birthday cake
birthday cake (Photo credit: freakgirl)


Today (Sunday 28th July) is my birthday, which naturally had me pondering birthdays in general. In July I have my birthday, my son’s birthday is a few days earlier and my granddaughter’s is next month. It turns out that a friend of a Facebook friend also has her birthday today, on my birthday. Jim Davis, the creator of Garfield was also born on my birthday. And finally, George, the latest addition to the royal family and third heir to the British throne was also born in July, the same month as me and my son.

The maths of birthdays is interesting. If you have 23 randomly selected people then the probability that at least two of them share a birthday is a shade greater than 50% (50.7297%). If you have 53 people the probability goes above 99%. This is known as the Birthday Problem or the Birthday Paradox, though it is not really a paradox, I believe. There are a number of simplification used in calculating the above. For instance, it assumes that all birth dates are equally probable, but they are not, and it also ignores leap days. Also mothers can sometimes, within bounds, select the day that their baby is born, especially for at risk babies and this potentially could cause a skew in the probabilities.

English: The birthday paradox: p(n) represents...
English: The birthday paradox: p(n) represents the probability that in a room with n people, some two (or more) will share the same birthday; q(n) represents the probability that in a room with n people, that at least one person will have the same birthday as a previously selected person. 中文: 生日悖论 2个人生日相同和跟某人生日形同的概率变化 (Photo credit: Wikipedia)

Some people have two birthdays. Well, the Queen has a real birthday and an official one, so that celebrations of her birthday would not fall too early in the year, but later, when the weather would hopefully be better. Unfortunately that means that in the Southern Hemisphere her birthday falls in the depths of winter!

2013 VividSydney on Queen's Birthday 10 Jun 2013
2013 VividSydney on Queen’s Birthday 10 Jun 2013 (Photo credit: hto2008)

Our years these days are defined in terms of “CE” or “Common Era” and “BCE” or “Before Common Era”. Older people can remember when it was “Before Christian Era” or even “BC” for “Before Christ” and “AD” for “Anno Domini” or “Year of Our Lord”.

I’m not going to argue whether or not Jesus really existed and whether or not he was divine, but if we assume for a moment that he was born, there is a lot of discussion on what year it was that he was born. Using the gospels and other historical information as a guide, many people believe that he was born 4 to 6 years before start of the Common Era. Or using the terminology, he was born up 6 years “Before Christ”! Humorous, I suppose.

Christ's Birth Orthodox church
Christ’s Birth Orthodox church (Photo credit: baswallet)

Most people view history as continuous and the dates as fixed and well known. That’s not the case of course – the calendar has been revised several times, and  even countries which are Christian may have different calendars. Other religions naturally don’t relate their calendars to the birth of Christ. I believe that some even count backwards.

Calendars have grown out of necessity. Tax collectors in particular love calendars. Calendars are used to keep track of one’s age. Before calendars were widespread years were kept track of by relating births and deaths to important events, like the installation of a particular ruler. For instance, the gospel writer Luke relates Jesus’s birth to a census taken at the time:

<code>In those days Caesar Augustus issued a decree that a census should be taken of the entire Roman world. 2 (This was the first census that took place while[a] Quirinius was governor of Syria.) 3 And everyone went to their own town to register. Luke 2 1-3</code>

This shows the way that all dates were reckoned, relative to fairly recent events. The possibilities for error are obvious. Even if the events are written down, going more than a few years into the past involves research and calculation. Such calculations lead to such absurdities as Bishop Ussher’s calculation of the age of the earth as around 6,000 years. Even the dates of events early in the Common Era  can be dubious. This seems strange to citizens of the modern world, who can measure time to the accuracy of the vibration of an atom, and can accurately date events for at least a hundred or more years into the past.

Animated version of the lead isotope isochron ...
Animated version of the lead isotope isochron that Clair Patterson used to determine the age of the solar system and Earth (Patterson, C., 1956, Age of meteorites and the earth: Geochimica et Cosmochimica Acta 10: 230-237). The animation shows progressive growth over 4550 million years (Myr) of the lead isotope ratios for two stony meteorites (Nuevo Laredo and Forest City) from initial lead isotope ratios matching those of the Canyon Diablo iron meteorite. (Photo credit: Wikipedia)

One last comment – people who were born in the same year as the Queen, but born after her actual birthday and before her official birthday can claim to be both older and younger than the Queen.

Queen of United Kingdom (as well as Canada, Au...
Queen of United Kingdom (as well as Canada, Australia, and other Commonwealth realms) (Photo credit: Wikipedia)

Where do ideas come from?

ideas (Photo credit: Sean MacEntee)

I was watching this on Youtube, and I found myself saying “Yes, but…”. What Stephen Johnson says in there is all true. I like his idea of a “slow hunch” that takes several years or decades to develop. Stephen’s environmental approach looks at the places that provide the environment where ideas flourish, such as coffee shops which flourished in the 17th century and later. The Wikipedia article notes that

Though Charles II later tried to suppress the London coffeehouses as “places where the disaffected met, and spread scandalous reports concerning the conduct of His Majesty and his Ministers”, the public flocked to them.

Apparently Charles did not like the new ideas emanating from the coffee shops and thought that doing away with them would do away with the ideas. I’m not so sure – the discussion groups from the coffee shops would almost certainly have moved elsewhere.

Lloyd's Coffee House
Lloyd’s Coffee House (Photo credit: Wikipedia)

Ideas certainly sprang from the coffee houses which mutated into or gave rise to the London Stock Exchange, Lloyd’s of London and some famous auction houses. I refer you to the Wikipedia article.

Stephen Johnson describes the environments that provide fertile ground for new ideas, and similar places have been invented and reinvented over the years. While Universities were, I believe, originally set up as places for the studying of religion, the concentration of bright people and the opportunities for discussion inevitably led to ideas which were not to the taste of the religious establishment.

Victoria University, Kelburn, Wellington, New ...
Victoria University, Kelburn, Wellington, New Zealand. (Photo credit: Wikipedia)

My “yes, but..” in relation to the Youtube article was not in relation to the matters Johnson discusses, which was the types of environments that favour new ideas, but how the ideas are formed in the human brain. Johnson talks about one person having “a piece of the puzzle” that completes a new idea, but I think that that is an oversimplification. I see it more like a huge floating jigsaw puzzle, with no edges and maybe many many puzzles. Each person gets millions of puzzle pieces and each person does his or her best to fit together as many pieces as possible and some of the pieces may be assembled incorrectly. I’m thinking of the “Intelligent Design” people when I write that.

a drawing of a 4 piece jigsaw puzzle
a drawing of a 4 piece jigsaw puzzle (Photo credit: Wikipedia)

An idea in that model is simply a realisation that that piece or pieces of the puzzle over here seem to fit with the piece or pieces over there. Any idea is based on innumerable prior ideas or realisations.

Ideas also seem to change over time. I think that I recall that when the idea that white light can be split into many colours was first put to me I accepted it with some reservations. Sort of “If you say so”. But today it seems obvious to me, though it can be that probes into the obvious turn up the un-obvious.

Classic Albums: Pink Floyd – The Making of The...
Classic Albums: Pink Floyd – The Making of The Dark Side of the Moon (Photo credit: Wikipedia)

So where do ideas come from? I’m uncertain. I’m not sure that there aren’t several sources of new ideas, but one that I keep coming back to is that there might be some process in our brains of which we are not conscious that continually and somewhat dumbly searches the puzzle pieces and tries to fit them together. It probably has guidance rules that say that, metaphorically, knobs must fit into sockets, there should be no gaps or space between puzzle pieces.

I call the process dumb because it seems to favour picking close by pieces, and it seems to repeatedly try the same configurations that have failed previously. I say this because sometimes, looking at a fact a new way or introducing a concept from another field may result in a totally new solution to a problem.

Visual Example of the Eight Queens backtrack A...
Visual Example of the Eight Queens backtrack Algorithm (Photo credit: Wikipedia)

I’m aware that I’ve used the word “idea” in a number of senses above, but I hope that it doesn’t detract too much from the argument. I’m also aware that I’ve stretched the jigsaw analogy well beyond the bounds!

As a final comment, I think that people misunderstand the Eureka Moment. The moment occurs not when one solves the puzzle, but the moment that one realises that the puzzle is solved. For instance, when a mathematician works on a proof he may get stuck on a particular step. He may try several solutions, proceeding from the solution under test through several other steps in the proof before he discovers the solution which works. The Eureka Moment happens when he discovers that the solution he is trying is the correct one, not when he chooses the solution. A subtle but definite difference.

archimedes (Photo credit: Sputnik Beanburger III)

Why are there always lemons?

English: Random copolymer brush
English: Random copolymer brush (Photo credit: Wikipedia)

I’m always interested in random happenings. Of course ‘random happenings’ always have a cause. Or less linearly, the whole field of the past results in the outcome at the point in question and all other points at the moment in time under consideration and at future moments in time. Or the space-time continuum is not mutable.

Multiverse (Photo credit: kevin dooley)

Whatever. We have recently had a couple of big storms. it being the winter season, and debris has piled up on the beach. This detritus is mostly of marine origin, mostly seaweed, with a sprinkling of other marine debris, such as mollusc shells, not to mention non-organic materials like rocks and sand.

Debris. Notice the small pieces of plastic.

There is a noticeable contribution of terrestrial origin of course, like tree trunks, limbs and even foliage. A significant portion is of anthropological origin, such as worked wood and plastic, and even concrete, tarmacadam, glass and metals.

The plastic is interesting. With the exception of the occasional chunks of polystyrene foam or similar, most of the plastic debris is small, like the rings from the necks of plastic topped containers or the teats from the tops of water bottles. (Aside: Why buy water when you can get it from the tap?) Whole bottles are rare for some reason.

Debris. If you look carefully you may see a few ballpoint pens.

To get back on topic, one of the things that I’ve noticed about the debris is that some objects tend to be found together – for instance left footed shoes may be found on one beach and right footed shoes on another. There is an unconvincing (to me) theory about this.

Right foot shoe

I’ve discovered that things appear to be washed ashore in groups. This may be a statistical aberration, but, for instance, after a recent storm I came across a group of toothbrushes scattered over a relatively small area. Now, there were about a dozen, which rules out a single source, like a flat or house, and they weren’t packaged in any way so that rules out a commercial source, so what could explain it?

Another time the flotsam consisted in part of  what was probably spoon worm corpses. In two particular areas there were hundreds of the disgusting looking things.

Debris with lemon

I don’t know the reasons for these groupings, but obviously some set of circumstances must have resulted in these happenings. Of more obvious provenance are the mass strandings of jellyfish at some times of the year which are no doubt related to the breeding cycle of these animals and particular wind direction. The occasional tennis balls or golf balls that I spot are easily explained too.

But…. But there are always lemons. Whenever I walk along the beach after a storm, I can almost guarantee that I will find at least one lemon. Why? I don’t tend to find apples, though apples float too. Nor, typically any other fruits. Maybe apples are softer and easily broken up?

Regardless, there are always lemons. Why are there always lemons?

Debris with lemon


Predicting the future

Future car!
Future car! (Photo credit: Little Black Cherry)

The farmer fed the chicken every morning at the same. The chicken realised this and ran up to the farmer every morning to be fed. One morning the chicken ran up to the farmer who grabbed it and chopped off its head. This demonstrates the dangers of inductive reasoning. The old turkey was a little more sophisticated however. When asked by a younger turkey when Thanksgiving was, he replied that it was on the fourth Friday in November. The younger turkey was incensed to find out that it was the fourth Thursday in November. The older turkey said to him “Boy, the humans celebrate it on the Thursday, but if I wake up on Friday morning, then I give thanks”.

Induction is looking at the past in a particular way to predict the future. Specifically, induction looks at a series of events in the past to predict the future. The sun has risen like clockwork every day, whether or not you can see it, for as long as anyone can remember and for as long as we can determine from reports from the past. Will it rise tomorrow morning?  I would put money on it because either it will, and I win, or it won’t and it won’t matter because we will almost certainly be dead. The argument comes down to “It has always happened in the past, so it will (or it is extremely like to) happen in the future.

Zabriskie Point at sunrise in Death Valley
Zabriskie Point at sunrise in Death Valley (Photo credit: Wikipedia)

The alternative method of reasoning is deductive reasoning. The deductive argument is that the rising of the sun is a consequence of the rotation of the earth. As the earth rotates, the sun appears to us on the earth’s surface to appear from beneath the horizon and travel across the sky. Actually, it is us who move, a good demonstration of relativity (but maybe I’ll go there another day). The argument goes stepwise from fact to fact and leads inevitably or logically to a conclusion.

Horus, ancient Egyptian God, the Sun God, depi...
Horus, ancient Egyptian God, the Sun God, depicted on papyrus (Photo credit: Wikipedia)

The trouble with this approach is that, for all its logical stepwise approach it is built on two things, a theory and a set of past observations. A scientist has a theory or decides to check a theory, so he does an experiment, and the results of his experiment support or do not support the experiment. The scientist assumes that the theory is true and bases his predictions on this. Unfortunately there is an inductive element to this – if the theory is true for the experiment, there is no guarantee that it will be true for subsequent experiments, even given that ‘ceteris paribus’ (all things remain the same). Some other unconsidered cause could affect the result. The argument is deductive, proceeding in logical steps from the theory, but the practise is inductive – the data has always supported the theory in the past, so it will continue to support the theory in the future.

New Scientist
New Scientist (Photo credit: Wikipedia)

To be fair to the inductivists, todays’ inductivists tend to specify the results of their arguments in terms of probabilities: the probability of the sun rising tomorrow is close to 100%, given that it has always risen in the morning for as far back as we can see, but there is a minute but finite possibility that it won’t for known or unknown reasons.

Let’s consider the case of the sun rising each day and suppose that the fact that the earth rotates is not known. To make the argument more deductive we can postulate causes and so long as the cause fits the facts, we can tentatively label the cause as a hypothesis. Suppose we conjecture that some deity causes the sun to rise each morning. This hypothesis certainly fits the facts and predicts with accuracy that the sun will continue to rise each morning. Such a hypothesis would not be accepted today, of course, except by some individuals.

Mathematical induction can be informally illus...
Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. (Photo credit: Wikipedia)

Is there any great difference between the theist and the scientist? The theist says “all things happen because of God”. The scientist says “all things happen because of the laws of nature”. They both explain things on the basis of their fundamental beliefs.

It is possible that people in the future may look at our theories of the sun rising and other things and consider them naive and consider our view of everything happening according to the laws of nature to be a quaint misunderstanding, in much the same way as many people would consider the “deity hypothesis” to be today.

cubed earth theory
cubed earth theory (Photo credit: Joelstuff V4)

In mathematics the situation is different. Induction is a much more formal process and is applied on top of an axiomatic system. Proved theorems are the results of the applying the axioms repeatedly to another proved theorem or the axioms themselves. Unproven assertions can be proved and turned into theorems or disproved and discarded (or possibly modified so that they can be proved). If something is proved in an axiomatic system, it is true for all time, and cannot be disproved in that system.

Specifically an inductive proof would go something like this: firstly the theorem would be proved for a generic case (eg if statement N is true, then statement N + 1 is true) and secondly it is proved for a specific case (eg statement 1 is true). Then all applicable statements are true because, if statement 1 is true, the generic case means that statement 2 is true, and so on for all cases. Because of the rigor of the argument and the undeniable conclusion of the argument, mathematical inductive proofs are of the same order of reliability as deductive proofs, that is, they are only wrong if there is an error in the logic.

English: Mathematical induction as domino effe...
English: Mathematical induction as domino effect, with text in Esperanto Esperanto: Matematika indukto kiel domen-efiko, kun teksto en Esperanto (Photo credit: Wikipedia)

Why the difference between scientific induction and mathematical induction? Well, I think that it is related to the fact that mathematics is axiomatic and therefore certain, whereas scientific induction is based on the laws of nature which are not and never will be, in my opinion, completely defined. If the basis of your argument is not certain, how can your conclusion be certain?

The End Of Certainty?
The End Of Certainty? (Photo credit: minifig)