Not all my thoughts on infinity involve trying to teach the concepts to a six-year-old.  My daughter’s questions are definitely what got me thinking about infinity again, but this is a question that has been intriguing me for years – since high school, in fact.  I’ve tried to pose it to others many times over the years.  Some “don’t want to think that hard”, and others, many of whom are much better at math than I am, dismiss it without giving me a satisfactory reason why.  Of course, what may be satisfactory to a math expert may not be satisfactory to me.

Anyone with knowledge of high school algebra should be able to follow this.  Whether you want to or not is another matter.  But, to those who do choose to take an interest, I have a request.  Can you tell me what you think?  If you have no opinion, can you refer this article to someone who might?  I’d love to have a genuine mathematical perspective on this, if one exists.

It’s a well-known fact that you can’t get a meaningful answer dividing any number by zero.  Even dividing zero by zero is problematic.  So, look at dividing 1 by any number greater than zero: the smaller the denominator (the number on the bottom), the greater the result.  It would appear that 1/x approaches infinity as x approaches zero.  Now, the fact that infinity isn’t a real-world number might be enough reason to question that answer.  You can’t fit an infinite number of anything in the known universe, and you can’t plot infinity meaningfully on any conventional graph.

But it’s worse than that.  If you approach x = 0 from the negative side, the result approaches negative infinity.  The graph of y = 1/x looks roughly like the drawing below.  It illustrates a bizarre result, scornfully challenging intuition to get any grasp on it whatsoever.  If you approach zero from the positive side, 1/x approaches positive infinity.  If you approach zero from the negative side, 1/x approaches negative infinity.  This is why, when you ask most people what 1/0 is, they’ll say that it’s “undefined”.  Math functions in microchips have a special error condition for dividing by zero – they won’t even attempt it.  Who can blame them?  What two “numbers” could be further apart than positive infinity and negative infinity?

Very rough plotting of y = 1/x

Now, let me digress for a moment, and talk about the most common way to produce flat glass.  It was developed in the mid-twentieth century, and involves letting the glass solidify on a pool of molten tin, or some other metal with a melting point lower than that of glass.  The glass comes out a uniform thickness, and very, very flat.  The glass isn’t actually flat, however.  It’s only as flat as the pool of tin, which has the curvature of the earth.  However, even on the scale of a very large window, the curvature of the earth is very slight.  By a quick calculation, you’d need a window about 70 feet long for it to dip a tenth of an inch.  That is very, very flat.

The point is, the greater the radius of curvature, the flatter the curve.  So, what might an infinite radius of curvature yield?  Might we not get a perfectly flat curve?  A circle with an infinite radius might be the same as a line.  A sphere with an infinite radius might be equivalent to a plane.  Certain non-Euclidean geometries might become Euclidean again.

To further clarify (I hope) the issue, consider if a two-dimensional space were mapped to the surface of a sphere, as shown below:

A Two-Dimensiona Space on the Surface of a Sphere

This becomes a finite two-dimensional space, but it’s one that I’ve seen quite often in lay discussions of non-Euclidean spaces.  The X and Y axes here are great circles on the sphere, perpendicular to each other, and they meet both at the “origin” (arbitrarily chosen) and at the maximum distance from the point, halfway around any great circle passing through that point.  We can’t call this point infinity, because it’s a finite space.  It should be noted that this is a two-dimensional space, equivalent to a plane (or part of a plane), and traveling through the sphere is not possible – entities in that space can travel only along the outside of the sphere.

Along the sphere, we can map a function like y = 1/x, which meets at the maximum distance on the other side of the sphere.  I won’t bother figuring out what the function is – we can define it very artificially if we want to.  But such a function would plot something like this:

The "plot" thickens!

Now, I think those of you who’ve kept reading probably know where I’m going with this.  The bigger the sphere gets, the flatter the curve gets, and the more the actual function mapped can resemble y = 1/x.  If the radius is infinite, then each great circle could be a straight line, and the sphere could be a plane.  Granted, this is a fudge.  I don’t think that the appearance of a totally flat surface is the only possibility.  Multiply the infinite radius by two pi to get circumference, and you get the exact same infinity.  It’s hard to get a definite shape from that.

This reminds me of problems I heard of in some quantum theory models, where infinities are canceled out by dividing them by other infinities.  It’s mathematically possible for them to work out, but not mathematically required – so it feels messy.

But, all messiness aside, if you do think of the number line as a circle of infinite radius, is it not possible for infinity and negative infinity to occupy the same point on a number line – and thus, in effect, to be the same “number”?  If we allow this, it either makes better intuitive sense of the y = 1/x equation as x approaches zero, or it wreaks havoc with the concept of infinity, or at least the intuitive sense of it.  Maybe it does both.

Is it possible that the transfinite numbers transcend positive and negative?  Is infinity just too big to have a plus or minus sign attached to it?  What other implications might such a trans-Euclidean geometry have?  Anyway, that’s about all I have for now.  So, what do you think?

Thrilled with senses one through five,
And happy just to be alive,
I skipped about the house with glee,
Quite heedless of velocity.

As joyful as you’ve ever seen,
I bounded through my set routine.
Down the stairs, as quick as light,
I jumped the last few – joyous flight!

Thump!

Pain!
Stars!
Where did that door frame come from?
Owwwwww!!!

I lay crumpled in a heap.
I’d failed to look before my leap.
I wasn’t limp, impaired, or dead.
A bleeding lump adorned my head.

I pondered then, and found it strange
How quickly happy moods can change.
Nothing else on earth can rain
On my parade like sudden pain.

If you always knew what you were getting into, there would be no real adventures in life. Not too long ago, I fell into an adventure, one of immense importance to a lot of people I never knew before this started. How I got involved might sound downright thoughtless and irresponsible, but I’m hoping it will turn out to be a good thing.

For several years now, my sister and her friend and employer, a breast cancer survivor, have been doing a three-day, sixty-mile walk to support breast cancer research. They live in northern Michigan, and have been doing the walk in Detroit, until last year when they missed the Detroit walk and decided, at the last minute, to do the walk in San Diego. They liked the travel experience and decided this year to try the Seattle walk, which is in my area of the country.

So far, so good – my sister is coming for a visit, and will be serving a very worthy cause, as well. But then I stepped in it. I signed up, too. At the time, I didn’t think much beyond joining in the fun, and serving that worthy cause. Now, the words “worthy cause” slip perhaps a little too easily from a person’s lips these days. They’re used to urge people to donate to a cause, or in statements of support from people who are not donating, or even as a preface to introducing some better, more worthy cause. It takes a shot of real life to give them meaning again.

I don’t know why I hadn’t given breast cancer research more thought. I have an aunt who’s a survivor, and I had a grandmother who was – plus, there’s my sister’s friend who, largely due to my new pursuit, is also becoming my friend. Just the number of people close to me whom this has touched should have told me this cause is different. Still, I registered for the walk and booked my orientation session without giving much more thought to it.

At that session, the group was invited to share reasons why they were walking. The first person to speak up was a woman who had lost her mother to breast cancer when she was young. She had signed up for the walk the previous year, and been diagnosed with breast cancer after signing up. She was unable to go on the walk, because she needed emergency surgery during the actual walk. But, THIS year, she is in remission, and, by God, she’s going. She was also the last person to speak up. Nobody felt up to following up that story. I left that orientation without speaking a word to anybody. I was beginning to see what I was in for.

Some guys may be thrilled to find a group so disproportionately female – not 80-20, not even 90-10, but 95-5, at the very least. But I’m shy by nature, and feel awkwardness more acutely than I should. I also have to work at asking people for money. You can’t walk the Susan G. Komen 3-Day For the Cure on good will alone – you have to raise substantial donations first. I’ll overcome both handicaps. I’ve been on several organized training walks, and the people I’ve met so far are truly wonderful people. Nobody thinks any less of me for being a man. I need to get over that.

The same people, some of whom raise the required funds year after year, have eased my fears there, too. I just have to get out there and do it. I’ll figure out how. If anything particularly noteworthy develops, I’ll be sure to let you know here.

So, is this cause any more worthy than any other cause that saves lives? It might not be. But this cause has many supporters at least partly because so many lives are at stake – hundreds of thousands a year die of breast cancer worldwide. So many, who have lost a loved one, look at new developments today and wonder if their mother or sister, their friend or only daughter, may have been saved by those treatments. How many, whose loved ones die this year, will wonder the same thing in a few years’ time?

The goal of Susan G. Komen for the Cure^® is no less ambitious than a complete cure for breast cancer in all its forms. Such a cure would undoubtedly help in the treatment of other cancers and save even more lives. In the mean time, each time someone’s wife or grandmother or cherished aunt lives even a few extra years, the world is a better and happier place.

Every adventure has its trials and tribulations, as well as its unexpected blessings and benefits. But most of them don’t benefit humanity in such an unambiguously positive way. By the time I’m wearing out a nice pair of shoes over three days in September, much of this work will be done, and the money we’ve raised will already be hard at work giving back life to many whose bodies, for no comprehensible reason, started destroying themselves.

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As I said before, I and my supporters will work out how to raise the necessary funds. I didn’t write this as a direct method of raising money, but mainly to tell others (and, to some extent, myself) what I’m doing here, and why. But who am I to make it difficult for those moved to contribute to do so right now? Anyone who wants more information can start here. To contribute, you can go here.

Remember, the money isn’t going to fund a fun hike and camping trip for an adult who can afford his own hikes and camping trips. It’s going to keep thousands and thousands of deeply cherished and fruitful lives from ending years too soon.

My favorite little girl in the world just asked me yesterday, “When you were negative infinity years old, were you happy?” She’s fascinated with negative numbers now, what you get when you subtract a larger number from a smaller one, say, five from two, and she’s also fascinated with infinity – so NEGATIVE infinity, less than any number, must be doubly fascinating.

Earlier the same day, she asked me, “What do you get when you add infinity and negative infinity together?” Does she have any idea how complex the answers to her simple questions are? I told her you can get anything – from negative infinity to zero, to positive infinity, and anything in between. I was preparing to explain why, but she was already aware of many strange properties of infinity, and was thus willing, for the time being, to take this one on faith. Instead, she asked, “What’s positive infinity?” so I had to explain that this is just another way of saying infinity, that positive meant “not negative”.

She has established in her mind that “there’s no number past infinity”, but I had to clarify that there are different sized infinities. So far, she hasn’t asked for an explanation of this, but I fear I’ll soon have to start figuring out how to explain Cantorian set theory to a six-year-old. How will I approach the diagonal argument before she understands infinite decimals – or is that the next step? Will I have to discuss non-Cantorian set theory, so we can talk about whether or not there are infinities between Aleph Naught and the Continuum? It seems to me she’s dangerously close to asking questions like that – and, if she gets any further, I’ll have to study just to keep up.

So, back to her question, she was reasoning that, since everyone is older than negative infinity, everyone must have been negative infinity at one time – just like every child in her school is older than one, and each was one year old at some point in the past. I guess the concept is that, infinity years ago, we were all negative infinity years old, and we all passed through our negative years, getting older and older, until we were zero, and were born.

I answered that I don’t know if I was happy, but I don’t think I existed infinity years ago. “Was the earth invented infinity years ago?” (She seemed to have made the conceptual shift between an age of negative infinity and “infinity years ago” rather seamlessly.)

“No, the earth wasn’t there infinity years ago.” (I opted not to get into who might have invented the earth.)

“Was NOTHING there infinity years ago?”

“I think that infinity years ago was so long ago, that not even NOTHING was there.”

“Whoa.” Her mind seemed sufficiently blown, and we moved on to a different topic.

I’m flattered that she thinks about my happiness over an infinite expanse of time. Was I happy forever ago? I hope I was. I hope she was, too. And I hope we will be happy forever from now, too. At least I know I’m happy now. How could I have a discussion like that, and not be?