# Making Use of the Armchair: The Rise of the Non-Expert

As with all news, when I heard about the Sochi skating controversy last week, I read multiple sources on it and let it simmer. From the comments, however, that I saw on Facebook, Reddit, and on the news websites themselves, one thing struck me—nearly everyone seemed to be have extensive knowledge of Olympic figure skating, from the names of the spins to the exact scoring rubric.

How could this be? Was I the only person who had no idea who Yuna Kim was, or that Russia had not won in the category before?

Much of this “everyone is an expert” phenomenon is explained by selection bias, in that those with more knowledge of skating were more likely to comment in the first place; therefore, most of the comments that we see are from those who are the most knowledgeable.

But it’s unlikely that there would be hundreds of figure skating experts all commenting on at once. Moreover, when you look at the commenting history of the people in the discussion, they seem to also be experts on every other subject, not just in figure skating. So another effect is in play.

Namely, the Wikipedia effect (courtesy of xkcd):

Of course, this effect is not limited to skating in the Olympics. When Newtown occurred, masses of people were able to rattle off stats on gun deaths and recount the global history of gun violence in late 20th- and early 21st-century.

Even so, not everyone does their research. There are still the “where iz ukrane????” comments, but undoubtedly the average knowledge of Ukrainian politics in the United States has increased drastically in the past few days. If you polled Americans on the capital of Ukraine, many more would be able to answer “Kiev” today than one week prior. For every conceivable subject, the Internet has allowed us all to become non-expert experts.

Non-Expert Knowledge

The consequences of non-expert knowledge range from subject to subject. The main issue is that we all start with an intuition about something, but with experience or training comes a better intuition that can correct naive errors and uncover counterintuitive truths.

• An armchair doctor might know a few bits of genuine medical practice, but might also throw in superstitious remedies into the mix and possibly harm the patient more than helping. Or they might google the symptoms but come up with the wrong diagnosis and a useless or damaging prescription.
• Armchair psychologists are more common, and it is easier to make up things that sound legitimate in this field. It is possible that an armchair psychiatrist will help a patient, even if due to empathy and not from psychiatric training.
• Armchair economist. Might say some insightful things about one trend that they read about in the economy, but could completely miss other trends that any grad student would see.
• Armchair physicist. Might profess to have discovered a perpetual motion machine, to be dismissed by a real physicist because the machine actually has positive energy input and is hence not perpetual. Or, might read about the latest invisibility cloak and be able to impress friends by talking about the bending of electromagnetic waves around an object by using materials with negative refractive index, but has no idea that it only works for a particular wavelength, thus making it practically useless (for now).
• Armchair philosopher. Perhaps the most common, the armchair philosopher notices the things that happen in life and takes note of them. The article that you are currently reading is armchair philosophy, as I basically talk about abstract stuff using almost zero cited sources, occasionally referencing real-world events but only to further an abstract discussion.

Going back to the physics example, we normal people might observe the drinking bird working continuously for hours and conclude that it is a perpetual motion machine. An armchair physicist might go further to claim that that if we attach a motor to it, we could generate free energy.

A real physicist, however, would eventually figure out the evaporation and temperature differential, and then conclude that it is not a perpetual motion machine.

Five minutes of reading Wikipedia will not allow you to match an expert’s knowledge. But having non-expert knowledge sometimes does help. It opens up the door to new information and ideas. If everyone spoke only about what they were experts in, the world would become boring very quickly.

In everyday speech, any topic is fair game except for, ironically, the one topic that everyone is deemed to be an expert in even without Wikipedia—(their) religion. But I digress. The point is, the way we talk about things on a day-to-day basis is very different from the way experts talk about them in a serious setting.

Some differences are very minor and just a matter of terminology. For instance, I was discussing the statistics of voter turnout in the 2012 election one time, and I had phrased it as “percentage of eligible people who voted.” At the time, I did not know that “turnout” was a technical term that meant precisely what I had just said; I thought it was just a loose term in that didn’t necessarily consider the difference between the electorate and the total population, hence why I phrased it so specifically. In this example, the statistics I presented were correct, and thus the conclusion was valid, but the terminology was off.

Other differences are more significant. In the case of medical practice, a lack of formal understanding could seriously affect someone’s health. Using Wikipedia knowledge from your smartphone to treat an unexpected snake bite in real time is probably better than letting it fester before help arrives. But it’s probably safest to see a doctor afterwards.

A non-expert discussion in a casual setting is fine, as is an expert discussion in a serious setting. But what about a non-expert discussion in a serious setting? Is there anything to be gained? If two non-physicists talk about physics, can any meaning be found?

My answer is yes, but you need to discuss the right things. For example, my training is in math, so it would be pretty futile for me to discuss chemical reactions that occur from the injection of snake venom into the human body. However, given that I had done my research properly, I might be able to talk about the statistics of snake bites with as much authority as a snake expert. Of course, it would depend on the context of my bringing up the statistics. If we were comparing the rise in snake deaths to the rise in automobile deaths, I might be on equal footing. But if we were comparing snake bite deaths between difference species of snakes, a snake expert probably has the intellectual high ground.

But even this example still requires you to use some area of expertise to relate it to the one in question. To the contrary, you can still have a legitimate discussion of something outside your area of expertise even without relating to an area of expertise that you already have. You only need to make a claim broad enough, abstract enough, or convincingly enough to have an effect.

Among all groups of people, writers (and artists in general) have a unique position in being able to say things with intellectual authority as non-experts. Politicians are next, being able to say anything with political power as non-experts. However, I’m interested in the truth and not what politicians say, so let’s get back to writers. F. Scott Fitzgerald was not a formal historian of the 1920s, but The Great Gatsby really captures the decade in a way no history textbook could. George Orwell was not a political scientist, but Nineteen Eighty-Four was very effective at convincing people that totalitarian control is something to protect against.

The Internet and the Non-Expert

On the other hand, Nineteen Eighty-Four was not crafted in a medium limited by 140 characters or by one-paragraph expectancy. If George Orwell were alive today and, instead of writing Nineteen Eighty-Four, wrote a two-sentence anti-totalitarian comment on a news story on North Korea, I doubt he would have the same effect.

It is usually hard to distinguish an expert from a non-expert online. Often, an expert prefaces oneself by explicitly saying, “I am an expert on [this topic],” but even this is to be taken skeptically. I could give a rant on the times people claiming to have a Ph.D in economics had no grasp on even the most basic concepts.

In addition to allowing us the sum total of human knowledge just a click away (well, maybe not all knowledge), the Internet allows us to post knowledge instantaneously and share it with millions of other users. We have not only the public appearance of non-expert knowledge, but also the virus-like proliferation of it. Since the dawn of the Internet, people have been able to acquire knowledge about anything, but there was a great divide between the few content providers and the many consumers. Only recently have we become the content makers ourselves. What is the role of armchair philosophy in the age of information?

Conclusion

Now is a more important time than ever to be an armchair philosopher, or an armchair thinker, precisely because of the overwhelming amount of information available to us. To deal with the data overload requires an abstract way to categorize information, to filter out the useless from the useful, the wrong from the less wrong, the less true from the true.

We are expected to deal with areas outside of our expertise, and as our knowledge of these areas grows from the age of mass information, our responsibility to use it correctly becomes greater. Forming opinions even on issues that you have no authority to form opinions on is now an imperative. We learned the capital of Ukraine in one week, and our googling of Kiev might prove useful in the future. To deal with a quickly changing world, we need to deal with all information, not just data that we are comfortable with, as effectively as possible.

# Why Math?

As a math major, I am often asked the question, “Why math?” In particular, why theoretical math, when it doesn’t seem to be related to anything?

I often have trouble coming up with a full and satisfying answer on the spot. Math is one of the subjects whose material and categorization can be confusing. It spans several fields that many have not even heard of. When I say “topology” or “analytic number theory,” it often draws blank stares; in fact, topology is often misunderstood as “topography,” the study of terrain.

Well, here is my more thought-out answer to why I study math.

Is It Relevant?

Take a good look at the following equation:

$\displaystyle \sum_{n} \frac{1}{n^s} = \prod_{p} {\frac{1}{1 - \frac{1}{p^s}}}$

Through the cryptic jumble of symbols, you might ask yourself, how is this useful?

An even more relevant question for most readers is, what does it even mean?

As it turns out, this particular equation has very little practical value. Yet it is one of the most fundamental equations in the field of analytic number theory, and it is a remarkable statement about the prime numbers.

It basically links the infinite set of natural numbers (1, 2, 3, 4, 5,…) with the infinite set of prime numbers (2, 3, 5, 7, 11,…). The proof is relatively simple, but I am not going to give it here. It is known as the Euler product formula.

There is virtually no “useful” information given by this highly abstract formula. It doesn’t help with daily finance. It doesn’t solve traffic congestion. It doesn’t even help in landing a rover on Mars. But it does is provide us with an insight into the fundamental truth of nature. In a way, this equation exceeds the known universe, as according to current theory, the universe is finite. The equation, by contrast, deals with the infinite.

In fact, modern mathematics is full of statements and theorems that have currently no practical use. There are entire branches and fields of study that are, in essence, useless. Sometimes, useless things have applications in the far future. Complex numbers, for example, were invented centuries ago, but didn’t really find any use until modern electronics and physics were developed.

Maybe everything we know today in math will be applied somehow. But this cannot happen forever. The universe is finite, after all, and knowledge is infinite. Sooner or later, or perhaps even now, we will have found knowledge that serves no use in our universe. This leads to the next question.

Is Knowledge Worth Seeking?

Should we seek knowledge for the sake of knowledge?

Is a culture with more knowledge inherently richer than one without?

Historically, knowledge in the form of technology had the power to save oneself, one’s family, and even one’s country. Entire civilizations were wiped out due to the technological superiority of the invaders. Knowledge has for a long time acted as a defense tool.

So perhaps we should embrace new knowledge for the sake of defending against a future alien force. But what about afterwards?

Assuming humans survive long enough to establish a galactic presence, and have enough technology to be virtually indestructible as a species, so that survivability is no longer an issue, what will be the point of further knowledge? What will be the point of knowledge for the sake of knowledge?

That picture above is the Mandelbrot set, a fractal generated by the fairly simple quadratic function

$z \mapsto z^2 + c$

where $c$ is a complex number.

There could easily be no purpose to this fractal, yet it certainly holds some value. It is aesthetically pleasing, and the ability to zoom in on the image forever raises some old philosophical questions. In this sense, it is almost like art, only the rules are completely different.

In essence, knowledge for the sake of knowledge is what math is all about. There is no intrinsic need for math to apply to the real world, nor does any topic in mathematics need an analogy in real life. Math is knowledge at the abstract level.

Recently someone asked me what classes I was taking, and when I mentioned topology, he asked if that was a map making course. Topology and topography sound quite similar, I suppose.

In any case, topology is a great example of what pure math is about. It is the underlying foundation behind geometry. Geometry is highly applicable in real life, because shapes, sizes, and angles of things all affect the way they work. But in topology, sizes and angles do not matter. A line is the same thing as a curve, a square is really the same thing as a triangle or a hexagon, and a sphere is really the same thing as a cube or amoeba.

And a donut is really the same thing as a coffee mug.

These fields of math are totally alien to the math taught at the pre-college level. Geometry, basic algebra, and calculus are about sizes of things and comparing objects to determine their shapes, lengths, volumes, etc.

But when you get to the higher fields, such as analysis, number theory, abstract algebra, and topology, everything completely changes. They feel like entirely different subjects than the math taught in middle school and high school.

Previously, you were told that dividing by zero is impossible and that it is pointless to think of infinity. But in complex analysis, you can actually “cancel out” zeroes and infinities provided certain properties are counted, and you actually care about where functions hit infinity and how often they do so. And in set theory, you discover that there are actually different sizes of infinity. These facts are much more interesting than, say, the quadratic equation, which is taught in every high school algebra course.

The fact that zeros can actually cancel out infinities, or that there are different sizes of infinities, is much more interesting than such a formula.

This graph, showing a region of the gamma function, generalizes the notion of factorial (i.e., 5! = 5 * 4 * 3 * 2 * 1) to complex numbers.

The gamma function is also closely related to the equation at the very top of the page, with the natural numbers on one side and the prime numbers on the other. Those two expressions also define the Riemann zeta function.

You might be able to see some relation between the two images. It turns out that the trivial zeroes of the zeta function, which can be seen as the strange color mismatches on a line going from the center to the left, are the result of the poles of the gamma function, which are the vertical spikes in the other picture.

Basically, that is why I study math. The point is not to memorize formulas or to calculate quickly. It is to discover fundamental truths out of ridiculous-sounding things, and to make sense out of them. In a way, this is what people do in other academic fields as well. Sometimes math goes over the top and seems completely useless. This is bound to happen. But some things, like art and mathematics, don’t need a practical purpose to exist. Such things are valuable in their own right.