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What Does Being Correct Mean?

In class, we were discussing the Parallel Postulate by Euclid. Basically it says that if you draw a straight line on top of two other lines so that they intersect, and if the angles on the same side of the first line are less than 2 right angles (180o), the two lines will intersect at some point on the same side.

Image: 6054, via Wikimedia Commons.

It’s weird learning about proving something that feels so elementary that I assumed it was just true by definition. I mean I can just look at the picture and it certainly looks like it should be correct just by careful inspection. But I guess that doesn’t really prove it without a shadow of doubt. What if what I was looking at was 179.999o and I just said they would never touch even though they would intersect given enough space. Granted, I would assume it was 180o so I would be correct based on the assumption being true.

When I look at this problem, I can’t help but reflect on the lessons, experiences, and “truths” that have instilled within me from previous mentors and teachers. It becomes very hard to try and think about other approaches or ideas other than “duh that true”.

What allowed me to think about this Postulate was learning about how other people through out history thought about the Parallel Postulate and created their own “new math”; their own pseudogeometry; their own imaginary geometry. Here I am unable to think “outside the lines”, but these other people created whole new systems from looking at the problem from a different angle. I have no problems creating weird parallels with my jokes and puns but can’t seem to do the same thing with math. (Yes, I love bad puns).

Poincare and Lobachevski were both people that worked in this pseudogeometry, which is now called hyperbolic geometry. (The former or “normal” geometry is considered “Euclidean Geometry”). In hyperbolic geometry it’s possible to have lines that would normally intersect in Euclidian space be considered parallel and non-intersecting in hyperbolic space. I think looking at the picture below will really help. I know it wasn’t until I built a hyperbolic plane by hand that it really sunk in for me. ( Make your own at )

A hyperbolic triangle. Public domain, via Wikimedia Commons.

Reflecting on the on hyperbolic plane I began to try to remember a time when what the instructor was teaching conflicted with something I already knew. As I thought I remembered something an art teacher told me about vanishing points. So imagine you’re standing on some railroad tracks that stretch straight forward for miles. As you look down the tracks, as you would if you were actually a train, at some point the individual components would become one whole line. Instead of seeing the left rail, the right rail, and everything else you would see a railroad track. At that point, the left and right rails have effectively become one, unable to tell them apart. Now what would happen if a train went down those rails that look like they became one? The train becomes smaller, or at least, it looks like the train is shrinking. At the time I could only think about how the teacher lost her mind. It wasn’t until I looked down a straight road that I realized how right she was.

After thinking about how perspective is everything I began to wonder what other things are different than they appear? I asked a friend, and she mentioned she actually had to unlearn some thing to be able to Fence (as in the sport) correctly. She told me that she had to change the way she extended her arm in order to be able to obtain the longest reach possible.

It turns out that a straight line with your arm is not the best way to have the longest reach. In all my learning, I had been taught that you to get the longest linear distance with line segments are to put each line segment end to end along the same axis. But in fencing, doing just that with your arm is not the longest. Why is fencing different?

When hold the sword in your hand, it seems that your muscles tighten to hold the load and your arm up. By tightening your muscles, you shorten your reach by as much as 2 inches for some people. When your muscles are relaxed, the joints can loosen allowing more space between the bones, which lengthens your arm. So by relaxing your arm a bit so it’s not parallel with the ground, your sword can reach just a little bit further.

Is math wrong when it comes to the physics of people and fencing? Absolutely not! In my case, it’s the model that the math was used on that was wrong. I assumed the arm was a rigid object with hinges at the shoulder, elbow, and wrist. Since I had modeled the arm in this fashion any math done to the model would never take into account the possibility of expansion of the hinges. Assumptions are the downfall of many people.

Isaac Newton and his Contributions to Mathematics

Sir Isaac Newton. Image: Arthur Shuster & Arthur E. Shipley, via Wikimedia Commons..

In class we discussed the Fundamental Theorem of Calculus and how Isaac Newton contributed to it, but what other discoveries did he make?

Sir Isacc Newton was born on January 4, 1643, but in England they used the Julian Calender at that time and his birthday was on Christmas Day 1642. He was born in Woolsthorpe-by-Colsterworth, a hamlet in the county of Lincolnshire. His father had already passed prior to his birth and his mother remarried after his birth and left Isaac to live with her mother. He went to The King’s School, Grantham from the time he was twelve until he was seventeen. His mother removed him from school after the death of her second husband, but later allowed him to return by the encouragement of the school’s headmaster. He rose to be at the top in rankings in his school, mainly motivated to get revenge towards a bully. He began attending Trinity College in Cambridge in 1661. After receiving his degree he developed his theories on calculus over the span of two years during the plague [1].

Newton’s work in calculus intitially started as a way to find the slope at any point on a curve whose slope was constantly varying (the slope of a tangent line to the curve at any point). He calculated the derivative in order to find the slope. He called this the “method of fluxions” rather than differentiation. That is because he termed “fluxion” as the instantaneous rate of change at a point on the curve and “fluents” as the changing values of x and y. He then established that the opposite of differentiation is integration, which he called the “method of fluents”. This allowed him to create the First Fundamental Theorem of Calculus, which states that if a function is integrated and then differentiated the original function can be obtained because differentiation and integration are inverse functions [2].

Controversy later arose over who developed calculus. Newton didn’t publish anything about calculus until 1693, but German mathematician Leibniz published his own version of the theory in 1684. The Royal Society accused Leibniz of plagiarism in 1699 and the dispute caused a scandal to occur in 1711 when the Royal Society claimed Newton was the real discoverer of calculus. The scandal got worse when it was discovered that the accusations against Leibniz were actually written by Newton. The dispute between Newton and Leibniz went on until the death of Leibniz. It is now believed that both developed the theories of Calculus independently, both with very different notations. It should also be noted that Newton actually developed his Fundamental Theory of Calculus between 1665 and 1667, but waited to publish his works due to fear of being criticized and causing controversy [1].

Newton not only discovered calculus but he is also credited for the discovery of the generalised binomial theorem. This theorem describes the algebraic expansion of powers of a binomial. He also contributed to the theory of finite differences, he used fractional exponents and coordinate geometry to get solutions to Diophantine equations, he developed a method for finding better approximation to the zeroes or roots of a function, and he was the first to use infinite power series.

His work and discoveries were not limited to mathematics; he also developed theories in optics and gravitation. He observed that prisms refract different colors at different angles, which led him to conclude that color is a property intrinsic to light. He developed his theory of color by noting that regardless if colored light was reflected, scattered, or transmitted it remained the same color. Therefore color is the result of objects interacting with colored light and objects do not generate their own colors themselves [1].

Sir Isaac Newton was a truly amazing mathematician and scientist. He achieved so much in his lifetime and the amount of discoveries he made can seem almost impossible. He helped make huge advancements in mathematics and created theorems that we still use heavily to this day.



Female Mathematicians, the Unsung Super-heroines

During most of my schooling I’ve generally only heard about male mathematicians and how they changed the way we do math. On occasion I have heard a woman’s name come up, but she is usually brushed over and not too much detail is given. I did some research and there are quite a few women throughout history that have contributed to mathematics and made numerous discoveries of their own. Why don’t we hear about them more often? It’s somewhat the same as how we don’t see much about super-heroines, even though comics and superheroes are all the rage right now. Women tend to get over looked and I believe this should stop. Maybe if I write a comparison of how female mathematicians can be compared to superheroes it will bring about a different view and we can get behind the power of the female brain.

Portrait of Sophie Germain.

Sophie Germain

Black Widow.

Black Widow

Female superheroes have been around since the early 1940’s, but with some caveats. They were still portrayed as being controlled by outside forces. Also why do we have to put them in the clothes we do; it’s just another way to classify them as less than a man. Wonder Woman had to have her bracelets to keep from going crazy, and this symbolized control. This reminds me of how early female mathematicians had to hide who they were and only communicate with fellow colleagues through letters and under different names. Sophie Germain was one such mathematician. She started teaching herself, at the young age of 13, when she ran across a book in which the legend of Archimedes’ death was recounted in her dad’s library. The story fascinated her and began her love for math, but like most stories about our female heroes, she was told she was wasting her time and her parents tried everything to keep her from studying on her own. To no avail, they couldn’t keep her from the love that bloomed over a terrible tale of how “Archimedes was so engrossed in the study of a geometric figure in the sand that he failed to respond to the questioning of a Roman soldier. As a result he was speared to death” (Perl 64). When she was 18 Ecole Polytechnique, a school, was built, but being female she was unable to attend. How have we been so blind to the contributions that females can make?

Her inability to attend any type of formal classroom to advance her knowledge makes me wonder how we have made advances in science and math at all. It reminds me of how we highlight male superheroes to be better and why female fans are so rare to come by in the world of comics. Portraying women to be less or just objects to admire doesn’t seem to be the way to advance in any area, but it happens, still to this day. Germain’s view in different fields has won her recognition as a wonderful mind and a fresh breath of air to the field of math and science. She wrote to Lagrange, Gauss, and Legendre under a pseudonym, surprising each of them that she was in fact a she. They helped her grow and learn more than she could ever on her own, but it saddens me that she would even need a man to obtain documents and texts for her to even expand her knowledge in the first place. Being left out is something I am all too familiar with and I don’t know what I would do if I had to solely rely on one or two men to get me the proper supplies to further my education. I value my education and see myself going far.

I’m very happy to not live in the time where a man had to vouch for a woman to be accepted into the scientific community like Emmy Noether had to deal with. Having to see her mentioned in comments with the pretense that Einstein said she was good so therefore she must be is quite ridiculous (Angier). With her inability to teach under her own name for quite some time and then to receive a ridiculously small pay when they did decide she was worth pay is shameful (“Emmy Noether Lectures”). I believe that a person’s merit is there whether someone says it is or not, especially if you’ve done the same research and studying as the next guy. Just because you’re a woman doesn’t make you any less intelligent nor should you have to rely on a man vouching for your intelligence. How often have we watched a TV show or movie where there was a man vouching for a woman? Or where a man is trying to keep a woman from getting credit or keeping her under control as if she was a crazy person who has no self-control of her own?

Take Iron Man 2, Black Widow. At first he assumes she’s just a pretty face, until she has to come in and save him with her awesome butt kicking skills. Not only is he shocked, but he takes a bit to adjust to the fact that she is her own hero. Female mathematicians have had to face this same stigma throughout the years. Germain eventually won her own right to her work and proved she wasn’t just a pretty face. The saddest thing is she wasn’t the only female that had to prove her worth.

Throughout the years, women did have to prove themselves, but this was also not always the case. There were a few women who had respect for their minds from the get go. One such woman was Hypatia, the first women to make a substantial contribution to mathematics, or so believed or documented. She was taught by her father and I believe that this might have been why she was accepted so easily in the circle of great minds of that age. She was led in by a great man and had very little to prove besides to show that his words about her were correct. She was the head of a school in her time, leading discussions on math and science to many students. She was a great hero in her time and I only wish that the many other female mathematician super-heroines throughout the ages had an easy path to their greatness.

Perl, Teri. Math Equals: Biographies of Women Mathematicians + Related Activities. Menlo Park: Addison-Wesley Publishing, 1978.

Angier, Natalie. “The Mighty Mathematician You’ve Never Heard Of,” New York Times Science section, page D4, March 27, 2012 (print edition). March 26, 2012.

“Emmy Noether Lectures,” Association for Women in Mathematics.

Random walk

George Polya was a Hungarian mathematician who made contributions in many branches of mathematics, among which was probability theory. In this blog we will discuss the “Random Walk” problem in probability. What is interesting is that George Polya actually first theorized this problem by accident.

George Polya was at that time a professor at a university in Zurich, Switzerland. The beautiful landscape there helped him develop a habit: taking afternoon walks in local gardens. One day, when he took his afternoon walks, one strange thing happened: he met a young couple six times. Well, I don’t know how large the garden was and how many paths were there, but this coincidence did surprise our mathematician. He was wondering how could this be possible, considering that he was then taking a random walk. After he mentioned this to his wife, he decided to do some research on the probability regarding random walk. His research actually established a new topic in probability theory.

Now let’s begin our discussion about the random walk problem with the simplest case. Consider one object moving along a straight line. Our assumptions are that this object can only move leftward or rightward, and the distance it moves each time is just one unit. For each time, it has a 50% chance to move leftward and a 50% chance to move rightward. A random walk model is thus created. And now we can ask: if when the object return to the starting point, the movement will end, then what is the probability that the object could return to the starting point? Thanks to George Polya’s work, the answer is 100%. Although some people may think this is unlikely, that’s only because when we think about this problem, we have a pre-assumption that the time is limited, or finite, which is always true in our real life. However, this mathematical model sets the time as endless, so it means when there is enough time and the object keeps moving endlessly, at last it will almost always return to the starting point.

When combining this theoretical conclusion with real life experience, mathematicians created the so-called “Drunken Man Going Home” problem. Assume there is a drunken man. He comes out of a bar and walks randomly along the line connecting his home and the bar. Once he gets home he will stop. Then what is the probability that the drunken man could finally get home? The answer to this funny question is still 100%. The math model of this problem is equivalent to the previous one. Because when time approaches infinite, the different positions of fixed points on the line actually make no differences. In my own words, any finite number in front of infinite vanishes to zero. So in this case, we just move the ending point from the starting point in the first example, to a new fixed point which denotes the drunken man’s home in the second example. The result keeps unchanged.

But mathematical problems always go from simple cases to complex cases. In the original random walk along a straight line, if we change the condition from “along a straight line” to “in a two-dimensional plane”, the complexity of this problem will definitely increase. If we change the condition to “in a three-dimensional space”, the complexity of this problem will increase dramatically. And the answers to this problem also change. The law is that, when the dimension increases, the chances that the object could get back to its starting point decreases, with the assumption that other conditions keep unchanged. For example, for a three-dimensional space, the probability is 34%; for four-dimensional space, the probability is 19.3%; when the dimension reaches eight, the probability that the object can get back is only around 7.3%. This means that there is only a lucky “drunken man”, who could always find a way home; there is no lucky “bird”: in a three-dimensional space, if the bird flies randomly, its chance to get back to its nest is much smaller! (Even if it could fly endlessly.)

Some people may cannot help ask, what can the theorem on random walk be used for. Well, obviously it is not developed to help drunken men feel confident when they need find a way home. It does have very wide applications in various areas. I will broadly list its applications in three different areas.

In economics, the theorem on random walks generates a hypothesis called “random walk hypothesis”. With the help of this hypothesis, economists could construct math models to research factors affecting shares price and the change of shares prices. It is quite understandable because randomness is q characteristic of the stock market.

In physics, the famous Brownian motion can apply the theorem on random walk to achieve the purpose to simplify the system. Since molecules move randomly to every direction, if we make an assumption that the molecules will only move to a finite number N directions, then the problem can be converted to a random walk problem in finite dimensional space. We could estimate the case in real physical world through our approximation in theoretical random walk models.

In genetics, a random walk could be used to explain the change of genetic frequency — a phenomenon that finally leads to genetic drift. Because gene mutation is always random, when we view the original gene pool as a point in a multi-dimensional space, then each gene mutation will generate a new gene pool, which can be viewed as a new point that forms through the original point’s random walk. After countless random walks, we could use the computer to draw pictures that show the trajectory of these abstracted points. The trend and some important characteristics of a gene drift could be described by this.

Randomness is everywhere in nature. At first glance, randomness connotes lack of order, unpredictable and uncontrollable. However, after mathematician’s great job, we do gain many laws and theorems on randomness. Random walk now has very wide applications in various disciplines, and we must attribute this partially to the coincidence happened to George Polya.



Two Pi or not Two Pi

The date I’m writing this blog post is March 14th, 2015. If you live in a very particular part of the world, you might represent that date as 3/14/15, which in turn might make you excited to see the first 5 digits of π all nicely lined up in a row. The rest of the world might be confused why you’re making such a big deal out of 14/03/15, and the Western engineers are just biding their time until it’s 3/14/16. The arbitrariness of π day not-withstanding, I’m here to talk about how π, and any days relating to it, pales in comparison to Tau (and any days relating to it!)

Still looks delicious, however. Image: Public domain, via Wikimedia Commons.

But first, some history: π, as we all should know, is the ratio between the diameter of a circle and its circumference. This is 3.141592…  for each and every circle EVER. Which is pretty cool! This usually marked an important discovery for each. Most early cultures didn’t get it quite right, but made their best rational number approximation. In ~1700 BCE, a Babylonian clay tablet uses a constant represented as 25/8, or 3.1250. Around that same time period, an Egyptian papyrus scroll approximates π as (16/9)^2, or 3.1605. One of the most successful techniques of approximating π during this time period is often attributed to Archimedes, where he calculates the perimeter of an inscribed N-gon. Archimedes used an 96-sided polygon, and calculated π to be somewhere between 23371<< 227 (roughly 13.1408 and 3.1429). At the same time, a famous Ancient Chinese mathematician, Liu Hui did the same algorithm on an 3,072 sided polygon, approximating π to be 157/50 3.1416. However, Liu Hui developed a similar, faster algorithm after noting that successive inscribed polygons formed a geometric sequence with a factor of four. Some 200 years later, another Chinese mathematician used this algorithm on a 12,288-sided polygon, calculating the π approximation out to be 31464625<100<31464625 +169625, which translates to roughly 3.141592920.

But enough about the history of π. What I couldn’t find, in all my research, is why ancient peoples were obsessed with the ratio of the circumference and diameter, rather than the circumference and the radius!? After all, a circle is literally defined by its radius as a distance from its origin. Why go through all that unnecessary doubling! I guess ancients needed their line segments to touch something ‘tangible’ on both sides, but thanks to them, let’s take a look at what we have to deal with:

The number π is such an entrenched constant that we developed a unit of measure to use it: radians (which are, of course, dimensionless). If we wanted to measure a full circle it is 2π radians. Wait, what? We have got this glorious constant that people constructed 12,000 sided polygons to calculate, and it only gets us halfway around the circle? So now π/2, π/4, π/8 don’t actually mean half of a circle, or a quarter of a circle, but half of a half of a circle, and quarter of a half of a circle. Enter Tau.

Why not use Tau to express the ratio of the circumference and the radius, that ever pivotal piece of circular information. Because the radius is exactly half the diameter, you can clearly see how Tau is 2π, or 6.28(ish). Why the letter Tau, though? Well my theory is that it is similar enough to Pi that folks don’t feel too threatened by it’s emergence. So what does Tau get us?

For starters, how many radians is a full circle using Tau? Just, Tau! And how about half a circle: Tau / 2! There is no need for a quick mental check of divide by two conversion (…or, wait, was it multiply by two? Which way are we going again? See how this is confusing!), what you want and what you’re looking for are simply in the constant you use.

See how much easier tau is? Image: Michael Hartl, via Wikimedia Commons.

“But Ryan,” you exclaim, “What about ei? Without π, how will Euler be identified!” And to that I say: never fear. While it is true that ei=-1, if you substitute in Tau for Euler’s Identity, you’ll find that cos() + isin() = 1 (Euler’s Formula substituting=) reduces down to a very tasty 1. Identity saved.

And the list goes on! A sine wave is still a sine wave, using instead of π. Using a constant derived from the actual construction of a circle, rather than a near arbitrary doubling of the radius that has been passed down from Ancient peoples and entrenched into our present day mathematical dogma, just seems to make more sense.

And best yet, our deliciou Pi(e) day of March 14th is only delayed a few short months until June 28th,which can be represented as 6/28 which as someone pointed out to me happen to be perfect numbers. This switch practically sells itself, to be honest…though we’ll have to come  up with a dessert called “Tau.”

Of course, there are alternative solutions…

Image: xkcd by Randall Munroe.


The Importance of ‘Nothing’

I’m a programmer. When I ask people their impression of what I do, the usual response is a long string of ones and zeros, said in a robot voice. Before I first started my Computer Science degree, I probably would have said the same thing. After my first semester, I would scoff at such binary answers, and feel powerful knowing I know how to write code. Halfway through my degree, I discovered that when you get down to brass tacks, zeros and ones are really all that comes down to. Finally, here at the end of my degree, I’m really happy that I don’t have to work in raw ones and zeros.

And it has always tickled my fancy that there is no Roman numeral representation for the number zero. I usually just pull this out for fun trivia, but after discovering in class that the Egyptians and Babylonians also struggled with the concept, I thought it might warrant a little extra research.

In this day and age, with our modern schooling, it seems as if zero is trivial. It literally means nothing, after all. It might have a few cool properties. For example, zero added to any number will result in the number as one example…but you can get the same behavior by just multiplying by one! For a computer scientist, zero is a boolean value. Zeroes also have a very friendly feel to them. If you see a lot of zeros at the end of a number, you know that number is a nice round one. And we like round numbers.

But being able to use zero is HUGE! Without it, we would either have an ill-defined positional notation for our numbers, or have to resort to an additive system like Roman numerals.  The lovely round number of 100,000 so cleanly represented here (with a little help from a comma) would require 100 M’s in a row using present day Roman Numerals. Even ancient cultures that used a positional notation would just use contextual clues to figure out if 216 meant 2016 or 2160 or what have you. Babylonians started to help with this problem by making two tiny stylus tick marks. So now, 2106 became 21”6. Interestingly enough, there was never any tick marks at the end of numerals, only in the middle. This leads scholars to believe that these tick marks were not an idea of zero; simply punctuation, much like our helpful little comma from before.

Zero is special in that it has two roles. It can be used for positional notation as we have just seen, but that was just as easily solved with punctation. Zero is also, of course, a number in and of itself, which brings on a whole barrel of troubles. Historically, numbers were thought of much more concretely. People used them to solve ‘real’ problems rather than abstract ones. It is a pretty far jump from for a farmer to go from five horses he owns, to five “things” in existence, to an abstract idea of ‘five’. If the farmer is solving the problem of how many more horses he needs, it is going to be “zero more horses.”

For this reason, perhaps it was lucky for earlier civilizations to miss out on zero. Working with zero can get you into a lot of trouble. There are cases of some of the brightest mathematicians of their time struggling with the concept of zero. And Indian mathematician has this to say about division:

“A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.” – Brahmagupta

You can tell he is reaching when he suggests a number divided by zero is N/0.

The first known zero, found in Cambodia. Image: Amir Aczel.

What’s even more mysterious is how there isn’t some clearly defined point in history where zeros are firmly established. There are some hints and teases in the nautical readings of Greeks and odd punctuation marks in Egypt, but nothing concrete. The earliest known writing of zero is famously from a stone tablet found deep in Cambodia, where it has the date of 605 in sanskrit, with a small dot to denote the zero between the six and five.

A clean rendering of the oldest known numeral using zero. Image: Pakse, via Wikimedia Commons.

It seems odd that such a powerful and tricky number wouldn’t have a more auspicious start. Instead, somewhere, someone in India put a dot on a tablet…and the world was changed forever.

I just hope something like this doesn’t happen:

Zach Wiener, SMBC-Comics 08/29/2012.



Mathematical Psychology

What if I told you I could explain every single one of your decisions and actions using math? WHAT? But humans are so unpredictable and random! We really don’t understand these strange creatures at all.

Enter mathematical psychology:

Ernst Weber. Image: Public domain, via WIkimedia Commons.

The Society for Mathematical Psychology says that mathematical psychology is “broadly defined to include work of a theoretical character that uses mathematical methods, formal logic, or computer simulation.” What does that mean? It basically means that mathematical psychology can be any type of psychological work that uses math to explain something or figure something out.

Gustav Fechner. Image: Public domain, via Wikimedia Commons.

The first traces of mathematical psychology go way back to Ernst Weber and Gustav Fechner in the 19th century. For the first time in history, these guys applied mathematical techniques to  psychological processes. The moment they stepped out from behind the desk and got their hands busy, both mathematical psychology and experimental psychology were born. These twins went hand in hand toward making psychology a quantified science.

Around the same time, researchers in astronomy were mapping the distance between stars by jotting down the exact time a specific star passed through the cross-hairs on their telescope. There were no automatic instruments, so the measurements were based on human response speed. The German astronomer Friedrich Bessel (who never attended a university) studied the differences between the measurements. Based on measurements of general human response speed, he used math to create personalized equations for each of his fellow astronomers. The equations would actually cancel out the personal differences from astronomer to astronomer.

Completely independently, the physicist Hermann von Helmholtz measured people’s reaction times to determine the speed of nerve conduction. Most of Helmholtz’s colleagues though that nerve signals passed along nerves immeasurably fast. Helmholtz used a freshly dissected sciatic nerve and calf muscle from a frog and attached an altered galvanometer as a timing device. He reported that the transmission speed is in the rage of 24.6 – 38.4 meters per second. Today we know that nerve impulses travel anywhere between 1 and 120 meters per second depending on the diameter of the fiber and the presence or absence of myelin, so Helmholtz was pretty on point.

The Dutch physiologist F.C. Donders and his assistant J. J. de Jaager combined these two branches of research to use the reaction times to figure out objectively the amount of time that it takes to complete elementary mental operations. He came to the conclusion that simple processes (automatic processes) are much faster than choice processes(things we decide to do), but his numbers were pretty far off due to lack of technology. However, a German psychologist named Wilhelm Wundt used Donder’s ideas to create the first psychological laboratory. It was hard to replicate the results from the lab, because of his method of introspection. In other words, he was observing his own results, so his own individual differences changed the results, but the lab was a huge turning point in the Mathematical Psychology.

Wilhelm Wundt in the first psychological laboratory. Image: Public domain, via Wikimedia Commons.

After the introspection method didn’t work out, two different schools of thought developed. People either gravitated towards studying the general conscious human experience or they were drawn to study the individual differences between humans.

Throughout the 20th century, the idea of behaviorism popped up in America and shifted the focus away from reaction times and more towards learning theory. Learning theory attempts to study the way people learn through objectively measuring their behavior (think Pavlov’s Dog). Throughout the Second World War, the military brought together a smorgasbord of mathematicians, engineers, physicists, economists and experimental psychologists in order to learn about human performance and limitations. These men and women essentially shaped mathematical psychology into what it is today.

In 1951, two guys named Bush and Mosteller published a paper titled “A Mathematical Model for Simple Learning” which presented the first detailed data on learning experiments.  From there, people began to publish more papers, teach courses, compile the mathematical models into volumes, create a Journal of Mathematical Psychology and otherwise take mathematical psychology seriously. Today mathematical psychology is a tried and true branch of psychology that many researchers could not work without. We’re taking baby steps toward understanding the brain one equation at a time.