I am a mathematical flirt. The Jesuit who taught me was far too clever to understand that I needed my sixth-form mathematics explained in words of one syllable, and preferably in terms of apples and oranges. But I was fascinated by a discipline which could not only describe the world yet could discover otherwise undetectable truths.
Are mathematics inherent in reality, or just our human grasp of the fundamental order of creation? I do not know, nor, as far as I can tell, does anyone else. But today I just want to highlight examples of the mathematics which I find especially appealing.
I am now about to give you the details of my credit card. It measures 55mm by 85mm. That means that Euclid and Plato would have approved, and so would the designer of the Acropolis, for these proportions are the key to its structures. The ratio of approximately eight to 13 is known as the golden ratio or, as Pacioli in the 15th century called it, the “divine proportion”. And that is a good name because that ratio is manifest in so many areas in art, mathematics and nature, suggesting its relationship to creation. If heaven has gates they will certainly satisfy the divine proportion.
Take a line and bisect at the point where the proportion of the original line to the longer section is the same as the proportion of the longer section to the shorter (got that clear?) and you have the divine proportion.
The human eye finds that proportion aesthetically satisfying. A look at Renaissance and traditional painting will show in how many ways the artist uses it to achieve a pleasing composition which draws our attention to where he directs us.
Take a line of numbers. This one is called the Fibonacci sequence. Here, the following sequence is discovered as the sum of the two preceding numbers. Thus: 0, 1, 1, 2, 3, 5, 8, 13, 21, etc. Only of casual interest? Not quite. Notice 8 and 13 – which is once more the ratio of the divine proportion. Indeed, successive Fibonacci numbers confirm this.
And it turns out to be everywhere. It appears in DNA and the hornplates of a turtle, the anatomy of a spider and the breeding of rabbits. It is the ratio involved in the division of tree branches and in the polyfurcation of veins. It is as if the mind of God, the structure of creation, and the mind of man were all connected by a single principle of beautiful harmony which lies at the heart of all existing things.
GIGO means Garbage In, Garbage Out. Wise words for many endeavours but not necessarily true. If you can remember how, solve this equation: i2 = −1 by taking the square root of both sides. The answer is i = √−1. Only it can’t be resolved because any number multiplied by itself is positive; it cannot be negative. √−1 is an “imaginary” number. But before you throw it away as garbage, just check on your digital camera. You will find that the equation which compresses the pixels contains such imaginary numbers. It has been discovered that many equations, almost too complex to be solved directly, can be simplified quickly by such numbers. Many of the operations in the modern world depend on using imaginary numbers.
Of course the answer would be useless if it contained √−1. But it doesn’t. In lay terms, imaginary numbers act as a scaffold brought on to assist, and then quietly removed. You and I know that the scaffold was there, we just didn’t see it.
What proportion of Catholics, at given ages, go regularly to Sunday Mass? Because I can never investigate them all I will have to use a sample. Leaving aside the (fascinating) issue of choosing the right sample and working out how the information will be obtained, I will still have to measure the accuracy of my answers. It may help me to go for an average, remembering that mean, median and modal averages may each be different. But I will also need to have a measurement of whether the distribution is broad or narrow. The standard deviation, which tells me that, will also indicate how my accuracy relates to sample size.
I may need to check my correlations, too. For example, is there a strong or weak correlation between age and Mass attendance? There is a calculation which tells us that, too.
I know that mathematics cannot give certainty in such matters but I am thrilled at the thought that we can calculate our degree of uncertainty. And when people say to me: “But they only asked 1,000 people. That can’t give a reliable answer.” I merely tap my nose and say: “It all depends.”
Yet ultimately all mathematical bets are off. Kurt Gödel may be one of our few contemporary personalities who will be remembered in 1,000 years. He showed that any formal system of mathematics that includes a modicum of arithmetic is incomplete, and there will always be true statements that cannot be proved within the system. Gödel died in 1978 of “malnutrition and inanition” caused by “personality disturbances”. Which shows that mathematics can drive you mad.
But the journey can be fun. I gave The Math Book and The Physics Book (by Clifford A Pickover and published by Sterling) to two grandsons (sixth form and undergraduate). They have reported with enthusiasm on these beautifully illustrated books – sufficiently interesting to them to be permanently at the bedside, and I pass on their firm recommendation to you.
secondsightblogdotnet 
Your mention of imaginary numbers brought back memories of my university days. We used to use Laplace transforms in electronics and automatic control. Laplace, by using imaginary numbers, allows you to solve complex differential equations using algebra. Sad creature that I am, I simply loved them. Now the world is digital it’s less obvious but you mentioned picture compression – a similar technique is used in the audio files in ipods, mp3 players etc. They expose an interesting mathematical construct – negative time. If you compress a sharp drum beat and listen to the compressed file you will hear a pre-echo. According to the maths (Fourier analysis) an impulse is made up of a series of waves that exist before the sound occurs, it just happens that they all sum to zero; as soon as you do anything to the sound that disturbs the balance the pre-sounds become audible. So how does mathematics know when I’m going to hit the drum? It’s absurd, but the maths all works. It’s how we got to the moon, fly aircraft, transmit moving images etc.
Anyway I suspect the more devout will tie themselves in knots about Mass attendance, so I thought I’d get in early! Perhaps somebody will suggest that evolution uses imaginary numbers!
As you mentioned the spider,It brought it to my mind as I found it fascinating. The Intelligence of Bees written by J Bell Pettigrew many years ago.He deals with the bees ‘knowledge’of the principles of solid geometry as shown in their building of that multi-hexagon, the honeycomb; It is a curious mathematical problem at which precise angle the three planes compose the botem of a cell ought to meet, in order to make the greatest possible saving, or the least expense of material an labour. This is one of the higher parts of mathematics,
The ingenious Maclaurin has determined precisely the angle required, and he found, by the most measuration the subject would admit, that it is the very angle in which the three planes at the bottom of the cell of(the double) honeycomb do actually meet.
Yet no one imagines that the bee performs mathmetical calculations. The solution to the mathmetical problems had to be built in together with the eye, the wing and the sting…just as a ma must design, build and feed a computer before it can do its tricks.
Fellow bloggers,
I am trying to decipher where we are going with all this?
Mathematics and applied Mathematics works on the level of the mechanical and repetitive, Mathemathics is also measurement, distance, volume weight, mass and so on hence we have come to understand to a limited degree the world and universe in which we live in that same mechanical and repetitive way. Mathematics has given rise in the last days, to great increase in technological knowledge and its myriad of applications.
Does God use Mathematics? In a sense He does, but, lest you think it is like our mathematics,
think again. His mathematic measurements are eternal in composition, cannot be measured, and like a snow flake, is not repeated.
The danger is we land up with God as some sort of mathematical equasion – which is rather silly
and nobody has dared to suggest such a thing -YET!
Does Mathematics therefore, for all its brilliant concepts that have been thought out and applied, carry with it, its own limitation and end?
I don’t think we have to go anywhere. If we each told our favourite story from the world of maths the readers of the blog might end up with a wonderful repository to browse and consider.
In the fictitious account of Bletchley Park, Enigma the central character, a mathematician, says that he knows that he is close to solving a problem when the equations look beautiful. Mathematics is part of how we access creation and so, in some way, it is part of the mind of God. If more people studied mathematics for this reason we might bridge some of the perceived divide between science and religion -and we might get more morality in science.
The mention of mathematics calls to mind the nonsensical statement which Dr John Robinson, then Bishop of Woolwich, made in ‘Our Image og God must go'(The observer, March 17th, 1963) Professor Bondi. ‘ he said commenting on the BBC, television programme, The Cosmologists, on Sir James Jean’s assertion that ‘God is a great mathematician.” stated quite correctly that what he should have said is that “Mathematics is God,” Reality in other words, can finally be reduced to mathematical formulae’ But Mathmatics is God ‘ is meaningless, since mathmatics is a system of reasoning, not a person. Nor is it exact to speak of God as a great mathematician since the word has for us connotation of one who with pencil and paper, has to work labouriousley to discover truth. It would be truer to say that God, in His construction of the universe, manifested the knowledge ,which we can only reach,in part,by mathematical reasoning. Every-where he has illustrated his ideas.!
‘Are mathematics inherent in reality, or just our human grasp of the fundamental order of creation? I do not know, nor, as far as I can tell, does anyone else.’
I don’t have an answer either. But humour me, if you will. Mathematics is a tool. It is a real entity. Like other tools, such as a hammer or a broom, it has a purpose or function. While it might be a real entity, nobody can point to it in a physical sense, because of its immateriality. Therefore, mathematics is paradoxical combination of power and immateriality.
Absolutes exist in our world such as life, death, matter, space in all of its dimensions, measurement, evolution (apologies to our creationists), infinity, finiteness, time, gravity, light, logic, and mathematics. Therefore, mathematics is a paradoxical trinity of absoluteness, power, and immateriality. In theology something of this description is commonly called God. Faith teaches us that God is behind the cosmos and everything that it contains. Therefore, God inescapably is the supreme mathematician.
Mathematics, is a theoretical construct which is developed through research. Paradoxically, numbers don’t physically exist, but are nonetheless real. Like God, mathematics is immaterial, perfect, real, powerful, absolute, authoritative, and ultimately unimpeachable. I think that mathematics is an immaterial reality that finds its expression throughout the cosmos. It is not only located in human understanding.
A list of some of the greatest mathematicians in history is provided by the following link. It is not an exhaustive collection due to its focus on historical notaries. http://fabpedigree.com/james/greatmm.htm
The next link is a lecture at Gresham College in the UK, and it deals with a book launch. The book is called ‘The Great Mathematicians’, which is written by Professor Robin Wilson, Honorary Fellow of Gresham College, and Dr. Raymond Flood, of Kellogg College, at the University of Oxford. http://fora.tv/2011/10/19/The_Great_Mathematicians
Fermat’s famous Last Theorem was solved by British mathematician Sir Andrew John Wiles KBE FRS, with the assistance of his former student, Richard Taylor. Wiles is currently a Royal Society Research Professor at the University of Oxford http://en.wikipedia.org/wiki/Andrew_Wiles
Richard Taylor is now the Herchel Smith Professor of Mathematics at Harvard University, and was also elected a Fellow of the Royal Society in 1995, along with Sir Andrew Wiles. http://en.wikipedia.org/wiki/Richard_Taylor_(mathematician)
Wiles was interviewed by PBS on how he eventually solved Fermat’s Last Theorem. You can read a transcript of the interview here. http://www.pbs.org/wgbh/nova/physics/andrew-wiles-fermat.html. It took Wiles seven years of research with the final solution needing 150 pages to demonstrate. In his lectures to fellow mathematicians from around the world, his proof needed several days of lectures in order to explain it to them.
He first came across the theorem when he popped into the Milton Road library in Oxford on his way home from school, as a ten year old boy. He was immediately obsessed with the theorem and resolved to solve it one day. A Google video documentary of how he solved Fermat’s Last Theorem is available here. http://video.google.com/videoplay?docid=8269328330690408516#
That’s very interesting, John. Do you think you could have a bash at introducing chaos theory? From a distance, I find that fascinating.
As mathematics is about order and pattern recognition, chaos theory has a similar vein. Despite any context or system being inherently random, unstable, and chaotic, chaos theory’s purpose is to find order in random data. The first chaos theoretician was a brilliant American mathematician and meteorologist called Edward Lorenz of the Massachusetts Institute of Technology. He passed away on the 16th April 2008 at his home in Cambridge Massachusetts, aged 90 of cancer. http://en.wikipedia.org/wiki/Edward_Lorenz
Around 1960, Lorenz was working in weather prediction. The meteorology of any area of the world is a random system. When he studied weather patterns, he observed that they did not change as foreseen, with the mathematics employed at the time. Lorenz decided to use a mathematical model with the aid of a computer. The equation that he employed for this task contained 12 variables. This computer model was to mimic the movement of air in the earth’s atmosphere.
He noticed that minute changes in the values of any of his 12 variables, would dramatically alter meteorological outcomes. This apparent sensitivity in initial conditions was first described by Lorenz as the butterfly effect. This meant in practice that any weather forecasts that were older than a week were to be considered extremely unreliable.
The story of the discovery of the butterfly effect revolved around an unusual result that he noticed on his computer printout one day. In 1961, he was dabbling in computer modelling, and wanted to examine a particular sequence of numbers again, however this time, he commenced the procedure during the middle of the computer’s operation. He was simply curious about what was going to happen. When he returned, he expected to see no difference to previous numerical patterns, but was surprised that the outcome was significantly different. He had set his computer originally to use decimals to six digits in length; he altered this practice by only including three decimals to see if there was any change, and in order to save paper.
He was surprised by the unexpected outcome. What should have occurred is that there should have been little change in the sequence of number patterns from the original pattern. Lorenz went on to demonstrate that even minute decimal values were significant variables in any practical experiment. It was this result that eventually became known as the butterfly effect. This observation was published as ‘‘Deterministic Nonperiodic Flow’, and is considered a seminal piece of work in chaos theory.
Despite my description above, the sum of my knowledge in chaos theory is zero. For those with a greater level of mathematical ability, these links below could spark your interest in chaos theory.
http://www.imho.com/grae/chaos/chaos.html
http://en.wikipedia.org/wiki/Chaos_theory
http://en.wikipedia.org/wiki/Edward_Lorenz
Thank you for coming back on this so quickly. Others may like to add to your start point. I hold in my mind a picture, gleaned from a book on the subject, of a column of cigarette smoke rising in still air. For a foot or two the smoke maintains a relatively ordered spiral. But at a certain height the spiral breaks up and the smoke goes every which way.Chaos is come again!
What I find fascinating here is that, under certain conditions, chaos occurs. But it is not simply random – there are rules even in the chaos.
The idea of mathematical chaos is splendidly described by John Candido.
The point of chaos theory is precisely that there are systems whose behaviour appears to be random and unpredictable but are in theory completely determinate. In practice however, although the overall pattern of behaviour is consistent, the detailed behaviour is unpredictable because the smallest external disturbance results in activity which, although conforming to the overall pattern, is grossly different.
A particular example of such a system is the electrical activity of the brain and probably the best known worker in this field is Walter J. Freeman (see http://sulcus.berkeley.edu/ ).
All these great Mathmaticians, but could they design anything as wonderful as the Universe.
If it were possible to construct a machine able to perform the same functions as the human brain,it would have to be largely electronic; if we brought together all the necessary component parts, and then in some miraculous way solve the vast problem of connecting them together, we should still be faced with the fact that even with the most modern components, several hundred would be faulty at any given moment,
Can a man accept that a television set without an electronic engineer could come to existence.
Then what conclusion are we driven to when we look at the vastly more brilliant design that no humanshas traced the blueprint?
Sorry Joh, I was unable to see all .your www s, my computer would not let me in. I could only see the 1st
“Paradoxically, numbers don’t physically exist, but are nonetheless real…”
John, Why is this a paradox?
“Paradoxically, numbers don’t physically exist, but are nonetheless real…”
‘John, Why is this a paradox?’
According to the ‘Concise Oxford Dictionary’, tenth edition revised, ‘Paradox’ means in its second listed meaning,
‘A person or thing that combines contradictory features or qualities.’
The features of reality, power, and immateriality, as exhibited by our number system, are similar to other phenomena such as electricity, the electrical, biochemical, and physiological workings of our brain and nervous system, the air that we breath, time, gravity, etc. etc. That is, they exist but they are not visible to the eye.
In terms of distinct numerical categories, does anybody know if the numbers of entities that exist and are visible to the eye, exceed the number of entities that exist and are invisible? If the categorical numbers of things that exist that are visible exceed the number of things that exist that are invisible, I would say that entities such as our number system are paradoxical in the above sense as outlined by the ‘Concise Oxford Dictionary’.
If the categorical numbers of entities that are visible do not exceed the number of entities that are invisible, there might be a case for saying that visible entities are paradoxical, rather than invisible entities. I can say this because the ‘Concise Oxford Dictionary’ definition states, ‘a person or thing that combines contradictory features or qualities’, points to or assumes that our human experience is the determining criterion for paradoxical attributes.
I admit that these questions might not be answerable, or its answer will be in flux as newer invisible entities are discovered by scientists. As a guess, I would say that the categorical number of visible entities far exceed the number of invisible entities. Ergo, invisible entities are a paradox in the above sense.
Many thanks, John, for all these links. So far I have looked only at the list of famous mathematicians, which was full of interest. One particular nugget, for readers of this blog, was an alleged proof by the mathematician Conway (inventor of the computer program ‘Life’) of the proposition “If experimenters have free will, then so do elementary particles.”
Quentin – Curiously (to my mind) the expression for the golden ratio includes the square root of five, and so the ratio itself is paradoxically an irrational number. I’ve just run a simple BASIC program confirming that the ratio of adjacent Fibonacci numbers converges quite rapidly towards the same value. It’s tempting to think that the coincidence must have some significance though I’m blowed if I can think what. Number theory is fascinating, but largely beyond me.
A possible point for discussion: is mathematics a part of Creation, despite being totally unrelated to matter except by application, or quite independent?
Peter.
This may not be what you mean, but I read that Alfred Noyes once wrote ‘under the scrutiny of the more philosophical science of our own day, “matter” itself is dissolving into the realms of ideas, and ideas appertain to a Mind.’ This is the mind that instilled the principles of solid geometry into thr bee; which installed the radar set of the bat, emitting and decoding two hundred squeaks a second as it closes on its prey, squeaks which last less than a thousandth of a second. ( Those who have read Leonard Dubkin’s
The White Lady will know how effecient that radar is, for he tells of a bat which flew repeatedly through the blades of an electric fan which was running at 800 reveloutions per minute-allowing the fan to have three blades,then ,in effect, forty blades slice past any given point each second!). So we come to Voltaire who stated in his Phisosophical Dictionary that ‘Either the stars themselves are great geometricians or the eternal Geometer has arranged.
I found that interesting at the time, many years ago.
the last word ‘them’ missed out.
Interesting question. Does mathematics have to be created – or just figured out? Does it make sense to ask such a question? What God created, He created freely – what He is, is determined – or at least, could not be different. It’s tempting to think that mathematics could not be different from what it is, and is not created, merely elucidated. I cannot imagine the non-existence of numbers (also, given the doctrine of the Trinity, wouldn’t it be heretical?).
Naturally all of us who contribute to this Blog have superior intelligence. So none of us will have any difficulty in solving this mathematical puzzle.
I deal you three cards, one of which is the Ace of Spades. I ask you to put, and keep, your finger on the card you are guessing to be the Ace of Spades.
I turn up another of the three cards and show you that it is not the Ace of Spades.
But I give you the opportunity to change your choice from the card under your finger to the remaining card.
So there are two cards face down on the table one of which is the Ace of Spades. Will you stay with your original choice or switch to the other card?
Which will give you the better chance: staying with the original, or switching? Or does it make no difference?
Some of you will have encountered this mystery before. If so, please keep quiet for 24 hours. Let’s say until Sunday morning. Then, we will welcome a correct answer with appropriate explanation. Meanwhile, the rest of you – get your brains into action. And, if you solve the problem, tell us all immediately. No doubt the winner will get an indulgence of 7 years and 7 quarantines. But not from me.
(This is not a trick question)
I suspect there is a subtlety that I ‘ve missed, but here goes.
Simple view: either face-down card might equally be the AoS, so it makes no difference whether I stay put or move.
More elaborately: with the third card out of the way, there are four equally probable situations:
1- Finger on AoS, stay put – win
2- Finger on AoS, move – lose
3- Finger not on AoS, stay put – lose
4- Finger not on AoS, move – win
There are thus two chances of winning and two of losing, so again it makes no difference.
Now show me where I’m wrong!
I thought we were to say nothing until Sunday.
So here goes.
There is no diffirence,It is just like tossing a coin,,, I think.
I have had a further thought- Am I sure that one of them ‘is’ the Ace of Spades?
Who knows? Perhaps you’re right. We’ll have to wait for Sunday morning
Traditionally this is a question about goats and game show hosts. The traditional answer is incomplete – maybe this is why Quentin has recast it?
If you have not met this problem before, you might like to do a series of trials and see what happens in practice. But don’t report before Sunday morning!
Of course people who have worked this out should certainly reply as soon as they are ready (and, at the time of writing, some already have). I just want those who have encountered the problem before and therefore know the correct answer, to lie low until Sunday morning.
I have a feeling this has to do with probability. When I chose one card at random it had a 1 in 3 chance of being the ace. Now one card is exposed the card I have my finger on still has a 1 in 3 chance but the other card now has a 1 in 2 probability so I should change my choice.
I am not a mathematician, but I will have a go at it. I think that the answer is in the end a paradox. In the first scenario, each card had a one in three chance of being the Ace of Spades. With one card being removed, we are left with two cards. Each card has got both a one in two chance of being correct, and a one in three chance of being correct. But that is crazy isn’t it! How can one card be both a one in two chance and a one in three chance? Frustrating to say the least! I think that I am going to be embarrassed and it will be higher, mathematical minds that will get this problem right.
I’ve long had a love/hate relationship with mathematics. Given the time, I can happily spend an hour or three following some idea wherever it leads, either in pure maths or some application in work or a hobby. Unfortunately my talent, for what it’s worth, lies in dogged persistence rather than quick results: which is not much use in examinations, where you are working against the clock and can’t just look things up in a textbook.
I can usually grasp mathematical ideas, but lack fluency in using mathematical language. All those formulae that one has to remember! It’s like trying to write in a language with which one has only a slight acquaintance. Anyone else had this problem? Or other problems?
I understand it precisely. I ave a very mathematical friend who I cycle with. He can explain all sorts of mathematical concepts to me and I can “understand them” with the eye of my mind from an aesthetic perspective- I can discuss things with him and even hazard an occasional question-but I couldnt do the language if my life depended on it-funny business the brain. So I share this problem…as to ‘other problems well how long have you got…? (joke)
“God made the integers – all else is the work of man” (sounds grander in German). That’s too simple. God sustains everything in being, and makes everything, including what we humans claim credit for. In mathematics you start from axioms – which are given, but you choose them in the first place. These lead you to conclusions which follow ineluctably (good word!) from the axioms. The axioms are typically plausible, but there is no need for them to correspond with reality. All that is necessary is that they be consistent. Science (as opposed to mathematics) is discovering what axioms are true in the real world – to what extent, and in what circumstances.
It’s sometimes said that sculpture is easier than painting. In sculpture, the statue is there inside the block of stone from the beginning. All the sculptor has to do is to chip off the irrelevant material surrounding it. Is mathematics like that?
tim,
‘Choosing the axioms’ recalls a bit of Chesterton: “All arguments begin wih an assumption that you cannot prove”. Like theism or atheism: each is plausible to some people, but neither can be proved.
The number of statues in the block is virtually infinite, and each sculptor will reveal a different one. I suppose the analogy in mathematics is that a great mathematician will reveal some quantitative relationship which was always there, but not explicitly recognised.
It’s accepted doctrine (I believe, but am ready to be corrected) that the existence of God can be proved. This bothered me for some time. I think it has to be understood in the sense that the existence of God can be proved from reasonable premises – but you can’t necessarily prove the premises.
Tim – “… the existence of God can be proved from reasonable premises …” I think God’s existence can be proved to the extent of showing that the contrary is incompatible with normal human ways of thinking, in particular that no event can be without a cause. The question for me is whether those ways are in this respect reliable.
That question is taken very seriously by some physicists (I’m a chemist with perhaps a more down-to-earth approach) who postulate that the entire universe sprang spontaneously into existence some thirteen billion years ago. I imagine that advocates of that idea would cite the common example of radioactive decay, which for any particular atom is a matter of pure chance although its statistical probability within a given time interval has a definite value for the isotope in question. Here, however, there is the possibility of a random fluctuation in energy levels happening to surmount the internal barrier to disintegration. The notion of a random fluctuation in nothing, leading to the vast observable universe, however strikes me as bizarre.
I suppose the Apostles when they saw Jesus after the Resurrection, and He showed them His wounds-they would have believed all He had told them about God the Father Also all the miracles they saw, would be now confirmed.
Thomas didn’t believe until he had seen!
The proof to me ,apart from other things are the witness’s in the Scriptures.The proof is in the belief that what they saw was true.
I am new to this fascinating blog, even though I read the Herald regularly. What stirred me to read and even reply is partly that last question, to which my answer is Yes. Hated maths in school, until someone opened my eyes to it (too late, since I had already chosen arts subjects for A level). Amazing what a good teacher can do! And this eventually led me to write a book about the problem (see top left at beauty-in-education.blogspot.com), and how to begin to overcome the fear of maths at the same time as bridging the gap between arts and mathematical sciences. One of the authors I found most helpful is also mentioned by Quentin in his column, namely Clifford A. Pickover. I recommend especially ‘A Passion for Mathematics: Numbers, Puzzles, Madness, Religion, and the Quest for Reality’. I am beginning to work with other people interested in the broader topic to find teachers and parents to help develop resources for a more ‘holistic’ Catholic approach to learning. Contact details can be found at my site, secondspring.co.uk. – S.C.
We don’t actually say that God can be proved but that “God can be known through the light of human reason” – Vatican I; that is as long as we don’t react cynically to the knowledge we receive. However Pope Paul VI added that the severity of fallen human nature may prevent some from discerning God’s existence -and the rest of us having our moments of difficulty.
May I recommend “Mathematics Made Difficult” by Carl E Linderholm? All but incomprehensible to non-mathematicians like me but nevertheless very funny. Unfortunately out of print (a copy is going on Amazon for a ridiculous sum, but the only easy way to get hold of it seems, unfortunately, to be a download of dubious legality) but you can read a review at http://www.goodreads.com/review/show/155087260 . Memorable quote:”It is not correct in logic to prove something by saying it over again; that only works in politics, and even there it is usually considered desirable to repeat the proposition hundreds of times before considering it as definitely established.”
Peter Wilson,
“..Here, however, there is the possibility of a random fluctuation in energy levels happening to surmount the internal barrier to disintegration..”.
I read your post with great interest. I have long been interested in the steadiness of radioactive decay -quite obviously since its steadiness seems to me a key factor in prediction and calculation. I think we may have spoken before on this topic and you seemed quite convinced of the certainty and reliability of half life. Are you talking about the possibility of an interruption to that mechanism?
Simple words in the answer please!!
Mike – sorry if my phrasing misled you. “… the possibility of a random fluctuation in energy levels …” is my slightly tentative explanation for the occurrence of disintegration in an individual atom. Where numbers are large (the number of nuclear units per gramme is about 6 followed by 23 noughts, and there are never more than a few hundred per atom so for practical purposes the numbers are always large) the randomness at the individual level is smoothed out and the progress of decay is quite regular.
You may perhaps find it helpful to look at the relevant part of my web site, http://www.peterwilson-seascale.me.uk/Nuclear.htm
Thanks Peter, I’ve visited your website, I particularly liked the churches forum page. Its not that I can’t get hold of the basics of these subjects-I did A level Chemistry nearly 30 years ago at Kitson College Leeds-you may remember it. I did that after training as a nurse at St James University Hospital. I still wanted to help people but was far too self centred to be a nurse…then I fell off the Cuillin ridge on Skye and it was an osteopath who restored me to my legs-A level Chemistry and 0 level maths were needed to apply
for the Bsc Hons Osteopathic medicine I eventually went on. I have a friend who was an MOD physicist and is now an electrical contractor. He told me the other day that he still had niggling doubts about laws observed in this solar system being universally assumed to be true beyond it. I’m a bit like that about those supposedly vast reaches of time we presume to stretch back into. Thanks though for your patient answers to my questions.
Fellow Bloggers,
You know, I was hopeless at Mathematics, I could never work out why 5 out of 2 people could not do fractions! (joke).
But if we may, can I pose a question or two to see what answers my fellow bloggers may have and where it may lead to along our discussion in Mathematics.
1. Would you say, Mathematics is complete order?
2. Is order part of disorder as we know it?
3, Can our contradictory, briused brain, mind find order?
When we poor tinkerers with bones analyse the human body we have to employ the use of imaginary vectors. Strain pattens in the human body are a fundamental background of poor health and osteopaths try to reduce them. The spine represents as an animate beam subject to turning forces at every turn which throws up complex torque. Though these patterns can be predicted by the shape of bones and joint surfaces sometimes we have to make up an axis which appears to exist because we can see the torque pattern operating around the would be axis. So the line exists but it does not, the force is expressed through the human matter but if you chopped up the body you would find no line. I have a vry good cycling friend who is also a mathematician, when he explains things like ‘series’ and ‘impossible numbers’ to me with great eloquence I think of the strain patterns I work with daily. Yes maths does exist in its own right -rathe as a musical score exists in order to intepret form.
mike Horsnall
So are you saying, these different forms of patterns that are breaking down the body are
disorder as the disorders are violating the order?
Is the pattern order or is it a form of disorder?
Is this a self imposed fixed pattern of order , in order to investigate?
Nektarios,
You have stumbled inadvertently upon the holy grail of osteopathic thinking!
The pattens generally used to act as blueprints for what might be termed musculo skeletal normality have been arrived at by a whole load of argy bargy , over about 150 years or so, between Zoologists, physicists, anatomists and us physical therapists. They are to do with the centre of gravity in the human body- living or dead, the clinical observations of our profession and the pure anatomy of the structural body-eg direction of movement by bony joint plane etc. These patterns are to a degree idealised and in part they are empirically arrived at by observation. SO we have all of your types present.
See my question above: Feb 10, 7:16pm)
This is the Monty Hall problem – so called after the American game show where it was presented and which made it popular.
No one who got an incorrect answer should be kicking themselves. Although the correct answer is certain, many people – including competent mathematicians – simply won’t believe it.
The correct answer is:
Staying with your first choice, chance of being correct is one out of three
Switching to remaining card, chance of being correct is two out of three.
Explanation.
When the cards were first dealt each card had a possibility of one third, adding up to a total probability of one. By discarding a card which is not the Ace, the total probability of one is now distributed over the remaining two cards, We know the odds of the first choice being the ace is one out of three, therefore the card remaining must have a probability of two out of three.
You can test it simply by trying it. I made 40 attempts, switching in each case, and I was right 26 times (65%). There was a little panic here because I lost the first four attempts. But switching soon overtook.
I then tried the computer version (which uses goats and cars) http://www.nytimes.com/2008/04/08/science/08monty.html
20 attempts. Switch proved correct13 times
This is an application of Bayes theorem. The equation here allows you to modify your probabilities according to your information. When the dealer turns over the non-Ace there is more information and that changes the odds.
If you want to look at the whole question thoroughly go to http://en.wikipedia.org/wiki/Monty_Hall_problem
Incidentally, under “simulation” you will find a quick way of practical testing.
A good and comparatively simple explanation is at http://betterexplained.com/articles/an-intuitive-and-short-explanation-of-bayes-theorem/
This uses two relevant examples: cancer screening and filtering for spam.
You may be left as I am: intellectually convinced but still feeling that it just can’t be so. But I am afraid that it is.
Bayes Theorem, which calculates conditional probability, is immensely important. It is essential for forecasting and prediction where the evidence is complex. Once again, mathematics helps us to establish a truth which might otherwise remain hidden.
(Con men can do well out of Monty Hall. Most people, perhaps through pride, tend to stick with their first choice. So most people lose.)
“Intellectually convinced but still feeling that it just can’t be so”: Yes, exactly! It called to mind the lines,
“Explaining metaphysics to the nation-
I wish he would explain his Explanation.”
(Byron, in ‘Don Juan’, referring to Coleridge).
Quentin – Thank you for the explanation. Probability theory was one of the mathematical topics with which I was never comfortable, being unable to see which of the conflicting arguments was actually correct. It takes great clarity of thought, which in other respects I thought I had, and it’s good to be reminded of my limitations once in a while.
When this is posed as a problem of how to win the car in a television quiz show, the answer may need to take into account other things besides the probabilities. The quizmaster who offers you the choice – is he on your side, or neutral, or against you? In most cases he will be neutral. If you watch the show regularly, and he always offers this choice, you may assume he’s neutral, and accept the swap. But maybe it’s a new show. In that case, you have no guide from previous behaviour – you don’t know if this offer is routine or exceptional. Maybe the quizmaster has instructions to hang onto the car if possible and will only offer the choice where you already have the winning card? If that is what you conclude, don’t take the swap. Personally, I think I’d rather go with the odds than rely on my judgement of character.
Having written the above, I looked at Wikipedia (‘the Monty Hall problem’) which analyses the question (and variants of it) much more thoroughly. Recommended if you’re interested – also a comfort to learn that serious mathematicians have got this wrong!
Just remembered another maths joke:
“There are 10 kinds of people in the world: those who understand binary numbers, and those who don’t”.
Ha ha ha -thats very funny and also very true. It took me 5 attempts to get 0 level maths, the last time I tried I had a Maths teacher who said to me, close to tears and in great frustration, “Well you either understand it or you don’t”
I hated maths and struggled hard to get an O-level pass in 1966 (three papers in those days, Algebra, Geometry, and Arithmetic and Trigonometry). My Latin master maintained that those who are good at maths are good at Latin, but I never saw the correlation. Similarly, the link between mathematics and music, so trumpeted (if that is the right word) by Albert Einstein is somewhat vitiated by the fact that Beethoven was functionally innumerate.
I think being good or bad at languages in general, is a myth. French was the only school subject I ever got a prize for. Latin was the only one I failed at the School Cert. exam.
But with maths and music, my weakness at maths did seem similar to my experience with an OU music exam, where I found it very hard do the essential four-part harmony question, without a piano.
I didn’t know that about Beethoven. But I do think the link between maths and music is interesting. A note which is an octave higher or lower than another note somehow “sounds the same”, – what we are hearing is a proportion; the note is produced by a string or column of air exactly half (or twice) the length of the original.
Music of the spheres.
I have a stinking cold, by the way, and it’s making my brain a bit fluffy. Ah, you’d noticed?
Presumably, maths as a tool used by early(ish) humans began with counting. Enumerating possessions, – sheep, or wives, for example. Only as they got deeper into it did humans realise that numbers followed rules of their own and weren’t just a useful way of totting things up. I don’t think we can say we invented maths. Rather, we discovered it. But to say we discovered it seems to suggest it had a prior existence, and that doesn’t seem right either.
We know from Beethoven’s account books that he couldn’t multiply and had to rely on sequential addition. When maths questions are asked on University Challenge I don’t even understand the question, let alone know the answer, but I take comfort from the fact that most of the contestants on that programme seem to have no Latin at all, nor any acquaintance with music (except popular music).
However, mathematical ability, like musical ability would seem to be a gift that in some individuals is manifested at a prodigiously early age. Fortunately one does not need to be able to write a fugue in order to appreciate it.
Are you numerate? That’s the equivalent of understanding the significance of numbers, just like the literate understand the meaning of words and sentences. If you are you are likely to make better decisions in many areas.
Why do shops often price good at say, £4.99 – when everyone knows that it really means £5? Because we read numbers from the left, in descending order of importance. And it works: shops simply sell more that way.
Pick up your newspaper and, very often, you will see important information expressed as a percentage. Yet around half the population over the age of 15 doesn’t understand percentages.
A study, published today, tells us that to the less numerate a one in a hundred chance sounds more dangerous than a 1% chance, while 76% correct answers sounds a better score than 26% mistaken answers.
“In general, people who are numerate are better able to bring consistent meaning to numbers and to make better decisions,” the study concludes.
I think it takes a particular aptitude to succeed in mathematics. It fascinates me, but as someone remarked earlier – I can’t speak the language.
Within my fascination, I can not decide if the subject is a science or an art.
Any takers?
John – I think this question has been argued for centuries. The subject has the attributes of both, needing strict mental discipline and an appreciation of beauty, symmetry in particular. Within my limited exploration of it I have generally found that an ugly result is wrong.
I would guess an art because I can apprehend it as an artist -I can ‘picture’ maths even though I don’t know the language.
I cannot resist the opportunity to split a hair. 76% correct answers will give you a better score than 26% mistaken answers (if all questions receive equal marks). However, 24% mistaken answers may also be inferior to 76% correct answers, unless you assume (and I can see why you might want to do this) that a failure to answer at all counts as a mistaken answer.
Mathematics was something that I was more interested in when I was younger. Is it a quasi-general rule, that youth are more open to very narrow and highly complex subjects? As a boy, it fascinated me. Alas, if only fascination could have turned into exceptional ability! As I have reached middle age, things like higher mathematics or very obscure, narrow subject matter, turn me off, but simple things like basic arithmetic and simple algebra, are both useful for helping me to pay off my monthly credit card bill, and a little enjoyable due to its obvious machine like perfection.
It is just so important that all mathematical and scientific talents are nurtured in all students. Our societies need the talents and research of mathematicians and scientists in order to advance our understanding of nature, and to harness the development of future technology. The same applies to all fields in the humanities.
Maybe Beethoven wasn’t so exceptional in not knowing multiplication tables. I haven’t read Pepys’s diaries, but happened to hear something read aloud from one of them on the radio recently. Pepys had just been introduced to the multiplication tables by a friend or acquaintance, and had set himself to learn them by heart, and was extremely excited and enthusiastic about this new discovery. Perhaps they just weren’t routinely taught, even to those who were “educated” throughout childhood and adolescence. Too busy learning Latin and Greek.
My favourite maths quote is from Bertrand Russell:
“We use maths not because we know so much about the world but because we know so little…”
or something along those lines…
mike Horsnall
Western philosophers like to play games from their ivory towers.
They like to make it appear they are saying something very profound when in fact a
child understands such things intuitively.
Philosphers are interesting to read, but the best book, a real must, an indespensible read, is the book that is YOU!
I am sure it would be a most interesting read for YOU. A journey of discovery, as you explore the depths, the heights and the widths that is YOU.
What wonderful things there are to discover; what awful things you will find there too;
what joys, sadness and miseries. What spiritual things will you really discover within, as
you read the book that is YOU?
Perhaps one important discovery reading the book that is YOU is, You are everyman, and everyman is YOU.
Nektarios,
Thank you for that kind exhortation-I’ve read a few chapters of:
” The Exploits of Horsnall Everyman” and ,like you say found it to be full of joys, sorrows and failed maths exams…perhaps you might like to act as my publishing agent?
mike Horsnall
The book that is YOU, Mike, is not for publishing, but LIVED. In the living of the BOOK THAT IS YOU, all around you will be able to read it! God and the angels read that book and your Guardian Angel knows it rather well, I would imagine. Humbling isn’t it?
The book that is You, is not just a select little bits of you, but all of you – warts and all.
You might not make any money from the BOOK THAT IS YOU, `but if you are quiet enough in yourself, still enough within, you could actually save thousands round about you.’ (St. Seraphim of Sarov).
What a blessing you may be YET!!
I think we fellow bloggers along the way have fallen speechless – I wonder why?
No stamina mate, no stamina….Have you saved thousands yet?
mike Horsnall
I don’t know – perhaps.
I have preached to thousands in my time and all
over Scotland.
One thing for sure, they are reading the Book that is I too.
Nektarios,
Yes -I’ve had a go at that too. I think its important to realise that we are a book for others to read and perhaps to see themselves in too. Rather like we find our ordinary view of others and the world is confirmed by literature; most of my best friends are men I have never met such as Paul the apostle, Moses and Peter etc. But also say the modernist poets etc, we are a book for others and we must not be afraid of being read.
I don’t think mathematics is primarily about number. It’s about patterns (of course, numbers display many many patterns) . Take the formula:
(x+1)squared = xsquared + 2x +1
Think of this as a pattern. Make a square of x cubes a side. Now add a further line of x cubes to each of two touching sides. Where they meet, there is space to put a single further block. Add in one cube there, and you get a larger square, of side (x+1) cubes.
Similarly, (xsquared – 1) = (x-1)(x+1).
Again, make a square of x cubes a side. Remove the cube in the top right-hand corner (or other corner of your choice). Now remove the remaining top row of cubes and place them vertically alongside the right-hand row of cubes. Now you have a rectangle, height x-1 cubes and width x+1 cubes: area (x-1)times(x+1).
These are two extremely primitive examples of patterns in maths. If you can see the pattern, you can see what the symbols mean and how they work. Is this why maths can be so satisfying?
That would be about right as far as I see it Tim. I failed O level maths four times but I can still see the patterns and grasp the underlying concepts-to be able to work the symbols must be marvellous.
It may be shaming to admit it, but I get great pleasure from the relation
x.squared -1 = (x+1)(x-1), even with the simplest numbers:
5×5=25: 6×4=24 6×6=36:7×5 =35 8×8=64: 7×9 =63 13×13 = 169; 12×14 = 168
1000×1000 =1,000,000: 999×1001 = 999,999 64×64=4096: 63×65 = 4095 …etc.
This is like the feeling one gets from the story of Gauss, as a schoolboy, being ordered as a punishment to add up the numbers from 1 to 100. He immediately produced the answer 5050. He saw that you could pair off the numbers from opposite ends of the list, so that each pair added up to 101 (1+100, 2+99, etc…). So the sum of the numbers is the number of pairs times 101, i.e., 50×101 = 5050.
Never mind Tim, Now spring is on its way you can get out a bit more…!!
For mathematical abilities in other species, see http://blogs.nature.com/news/2012/02/alex-the-parrots-last-experiment-shows-his-mathematical-genius.html?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+news%2Frss%2Fnewsblog+%28News+Blog+-+Blog+Posts%29&WT.ec_id=NEWS-20120221
One attraction of maths is the elegant solution – a simple, unexpected, convincing answer to a tricky problem. We’ve had Gauss’s method of adding up the numbers from one to a hundred. A few other elementary examples:
How many matches do you need to arrange for a knock-out competition with x entrants?
You can tile (completely cover) the 64 squares of of a chessboard with 32 dominoes. Cut out the squares from two opposite corners. Can you now tile the remaining 62 squares with 31 (intact) dominoes?
Euclid’s proof that there is no highest prime number.
The visual proof of Pythagoras’s Theorem – by inscribing one square at an angle inside another.
Other suggestions? . .
Tim – For ingenuity, I would add the proof that the square root of 2 cannot be a rational number. Does anyone know if it can be extended to the square root of 3 etc.? I suspect not, at least in anything like the original form.
Monty Python would have simply loved that parrot!
I find most interesting The Great Pyramid as a mathematical perfection,
and one of the seven wonders of the World.
In its relation to the Bible is fascinating.
How true is it, does anyone know?
Sorry, St Joseph, I haven’t understood you. How true is what?
Tim it is so interesting. if you look on the web The Great Pyramid.
It dates the Crucifion, the Baptism & Christs Birth and reading the mathmetical measurments etc was just amazing, so I wondered what others thought.
I found a book a couple of weeks ago, that someone sent my late husband about 20years ago called The Delicate Balance. I have been reading it and was quite intrigued. By a John Zajac 1989.
Ah. I’m afraid I’m programmed to be a sceptic about things like that.
Tim thank you.
And me too.
However the thought came to me that if we were born B C and reading the Bible (I am not considering the scientific findings of the Grt Pyramid to be Holy Scripture) would we have been sceptical about the Messiah?
Just a thought.
We recite the second coming of Christ at Mass,do people think of that or are they sceptical? I’m not. He could come any day, sooner the better for me.
!
Peter, excellent suggestion. Proof that the square root of 2 is irrational should certainly be added to the list. Your query about a similar proof for other square roots is also attractive. My instant reaction was although this proof depends on a specific property of even numbers, it must be possible to generalise it – a little more thought suggested it wasn’t that simple (if you generalise it wrong you may end up proving that the square roots of all integers are irrational, which is rubbish). I’m too confused to answer your question – maybe someone else can help?
I doubt that Euclid or Plato would approve of the decimalisation of the system of measurement.
And there are easier ways to explain the Golden Mean (divine proportion)-
http://nourishingobscurity.com/2011/02/03/the-mathematical-precision-of-the-universe-part3/
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Wow that was strange. I just wrote an very long comment but after
I clicked
submit my comment didn’t appear. Grrrr… well I’m not writing all
that over again.
Anyhow, just wanted to say great blog!