## Oddities

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Humans and not very good at calculating odds and how probabilities work. For instance, if we are tossing coins and we get six heads in a row, the probability of getting yet another head is still fifty-fifty. Yet people feel that after a series of heads that it is more likely that more tails than heads will turn up for a while, so that the ratio of heads to tails returns to the expected one to one ratio.

But the expected ratio of heads to tails for all subsequent tests is one to one. It’s as if a new set of tests is being started, and so any lead that has already built up is, in all probability, not going to be reduced.

This seems odd. If we have done one thousand trials and have turned up 550 heads to 450 tails, the ratio of heads to tails is about 0.818 and the ratio of heads to the number of tests is 0.55. Surely more tests will take the ratios closer to the expected values of 1.0 and 0.5? Surely that means that there will be more tails than heads in the future?

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Well, the answer to both questions is no, of course. The ratios for the whole test may move closer to 1.0 and 0.5, but equally, they may move further away. In the extreme case, there may never be a tail again. Or all the rest of the throws may result in tails.

Interestingly, if the subsequent tests produce a series of heads and tails, the difference between the number of heads and tails stays at around 100, but the ratio of tails to heads for the whole test slowly creeps closer to 1.0 and the ratio of heads to the total number of tests closes in on 0.5 as more and more trials are done. By the time we reach two million tests, the two numbers are not very far from the expected values, being 0.9999 and 0.5000 respectively.

So, if you think to yourself, as you buy a lotto ticket “Well I must eventually win, if I keep buying the tickets”, it doesn’t work like that. You could buy a lotto ticket forever, literally, and never ever win. Sorry.

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Lotto and sweepstakes are, I believe, a different type of gambling from other forms, such as betting on horses or poker and other gambling card games. Lotto, sweepstakes and raffles involve no element of skill, and the gambler’s only involvement is buying the ticket. Betting on horses or cards involves skill to some extent, and that skill comes down to things like working out the probabilities of a particular card coming up and the probabilities of other players having certain cards in their hands.

Both types of gambling encourage the gambler to gamble more. If a gambler doesn’t win on the Lotto he or she might say to his or herself “Better luck next time.” Of course, luck does not exist, but probabilities do, and this is a mild form of the Gambler’s Fallacy described above. Nevertheless, people do win and the winners appear on television for us all to see and emulate.

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There’s two sorts of strategy for winning the Lotto. First there’s the “always use the same numbers” strategy, and then there’s the “random numbers” strategy. If you always use the same numbers, goes the theory, then eventually there must be a match. That’s wrong of course, since the number combination may not appear before the end of the universe.

The random number strategy argues that there is no pattern to results so it is silly to expect a particular pattern to eventuate. This strategy acknowledges the random nature of the draw, but doesn’t give the gambler any advantage over any other strategy, even the same numbers strategy. It is certainly easier to buy a randomly generated ticket than to fill in a form to purchase the same numbers every time.

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Some people experience a run of luck. They might have three things happen to them, so go and buy a lotto ticket while their luck holds. Then is they win they attribute it to their lucky streak. It’s all nonsense of course. They conveniently forget the many, many times that they bought a ticket because of a lucky streak, only for the ticket to be a loser.

The proceeds from the sales of lotto tickets don’t normally all go to holders of winning tickets. Firstly the operators of the system need to recoup their costs. It’s not cheap to own and operate those fancy machines with the tumbling balls and it also costs to employ the people to check that the machines are fair.

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If one of the balls is dented, will that affect the probability of that ball being selected? Maybe, just a little, but the draw should be fair so those providing the lotto equipment spend a large amount of effort to ensure that they are fair, and the costs of that effort must come out of the prize funds.

Secondly, the state or maybe the lotto organisation itself will often withhold part of the lotto sales takings for local or national causes, such as cancer research, or societal things, like the fight against teen suicide. The money for humanitarian causes is deducted from the prize funds.

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One of the humanitarian causes is often the fight against problem gambling. It’s ironic and somewhat appropriate that funds from gambling are used to combat problem gambling. It seems that some people get such a thrill from gambling that they use all their, then borrow or steal from others to continue to gamble.

They invoke the Gambler’s Fallacy. They suggest that their luck must change sooner or later. It doesn’t have to, and may never change, but they continue to spend money on their gambling. They also don’t take account of the fact that they might win, eventually, by sheer chance, but it is unlikely that their winnings will cover what they have already gambled away. They have a tendency to believe that one big win will sort things out for them. It won’t of course.

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So, the only true fact about Lotto and similar draw is that you have to be in to win. But just because you are in doesn’t mean that you will win. You probably won’t. The best way to treat Lotto and other similar games is that you are donating to a good cause and you might, but probably won’t get something back. So, I’m off to buy a lotto ticket. I might win thirty million dollars, but I won’t cry if I don’t.

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## Looking for Inspiration

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I suppose that everyone has seen the so-called “Inspirational Quotes“. If you haven’t, it is unlikely that you have been using the Internet a lot! Inspirational Quotes are short sentences, usually totally devoid of context that, supposedly, provide guidance or inspiration for those who need it. Usually the quotation is in large font applied over the top of a sunset, or a couple hand in hand, or a cute puppy or other animal.

Since the quotation is usually without context, the reader is free to apply it however he or she wants. You can apply it to your own situation, whatever that might be. A large portion of the quotes exhort the reader to just get up and do it, whatever it might be. The idea is that one should take one’s chance and go for it.

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This is all well and good if the advice is appropriate. The original writer has no way of knowing this. Someone might take the message as a sign to get out of a situation where they are safe and comfortable and to take risks. Unfortunately, if this turns out to be a mistake, there is usually no way back.

Many of the inspirational quotations have a religious slant to them.  Søren Kierkegaard reportedly said “Now, with God’s help, I shall become myself.” It’s easy to make fun of inspirational quotes, both religious and secular, such as the foregoing. After, if he wasn’t himself when he made the quotation, what was he? It is so devoid of context that one can’t help asking oneself what one is supposed to do to become oneself?

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Can the quotations be dangerous? I suppose that if one is depressed or suicidal it would be unfortunate to come across a quotation that said, basically, “just do it,” but it is unlikely that a simple quotation like that would actually incite suicide.

I suspect that most of the inspirational quotations are pretty benign. People look at them and are momentarily uplifted or cheered up by then, but then just carry on with their lives. The quotations may help them cope with a difficult situation or help them be happy in the situation that they find themselves in. I doubt that the motivation goes deep enough to completely change their lives, but I don’t know if anyone has ever checked or studied the phenomenon.

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After I started thinking about inspirational quotations, I wondered who it is who writes the things. Someone must spend a lot of time either extracting them from online books and pages and maybe they even type them up from paper books! In many cases they then paste the text onto pretty pictures of all sorts of things. Sunsets seem to be a favourite.

Then I discovered the on-line generators for these things. Some of them just allow you to type in whatever you like, but some of them will generate the whole thing for you. One that I’ve played with a bit is InspiroBot, which produces quotations using some sort of algorithm, and calls itself an Artificial Intelligence. It produces image/quote combinations which range from ones which seem sense free to those that seem like they mean something.

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I was wondering how the meme arose, then I though back to the times when computers were just entering the workplace. Way back when printers could only print letters and numbers people would draw something using just letters and numbers. If you went up close you could see the letters and numbers but from a distance the different densities of the letters looked like a image of something, so people covered whole walls with, say, a picture of an astronaut, or a pinup.

When printers could print images these were replaced with smaller pictures of astronauts or pinups or someone’s kids. Then someone somewhere decided to inspire their staff with a poster or picture with an inspiring caption. Naturally spoof and satires of these soon appeared, and also people started putting up quotations that had inspired them, and spoofs and satires of those also appeared.

Nowadays of course, the whole thing has moved to “social media”. People spot a quotation which appeals to them and post it on Facebook. This quite often means that you might see the same “inspirational posting” several times, as other people share it with their friends which might include you!

I’m intrigued by the programs that produce the quotations by algorithmic means. Since they produce only a short sentence, there’s more chance that you can see sense in the result, than there would be if the algorithm produced a whole article or something. I’ve found one site where an algorithm produces a small article on each refresh, and the results seem to me to be a bit odd when I try to make sense of them.

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It reminds me of a famous hoax perpetrated by Alan Sokal on the unwary editors of an academic journal. Sokal wrote an article which was composed of buzzwords and references to Post Modern writers, since he believed that all that was required of an article to get it published was the buzzwords and the gratuitous references to Post Modern writers.

He succeeded in getting it published, which ironically gives the article meaning of exactly the sort that he was ridiculing. While it had no meaning in the context of an academic article, it was an unfavourable commentary on the meanings and lack of rigour espoused by the Post Modern movement. If you are interested in producing your own Sokal-type article, there is a web site called “The Post Modern Essay generator, which will do it for you.

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So, are all, or the majority of inspirational quotations generated by an algorithm or do people create them and post them themselves? I think that most are created by people. At least the quotes are, but the actual postings may not be. The quotes seem to, in most cases, almost make sense, but they don’t always seem to match with the pictures. I’d guess that people are using a generator but posting their quotes, whether gleaned from elsewhere or created by themselves, and the picture is more or less random and may not match the quotation.

## Round it up!

Quite often a visit to Wikipedia starts of a train of thought that might end up as a post here, and often I forget the reason that I was visiting Wikipedia in the first place. However in this case I remember what sparked my latest trip to Wikipedia.

I was looking at the total number of posts that I have made and it turns out that I have posted 256. This is post number 257, which is a prime number incidentally. To many people 256 is not a particular interesting number but to those who program or have an interest in computers or related topics, it is a round number.

A round number, to a non-mathematician is a number with one or more zeroes at the end of it. In the numbering system with base 10, in other words what most people would considered to be the normal numbering system, 1000 would be considered to be a round number. In many cases 100 would also be a round number and sometimes 10 would be as well.

In the decimal system, which is another name for the normal numbering system, the number 110 would probably not usually be considered a round number. However, if we consider numbers like 109, 111, 108 and 112, then 110 is a round number relative to those numbers. Rounding is a fairly arbitrary thing in real life, usually.

We come across round numbers, or at least rounded numbers in the supermarket on a daily basis, if we still use cash. Personally I don’t. I recall when the one cent and two cent coins were introduced people were appalled that the supermarkets would round their bills to the nearest convenient five cents.

So a person would go to a supermarket and their purchases would total to, say, \$37.04. The cashier would request payment of \$37.05. Shock! Horror! The supermarket is stealing \$0.01 off me! They must be making millions from all these \$0.01 roundings. In fact, of course, the retailer is also rounding some amounts down too, so if the bill was \$37.01 the customer would be asked to pay only \$37.00. So the customer and the supermarket, over a large number of transactions, would end up even.

Then of course the 5 cents coins were removed and this added an extra dilemma. What if the total bill was \$37.05? Should the customer’s bill be rounded to \$37.00 or to \$37.10? This is a real dilemma because, if the amount is rounded up, then the supermarket pockets five cents in one ten cases, and if it is rounded down the supermarket loses five cents in one in ten cases. If the supermarket a thousand customers in a day, one hundred of them will pay five cents more than the nominal amount on their bill, meaning that the supermarket makes a mere five dollars.

The emotional reaction of the customer, though, is a different thing. He or she may feel ripped off by this rounding process and say so, loudly and insistently. Not surprisingly most supermarkets and other retailers choose to round such bills down. Of course, all the issues go away if you don’t use cash, but instead use some kind of plastic to pay for your groceries, as most people do these days.

There are degrees of roundness. In one context the number 110 would be considered round, if you are rounding to the nearest multiple of ten. If you are rounding to the nearest multiple of one hundred, then 110 is not a round number, or, in other words a rounded number. If we are rounding to the nearest multiple of three, then 110 is not a rounded number but 111 is (111 is 37 multiplied by 3).

Real numbers can be rounded too. Generally, but not always, this is done to eliminate and small errors in measurement. You might be certain that the number you are reading off the meter is between 3.1 and 3.2, and it seems to be 3.17 or so, so you write that down. You take more measurements and then write them all down.

Then you use that number in a calculation and come up with a result which, straight out of the calculator, has an absurd number of decimal places. Suppose, he said, picking a number out of the air, the result is 47.2378. You might to choose to truncate the number to 47.23, but the result would be closer to the number that you calculated if you choose to round it 47.24.

A quick and easy way to round a real number is to add half of the order of the smallest digit that you want to keep and then truncate the number. For the example number the order of the smallest digit is 0.01 and half of that is 0.005. Adding this to 47.2378 gives 47.2428, and truncating that leaves 47.24. Bingo!

Another way of dealing with uncertain real numbers such as results from experiments is to calculate an error bound on the number and carrying that through to the calculated result. This is more complex but yields more confidence in the results than mere rounding can.

To get back to my 256th post. Why did I say that this is a round number in some ways? Well, if instead of using base 10 (decimal), I change to using base 16 (hexadecimal) the number 256 (base 10) becomes 100 (base 16), and those trailing zeroes mean that I can claim that it is a round number.

Similarly, if I choose to use base 2 (binary), 256 (base 10) becomes 100000000 (base 2). That is a really round number. But if I use base 8 (octal), 256 (base 10) becomes 400 (base 8). It’s still a round number but not as round as the binary and hexadecimal versions are, because it start with the digit 4. As a round number its a bit beige.

It’s interesting (well it is interesting to me!) that there are no real numbers in a computer. Even the floating point numbers that computers manipulate all the time are not real numbers. They are approximations of real number stored in a special way (which I’m not going to into).

So when a computer divides seven by three, a lot of complex conversions between representations of these numbers goes on, a complex division process takes place and the result is not the real number 2.333333…. but an approximation, stored in the computer as a floating point number which only approximate, while still being actually quite accurate.

## Once a week

I’ve been pondering the topic of ‘the week‘. Not the ‘topic of the week’. The week, as in seven days. It’s an unusual number to use as a unit for a length of time, as it is a prime number of days, and this makes using fractions of a week a bit tricky. Half a week is 3 and a half days long, so it’s not usual to, for instance, agree to meet someone in ‘half a week’.

No, we say ‘See you in three days’, or four days. We might say ‘this paint will take 2 and a half days to fully dry’, but this is a bit odd. We’d usually say something like ‘this paint will take between 2 and 3 days to fully dry’. We usually treat days as ‘atomic’ when counting days. The number of days is usually an integer, although we could break days down and use fractions or real numbers with them.

The fact that the number of days in a week is a prime integer also makes converting from weeks to days and days to week interesting. Quick, how many days in seventeen weeks? The answer is 119. How many weeks is 237 days?  The answer is 33 with six days left over. It’s not easy.

Four weeks is 28 days, which is approximately a lunar cycle. It is also very approximately one month. There are approximately thirteen 28 days period in a year, assuming a 365 days year which is approximately correct. This is probably why some calendars have thirteen months.

The lunar cycle is around 29 and a half days, whereas the month defined as one twelfth of a year is around 30 and a half days. Nothing fits! The month is based on the lunar cycle, and the ancients noticed that that the twelve lunar cycles is 354 days which was close to the 365 and a bit days that comprise a year.

So, they decided to make it fit. They divided the year into 12 months, which left them with bits of days just lying around. This was obviously untidy so they scrunched up the bits into one days and tagged them onto the various months more or less at random. The final left over bit that they ended up with they ignored.

That’s how we ended up with mnemonic rhyme “30 days hath September, April, June and November…” with that horrible line that doesn’t scan. That’s rather appropriate really, as the reason that the rhyme is needed is because the days don’t fit properly into the months. It’s an uneven rhyme for an uneven scheme.

The ancients ignored the odd bit of a day that was left over until someone noticed that the year was still sliding out of synchronisation with the seasons. So they added or took away a day or two here and there in special, short or long years. Problem solved.

Well sort of. They ended up with a super complex list of rules for working out how many days there are in a month, where to fit extra days into the calendar, and when to fit them in. Horror!

Finally scientists decide to cut through all this confusion and define a second by using an atomic clock. Providing you don’t accelerate the clock to a significant fraction of the speed of light and keep it at absolute zero. Easy!

Again, sort of. The standard second times sixty give a standard minute. The standard minute times sixty gives the standard hour. The standard hour times twenty four gives the standard day and the standard day times seven gives the standard week. Yay, you might say.

Unfortunately the actual day and therefore the actual week is not exactly equal to the standard day or week. It would be quite legitimate to claim “Wow, this is a long week, it’s 0.608111.. standard seconds longer than a standard week”. But don’t expect much sympathy.

Seven days is actually a pretty reasonable length to a week. We divide it into “the weekend” and “the rest of the week”. If it was a couple of days longer, it would be a long time between weekends. We’d probably be tempted to add an extra day to each weekend, or maybe alternate weekends…. But now we’re getting complicated again.

If the week was shorter, we’d probably get less work done. If the week was five days and we still had a two day weekend then time available for work would be about 17% less. Of course five day working weeks are fairly recent in historical terms, but I’m not going to work out the numbers for a 6/7 working week and a 4/5 working week.

Speaking of work, and assuming that most people would not work unless they have to, we have developed various coping strategies. We count the days to the weekend. “Only three more days to the weekend.” Tomorrow is Thursday and that means only one more day to the weekend.”

We designate Wednesday as “Hump Day”, since it is the middle of the week and if we reach Hump Day before having a breakdown or perhaps killing someone, that’s a win. There’s only half the week to go and we’ve broken its back.

We celebrate Fridays, often with a quick drink, then shoot off to enjoy the weekend. We come in on Mondays, faced with five more days of toil. On Tuesdays, we’ve at least knocked off one day, but it’s still a bit beige. Wednesday is Hump Day and we’re halfway there! When Thursday comes we’re almost there, and Friday is relatively easy. It’s practically the weekend, when we block out the thought of Monday all together if we can.

The week has a sibling called the “fortnight”. Two weeks, as a chunk. At one time the fortnight was usually reserved for a summer holiday. A fortnight at the beach or the bach. Time away with the kids. Idyllic golden weather by the sea. Of course, we only remember the good times, and forget the bad ones, but still it would be summer, it would be fairly warm, and the weather is usually better in the summer.

Weeks are the medium sized sections of our lives, often used to split up the humdrum from the pleasant parts of our lives. We should appreciate our weeks, no matter how many standard seconds long they are.

## Choose! Choose now!

Life throws many choices in our way. One view of the world is that it is like a many branched pathway, with our every day choices causing us to thread a particular path though this maze of branches, to reach the ever growing tip of the tree of events that is our past.

The future is yet to come into being but we can see dimly into it, and we use this limited view to inform our choices. The view into the future is like a mist. Things appear dimly for a while only to fade and be hidden from view. Sometime in the future is the instant of our demise. We know it’s coming but we do not usually know how and when.

We try to compensate for our inadequate view of the future by trying to cater for all possibilities, and one way we do this is by making a will, to prescribe how we would like our things, our assets, to be distributed when we are dead.

Some people try to predict the future. Some people gamble, on horses or whatever, trying to guess the winner of a race. There are two sorts of such people, those who estimate the odds and then build in as much of a safety margin as they can. These are usually the ones who run the books, while the other sort take a more optimistic view and gamble that the bookmakers are wrong. The first group is generally happy to make small profits while the second group want high returns. Generally the first group does a lot better than the second group over a reasonably long time frame.

The interesting thing about choice is that it is a discrete thing. We choose from one or more possibilities and the number of those possibilities is an integer. Often it is a choice between option one or option two. Pretty obviously it isn’t option one point five.

If we have two possibilities, call them A and B, then the probability of A occurring might be thirty percent. This means that the probability of B happening is seventy percent. The two must always add up to one hundred per cent.

So there is a mapping here between discrete events and continuous probabilities. Between integers and real numbers. One way of looking at this is that “event A” is a sort of label to the part of the probability curve that represents the event. Or it could be considered that the probability of the event is an attribute of the event.

It could be that when a choice is made and the probability of making that is more probably than making the other choice then that it is similar to making a choice of road. One road is wide and one is narrow. The width of the road could be related to the probability of making that choice.

The width of the road or the probability of the choice may well be subjective of course. I might choose to vote for one political party because I have always voted for that party. The probability of me voting for that party is high. The probability of my voting for another party would be quite low. However for someone who is the supporter of another party, the road widths are the other way around.

Is it true that when I vote for the party that I usually vote for that I exercise a choice? Only in a weak way. Merely doing things the way that one has always done is just taking the easy way and involve little choice. The reason for taking the easy choice may be because one has always done it that way and there is no reason to change. Habit, in other words.

Most choices we make are similar. We have a set of in-built innate or learned reactions to most situations, so that we don’t have to trouble to make a choice. If you make a choice, if you drill down far enough you will find that there are always reasons for a choice that you make. Your father always voted for the party, so you do out of loyalty and shared beliefs.

Every choice, when you examine it, seems to just melt away into a mass of knee jerk reactions and beliefs. When you examine choices you find that there was in fact no other way that we were likely to choose and free choice doesn’t really exist.

We have all been to a fast food restaurant only to find that the person before us is unable to make up their mind. This is probably because they do not have strong preferences so that they don’t have any reason to choose one dish over the other, or they dislike all the dishes equally.

If we put people in a situation where they have no reason to prefer one course of action over another and we force them to make a choice, they will often think up ludicrous reasons for making the choice that they finally make.

For instance on game shows where they have to make a selection from a multiple choice question in a limited amount of time, quite often they will say something like “I haven’t pressed B in a while”, or “I guessed A last time and it worked out for me so I did it again”, even something like “It’s my boyfriends favourite colour.” It’s hard to know if they really used that reasoning or whether they are justifying their choice after the event.

Another way to cause people to make a random choice is to try and remove all distractions. I can envisage an experiment where people are placed in a room with a screen and two buttons. They are then told by a message on the screen to press the correct button within ten seconds and a count down starts. Since they have no knowledge of which is the correct button they will be forced to choose any button to press or to let the timeout expire. Then they will asked why they chose that particular button. The results of such a test would be interesting.

## Parallel worlds or a Continuum?

A cursory search on the Internet doesn’t tell me one way or another if Erwin Schrodinger owned a cat. Nevertheless he could have owned a cat, so the existence of Schrodinger’s actual cat is unknown to me. David Deutsch might possible argue that Schrodinger’s decision to own a cat or not own a cat resulted in two parallel worlds.

The above is obviously a play on the original scenario outlined by Schrodinger, the famous Schrodinger’s Cat thought experiment. The cat’s state before the box is opened is a strange state, referred to as a superposition, where the cat is both alive and dead. When the box is opened it is argued that this state is somehow resolved with cat being definitely alive or dead.

Suppose that we install a detector in the box with the cat which determines whether or not the cat is dead and notes the time when  it dies. Does this resolve the paradox? After all, if the detector says that the cat died three minutes ago, then we now know exactly when the cat died.

This doesn’t resolve the issue, though, as the detector will also be in a superposition until the box is opened – we don’t know if it has been triggered or not. Of course, some people, including Schrodinger himself, are not happy with this interpretation, and it does seem that, pragmatically, the cat is alive until the device in the box is triggered and is thereafter dead.

However the equation derived by Schrodinger appears to say that the cat exists in both states, so it appears as if Schodinger’s “ridiculous case” (his words) is in fact the case. Somehow the cat does appear to be in the strage state of superposition.

If we look at the experimenter, he (or she) has no clue before opening the box whether the cat is dead or not. Nothing appears to change for him (or her), but in fact it does. He (or she) is unaware of the state of the cat, so he (or she) is in the superimposed state : He (or she) is unaware whether or not the cat is alive or whether it is dead, which is a superposition state.

Yet we don’t find this strange. If we remove the scientific gadgets from the box, this doesn’t really change anything – the cat may drop dead from old ages or disease before the box is open. Once again we cannot know the live/dead status of the cat until we open the box.

So, what is special about opening the box? Well, the “when” is very important if we consider the usual case with the scientific gadgets in the box. If we open the box early we are more likely to find the cat alive. If we open it later it is more likely that the cat will be dead. Extinct. Shuffled the mortal coil.

So it is the probability of atomic decay leading to the cat’s death that is changing. It may be 70% likely that cat is dead, so if we could repeat the experiment 1000s of times 7/10th of the time the cat is dead, and 3/10th of the time the cat is still alive. Yeah, cat!! (There’s also a possibility that the experimenter gets a whiff of cyanide and dies, but let’s ignore that.)

But after the box is opened, the cat is 100% alive or 100% dead. Apparently. How did that happen? Some people claim that something mysterious called “the collapse of the waveform” happened. I don’t think that really explains anything.

The same thing happens in the real world. If I don’t check my lotto tickets I’m in a superposition state of having won a fortune and not having won a fortune. When I check them I find I haven’t won anything. Again! I must stop buying them. They are a waste of money.

The many worlds hypothesis gets around this by postulating the splitting of the world into two worlds whenever a situation like this arises. After I check my lotto ticket there are two worlds, one where I am a winner and one where I am not. How can I move to the world where I’m a millionaire? It doesn’t seem fair that I stuck here with two worthless bits of paper. does it?

And what does the probability mean? In the lotto case it is 1 in an astronomical number that I come out a winner and almost 1 that I get nothing. In the cat case it may be 60/40 or 70/30, and in the cat case it changes over time.

If the world splits every time a probabilistic situation arises, then the probabilities don’t actually mean much. What difference does it make if a situation is “more probable” than another situation if both situations come about in the multiverse regardless? It doesn’t seem that it is a meaningful attribute of the branches. What does it mean, in this model that branch A is three times more likely than branch B? Somehow a continuum (probability) reduces to a binary choice (A or B).

We could consider that the split is not a split at all, but that reality, the universe, whatever, has another dimension, that of probability. Imagine your worldline, a worm travelling through the dimensions of space and the new one of probability. You open the box and lo! Your worldline continues, and the cat is now dead or alive, but not both.

But which way does it go? That is determined purely by the probabilities, by the throw of the cosmic dice, but once it chooses a path, then there is no other possibility. In the space dimensions you can only be in one place at a time. If you are at A you cannot be at B, and similarly in the probability dimension, if you are at P you cannot be at Q.

However any point P (the cat is still alive!) is merely a point on the probability line. There are an uncountable number of points where the cat is alive and also an uncountable number of points where the cat is deceased. But the ratio between the two parts of the line is the probability of the cat’s survival.

## A Sum of All the Parts

Consciousness is fascinating and I keep coming back to it. It is personally verifiable in that a person knows that he or she is conscious, but it is difficult if not impossible to tell if a person is conscious from the outside.

When you talk to someone, you and that person exchange words. You say something, and they respond. Their response is to what you say, and it appears to show that the person is a conscious being.

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It’s not as easy as that, however, because it is conceivable that the person is a zombie (in the philosophical sense) and his or her responses are merely programmed reactions based on your words. In other words he or she is not a conscious being.

It seems to me that the best counter to this suggestion is that I am a conscious being and I am no different in all discernible ways from others. It is unreasonable to suggest I am the only conscious being anywhere and that all others are zombies.

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Of course this leaves open the suggestion that some people may be philosophical zombies, but that then raises the question of what the difference is, and how can one detect it. William of Occam would probably wield his razor and conclude that, if one can’t tell, one might as well assume that there are no zombies, as assuming that there are zombies adds a (probably) unnecessary assumption to the simple theory that all humans are conscious beings.

It follows that consciousness is probably an emergent phenomenon related to the complexity and functioning of the brain. It also follows that lower animals, such as dogs, cats and apes are also probably conscious entities, though maybe to a lesser extent that we are.

The only way we can directly study consciousness is by introspection, which is more than a bit dubious as it is consciousness studying itself. We can indirectly study consciousness by studying others who we assume to be conscious, maybe when they have been rendered unconscious by anaesthetics and are “coming round” from them.

In addition, consciousness can be indirectly studied using mind altering drugs or meditation. However we are mainly dependant on verbal reports from those studied this way, and such reports are, naturally, subjective.

When we introspect, we are looking inwards, consciously studying our own consciousness. There are therefore limits on what we can find out, as the question arises “How much about itself can a system find out?”

A system that studies itself is limited. It can find out some things, but not all. It’s like a subroutine in a bigger program, in that it knows what to do with inputs and it creates appropriate output for those inputs. Its sphere of influence is limited to those processes written into it, and there is no way for it to know anything about the program that calls it.

A subroutine of a larger program uses the lexical, syntactical and logical rules that apply to the program as a whole, though it may have its own rules too. It shares the concept of strings, number, and other objects with the whole program, but it can add its own rules too.

The Universe is like the subroutine in many ways. The subroutine has inputs and outputs and processes the one into the other. In this Universe we are born and we die. In between we spend our lives.

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An aware or conscious subroutine would know that it processes input and creates outputs, but it would have no idea why. We know that we are born, we live and we die. Apart from that we have no idea why.

This sort of implies that while we may use introspection to investigate some aspects of consciousness we will always fall short of understanding it completely. We may be able to approach an understanding asymptotically however – we might get to understand consciousness to the 90% level, so it would not be a total waste of time to study it.

Consciousness seems to be more than a single state, and the states seem to merge and divert without any actions on our part. For instance, when I am driving there is a part of me that is driving the car and a part of me that is route planning, and maybe a part of me that is musing on the shopping that I intend to do.

The part of me that is driving is definitely aware of what is happening around me. I don’t consciously make the decision to slow down when other traffic gets in the way, but the part of me that is driving does so.

Similarly the part that is route finding is also semi-autonomous – I don’t have to have a map constantly in my mind, and don’t consciously make a decision to turn right, but the navigator part of my consciousness handle that by itself.

Those parts of my mind are definitely conscious of the areas in which they are functioning, because if they were not conscious, they would not be able to do their job alone and would frequently need to move to the front of my consciousness disrupting my musing about my shopping.

It’s like part of my consciousness are carved off and allowed to perform their functions autonomously. However if an emergency should arise, then these parts are quickly jolted back into one.

The parts of my mind are definitely conscious as, at a low level, I am aware of them. I’m aware of the fact that I’m following that blue car, and I’m aware that I have to turn left in 200m or so. I’m also aware of my shopping plans, while I’m aware of the music on the radio.

While it sounds scary that I’m not totally concentrated on my driving, I believe that this sort of has to happen. If I was totally concentrated on my driving, I would need to stop at every intersection so that I could decide which way to turn.

I would need have my shopping list completely sorted out, to the point of knowing which stores I am going to before even getting the car, and I would have to plan my route precisely. This would not allow for those occasions when passing by something or some shop reminds you that you need something that is not on your shopping list.

This splitting of consciousness allows us to perform efficiently. The only downside is that splitting things too much can result in us becoming distracted. And that is the reason we shouldn’t fiddle with the radio or use cellphones when driving.

## Simple Arithmetic

There periodically appears on the Internet an arithmetic type of puzzle. Typically it will be a string of small natural numbers and a few arithmetic operations, such as “3 + 7 x 2 – 4” and the task is to work out the result.

The trick here is that people tend to perform such a series of calculations strictly from left to right, so the sequence goes:

3 + 7 = 10, 10 x 2 = 20, 20 – 4 = 16. Bingo!

Most mathematicians, and people who remember maths from school would disagree however. They would calculate as follows:

7 x 2 = 14, 3 + 14 = 17, 17 – 4 = 13. QED!

Why the difference? Well, mathematicians have a rule that states how such calculations are to be performed. Briefly the calculations is performed from left to right, but if a multiplication or division is found between two numbers, that calculation is performed before any additions or subtractions. In fact the rule is more complex than that, and a mnemonic often used to remember it is “BODMAS” or “BEDMAS” (which I’m not going to explain in detail here. See the link above).

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This rule is only a convention and so is not followed everywhere, so there are various “correct” answers to the problem. Also, there are still ambiguities if the conventions are applied which could cause confusion. However, most people with some mathematical training would claim that 13 is the correct answer.

Interestingly, computer programming languages, which are much stricter about such things, codify the precedence of operations in a calculation exactly, so that there can be no ambiguity. It is the programmers task to understand the precedence rules that apply for a particular language.

In most cases the rules are very, very similar, but it is the documentation of the language which describes the rules of precedence, and wise programmers study the section on operator precedence very closely.

There are ways of specifying an arithmetic problem uniquely, and one of those (which is sometimes of interest to programmers) is “Reverse Polish Notation“. Using this my original puzzle becomes “3 7 2 x + 4 -” which looks odd until you understand what is going on here.

Imagine that you are traversing the above list from left to right. First you find the number “3”. This is not something you have to do, like “+”, “-“, “x” or “/”, so you just start a pile and put it on the bottom. The same goes for “7” and “2”, so the pile now has “3” on the bottom and “2” at the top and “7” in the middle.

Next we come across “x”. This tells us to do something, so we pull the last two things off the pile and multiply them (7 x 2 = 14) and stick the result, “14” back on the stack which now contains “3” and “14”. The next thing we find is “+” so we pluck the last two things off the pile “3” and “14” and add them, putting the result “17” back on the (empty) pile.

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Next up is “4” which we put on the pile, and finally, we have “-“, so we pull the two last elements from the pile (“17” and “4”) and subtract the second from the first, giving “13” (Yay!) and that is the answer which we put back on the stack. The stack now contains nothing but the answer.

This looks confusing, but that may be because we are used to the conventional left to right way of doing things. It is actually easier for a computer to understand the RPN version of the puzzle and there are no ambiguities in it at all. Technically, it’s a lot simpler to parse than the conventional version.

Parsing is what happens when you type a command into a computer, or you type something complex, such as a credit card number into the checkout section of a web site. The computer running the web site takes your input and breaks it up if necessary and checks it against rules that the programmer has set up.

So, if you type 15 numbers or 17 numbers into the field for the credit card number, or you type a letter into the field by mistake, the computer will inform you that something is wrong. Infuriatingly, it may be not be specific about what the trouble is!

Anyway, back to the arithmetic. It grates with me when people make simple arithmetical errors and then excuse themselves with the phrase “I never was much good at maths at school”! That may well be true, but to blame their problems with arithmetic of the whole diverse field of mathematics.

It’s like saying “I can’t add up a few numbers in my head or on paper because I missed the class on elliptic functions“! It’s way over the top. For some reason people (especially those who can’t get their head around algebra) equate the whole of mathematics with the bit that they do, which is the stuff about numbers, which is arithmetic.

As we evolved, we started counting things. It’s important to know if someone has got more beans than you or that you have enough beans to give everyone one of them. We invented names for numbers and names for the things (operations) we did on them.

We did this without much thought about what numbers actually are. We as a species have only relatively thought deeply about numbers fairly recently, and we only discovered such things as real numbers and geometry in the last couple of thousand years so it is not surprising that the average brain has yet to expand to cope with the more advanced mathematical concepts.

This could be why so many people these days equate fairly simple arithmetic with mathematics as a whole – our brains are only now coming to grips with the concept that there is more to maths than simply manipulating numbers with a very few simply operations.

It may be that the average human brain never will get to grips with more advanced maths. After all, people can survive and thrive in the modern world with on a rudimentary grasp of mathematics, the arithmetic part.

Some human brains however do proceed further and much of modern society is the result of mathematics in its wider sense applied to the things that we see around us. For instance,  we could not have sent men to the moon without advanced mathematics, and technology relies heavily on mathematics to produce all sorts of things. It’s a good things that some brains can tell the difference between the field of arithmetic and mathematics as a whole.

## Let’s be Rational – Realer Numbers

Leopold Kronecker said “God made the integers, all else is the work of man”. (“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk”). However man was supposedly made by God, so the distinction is logically irrelevant.

I don’t know whether or not he was serious about the integers, but there is something about them that seems to be fundamental, while rational numbers (fractions) and real numbers (measurement numbers) seem to be derivative.

That may be due to the way that we are taught maths in school. First we are taught to count, then we are taught to subtract, then we are taught to multiply. All this uses integers only, and in most of it we use only the positive integers, the natural numbers.

Then we are taught division, and so we break out of the world of integers and into the much wider world of the rational numbers. We have our attention drawn to one of the important aspects  of rational numbers, and that is our ability to express them as decimal fractional numbers, so 3/4 becomes 0.75, and 11/9 becomes 1.2222…

The jump from there to the real numbers is obvious, but I don’t recall this jump being emphasised. It barely (from my memories of decades ago) was hardly mentioned. We were introduced to such numbers as the square root of 2 or pi and ever the exponential number e, but I don’t recall any particular mention that these were irrational numbers and with the rational numbers comprised the real numbers.

Why do I not remember being taught about the real numbers? Maybe it was taught but I don’t remember. Maybe it isn’t taught because most people would not get it. There are large numbers of rational accountants, but not many real mathematicians. (Pun intended).

In any case I don’t believe that it was taught as a big thing, and a big thing it is, mathematically and philosophically. It the divide between the discrete, the things which can be counted, and the continuous, things which can’t be counted but are measured.

The way the divide is usually presented is that the rational numbers (the fractions and the integers) plus the irrational numbers make up the real numbers. Another way to put it, as in the Wikipedia article on real numbers, is that “real numbers can be thought of as points on an infinitely long line called the number line or real line”.

Another way to think of it is to consider numbers as labels. When we count we label discrete things with the integers, which also do for the rational numbers. However, to label the points on a line, which is continuous, we need something more, hence the real numbers.

Real numbers contain the transcendental numbers, such as pi and e. These numbers are not algebraic numbers, which are solutions of algebraic equations, so are defined by exclusion from the real numbers. Within the transcendental numbers pi and e and a quite large numbers of other numbers have been shown to be transcendental by construction or argument. I sometimes wonder if there are real numbers which are transcendental, but not algebraic or constructible.

The sort of thing that I am talking about is mentioned in the article on definable real numbers. It seems that the answer is probably, yes, there are real numbers that  are not constructible or computable.

Of course, we could list all the constructible real numbers, mapped to the real numbers between 0 and 1. Then we could construct a number which has a different first digit to the first number, a different second digit to the second number and so on, in a similar manner to Cantor’s diagonal proof,  and we would end up with a number that is constructed from the constructible real numbers but which is different to all of them.

I’m not sure that the argument holds water but there seems to be a paradox here – the number is not the same as any constructible number, but we just constructed it! This reminds of the “proof” that there are no boring numbers.

So, are numbers, real or rational, just labels that we apply to things and things that we, or mankind as Kronecker says, have invented? Are all the proofs of theorems just inventions of our minds? Well, they are that, but they are much more. They are descriptions of the world as we see it.

Whether or not we invented them, numbers are very good descriptions of the things that we see. The integers describe things which are identifiably separate from other things. Of course, some things are not always obviously separate from other things, but once we have decided that they are separate things we can count them. Is that a separate peak on the mountain, or is it merely a spur, for example.

Other things can be measured. Weights, distances, times, even the intensity of earthquakes can be measured. For that we of course use rational numbers, while conceding that the measurement is an approximation to a real number.

A theorem represents something that we have found out about numbers. That there is no biggest prime number, for example. Or that the ratio of the circumference to the diameter is pi, and is the same for all circles.

We certainly didn’t invent these facts – no one decided that there should be no limit to the primes, or that the ratio of the circumference to the diameter of a circle is pi. We discovered these facts. We also discovered the Mandlebrot Set and fractals, the billionth digit of pi, the bifurcation diagram, and many other mathematical esoteric facts.

It’s like when we say that the sky is blue. To a scientist, the colour of sunlight refracted and filtered by the atmosphere, peaks at the blue wavelength. The scientist uses maths to describe and define the blueness of the sky, and the description doesn’t make the sky any the less blue.

The mathematician uses his tools to analyse the shape of the world. He tries to extract as much of the physical from his description, but when he uses pi it doesn’t make the world any the less round as a result. Mathematics is a description of the world and how it works at the most fundamental level.

[I’m aware that I have posted stuff on much the same topic as last time. I will endeavour to address something different next week].