Wikipedia – Me on the net

Crime and Punishment

I’m convinced that most people go through life making few real choices. Oh, we all can all look back and say “Oh, decided to do so-and-so”, but I’m convinced that we didn’t make a choice in the sense of sitting down, making list, considering alternatives, options and consequences. I guess that the nearest that we would come to doing that would be when we are budgeting, or deciding where to go on holiday.

No, our “choices” are driven by needs (“We need to go to the mall to buy….”) or desires (“Let’s eat at the Peppermill today. I had a great omelette there last week!”). Someone comes up with a need or desire and we go along with it or we don’t.



Embed from Getty Images

My point is that there is a thing, which I believe doesn’t exist, called “Free Will” which allows a free choice between alternatives. There is a philosophical war going on between the believers in “Free Will” and those believing in “Predestination” for millennia.

It seems to me that the closer you look at the Free Will/Free Choice thing, the more you discover the reasons that people make the choices that they do. The more reasons, obviously the less “free” the choice will be, and the more you dig the more reasons you find and the less free the choice becomes. I contend that eventually, the room for freedom of choice shrinks to nothing.



Embed from Getty Images

An interesting test would be to put people into a box with a screen and two buttons, and not give them any instructions except “Go into the box and sit down”. Maybe play them some elevator music to set the tone. When you pull them out after 10 minutes or so, they will have pushed zero, one or two buttons. If you then say, in a neutral tone, “You pushed zero, one, two buttons”, they will immediately begin to tell you their justifications for their action or actions.

Justification are not reasons. People often something like “Well, you left me in there with no instructions. Buttons are for pushing, So I thought that I would push one and see what happens” or “Nobody told me to push the buttons, so I didn’t”.

These statements say little about the reasons for the person’s actions or inactions. The reasons that they press or don’t press the buttons relate more to a person’s character and state of mind at the time than the justifications given. For instance, the person may be a rule follower, and without rules, would do nothing. Another person may be a rule breaker and, without rules, feels free to do whatever they wish. We all are a mix of both types of course.

People don’t think “I’m a rule-breaker, I’ll push a button”, so they can’t really claim this as a reason for their choice, and they can’t be said to have made a free choice if constrained by this innate or learned facet of their behaviour.



Embed from Getty Images

Some people believe that, in spite of the postulated fact that there is only one possible outcome when a choice is made, that a choice has in fact been made, since if the circumstances had been different a different choice would have been made.

This seems to me to be dodging the question. (It’s not “begging the question” in the strict usage of the phrase). I look at it like this: if we were to roll back time to before the moment that a choice was supposedly made, such as the point when the door of the box closed, and we let time roll forward again, could anything different happen. It is my contention that since all other factors remain the same, that the same thing would definitely happen.

תרשים כללי של פנופטיקון, מבנה הטרוטופי (פוקו) (Photo credit: Wikipedia)

Which brings me to the point of this post, which is, how do we justify meting out punishment for a crime, when the criminal was unable to choose not to commit it. Take away the concept of free will and punishment of the criminal seems cruel, unnecessary and unethical at the first glance. Wikipedia gives four justifications for punishment.

Justifications for punishment include retribution, deterrence, rehabilitation, and incapacitation.

Of those justifications the first, retribution, is problematic in a predestined world. The criminal could not have not committed the crime, so revenge or retribution loses most of its point.

However, retribution is rolled up into deterrence. If other criminals see what happens to the criminal in question, they will possibly be less likely to commit similar crimes. In other words the reluctance to suffer the consequences becomes part of their character which results in them not doing similar. When the chance to commit a similar crime arises then this factor becomes part of the character and they do not do it.

Similarly the criminal in question will be deterred (one hopes) from committing the crime again. He will hopefully be rehabilitated and the punishment for his current crime will influence him when the possibility of committing a similar crimes turns up. The punishment is in his memory and is a part of his personality and could be a reason for not committing the crime in the future. He may claim, in the future, that he “chose” not to repeat his crime, but in fact he could not chose to do it because of his personality and his memory of his punishment.

If the punishment results in a prison sentence then of course he cannot commit the crime or similar crimes. Wikipedia uses the term “incapacitated” and indeed that is so if he is imprisoned. An execution is a pretty final way of “incapacitating” a criminal and for many justice systems it it the ultimate punishment for severe crimes.

In the past in many countries, the criminal was tortured before execution, a process which horrifies us these days, but which seemed justified at the time. It at least some of these cases the intent was “drive out” evil influences.

The past crimes of others and their subsequent treatment, whatever it was, also serves to warn and influence others who might also have otherwise committed similar crimes. So even in a predestined world punishment would have a deterrent effect on others as it will influence others. In fact the only difference between a universe which allows for free will (somehow) and a predestined universe is the idea of “blame”.

The “free will” universe blames the wrongdoer, but the predestined universe doesn’t as the wrongdoer could not do otherwise than he did. There are still reasons for punishment in a deterministic universe in spite of that.

How do I get from A to B?

We had reason to visit another suburb today. It wasn’t until I was sitting waiting for some traffic lights to change that I thought about how I was navigating from home to destination.

We just got into the car and drove there. I didn’t consider the route in advance, and it seemed that I just pointed the car and we got there. Obviously I knew the way, as we had been there or through there a number of times in the past. But I didn’t have the destination in mind from start to finish, at least not consciously. I’m not sure that I had it in the forefront of my mind at all.

I knew that it was in that direction though, and that did not leave a lot of route options. I did have a general feeling that I should go south, in this instance and that really only leaves two options, the back road, or the motorway. The back road is a lot prettier!

I made the choice to take the back road but it was not, as I said, at the forefront of my mind, as I was doing other things at the time, like finding my keys, my phone, my wallet and these things occupied the forefront, while the decision about which route to take was more background.

So the route was chosen more of less in the background, but not subconsciously. Much the same process happened on the way there, and at each junction or turning point, I didn’t have to consider at the front of my mind which direction I should drive. I just did it. Some part of my mind knew that to get to our destination I had to turn right, or go straight on or whatever.

This is good because the front of my mind was doing the driving, keeping the car on line, signalling, accelerating or braking, keeping us safe on the road. Except that it wasn’t right at the front mind, since I was also talking to my wife about various things. Christmas things from memory.

English: Two motorcycle trailing off the brake... — English: Two motorcycle trailing off the brakes through Tooele Turn at Miller Motorsports Park. Rider on the white bike is Warren Rose, Rider of the green bike is Dave Palazzolo (Photo credit: Wikipedia)

I’ve been driving for many years and I’m confident that if needed the driving part of my mind can instantly oust the things currently in my mind should the unexpected happen. Many year ago, when I used to smoke, I was driving with a friend and an emergency happened. When it was over I realised that I was no longer holding my cigarette. Meantime my friend was scrabbling between his legs where my cigarette had ended up when the driving part of my mind grabbed precedence and the cigarette holding part was temporarily ousted.

The route planning part of my mind would not suddenly get control like that, fortunately. That would be highly dangerous. I could if I had wanted have brought the route planning part of my mind to the front, but it wouldn’t say much except “turn left at the next junction”.

Turn Left, Turn Right (Photo credit: Wikipedia)

I have on occasion made a navigating mistake. I’m going to A and the route to B is the same in part. Suddenly I realise that I have missed a junction and will have to backtrack. It seems that the route finding part of my mind spends much of the time dozing and checks in only infrequently, sometimes missing the turning point or ritually following a more usual route.

It also seems that the information it keeps is like an instruction to take an action at each decision point rather than the whole route from home to destination as well as a general direction, less well specified. GPS guidance systems seem to work this way too in that they instruct you to take an action at each junction without setting out the whole route each time.



Embed from Getty Images

The model of the mind that I’ve used above, of various parts of the mind at various levels of “forefrontness” or consciousness is nothing new. The need to make a part of the mind the one at the top of the conscious levels, suddenly as a result of a danger, or selectively by choice, as in route following reminds me of the way that computer programs

Computers have several methods for navigating through programs and reacting to things that happen when they are running. One big part is called “handling errors”. Dividing by zero is an error and if the computer reaches a point where it has to divide by zero something needs to be done. The program can report the error and gracefully stop, or it can take some action to fix the error and then carry on.

Computers handle error by means of “interrupts”. Whether the errors is software (eg divide by zero) or caused by hardware connected to it (eg input/output errors) the computer stops what it is doing and runs a bit of program that handle the errors by sending a message or fixing things up. The bits of program that were running are suspended and after the error is handled the bits that were running may be given back control.

The mind seems to work in a similar way. When an emergency arises the current part of the mind that is at the forefront gets suspended and the emergency is handled by another part of the mind. A pedestrian steps into the road and you react by standing on the brakes “before you know it” as the saying goes. As soon as the emergency is over, the conscious mind takes over again.

a short .gif of the Taiwanese animated pedestr... — a short .gif of the Taiwanese animated pedestrian road crossing sign (Photo credit: Wikipedia)

You do indeed react “before you know it”, one might say instinctively. But humans have not been driving cars for much more than one hundred years, so it appears that the reaction is not instinctive in itself, but is an instinctive reaction to a danger that has been learned. We seem to have this fast reaction to events which is instinctively based but can be applied to learned situations, which is much more flexible than hard-wired instincts would be.

So, pondering on how I get from A to B has led me to conjecture that there are parts of the mind which are forefront in our minds and other parts which are not directly in the forefront but which can be brought to the forefront in an instant, when an event happens. It is evident that these parts are only partially backgrounded as the mind as a whole has some aware of the location at the time, but they do act semi-autonomously, that is until the pedestrian steps out onto the roadway.



Embed from Getty Images

Evidently there are parts of the mind that are less foregrounded and more backgrounded. When the part of the mind that is concerned with driving wants to signal or change gear, another part of the mind which controls the arms and legs wakes up and make the limbs move as needed.

I’ve spoken above as if all these different levels are discrete states, but I think it more likely that is a continuum from the foreground of the mind to the background or a least the series of levels of consciousness are close enough togerther to appear so. The mind is a complex and wonderful thing.



Embed from Getty Images

[Comment: After finishing this post I went looking for other discussions of the same topic. I first found this Wikipedia article which has the issues mentioned in the article’s header. Interestingly the implication in the article is that there is a single level of consciousness at any one time. This I do not agree with. Another article I found was a little better, I feel, but only because it acknowledges that several levels may be active at the same time, but divides them into three levels with well defined scopes. I feel that it is a lot more complex than that, with all sorts of sections of the mind at all sorts of levels being active at the same time. Neither article deals with the issue of one section of the mind apparently seizing the highest level when an external event triggers it.]

I forgot!



Embed from Getty Images

Well, I almost forgot to write my weekly post. My excuse is that Monday is a public holiday and this upset my schedule. Of course this is only an excuse.

It did start me thinking about forgetting things. We have probably all forgotten appointments at one time or another, though with cell phones being ubiquitous and possessing calendars, we probably should never do so in the future. Yeah, right!

A page of a calendar. (Photo credit: Wikipedia)

The reasons why we forget appointments can be myriad, but I’d suggest that at least some of the appointments that we miss are ones that we would prefer to miss, such dentist appointments. I’d guess that we would be much more likely to remember lunch appointments or some other types of appointments that we would enjoy.

Some people seem to forget things that I am unlikely to forget, such as plane flights. It always amazes me that people can turn up for flights at the last minute. I’m usually in the airport, waiting, well before the official check-in time starts.



Embed from Getty Images

It is probably a consequence of our busy lives that we need calendars and other “aide memoires” like shopping lists. One can imagine that in earlier times to-do lists were shorter and the number things that one needed to carry on a normal life was a lot smaller and our memories were able to cope. But then again, that may be an illusion. Was life really simpler in the past?

If you miss an appointment that affects not only you, but the people with whom you have the appointment. The dentist will be looking at an empty chair, and while he may enjoy the break, it will cost him money. That’s why some service providers like dentists may charge you a fee if you forget an appointment. Some places get the receptionist to send you a text the day before as a reminder.

Although I’ve suggested above that maybe in simpler times the need for a calendar was less, the need was not non-existent. Farmers, for example, need to know the best time to plant seeds to ensure a good crop.

The Babylonian calendar for example was based on both lunar and solar cycles which would enable the priests to suggest the correct time for planting crops. This calendar has its origins around 2000 BCE according to Wikipedia.

The Mayan calendar is even older, with roots in the 5th century BCE. The Mayan calendar came to popular attention in 2012, when it was conjectured that the Mayan calendar would end on 21 December 2012, and that this would signal a global catastrophe.

Of course nothing significant happened and the Wikipedia article on the topic explains that the Mayan calendar did not end on that date. Surely very few people seriously thought that it would. Even the ancient Mayans did not predict a calamity at that time, so far as scholars are aware.

Forgetting important dates is a recipe for matrimonial disharmony. There seems to be a gender bias here as most males do not worry too much if you miss a birthday, but it seems more important to females. Whether or not this is real gender difference or merely a societal one, I’m not prepared to guess. Whatever the cause, it is best to remember one’s spouse’s birthday. I personally find it very difficult to remember dates other than the usual wedding anniversary, spouse’s birthday and my children’s birthdays.

There is another sort of forgetting though, one that creeps up on one as time passes. Some of that can be attributed to loss of facilities due to age, but memories do seem to fade regardless of ageing. Even in your twenties your recall of events ten years earlier can be faulty, though for some reason some things can be said to stick in your mind.

It’s difficult to work out what we have forgotten and what we have simply had no reason to remember. Something may, for some reason or other, trigger a memory, and we say “Oh, I had forgotten that!” when obviously, we hadn’t.

Equally, memories may be replaced by false memories. For instance, for years I believed that I had a memory of an event at a particular place involving my sister’s pushchair. When I visited the place many years later, it was apparent that the event could not have happened as I remember it. The slope of the path, the flower beds were all wrong.

It seems to me that memories can be true, or totally false, and that a memory of events may be completely lost, and this would make the study of memory very difficult and that relying on memory for veracity is impossible.

There are people who claim to have memories of a previous life. I don’t believe such claims, as I don’t believe that there is such a thing as reincarnation. If there were such a thing, one wonders why those who remember their previous lives were always important and powerful people in past lives.

But memory loss definitely gets worse as one gets older. Some unfortunate people suffer so badly from memory loss that they cannot recognise their own families and get lost in places with which they were once familiar. As people live for longer this phenomenon is becoming a big problem.

Of course there is research into both ageing and memory, so this situation will hopefully improve, but at the moment, it doesn’t seem attractive to live to 100+ and lose all one’s faculties.



Embed from Getty Images

Memory is one of the things that makes you you. You can remember many things from your past, and while you can’t completely remember being the person that you were, you can remember some of it. Maybe you can remember what you were feeling at the time, maybe not.

But you believe that the person that you remember was you, in spite of the forgotten things, the things that you remember wrongly, because there is a continuous thread of memory between the person that you remember and you in the here and now.

Yet in another sense, that person was not you. His memories only encompass the time up until his now. Your memories of the time from then on are of events that have changed you and your memory of events before that person’s now have also changed, or been lost. How could you be the same person?



Embed from Getty Images

Grand kids

English: A typical Deutsche Bahn railway stati... — English: A typical Deutsche Bahn railway station clock (Photo credit: Wikipedia)

This post may be a little late this week, as we took the grand kids to a local “wildlife park”. Which leads me not so subtly into the topic of the week.

As kids we used to go to Granny and Grandpa’s house quite frequently. I recall sleeping over at one time, and in later years my sister lived with my Granny. I recall it being fun, but that was mostly because we could poke around in their house and garden. They had a long garden with a goldfish pond, a few apple trees, a garden and a large wooden shed.

What I don’t remember is being taken out to “wildlife parks” or similar by my grand parents. In fact, I don’t remember going out with my grand parents anywhere. I’m pretty sure that we must have gone out with them, but it must have been rare and we probably only walked around the block or something.

We did use to have big family parties and I do remember my Granny being at my Uncles and Aunties’ houses during family parties. My Grandpa died fairly young so I don’t remember him at them.

It’s different with our grand kids. I don’t know if it is because we are much more active than my grand parents were or because we are able to drive them to places. Neither of my grand parents drove that I can remember.

In contrast we fairly often drive our grand kids on trips to various places, sometimes in conjunction with their parents, sometimes without.

On Sunday, their uncle (my son) and his wife decided to take my grand kids to a local “wildlife park”, called Staglands. I went along and so did my daughter, their mother. Staglands is a fair way out of town, in a beautiful picturesque valley. My daughter drove the kids there, and I drove my son and daughter-in-law.

The sole survivor of presumably a brood of ducklings.

We didn’t stop to take photos of the valley on the way there, as we travelled independently and didn’t want to keep them waiting. As it turned out, we got there first. The road is a secondary route between a major north-south valley (the Hutt Valley) and the Coast Road.

It is a fairly quiet road, most of the time, so is a favourite road for serious cyclists, the ones with lycra suits who are not afraid of some fairly serious hills. I’ve no problem with them, but they did make it difficult on the narrow road at times.

Staglands is 17km up this road, so that part of the trip took a while, but we got there and parked, waiting for my daughter and the grand kids. They all hopped out the car and admired the guinea fowl and peacocks which roamed the car park.

I was informed that the brown peacocks were the female ones then we paid the entry fees and went in. The path bypassed the café and wound down to a couple of small lakes, with the usual wildfowl, mainly ducks, including one small survivor of a presumably larger brood.

On the way it passed a small cul-de-sac with a “cave”, Tracey’s Cave, with a constant splash into a pool of water which created interesting ripples.

Kea, too busy tending the plumage to chat.

We then followed the main path up to a barn where there was a fire burning in a barrel. Did I forget to mention that we bought small packets of marshmallows and sticks at the gatehouse? Ooops. Great fun was had toasting the marshmallows, something that I’ve never seen the point of, until now.

Just down from the barn was a small paddock with a number of Kune Kune pigs. These are fairly small, hairy pigs. Kune Kune are friendly and docile animals although I would not like to be in the paddock with them. The kids, Hamish, Duncan and Louise, loved feeding them out of the small packets of feed that we bought at the gatehouse.

Tim, Hamish, Duncan, Louise and Kaz, with parakeet

A walk-in aviary was next, which contained Kea. These are a large alpine parrot, which in the wild are attracted to people and their cars. Naturally people stop to see them and the Kea respond with thievery and destruction. They pinch sandwiches and pull the rubber bits from cars.

The Kea in the aviary were less exuberant than that preferring to preen out of reach, but other birds, small parrots, were friendlier and would sit on one’s hand. A passing sparrow stopped by overhead on the chicken wire roof.

The stables contained horse, donkeys and small lambs and goats, all of which got a ration out of the small bags of feed. It almost seemed that these larger animals were taking the small portions simply to be friendly.

Next was a larger aviary, planted with toe toe, a large tussocky grass with plumed flower heads, much like “Pampas grass” which is well known in some other countries. Small birds, such as finches, cockatoos and budgerigars live in this aviary.

The kids all love the “swing bridge” which connects one half of the park with the half on the other side of the stream. The first area is a mock recreation of small settlement from around the 1900s, which has been used as a set for films. Up from there is a large pond with a walkway, with on one side, a wooden railway with a push cart, usually a hit with kids, but for some reason unused today. Though I did push Duncan along in it.

The inevitable scars of forestry, which should quickly heal.

After the pond was a large open paddock with deer, sheep and goats. A notice on the gate mentioned that the animals in the paddock could be “quite pushy”. A quick scramble and Hamish, Duncan and I had a glorious view of the valley, only slightly spoiled by a large logged area on the opposite side of the valley.

Then it was back down to the gatehouse and the café for re-fuelling. Hamish managed a couple of large sausage rolls (a mistake – I got the order wrong, it should have only one!) and was still hungry. Duncan went for a more sophisticated hot dogs and chips, while Louise claimed to be satisfied with an ice-cream though she did help herself to he mother’s chips. Tim (my son) and Kaz (his wife) both had nachos. I had a bacon and egg panini.

Staglands is a great place for that sort of trip and we thoroughly enjoyed ourselves. It is great to be able to spend time with the grand kids, and it a shame that, for whatever reason, we didn’t get to do similar with our grand parents.

Sports by proxy

San Diego representin' — San Diego representin’ (Photo credit: San Diego Shooter)

We sit in our chairs and watch sport. We watch the news. We catch up on what friends are up to on Facebook and Twitter. Very few of us actually do anything. This is both worrying and amazing.

I’ve been watching the Commonwealth Games from Glasgow in Scotland. Sitting on the sofa I see the Silver Ferns of New Zealand play England at netball in the semi-final. (I’m glad the Silver Ferns won, of course, but you can’t help feeling sorry for the English team.)

The Official Logo of the Fiji Commonwealth Games Association (Photo credit: Wikipedia)

The team members are healthy, fit and professional in their attitude towards the game, and I include both teams in that assessment of course, and they take it seriously. You can see the joy on the faces of the New Zealand team and despair on the face of the English team who came so close.

Then I look at myself. I’m reasonably fit, reasonably healthy, but I’ve not dedicated my life to a sport. I possibly could have been a reasonably good runner had I pursued my inclinations at school, but there was always someone better than me. I did enjoy running for its own sake though.

So given that not everyone can be a top sports person, why do we spend so much time watching sport? We hope that our support and the fact that the sports people that we are supporting are aware of our support and that it helps them.

Hmm, I’m not so sure. When a Silver Fern player is trapped after the event by a media person, they will quite often acknowledge the support of the “people back home”. On occasion a sports person will dedicate a match to a relative who may have recently died, or to close family, wife and babies, or even the general supporter who stays up late to catch the broadcast of the event.

But I still wonder. When I have played in sport in the past, the sheer involvement of playing the game drives out all thoughts of supporters or other spectators. The roar of the crowd is not heard, the sideline cameras are invisible and the only thing that is experienced is the game, team mates, opponents, match officials. OK so I’m extrapolating more that a little here, but I think that it is true!

I’m not saying that support does not help a team. It definitely seems to but not at a conscious level as the conscious level of the brain is fully occupied by playing the game or it should be.

Sports people who are interviewed after a game do credit the support from people “back home” and friends and families, and I believe that in a way the support does help a team play well. A visiting team is always at a disadvantage because of the home support.

Sports men and women, after a game are asked for a snap diagnosis of the game. The best will credit the supporters for helping them win, and the best losers will accept the responsibility for the loss, which is slightly unfair if you think about it, as they cannot pass any responsibility for the loss to the supporters.

It is true that the best teams attract the most fanatical supporters, but equally, fervent supporters can help to create success for a team. So what do the supporters get out of it?

Well, there are active supporters and passive supporters, by which I mean that some supporters actually go along to the game, and some watch it on television. Both types of supporter may collect “memorabilia” of the team and this can be very lucrative for the team, and possibly help to provide the team with equipment to help them succeed.

English: Fourth quarter of a college football ... — English: Fourth quarter of a college football game between the visiting No. 6 USC Trojans and the Stanford Cardinal at Stanford Stadium on November 15, 2008. It was the first sold out game at the new Stanford Stadium, which opened in 2006. USC would win, 45–23. (Photo credit: Wikipedia)

But the “memorabilia” and the support provides the supporter with a feeling of belonging or association with the team. Some supporters purchase team shirts with name and number of their favourite player, thus associating themselves with the player. I believe that this is not really an attempt to be that player, which would be delusional, but to tell the world of their association with their favourite player in a way that is instantly recognisable to another fan.

However, what televised sports give to the supporter is an idea of what it is to be a football player or a racing driver, or even a manager or a coach. Every change to a team or the support staff is avidly reported in the press or television. Supporters then take the information and discuss the pros and cons.

For the supporter, it is almost like being a coach or manager. He or she sees what it is to be a coach or a manager, at least to some extent. An avid supporter of the team may know almost about as much about the team as some people on the staff of the team, or so they would like to think.

A good manager knows this and may put out bulletins on an injured player’s recovery and coaching staff may provide limited access to training sessions. This is done to provide a rapport with the fans, so that the fans support the manager.

When the team takes the field, spectators at the ground get a feeling of participating at an event. There is a crowd noise, a sort of voiceless roar, modulated at times by team songs and cheers and groans when points or a goal are scored. I recall being at a netball match, though I can’t recall why. The crowd noise there was headachingly high-pitched.

The way to become the centre of the event is to become a player, but most people don’t have the skill to take part in sport at a high level. With the television cameras and other technology that is available today, a spectator at home can almost come to believe that they are on the field of play.

In car racing this is taken to the next level, with cameras in cars that show, pretty much, the view that the driver sees. In field and other sports slow motion replays allow the viewer to see more than even the on field officials can see. An official might have a tenth of a second of action on which to base his decision, but the slow motion replay allows the armchair supported the luxury of several seconds and several different angles from which to view the incident.

Formula 1 Rd5 Barcelona 2011 (Photo credit: julien.reboulet)

All of which give a supporter the feeling of being at the event and of being part of it. Not everyone can become David Beckham or Usain Bolt, but being a supporter can almost get you there in a way.

The Negative Universe

Dandelion(negative) (Photo credit: tanakawho)

If cosmologists are to be believed the Universe came from nothing and is likely to return to nothing. This is odd as there seems to be an awful lot of it! There are between ten to the power 78 and ten to the power 82 atoms in the observable universe, according to some estimates. There’s also a huge amount of energy out there in the universe, and as Einstein said, this is equivalent to matter, according to his famous equation.

Maxwell's Equations — Maxwell’s Equations (Photo credit: DJOtaku)

It is likely that the enormous amount of matter and energy that we see out there is balanced by an equivalent amount of “negative” matter and energy somehow. “Negative” is in scare quotes because it may not describe what is actually going on. Anyway, the negative matter and energy may be incorporated into this universe somehow, which means that on average half the universe is this sort of energy. We can’t see it anywhere so far as I know, so it is a bit of a puzzle.

Large Format Doha Panorama Portra 400 (Photo credit: Doha Sam)

We can see evidence everywhere for “normal” matter and energy, and we should be able to see evidence of “negative” matter if it is anywhere near us. As I understand it, “negative” matter would behave differently to “normal” in various ways and should be detectable. I’m not sure in what ways it would be different – I can guess that there could be a gravitational attraction between particles of “negative” matter, just as there is between particles of “normal” matter, but there could be a gravitational repulsion between “negative” matter and “normal” matter for example.

Galaxy NGC 720 (NASA, Chandra, 10/22/02) (Photo credit: NASA’s Marshall Space Flight Center)

But my ignorance is almost total. I do believe it is true that “negative” matter should be detectable.) Since we can’t see or detect “negative” matter within our locality (ie “the observable universe“) it may be grouped elsewhere in the universe. If so, it may not have any observable effect in our neck of the woods, but inevitable it will have an effect at some time in the astronomical future.

Español: es la misma imagen que aparece en el ... — Español: es la misma imagen que aparece en el articulo en ingles: http://en.wikipedia.org/wiki/Sloan_Great_Wall (Photo credit: Wikipedia)

The reason that I say this is that the universe doesn’t seem to be expanding faster than the speed of light so any effect such as (possibly) gravity which does appear to have a “speed of light” effect will eventually affect our corner of the universe. But the situation is complex, and as the Wikipedia article says,

Due to the non-intuitive nature of the subject and what has been described by some as “careless” choices of wording, certain descriptions of the metric expansion of space and the misconceptions to which such descriptions can lead are an ongoing subject of discussion in the realm of pedagogy and communication of scientific concepts.

In other words, there are many misconceptions and misinterpretations around this topic. However any effect of the possible existence of “negative” matter on our little neck of the universe is likely to be felt a long time in the future, even on a cosmological time scale, I feel. “Negative” matter could have created a negative universe, I guess, which mirrors this universe.

In a negative universe at least one dimension would be reversed but all other dimension would have the same polarity as our universe. If an odd number of dimensions were reversed, would all but one cancel out? I’m not sure but a cursory mathematical examination would indicate that this would not be so, but I lack the time to explore the concept in depth.

In our universe things tend to a state of disorder. If one partition of a closed system contains all the matter (in the simplest case, as a gas) and the partition is removed then eventually the matter is eventually dispersed through the whole system. In a negative universe, possibly the opposite would apply – gas dispersed through a system could tend to bunch up in one part of the system. Maxwell’s demon could watch benignly without lifting a finger.

Our view of this is that it is extremely unlikely – would a glass spontaneously rise from the floor, gathering the scattered wine and land on the table complete with wine? Perhaps this is a parochial view, only true in our universe. In some alternate universe, this may be the normality – entropy may tend to decrease, order may tend to increase. Such an entropy twin may simple be the time reversed twin of the original universe. Or the original universe perceived from a time reversed perspective.

The Grand Canyon Time-Zero Project (Photo credit: futurowoman)

If the universe sprung out of nothing, then the sum of the universe is zero. Any object has its anti-object. Any event has its anti-event. Maybe the universe has a partner, an anti-universe if you like, or even a mirror universe. Time in our universe runs from the zero moment into the positive future. In a mirror universe would presumably (and debatably) run in the opposite direction from the zero moment and all spacial dimensions would be reversed.

Plus-Reversed,-1960 (Photo credit: Wikipedia)

This would correspond to a point reflection in time and space, which may or may not be the same as a rotation in time and space. I’m not sure. Some of complexities can be seen in this Wikipedia article on “parity”. In particular some interactions of elementary particles may display chirality, which means that they come in left and right handed versions, like gloves or shoes. All of the above means that if a person were to be point reflected into an anti-universe and all the elementary particles of his body were switched with their anti-particles, there may be no way for the person to tell that the switch had occurred.

different flowers from same plant (Photo credit: ghedo)

Sure, time would be reversed, but so would literally everything else, so a left-handed glove would appear, in the point reflected world, to still be a left-handed glove even though, if we could see the glove it would appear to us, from our point of view to be right-handed. Of course I’ve assumed for much of the above that the reflection that transforms a universe to its anti-universe is a point reflection.

In mathematical terms that means that all variables are reversed. That is x is replaced by -x, y by -y, z by -z and so on. It may be that the reflection may be in a line and the x dimension stays the same while the others are negated. Or it may be a reflection in a plane (a mirror reflection) where 2 dimensions are unchanged. Or it may be a reflection in a higher number of dimensions.

As you can see, the subject is complex and I’ve not got my head around it (obviously!), but I believe that if we were switched into the anti-universe (including all out particles) it would not look any different from this universe. In fact we would probably find ourselves discussing our anti-universe, which would be our original universe. In fact it would not matter which universe we called “the original” because they both would have come into existence at the same time and there would be no meaning to the term “original”.

A face.. (the original OMG Wall) (Photo credit: eworm)

The Start of New Year

an old post card (Photo credit: Wikipedia)

According to sources on the Internet, the winter solstice in the Southern Hemisphere occurs at 10:51 am UT on 21 June (this year, 2014). That translates to 10:51 pm in New Zealand. Just as in the Northern Hemisphere the start of the year corresponds roughly to the winter solstice there, I like to think that the winter solstice in the Southern Hemisphere corresponds to the start of the year in the Southern Hemisphere. I don’t think that I would get much support to the start of the year officially changed, though!

The year can be divided into halves by the solstices, the winter solstice marking the sun’s most negative elevation with respect to the South Astronomical Pole since the previous June. From that moment in time the sun starts to move higher into the sky until, at or around 21 December, when the summer solstice occurs.

Midway between the solstices falls a time when the day and night are roughly equal in length. Around this time the sun crosses the celestial equator, and this time is called an equinox. There are two in the year, one when the sun is apparently moving south in the sky (the vernal equinox in the Southern Hemisphere) and one when it is moving north in the sky (the autumnal equinox in the Southern Hemisphere).

The Sun & the ecliptic rotation around the Ear... — The Sun & the ecliptic rotation around the Earth : The green Sun is the one of the vernal equinox (march), it is followed by a summer solstice Sun. Then automn equinox and winter solstice. The ground plane (latitude 50°N) is green, the rotating ecliptic plane is blue. Also represented are the celestial equator, the two tropics and the rotation axis. (Photo credit: Wikipedia)

Each of these quarter points of the year is or was celebrated with a festival of some sort, some of which, particularly the winter solstice were supposedly characterised by “unrestrained revelry“. The summer solstice was comparatively restrained, the vernal equinox was a celebration of new growth, and the autumnal equinox was a harvest festival, a gathering in and celebration of bounty produced by the year’s hard work.

What I wasn’t aware of is that there were other events called “Cross Quarter moments”. These are moments halfway between the equinoxes and solstices, and they are known as Embolc, Beltaine, Lughnasad, and Samhain. The Cross Quarter moments. the solstices and the equinoxes are set out in order for 2014 in the chart referenced here.

Two of the Cross Quarter moments I have heard of, Beltane and Samhain. Beltane falls between the vernal equinox and the summer solstice and is roughly at the beginning of May, so corresponds roughly with May Day. It is astronomically the beginning of summer, but seasonal lag means that the season starts a little later than this.

Children (usually girls) still dance around the maypole or maytree, but few of them, and probably few of the adults have any idea of the origins of this ritual. Although it probably is related to Beltane or the start of summer, the significance and symbolism of the maypole is still debated. Some of the possible suggestions seem dubious and far-fetched, and I don’t think that is wrong to suggest that they reflect the prejudices of the people that make them. In particular it appears that Puritan Christians may have over-emphasised some aspects of the dance and celebration to argue for its banning.

Morris dancing is also associated with a spring festival, usually Whitsun. It may possibly have been associated with Beltrane, but I don’t know the history of morris dancing, Whitsun and Beltrane or spring festivals in general well enough to assert this. There is a long tradition of ancient non-Christian rituals being adopted and given a Christian slant, so this may be possible.

Cotswold-style morris dancing in the grounds o... — Cotswold-style morris dancing in the grounds of Wells Cathedral, Wells, England — Exeter Morris Men (Photo credit: Wikipedia)

Samhain also has a long history and probably pre-dates Christianity. It is associated with the beginning of winter and marks the point where all crops are gathered and animals prepared for winter. Once again the Christian church has adopted the festival and the roots of “harvest festivals” are to be found in Samhain’s pre-Christian traditions.

The Christian church adopted the festival as All Saints (Hallows) Eve or Halloween. I note from the Wikipedia article that I linked to that some people consider that Halloween has no relationship with Samhain, but considering the similarities of the two traditions which happen at the same time of the year, I think that this seems unlikely.

Bonfires form a great part of the Samhain festival, maybe as an attempt to ward off the coming darkness of winter. I don’t think that it is a coincidence that there are still “celebrations” on 5 November, otherwise known as Guy Fawkes Day. An effigy of Guy Fawkes is burnt on a bonfire, in spite of the fact that Guy Fawkes was actually hanged.

All of the example above refer to the “Gaelic versions” of the various dates and festivals. It’s a bit simplistic to refer to a single “Gaelic version” as the dates and festivals have, naturally, changed over the years. Other cultures of course have their own versions of the various festivals. In the Tropics (between the Tropic of Cancer and the Tropic of Capricorn the sun is overhead at least once in the year, an obvious time for a festival!

Since we have just passed the winter solstice, we can look forward to longer days and shorter nights from now until the summer solstice, which for us in the Southern Hemisphere comes around 21 December. So far this year winter has been fairly mild and a little wet. As we move towards the vernal equinox we still have the bulk of winter to come, as the astronomical year does not match the climatic year because of the seasonal lag.

Nevertheless it is a time to look forward and one can understand why the winter solstice is a such a time. It is a time of feasting, of using up some of the stores put away at the time of the autumnal equinox, the salted beef and cured hams. It is a time to relax, for mending and repairing, and for staying out of the weather as much as possible, as the weather of winter means that essential tasks only will be undertaken and the rush of springtime is still ahead. While the end of winter may bring shortages , it is still near the beginning and the stores are still full.

Is “schooling” an education?

Well, schooling should be an education. It should prepare the pupil for life. Dictionary.com has this as a prime definition of education:

The act or process of imparting or acquiring general knowledge, developing the powers of reasoning and judgement, and generally of preparing oneself or others intellectually for life.

Schooling doesn’t always do this – Greek history is probably of little use to a car mechanic and scientists are only interested in Greek history in so far as it has cool cast list of names and an alphabet from which they can plunder names for obscure fundamental particles or asteroids.

View from one end of Eros across the gouge on ... — View from one end of Eros across the gouge on its side towards the opposite end.(greyscale) (Photo credit: Wikipedia)

Arguably, though, Greek history is a fascinating window into an early culture, and studying events in Greek history can provide insights into contemporary society and while it may not be of obvious direct benefit to the mechanic and the scientist, such studies can inform sociologists, political studies specialists and many others, and it is worth remembering that mathematics, science, logic, philosophy, politics and many other fields of human endeavour have their roots in ancient Greece.

Temple Statue of Poseidon (Photo credit: greekgeek)

But back to schooling. Everyone has been bored at school, for a number of reasons. The subject could be more than the student can handle, or it could be too simple, or it may not be a subject in which the student has no interest.

One of the issues with schooling is that we are taught, well, “subjects”. Well, we are taught “maths” or “biology” or “French”, or “Woodworking” or whatever. We are taught “English”, which is about how sentences are formed and we are drilled in verbs, nouns, adjectives and more esoteric beasts of the English language. Then there is “English Literature”, which largely consists of forcing pupils to read and “study” relatively old English language texts ranging from Shakespeare to Dickens. Rarely anything more modern.

English: Literature (Photo credit: Wikipedia)

There is a syllabus, specifying what we are to be taught. This is used to constrain the teachers and students, so that they can be set examinations to see, basically, how much the teachers have been able to force into usually unwilling minds.

writing/editing my environmental sustainabilit... — writing/editing my environmental sustainability cornerstone seminar syllabus at nabolom bakery in berkeley (Photo credit: davidsilver)

This all seems mechanical and soulless, but a good teacher will try to insert into the gaps and voids of the subject and the syllabus a little education. He or she will try to convey the beauty of the English language as used by Shakespeare and the other authors, he or she will try to make Romeo and Juliet into real people for the students, he or she will explain the societal background of the Dickens tales.

The good teacher will teach something more – how to look beyond the surface story to the people and the societal background, not just in the set books or any books, but in all the situations that life may throw at the student over the years.

A study of literature can not only give the student the knowledge of what is in the books, and maybe an appreciation of the era in which the books are set but may also provide the student with the ability to look critically at the era in which they are living. For some, maybe more than a few of the students, this will provide them with the tools to examine sources like the media and consider such things as bias and veracity.

A teacher of maths will try to not only enable the students to pass their maths exams but also to prepare them, a little, for life. The simple techniques of addition and subtraction may be all that they need, but sometimes they may need a bit more. Some of the students may go on to be mathematicians, to study the subject in its own right. But many more may acquire the tools to understand some of the numbers that surround us all in our daily lives.

Day 304: Problem Solving Strategies for Math (Photo credit: Old Shoe Woman)

For instance, when a poll result is given on television, often they also quote a ‘margin of error’. A small but significant number of people will have some idea of what that actually means from some long ago statistics class. The vast majority doesn’t have a clue as to what it means, but the brightest might gather that it relates to how accurately the poll represents the wider population.

Another example of a mathematical tool that could be useful is contained in an episode of the British sitcom, “Please Sir!”. This is a comedy about an inspirational teacher and a class of pupils who are rejects from other classes. The teacher follows an informal teaching agenda as it is evident that his class is not going to pass any exams.

He tries to instil some mathematics into his students, using as an example a bet on a horse race. He calculates the odds only for one of his students, the son of a bookie, to correct him. The teacher is astounded that the student can calculate the odds so accurately in his head, saying to the student that he didn’t know that the student was good with maths. The student replies that this wasn’t maths, it was “odds”.

At the bookies (Photo credit: Phil Burns)

Science, likewise, has ramifications beyond the bland and often boring stuff a student learns at school. While he or she may come close to disaster in a lab, he or she may take away the concept of analysis and the scientific method that may help him or her in later life. At least when one of the TV detectives grabs a scrap of clothing or a sample of blood or something and sends it for analysis, he or she may have an inkling of what is happening. Though these shows are an education of a sort in themselves.

Day 53 - West Midlands Police Forensic Scene I... — Day 53 – West Midlands Police Forensic Scene Investigators Lab (Photo credit: West Midlands Police)

So why is the educational system focussed on schooling rather than educating? Well, for one thing it is easier to measure schooling rather than education. Facts trotted out for an exam yield a measurable yardstick to judge both student and teacher. It’s altogether more difficult to measure education.

Seal of the United States Department of Education (Photo credit: Wikipedia)

That’s because an education is not about facts learned. It’s about facts learned and a deeper understanding of how the facts interrelate within the system, be it Greek history, English literature, maths or science. Nevertheless, the best teachers provide an education as well as schooling. They should be applauded for it.

The process of Philosophy

Philosophy & Poetry (Photo credit: Lawrence OP)

Philosophy is a strange pastime. Scientists measure and weigh. Mathematicians wrangle axioms and logical steps. All other disciplines draw on these two fields, which are probably linked at deep level, but philosophy draws from nothing except thoughts and the philosopher’s view of the Universe.

Well, that’s not completely accurate because philosophy has to be about something and the only something we have is the universe. But philosophy does not have to be about the universe as we know it. What if there was no such thing as electrical charge, or, the prudent philosopher thinks, what if there was no such thing as the thing we call electrical charge. At a more basic level, what is electrical charge.

Philosophers are always getting pushed back by scientists as scientists figure what they think is the case. If there is a scientific consensus on what comprises an electric charge then that question no longer interest philosophers to any great extent. Philosophers mentally travel through the lands marked “Here be dragons”.

Philosophy is also interested in less “physical” things like ethics and morals, what comprises identity, predestination or free will, what can we know and what knowing is all about. How did the Universe come to exist, or more basically, why is there something rather than nothing?

If you look at this list it comprises extensions to or extrapolations from physics, psychology, physiology, medicine, biology, and other fields of science. Philosophy doesn’t use mathematics (usually), but it uses logical argument or should. It not (usually) built on axioms, so doesn’t have the rigid formality of mathematics.

Illustration of Plato's Allegory of the cave. — Illustration of Plato’s Allegory of the cave. (Photo credit: Wikipedia)

Philosophers are big users of metaphor, such as Plato’s “Allegory of the Cave”. A metaphor of the expansion of a balloon was used as a philosophical explanation of the expansion of the Universe discovered by Edwin Hubble. Philosophers also imagine physical machines which do not yet exist and which may never exist, such as the ‘teleporter’ which makes a material object at point A disappear and reappear at point B.

Star Trek - Enterprise D Transporter — Star Trek – Enterprise D Transporter (Photo credit: tkksummers)

Quantum physicists have teleported quantum information from one point to another, but this is not the same as teleporting atoms. So far as I can gather from the Wikipedia article, what is teleported is information about the state of an atom, so the same atoms must already be at point B before the teleportation event, and the event is a sort of imprinting on the target atoms. It sounds like the atoms at point A remain in situ, so it is more of a tele-duplication process really. However I don’t really understand the Wikipedia article so I may be wrong.

The philosopher is not interested in the quantum nuts and bolts though. He or she would be interested in the process – is a person walks in to the teleporter at point A the same person as the person who walks out of the transporter at point B? Unless his actual atoms are transported by the process, which seems an unlikely implementation, the person at point A shares nothing with the person at point B except a configuration of a second set of atoms. Is the person at point A destroyed by the machine and recreated at point B? What if something goes wrong and the person at point A does not disappear when the button is pressed? Then we have two instances of the person. Which is the real instance?

Unknown Person (Photo credit: Wikipedia)

Notice that the philosopher takes a physical situation of travel from point A to point B and considers a special case, that of travelling between the two point without travelling the old slow way of travelling between all the intervening points and doing it quickly. There is no physics which can currently perform this task, but as usual, scientists are working to, one might say, fill in the gaps.

The Sci-Fi Fly! (Photo credit: Carolyn Lehrke)

Many times the scientist is also a philosopher – he may have at the back of his mind the concept of teleportation when he creates his hypotheses and does his experiments, but he probably doesn’t concern himself with identity. That is still the realm of the philosopher at present, but if a teleportation device were ever created, it would stop being a philosophical matter, and become a matter of law and psychology and maybe some field that does not exist yet, just as the field of psychology did not exist at one time.

General Psychology (Photo credit: Psychology Pictures)

I’m trying to paint a picture of the area that a philosopher is interested in. If the whole of human knowledge is a planet, then physics and maths are part of the outer most layers of the atmosphere, the exosphere, and this merges with the depths of space are the domain of philosophy. At lower levels are things like chemistry, biology, psychology and other more applied sciences. Don’t look too closely at this analogy because I can see two or three things wrong with it, and I’m not even trying.

But the main point I am making is that philosophy purposefully pokes and prods the areas beyond the domain of current mathematics and physics. Of course the line is not a definite line and there is a grey area. Some physical hypotheses verge into philosophy and some philosophical ideas are one step from becoming physical hypotheses. The suggestion that there be many universe like and unlike ours is one such suggestion that physicists are taking seriously these days.

2-step branching in many-worlds theory (Photo credit: Wikipedia)

Many of these ideas are not new and many have been used in what has been called “science fiction” for many years, especially the parallel universe theory. Time travel is another common science fiction theme. Although these ideas are used and developed by authors of fiction, physicists have adopted such ideas to advance science, though I don’t mean to suggest that scientists have directly borrowed the ideas of science fiction authors. It is probable that many ideas actually travelled in the opposite direction, from science to fiction.

Since philosophy is at heart discursive and not rigidly analytical (in most cases), there is more freedom to expand on ideas that are not what is called “mainstream”. Because of this freedom it is likely that (like economists) no two philosophers will agree on anything, but they will have fun arguing about it.

Two-ness

Lane #2 Swimming (Photo credit: clappstar)

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.

Directions (Photo credit: Gerry Dincher)

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.

English: : A mirror, reflecting a vase. Españo... — English: : A mirror, reflecting a vase. Español: : Un espejo, reflejando un vasija. (Photo credit: Wikipedia)

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.

Flowers in Mirror Image (Photo credit: ClaraDon)

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!

"The Strange Case of Dr. Jekyll and Mr. H... — “The Strange Case of Dr. Jekyll and Mr. Hyde” (Photo credit: Profound Whatever)

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 of Arts - 2000 Visual Mashups — Two of Arts – 2000 Visual Mashups (Photo credit: qthomasbower)

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.