## Two-ness

Last week’s post was going to be about the number two, but I got diverted into talking about existence/non-existence instead. Existence/non-existence is only one of the many attributes that comes in only two possible varieties or types. Up and down, left and right, in and out, positive and negative.

These attributes might be associated with another attribute representing a magnitude, such as distance, weight or other attribute. So we may say 20 metres to the left, thus locating the object or event in relation to the datum or origin. Both attributes are required in such circumstances, since the directional attribute (left/right) does not completely locate whatever it is, event or object. Neither does distance, by itself, locate the event or object.

Relative to datum, in a three dimensional world, any three axes define direction and the datum itself divides the direction into two opposite parts. If you include the fourth dimension of time, the datum, now, still divides the direction into two parts, before and after. This of course can be extended to as many dimensions as you may choose to conjecture.

One interesting two-ism is the two-ism of a mirror. When you look in the mirror you see an image of yourself. When you move your left hand, the image appears to move its right hand, and the image’s hair parting appears to be on the opposite side to yours. This is a mind trick, since if you see a person raise the hand on their right as you look at them, your mind says that it is their left hand that has been raised. If they have a parting on the left as you look at them, your mind tells you that their parting is on their right.

This illusion is so strong that people misunderstand the reason why words appear reversed in the mirror, and why it is hard to trim your moustache, or pluck hairs in the mirror.

Many people are puzzled because a mirror appears to reverse things left-to-right but not up-to-down. It doesn’t – your left hand is still on the left, and your right hand is still on the right, your head is still at the top and your feet are at the bottom.

The trick is that your nose is closer to the mirror than the back of your head and the same is true of the image. The image’s nose is closer to the mirror than the back of the image’s head. If you draw a map of yourself, the mirror and the image, you will see that the mirror reverses the axis between the original and the image. The front/back axis. Once you see that, it is obvious, and it is hard to see how you could have thought otherwise. It doesn’t help your coordination when you part your hair though!

When we consider the number two, it is an interesting integer, the second of the natural numbers. Interestingly we use the second ordinal number to describe the second natural number, and we use the second ordinal number in that definition too. I’m sure that the circular nature of this description is apparent.

I’m a fan of the axiomatic approach to number theory. An axiomatic system consists of a set of axioms that are used as the basis of reasoning. A theorem in such a system is a set of steps leading from a premise to a conclusion. A premise should be the conclusion of a previous theorem.

Skipping a lot of details, one axiomatic approach is to define a function S, the successor function. S(x) then refers to the successor of x, where x is a natural number. So S(7) is 8, S(1,000,000) is 1,000,001. S(1) is 2, and we have a non-circular definition of the number 2. Erm, almost. The number and its successor form a pair and a pair has how many members? Two. There’s still a whiff of circularity there, to my mind.

Two is an even number and the first of them. An even number is a number which can be split into two in such a way that the two parts are the same number. To put it another way, if you take an even number of stones and put them alternately into two piles, you will be left with two piles each with the same number of stones. If you take an odd number of stones, and perform this test, you will find that the two piles have a different number of stones.

If you consider the set of even number and the set of all natural numbers you might conclude that there will be less even numbers than natural numbers. Paradoxically, there are as many even numbers as there are natural numbers.

It is possible to demonstrate this by a process of mapping the even numbers to the natural numbers. 1 then maps to 2, 2 maps to 4, 3 maps to 6 and so on. This mapping process is also called ‘counting’. For each and every natural number there is a corresponding even number and for each and every even number there is a natural number. The two sets of numbers map one to one. If two sets map one to one, it is said that their cardinality is the same, or in common language, they are the same size.

We are more used to finite sets of things (like the set consisting of a pack of cards) than infinite sets of things (like the set of even numbers or the set of natural numbers). If you take half the members of a finite set away, you have a smaller set of things. For example if you take all the black cards out of a set consisting of a pack of cards, the resulting set is smaller, but for infinite sets of things like the natural numbers this is just not true. If you take the odd numbers from the set of natural numbers, the resulting set of even numbers is the same size as the original set, not smaller.

Much of the above is far from rigorous, and I’m aware of that. However, the main thrust of the arguments is still, I believe, valid. Numbers are fascinating things, with each one having unique properties, and a whole lifetime could be spent considering just one number.

## Love Actually….

(I’m running late this week. I hope to be on time next week).

Love. The word is thrown around with gay abandon, people claim to be motivated by it when they do extraordinary things. It’s been the subject of literature, from classics to stuff which perhaps should never have been written. Atrocious poetry attempts to define it and celebrate it, and sublime poetry achieves its heights for the same reasons.

There are many sorts of love, man for woman, woman for man, and also the love of a person for another of the same sex. People love animals and people love children, though there are certain loves of these kinds which are strange, bizarre, or wrong. There are the loves for team mates or squad members which strengthens the team or squad, to the extent that in wartime a squad member may sacrifice himself for the sake of the others in the squad.

People love things. The new Maserati, Holden or Ford. The latest iPhone. An iPad, or other tablet. A new dress, new shoes, new Gucci bag. A new hairdo, new sneakers or a nice juicy steak.

I love a good lie-in in the Morning. Some (who in my opinion are slightly insane) love to be up with the lark. Some love a tropical beach, others an alpine traverse. Some love to run, to the extent where sometimes they will run for hundreds of kilometres.

Most people love a challenge, a crossword puzzle or Sudoku. Some love competing with others, some love to challenge themselves by jumping out of a plane, or climbing a high peak in the mountains.

Maybe some would quibble that I have used the word ‘love’ above where others would have used ‘like’. I make no apology for that as the one shades in to the other. But what are the characteristics of love? I’d say that no one definition fits all cases, as is common with any human characteristic.

Love primarily is understood to be most simply defined by the (usually) male/female couple. In a marriage ceremony, at least in the Christian rites, the couple pledge to love each other, (amongst other things). In other religions where marriages may be arranged by the parents and the couple may not know each other very well at the time of the marriage, I do not know what pledges are made. Of course, it is not unknown for marriages to be arranged in countries that have adopted the Christian religion. If fact it is probably more common than people realise. The opposite probably holds in countries where the religion is not Christianity.

I’m aware that the above is horribly full of unwarranted assumptions and suppositions about culture of which I know little. Someone once joked that in a Hollywood love story, boy meets girl, they fall in love, and get married. In a Bollywood (Indian film industry) love story, boy meets girl, they are married, they fall in love.

A quick scan of the synopses of both Hollywood and Bollywood shows that the truth in much more complex. The Internet Movie Database like 50 Best Bollywood films, and the teasers for the films show plots which would not be out of place in a similar list of Hollywood love films, with only minor amendments. The Hollywood films tend to replace parental pressure with a societal one, but much else remains the same. The gloriously over-generalised Hollywood love story is boy meets girl, they are forced apart because perhaps one is one is from the wrong side of the tracks (a Hollywood favourite), or they initially hate each other, or they are about to be married to others, or they somehow misplace each other. Finally they resolve whatever difficulties separate them.

Perhaps there is only one love story, across all of mankind – boy meets girl, difficulties keep them apart and either they resolve them (happy ending) or they don’t (tragedy). Now I come to think of it, most love stories apart from fairytales are of the second sort – the films “Love Story”, “Titanic” and perhaps “Gone with the Wind” bear this out.

But what is love? It is something some people spend all their lives looking for, and something which others find easy to find. Spousal love seems to be a binding force. It creates an unbreakable team and gives a couple extra power over adversity through synergy. Spousal love can lead to a long lifetime together and passing on within hours of the second spouse when the first dies.

Love in unidirectional. One can love someone without being loved back, but if one is loved back a positive feedback is achieved. Love is happiness, except for the case where the love is not returned, not even recognised. Love is eternal, except when the love ends. That is, if one loves one cannot conceive of the love ending. Love is generous, gentle and giving, unless the love of the loved one is claimed by another, in which case love engenders hate and loathing. Love is unselfish, unless the love is such that family and friends are cast aside. Love is selfish when the lovers are so engrossed in each other that worldly events pass them by.

The Bible says this about love:

4 Love is patient, love is kind. It does not envy, it does not boast, it is not proud.
5 It does not dishonor others, it is not self-seeking, it is not easily angered, it keeps no record of wrongs.
6 Love does not delight in evil but rejoices with the truth.
7 It always protects, always trusts, always hopes, always perseveres.

This verse from 1 Corinthians 14:4-7, which is often used in marriage ceremonies in Christian churches only gives the positive aspects of love which is returned. Misdirected love has caused everything from domestic disputes to full scale wars. The mythical Trojan Wars were supposed to have been about the love of Helen who was already married to Menelaus for Paris. The story may have been based around true events which may have separately happened over some time around 1190BCE.

One can paraphrase the writer of the Epistile and say “Love is impatient, love is unkind. It is envious, it boasts and is proud”, because often it is. Love is not always good. But when it conforms to the Epistile writer’s definition, it is the oil that helps to make the world go around.

A self-imposed deadline is, I think, the worst sort. I set my phone to alert me once a week to blog something. And it just went off. I have no definite plan so this is going to be off the top of my head. Call it “philosophy” or call it “miscellaneous”. I don’t care!

Today my daughter texted me a picture of a smallish fungus. (Aside – the spell checker doesn’t recognise “texted”. The spell checker needs to be updated!) Apparently my grand-daughter, who knows of my interest in fungi, thought that Grandad might like this “interesting fungus”. Who knows, because of me she might grow up to be a famous mycologist, the one who saves the human race from the mutant spores. Hmm!

Which made me wonder about the effect we have on our descendants. Obviously there are two ways we can do that -the genetic and the environmental. Our genes may predispose our offspring to certain tendencies and our personal influences plus the environments (in the widest sense) in which we live effect the way that we and our children behave.

Of course I don’t know where Loulou will head in life, but I do recognise some traits in her which I see in myself, but I can’t tell whether or not these are inherited traits or learned ones. Computers are no mystery to her, and she knows that it is possible to take a photo (on a phone) and send it to Grandad. She knows how to unlock the iPad and has no fear of the Android tablet. She and I have a liking for a particular on-line game.

Loulou’s Dad works with computers. She has two older brothers who are computer literate. Her mother grew up  with Commodore 64s and computer games loaded from tape. Loulou’s mother’s Dad (me!) worked in computers from before Loulou’s mum was born, but I don’t think I bought the job home, though we did get a Vic 20 and later a Commodore 64.

But Loulou is not a geek. When we went on a walk Loulou decided that tights, t-shirt, a tutu, and gumboots were the appropriate wear. (I think that, stylistically, it worked). She’s a fan of Dora the Explorer but is not a fanatic. Maybe she will grow into that. However, her Mum and Auntie weren’t particularly doll orientated.

Hmm, this article seems to be all about Loulou. It’s too early to tell, but I won’t be disappointed if she goes down the geeky route. I’d be interested in how she gets on in that world. I won’t be disappointed if she *doesn’t* either. Either way she’s the most wonderful granddaughter.