Tag: Math

On the Video Games and Violence Discussion

With three recent deadly shootings (one in Isla Vista and the second in Seattle; a third in Las Vegas as I was writing this post), I’ve once again heard many ignorant statements thrown around regarding video games and violence. Much of the ignorance comes from making blanket statements completely lacking in nuance, from both sides.

huffpost-videogames-vs-guns

Here is what’s wrong with the current discussion:

1. The anti-video game side ignores the actual crime statistics.

Whether you look at the past decade or past two decades (when video games arose and flourished), you see that general crime, violent crime, and juvenile crime are all down significantly.

video-games-crimes

Violent juvenile crime in the United States has been declining as violent video game popularity has increased. The arrest rate for juvenile murders has fallen 71.9% between 1995 and 2008. The arrest rate for all juvenile violent crimes has declined 49.3%. [1]

Of course, this does not mean that (violent) video games are causing the reduction in violence. Here is a graph that goes forward by several more years [2]:

video-game-sales

The point is that even if a study comes out demonstrating a link between video games and aggression, it is another step to go from aggression to actual violent crime, which is hard to measure because we can’t just run experiments on violent crime. To show that video games have a strengthening effect on the crime rate, you must show that in the absence of video games, the crime rate would be decreasing faster than it already is (or something equivalent to that).

2. Both sides have a wrong assumption about overall crime.

Because our media gives plentiful attention to violent crimes—the more deaths, the better—we get a sense that the nation is becoming more violent, and we desperately look for any changes that could have caused this increase in violence.

In fact, the violence rate was fairly constant until 1994, when it began dropping steadily [3]:

gallup-violent-crime-rate-graph

The public does not see it this way. According to the same Gallup poll [3]:

Despite a sharp decline in the United States’ violent crime rate since the mid-1990s, the majority of Americans continue to believe the nation’s crime problem is getting worse, as they have for most of the past decade. Currently, 68% say there is more crime in the U.S. than there was a year ago, 17% say less, and 8% volunteer that crime is unchanged.

Not as relevantly, but shockingly, even our long-term historical assessment is wrong. A poll was done in the UK on perceptions of violence [4]:

When I surveyed perceptions of violence in an Internet questionnaire, people guessed that 20th-century England was about 14 percent more violent than 14th-century England. In fact it was 95 percent less violent.

This flawed assumption significantly changes the way we approach the video games and violence discussion. Instead of asking, “What is responsible for the recent rise in crime rates?” and noting that video games exist now whereas they didn’t exist before and then drawing the facile conclusion, we should ask, “Do video games hold back an even greater decline in violence?”

3. The pro-video game side ignores the link between video games and aggression.

Just like ignoring crime statistics, one can also ignore psychological effects of violent video games.

In a meta-analysis of the psychological literature, Craig A. Anderson and Brad J. Bushman, violent video games were generally found to be associated with aggression [5].

One concern of violent video games is that violence is often rewarded. A study [6,7] shows a difference in player aggression between a game where violence is rewarded and one where violence is punished.

videogames-aggression

It would be nice if psychological results were not ignored by the pro-video game side. On the other hand, psychological results are often tenuous and likely to be wrong. So it would also be nice if the anti-video game side took these results with a bit more caution. After all, some studies are skeptical of the video game-aggression link [8,9].

Finally, even if we assume that violent video games definitely lead to increased aggression, this is one step removed from deducing that video games actually lead to violent crimes such as shootings.

4. Mechanisms are argued instead of statistics.

I wrote about this topic before in my blog post “Mechanisms vs Statistics,” which incidentally used video games and violence as the example.

The gist is, if you don’t use statistics or real evidence, then you can argue anything you want. If you are anti-video games, you could argue that gamers imitate the characters they play, hence they become more prone to going on shooting rampages. If you are pro-video games, you could argue that someone who otherwise would have committed a violent crime satisfied their aggression in video games instead of in real life, thus decreasing crime. Without data, it’s hard to say which of these stories is more correct, or correct at all. (And you could come up with dozens of such plausible-sounding stories for either side.)

Even with statistics, we have to make sure to interpret the data carefully. Being relaxed with statistics will lead us to believe the wrong things.

[1] http://videogames.procon.org/

[2] http://marketshadows.com/2013/04/23/dear-america-heres-why-everyone-thinks-you-have-a-problem-with-guns/

[3] http://www.gallup.com/poll/150464/americans-believe-crime-worsening.aspx

[4] http://stevenpinker.com/publications/better-angels-our-nature

[5] Anderson, C.A. & Bushman, B.J. (2001). Effects of Violent Video Games on Aggressive Behavior, Aggressive Cognition, Aggressive Affect, Physiological Arousal, and Prosocial Behavior: A Meta-Analytic Review of the Scientific Literature Psychological Science September 2001 12: 353-359.

Click to access VideoGames1.pdf

[6] Carnagey, N.L., & Anderson, C.A. (2005). The effects of reward and punishment in violent video games on aggressive affect, cognition, and behavior. Psychological Science, 16(11), 882-889. http://pss.sagepub.com/content/16/11/882.abstract

[7] http://scienceblogs.com/cognitivedaily/2005/11/23/punishing-video-game-violence/

[8] Williams, D. & Skoric, M. (2005). Internet fantasy violence: A test of aggression in an online game. Communication Monographs, 72, 217-233. 

Click to access CMWilliamsSkoric.pdf

[9] http://scienceblogs.com/cognitivedaily/2005/08/31/more-on-video-game-violence/

Can Geometry Be Racist?

I recently stumbled upon this article by the Daily Mail: “Why every world map you’re looking at is WRONG: Africa, China and India are distorted despite access to accurate satellite data.” The article’s main beef is with the Mercator projection, a map which you have definitely seen and which looks like this [from Wikipedia]: Mercator_projection Here is a political map in the style of Mercator [source, with watermark]: MercatorProjection-2 The point of the Mercator projection is to preserve straight lines and compass orientation (i.e. very useful for navigation). For example, Atlanta and Los Angeles have roughly the same latitude, and are separated by 2173 miles. However, if you go up straight north from both cities, the distance starts getting smaller and smaller, until it eventually reaches zero when you’re at the north pole. To account for this change in distance, the Mercator projection exaggerates areas that are far from the equator. Here’s a visualization of this distance getting smaller as you go further from the equator [source; just focus on the triangle on the globe; the picture was demonstrating non-euclidean geometry where the angles of a triangle don’t have to add up to 180 degrees]: triangle-globe So what’s the point? Now that the geometry lesson is out of the way, here is the point of the Daily Mail article. If you look at the Mercator maps, you’ll note that Greenland looks at least as big as Africa, when it is actually 14 times smaller (836,000 sq miles vs 11,670,000 sq miles). It also notes that the Scandinavian countries look bigger than India, when, in fact, India is 3 times larger. These are all great points. However, one statement sounds strange: “It gives the right shapes of countries but at the cost of distorting sizes in favour of the wealthy lands to the north.” Another statement is, “Much of this is due to technical reasons, said Mr Wan, while other inconsistencies are caused by ideological assumptions that can change the way we see the world.” “The wealthy lands to the north”? Ideological assumptions? I’m not sure if the author is just using these phrases sensationally, but there is an issue here. The Mercator projection is not racist or imperialist or north-ist. It is simply a geometric application. In fact, it is physically and mathematically impossible for a 2-dimensional map to accurately portray the globe.

The author even concedes this point in the article: “The biggest challenge is that it is impossible to portray the reality of the spherical world on a flat map – a problem that has haunted cartographers for centuries.” Then hasn’t the author figured out the solution to the title? “Why every world map you’re looking at is WRONG: Africa, China and India are distorted despite access to accurate satellite data.” Answer: because of geometry, it is, has always been, and always will be, impossible. The only way to look at the globe truly accurately is via… a globe. Or Google Earth. So what then is the point of the Mail article? A refresher course on history?

Anyways, it is an interesting topic to think about; I just thought the implied arguments were severely flawed. Namely,  the statement that “other inconsistencies are caused by ideological assumptions that can change the way we see the world” implies that perhaps the Mercator projection causes us to think more of “big” countries far from the equator, which happen to be richer, and less of “little” countries nearer the equator, which happen to be poorer. Also, the fact that this pseudo-explanation is even implied seems to weaken the real answer to the question, why every world map is wrong. Look at the phrasing again: “Why every world map you’re looking at is WRONG: Africa, China and India are distorted despite access to accurate satellite data.” The northern racism explanation (latitude-ism?) makes it seem like we can make accurate maps because of accurate satellite data but we don’t because we want to perpetuate northern superiority and oppress the southerners. (Of course, the Mercator projection equally distorts southern countries, but most of Earth’s landmass is in the Northern hemisphere.)

Thus, the article is extremely misleading and is another example of taking some of the views of postmodernism too far while simultaneously discounting mathematical knowledge. The objective facts—the impossibility of accurately representing a sphere on a plane—are right there and we even see them, but some of us just choose to ignore them. Also, the comment section of the Mail article seems to share this sentiment of critique. Plenty of factors contribute to racism, but geometry is not one of them.

A Single Cause For Everything

Our society loves to pin each problem on one cause. The most recent example is Elliot Rodger. Some say he was a misogynist (Huff Post) and others that he was mentally insane (TIME). Others blame the system instead, claiming that he was operating under a grander systematic male privilege (Salon) or that therapists and law enforcement are inadequate to detect signs mental illness (Slate). And here is yet another pair of conflicting reports in the misogyny (Washington Post) vs mental illness (National Review) debate. Despite the variety of voices in the debate, they all seem to agree on one thing: their reason is the only reason.

The title of the TIME article says blatantly, “Misogyny Didn’t Turn Elliot Rodger Into a Killer,” and the first sentence reads, “Yes, Elliot Rodger was a misogynist — but blaming a cultural hatred for women for his actions loses sight of the real reason why isolated, mentally ill young men turn to mass murder.”

Besides this acknowledgement, the articles all present evidence that furthers their own theories while not considering evidence that might support other theories. It’s very difficult to dig up an article that discusses, for instance, with nuance how much of it was caused by misogyny and how much by mental illness, or how the two factors behave in tandem. (Or whether there is a third factor: this article (Salon) talks about the role of race in Rodger’s motives.)

In case you’ve already made your mind on which side of the misogyny vs mental illness debate you fall on, here is a simpler, non-politically-charged example. Suppose we want a theory to predict where there is snow and where there isn’t snow. The first theory I’ll propose is the latitude theory: higher latitudes are colder and should thus have more snow (assuming we’re in the Northern hemisphere).  If this theory were completely true, the snow distribution might look something like this.

latitude-us-map

Everywhere north of the latitude line, there is snow, and everywhere south, there is no snow. Clearly this isn’t true.

Here is another theory: water proximity theory. Snow needs water to freeze, so snow will form near bodies of water. If this theory were completely true, then we should only find snow near water. Clearly this isn’t true either.

Here is an actual picture of snow cover from NASA:

snowcover

And here is an animated gif of world snow cover:

Earth-satellite-snow

As one can see, neither theory is true as an absolute statement. The correct way to think of these theories is as probabilistic theories. That is, the more north you go, the higher the chance you will encounter snow. The same goes for being near bodies of water, to a lesser extent. Even then, snow cover cannot be explained as a combination of these two factors alone: mountainous regions have more snowfall as well.

The debates in our current-day media are akin to one side saying that latitude determines everything and the other side that proximity to water determines everything. Neither side is willing to look rationally at the cold facts around them.

History is another subject where it is more clear that everything has multiple causes. In just less than two months from today, it will have been 100 years since the beginning of World War I. One might argue that the cause of WWI was the assassination of an archduke, but this simplistic explanation misses all the political tensions and alliances at the time. Similarly, one could argue that it was purely due to the political landscape and that war would have broken out regardless of the assassination. Both causes were necessary to an extent. If Franz Ferdinand had been assassinated in a less tense time, war might have been averted. Similarly, if no assassination had occurred, the great powers might not have had a proper excuse to actually go to war.

So why can’t we use scientific or historical reasoning on sociological issues?

Religion is a great example of this single-cause mentality. The honor killing of Pakistani woman Farzana Parveen last week was unanimously condemned in the US, similarly to the Elliot Rodger shooting. However, whenever someone tried to posit a cause that could have contributed to the honor killing, the other side would knock it down, saying it couldn’t be the right cause, and they give examples. For instance, if you go to the comment section of any major news story about this event, you’ll invariably find that someone criticizes Islam for condoning honor killings and promoting misogyny, and then someone else responds by pointing out that honor killings sometimes happen in other cultures (e.g. Hindu) as well.

Both sides make decent points but such conversations are useless since they are both saying true things but ignoring what the other side is saying. Just as “more north = more snow” is not always true, it is also not false. So sure, Islam might not be the only reason that honor killings occur so much in Pakistan, but it’s a pretty strong factor. Just because a cause is not the only cause does not mean that it is not a cause at all.

With religion in general, people very often make absurdly simplistic statements themselves and assume other people’s views of religion are absurdly simplistic (perhaps by projection). This might also be reflected in the general media and American culture as a whole. We love simple answers to complex problems. I’m not advocating that we personally conduct full academic research for every problem we face, but we are clearly too far on the simplistic side. The problem is that we’re thinking too little, not too much.

Elliot Rodger’s event, just like any other event, has a variety of causes. Both misogyny and ill-handling of mental illness are to blame. Snow cover depends on several conditions. World War I had a complex background, as do honor killings and suicide bombings.

Solutions to oversimplification of causes?

  • Prefer depth of news, not breadth. Instead of gaining a superficial understanding of many stories, try to understand one story really well. Read 10 different articles on Elliot Rodger and look at the issue from all sides.
  • Look at the statistics yourself. Numbers don’t oversimplify themselves.
  • Acquire more information. Have an opinion on Russia’s involvement with Ukraine? See if your opinion changes if you read up on past involvements.
  • Read the comments section of the article. While 90% of it may be trash, someone might point out something worthwhile.

Observer Selection

Today was my graduation from Cornell, but since I’m not a fan of ceremony, the topic for today is completely different: a subset of selection bias known as observer selection.

Selection bias in general is selecting particular data points out of a larger set to distort the data. For example, using the government’s own NOAA website (National Oceanic and Atmospheric Administration), I could point out that the average temperature in 1934 was 54.10 degrees Fahrenheit, while in 2008 it was 52.29. Clearly from these data points, the US must be cooling over time. The problem with the argument is, of course, that the two years 1934 and 2008 were chosen very carefully: 1934 was the hottest year in the earlier time period, and 2008 was the coolest year in recent times. Comparing these two points is quite meaningless, as the overall trend is up.

us_temperature_thru_2013Observer selection is when the selection bias comes from the fact that someone must exist in a particular setting to do the observation. For instance, we only know of one universe, and there is life in our universe—us. Could it have been possible that our universe had no life?

The issue with trying to answer this question is that if our universe indeed had no life, then we wouldn’t exist to witness that.

“The anthropic principle: given that we are observing the universe, the universe must have properties that support intelligent life. It addresses the question “Why is our universe suitable for life?” by noting that if our universe were not suitable for life, then we wouldn’t be here making that observation. That is, the alternative question, “Why is our universe not suitable for life,” cannot physically be asked. We must observe a universe compatible with intelligent life.”

the multiverse

The point is, there may be millions, billions, or even an infinite number of universes. But even if only one in a trillion were suitable for life, we must exist in one of those. So our universe is not “fine tuned” for life, but rather, our existence means we must be in a universe that supports us.

A list of observer effects:

  • The anthropic principle, as above. Our universe must be suitable for life.
  • A planet-oriented version of the anthropic principle: Earth has abundant natural resources, is in the habitable zone, has a strong magnetic field, etc.
  • A species-oriented version of the anthropic  principle: Our species is very well adapted to survive. If we weren’t, then we wouldn’t be thinking about this.
  • There are no recent catastrophic asteroid impacts (the last one being 65 million years ago). If there were, then we again wouldn’t be observing that.
  • The same goes for all natural disasters. No catastropic volcano eruptions, no nearby supernovae or black holes, etc.
  • The same goes for apocalyptic man-made disasters. Had the Cold War led to a nuclear exchange that wiped out humanity, we would not be able to observe a headline that said, “Nuclear Weapons Make Humans Extinct.” Thus, we must observe non-catastrophic events in the past.
  • Individual life follows this as well. Say you had a life-threatening illness or accident in the past, but you’re alive now (of course, given that you’re reading this). Given that you’re alive now, you must have survived it, so to the question, “Are you alive?,” you can only answer yes.

All of these are strong observer effects, in that they are absolute statements and not probabilistic ones, i.e. “Our universe must have life,” and not “Our universe probably has life.”

There are numerous other observer effects that are probabilistic but can be still very significant. For example, given that you are reading this, you are more likely in a literate country than in less literate one. Moreover, the probability would be higher than that if I did not know anything about you.

In this post, I mentioned the example of democracy in political science. In summary, political science has a lot more to say on democracy than on any other form of government. Is this because we are personally biased towards democracy? Not necessarily. In a less open system, fields like political science might be forbidden from research (or academia is rated less important), and hence there are no (or few) pro-totalitarian political scientists. Hence, we end up seeming to favor democracy.

We also know that history is written by the victors. But a related historical example is the rise of strong states combined with the rise of liberalism  and progressive thoughts in the Modern era. Namely, states in which liberalism arose (England, France) tended to be strong states. A weak state adopting progressive measures would be wiped out by a stronger one. Hence, history is also analyzed by the victors.

So what can you do about observer selection? All we can do is try to be aware of it and introduce corrections to study a full set of possibilities rather than the subset we are in by being a particular observer. For instance, if we were just using historical data of natural disasters, we would be underestimating the actual probability of a catastrophic disaster, as we live in a time where none could have occurred for a while.

Mechanisms vs Statistics

Last semester, our apartment had a debate over whether video games cause violence. It came down to arguing logical mechanisms, but without any use of statistics by either side. The argument basically turned into my word vs your word, since there was no objective basis on which to judge anything.

If your answer were yes, you might propose the mechanism: “People who play violent video games are likely to imitate the characters they play, thus becoming more aggressive in real life.” This statement might be logically sound, but without any supporting evidence, it has little credence.

You could easily propose a counter-mechanism: “People who would otherwise commit violent crimes satisfy their urges in video games and not in real life, thus decreasing the crime rate.” Again, this seems plausible, but without any data, we simply don’t know whether this effect outweighs the other. We need real stats.

Naively looking at statistics does not help either. Depending on which stats you look at and how they are presented, the conclusions can go either way (graph 1 and graph 2):

video-games-crimes

video-game-walsh

In any subject, one important concern is matching theories with empirical data. In the hard sciences, one tests the theory by experiment, and it is often possible to verify or deny claims with empirical data. But in the social sciences, experiments are sometimes impossible. To see what would happen if Germany had won World War II, we cannot simply recreate the circumstances of the war in a petri dish. So we must do the best we can with the limited data we have.

This lack of statistics affects many other issues, perhaps more important ones. For instance, in the public debate over gun control, there are clearly two competing mechanisms: “More guns = more shootings” and “More guns = more protection.” Each makes logical sense on its own, but the way to figure out the more accurate one is not by purely logical argumentation (which will lead nowhere), but by use of statistics, i.e. show the real effects of implementing or not implementing gun control laws. This would be much more fruitful than mindlessly yelling mechanisms across the void.

Statistics in the Social Sciences

I’ve always wondered whether the rigorous application of statistics is underutilized in the social sciences. This is less so a problem in economics, where the subject is, by nature, highly quantitative. But in fields like psychology, sociology, and political science, where a background in mathematics is not common (unlike for biology, chemistry, and physics), researchers can intentionally or, very often, unintentionally (this is a really good Economist article) produce wrong results by abuse or misunderstanding of statistical inference.

economist-research-statistics

As an onlooker whose training is in mathematics, I cannot help but to feel frustrated by the lack of numeracy in our “scientists.” The Economist article does a good job at showing how failure to understand statistical concepts leads to false results being published, even past peer review.

What triggered me to write this post was an assigned reading for a comparative politics class. In it, Adam Przeworski discusses the inherent selection bias in matching countries for experimentation. Noting that democracies have higher economic growth rates than authoritarian regimes, Przeworksi brings in the relevant data that democracies have a significant chance to die off when faced with economic failure whereas authoritarian regimes are not as affected. Hence, observing that democracies have higher growth rates does not signify that democracy leads to economic growth, but rather that economically failing democracies are not observed because they tend to disappear.

“What we are observing here is what the statistical literature calls ‘selection bias.’ Indeed, I am persuaded that all the comparative work we have been doing may suffer potentially from selection bias.”  (p. 19, stable JSTOR link)

In context of a comparative politics theory symposium, this makes a lot of sense to state. But the phrasing is really interesting to a math person: selection bias is a given, and is one of the tools we use to analyze anything. My instinctual reaction to the reading was “Duh, obviously there is selection bias.” While I am sure the field of comparative politics is more aware of selection bias than Przeworski makes it appear to be, the fact that Przeworski framed it as such (“what the statistical literature calls ‘selection bias'”), as if to imply that the formal tools of statistical inference are generally beyond the scope of comparative politics theory, is a bit unnerving.

Przeworski, Adam in The Role of Theory in Comparative Politics: A Symposium, World Politics, Vol. 48, No. 1 (Oct., 1995), pp. 1-49.

10 Surprising Mathematical Facts

Since “10 Mind Blowing Mathematical Equations” is one of my most popular articles, I decided to write another math list.

Plenty of things in math are downright uninteresting. Who cares that the area of a circle is πr², or that a negative times a negative is a positive? Why should this interest us at all? Perhaps the answer can be found in the most unexpected results, the counterintuitive facts that have sometimes eluded even the best mathematicians.

1. Birthday Paradox

Birthday_Paradox

The birthday paradox says that if there are 23 people in a room, there is a more than 50% chance that two people have the same birthday. It seems counterintuitive because the probability of having a birthday on any particular day is only 1/365.

But the difference relies on the fact that we only need two people to have the same birthday as each other. If, instead, the game was to get someone with a birthday on a particular day, such as March 14, then with 23 people, there is only a 6.12% chance that someone will have that birthday.

In other words, if there are 23 people in a room, and you choose one person X, and ask, “Does anyone else have the same birthday as X,” the answer will probably be no. But then repeating this on the other 22 people increases the probability every time, resulting in a net probability of more than 50% (50.7% to be more precise).

2. Mandelbrot Set (Looks Like This)

Mandelbrot Set

The Mandelbrot set is a set of complex numbers that, when iterated according to a certain formula, do not escape to infinity. Based on the simplicity of the formula itself, which is z -> z² + c, you would not expect such a complex figure to arise.

When you zoom in on the Mandelbrot set, you get an infinite number of smaller Mandelbrot sets, which in turn have infinitely more… (This kind of behavior is typical among fractals.)

Mandelbrot Zoom

It really captures the idea of worlds within worlds, universes within universes. Here is a video of a zoom (among many on YouTube). I think it’s absolutely mind blowing.

If you still don’t think theoretical math is awesome after seeing that video, I don’t know what to say.

3. Banach-Tarski Paradox

Banach-Tarski_Paradox_Wiki

The Banach-Tarski paradox says that you can split one shape into two perfect copies of itself. More specifically, it says that given a solid ball in 3-dimensions, it is possible to break it into a finite number of pieces and then arrange them back into two identical copies of the original ball.

Of course, it’s highly counterintuitive, and it’s considered by many to be the single most paradoxical result of mathematics. After all, in real life, we never see one object suddenly turning into two copies. In fact, it seems to defy the conservation of mass in physics, which says that mass should be preserved; shouldn’t the result, with two objects, have twice the mass of the original?

Well, not if the original mass was infinity. Then doubling infinity is still infinity, so there is technically no breaking of laws. For a layman explanation of the Banach-Tarski paradox, see this article I wrote in 2010.

4. Monty Hall Problem

Monty_hall_door

This infamous problem is stated as follows:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

No one I know has gotten the correct answer on the first try. Surprisingly, the answer is that it’s better to switch!

Rather than trying to explain the details of the problem here, I will refer you to the Wikipedia article, which does a very good job at exposition. The story is pretty funny too:

Many readers of vos Savant’s column refused to believe switching is beneficial despite her explanation. After the problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine, most of them claiming vos Savant was wrong (Tierney 1991). Even when given explanations, simulations, and formal mathematical proofs, many people still do not accept that switching is the best strategy (vos Savant 1991a). Paul Erdős, one of the most prolific mathematicians in history, remained unconvinced until he was shown a computer simulation confirming the predicted result (Vazsonyi 1999).

The lesson is, don’t trust your intuition.

5. Gabriel’s Horn and the Painter’s Paradox

GabrielHorn

Familiar perhaps to calculus students, Gabriel’s horn is a shape that has a finite volume but an infinite surface area (both are straightforward to check with integral calculus).

A popular way to make this into a real-world problem is to imagine painting the shape. The painter’s paradox states that it is possible to completely fill the horn with paint (finite volume), but it is impossible to completely paint the horn’s inside (infinite surface area).

Koch_Snowflake

The Koch snowflake is a shape, along similar lines, that has finite area but an infinite perimeter. In fact, the Mandelbrot set, from #2, also has finite area and infinite perimeter!

6. Basel Problem

\displaystyle 1 + \frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\cdots =\frac{\pi^2}{6}

The only item to appear both in the 10 equations list and in this list, the Basel Problem says that if you take the reciprocal of all the square numbers, and then add them all together, you get pi squared over six.

If you’re a normal, sane human being, it was probably completely unexpected that the stuff on the left side has anything to do with pi, the ratio of a circle’s circumference to its diameter.

7. Abel’s Impossibility Theorem

\displaystyle x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}

Most of you in high school have seen the quadratic equation, which tells you how to solve the degree 2 polynomial equation ax² + bx + c = 0.

But the story doesn’t end there. In the 1500s, mathematicians solved the cubic equation (degree 3), which is just one step up: ax³ + bx² + cx + d = 0. The corresponding solution is far more complicated:

Cubic_Formula

Thank heavens you didn’t have to learn that in high school. But let’s go one step further. How do you solve a quartic equation (degree 4): ax⁴ + bx³ + cx² + dx + e = 0? At this point, the formula is absolutely ridiculous:

Quartic_Formula

I dare you to click on that and scroll through the whole thing.

Now breathe a sigh of relief, because I’m not going to show you the formula for the next step up, the quintic equation (degree 5), ax⁵ + bx⁴ + cx³ + dx² + ex + f = 0, because it doesn’t exist! It’s not that we haven’t found it yet; we actually proved it’s impossible! In fact, for any polynomial with degree 5 or higher, there is no solution in roots.

8. There Are Different Levels of Infinity

Ordinal Numbers Omega Exp Omega

Yes, some infinities are bigger than others. Technically, infinities have a property called cardinality, and an infinity with a higher cardinality than that of another infinity is the larger one. (Regular numbers have cardinalities too, but the cardinality of an infinity is always higher than that of a mere number.)

There are still many counterintuitive facts about cardinalities of infinity. For example, are there more integers than even integers? You would think that there are, since you’re missing all the odd integers. But the answer is no, they have the same cardinality. Are there more fractions than integers? Nope, there are just as many integers are there are fractions.

However, Georg Cantor showed that there are actually more real numbers than there are fractions. The real numbers are often referred to as the continuum, and for a long time, it was conjectured, but not known, that there is no level of infinity between integers and the continuum; this conjecture became known as the continuum hypothesis.

It turns out that the continuum hypothesis is neither true nor false in the normal sense. It was proved that it can be neither proved nor disproved. (Read that sentence again.) More precisely, Paul Cohen proved that the continuum hypothesis is independent of ZFC, the standard set of axioms for mathematics.

9. Gödel’s Incompleteness Theorem(s)

true-false-provable-refutable

Basically, it was proved that some true things cannot be proved. There are various layman formulations of this result, and I’ll list a couple here:

  • Any sufficiently powerful system has statements which can neither be proved nor disproved. (E.g, continuum hypothesis.)
  • Any sufficiently powerful system cannot prove itself to be consistent, even if it is consistent.

These became known as Gödel’s incompleteness theorems. Not surprisingly, these had huge implications in not just math but also philosophy.

10. Fermat’s Last Theorem

pythagoras-3-4-5

The Pythagorean theorem says that in a right triangle, a² + b² = c². Now suppose we force the variables to be integers. So the solution a=3, b=4, c=5 is allowed, but a=1.5, b=2, c=2.5 is not allowed, even though it fits the equation. It can be shown that there are an infinite number of solutions with a, b, c all integers.

But what happens if we take this one step up? How many integer solutions are there to a³ + b³ = c³? The answer is none. The same happens with a⁴ + b⁴ = c⁴: no solutions.

\displaystyle a^n + b^n = c^n

In fact, Fermat’s Last Theorem states that for any exponent higher than 2, this equation has no integer solutions. This famous problem, conjectured in 1637, took nearly four centuries to solve, being proved finally by Andrew Wiles in 1995.