, straight lines, a mathematical limit and so on. This is probably the view that most people hold about what mathematics is.

In this scenario, we can say without a shadow of doubt that “2+2=4” and it always will be. Because the integers 2 and 4 are said to exist in a place outside of our brains.

Mathematical formalism, on the other hand, is essentially the idea that math exists only insofar as we dream it up. Whereas mathematical Platonism says that the concept of the derivative always existed in some abstract space and humans merely ‘discovered’ it, formalism tells us that the differentiation did not exist in our universe before Newton and Leibniz thought it up.

One of the interesting ideas brought up by the conflict between these two schools of thought is whether unsolved problems are “True” in some sense. Most of modern math is built from 9 axioms known as the ZFC. Every theorem is practically built FROM these base axioms using pure logic. However, we know some problems, such as the Continuum Hypothesis, are undecidable from these axioms, meaning it cannot be proved or disproved from those 9 statements alone. Now, some Platonists might argue that there exists an abstract universe of sets where the hypothesis is determinately true or false, and our axioms simply don’t reflect that well enough. A formalist would say that the hypothesis is simply undecidable and remains that way, and we can either add an axiom to make it true or false, but we hold that power.

The question of whether “2+2 = 4”, then, is really up to us to decide. If you are a mathematical Platonist, the statement is simply true. Given the existence of “2”, “4” and the addition operation in some higher space, obviously 2+2 is equal to 4. However, if you are more of a formalist, you might be of the opinion that “it depends”. Depends on the axioms we use and the system of arithmetic we are working in. Sometimes it may equal 4, sometimes it may equal 5, other times it may equal 0.

Game Review | Shadow of the Colossus

So this is one of those classic games from the PS2 whose legacy has somehow endured to this day. I was too young to play the original but I’ve heard nothing but good things about this game and so I was VERY excited to play the PS4 Remastered version.

Boy was I disappointed.

Now let’s set the record straight here: this is not a necessarily bad game by any means. I just don’t feel I enjoyed it nearly as much as I was expecting to. And by that I mean this is one of the most frustrating games I’ve ever played.


Let’s start off with the good. The premise of this game is that you’re a young boy trying to bring a deceased girl back from the dead, and in order to do so you seek the help of an ancient spirit/god, who commands you to slay 16 colossi in return for her life. And that’s pretty much it.

It’s not an elaborate story but it’s elegant in its simplicity. It’s surprisingly emotional at times and though the ending felt a bit out of nowhere to me,(No Spoilers) I still found it quite moving on the whole. A big part of this game’s ability to make you feel is the score and visuals.

Audio-visual Beauty

Visually this game is just stunning. Although much of the world you travel through consists of bleak wetland, mountain and desert landscapes, the game still does a surprising job of making them look beautiful. Interspersed between those large open areas you have pockets of astounding beauty where the developers do some gorgeous stuff with greenery and lighting.

Not to mention the colossi themselves. There’s some brilliant textures that really gives these massive beasts a formidable prowess that makes (almost!) every battle feel like the daunting task that it is.

The visuals, however, are small potatoes compared to the MUSIC. Team Ico, the developers of this game have succeeded in crafting a musical masterpiece where every colossus fight is an adventurous score in and of itself. Often I found myself running along the back of these massive creatures as the music swelled and few moments in gaming have felt more cinematic to me.

The game can also be quite somber at times, as you slay off the last remnants of life roaming this vast empty lands. All in all, the visuals and music perfectly capture the emotions and feelings this game is trying to get at. Unfortunately, they try to do the same with gameplay

Goddamn Gameplay…

Your character controls terribly. The movement is clunky, he can barely jump the distances you want him too sometimes, he stumbles all the time. The horse that you ride is a complete wimp she will come to grinding halt if the slightest pebble in terrain comes your way so you have to spam the triangle button to get her to move again which sucks because she takes FOREVER to accelerate. Whichever way you framed it, I found the physical act of playing this game very frustrating. Half the time, I had little idea what I was meant to be doing.


Just to give an example, there’s a boss fight where you have to pick up a burning stick in order to defeat the colossus. However, the game never ONCE tells you that picking up an object is a thing you can actually do, much less tell you the control for it. It took me like half an hour of trying the most random things before I realized you could actually pick stuff up in this game. Why would it not tell you that???

I read a few reviews online and from what I could glean these features are often praised by players because it complements the character. You play as an inexperienced boy who is way in over his head, and the controls reflect that. And to be honest, as an artistic decision, I totally agree. It’s purely brilliant and fits the character perfectly. At the same time, I still had to actually play the game and knowing this fact did not make it any less frustrating for me.


It took me around 7 and a half hours to finish this game, but I’d say around 4 hours of that is just struggling with finicky controls and dealing with unintuitive puzzle designs and trying to get the colossi to where you need them to be. The story, soundtrack and scenery elevate this game to incredible heights but the actual gameplay brought it down considerably for me.

Rating: 4.5 out of 7.

I give this a 4.5/7. Shadow of a Great Game.

Click here to see how my rating scale works.


Practically everyone who’s aspired to become a scientist or has delved into scientific research of any kind is aware of the Nobel Prize. It is widely deemed the cynosure of accomplishment in a field, and is awarded to those who make massive breakthroughs in cutting-edge science.

The Nobel Prize

Of greater interest to me personally, however, is the Ig Nobel Prize.

A pun based on the word “ignoble” this prize recognizes and honors the unusual, the niche, the surprising and quite often the outlandish achievements in science. The following is the brief list of my favorite winners, and why I love them.

1. Pizza Protection

The 2019 Medicine Prize winner was Silvano Gallus, for his numerous papers about the effect of pizza consumption on the risks of cancer and death from disease. In my book, anyone who gives a solid scientific reason for why we should eat pizza is automatically one of the greatest human beings ever.

Silvano Gallus himself, smiling gleefully in front of what I am sure is a delicious authentic Italian pizza.

2. Miscalculations

The 2011 Mathematics Prize was awarded to a number of different people from different time periods. A couple of example include: Dorothy Martin, who predicted the world would end in 1954; Lee Rang Jim, who predicted the world would end in 1992, and Harold Camping, who predicted the world would end in 1964 and then later he changed his prediction to 2011. These aren’t all of the winners, but I’m sure you can see what the others’ achievements looked like.

I thought this image seeemed fitting

To quote the givers of the prize, they were recognized for “teaching the world to be careful when making mathematical assumptions and calculations.”

3. Pitch Drop

This award isn’t just to an interesting result, it is one of my personal all-time favorite experiments. The 2005 Physics Award went to a pair of researchers in Queensland University for the Pitch Drop Experiment, which was started in 1927. Essentially, it’s a glob of black tar that is being dropped through a funnel. And it’s still going

Nobody has ever been in the room to observe a drop fall.

The experiment averages one drop every 9~10 years. It fascinatingly blurs the line between solid and liquid, having a viscosity 100 billion times that of water. You could observe this for hours, days, weeks, even and see no discernible change in the structure of the pitch. Slowly but surely, however, it flows.

4. Inflated Numbers

So now Zimbabwe is mostly getting by using the US dollar, but remember the time when their currency was so absurdly inflated it jumped up into the trillions? The 2009 Mathematics Prize went to the Governor of Zimbabwe’s Reserve Bank, for “giving people a simple, everyday way to cope with a wide range of numbers.”

Roughly equivalent to 130 USD at July 2008.

Many developing countries target inflation to be around 5-10%, and anything above 15% is serious cause for concern in any economy. In summer of 2008 Zimbabwe’s inflation was an estimated 250 MILLION percent. That’s 250,000,000%. This, kids, is why learning basic economics is important, especially if you’re in charge of an entire country’s money supply.

5. Insensitive Incentive

This is probably the funniest one yet. The 2015 Economics Prize was awarded to the entirety of the Bangkok Metropolitan Police. The reason?

Offering police officers extra cash if they refused to take bribes.

I’ll just let you unpack that statement yourself.

6. How much wood would a woodpecker peck?

The 2006 Prize in Ornithology went to Ivan Schwab from UC Davis. Him and his partners conducted research into the headaches that woodpeckers may or may not receive from constantly, well… pecking wood.

The answer has to do with a small brain and a dense cranium that prevents it from shock and therefore headaches!

Mostly this one appealed to me because I like the idea of a bird-lover caring enough to check if a bird’s head hurt after a long day of knocking on wood. Really shows you the lengths of human compassion.

The Importance of Nonsense

The motto of the Ig Nobel Prize is to honor “achievements that first make people laugh, then make people think. They’re a fun, goofy event that acknowledges fun, goofy projects and occasionally blunders in the name of science. But even though these may seem unimportant, sometimes it is the pursuit of the trivial and nonsensical that lead to the biggest breakthroughs. After all, Fleming discovered Penicillin because he was too lazy to keep a clean lab.

Similarly, 2006 Ig Nobel Prize winner was conducting experiments on mosquitoes to see how they responded to different cheeses. In the pursuit of this seemingly inconsequential endeavor, he found limburger cheese and human feet are practically indistinguishable to malaria mosquitoes.
Now, this result is being used and cheese traps are placed in strategic locations across Africa to combat malaria.

So go out there and experiment with something stupid! You never know where it may lead…

Game Review | Uncharted 2: Among Thieves

I’m starting a new “series” here where I review video games I’ve played and talk about game design and stuff. Mostly because I don’t have actual friends to talk to. These will just be games I’ve played recently and stuff, so it’s not like an update on the latest releases or whatever.

I know I’m about 11 years late on this, but Uncharted 2 is one of the greatest games ever made. Incredible spectacle, bombastic music and characters that feel more human than most video games have ever achieved. I mean just look at the opening shot of the game, where you are hanging off a TRAIN ABOUT TO FALL OFF A CLIFF.

Sony Messed up the PS3 box art - Uncharted 2: Among Thieves ...

Not only is this moment a mind-blowing scene right off the bat, but also works as a tutorial for the game’s climbing mechanics and inspired the game’s legendary cover art. If you haven’t played this game yet please do yourself a favor and find a way to do so, because I promise you it is worth it.


You play as treasure hunter and thief Nathan Drake, in his quest to find the lost city of Shambala, racing against other thieves as he does so (hence the title). It’s a relatively simple premise, one seen in many other franchises and characters such as Indiana Jones and Tomb Raider, but Uncharted 2 brings that premise to video games in a way no other series has. 

First off, just the writing and dialogue in this game is brilliant. Drake’s wit and relatability makes him a more interesting protagonist than most films and TV shows manage to pull off. And every character is distinct, with their own flairs and idiosyncrasies. Just watch this scene below.

See how these people have more personality than anyone on Riverdale?

Not only that but once you hit Chapter 5 this game just does not stop. You go from spectacle to spectacle, jumping through rooftops in a war zone, unlocking hidden chambers, and rescuing people around you. It’s a simple enough story on the macro scale, in the sense that you are trying to find a lost city and you follow a trail of clues to reach there. But what truly distinguishes this game is the spectacle that comes in between.

Visual Spectacle

I think those screenshots speak for themselves, but even they cannot convey the intensity of the adrenaline rush you feel when actually PLAYING this. All I can say is, you have to play it to experience it. Which brings us to gameplay.


There are 3 main facets to the gameplay here: climbing, puzzle solving, and shooting. The climbing is incredibly cool, jumping around buildings and across cliffs and up a derailed train. There are a few interesting environmental puzzles to solve but nothing that particularly blew me away. The shooting, though, is where this game shines.

This is a classic third-person cover based shooter. You have two guns, and you crouch behind cover and pop out to shoot at enemies. It’s a simple gameplay loop, but what Naughty Dog does here makes it ridiculously fun.

The subtitle says it all…

Every fight, encounter, is set in a different environment, a different level that you’ll have to traverse. Sometimes you might have to fight enemies on rooftops, other times it’ll be in a moving train, or in a secret cave. You’ll have climbing sections interspersed with shooting, which leads to some places where you’ll have to fire at an enemy while hanging from a signpost.

Also of note here is the way your character moves. If you’ve played Mario or Minecraft or Modern Warfare, all those games share one thing in common apart from the fact they start with M: the characters move like super humans.

PS4 Uncharted: The Nathan Drake Collection vs PS3 Uncharted 2 ...
The movement is human-like but not clunky in any way. Naughty Dog found the perfect balance.

In those games you can jump and turn and crouch at the turn of a dime, and yes it’s incredibly satisfying to do but it’s not how any living beings move. In Uncharted 2 Drake moves like a person. Sure, he can leap incredible distances and has infinite grip strength, but his strides and jumps and and turns have weight to them. Combined with the well-written dialogue, it just adds a level of immersion to the game where you feel closer to Drake himself as a character; you can relate to him more.


It is not an exaggeration to say this game perhaps single-handedly turned the tide in PlayStation’s favor in the midst of the console war. The astounding variety, spectacle and unbelievably fun gameplay is why this game is revered as the bar to which cover-based shooters should be compared to.

Uncharted 2 Gameplay HD - YouTube
If it’s your first cover-based shooter, I’d recommend Normal, and if you’re an expert than Crushing difficulty is the way to go.

I’d recommend playing this game on Hard difficulty, to heighten the tension and make you really think about how you shoot and use the environment around you. There’s a slow section near the beginning and a slightly abrupt ending that just barely pull this game away from a perfect score, but in my opinion from chapter 5 to 25 you will experience one of the greatest shooting games ever made.


Rating: 6.5 out of 7.

See my rating scale for persepctive.

Trolley Problems

1. The Classic Problem

A trolley is heading down a track with 5 people tied down to it. There is an alternate track it can go down with only 1 person tied down to it. You’re standing next to a lever which can redirect the trolley to the track with only one person. Do you pull the lever and make the switch, killing one person to save 5?

My answer: Of course you do. People who say don’t pull the lever are afraid that you pulling the lever will put that person’s blood on your hands, whereas you doing nothing means the death of the 5 people wasn’t necessarily your fault. But for me, you’re still making the choice to do nothing when you have the ability to pull the lever, so it’s just as bad as pulling the lever to kill 5 people instead of 1. So the obvious answer is to pull the lever.

Now let’s get onto some interesting ones. I promise they won’t be the classic alternate versions you’re used to.

2. Theseus’ Trolley

Same setup as before, except now there’s about 10,000km between where the track splits and where the second man is tied down. If, during the long journey, a team of engineers on the trolley replace each part bit by bit – replace a plank of wood here, a nail there – until every single original piece has been replaced with a new one, is the trolley that kills the man the same one that you diverted? 

The key here is that the trolley is moving as a single entity the entire time and any given replacement at one time is only a small part. If you replace a car’s tire, for example, you can’t say it’s an entirely new car. But what if you replace the windshield wipers next month? Is it a new car now? What about if you replace the seats inside with leather? Taking one replacement at a time, there seems to be no change in the entity called ‘the trolley’ at all. However, over time, once every single original piece has been replaced, is it still the original

3. Driverless Trolley

The trolley system has been privatized and monopolized by the Edison corporation and all but automated away using AI. You drive an automated trolley down a track and see a track straight ahead that has a person on it.

Very energy-efficient

The trolley is programmed to brake when it detects an obstacle and so you sit back and decide to let the trolley stop itself. Too late, you realise the detection system hasn’t worked, and you go to slam the brakes, but the damage is done. The trolley runs over and kills the person. Who is responsible for his death?

Should the company be held liable for a detection software that didn’t work in this one circumstance? Or are you to blame for not stopping the trolley when you had the chance to? Often these trolley problems are criticized for not actually being applicable or relevant in the real world, and thus the conclusions are not based on anything tangible. This one, however, will only become increasingly pertinent with the rise of automated vehicles, and how we as a society choose to answer this question will have very real legal implications.

4. The Riemann Trolley

A trolley is heading down a track with a countably infinite number of people on it. If it continues down that path, it would literally kill infinitely many people. 

There is another track, however. For every nth individual on the first track, there are n individuals tied together on the second track. In other words, you have 1 + 2 + 3 + 4+… people on the second track.

According to the analytical continuation of the Riemann Zeta Function, this expression is equivalent to -(1/12). Do you trust this track to continue down the path of infinitely many people or do you pull the lever to revive one twelfth of a person? 

If you don’t know about the this expression or the Riemann Hypothesis at all, 3blue1brown has an AMAZING video on this topic which, if you take your time to watch through and understand, is immensely rewarding even for mathematical laymen.

5. The Trolleyologist

A trolley is headed down an empty track. However, you have the option to divert the trolley to another track where a philosopher who constantly asks you trolley questions is tied down. How fast do you pull the lever?

Prometheus and Divine Fire

And in a single word to sum the whole-
All manner of arts men from Prometheus learned

– Aeschylus, Prometheus Bound

This is the first comic I ever made, in it’s terribly pixelated glory:

Clearly I made good use of all the available space

The Greek myths and stories have always held a special place in my heart, ever since I was but a young 9-year-old reading about the adventures of the demigod Perseus Jackson. 5 years after finishing the much-treasured series, I retain that fascination with Greek mythology to this today, and one of my favorite stories is that of Prometheus. 

In case you’ve been chained under a rock and have never heard of this story, this video provides a quick animation of the tale. Short version is, Prometheus steals fire from the gods and gives it to the humans. Zeus was very much unhappy with this, and so he sentenced Prometheus to an eternity of torment chained to a rock. It’s a fun (the Greeks had a different definition of fun) story about the origin of fire that you tell your child before bedtime, and all’s well.

Except I don’t think that’s all there is to it. 

The Ancient Greeks were a society of poets and philosophers. In fact, the words themselves derive from the Greek language (“poet” comes from “poiētḗs” – inventor; “philosophy” comes from “phil” – love and “sophos” – wisdom). Their mythology wasn’t simply an assortment of bedtime stories, nor was it a set of doctrines to live by. Instead, many of their myths were attempts to personify and characterize the forces that governed the world around them.

It is not so surprising, then, that the Greek Gods and Titans were so… human. They weren’t perfect, omnipotent beings like Yahweh of the Abrahamic religions, nor were they abstract spirits or ideas like those present in Aboriginal myths*. Instead, they were embodiments the best of us and the worst of us. They were powerful and intelligent and loving and caring but they could also be zealous and cold and ruthless and tempestuous and prideful and petty. 

*this isn’t to say the Aboriginal and Abrahamic myths aren’t interesting in their own ways, of course.

The other half is attractive young men

And so with this in mind, Prometheus’ myth takes on a new meaning. The story tells us that humanity began to prosper and grow bountifully after receiving fire stolen from the Gods. It allowed us to transform from docile beings of dirt and clay to people of passion, ambition and aspirations. The “fire” then can now be seen as so much more than a flame; it’s a representation for our innate drives that compel us to think and feel and love and hate. 

Here the myth becomes less a tale about the origin of fire and more of a declaration by the Greeks that we hold this Godly flame. In other words, if there is anything that can be called divine in our universe, it is us. If there are beings that can be called Gods out there, then we have the same drives and capacities and dreams as them. Prometheus is said to have fashioned humans in the image of the Gods, but the Greeks fashioned their Gods in the image of humans.

Creation of Man by Prometheus
Prometheus making man out of mud.

There’s something about this idea that is fascinating to me. The Ancient Greeks seemed to understand and capture what I feel many other societies did not: the search for the divine is misguided, for it is inside us. Not in some temple or altar or in some mountain in the sky. We are free to be masters of our own lives and captains of our own ships. 

The very same fire that fuels the Gods fuels us. That’s why we can create, craft, inspire and imagine and it’s also why we can be mean and cruel and rapacious and irascible. We take pride in humanity’s accomplishments, and we take fault for its atrocities. 

And there’s my take on one of my favorite myths! If anything I hope I’ve given you a motive to look more into the fascinating world that is Greek mythology, and feel a little better about yourself, knowing you have divine fire in you. To top it off, here’s a redraw of that very first comic:

Now that looks so much more presentable…

Other latest posts

  • Motivation and School
    I haven’t posted in a while, and the reason is schoolwork has snatched all the time out of my hands. It just feels like I’ve constantly had something or the other on my plate at […]
  • Does 2+2=4?
    While scrolling through my twitter feed I came across a series of threads all essentially asking the same thing: what is the meaning of the phrase 2+2=5? If you’ve read Orwell’s 1984, you’ll know this […]

Newcomb’s Paradox

I feel like math and english are often seen by people on two opposite ends on a spectrum of subjects. On one side you have the “strict” and “rigorous” math, where there is only “one right answer.” On the other hand, you have “free” English Literature, where any answer can be correct “as long as you can justify it.”

But this had always seemed just plain incorrect to me. Literature is NOT a wishy-washy dream subject where every answer can be right. If you think J.B Priestley’s An Inspector Calls is a pro-capitalism play, you’re wrong. If you think Steinbeck’s Of Mice and Men is a celebration of opportunity and freedom in America, you’re just plain wrong. There’s no way about it, it’s just wholly incorrect.

And math questions don’t always have only one right answer. Quadratics have 2 solutions, and trigonometric identities can have infinite solutions within an unbounded domain. And in some, strange, strange cases, we can have 2 solutions that are contradictory. The existence of either one should disprove the other, but both seem to be true. This is the crux at the heart of Newcomb’s Paradox.

Now, we consider a scenario. Two closed boxes lay in front of you, labelled A and B. Now, you can either take only Box B or both boxes A and B. You can’t choose only Box A because of… reasons. Anyways, overseeing your choice is a master psychologist and human reader. Let’s call this guy Kevin because… you know… reasons. He tries to predict your decision, and allocates money into the boxes accordingly before you actually make the choice.  

Now, before anything begins Box A contains a $1,000 cheque. Sweet, a brand new PC, I hear you think to yourself. But here’s the thing. If Kevin predicts you’ll take both Box A and B and guarantee yourself that sweet grand, he’ll leave Box B empty. However, if he predicts you’ll take only Box B, he’ll slip in a cool $1,000,000 cheque. Yeah, forget the brand new PC, you can go buy a brand new MANSION with that kind of money. And Kevin is pretty smart, so his predictions are right 90% of the time.

To recap: if you choose both boxes, you’re guaranteed $1,000 but only have a 10% chance of getting that extra $1,000,000 in Box B. If you choose only Box B you’ll have a 90% chance of getting a quick one million, but you’re missing out on that $1,000 in Box A. The table below nicely summarises the possibilities:

What do you do to maximise your winnings?

The correct answer is…non existent.

Ok, well looking at the table the answer may seem obvious. No matter what the Kevin predicts, your winnings are ALWAYS higher if you choose both boxes than if you don’t. After all, he makes his prediction and puts the money in the boxes before your choice. So whatever is in Box B, your earnings are greater if you choose Box A as well. No-brainer. Both boxes is CLEARLY the right choice. And you’re right!

Except… the thing is Kevin is 90% accurate. You taking both boxes means there’s only a 10% chance of Box B having a million bucks. But Box B would have a 90% chance of being filled with money if you don’t take Box A. Let’s use Expected Value to calculate our expected earnings. Now EV = k=0nakP(ak)

What that mess of symbols means is that you multiply the value of each outcome by the probability of that outcome occurring, and then add all of them up.

So if we choose both boxes: $1000 x 1 + $1,000,000 x 0.1 = $101,000

This is because the probability of getting $1000 in Box A is guaranteed, but we’re relying on the 10% chance that Kevin is wrong and predicted we’d choose only one box.

But if we choose one box: $1,000,000 x 0.9 = $900,000

Because there’s a 90% chance that Kevin predicted we’d choose just one box. And since the expected earnings for choosing just Box B are much higher than choosing both boxes, we choose Box B. No brainer. And you’re right!

Except…the dominant strategy clearly dictates that both boxes have more value than just one. And herein lies the crux of the Newcomb Paradox. Two different, mathematically valid lines of reasoning that result in contradictory conclusions. And down comes crumbling the facade of math’s indestructible logic and rigor. 

While it may seem to nullify the beauty that comes from the universal truths of math, I prefer to think it somewhat adds to it. To know that math isn’t just this black or white, yes or no, right or wrong. It can be mysterious, nuanced, paradoxical even. That, to me, is the true beauty of Newcomb’s Paradox.

Shuffling Cards

So last week I entered this programming competition and one of the questions was about permutations of letters. Essentially, there was some string of letters and we had to count how many permutations of that string were substrings of a larger string we were given.

The details don’t really matter, the main point is that at some point pretty much everyone in the competition thought about using the permutations function to solve it. Basically what it does is it takes some string as an input and outputs all of the permutations of that string. For example if you gave it ‘abc’, it would output: [‘abc’, ‘acb’, ‘bac’, ‘bca’, ‘cab’, ‘cba’].

Now this was all well and good for small 3 letter strings, but what a lot of us didn’t initially realise was the sheer number of permutations that came with even slightly longer strings. Even when you just increase the input to strings with 10 characters, the function outputs over 3 and a half MILLION strings. Somehow none of us realised how impossible using this function would be, and I think it comes down to the fact that humans have a pretty terrible idea of how fast factorials grow. So let’s talk about it using cards.

Alright, let’s suppose you’re shuffling a deck of cards. Every time you shuffle, the 52 cards fall in some arrangement. We can count the total number of arrangements possibly quite easily. For our first card, we have 52 choices. For the second, we have 51 (since we’ve already used one card), for the 3rd, 50, and so on until we reach the final card. The total product is 52 x 51 x 50 x… 3 x 2 x 1 = 52! (That’s 52 factorial, I’m not just really excited about the number 52) This is what we mean by the number permutations of the deck. This is also the number of strings that the ‘permutations’ function would’ve output if we had given it a 52-letter input. 

The number of possible permutations of a deck

Now, we can write 52! in standard form; it’s 8.06 x 10^67. Simple enough. But that doesn’t even come CLOSE to representing how massive that is. So let’s try and visualise it.

We’re gonna make some assumptions. Let’s say around the world, there are 5000 decks of cards being shuffled every second. Doesn’t seem like too unreasonable an estimate. Supposing every new shuffle made is an arrangement that’s never been made before, how long do you reckon it’ll take to get through all of them at that rate? A couple of years? A couple of centuries? Millennia? The lifetime of the UNIVERSE?

Not. Even. Close.

5000 shuffles a second means 1.61 x 10^64 seconds. That is a LONG time. By contrast, the age of the universe is only about 4.3 x 10^17 seconds. 

Let’s do a thought experiment in an attempt to visualise this length of time. Suppose you’re waiting until that momentous day when, at last, every single possible arrangement of 52 cards will have been done. What can you do to pass the time? I mean, apart from reading Astronomical Blunder Comics? 

Well, you could type out every single word in the Oxford English Dictionary, of course! But don’t worry about being a bad or slow typer. For our purposes, you’re just gonna have to type at the gentle pace of one word every billion years.

Yes, you read that correctly. One word. Every billion YEARS. The sun will swallow up the Earth in its entirety and collapse into a white dwarf long before you’ve finished your first page. And remember while all this is going on, you’ve still got 5000 new arrangements a SECOND being made.

Roughly equal to one word every billion years.

So, there’s about 600,000 words in the OED as of now, and that’s the copy we’ll use. After painstakingly typing out all 600,000 words at that snaillike rate, go take a trip to the Sahara Desert and remove a singular grain of sand (Let’s just pretend the Earth still exists after this time. Nothing else about this scenario is normal either). Delete all those words you spent all that time typing up, and restart from the beginning, at the same rate. When you’ve once again compiled the entirety of the OED, take out another grain of sand from the Sahara.  Repeat until the Sahara is no longer a desert, just a landscape of bedrock. That should be about 1.5 septillion grains of sand, which means 1.5 septillion copies of the OED, all typed at 1 word every billion years.

Now that you’ve reduced the Sahara to just large rocks and gravel, let’s help out the environment a bit. Let’s remove one kilogram of CO2 from the atmosphere (we’re also going to assume no-one is going to be polluting earth anymore because they’ll all be too busy waiting for all the card arrangements to be reached). So, re-fill the Sahara, delete all the words and start typing again. 

It’s a thought experiment, don’t read too deeply into it…

Finish the OED, then remove a grain of sand until you drain the Sahara, then remove a kilogram of CO2. Once ALL of the world’s CO2 has been removed and the atmosphere is completely clear of any carbon…

There will be 1.60 x 10^64 seconds left on the clock. We won’t have even gone through 1% of all the possible combinations. Only once we do that entire process 160 more times, will we have gone through every single permutation. Which means putting back all the carbon in the atmosphere, filling the sahara with sand, starting typing on a blank new document at the rate of 1 word every billion years, and repeat. One hundred and sixty times. 1.61 x 10^64 seconds is a LOOONG time.

This seems almost ridiculously false, because a deck of cards really isn’t all that much; it’s something you can fit in the palm of your hand. Yet the combinations that can be derived from it are barely comprehensible or countable from a human perspective. 

So just know that every time you rearrange your 52 cards, there’s a pretty good chance that it will be the first, and probably only, time that specific arrangement is made for all of human history. If you’re ever feeling unspecial and want to create a historic new moment that will likely never occur again…

Shuffle a deck of cards. 

Sex, Graph Theory, and why you’re taught math

Math, am I right? 

Calculus, trigonometry, geometry, algebra, statistics, proofs. Who needs all of that, and why is everyone forced to learn it at school? Except for mathematicians, of course. And perhaps the odd theoretical physicist or computer scientist. 

Well, ask the sociologists behind this book. More specifically, a claim they made based on surveys saying that men in the U.S tend to have more opposite-sex partners than women. By a LOT. Supposedly the average American male has 74% more female partners than females have male partners. Ugh, men and their promiscuous playboy ways. Typical.

Actually, not quite so. In fact, there’s no way the ratio can be that different, and perhaps if the sociologists knew a bit of math beforehand, they’d be able to dismiss the results of the surveys much more easily. Let’s go through it with a little bit of graph theory.

If you’ve ever done one of those connect-the-dot puzzles, you’ve already indulged in what graph theory is all about. Formally, a graph is a pair G = (V, E) where V is a set of vertices and E ⊆ {(u,v) | u, v ∈ V}. But really, it’s all about dots and lines connecting them. 

ProTip if you want to make your child seem like a prodigy, just say this every time he does a connect-the-dots puzzle.

We call a dot on a graph a vertex and a line from one dot to another (or from one dot to itself!) an edge. So to visualise the population of sexually active men and women, let’s create two columns of vertices as below:

Note that the diagram doesn’t show exactly how many men and women there are.

Now, if a man and a woman are partners, let’s connect them with an edge (for now, we’ll just assume random, arbitrary partnerships and disregard same-sex relationships). Each vertex can have one or more edges connected to it.

Notice that every edge is connected to BOTH the men and women column

OK, so now we’ve visualised the scenario using graphs. Going back to our goal, we’re trying to find the average number of partners of the opposite sex for both men and women. Converting that into our graph theory lingo, that means finding the average number of edges connected to a vertex for each column. Well, to do that you just add up the number of edges connected to all the vertices in one column, and divide by the number of vertices. 

Let’s call the total number of men Vm and the total number of edges connected to the men column Em. Similarly, the total number of women will be Vw and the total number of edges connected to the women column will be Ew. So the average number of opposite partners for men is Em/Vm and for women it’s Ew/Vw.

So the ratio of average partners of men to average partners of women is (Em/Vm)/(Ew/Vw). But Em and Ew are the same. Obviously. A five-year-old looking at the graph could tell you that, since each edge is connected to one vertex from the men column, and one vertex from the women column. In simple terms, it takes two to tango. So the ratio of opposite partner relationships between men and women is simply the ratio of the NUMBER men to women: Vm/Vw

When you look at it this way, it seems ridiculous that men have 74% more opposite sex partners on average. If that were true, it would imply that there are 74% more sexually active men than women in the United States. And even if THAT were true, the number tells us nothing about the promiscuity of men or women at all, it’s just a ratio between the NUMBER of men and women present.

Now, it seems almost comical that the sociologists didn’t notice this simple fact, but that’s the issue: people largely underestimate the role of mathematics, in everything. Math isn’t just adding and multiplying numbers so you can calculate your bills or fill out your taxes. It’s a way to build rigorous arguments and present and analyse things in different ways. We aren’t taught math because the formulas will come in useful one day; most of them won’t. It’s necessary because it strengthens our logical thinking and reasoning abilities, something you need regardless of your interests or profession. 

So maybe don’t be so hard on math. And this may be a math nerd saying that so you can call me biased, but all you have to do to prove me wrong is build a rigorous argument using logic and base axioms, and we’ll agree to disagree.