Why Are College Students Not Choosing Math/Science?

microscope

From the Wall Street Journal in 2011:

Although the number of college graduates increased about 29% between 2001 and 2009, the number graduating with engineering degrees only increased 19%, according to the most recent statistics from the U.S. Dept. of Education. The number with computer and information-sciences degrees decreased 14%

After coming up with the topic for the post, I found this article from 2011 with a similar title and citing the same WSJ story. It argued that the high school teaching environment was not adequate in preparing students for rigorous classes in college. 

In addition, the article includes the argument that in the math and sciences, answers are plain right or wrong, unlike in the humanities and social sciences.

I can agree with these two points, but I want to add a few more, with the perspective of year 2013. Also, I am going to narrow down the STEM group a bit more, to just include math and science. The main reason is that in the past years, the number of CS majors has actually increased rapidly. At Cornell, engineering classes can be massive and there does not seem to be a shortage of engineers. Walk into a non-introductory or non-engineering-oriented math class, however, and you can often count the number of students with your fingers. So even though STEM as a whole is in a non-optimal situation, engineering and technology (especially computer science) seem to be doing fine. So then the question remains.

Why Is America Leaving Math and Science Behind?

I mean this especially with regards to theoretical aspects of math and science, including academia and research.

In this situation, money is probably a big factor. The salary of a post-grad scientist (from one article at $37,000 to $45,000) is pitiful compared to that in industry (which can a median early-career salary of up to $95,000, depending on the subject, according to the same article). Essentially there is a lack of a tangible goal.

There are other factors besides money. Modern math and science can be quite intimidating. All major results that could be “easily” discovered have already been discovered. In modern theoretical physics, for instance, the only questions that remain are in the very large or the very small—there is little left to discover of “tabletop” physics, the physics that operates at our scale. Most remaining tasks are not problems in physics, but puzzles in engineering.

Modern mathematics is very similar. While there are many open questions in many fields, the important ones are highly abstract. Even stating a problem takes a tremendous amount of explanation. That is, it takes a long time to convey to someone what exactly it is you are trying to figure out. The math and science taught in high school is tremendously unhelpful in preparing someone to actually figure out new math and science, and it is thus difficult for an entering college student to adjust their views of what math/science are.

Even the reasons for going to college have changed. More than ever, students list their top reason for going to college as getting better job prospects rather than for personal or intellectual growth.

In addition, society seems more than before focused on immediate gain rather than long term investment. Academia’s contribution to society, especially in math and science, is often not felt until decades or even centuries after something was invented. Einstein’s theories of relativity had no practical application when he made them, but our gadgets now use relativity all the time. Classical Greece knew about prime numbers, but prime numbers were not useful until modern-age data encryption was required. Even a prolific academic could receive very little recognition in one’s own life.

However, with the rise of online social networks in the last several years, you can now see what your friends are up to and what they are accomplishing in real-time. This should at least have some psychological effect on pushing people towards a career where real, meaningful progress can be tracked in real-time. Doing something that will only possibly have an impact decades later seems to be the same as doing nothing.

Considering the sentiment of the last few paragraphs, it might sound like I am talking about the decline in humanities and liberal arts majors. Indeed, while the number of math and science majors is increasing (though not as much as in engineering/technology), it almost seems like the theoretical sides of math and science are closer in spirit to the humanities and liberal arts than they are to STEM. The point is not for immediate application of knowledge, but to make contributions to the overall human pool of knowledge, to make this knowledge available to future generations.

Is this just a consequence the decline of education or the fall of academia in general? STEM is not really education in the traditional sense. It is more like technical training.

In all, the decline of interest in theoretical math/science is closely correlated with the decline of interest in the humanities/liberal arts. Our culture is fundamentally changing to one that values practicality far more than discovery. (For instance, when is NASA going to land a human on Mars? 2037. JFK might have had a different opinion.) Overall this is a good change, mainly in the sense of re-adjusting the educational demographics of the workforce to keep America relevant in the global economy. But, we should still hold some value to theory and discovery.

Additional resources:

  • National Science Foundation statistics – [link]
  • National Center for Education Statistics – [link]
  • Pew social trends – [link]

Stress and GPA-centrism in College

studying

Every time papers, projects, and prelims come around, the campus stress level rises dramatically. Sleep is lost (or outright skipped), meals are avoided, and all activities other than studying are brushed off. This happens not once a semester but throughout, corresponding to large assignments for every class.

And every time this happens, it seems that many students are focused not on actually learning the content, but on scoring higher grades than others. Of course, this phenomenon occurs in certain majors (engineering) much more than others. And I would guess that it happens at Cornell at an above average rate compared to that of the typical university. But it raises some questions that I want to explore.

Just a couple of notes. First, this article will mainly focus on the math/science/engineering side. And second, I do not think I need to mention why Cornell should be concerned about student stress.

Should Competition Be GPA-Focused?

Competition to a certain degree is beneficial, and I think no one would argue with that. As a math major I know very well that competition leads to efficiency. But there is a line where the marginal benefit in efficiency is not worth the huge increase in stress levels, and in this respect I think Cornell has crossed the line for good.

In addition to the GPA competition, there is the additional factor that students are competing for jobs, internships, and research positions. I think the competition here is mostly fine (except regarding salaries vs societal contribution; this topic deserves its own post). Combined with academic competition, however, this induces a vast amount of anxiety and stress in the students.

Without mentioning names, I will list some of the extreme behaviors I have observed of people I know:

  • A student pulled multiple all-nighters in a row to finish a project. While this might be plausible in real life for a rare occasion such as a doctoral thesis or a billion dollar merger, the student did this regularly for his classes.
  • A student has problem sets due on four out of five of the weekdays, and spends literally all his time outside of class eating, sleeping, or doing problem sets. In one particular class, the problem sets he hands in are 10-20 pages per week.
  • A student took 50+ credits in one semester, though he claims to know of someone who took 61.
  • A student brought a sleeping bag, refrigerator, and energy drinks to one of the school computer labs, and pretty much lives there, returning to his living place once every few days to shower.

Interestingly enough, I think these particular students will do fine, as they seem to know their own abilities and limits, and more importantly, they are all aware of what they are doing. They are also all very smart people who can actually learn the material. This “top tier” of students is not really adversely affected by competition, since they are smart enough to excel regardless of whether competition exists. Moreover, these students don’t seem to be grade-focused: they learn the material, and the grade comes as a byproduct of learning.

The group I am actually concerned about is the second tier. (Note: I just realized after typing this how judgmental that statement sounds, but hey, from statistics, unless you define the first tier to include everyone, there must be a second tier.) This group I would define to be the smart people who don’t seem to understand how the first tier operates. They see the students in the first tier getting high grades and know those students are smart, so they think that if they prepare the tests well and get high grades, they will become as smart.

What they don’t realize is the difference in cause and effect. The first tier prioritizes understanding first and the grades come as a byproduct, whereas the second tier prioritizes grades first and hope to gain some understanding as a byproduct.

Again, just as a disclaimer, these are just arbitrary definitions for first and second tier I made up for this particular observation. I am not saying that this criterion is the final say, and of course, there are numerous other factors regarding how well one does in college.

But from my experience, it is precisely the students in this second tier who are stressed. They are the ones trying so desperately to beat the test instead of to learn the material. And they are the ones who make college seem so competitive, as you can always hear them talking about tests and what their friends got on the tests and how they are being graded and what the format of the test will be.

On the other hand, the student I mentioned above who lives de facto in one of Cornell’s computer labs—I have not once heard him talk about anything specifically regarding a test.  The closest was talking about the material that was to be covered, but he was talking about stuff that was beyond what the class taught for the topic, stuff that he knew was not going to be on the test.

Some of you might be thinking, “That’s great, but what kind of job is he going to get if he is not grade-focused?” Good question. After working there for a summer internship, this same friend rejected a return offer from Goldman Sachs.

How to Break from GPA-centrism

I am not worried about this person’s career at all. I am worried about the second tier, the smart people doing mundane tasks, wasting a lot of potential creative brainpower that the world needs more than ever. Renewable energy, bioengineering, artificial intelligence, space exploration and colonization, nanotechnology—there are so many people here who would be excellent for these fields, yet many of them seem too bogged down by current competition-induced stress to even think into the future.

Indeed, this GPA competition is a force to be reckoned with, as it really is self perpetuating. Those students in other groups or who are apathetic to grades will tend to become more grade focused just from sitting in lecture, as there are always people who ask for as much details of the test as possible. I feel that this defeats the purpose of a test, which is to measure how well you know the material or how well you can apply a certain skill, not how much of the test structure you can memorize or how much content can you cram the night before.

Overall though, there are some measures that can be taken to reduce this kind of stress.

  • Reduce the importance of the GPA. I do not know if I would go so far as to remove it, though. For example, at Brown University, the GPA is not calculated. Somewhere in the middle ground should be best.
  • Stop showing score distributions, or show them only for major tests like a midterm/final. In many engineering classes at Cornell, the first thing that is requested after a test is graded is to see the score distribution (often in graph form), along with the mean, median, standard deviation, etc. In fact, this has become so common that it is now the first thing the professors put on their lecture slides. Moreover, the computer science department uses an online course management system which automatically tells students the mean, median, standard deviation, etc. for every single assignment, not just tests. Being a math major who would normally love extra statistics, I thought this was cool at first. But now I despise it—it is just too much information that I don’t need in order to learn the material, and it only detracts from my learning experience. And the way the page is setup, it is not something you can just ignore.
    • A side note: One of my classes in the CS department actually does not list grades, and I definitely feel more pressured to actually learn in that class, not more pressured to beat other people at grades like in other CS classes. Props to Professor John Hopcroft.
  • Teach better math/science/engineering/CS much earlier in the education system. A friend showed me this article today, a comparison between the CS education systems of the US and Vietnam, a comparison that is horrific for the US. If students already knew the foundations, then college would be what it was supposed to be: going really deep into a topic in a learning atmosphere, not treating us like elementary school children because, well, frankly that’s the level of engineering/CS of many college entrants. For instance, I think it’s great that students are trying to take Calculus 1 freshman year and then do engineering. Hard work is certainly a virtue. But wouldn’t it be much better for both the student and the college if they mastered calculus in high school? Imagine how much better our engineers would be.

What I envision is a class where students are trying to learn, not to beat each other on a test. I hope this vision is not too far-fetched.

College and Smartphones

Photo from Newegg.
Image from Newegg.

Last December I obtained a Samsung Galaxy S3, my first ever smartphone. Yes, I’m only about 6 years late to the smartphone party. Before this, I had been using a Motorola Razr flip phone for years and didn’t really think a smartphone was necessary. But after just three months, it is already difficult to imagine not having one.

Smartphones in College Life

According to various reports I found, somewhere between 50-70% of college students have a smartphone. But statistically, at a school like Cornell, whose students come from families that more affluent than average, it would be reasonable to assume the percentage should be much higher. In fact, almost everyone I know here has a smartphone.

According to one source, the percentage of all college students who had a smartphone in 2009 was about 27%. Another source claims that the score in 2008 was 10%. Yet at Cornell, the figure was already 33% by 2008. I think it would not be unreasonable to estimate that Cornell’s smartphone usage is about a year (and a bit more) ahead of the average trend, and I would bet that currently between 80% and 95% of students at Cornell have a smartphone.

The social implications of having a smartphone here are significant. To have the Internet at your fingertips is to have all the knowledge you need on demand about activities, people, or just random information in a conversation. It is also to check email, respond to messages, or to share videos. Since the social norm is to have one, and the students expect other students to have them, most things are done with a smartphone in mind. At Cornell, to not have a smartphone is to be technologically behind. The Red Queen analogy comes to mind: “Now, here, you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!”

The “Superhuman” Extension

I resisted getting a smartphone for a long time because I thought (1) having a laptop was sufficient, and (2) not having a smartphone brought more peace of mind. However, the utility gained from having one far outweighs the silence of not having one. I can check my email at any time, look up anything I want, and a few weeks ago in NYC for an interview, I used it as a GPS.

There is an interesting article by John D. Sutter from CNN Tech last September, which is titled, “How Smartphones make us Superhuman.” It obviously makes the case supporting them. An excerpt:

In addition to enabling us to video events on a second’s notice, potentially altering the course of global politics, these high-tech human “appendages” increasingly have become tools for fighting corruption, buying stuff, bolstering memory, promoting politics, improving education and giving people around the world more access to health care.

While I don’t use the smartphone for any political reasons as Sutter might admire, I find it invaluable as a college student to be able to access information from anywhere. In classes, I still use my laptop as it is faster to type things, given that it is set up. But on the go, and for other occasions, using a laptop is impossible. It may sound strange, but without the Internet, I would feel disconnected from the world.

In fact, last May, my laptop broke down and would not even get to the boot menu. While it was being repaired (for about a month), I had no immediate Internet connection for the first time in years, and it felt that something was deeply missing. I had to go to the school libraries multiple times a day, and I would be there at odd hours. (Though, I did manage to write a 23-page math paper saved between various emails and flash drives.) Not having a computer was certainly survivable, but at a huge inconvenience.

Similarly, for a smartphone it is obvious that anyone can survive without having one. However, the productivity, convenience, time-efficiency, and omniscience are clearly worth it for any student living in 2013.

Math or Computer Science?

Well this is an interesting situation. Just a month ago I announced that I was adding a computer science degree, so that I am now double majoring in math and computer science. The title of the post, after all, is “Computer Science AND Math.” Given the circumstances at that time, I think it was a good decision. My work experience had been mostly in software, and a CS degree from Cornell should look pretty good. In addition, I was wanting a more practical skillset.

decisions-2

In the past week, however, things have changed. I received and accepted an internship offer from my dream workplace, based on my background in mathematics and not in CS (though my prior CS experience was a plus). Based on this new situation, I have considered dropping the CS major (next semester) and taking more advanced math:

  • The CS degree has some strict course requirements, and I am afraid that if I go for the degree, I may be forced to skip certain math classes that I really want to take. For instance, I may have to take a required CS class next semester that has a time conflict with graduate Dynamical Systems, or with Combinatorics II. And given that I am currently a second-semester junior, I don’t have that much time left at college.
  • Even this semester, I am taking Algorithms, which meets at the same time as graduate Algebraic Topology. While Algorithms is pretty interesting and the professor is excellent, I am already very familiar with many if not most of the algorithms, extremely familiar with the methods of proof, and I feel that the experience is not as rewarding as possibly taking Algebraic Topology with Allen Hatcher, who wrote the textbook on the subject. I feel that I could learn algorithms at virtually any time I want. But learning algebraic topology with Allen Hatcher is a once-in-a-lifetime opportunity that I am afraid I am missing just because I want to get a CS degree to look good.
  • Even not being a CS major, I will still be taking some CS classes out of curiosity. However, these classes will no longer feel forced, and will not restrict me from taking the higher level math courses that I want to take.
  • My risk strategy for grad school is different now because of the internship. In the past, I would have been willing to take a decent grad school in math or really good job. (I would prefer grad school over getting a job, but of course, a good job is better than a mediocre grad school.) However, now that I have my dream internship, I am willing to play the grad school game with more risk.
  • But whether for grad school, trading, or just for curiosity, I would prefer taking advanced (graduate) math classes over undergraduate CS classes. In a sense, my taking of the CS degree was a hedge bet, as I wanted to reduce the possible cost of the worst case scenario. I knew that it would directly inhibit my ability to take advanced math classes via class time conflicts, but the thought was that if I couldn’t get into a good math grad school or get a good job using math, at least I would have a CS degree from Cornell. But, in this new situation, I think the risk is significantly reduced and the hedge is no longer necessary.

Interestingly enough, the primary motivation for dropping CS wouldn’t be to slack off, but to be able to explore more advanced math. (At least, that’s what I tell myself.)

I think this might be the second time in my life where I have had to make an important decision. (The first time was deciding where to go to college, and I certainly think I made the right choice there.) Unfortunately, I really can’t be both taking as many interesting math courses as I can, and at the same time be pursuing a CS degree. As much overlap as there is, I can’t do both. In an ideal world this might be possible, but not currently at Cornell.

So instead of the idea of having math and computer science, I am now having to think in terms of math or computer science. I am currently in favor of going with math, but I am not completely sure.

Edit: Thanks for the discussion on Facebook.

Thoughts on Classes, Spring 2013

In a previous article, I posted my schedule and about my decision to double major in mathematics and computer science. The computer science department seems to be quite backed up at the moment, so I have not received any official response yet.

I can see why the CS department is really backed up. In most of my experience at Cornell, I had class sizes of 10-30, with larger class sizes (150-250) only at introductory level courses such as Sociology 1101 or Astronomy 1102. It would be quite rare to have an advanced level course with that many people in them.

goldwin smith 132

But, CS easily has 150-250 people in each class. In the first few days of class, even in large lecture halls, there were no seats left and the late-arrivers had to sit in the aisles. I think students here see CS as too lucrative of a skill to pass up. Some difficult or otherwise time-consuming homework assignments have caused class sizes to drop significantly, but there are around 150-250 people in the CS classes. On the other hand, my math classes have 14 and 6 people respectively.

Math 4340 – Abstract Algebra

Professor: Shankar Sen

Cayley-Graph

This is a fairly trivial class so far. We are covering basic group theory and it is quite a relief compared to some of the more intense math I did last semester (*cough* topology). The course is supposed to move on to rings and modules later; however, in linear algebra we actually covered much of the foundations of ring theory and modules.

However, given the lack of difficulty of the topic so far, the homework grading has been quite harsh. I usually skip writing down every rigorous step if I think some part is obvious. Learning the material is more important than writing down every detail of the proof, in my opinion.

Math 7370 – Algebraic Number Theory

Professor: Shankar Sen

Relatively-Prime-Grid-Points

There are no exams, no prelims, and no homework. However, it is a graduate level seminar-type class and it is pretty insane. I have put up my lecture notes on Scribd, and even if you know nothing about college math, if you click that link, you can probably see how much more difficult 7370 is than 4340.

It is a really good thing I had a basic introduction to ring theory and modules before taking this course. Knowing what PID (principal ideal domain) and UFD (unique factorization domain) mean, knowing the difference between prime and irreducible, etc., was extremely helpful.

This class is even more difficult than the graduate Complex Analysis course that I took last year. Before I took complex analysis, I actually knew quite a bit about complex variables, complex functions, and contour integrals. I had even studied the Riemann zeta function in high school. And on top of that, I was not the only undergraduate in that class—there were at least 3 others.

But for algebraic number theory, this is really new material, most of which I haven’t seen or even heard of, and moreover, I am the only undergrad in the class. However, I talk with the professor outside of class and I am confident that I can learn the material if I really try.

Math 4900 – Independent Research/Reading – Elliptic Curves

Elliptic-Curve

Since I felt that I was doing too much CS and not enough math, I decided to add on an independent reading class. The book is The Arithmetic of Elliptic Curves by Joseph Silverman.

I have seen elliptic curves in complex analysis in the form of the Weierstrass P-function and equating points in the complex plane by a lattice. To see the algebraic side of it will be interesting though, especially because I am interested in number theory for possible research.

In addition to this official reading, I am also reading and doing problems from Tom Apostol’s Introduction to Analytic Number Theory, so that I can get both the algebraic and analytic sides to it.

CS 4820 – Introduction to Algorithms

Professor: Dexter Kozen

Minimum-Spanning-Tree

This is a really fun theoretical and mathematically oriented class. After all, Kozen is practically a mathematician.

Given my mathematical background, especially the combinatorics class I took last semester, this algorithms course is not too difficult and in fact fairly trivial so far. But, I expect it to get more sophisticated once we get over the introductory stuff. For instance, on our discussion board on Piazza, one student asked how to use a contradiction proof. In just topology alone, I probably used about a hundred.

In addition, Kozen shares some very interesting stories during lecture. Just last Friday, he was talking about dynamic programming and discussed a project using body scan data to analyze the number of dimensions it took to store the size information of a human body. “Are women 2-dimensional? I don’t think so,” said Kozen. In fact, he recalled from the study that women were around 5-dimensional and men were fewer.

Also, when he was explaining the growth of the Ackermann function A(n), he noted that even A(4) was an extraordinarily large number, and in fact that it was “even higher than Hopcroft’s IQ.”

CS 4850 – Mathematical Foundations for the Information Age

Professor: John Hopcroft

Large-Graph

From the title of this course, one might think it is really easy, but even as a math major, I find it nontrivial (that means hard, in math terms). In fact, I’d say at least 30-40% of the class has dropped since the first day. The fact that Hopcroft won a Turing award makes the class no easier.

It is essentially a mathematical and statistics course with applications. We proved the Central Limit Theorem on the first day of the class and then looked at spheres in high dimensions, with the intent of generating random directional vectors in high dimensions. As it turns out, most of the volume of a high-dimensional sphere is on a narrow annulus or shell, and when a given point is taken to be the north or south pole, the rest of the volume is located at the equator.

Currently we are studying properties of large random graphs, in particular, properties that appear suddenly when the edge saturation of the graph passes a certain threshold. For instance, below a certain number the components of the graph are all small, but above that number, a giant component arises. For an assignment I showed how this giant component phenomenon arises in connections of the Reddit community.

CS 3410 – Computer System Organization and Programming

Professor: Hakim Weatherspoon

32-bit-ALU

In contrast to the high-level programming I have done in the past, this course is about low-level programming and the hardware-software boundary. The programming language for this course is C.

We are building up a processor from the ground up, one could say, with basic logic gates to begin with. The first project was to design a 32-bit arithmetic logic unit (ALU) using Logisim, a circuit simulation program. For instance, for a subcircuit we needed to create a 32-bit adder with overflow detection.

The above picture is actually a screenshot of the overall ALU that I designed for the class. The subcircuits are not shown (this project is not due yet, so it would break academic integrity to show a more coherent solution).

Computer Science and Math

At the end of last semester, I decided to double major in computer science and math, rather than just in math. This decision was based on several reasons:

  • Practicality. As much as I love theoretical math, most of it is totally irrelevant to the real world. CS is closely related but far more useful.
  • Opportunity. Cornell has a top-rate CS department, and it would be a shame for me to not take advantage of it. I am virtually done with my math major as well, so it does not cut into that.
  • Expanding my skill set. I think CS is a strong backup in case I didn’t get anywhere with math.

The catch is, being a junior already, I need to rush the major in my remaining 3 semesters (including this one). This will require quite a bit of work, but due to my math major, I have much of the foundation done. I also have every liberal arts distribution requirement out of the way. In addition, during my sophomore year I took CS 2110, so that is another requirement done. My schedule for this semester is below.

2013 Spring Schedule

The most interesting class will be Math 7370, which is Algebraic Number Theory at the graduate level. I have some background in analytic number theory but not in algebraic number theory, so it will be interesting to see the differences. Also, the same professor is teaching 4340 and 7370, so I should have ample opportunity to ask any questions about algebra.

As for CS classes, I hear CS 3410 has a lot of work, but I am prepared. Also, given that I did decently in Combinatorics (Math 4410) last semester, CS 4820 should not be too hard. And given the knowledge from a math major, I doubt 4850 will be that difficult.

In addition, I have planned a more consistent posting schedule for this blog. There won’t necessarily be more posts, but they should be spaced more evenly. Also keep an eye on my math blog epicmath.org—I will continue to update it this year.

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.