Posted in History & Literature

Stigler’s Law

In 1980, statistician Stephen Stigler suggested that in the history of science, no scientific discovery is really ever named after its original discoverer. This is because eponymous laws and discoveries tend to be named after the person that made it widely known.

Take the example of the famous Pythagorean theorem, which was known to Babylonians before Pythagoras was even born. Halley’s comet had been documented by astronomers since 240 BC. Fibonacci numbers were well-known to Indian mathematicians since 200 BC – 1400 years before being described by Fibonacci.

You could make the argument that these discoveries could not be traced to the original discoverer as it was too long ago and the discoverer’s name was never documented. This is certainly one reason for Stigler’s law being true. Usually, scientific discoveries tend to be popularised and named after the discoverer has already died, meaning that there is a chance they could be forgotten already.

There are plenty of examples where the original discoverer is known, thanks to historians of science, but it is too late to reverse the eponym as the name has firmly rooted itself into people’s vocabulary.

Take Alzheimer’s disease, which was discovered by Beljahow in 1887, but named after Alois Alzheimer in 1901. The bacteria Salmonella was identified by Theobald Smith in 1885, but as he was a junior inspector, his boss Daniel E. Salmon took the credit instead.

It is also worth noting that throughout history, there have been many cases of discoveries being made simultaneously by independent scientists.

Sometimes, the scientists credit each other and share the fame, such as Charles Darwin who decided to co-present his theory of natural selection with Alfred Russel Wallace, another scientist who came to similar conclusions at the same time.

Other times, scientists may fight aggressively to assert their credit, such as Isaac Newton and Gottfried Leibniz, who both discovered calculus around a similar time, but fought to claim that they were first.

An important lesson to learn here is that as much as we love stories where a brilliant individual changed the course of history, most advancements in human history happen thanks to collaboration and inherited knowledge over time. Things rarely happen in a vacuum and we all rely on each other’s experiences and knowledge, building on our predecessors to achieve greatness.

Ironically, Stigler’s law follows its own law, as Stigler identified sociologist Robert K. Merton as the original discoverer of this law.

Posted in Life & Happiness

Shoot For The Moon

A common saying goes:

“Shoot for the moon: even if you miss, you’ll land among the stars”.

The saying was coined by author Normal Vincent Peale, who was a minister famous for his books and work on the power of positive thinking. He was also widely criticised by many psychologists and mental health experts, who noted that his style of positive psychology was not founded in evidence and realism, but in naive optimism.

The saying sounds lovely at first, because it seems to be a beautiful metaphor for trying your best at everything. It says that whatever happens, you will land on another beautiful opportunity and good things will happen.

But of course, life does not work that way. As important as it is to make an effort to try and take action, you will not always be positively rewarded for it.

As it is with everything, science can help us break down the flaws with the philosophy of this saying.

Firstly, the Moon is 384,400km away from Earth. It took brilliant scientists and mathematicians with a significant amount of NASA budget 6 years on the Apollo program to put astronauts on the Moon.

Dreams are certainly achievable, but we cannot ignore that sometimes we have to pour in much time, resources and energy to achieve them. When we look upon someone’s success, it is important to consider how much effort they may have put in. Furthermore, it is paramount that we be realistic with our goals and dreams, in that we need to be patient and accept that it could take a series of failures, sacrifices and heartbreak for us to land on the Moon.

Secondly, space is unimaginably massive. If you shoot for the moon and you miss, there is a very high chance that you will float along the lonely, vast emptiness of space for the rest of eternity in a vacuum before you hit anything else (realistically, you will die of suffocation, thirst, starvation or being frozen first). The nearest star to us is the Sun, 150 million kilometres away. The second closest star – Proxima Centauri – is about 4.24 light years away. This means that even if you travelled at the speed of light, it would take 4.24 years, covering a distance of 40 trillion kilometres.

This fact teaches us that we have to be prepared for the fact that when we chase our dreams, there is a chance of things catastrophically failing. That is just life.

Lastly, even if by some miracle you survived the journey and landed among the stars, it would not be what we expect. As romantic as it sounds to land and live on a star like the Little Prince, in reality, stars look much like the Sun – a gigantic, glowing ball of fire. You will be incinerated even before you land on it.

And there is our final lesson from this saying: even if you achieve your goals, the end result may be completely different to what you expected. You may not even be happy with the outcome. So avoid pinning all of your hopes and happiness on achieving a single dream. Make sure to diversify your goals and identity.

As factually wrong as the saying may be, we can still learn valuable lessons from it, albeit completely the opposite message. But perhaps this is the more important truth in life: sometimes, we fail to achieve our dreams.

That said, we must continue to try for our goals and dreams, just with realistic expectations of how life can go. Had NASA given up after the tragic fiery accident of Apollo 1, we may have never been able to experience the glorious moment of humanity setting foot on another celestial body.

Shoot for the moon, but maybe have a backup plan. And if you fail, don’t lose heart and give up, but instead try again and try new, different things constantly.

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 Science & Nature


If you mix 1 part water to 1.5-2 part corn starch, you create a strange mixture called “oobleck“, named after a Dr. Seuss story. It is so simple to make, yet it exhibits some very strange properties that makes it a popular science experiment.

Oobleck is what is known as a non-Newtonian fluid, where the viscosity (or “thickness”) changes with how much stress it is under. If you press your finger gently into it, it will feel like water, but if you strike it with a hammer, it will behave as a solid. It will stiffen when you stir it, but run when you swirl it.

Related image

You can even run over a tub of oobleck as long as you change steps quickly enough to apply enough pressure to keep the fluid under your feet solid. This is because oobleck becomes very viscous under high stress, making it behave more solidly (shear thickening).

We can learn from oobleck not only some interesting physics principles, but also how to interact with people.

Much like a non-Newtonian fluid, people will tend to react stiffly and with more resistance if you apply stress or force. But if you apply gentle pressure and be assertive, you will find people generally react more softly and fluidly.

This simple change in your approach will lead to much better conflict resolution and constructive outcomes when dealing with other people.

Image result for oobleck run gif

Posted in Science & Nature

Death Pose

When a dinosaur fossil is excavated, it is not uncommon to find the dinosaur in what is known as the death pose. The long neck is bent dramatically backwards and the mouth is gaping open, as if the dinosaur is letting out one final bellow.

For a long time, palaeontologists believed that dinosaurs found in this pose had remarkable neck flexibility. For example, the Elasmosaurus was originally thought to have a snake-like neck that could bend and curl around, even being able to lift its head above the water, as seen with the image of the Loch Ness Monster. However, in reality, the neck would have been too stiff and heavy to move around like that, meaning that Elasmosaurus would have swam around with a straight neck, barely lifting its head above water.

It is still unclear exactly why dinosaurs are often found in the death pose.
Traditionally, it was believed that the strong ligaments holding the neck bones (vertebrae) contracted as they dried out, bending the neck backwards where there are more ligaments.
Others refute this theory, instead suggesting that the dinosaur remains would be rearranged by water currents, or that the carcass would naturally bend backwards when floating in water.
Finally, another group of scientists believe that the pose happens in the final moments of the dinosaur’s death throes, suggesting that they experience opisthotonus (arching of the back muscles, as seen in tetanus) either due to lack of oxygen in the brain, or poisoning.

It is fascinating to think that although these dinosaurs have been dead for 66 million years, we still have so much to learn from them.

Posted in Science & Nature


Sudoku is a mathematic puzzle that has gained considerable popularity in the 21st century, rivalling the classic puzzle that is the crossword. You are given a 9×9 table divided into 9 equal squares, filled with a certain number of digits. Your goal is to fill in the table so that each row, column and subsquare (of 9 small squares) contains every digit from 1 to 9. You are not allowed to have the same number appear on the same row, column or subsquare, as there are not enough spaces for spare digits.

The more digits (“clues”) that you are given at the start of the puzzle, the easier it is to solve it. This begs the question: what is the minimum number of clues that you need to solve a sudoku puzzle?

Sudoku puzzles with 17 clues have been completed traditionally. We know that 7 clues is not enough as the last 2 digits can be interchanged, creating puzzles with more than one solution. Using mathematics, we know that if we can solve a puzzle with n clues, then a puzzle with n+1 clues can be solved as well. Ergo, the answer lies somewhere between 8 and 16.

In 2012, Gary McGuire, Bastian Tugemann and Gilles Civario tackled this problem using one of the oldest tricks in mathematical analysis: brute force. The total number of possible sudoku puzzles that can be generated is 6,670,903,752,021,072,936,960, or 6.67 x 10²¹. After accounting for symmetry arguments (meaning that two puzzles may be essentially identical, but just rotated or flipped), we are left with 5,472,730,538 possible unique solutions.

The team used supercomputers to analyse all of these possibilities to see if any puzzle can be solved with just 16 clues, as the conventional thought was that 17 was the minimum number of clues possible from traditional methods. After a year of calculations, the computer found no sudoku puzzle could be solved with only 16 clues. This was confirmed by another team from Taiwan a year later, proving that the minimum number of clues required for sudoku is indeed 17.

Posted in Science & Nature


Brontosaurus (“thunder lizard” in Latin) is one of the most well-known dinosaurs. It is the poster child of the sauropods, a group of massive four-legged dinosaurs with very long necks and tails, known as some of the largest animals to ever walk on land.

After going extinct around 66 million years ago, the Brontosaurus was rediscovered in fossil form in 1879 by palaeontologist O.C. Marsh, who is infamous for his rivalry with another palaeontologist called Edward Drinker Cope as part of the “Bone Wars”. The Bone Wars was the fierce competition between the two palaeontologists, involving aggressive digging to discover as many dinosaurs as possible, while both tried to slander and impede each other through dishonest, unprofessional means. This dispute resulted in rushed announcements of new discoveries sometimes, leading to fascinating stories such as Cope accidentally putting the skull of the Elasmosaurus on its tail instead of the neck.

So what does the historical context of the Bone Wars have to do with the Brontosaurus? In 1903, another palaeontologist argued that the Brontosaurus was actually a specimen of the already discovered Apatosaurus. Two years later, the American Museum of Natural History unveiled the first mounted sauropod skeleton and named it a Brontosaurus. However, they had accidentally used the skull of a different dinosaur called Camarasurus, mounted on the skeleton of an Apatosaurus. With no further evidence supporting Brontosaurus as a separate genus, the scientific community agreed that the Brontosaurus was really just an Apatosaurus.

Despite this news, Brontosaurus remained hugely popular amongst the general population thanks to its early publicity. At the same time, Brontosaurus not being a real genus of dinosaur became a popular factoid (false information accepted as fact due to popularity). In a field such as palaeontology where evidence can be scant or incomplete, such misclassification is common. For example, the Triceratops is in fact simply the juvenile form of another dinosaur named the Torosaur.

But then in 2015, a group of scientists used computer modelling to analyse sauropod fossil data including the original fossil discovered by Marsh. What they discovered was that there were enough differences between the Brontosaurus and Apatosaurus, such as differences in pelvic bone structure, to classify Brontosaurus as its own genus. After more than a century, the Brontosaurus has had its name cleared and restored to its former glory.

The story of the Brontosaurus is a great example of one of the principles in science: nothing is 100% true. Science never proclaims something as the one truth. We can hypothesise, support it with evidence and construct a theory that makes sense of the cosmos, but we can never be sure that we definitely have the answer. In the face of new evidence and re-examination of the analysis, what was once regarded as “truth” can easily be proven to be wrong.

This is an unpopular aspect of science, because people tend to want security and certainty to soothe their anxieties about not knowing. But instead, we get to stay curious and continuously question the nature of the universe and how everything works, making fascinating discoveries and learning something new every day.

For how boring would life be if we had nothing more to learn?

Posted in Science & Nature


Nature is surprisingly balanced. For every action, there is an equal and opposite reaction (Newton’s Third Law of Motion). Energy can change forms in an isolated system, but cannot be created or destroyed as the total energy must remain constant (Law of Conservation of Energy). Similarly, matter is balanced by the existence of antimatter.

Antimatter is a substance that is the polar opposite of matter. For example, instead of positively charged protons and negatively charged electrons, anti-protons are negative and anti-electrons (or positrons) are positive. Much like matter, antimatter particles can interact with each other to form more complex particles, such as an anti-atom, meaning that it is conceivable that an entire world could be made out of antimatter.

When antimatter and matter collide with each other, they annihilate. Much like the equation 1 + -1 = 0, the two opposites cancel each other out. Conversely, to create matter out of nothing, you must create an equal amount of antimatter to balance it out. Strangely though, physicists have noted that there is a great imbalance between the two in the observable universe. There seems to be far more matter than antimatter, which does not make sense. The question of why this imbalance exists is one of the biggest unsolved mysteries in physics.

An interesting lesson we can take away from antimatter is the concept that to create something out of nothing, you must balance it out with “anti-something”. If you borrow money from the bank, you may have $1000 now, but you have also created a -$1000 debt. The total balance is still 0.

The same concept can be applied to happiness. If something makes you happy, then the possibility exists that the same thing can cause you an equal amount of grief. Let’s say you find a fulfilling relationship with a significant other who brings you extreme joy. This is balanced by the extreme grief that will be brought to you if the relationship is strained or ends abruptly. Ironically, the pursuit of happiness creates more room for potential misery, as grief comes from the loss of something we care about.

So what does this imply? Does it mean that we should avoid falling in love or caring about anything, because it will only hurt us in the end? Should we even bother trying to live a happy life if it is cancelled out by all the sadness that it can bring along the way? Of course, these are silly thoughts. How dull life would be if we did not have any ups or downs.

Instead, the lesson here is that we should be mindful that happiness is not free. Grief is the price we pay so that we can experience the wonderful moments of joy, love and connection that life can give us only if we reach out. If you avoided connecting with someone or taking a leap of faith due to fear of failure or loss, then your life would be empty. This philosophy allows us to be grateful for the joyful moments, while helping us endure grief as we know that is the price we must pay for true happiness.

You can’t let fear steal your funk. To quote Alfred Lord Tennyson: 

“‘Tis better to have loved and lost than never to have loved at all.”

Posted in Science & Nature

Car Keys

There are times when you park your car, start walking away and you remember that you forgot to lock the doors. You click your remote car keys but you are already just far enough that the signal does not reach your car. Fortunately, there is a lazy way to extend your car remote’s range.

If your hold your remote against your head (such as next to your chin or your temple), you will find that suddenly, the remote works from a longer distance like magic. How can this be?

There are two explanations that factor in.

The first is very simple: height. The higher you hold your remote, the less barrier there is between you and the car, making the signal more likely to reach it. But this cannot be the only answer as the trick works when there is nothing between you and the car.

The second explanation is more technical. When you press the key to your body and click it, the electromagnetic waves that comprise the signal can cross past your clothes and skin into your body, which is mostly composed of water. The water acts as a capacitor as the signal starts to “charge” you, all the while the signal is being rapidly bounced back and forth between the remote and you. In essence, your body acts as a giant aerial that amplifies the signal, almost doubling the range of the remote.

Arthur C. Clarke once wrote: “Any sufficiently advanced technology is indistinguishable from magic”.
But even the simplest scientific principles can seem like magic until we bother looking under the hood.

Posted in Science & Nature

Dinosaur Meat

Did dinosaurs have red or white meat? Typically, we think of white meat as coming from poultry, such as chicken, duck, turkey, while red meat come from large mammals such as cows, pigs and deer. So if you were to hunt down a stegosaurus or a triceratops and cooked it over a barbeque, what colour would their meat be?

The redness of meat comes from a protein called myoglobin, which carries oxygen from the blood to the muscle cells. It is similar to haemoglobin, which gives blood the characteristic red colour. An important note is that when you see reddish water drip from meat from the butchers, you are seeing myoglobin, not blood (the blood is drained when the meat is prepared).

The difference in colour between red and white meat come from the type of muscle fibres and their myoglobin content.
Red meat is made from slow-twitch fibres, which are useful for sustained activities such as walking or to keep standing. They exert a smaller force over a longer period of time, meaning they require more oxygen for aerobic respiration (a more efficient way of burning fuel using oxygen). Ergo, red meat is full of myoglobin, hence its deep rich red colour.

On the other hand, white meat is made of fast-twitch fibres. These fibres are better suited for quick bursts of energy, such as flying or quickly responding to a threat. These fibres use anaerobic respiration (no oxygen), which allow for a quicker, faster burn of energy, but only for a short time. In birds, the breast muscles are typically very white, but they do have some slow-twitch fibres in other muscle groups such as their wings and legs, which is why there is a distinction between light and dark meat.

So how about dinosaurs? Dinosaurs are the ancestors of birds and reptiles, so it would make sense for them to have had white meat. However, the majority of dinosaurs, especially large ones such as sauropods, would have had very powerful muscles with slow-twitch fibres, making their meat quite red. A good example are ostriches. Even though they are birds, their meat is as red as beef because they have powerful leg muscles for running.
Smaller animals such as raptors probably had more white meat akin to modern poultry, as they would require sudden bursts of energy for ambushes.

As for how they would taste, that is something we could not answer until Jurassic Park becomes a reality.