Category Archives: Teaching

Freedom From Mathematics

Growing Up To Be Big And Strong

Self-reliance is an interesting idea and one that we comfortably accept as a form of strength. Our cultural overtones tell us that dependence is a weakness, or at the very least, independence is a strength.

We have been praised since childhood on our ability to do things on our own.

“She took her first step by herself!”,
“You went potty all by yourself!”,
“You tied your shoelaces all by yourself!”,
“You’re riding a bicycle without training wheels!”,
“Hey, you’re in Little League now! No more tee-ball!”,
“Wow, your retirement portfolio is not completely dependent on one asset type!”

and so on.

We using words like “crutch” to connote dependence. As a result, a subtle bias creeps into our mindset from this type of repeated metaphoric language use. Consider these statements, for example:

The bully: “Aww, you want to go and run back to your momma?”
A political view: “You need to get off welfare, and get a job and support yourself.”
A social view: “She’s so clingy.”
The deadbeat’s nemesis: “Must I do everything for you? Can’t you do anything for yourself?”

But what is dependence? Is it a crutch for the feeble and invalid or is it more in the sense of ‘robust redundancy’ like we have when discussing, say, computer networks?

It’s both — there’s good dependency and bad dependency once we condition on a specific event. And this goes back to a general principle I’ve maintained, which is that the vast majority of things are neither unconditionally good nor unconditionally bad. Unconditionally, certain things have a high likelihood of being good or bad — for example, it is a bad idea to drill a hole in your skull. But given a certain type of brain trauma, a qualified surgeon can legitimately make the (good) decision to drill a hole in the skull. “Bad idea” presupposes that death is a bad thing.

Unfortunately, we, as a society, have become, at least it feels like, ‘expected value’ maximizers. In other words, if something has a higher likelihood of being bad than good, then we’ll just classify that thing as ‘bad’ without context. Guidelines eventually become unbreakable rules.

In some cases, the ‘guideline-to-rule’ paradigm works out well because we’ll tolerate the low probability inefficiency where the rule really did need to be broken. But, in a somewhat meta-argument, most guidelines oughtn’t be ‘rule-ified’ (and of course, the meta-argument is that I’m providing a ‘rule’ to keep things as guidelines. That rule is a result of an expected value maximization.).

We, as a society, also engage in dualistic thinking. “It is this or it isn’t this.” We like binary states, I think. Look at how much we can do with on-off switches (computers). But our dualistic thinking pre-dated modern computers. We look for opposites because we understand one concept better when we have a frame of reference. That frame of reference happens to be an ‘opposite’ — everything that the first concept is not. It’s good for set theory, but it’s bad for language — though I should say, “it’s not necessarily good for language”.

What is the opposite of weak? Strong. What is the opposite of dependent? Independent. Combine this with a little word association and we have what we understand as ‘cultural’ values. If a crutch is a form of dependency, and being dependent is being weak, then a crutch is a weakness. And ‘a’ crutch turns into ‘any’ crutch — guidelines to rules. We also have other word associations: independence is freedom, freedom is good. Thus, using the dualistic mindset, we have ‘dependence is enslavement, enslavement is bad’.

But there is another view on dependence. It’s called ‘support’. People who are recovering from an illness or disease be it physical or mental, may point to their ‘support’ group — their robustly redundant network — for providing strength. Never once, though, is having that support group a weakness in an absolute sense. The person who is recovering, will ultimately want to be independent, but it’s not an independence of support, but an independence from that which was debilitating.

Friends are an example of a support network. We oughtn’t try to become independent of them nor should we view our emotional gratification that we receive from friendship as a form of weakness. Of course, this perhaps should come with the understanding that we are individuals and we do our own thing — so find the balance.

The attraction towards independence is understandable. It’s about control. The fewer things we are dependent on, the more control we have. More control means more freedom. And freedom is good.

If a person needs no one, needs nothing, then what is he? Is he a fountain of giving? Can he actually have friendships, relationships, etc.? There is some good theology and philosophy surrounding this notion and I won’t pretend to provide any radically deep insight here. But a practical level, that does not have us ascending to the heavens, when pushed to an extreme, this type of independence leads to isolationism. This is the bad kind of independence, much in the same way that relying on someone or something entirely is bad. They both sap us of the human experience.

Why do we want freedom?

Is that even a question to ask? Isn’t it just obvious that we want to be free? Actually the question itself is ill-posed. Freedom from what? Freedom from that which is bad. But what is bad? And what bad is, is generally unclear. So, I’ll say that we, as individuals, want to be free from the things that we, as individuals, find individually to be bad. My bad isn’t necessarily your bad, in other words. We want to be free from the things that hold us back, the things that are debilitating to us.

These things are a matter of personal preference. The majority of us will find that ‘how-to’ independence is a good thing. Many of the examples I gave at the beginning were examples of ‘how-to’ independence. We don’t need anyone else to tie our shoelaces for us, for example. But what about emotional independence?

I’d venture to guess that many of us don’t really consider the idea of emotional independence as a sensible one, especially when it comes to the ‘good’ emotions: love, joy, happiness. We want to share those emotions with others and want it to be shared with us. We do, however, want emotional independence from the negative emotions: fear, hate, anger, jealousy because we don’t want to be controlled in that way. We are content to be controlled in a good way.

Seeking external feedback can be viewed as weakness. We’ve all been asked, have probably asked ourselves, or heard, “Why are you seeking his/her approval?”. But seeking external feedback isn’t a weakness, it’s about what we do with that feedback that will define strength or weakness.

There’s the oft quoted entrepreneurial motto: “Don’t worry about what the critics say, prove them wrong!” or something to that effect. Do we buckle under the criticism or do we stand strong and mighty and push forward regardless? This is the type of imagery that the independence-dependence, good-bad story gives. But again our guidelines-to-rules habit gets in the way and sends the pendulum into “I’ll do whatever I want and not heed any warnings of any sort” territory.

In this way, we drain ourselves of the human experience by going too far, too often, in extreme directions. Somewhere in the middle, is the “Goldilocks Zone”.

Some Irony And The Education Tie-In

What’s simultaneously amusing and sad is the dependence we have on mathematics and our (societal) denial of it. I am often the recipient of statements like “I’ve never needed <math topic>” or “When will I ever use this?” or the really delusional “We’re in the 21st century, why do we insist that kids know their times tables when there’ll always be a calculator?”.

I understand, to some extent, where people are coming from when they make these statements since at their personal level, this is knowledge they are comfortable not knowing — that is, they are comfortable with having given up the freedom they could have if they understood mathematics better. This is fine — how far into the sand one wants to bury their head is personal preference. There are plenty of things I don’t know in other areas (and in mathematics as well) and I’m sure I’m constrained by that lack of knowledge.

On the one hand, if you are part of modern society in any capacity, you are a beneficiary of mathematics. That smart phone you have, the viewing device through which you are reading this article, the electricity that runs through your home, etc. are made possible because of mathematics coupled with other areas of discipline. This is undeniable. The modern conveniences that we have, didn’t just fall from the sky.

On the other hand, if one is part of modern society, it’s easy to take the stance of “somebody else will figure it out.” It is, practically, cultural fact that mathematics is hard. Mathematics is also one of the few subject areas that many people will have a shared common experience of misery. And since it’s a bad experience, people want to find a way to be free of it. Freedom from mathematics means denying its value at a personal level and extrapolating it to a societal level.

If at the root of the desire to be free from mathematics is indeed that learning it was a miserable experience, then that’s perhaps where we should begin. And I understand the naivety in the statement. But it’s a simple philosophical starting point to guide us.