## Why Study Mathematics?

In my job, this question is one I get asked very often. To be honest, it usually comes in a slightly different form …

#### “When am I ever going to use this? What is it good for?”

As a high school math teacher for 15 years, this is one of the most common questions I received. When I began lecturing at university, I was surprised to find that I still sometimes get asked variations of this question. I suppose it’s a good question, if the idea is that at some point someone will say to you

“Determine

and have your answer on my desk by 5pm today. And don’t get any funny ideas about using WolframAlpha!”

Because the truth is, that rarely happens.

I often give a joking answer, and say flat out, “You won’t,” and then go on a rant about how math doesn’t need to be good for anything, because it is just good. Nobody ever stood in the Sistine Chapel, staring at the ceiling, asking what it was good for! They just appreciate the inherent beauty, because it speaks to their soul. Math is the same.

I think that’s a perfectly good answer, to be honest. But in a more serious light, I find the answer to the question is actually another question: “When are you not going to use this?”

Of course, there are direct applications of many branches of math. But those tend to be very specific, and these days depend heavily on software to do the heavy lifting, so I tend not to think of those. Instead, consider that football players perform bench press as part of their training, to the point that the ability to bench press 225 pounds for as many reps as possible is tested at the NFL combines. Yet not once have I ever seen a football player perform the bench press during a game. Why do they do it then? Couldn’t they just practice the skills they will actually use in a game? I can promise you that at no point during a football game does a player think “oh, this situation is just like bench pressing 225 pounds – I will apply that same skill now.” And I imagine there are very few football players who complain while lifting weights that they will “never use this in real life”. Of course, we know that the reason they train the bench press is that it increases strength and power, so that when the time comes that they need it, it will be there without consciously calling upon it.

Studying mathematics is the same. Math teaches so much if we are awake to the lessons. Here are some things I have learned, continue to learn, and apply regularly from my math studies, along with some examples of how they have impacted me personally.

#### Scale simple solutions to solve large problems

It is almost always the case that large problems can be effectively solved by breaking them into smaller problems, or by developing scalable solutions to simpler problems. For example, about 3.5 years ago I decided I wanted to learn to draw, so I took a piece of white printer paper and a mechanical pencil and drew a superhero-esque muscle man. It sucked. Like a lot. But I was not discouraged in the least by that. I was fueled by it. Why does this suck so much? I know how I want it to look, why can’t I make it look that way? I was excited by the fact that I could recognize how much it sucked, and by the prospect of working to slowly strip away the suckness. I spent hundreds of hours, solving small problems that were contributing to the suckyessence, and slowly scaling them up. Want to draw a heavily muscled arm? Learn to draw a cylinder. Then learn to draw little cylinders that lie on the main one. Then learn to draw “twisted” cylinders and tubing that changes diameter as it twists. Learn anatomy. Now put it all together. I intuitively understood platonic solids and how they interact with and reflect light. I applied these understandings to understand the types of skills I needed to hone with the way I held and manipulated pencils. I started looking closely at things I never paid attention to before. I still do this, and at no point during this process do I ever consciously say “Oh, that’s just like <fill in math course here>”, but at every point I feel exactly the way I feel when I am working on difficult math problems.

#### Being right also means proving you are

Math is really never about just “getting the right answer”. It’s about proving that an answer – or a result – is correct. The emphasis on proof is critical. In the real world, being right is rarely enough if you can’t convince others that you are. Careful, methodical, and audience-appropriate explanations are invaluable in this regard. Developing and writing proofs in mathematics is as much an art form as it is a science (perhaps even more so), and my studies in mathematics immeasurably improved my approach to constructing an audience-appropriate argument or explanation. This has had a profound impact on my communication skills, as well as my approach to confrontation. I have used this skill in more ways than I can list, but some examples are: when I have been in contract negotiations, when I deal with sales people when buying big-ticket items (and even when I bargain at markets), when I find myself moderating arguments between friends, family, colleagues or students, and when I used to work as a personal trainer and had to motivate and justify the kinds of exercise and diet choices I wanted my clients to make. In every single one of these situations, and more, I am really constructing proof. In fact, I would say that proof dominates almost all my communication.

#### Emotional attachment to a belief is irrelevant

Not wanting to be wrong about a belief, especially if it has been long-held, is normal. It is, however, illogical and possibly even dangerous in the face of proof to the contrary. Mathematics trains us to seek, understand and ultimately accept proof on its own merit, and not on any emotional yearning. It also trains us to be grateful when proven wrong, since it makes little sense to want to be wrong for even one moment longer than necessary. My training in math has led to a much more open-minded approach to new thoughts and ideas, and after careful consideration – which involves listening to argument dispassionately, asking relevant questions and weighing evidence – I find myself either happily embracing a new thought, or else more confident in the one I already had, having had the opportunity to test it rationally against a differing viewpoint.

#### Creativity and math are NOT mutually exclusive.

Not even close. Deep study of mathematics reveals and refines a strong creativity that aligns with and is mutually supportive of logic. This fusion is relatively rare, and people who have it are prone to what seem to be exceptional accomplishments. In truth, the exceptionality of it is not the accomplishment itself but the relative scarcity of people who can do it. One of my favourite examples is Leonardo da Vinci, who most people think of as a great artist, but who was also an accomplished mathematician and scientist, and who did not consider these as separate pursuits or modes of thinking. I find the same is true in my own life, although there are many people who wonder how a mathematician could be artistic.

#### Clarity lives just on the other side of contemplation

The journey math students regularly take from being completely mystified and often intimidated, to understanding and comfort is a lesson in overcoming that serves us well in all the challenges the future can bring. It instills a confidence that says, “I may not understand this right now, or even feel like I ever could, but I know I can do it.” General wisdom suggests that “easy” might seem gratifying in the moment, but true satisfaction comes from overcoming a challenge. Many people shy away from challenge for fear of failure, but studying mathematics teaches us that we can tackle large problems, even if they seem overwhelmingly daunting at the outset. An example that makes me laugh is the time I purchased a large and intricate piece of exercise equipment for my home gym (a functional trainer/smith machine combo). I bought it used, so it did not come with any assembly instructions, and perhaps embarrassingly, it didn’t occur to me to use Google. When I picked it up the seller had already “helpfully” disassembled it into n pieces, where n is large. I was completely baffled at how to reassemble it when I got it home. But I was not daunted. I laid all the pieces out on the floor, shuffled them around into sensible groups, and slowly assembled sections that made sense. I made mistakes and discovered them when they led to chaos. I backed up, took a different approach, and eventually put it together. The process was not “clean” – I hurt my hand trying to brace a nut while tightening a bolt, and cursed myself for not taking the time to get a wrench to hold it in place. But the result looks like it was assembled by a pro. I’ve had it for many years now, and it still works perfectly. I am fully aware that my engineer friends would consider this a trivial exercise, but for me it was a hard-fought and well-earned victory. This type of approach has stood me well time after time.

#### You don’t always have to see the whole path to the goal

How often have you been working on a difficult proof or problem, not really knowing if you were getting anywhere good, nevertheless continuing to take small, logical steps – always forward, occasionally pausing to reorient yourself to see if the direction made sense – when suddenly you found yourself having completed the entire thing? Some mathematicians call this the “follow-your-nose” principle of proof. A leads to B which leads to C etc. This might be the most important lesson of all. If you have a long-term goal that seems incredibly distant and perhaps overly ambitious, consider that if you just point yourself in the right direction and take small steps, occasionally reorienting yourself, you do eventually get where you want to go. Plus, the journey is so rewarding. In my life I have used this principle I learned from proof over and over and is in fact how I ended up lecturing at university, something that has been a dream of mine since the 12th grade.

And that concludes my very long answer to the common question! I hope you found something of value.

Rich

## Open-Mindedness

Recently I have been thinking a lot about why so many people seem inconvincible of certain things which I hold to be true. And while I could certainly make a list of some of these things, that is not the intention of this blog entry. Instead, I have been reflecting on open-mindedness and wanted to share.

Many people – myself included – often enter into discourse with someone of a differing opinion with the intention of convincing them to change their mind. For example, maybe your friend Paul thinks all trees in your neighborhood that are taller than 12 feet should be pruned to 12 feet or less, so as not to obstruct anyone’s view of the lakefront. You know that he’s clearly wrong! You get into a discussion. Only it’s not really a discussion – it’s an argument each of you is trying to win. Maybe out of frustration you start incorporating personal attacks. Maybe you get so angry at Paul’s refusal to capitulate, as well as the horrible things he is saying about you, that it ends your friendship. Maybe in the middle of the night, Paul prunes all of your tall trees. Maybe the next night you erect a 30 foot statue on your lawn directly in Paul’s line of sight to the lake … and so on.

It’s sad, and you don’t even like the statue, but what choice is there? Paul must be taught a lesson.

I wish this was hyperbole. Sadly, it is not. And the conclusion is clearly suboptimal.

Well … let me construct a basis for discussion with some (hopefully) fair assumptions. In doing so I’m going to have to use a little bit of math terminology, and it occurs to me that some people might not know precisely what I mean, or even be put-off by some of my more mathematical references. If you think this might be the case, I ask you to bear with me. The concepts and symbols I use are the best way for me to illustrate my point, and I’ve included here a bit of a math lesson, in case it is not something you’ve encountered in your life – it will clarify some of the words and concepts I use for the rest of this article. Of course, if you feel there’s no need for you to read this section, by all means scroll past it and keep reading (I won’t feel bad).

Some Math concepts

Sets
Mathematicians like to talk about collections of values that are somehow related, and when they do, they use the word set. We use curly brackets to list the objects (known as elements) of a set. So for example the set $F=\{apple, orange, banana, kiwi, peach, nectarine\}$ is a set I have named $F$, and just so you know, it is the set containing all the fruits I might bring to work with me in my lunch. A subset of a set $S$ is another set that only contains elements from $S$. So for example $M=\{apple, kiwi\}$ is the set of fruits I brought to work in my lunch on Monday, and is a subset of $F$. On the other hand, $A=\{apple, pineapple, banana\}$ is not a subset of $F$.

A Little Bit of Algebra (Apologies to the Arithmophobic)
Consider this simple algebra equation:
$\displaystyle 3x+4y=7$
The $x$ and $y$ are understood to be symbolic of numbers, but the use of symbols mean that they vary – which is to say, they are variable. The equation is a statement. In this particular statement,
$x = 1, y=1$
would be a valid solution (i.e., the equation becomes true), since
$3\times 1 + 4\times 1=7$.
So would
$x = 5, y=-2$,
since
$3\times 5 + 4\times (-2)=7$.
However
$x = 5, y=2$
would not be a solution (i.e., the equation becomes false), since
$3\times 5 + 4\times 2=23$,
which is not 7.

Statements
In math and philosophy, a statement is a sentence that must either be true or false (but not both, and not maybe). Very often the truth value (i.e., “true” or “false”) of the statement depends on values for variables contained in the statement. The algebra equation above is a statement. Another example is the statement “I like cheese”, which contains two variables: “I”, and “cheese”. If the “I” refers to “Rich Dlin” (i.e., it is me speaking and not you), and the “cheese” refers to “Havarti”, then the statement is true. If the “I” is “Rich Dlin”, and the “cheese” is “Cambozola”, the statement (I promise you) is false. Notice that if the “cheese” were to refer to “gingerbread cookie” the statement would be nonsense, since “gingerbread cookie” is not a cheese – even though it is true that I like gingerbread cookies, it is irrelevant in the context of this statement. A mathematician would say “gingerbread cookie” is not an element of the set of all cheeses. Going back to the algebra example, {(1,1),(3,-2)} is a subset of the set of solutions to the equation given. The actual set has an infinite number of solutions in it, but that’s more than I need to talk about here. What I will say is that the truth value of the statement “Three times John’s favorite number plus four times Gail’s favorite number will yield seven” is:

True if (“John’s favorite number“, “Gail’s favorite number“) belongs to the set of solutions of 3x + 4y = 7,

False if (“John’s favorite number“, “Gail’s favorite number“) does not belong to the set of solutions of 3x + 4y = 7, and

Nonsense if, for example, John claims his favorite number is “cinnamon“. Be on the lookout for nonsense – it is surprisingly common.

The Assumptions

Ok. Welcome back. Here are the assumptions I was talking about:

All questions have a right answer
… when the answer is justified properly with a well framed statement.
The truth value of the statement may be subject to variables that change which answer is correct, but with a fixed set of values for the variables, there is a right answer. For example, the question “Should all trees taller than 12 feet in our neighborhood be pruned?” could be answered “Yes”, justified with the statement “It is unacceptable for some trees in our neighborhood to block sight lines to the lakefront”. Note that here the answer to the question is “yes” if the statement is true, and “no” if the statement is false, and may reasonably depend on whether or not the tree is also so wide, or part of a grove, as to make it impossible for a resident to see the lakefront from a different angle standing on the same property. It may also depend on whether 12 feet is a reasonable height with respect to whether or not sight lines get blocked. In this case these variables need to be introduced into the statement, or else agreed upon as not being variable.

The right answer may well not be knowable
… even with the variable values fixed – which doesn’t mean there is no right answer!
As an example, consider the question “How many humans are alive on Earth right now?”

• The number changes many times in a short span of time. So the truth value of the answer depends on what time it is indexed to.
• The answer is subject to a definition of “alive”, and the answers to whether or not some organisms are living humans are in dispute.
• “On” Earth needs to be defined. If I am in an airplane, am I on Earth? What if I am in low orbit?
• However there is an answer, if we fix the variables.
• There is currently no way, even with the variables fixed, to know the answer.

Knowing the truth is inherently valuable.
This is a big one. Many people demonstrate by their behavior that they do not adhere to this assumption. A simple example is the person who refuses to go to the doctor about a problem because they are afraid of what they might find out. In some ways, not wanting to know the truth is a human quality, especially in situations where a false belief has spawned an entire tree of values and beliefs we have been living by. If the root belief is false, what happens to the tree?

When it Comes to Truth, What We Want Doesn’t Matter
So with these assumptions, my position is that for any belief I hold, I am either right or wrong, and that I may not be able to tell. So then what am I to make of someone who disagrees? Can I immediately conclude that they are wrong? Clearly not. However I freely admit I want them to be wrong, so that I don’t have to be. After all, being wrong has some negative implications. On a fairly benign end it means I have been somehow deluded, which injures my pride. On an extreme end it may mean I have to discard an entire tree of conclusions that were premised on my error, leaving behind a buzzing hive of uncomfortable questions and observations about my previous behavior. But if the root belief is actually wrong, what choice do I really have? Since it is rooted in falsehood, the whole tree is an illusion anyway.

Here is a hard truth: What we want has nothing to do with what is true. I want there to be peace in the Middle East. But there is not peace in the Middle East, and no amount of wishing on my part, no matter how fervent, can alter the truth value of this or any other statement. On the other hand, what is true can and should definitely impact what I want. What we all want.

Ok. Here is another statement that is tautologically true: In the set of things I hold to be true, some might be false. And from a probability perspective, I am also comfortable saying that in the set of things I hold to be true, some are true, and some are false. I want to say “most are true and some are false”, but I am honestly not sure I have a reasonable argument to claim that, so we’ll leave it there as a desire more than a fact.

Now I will focus on statements where the truth depends on fixing values for the variables in the statement., which to me is the core of the shades of gray argument: In cases where there is a continuum of possibilities between true and false, almost everything in the set of things I hold to be true lies somewhere within the boundaries of the continuum, rather than on one of the ends.

Here a philosopher or mathematician might (and should!) argue that there can be no continuum between true and false, since those are binary options. My response is that I am talking about a sphere of reasonable answers centered on the truth, where anything outside the sphere is easily demonstrated to be false (or worse, nonsense), but things get a little more touchy inside the sphere. This is a consequence of my point about the truth of a statement depending on fixing values for variables the statement depends upon. To elaborate on this, I am going to define something called an assumption set.

Assumption Set
Suppose a statement depends on a set of variables. For example, consider the statement “Running is good for you.” The truth of this is not absolute. It depends on some variables:

• How much running (the quantity of the running)?
• How intense (the quality of the running)?
• What preconditions do you have that running would exacerbate (e.g, bad knees, asthma, heart problems)?
• Where do you plan to do your running (road, track, beach)?
• and many more.

So before we could discuss whether the statement is true, we would have to fix values for these variables. I call these fixed values the assumption set. So for example an assumption set for this statement could be
$R=\{45 minutes per day, at 80\% of maximum heart rate, \{sensitive to sunlight, plantar fasciitis\}, track\}$.
Notice that one of the elements (the preconditions) in this assumption set is itself a set – that’s completely acceptable. On the whole, I would judge this assumption set to be a reasonable one – which is to say, the elements of the set have a probability associated with them that makes them not unexpected in the context of discussing the claim that “Running is good for you.”
Another assumption set could be
$S=\{15 hours per day, at 120\% of maximum heart rate, \{multiple hip replacements, torn Achilles tendon\}, Interstate Highways\}$.
On the whole, I would judge this assumption set to be very unreasonable – which is to say, it is highly improbable that this would be an assumption set on which the claim “Running is good for you” would be a relevant discussion.

Reasonable Answers (Approximately True?)
A reasonable answer to a question can be defined as a statement that is true when evaluated with a plausible assumption set. That is to say, the assumption set is comprised of elements that have probabilities high enough that if we observed them we would not be surprised. In situations where the variables are in constant flux, the approximate truth value of a statement may be argued as the one that holds given the most likely assumption set. In cases like this, we may generalize a statement as true, while being willing to challenge it in the face of a game-changing assumption set. We maybe won’t talk about who gets to define “plausible”, even though there are times when that becomes the most relevant thing.

Arguing(?) With an Open Mind
Here I have chosen to use the word “arguing”, even though in truth I prefer the word “discussing”. That’s because most people seem to think that discussions between people in disagreement need to be arguments. I disagree. Remember the assumption that we are not right about everything? And remember the assumption that knowing the truth is inherently valuable? These two should premise every discussion we enter into. So when discussing the answers to questions, or the truth about statements, we need to do our best to remember that what we are trying to do is get as close to the center of the sphere as possible, because that is a good thing to do, and because we may not be there yet.

Of course, we all think we are closer than an opponent. If not, we wouldn’t be having the discussion in the first place. But keeping in mind that if two people are in disagreement, one of them must be wrong, a productive conversation is one where at the end of it the parties have converged on something they both hold to be as close to true as they can see getting. When this happens, the world gets a win. I’ll list some techniques for true open-mindedness.

Discussing With an Open Mind

1. Remember that you might be wrong.
Put another way, be willing to change your mind, or adjust the approximate truth of what you believe.
See, you believe that you are probably right. You may even believe that you are certainly right (although for the truly reflective, certainty is a pretty difficult thing to attain). But your opponent has the same thoughts. Both of you probably have many reasons for these. And they probably have a lot to do with assumption sets, and which one of you is applying the most plausible set. Sometimes the discussion is not about the truth of the statement but on the plausibility of the assumption set. Keep that in mind. Yours may be the less plausible. Or maybe both assumption sets are equally plausible, in which case the statement can be split into two (or more) more detailed statements that include some of the differing assumptions explicitly. But keep in mind that emotional attachment to an assumption set can and will blind you to the plausibility of an alternate set, and ultimately cause you to refute a statement with unreasonable (even fanatical) obstinacy.
2. Have higher expectations for yourself than you do for your opponent.
This means you need to challenge yourself to inspect the assumptions and claims of yourself and your opponent objectively, even if they are not doing the same thing. When you do this – and do it out loud – they hear that. Look at elements of the assumption sets and objectively evaluate their probability. Also evaluate whether they change the truth value of the statement or not. And be prepared to evaluate whether or not they render the statement as nonsense – this happens surprisingly often but it’s not obvious until it is isolated. Discussing things this way models a behavior that is necessary for the two of you to converge on a conclusion you both agree with. And if you are consistent with it, your opponent will often adopt the same style, if only because they think this is the way to convince you they are right.
3. Thank your opponent, regardless of the outcome.
I don’t mean this as a politeness. I mean this in the most sincere sense. Any opportunity we get to reflect on our set of beliefs is valuable. Sometimes your opponent and you will converge. Sometimes you will not, and they leave the exchange completely unmoved, perhaps even claiming “victory”. This is sad, since the only true victory would be a convergence of opinion, but ultimately it is not relevant to your own experience. Make it so that if you have moved on a topic, it is because you discovered something you were not considering, or were considering incorrectly, and now you are closer to the center of the sphere of truth. If you do not move, make it because you were not presented with any strong evidence that you needed to. In either case your beliefs will have been strengthened in some way, either because you changed to something as a result of new insight, or because you were challenged in some way, and it was unsuccessful. For this you have your opponent to thank.

How to Spot Real Open-Mindedness
Many people claim to be open-minded. It may be true, or it may be a trick (some people say it so that when you fail to convince them of something it will prove they were right). True open-mindedness doesn’t mean you are ready to believe anything. It means you are willing to change your mind when presented with evidence that objectively compels you to do so. If you know of (or are) someone who has changed their mind in the moment, during rational discourse, but who was fairly slow to do so, they are probably the type of person I am describing. This goes back to my point that we are probably not right about everything we believe. Which means mind-changing can occur. Which means if you’ve seen it occur, it occurred in someone with an open mind.