What Is a Proof?

I’ve talked a fair amount in some of the earlier posts about the idea of a proof. Now that we’ve developed a conceptual underpinning of what that means, we can see one in action. I hope my readers enjoy this as much as I do, as what we will discuss here is among my favorites.

In case you’re worried, I’m not going to start talking about the proofs you may have seen from high school geometry. While these technically count as proofs, they are a far cry from what most proofs today look like. Every proof will justify what it says, but almost never is this done in the painful and boring ‘two column’ setup that I have heard many talk about, or in some kind of highly structured list. Proofs are usually presented in more of a narrative form, actually.

Just because a proof is highly logical does not mean it is not beautiful. Just as in every other form of art, precision does not stand at odds with beauty. Most mathematics today is done with sentences and paragraphs explaining and guiding the reader through the ideas. When this is done well, the person reading not only can understand why each step in the progress is correct, but is able to perceive the overall direction of the proof. As a graduate students, one of the ways I know I understand a proof is if I can give it a “plot summary,” capturing the progression of ideas with minimal reference to details. As you read through and study a proof, things should begin to make more sense as you go, or at least they should come together at the end. The best proofs will have a sort of climax that represent a very powerful or surprising flash of insight, and the experienced reader will immediately notice one of these when it occurs, just as you can tell when a movie is reaching its climax.

Just as someone who reads a lot of literature can perceive different sorts of depth and beauty in writing, so an experienced reader of mathematics sees different kinds of beauty in a proof – the most common words of this sort that I hear are beautiful, elegant, and slick. I haven’t every actually asked other mathematicians about whether they consider these aesthetic terms synonyms or not (I’d love to hear some comments from others on that!) but I personally think these are all subtly different. Maybe one day I’ll be able to put into words what I mean by each of those. But regardless of whether these words actually have different aesthetic meanings, my point is that math is not merely black-and-white. There is rich color to be found, if you know where to look and keep an open mind. There’s a lot of beautiful math out there. One of my primary career aspirations is to make this beauty accessible to everyone I come across.

So, to begin on that mission, I will now try my hand at a presentation of one of the most famous equations of all – the Pythagorean Theorem.

This is a very ancient piece of mathematics, that was proved by a civilization that developed mathematical proof in a way that had lasting impact, ancient Greece. The theorem is named after Pythagoras, who either proved this himself or whose school of pupils brought forth a proof. Today, this is how we normally phrase the theorem:

Pythagorean Theorem: Suppose that the sides of a right-angled triangle have lengths a, b, and c, with c the length of the longest side. Then the following equation is always true:

a^2 + b^2 = c^2.

Since the Greeks phrased all of their mathematical work directly in terms of geometry, they would have used the areas of actual squares as the parts of this equation. I don’t want to try to actually use their phrasing, because that might lead to confusion, but the following pictures should give a good illustration.

This is the framework within which a Greek mathematician would have thought about the Pythagorean theorem. It is worth saying that it is far from obvious that this should be true, and I think that ought to make it interesting to us. Why should there be any equation relating these three sides, and why this one in particular? In all areas of life, we get bored by something that is too obvious, but less obvious things catch our attention. It seems to me that non-obviousness is at least part of why we like comedy, magic, and sports – the punchlines we don’t see coming make us laugh the most, we love magic because it amazes and surprises us, and we love the upset victory, the Cinderella story, and the impossible goal in sports in large part because they surprise us.

To me, something similar should happen with great mathematical ideas. We don’t talk about the fact that 1+1=2 very often – it is just as true as the Pythagorean theorem, but it is too obvious to be of much interest. Another reason this is more surprising than 1+1=2 is because it is a broader statement. This is not a mere calculation. Your calculator can’t do this. This is not a calculation about a particular shape, this is a statement about the way that all right triangles are. So we learn far more from the Pythagorean theorem than we ever could from building lots of triangles and measuring their sides, because there would always be more triangles we hadn’t measured yet.

But how do you know this is true? Well, this is where proofs come in. There are literally entire books published that contain nothing but hundreds of different and creative proofs of this statement, including a proof by Albert Einstein and one by American President James Garfield, plus the original proof discovered and written down by the Greeks themselves. The proof that I have chosen to present is the one that I find the simplest and most enjoyable.

Proof of the Pythagorean Theorem:

This proof centers around two key ideas and a little cleverness. The two main ideas we will need are (1) Two shapes that have the same dimensions have the same area, and (2) if two shapes have the same area, and if we remove from both shapes identical pieces, then the new shapes still have the same area. These two statements are just part of what area is, so this is a good starting point. Now, we need to use these ideas in a clever way. We will do so with two conveniently-drawn squares. Consider first “Square 1”:

The green triangles in this image are all identical to each other – in fact, these are right triangles with side lengths a, b, and c. There are also two smaller squares, with side lengths a and b respectively. Square 1 itself has side length a+b. We will use this information to work out some areas later.

Now, lets look at a different square, which we will call “Square 2″:

Square 2 also has four green triangles, which are identical to the green triangles in Square 1. There is also a square of side length c in the inside of Square 2. Finally, notice that Square 2 also has side length a+b.

Now, we can compare these two squares. Square 1 and Square 2 are both squares with side length a+b, and so are identical shapes. So, using idea (1) from the beginning, we know that the area of Square 1 is the same as the area of Square 2. We can also look at the pieces of these two squares. Both squares have four identical green triangles, and these have the same area. Since Square 1 and Square 2 have the same area, and idea (2) tells us that we can remove the four green triangles from each and the resulting shapes will still have the same area as each other. But what happens when we remove the green triangles from Squares 1 and 2? The remaining portions of Square 1 are the two small squares with side lengths a and b, which has total area a2+b2 and the remaining portion of Square 2 is the square with side length c, which has area c2. Since these total areas must be the same, we now know that a2 + b2 = c2.

There it is! I encourage you to reread the proof a few times if you don’t quite feel like you understand. It often takes multiple readings to get a handle on what is happening. Take a look at the logic I used – are there gaps you can fill in? (For example, I opted to not explain why Square 1 and Square 2 actually are squares – can you explain this on your own?) When all is considered, this argument truly is airtight. I hope you will appreciate the beauty of the reasoning, it’s simplicity and cleverness. If not, then perhaps a different idea from math will be more appealing to you.

I hope people walk away with a greater understanding of what math is about, and perhaps some joy too. And feel free to write me about anything from your math class you’re curious about and would like to see proved!

From Numbers to “Pure” Math: Ancient Greece

In my previous post, I talked about the conceptual progression from the most basic intuitions about counting to the more general idea of number that transcends any particular physical situation.

As big a development as this is, I would say this not mathematics proper. In most civilizations where the number concept developed, the most complicated uses of number were related to measurements – something like geometry. Many ancient civilizations, for example, knew that a triangle formed out of ropes of lengths 3 units, 4 units, and 5 units formed a right angle (that is, a perfect 90 degree corner), and so that was often used to help place markers for rectangular fields. Many ancient peoples also had numerical approximations for some important numbers, though they probably did not think of them as “approximations” in the same way we do today. Details about ancient mathematics can be found without too much trouble via a search engine or a book on math history, it is fascinating to read about, but I won’t go into much more detail here.

The idea of precision we are used to today took a long time time to develop. Ideas involving number were presented until comparatively recently only as “word problems.” This is because the written symbols for numbers largely had not been developed, or were very cumbersome. (To see what I mean by cumbersome, imagine trying to multiply or divide using Roman numerals!) Most ancient peoples viewed numerical work as strictly practical, or in some cases spiritual elements may have been involved (as in some ancient astronomy, for instance). In any case, maximizing the precision was not really the goal, the goal was to be ‘good enough’ to use in everyday life. For example, consider the number we call pi today:

\pi = 3.1415926535...

This constant is equal to the circumference of a circle divided by its diameter, and is very important in geometry and in every area of math. There were some civilizations, like ancient Babylon, that seem to have treated this number as being exactly the fraction 25/8, which written as a decimal is equal to 3.125 [1]. We know today that pi is not equal to 3.125. In fact, today we know that the constant pi is something called an irrational number, which means there is no fraction equal to pi. But an ancient mathematician either didn’t know that 25/8 was not equal to pi, or they didn’t care that much. What they considered important was that the numerical methods got “close enough” to make something work correctly, and little else mattered. It is true that we use approximate values for pi all the time in calculations – computers cannot store irrational numbers exactly, and so they use approximations, and engineers will frequently round off their solutions. But even here, there is precision, because when an engineer does this, they are aware that they are using an inexact value, and they even have ways of keeping track of how much they have rounded off their answers. The awareness of the ‘inexactness’ of the rounded values either was not in the mind of these ancient civilizations, or they simply did not care.

While many peoples did make huge developments in advanced mathematics in ancient times, China and India stand out to me, the way mathematics was done in ancient Greece was a highly influential and new approach. Speaking as a mathematician, I would say that the huge change that began in Greece was the introduction of rigorous argument to mathematics. What this really means is, unlike peoples before them, the Greeks did not content themselves with application and approximation. They wanted to know why certain facts about numbers are as they are, and they wanted to know beyond any possible doubt. This was a revolutionary approach.

The way Greek mathematicians started this was by making unambiguous definitions of what objects like circles and lines actually are. They also laid out clearly a set of basic assumptions that serve as the rules of geometry. In that day, the tools of geometry were a compass – which is a tool for drawing circles, and a straightedge – which is any perfectly straight object (like a ruler, except a straightedge does not have markings on it like a ruler does). The straightedge tool came along with some rules – for example, every straight line segment can be extended. The compass also has an associated rule – every circle is defined by a point (its center) and a distance (its radius). These rules bring clarity about what circles and lines are. A circle isn’t some vaguely round thing, a circle is a shape all of whose points are exactly the same distance from some central point. In a similar fashion, the Greeks made explicit certain key concepts about angles, areas, and volumes.

Within a clear, structured framework, the Greeks could then find theorems, and used proofs to explain them. In mathematics, a theorem is just any true statement about math, and a proof is a bulletproof explanation for why a certain statement is true. It is worth noting that no other discipline, including physics and other sciences, have a concept of proof anywhere near this powerful. A scientist would usually be content to say a scientific theory is proven, roughly speaking, if it has a large body of empirical evidence in its favor, and very little evidence against it by comparison. This does not count as a mathematical proof, because empirical evidence could always have an exception that we just don’t know how to find. While the mathematician will frequently use empirical data to help them formulate their ideas, a mathematical proof relies only on logical argument. For this reason, a genuine mathematical proof literally cannot be mistaken, in the sense that if you accept that words like true and false actually mean something, and if you accept the conceptual framework of Greek geometry, then you cannot deny any theorem of Greek geometry.

To be fair on this point, you can always reject the framework. You can start with different definitions and assumptions if you want to, there is nothing to stop you from doing that. But by making different assumptions, you of course aren’t really doing Greek geometry anymore, you are doing some other kind of geometry that might turn out differently than Greek geometry, but it does not invalidate what the Greek geometers did.

Up to this point, I have yet to actually present a proof. I will do this in my next post, where we will talk about the most famous theorem of Greek mathematics today, the Pythagorean theorem. But before we can truly understand what the Pythagorean theorem is and what its proof means, it is important to have in mind this landmark intellectual achievement of the ancient Greeks, to which all who study mathematics today are indebted for their great contribution.

[1] Heaton, Luke. A Brief History of Mathematical Thought. London, Constable & Robinson Ltd. 2015.

Math is More Than Calculation

Before attempting to start talking about what math is and why I am so interested in it, I want to clarify what it is not. What I’d call math is probably is not what you learned in school. I think a better label for what most people think of when they hear “math” is calculation. What is taught in school, or at least the way a lot of people approach what is taught in school, is memorized methods for arriving at results. You memorize your times tables, you memorize how to add fractions, you memorize the quadratic formula. Note the repeated use of the word “memorize.” You learn that certain techniques are right, and try to replicate them. And that’s the end of it. No questions asked. When we do calculation, we do essentially the same thing a computer does. We receive instructions and execute those instructions.

So if calculation is not mathematics, what is? I don’t really take issue with calculation; memorizing certain techniques is quite useful. There is a place for that in math, just like learning basic vocabulary has an important place in English class. But in English, you don’t stop there. The problem with math classes, or the way people approach them, is that most people stop at calculation. I think the best way to express what I mean is to share how I first fell in love with math.

I was in second grade. We had already learned our times tables, and we were beginning to learn problems like 32 times 28 using the following method:

\begin{aligned} & 32 \\ \underline{\textnormal{x}} & \underline{28} \\ 2 & 56 \\ +\underline{6} & \underline{40} \\ 8 & 96\end{aligned}

In case you didn’t learn this method, the general idea is that you multiply 8 by 32, and make that result a new row, you then place one zero in the next row, and then place 2 times 32 to the left of the zero. The last step is to add together the two rows.

I had already been curious about multiplication when I learned how to do something like 32 times 8, but got even more curious when I learned this way to multiply numbers like 32 and 28. It caught my imagination. I wanted to know why we did things this way. I thought about it, and it took a while but I eventually came up with some ideas. I knew that multiplication was about grouping things together, that is, the sentence “2 times 4 equals 8” can be rephrased as “2 groups of 4 make a total of 8”. Now, say we have 28 groups of 32 people. This is really the same thing as 20 groups plus 8 more groups, and I realized this is why you had the two rows, one for the 8 and one for the 20. I also knew that 20 = 2 x 10, and so 32 x 20 = 32 x 2 x 10. So, the 0 at the right side of the second row is there because of the 10, and the other work is just 32 x 2. So I started to see why we were doing things this way.

Around this time, I had an idea that made me even more curious. I thought that, perhaps, the same thing will work for bigger numbers. If I wanted to do 789 x 123, for example, couldn’t I make add together three rows, one for 100, one for 20, and one for 3? And since the row for 10 had one zero, I should put two 0’s on the row for 100. In the notation from earlier, I thought that maybe this would work:

\begin{aligned} &789 \\ \underline{\textnormal{x}} & \underline{123} \\ 2 & 367 \\ 15 & 780 \\ +\underline{78} & \underline{900} \\ 97 & 047 \end{aligned}

We hadn’t learned this yet, so I went to my teacher after school, very excited, and asked her if that was correct. She told me it was right! I was really happy and proud, but it got better. I could see now that the size of the number didn’t matter, I could use the same trick! This teacher graciously helped me learn more math after school when she could, and let me use her whiteboards to multiply huge numbers together for fun. What I had noticed I later learned had a name – the distributive law. We normally write this as

A \times (B + C) = A \times B + A \times C

What I had noticed in multiplying numbers like 32 and 28 is that the “standard method” we were being taught was taking advantage of this rule in the following way:

32 \times 28 = 32 \times (20 + 8) = 32 \times 20 + 32 \times 8

The additional step I noticed is that you can use the same rule for larger numbers like this:

789 \times 123 = 789 \times (100 + 20 + 3) = 789 \times 100 + 789 \times 20 + 789 \times 3

Challenge for the Reader: Don’t just take my word for it! Convince yourself that the distributive property still work when more than two numbers are being added together.

This may sound weird to some, but I would probably rank this moment in the top five or ten most joyful moments of my life, and spurred a strange excitement with doing long and arduous multiplication problems. It was never calculating in and of itself that I enjoyed, though there is a sort of childish glee looking at incomprehensibly large numbers written on a whiteboard. The reason this moment had such an impact on me is because this wasn’t just something I’d been taught, this was something I discovered on my own. I understood where it came from, I knew how to explain it, and most importantly, I found it on my own.

This, I would say, is what math is really about. It is about understanding not just how things are, but why they are that way. It is about finding patterns, asking questions, and discovering why we see the patterns we do. Even more than this, math is an art form that transcends time and language. In the words of G.H. Hardy in his classic essay A Mathematician’s Apology, “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”

The difference between a calculation and true mathematics is like the difference between a road sign and Shakespeare. Both are written word, but the first is dull and the second is artistry. A simple calculation on a receipt is dull, but the writings of the great mathematicians of the past are works of art. To quote Hardy again, “there is no permanent place in the world for ugly mathematics.” We hardly glance at road signs, but we still read Shakespeare hundreds of years after his death. So it is with great mathematics.

So what does real mathematics look like? What does it mean to show why a pattern is true? Why do I and so many others find mathematics so incredibly fascinating? These will be topics for posts soon to come. But if you learn anything from this post, calculation and mathematics are not the same thing!

Where Did Numbers Come From?

In my last post, I argued that a distinction needs to be made between calculation and math. But certainly these are related, because calculation is about numbers, and math uses a lot of numbers. So what exactly is the connection?

Working from my own experience being trained to be a mathematician and educator, as well as from all I have learned about the history of math through the years, it seems fair to say that math arose primarily out of calculation. But these are not identical – it could equally well be said that poetry arose from the most primitive forms of communication, but that in no way means that literature can be reduced to the musings of a civilization that only just begun to develop language. So it is with mathematics.

A particular paradigm shift within the development of language will be useful to talk about. Language most naturally would have began as a tool for describing things around us and visible to us. Whenever words capable of going outside these limits began to emerge, the power of language would greatly increase. In particular, the ability to tell a story is revolutionary. Stories necessarily require some degree of abstract language. You need to be able to talk about times other than right now, places other than right here, things and people that you cannot see, or have never seen, or that might even not exist at all. Development from specific, immediately present events and objects to more abstract, broader concepts would greatly increase the power of language. When you can tell a story, you can also explain why things are the way they are, not just what they are.

Mathematics has a nearly identical progression. Naturally, counting physical objects is where math started. The very most ancient form of counting would not have looked much like we think of it today, either. What we do is more abstract. Say, for instance, that you have some tools that you keep track of, and that you nor anyone around you has yet developed any idea of counting things or of numbers. You still want to keep track of your tools, so you probably want to come up with a way to check whether you have them all. One way to resolve this dilemma would be to learn some verbal pattern that you’d rattle off as you pick up your tools, some pattern of sounds or movements you remember well. These rituals would most likely have been be something rhythmic, like a chant at a sporting event or musical lyrics. Perhaps you pick up one tool for every line of your jingle, or in some way you can remember. Every day, you finish picking up your tools as soon as your memorized ritual is over. If one day, you finished gathering all your tools but you hadn’t finished yet, you’d conclude that someone took something from you!

That is likely something like how things began. Notice that nowhere is the idea of a number mentioned. You don’t need number to do this. All you need is some vague notion of “more” and of “less,” and for day-to-day life that would serve you very well. In fact, this is also how we count today. We all memorize the same ritual. Our ritual sounds like “one, two, three, four, …” and so on. This is the essence of counting at its beginnings. With this comparison, we can already begin to see how things might develop. Perhaps disputes that involve what we now call counting led people to realize that having a shared ritual for counting objects would be beneficial for solving disputes. I certainly don’t claim to be an expert on these things, it is quite possible some of what I said is not quite right. But there is some reading behind this, and it does seem like this is a quite plausible account of how humans could gradually develop a counting system.

Like with language, the paradigm shift here occurs when things get more abstract. Today, we know what “two” means, even if we are not told what it is there are two of. In terms of the previous discussion, this is because we understand where “two” belongs in our ritual. But over time, this understanding develops beyond the ritual, and we start to think of the number 2 as a concept in and of itself. Concepts like number and shape, just slightly more abstract than what we see with out senses, radically change how humans were able to socialize.

With this background, it is now easier for me to explain what math is. One possible attempt at defining mathematics is the effort towards understanding concepts like number, shape, and measurement. It is not really about any particular number of trees or cows, or the shape of any particular cave or mountain, but about the ideas of number and shape themselves. For example, there are many round things, and when we do mathematics, we want to understand what roundness is. When we study number, we want to discover rules that we can use to count things.

Hopefully, this has provided something of an introduction to what math is and its history. Later on, I hope to go more in depth into some of the mathematical techniques of ancient civilizations like Egypt, Babylon, and China, because they truly did develop remarkable insights for their time. The next big transition in the history of mathematics came with the ancient Greeks. Even though Greek mathematics is roughly 2000 years old, the work done in this marvelous ancient civilization is in many important ways very similar to what is done in universities today. That key development will be the topic the next post.

What is This Blog?

Welcome! My name is Will Craig. As of this writing, I am a PhD student in mathematics at the University of Virginia. My current aim is to be a university professor. I greatly love both teaching people about various kinds of mathematics as well as working on modern research topics in number theory. I also enjoy reading and learning about pretty much anything, but especially philosophy, theology and religion, and the various sciences.

The title I’ve started off with for my blog is “Mathematical Apologetics”. To unpack what this means, we first must understand the word apologetics. This word has little to do with the English word apologize; rather, it derives from the Greek word apologia, which is a term which means “to give a defense.” This English term apologetics is a noun form of apologia, and an apologist is a person who does apologetics. It is used to denote someone defending a position or belief using some kind of reasons and evidence. This term is most commonly used in the context of defending a belief related to philosophy or religion; so you may have Christian apologists, atheist apologists, Muslim apologists, and so on.

The reason I’ve given the blog this name is two-fold. Firstly, I have loved math for nearly my whole life. And I don’t mean “math is my favorite subject” kind of love, I mean in the same way that a singer loves music or a painter loves a masterpiece. It is one step short of obsession. Math can be breath-taking, subtle, deep, and remarkably beautiful if you know where to look. Most professional mathematicians think of their mathematics as something comparable to an art form, but this idea has been portrayed most famously and poignantly in the great twentieth century mathematician G.H. Hardy’s classic essay entitled A Mathematician’s Apology. In this essay, Hardy takes the role of an apologist for the discipline of pure mathematics (that is, math that does not necessarily apply to the real world). While I do not fully agree with everything in the book, I highly recommend it to anyone who wants to know what it is like to be a mathematician.

I will say I agree with the core of his essay, which views mathematics as primarily a form of creative art, and since I also consider myself an apologist for this idea, I gave the blog this title as a way of paying homage to that great essay. It has been very frustrating that this side of mathematics is never seen, with all emphasis placed on “application to the real world.” It takes the fun out of it. Imagine if in art class you just learned how to paint walls one solid color, or perhaps you learn how to paint a stop sign. How much would everyone hate art class if that’s all you did? Sadly, math class today is a lot like that. One of my goals here is to present as best I can the beautiful, interesting, fun side of mathematics.

The second reason for this name is that I hope to discuss issues important to me outside mathematics in the manner of an apologist, using my background as a mathematician. Just as I believe most people misunderstand what mathematics is truly about, I think most of what is important to me is also greatly misunderstood. Modern “debate and discussion” is not only mostly unproductive, but harmful both intellectually and emotionally. And quite often, it seems like people have never tried to carefully understand those they disagree with, or to really understand deeply what they believe. There are plenty of exceptions to this, of course, but that we have any public figures at all engaging in such nonsense is saddening.

I hope to take the skills I’ve been developing that are important to mathematical thought and carry them into other areas in which I am interested. This will focus on areas outside of math about which I am most interested, and which matter the most to me. I plan on including issues of emotional/mental health, religious and philosophical topics, and some of the hard and soft sciences, as these are all important and interesting to me. I hope by taking the approach of a mathematician, I can try to calmly understand both sides of important issues, and to avoid common misconceptions. I hope that taking the perspective of a mathematician will bring a point of view that is not often seen or considered.

As a closing note, I also hope to have a Q&A aspect to the blog in addition to talking about what interests me. I have set up an email address for the blog, mathematicalapologist@gmail.com so anyone who wants to know more about the blog, about what I do as a mathematician, or has general suggestions or questions can contact me from the “About the Blog” page and I’ll reply as quickly as I am able to. Hope everyone enjoys the blog, and hopefully learns something.