On Mathematics: Approximation and Exactitude

The first chapter of Here’s Looking at Euclid (which is numbered at Chapter Zero) deals with the cultural differences between dealing with numbers and instinctive human reactions to dealing with numbers, which leads to a lot of discussion of approximating numbers versus exact counting.

The author ends the chapter with the conclusion that numbers are a human construct imposed on the outside world as a way to try to make sense of it. I find this conclusion baffling as it follows on the heels of this question, “If our brains can represent numbers only approximately, then how were we able to ‘invent’ numbers in the first place?” Perhaps the whole thing would make more sense if one assumed that God created numbers and mathematics, and that even our attempts at exactness are derivative from His truly exact calculations. (Pi, anyone?)

To me, the most interesting part of this exploration of the human brain and numbers was the idea that we innately tend to think logarithmically rather than linearly. That is, we tend to think in terms of comparisons and ratios rather than exact numbers as laid out on a number line. Don’t believe me? Which sounds more drastic, the difference between one and a million or the difference between one million and two million?

See what I mean?

Even those of us who lean toward logic and precision of calculation still have a human inclination to view numbers in an approximate and comparative way. (Possibly because this is more useful in everyday life, as people who tend to get caught up on precise calculations are often reminded. Counting how many items are in the carts of each person in each line of the grocery store isn’t going to save you any time, even if you do manage to calculate which line is mathematically shortest, but a quick estimate and comparison of heaping full carts vs one nearly empty cart might save you quite a bit of time.)

Now, here’s one of the interesting bits: Teenagers who were tested on their ability to rapidly compare groups of dots and accurately estimate the differences in sizes of the groups varied greatly in their ability to make these estimates. The ones who scored highest on these tests correlated to those who tended to score highly  on their school test in the precise calculations of formal mathematics. In other words, the better you are at estimating and comparing, the better you likely are at precise calculations.

This brings to mind teaching approaches that focus on the natural developmental stages of children. Perhaps rushing children past the early, colorful, comparative stages of learning math into ‘proper’ academics actually slows down their progress in the long run. I have no idea off the top of my head what that means about teaching math as specific ages, but it does seem to lend general support in the direction of allowing younger children time to focus on creative play instead of formal academics.

This chapter of the book sparked one last ponderable thought for me: If most of our formal mathematics are based on a logical, linear scale, are there similar levels of advanced mathematics yet to be discovered along the path of more intuitive, logarithmic scale?

Daily Thanks #4: Stories

I like a good romance. Let’s face it, I even like a dumb cheesy romance. As long as the guy gets the girl I don’t care if they’re epic heroes or cartoon umbrellas, I’m sold. (Seriously, I shed tears over the cartoon umbrella romance. It’s a little embarrassing.)

I like buddy stories, where a couple of guys survive an impossible number of explosions together while killing a ridiculous number of bad guys, and I like stories with so many twists and turns that I can’t figure out where the story will end up.

I like historical stories, futuristic stories, post apocalyptic stories and time travel stories. Alternate dimensions can be fun too.

For the first few months of marriage I couldn’t really get into any fiction. Living my own personal fairy tale somehow overrode any other stories for a while. But stories are so built into me that it didn’t take so very long before I decided that even fairy tales endings are better with extra stories piled into them. 🙂

Today I’m thankful for the stories that make life seem simpler, the ones that leave me chewing on questions, the ones that look at the world sideways and give me a glimpse of what might be, the ones with brave heroes and epic battles, the ones with tiny victories  and (I guess) even the dumb ones that make me cry.

 

Writer

I have a plan. This plan involves getting lots of writing done, but you may or may not see any of it here on my blog. For once in my life I’m going to finish writing a book. (Technically it’s the second time, but since I was co-writing the first time, I still only wrote half of a book.)

At Colton’s nudging, I have a plan for this month that significantly simplifies the amount of time I normally spend on grocery shopping, and might even cut down a little on cooking and dishwashing time.  I’m finally actually making space in my life to write.

The only problem is that almost coming down the stomach flu while actually coming down with a cold right before leaving on a weekend trip leaves a messy house full of dirty dishes and a fridge that desperately needs to be cleaned out but even more desperately needs to be restocked. As a result, this week is being spent trying to reshape the house into livable condition while simultaneously trying to recover from the energy drain of a weekend trip right after being sick.

The extended afternoons of writing time are so close I can almost touch them, but right now all of the small amounts of energy I can muster are being funneled toward reducing the giant pile of dirty dishes next to my sink. I’ve made progress in that the pile no longer extends to the floor by the sink, and doesn’t even look like it’s going to tip over at any moment.

I have this lurking feeling that as soon as I get the house in decent order some kind of urgent disaster that I dare not ignore will strike. In that case I’ll be very glad to have lots of easy meals all ready to prepare…

I’m going to finish writing a book.

Someday.

Things I learned from Agatha Christie: Times Change

Agatha Christie wrote sixty-six mystery novels over the course about fifty-six years. The world changed in the course of the those fifty-six years. As a teenager, getting a smattering of impressions from different points in those years, in no particular order, was sometimes startling and even a little sad.

Over that entire span of time there were always elderly people who complained about how times had changed, but at the end you could feel that they really had.  Even with less stringent definitions a ‘proper housemaid’ made fewer and fewer appearances, and while young people might gallivant around solving mysteries in any era, they moved from nudging at boundaries to barely being aware earlier boundaries had existed. The feel of the entire world changed.

And yet, however much the feel of the world changed, the bones of it never did. From beginning to end of Agatha Christie’s stories, proper housemaids or no, people fall madly in love, do crazy stupid things that sometimes end well and sometimes end very badly indeed, eavesdrop on strangers, ignore close relations, throw parties to impress distant acquaintances, poison elderly aunts, bludgeon rivals, and sometimes drop everything and risk their lives to prevent the murder of another human being.

As a teenager I might have a grasped in a distant sort of way the concept that times change but human nature remains the same. In Agatha Christie I saw it happening, and never even knew it might be profound.

Things I learned from Agatha Christie: Never Marry a Dull Person

Something else I learned from Agatha Christie is that merely sharing similar literary taste and general good sense with a person does make them a good choice for marriage. When people in your village start collapsing with alarming rapidity, when the servants start slipping on ladders more frequently than their previous clumsiness level suggests reasonable, and when your extended family has alarming rash of car accidents, you want to be in a relationship with someone who accepts the implications.

A dull person may well just wave aside your concerns with the very sensible suggestions that you’re blowing things out of proportion.

Obviously, your chances of survival are much better if you’re working with someone who’s willing consider the clues and jump into tracking down the murderer with you.

In other words, marriage is aided by a sense of shared adventure.

This particular advice has borne out in many other stories and genres as well. If you are attacked by any type of undead creature, or transported in time or between dimensions, or start developing a superpower, you’re much better off being able to trust the person you love with this startling information.

Of course, I have the best of both worlds, as Colton and I share at least an overlapping (if not quite identical) taste in literature and movies *and* a sense of adventure. I’m pretty sure being married to Colton astronomically raises my odds of survival in case a catastrophe of any genre suddenly breaks out around us.

Things I learned from Agatha Christie: Imagination is Vital

I’ve started rereading my Agatha Christie books. As a teenager they were one of my reading staples, and thanks to a thrift store with a good book selection and prices even I could afford, I accumulated a couple dozen of her books. Coming back to them after ten or twelve years of only sporadic reading makes certain themes a lot more obvious.

For one thing, I remember why I used to be concerned about eating food that was just sitting around, because you never know when someone might have poisoned a sandwich or a cookie or an apple in the assumption that a particular person was going to eat it, and by coming along and eating it you would throw off their plans entirely, not to mention dying in the process.

Another thing that stands out to me is the connection she makes between imagination and sympathy. A person with no imagination is pretty scary in her world, because that person has absolutely no idea what other people want or like. The kindest person, without the imagination to know what kindness is needed, becomes a frightening force.

This kind of person relentlessly bakes fruitcake for neighbors who hate fruitcake, knits ugly sweaters for relatives who are never cold and cannot comprehend a friend’s disappointment at being unable to travel, because traveling is so inconvenient, surely the friend wouldn’t have enjoyed it anyway.

And naturally, any person this annoying is eventually going to be knocked in the head with a poker or poisoned with arsenic laced fruitcake. (I learned that from Agatha Christie too.) So you can see how dangerous it is to be a person who lacks imagination!

I must recommend that every person cultivate great powers of imagination. Not only does it save the people around you great tedium and annoyance, but it may just be vital to a long and healthy life as well.