# THIS INFINITY IS TOO SMALL – Marilyn Armstrong

College was not, as it turned out, particularly useful for practical stuff. Although I learned a reasonable amount, it had a tendency to be the kind of thing that makes great conversation while playing Trivial Pursuit rather than while trying to figure out your household budget for the month.

Consider the subject of infinite sets. I am not a mathematician. I’m okay with arithmetic and I can figure out a basic, algebraic equation if you give me enough time and scratch paper … but otherwise? Unless it’s part of a computer language, I’m at a loss.

Finite versus infinite sets. Equipotent sets. Countable sets. Example!

I remember infinite sets because it was similar to trying to understand time travel.

An infinite set is any combination of numbers that has not end. There are lots and lots of them. All positive numbers, like 1,2,3,4,5,6,7 … and obviously, you can keep counting until the moon turns blue and the world is exhausted.

But what about an infinite set of all negative AND positive numbers, so that they go back forever into the minuses as well and infinitely forward into the positives. Forever and a day. With no end. That would be twice as big as all positive number … but equally infinite.

There can be infinite sets of only numbers which divide by three or cardinal number and any bizarre combination of fractions. They are all infinite, but some are bigger than others.

Finite and infinite sets. Two sets have the same cardinality when there is bijective function associating them. Cardinality it is reflexive, symmetric and transitive. Countable sets: the set of all integers, set of even numbers, positive rationals (Cantor diagonalization). Set of real numbers between 0 and 1 has same cardinality asset of all reals. Computability of functions.

How can one infinity be bigger than another infinity? Apparently, universes are sort of like that and now, my brain is due to explode because I can’t keep this kind of information in there.

Our personal numeric world consists of shockingly finite numbers. That’s one of the amazing parts of retirement. You have what you have and you will never have more unless you hit the lottery or have an extremely rich relative planning to die and leave his fortune for you. Retirement income just “IS.” It won’t get bigger. Retirement income pretty much stays the same while the world trundles on.

Life and the universe may be infinite, but retirement income is not.

It’s just a thought to ponder. If you feel like pondering.

# THE TRUTH OF SCHOOL

I always find myself defending school to kids. They complain it’s dull. That there’s nothing in it that “grabs” or fascinates them — and nothing they will find useful in life.

I find myself trying to explain that school wasn’t fascinating, but that many of the boring stuff you learn in it is indeed going to be useful. Like arithmetic, the ability to add and subtract mentally without a calculator or even a piece of paper and a pencil. The point of school wasn’t only to intrigue or titillate us but to make us ready to face the real world in which we all must live.

High School, really

Some studies were dull, but you needed to know it because while there’s creativity, there is day-to-day life too and unless you are one of the entitled few, you will have to do your share of it.

I was the kid who had a book in my lap so when no one was looking, I would read. Although I love science today, in school, it wasn’t interesting. Maybe it was the teachers who were dull. In high school I had a double period of botany beginning at eight in the morning when I was already half asleep. The class went on for two hours. We had a teacher who knew her stuff, but talked in a monotone. She’d start to talk — and I’d black out. Gone.

I did not do well in that class. A pity because I was interested, but she was better than a sleeping pill. Twice as good, really. Nothing I ever took knocked me out as well as she did.

Social studies which would today be … what? Social science? History? Some weird version of both? It consisted of everything that wasn’t English, math, or science. What we called “the rest of the stuff.” I was a passionate, ardent, enthusiastic reader.  I loved history and the world. But social studies? With those stupid work books where you would answer a question and then you had to color the pictures. Seriously? Color the pictures?

I flunked coloring.

English was dull, too. We had to read books that were of no interest to anyone. I suspected the teachers found them dull too, but it was in the curriculum and that’s what they were supposed to teach. They did. We yawned. I drew pictures of horses in my notebooks. Sometimes, when I got tired of horses — I never got the feet right — I moved into castles. I was better at castles.

If they let us write, I was good at that. But being good at it didn’t make it interesting. My summer vacation wasn’t the stuff to brighten my week.

The teachers droned on and on. Those of us who intended to go to college hung in there. It never — not once, not for a split second — crossed my mind that I should drop out and work at an entry-level jobs for the rest of my life because I was bored at school.

1893 Thayer Library Photo: Garry Armstrong

For me, going to college was exactly the same as going to heaven. I would go to college because I knew I could learn. I never doubted my ability to think. I was sure if I made it to college, the rest would follow. And so it did.

I learned a lot of things in college. Ultimately, the really interesting parts of my education were learned at work, when math, science, and statistics were relevant and meaningful.

When you are working, the things you learn are in a context. You discover science has a purpose. Numbers are not random shapes which you jiggle around until you get the answer or sit with empty eyes wondering what this is supposed to mean. I did stuff at work I had found impossible in a classroom.

It wasn’t my fault. It was their fault. They taught the material so poorly no one who didn’t have a special fervor for it figured it out. What a pity for everyone. Worst of all, they meant well. They genuinely did the best they knew how.

College had its share of drones and bores … but there were enough wonderful teachers — maybe a dozen — who were inspirational.
They were was enough.  For each year of school, there was at least one or two teachers who made a difference in my life. Plus, I was in an environment where everyone wanted to learn. We needed to learn.

We chose it.

I have never properly explained the whole school thing to my kid or granddaughter. I told them “Oh, it’s not that bad.”

PS 35, Queens

Except, it really can be that bad. Sometimes, it’s even worse and comes with boring teachers and brutal classmates. That is very bad. Whether they are teasing you because of your color or because you are smart and they aren’t … cruelty is cruelty and kids can be cruel.

The thing is, you don’t stay in school because it’s fun. Or because the quality of education is uplifting. You are there because you know that this is what you must do if you want to have a real life.

If you also get wonderful, inspiring, enlightening teachers, that’s better. But even if they are dull, you still need to be there.

School is the work of childhood. It’s the “why of the how” of growing up.

# How Big Is Infinity?

I’ve always loved this concept. May the bigger infinity win!

Infinity is a concept that refers to something that grows without bound. But, is there any definitive explanation of how big can it grow? In this TED-Ed video, Dennis Wildfogel explores the mind-bending concept of the “infinity of infinities” and explains how it led mathematicians to conclude that math itself contains unanswerable questions.

### THIS IS COOL. I WANT TO LEARN SOMETHING ELSE, TOO!

Video via – TED-Ed
Further Readings and References @ Infinity (Wikipedia), Skulls in the Stars, Stack Exchange

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# WHEN THIS INFINITY IS BIGGER THAN THAT ONE

###### FINITE RESOURCES IN AN INFINITY OF UNIVERSES

College was not, as it turned out, particularly useful for practical stuff. Although I learned a reasonable amount, it had a tendency to be the kind of thing that makes great conversation while playing Trivial Pursuit rather than  while trying to figure our your household budget for the month.

Consider the subject of infinite sets. I am not a mathematician. I’m okay with arithmetic and I can figure out a basic, algebraic equation if you give me enough time and scratch paper … but otherwise? Unless it’s part of a computer language, I’m at a loss.

Finite versus infinite sets. Equipotent sets. Countable sets. Example!

I remember infinite sets because it was similar to trying to understand time travel.

An infinite set is any combination of numbers that has not end. There are lots and lots of them. All positive numbers, like: 1,2,3,4,5,6,7 … and obviously, you can keep counting until the moon turns blue and the world is exhausted.

But what about an infinite set of all negative AND positive numbers, so that they go backward forever into the minuses as well and forward into the positives. Forever and a day. With no end. That would be twice as big as all positive number … but equally infinite.

There can be infinite sets of only numbers which divide by three or cardinal number and any bizarre combination of fractions. They are all infinite, but some are bigger than other.

Finite and infinite sets. Two sets have the same cardinality when there is bijective function associating them. Cardinality is is reflexive, symmetric and transitive. Countable sets: set of all integers, set of even numbers, positive rationals (Cantor diagonalization) Set of real numbers between 0 and 1 has same cardinality as set of all reals. Computability of functions.

How can one infinity be bigger than another infinity? Apparently universes are sort of like that and now, my brain is due for explosion because I can’t keep this kind of information in there.

Our personal numeric world consists of shockingly finite numbers. That’s one of the amazing parts of retirement. You have what you have and you will never have more, unless you hit the lottery or have an extremely rich relative planning to die and leave his fortune behind for you. Retirement income just “IS.” It won’t get bigger. Retirement income pretty much stays the same while the world trundles on. Life and the universe may be infinite, sort of, but retirement income is not.

It’s just a thought to ponder. If you feel like pondering.

# WHERE IT ALL COMES TOGETHER

## Converge

Photos are visual spaces where shapes and lines, objects, and people come together. Geometrically rich photos bring together distinct forms into a single visual surface.

In other words, shooting towards a vanishing point. Which is how I do most of my photographs. That’s what makes them visually interesting. Of course, it also makes this difficult to respond to because probably 75% of all of my pictures include angles that converge to a point.

But here are a sampling that include some of my favorites.