Author: IMC

You may have read the British Government is being petitioned to grant a posthumous pardon to one of the world’s greatest mathematicians and most successful codebreakers, Alan Turing. You may also have read that Turing was convicted of gross indecency in 1952 and died tragically two years later.

But who, exactly, was he?

Born in London in 1912, Turing helped lay the foundations of the “information age” we live in.

He did his first degree at King’s College, Cambridge, and then became a Fellow there. His first big contribution was his development of a mathematical model of computation in 1936. This became known as the Turing Machine.

It was not the first time a computer had been envisaged: that distinction belonged to Charles Babbage, a 19th century mathematician who designed a computer based on mechanical technology and built parts of it (some of which may be seen at the Science Museum in London or Powerhouse Museum in Sydney, for example).

But Babbage’s design was necessarily complicated, as he aimed for a working device using specific technology. Turing’s design was independent of any particular technology and was not intended to be built.

It was very simple and would be very inefficient and impractical as a device for doing real computations. But its simplicity meant it could be used to do mathematical reasoning about computation.

Turing used his abstract machines to investigate what kinds of things could be computed. He found some tasks which, although perfectly well defined and mathematically precise, are uncomputable. The first of these is known as the halting problem, which asks, for any given computation, whether it will ever stop. Turing showed that this was uncomputable: there is no systematic method that always gives the right answer.

So, if you have ever wanted a program that can run on your laptop and test all your other software to determine which of them might cause your laptop to “hang” or get stuck in a never-ending loop, the bad news is such a comprehensive testing program cannot be written.

Uncomputability is not confined to questions about the behaviour of computer programs. Since Turing’s work, many problems in mainstream mathematics have been found to be uncomputable. For example, the Russian mathematician and computer scientist, Yuri Matiyasevich, showed in 1970 that determining if a polynomial equation with several variables has a solution consisting only of whole numbers is also an uncomputable problem.

Turing machines have been used to define measures of the efficiency of computations. They underpin formal statements of the P vs NP problem, one of the Millennium Prize problems.

Another important feature of Turing’s model is its capacity to treat programs as data. This means the programs that tell computers what to do can themselves, after being represented in symbolic form, be given as input to other programs. Turing Machines that can take any program as input, and run that program on some input data, are called Universal Turing Machines.

These are really conceptual precursors of today’s computers, which are stored-program computers, in that they can treat programs as data in this sense. The oldest surviving intact computer in the world, in this most complete sense of the term, is CSIRAC at Melbourne Museum.

It seems a mathematical model of computation was an idea whose time had come. In 1936, the year of Turing’s result, another model of computation was published by Alonzo Church of Princeton University. Although Turing and Church took quite different routes, they ended up at the same place, in that the two models give exactly the same notion of computability.

In other words, the classification of tasks into computable and uncomputable is independent of which of these two models is used.

Other models of computation have been proposed, but mostly they seem to lead to the same view of what is and is not computable. The Church-Turing Thesis states that this class of computable functions does indeed capture exactly those things which can be computed in principle (say by a human with unlimited time, paper and ink, who works methodically and makes no mistakes).

It implies Turing Machines give a faithful mathematical model of computation. This is not a formal mathematical result, but rather a working assumption which is now widely accepted.

Turing went to Princeton and completed his PhD under Church, returning to Britain in 1938.

Early in the Second World War, Turing joined the British codebreaking operation at Bletchley Park, north-west of London. He became one of its most valuable assets. He was known by the nickname “Prof” and was described by colleague Jack Good as “a deep rather than a fast thinker”.

At the time, Germany was using an encryption device known as Enigma for much of its communications. This was widely regarded as completely secure. The British had already obtained an Enigma machine, from the Poles, and building on their work, Turing and colleague Gordon Welchman worked out how the Enigma-encrypted messages collected by the British could be decrypted.

Turing designed a machine called the Bombe, named after a Polish ice cream, which worked by testing large numbers of combinations of Enigma machine configurations, in order to help decrypt secret messages. These messages yielded information of incalculable value to the British. Winston Churchill described the Bletchley Park codebreakers as “geese that laid the golden eggs but never cackled”.

In 1945, after the war, Turing joined the National Physical Laboratory (NPL), where he wrote a report on how to construct an electronic computer, this time a general-purpose one unlike the machines dedicated to cryptanalysis which he helped to design at Bletchley Park.

This report led to the construction of an early computer (Pilot ACE) at NPL in 1950. By then, Turing had already moved on to Manchester University, where he worked on the first general-purpose stored-program computer in the world, the Manchester “Baby”.

In their early days, computers were often called “electronic brains”. Turing began to consider whether a computer could be programmed to simulate human intelligence, which remains a major research challenge today and helped to initiate the field of artificial intelligence.

A fundamental issue in such research is: how do you know if you have succeeded? What test can you apply to a program to determine if it has intelligence? Turing proposed that a program be deemed intelligent if, in its interaction with a human, the human is unable to detect whether he or she is communicating with another human or a computer program. (The test requires a controlled setting, for example where all communication with the human tester is by typed text.)

His paper on this topic – Computing Machinery and Intelligence – was published in 1950. The artificial intelligence community holds regular competitions to see how good researchers’ programs are at the Turing test.

The honours Turing received during his lifetime included an OBE in 1945 and becoming a Fellow of the Royal Society in 1951.

His wartime contributions remained secret throughout his life and for many years afterwards.

In 1952 he was arrested for homosexuality, which was illegal in Britain at the time. Turing was found guilty and required to undergo “treatment” with drugs. This conviction also meant he lost his security clearance.

In 1954 he ingested some cyanide, probably via an apple, and died. An inquest classified his death as suicide, and this is generally accepted today. But some at the time, including his mother, contended his death was an accidental consequence of poor handling of chemicals during some experiments he was conducting at home in his spare time.

The irony of Turing losing his security clearance – after the advantage his work had given Britain in the war, in extraordinary secrecy – is clear.

The magnitude of what was done to him has become increasingly plain over time, helped by greater availability of information about the work at Bletchley Park and changing social attitudes to homosexuality.

Next year, 2012, will be the centenary of Turing’s birth – with events planned globally to celebrate the man and his contribution. As this year approached, a movement developed to recognise Turing’s contribution and atone for what was done to him. In 2009, British Prime Minister, Gordon Brown, responding to a petition, issued a formal apology on behalf of the British government for the way Turing was treated.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Graham Farr, Monash University

Momentum is gathering behind calls to pardon the father of computer science. BinaryApe

You may have read the British Government is being petitioned to grant a posthumous pardon to one of the world’s greatest mathematicians and most successful codebreakers, Alan Turing. You may also have read that Turing was was convicted of gross indecency in 1952 and died tragically two years later.

But who, exactly, was he?

Born in London in 1912, Turing helped lay the foundations of the “information age” we live in.

He did his first degree at King’s College, Cambridge, and then became a Fellow there. His first big contribution was his development of a mathematical model of computation in 1936. This became known as the Turing Machine.

It was not the first time a computer had been envisaged: that distinction belonged to Charles Babbage, a 19th century mathematician who designed a computer based on mechanical technology and built parts of it (some of which may be seen at the Science Museum in London or Powerhouse Museum in Sydney, for example).

But Babbage’s design was necessarily complicated, as he aimed for a working device using specific technology. Turing’s design was independent of any particular technology, and was not intended to be built.

The now iconic shot of Alan Turing.

It was very simple, and would be very inefficient and impractical as a device for doing real computations. But its simplicity meant it could be used to do mathematical reasoning about computation.

Turing used his abstract machines to investigate what kinds of things could be computed. He found some tasks which, although perfectly well defined and mathematically precise, are uncomputable. The first of these is known as the halting problem, which asks, for any given computation, whether it will ever stop. Turing showed that this was uncomputable: there is no systematic method that always gives the right answer.

So, if you have ever wanted a program that can run on your laptop and test all your other software to determine which of them might cause your laptop to “hang” or get stuck in a never-ending loop, the bad news is such a comprehensive testing program cannot be written.

Uncomputability is not confined to questions about the behaviour of computer programs. Since Turing’s work, many problems in mainstream mathematics have been found to be uncomputable. For example, the Russian mathematician and computer scientist, Yuri Matiyasevich, showed in 1970 that determining if a polynomial equation with several variables has a solution consisting only of whole numbers is also an uncomputable problem.

Turing machines have been used to define measures of the efficiency of computations. They underpin formal statements of the P vs NP problem, one of the Millennium Prize problems.

Another important feature of Turing’s model is its capacity to treat programs as data. This means the programs that tell computers what to do can themselves, after being represented in symbolic form, be given as input to other programs. Turing Machines that can take any program as input, and run that program on some input data, are called Universal Turing Machines.

These are really conceptual precursors of today’s computers, which are stored-program computers, in that they can treat programs as data in this sense. The oldest surviving intact computer in the world, in this most complete sense of the term, is CSIRAC at Melbourne Museum.

 

CSIRAC was Australia’s first digital computer, and the fourth “stored program” computer in the world. Melbourne Museum

It seems a mathematical model of computation was an idea whose time had come. In 1936, the year of Turing’s result, another model of computation was published by Alonzo Church of Princeton University. Although Turing and Church took quite different routes, they ended up at the same place, in that the two models give exactly the same notion of computability.

In other words, the classification of tasks into computable and uncomputable is independent of which of these two models is used.

Other models of computation have been proposed, but mostly they seem to lead to the same view of what is and is not computable. The Church-Turing Thesis states that this class of computable functions does indeed capture exactly those things which can be computed in principle (say by a human with unlimited time, paper and ink, who works methodically and makes no mistakes).

It implies Turing Machines give a faithful mathematical model of computation. This is not a formal mathematical result, but rather a working assumption which is now widely accepted.

Turing went to Princeton and completed his PhD under Church, returning to Britain in 1938.

Early in the Second World War, Turing joined the British codebreaking operation at Bletchley Park, north-west of London. He became one of its most valuable assets. He was known by the nickname “Prof” and was described by colleague Jack Good as “a deep rather than a fast thinker”.

One of the famous Enigma machines decrypted at Bletchley Park. Keir David

At the time, Germany was using an encryption device known as Enigma for much of its communications. This was widely regarded as completely secure. The British had already obtained an Enigma machine, from the Poles, and building on their work, Turing and colleague Gordon Welchman worked out how the Enigma-encrypted messages collected by the British could be decrypted.

Turing designed a machine called the Bombe, named after a Polish ice cream, which worked by testing large numbers of combinations of Enigma machine configurations, in order to help decrypt secret messages. These messages yielded information of incalculable value to the British. Winston Churchill described the Bletchley Park codebreakers as “geese that laid the golden eggs but never cackled”.

In 1945, after the war, Turing joined the National Physical Laboratory (NPL), where he wrote a report on how to construct an electronic computer, this time a general-purpose one unlike the machines dedicated to cryptanalysis which he helped to design at Bletchley Park.

This report led to the construction of an early computer (Pilot ACE) at NPL in 1950. By then, Turing had already moved on to Manchester University, where he worked on the first general-purpose stored-program computer in the world, the Manchester “Baby”.

The remade Bombe machine at Bletchley Park, England, features miles of circuitry. Keir David

In their early days, computers were often called “electronic brains”. Turing began to consider whether a computer could be programmed to simulate human intelligence, which remains a major research challenge today and helped to initiate the field of artificial intelligence.

A fundamental issue in such research is: how do you know if you have succeeded? What test can you apply to a program to determine if it has intelligence? Turing proposed that a program be deemed intelligent if, in its interaction with a human, the human is unable to detect whether he or she is communicating with another human or a computer program. (The test requires a controlled setting, for example where all communication with the human tester is by typed text.)

His paper on this topic – Computing Machinery and Intelligence – was published in 1950. The artificial intelligence community holds regular competitions to see how good researchers’ programs are at the Turing test.

The honours Turing received during his lifetime included an OBE in 1945 and becoming a Fellow of the Royal Society in 1951.

His wartime contributions remained secret throughout his life and for many years afterwards.

In 1952 he was arrested for homosexuality, which was illegal in Britain at the time. Turing was found guilty and required to undergo “treatment” with drugs. This conviction also meant he lost his security clearance.

In 1954 he ingested some cyanide, probably via an apple, and died. An inquest classified his death as suicide, and this is generally accepted today. But some at the time, including his mother, contended his death was an accidental consequence of poor handling of chemicals during some experiments he was conducting at home in his spare time.

Dino Gravalo.

The irony of Turing losing his security clearance – after the advantage his work had given Britain in the war, in extraordinary secrecy – is clear.

The magnitude of what was done to him has become increasingly plain over time, helped by greater availability of information about the work at Bletchley Park and changing social attitudes to homosexuality.

Next year, 2012, will be the centenary of Turing’s birth – with events planned globally to celebrate the man and his contribution. As this year approached, a movement developed to recognise Turing’s contribution and atone for what was done to him. In 2009, British Prime Minister, Gordon Brown, responding to a petition, issued a formal apology on behalf of the British government for the way Turing was treated.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Graham Farr*

 

Does the thought of p-values and regressions make you break out in a cold sweat? Never fear – read on for answers to some of those burning statistical questions that keep you up 87.9% of the night.

• What are my hypotheses?

There are two types of hypothesis you need to get your head around: null and alternative. The null hypothesis always states the status quo: there is no difference between two populations, there is no effect of adding fertiliser, there is no relationship between weather and growth rates.

Basically, nothing interesting is happening. Generally, scientists conduct an experiment seeking to disprove the null hypothesis. We build up evidence, through data collection, against the null, and if the evidence is sufficient we can say with a degree of probability that the null hypothesis is not true.

We then accept the alternative hypothesis. This hypothesis states the opposite of the null: there is a difference, there is an effect, there is a relationship.

• What’s so special about 5%?

One of the most common numbers you stumble across in statistics is alpha = 0.05 (or in some fields 0.01 or 0.10). Alpha denotes the fixed significance level for a given hypothesis test. Before starting any statistical analyses, along with stating hypotheses, you choose a significance level you’re testing at.

This states the threshold at which you are prepared to accept the possibility of a Type I Error – otherwise known as a false positive – rejecting a null hypothesis that is actually true.

• Type what error?

Most often we are concerned primarily with reducing the chance of a Type I Error over its counterpart (Type II Error – accepting a false null hypothesis). It all depends on what the impact of either error will be.

Take a pharmaceutical company testing a new drug; if the drug actually doesn’t work (a true null hypothesis) then rejecting this null and asserting that the drug does work could have huge repercussions – particularly if patients are given this drug over one that actually does work. The pharmaceutical company would be concerned primarily with reducing the likelihood of a Type I Error.

Sometimes, a Type II Error could be more important. Environmental testing is one such example; if the effect of toxins on water quality is examined, and in truth the null hypothesis is false (that is, the presence of toxins does affect water quality) a Type II Error would mean accepting a false null hypothesis, and concluding there is no effect of toxins.

The down-stream issues could be dire, if toxin levels are allowed to remain high and there is some health effect on people using that water.

Do you know the difference between continuous and categorical variables?

• What is a p-value, really?

Because p-values are thrown about in science like confetti, it’s important to understand what they do and don’t mean. A p-value expresses the probability of getting a given result from a hypothesis test, or a more extreme result, if the null hypothesis were true.

Given we are trying to reject the null hypothesis, what this tells us is the odds of getting our experimental data if the null hypothesis is correct. If the odds are sufficiently low we feel confident in rejecting the null and accepting the alternative hypothesis.

What is sufficiently low? As mentioned above, the typical fixed significance level is 0.05. So if the probability portrayed by the p-value is less than 5% you reject the null hypothesis. But a fixed significance level can be deceiving: if 5% is significant, why is 6% not?

It pays to remember that such probabilities are continuous, and any given significance level is arbitrary. In other words, don’t throw your data away simply because you get a p-value of 6-10%.

• How much replication do I have?

This is probably the biggest issue when it comes to experimental design, in which the focus is on ensuring the right type of data, in large enough quantities, is available to answer given questions as clearly and efficiently as possible.

Pseudoreplication refers to the over-inflation of degrees of freedom (a mathematical restriction put in place when we calculate a parameter – e.g. a mean – from a sample). How would this work in practice?

Say you’re researching cholesterol levels by taking blood from 20 male participants.

Each male is tested twice, giving 40 test results. But the level of replication is not 40, it’s actually only 20 – a requisite for replication is that each replicate is independent of all others. In this case, two blood tests from the same person are intricately linked.

If you were to analyse the data with a sample size of 40, you would be committing the sin of pseudoreplication: inflating your degrees of freedom (which incidentally helps to create a significant test result). Thus, if you start an experiment understanding the concept of independent replication, you can avoid this pitfall.

• How do I know what analysis to do?

There is a key piece of prior knowledge that will help you determine how to analyse your data. What kind of variable are you dealing with? There are two most common types of variable:

1) Continuous variables. These can take any value. Were you to you measure the time until a reaction was complete, the results might be 30 seconds, two minutes and 13 seconds, or three minutes and 50 seconds.

2) Categorical variables. These fit into – you guessed it – categories. For instance, you might have three different field sites, or four brands of fertiliser. All continuous variables can be converted into categorical variables.

With the above example we could categorise the results into less than one minute, one to three minutes, and greater than three minutes. Categorical variables cannot be converted back to continuous variables, so it’s generally best to record data as “continuous” where possible to give yourself more options for analysis.

Deciding which to use between the two main types of analysis is easy once you know what variables you have:

ANOVA (Analysis of Variance) is used to compare a categorical variable with a continuous variable – for instance, fertiliser treatment versus plant growth in centimetres.

Linear Regression is used when comparing two continuous variables – for instance, time versus growth in centimetres.

Though there are many analysis tools available, ANOVA and linear regression will get you a long way in looking at your data. So if you can start by working out what variables you have, it’s an easy second step to choose the relevant analysis.

Ok, so perhaps that’s not everything you need to know about statistics, but it’s a start. Go forth and analyse!

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Sarah-Jane O’Connor*

 

Statistical databases (SDBs) are collections of data that are used to gather and analyse information from a variety of sources. The data may be derived from sales transactions, customer files, voter registrations, medical records, employee rosters, product inventories, or other compilations of facts and figures.

Because database security requires multiple processes and controls, it presents huge security challenges to organizations. With the computerization of databases in healthcare, forensics, telecommunications, and other fields, ensuring this kind of security has become increasingly important.

In a paper published Thursday in the SIAM Journal on Discrete Mathematics, authors Rudolf Ahlswede and Harout Aydinian analyse a security-control model for statistical databases.

“Providing privacy and confidentiality in SDBs is not a new issue,” Aydinian points out. “Privacy interests have evolved from the very first census in the United States. Recorded protests until the mid-20th century reflect constitutional issues resulting from the requirement for U.S. residents to provide sensitive personal information. Questions on census forms about diseases, mortgage values, and other items have raised many concerns.”

While such databases are very helpful in aggregating data, there is a risk that confidential information about an individual’s record may be deliberately compromised. “Since such data sets also contain sensitive information, such as the disease of an individual, or the salary of an employee, it is necessary to provide security against the disclosure of confidential information,” says Aydinian. “Even in cases where a user has no direct access to sensitive information, sometimes confidential data about an individual can be inferred by correlating enough statistics.”

Typically, statistical databases are designed to only accept queries that involve specific statistical functions (such as sum, average, count, min, max, etc.). However, the use of these queries may render databases susceptible to compromise. For instance, it may be possible to infer information about specific individuals by putting together data from a sequence of statistical queries, using prior knowledge of an individual, or through collusion among users.

An SDB is considered secure if no protected data can be inferred from available queries. “In the literature, many scenarios of compromise and inference control methods have been proposed to protect SDBs,” Aydinian says. “However, to date no one security control method is capable of completely preventing compromise.”

Query restriction is one of several general approaches used for security control. A “query request” retrieves a subset of data from a database that meets a set of conditions. In query restriction, the kind and amount of data that can be retrieved by such queries is limited, for example, the size of the data, or the amount of overlap between data that is returned.

In one type of query restriction method, only certain sums of individual records (called “SUM queries”) that meet a minimum specified size or number, and satisfy a specified set of conditions, are available to users.

Aydinian explains with an example. “Consider a company with a large number of employees. Suppose that for each member of the company, the sex, age, rank, length of employment, salary etc. is recorded. The salaries of individual employees are confidential. Suppose that only SUM queries are allowed, i.e. the sum of the salaries of the specified people is returned. Then one might pose the query: What is the sum of salaries for males, above 50, and during the last 10 years?”

The task addressed in the paper is to provide an optimal collection of SUM queries that prevents compromise of confidential information—such as individual salaries, for instance. A natural solution is to maximize the number of available SUM queries. The authors obtain tight bounds for the maximum number of such queries that return subsets of data without compromising groups of entries.

“Future work in the query-restriction approach includes evaluation of new security-control mechanisms, which are easy to implement and guarantee absolute security,” says Aydinian. “At the same time, it is desirable that these methods satisfy other criteria like richness of available queries, consistency, cost etc. It also seems promising to develop methods combining different security control mechanisms.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Society for Industrial and Applied Mathematics

The notebooks of Sir Isaac Newton, who was famously reported to have suffered a (scientifically) earth-shaking blow to the head from an apple, are being scanned and published online by the University of Cambridge.

Newton, a Biblical numerologist when he wasn’t developing calculus or building the first reflecting telescope, founded classical mechanics with Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), which was first published in 1687. In the book that made his name, Newton set out his three laws of motion, and his theory of universal gravitation (prompted by pondering what force plummeted the fruit straight down onto his head, or so goes the apocryphal tale).

Newton studied and later held the Lucasian Chair of Mathematics at Cambridge, which was given numerous manuscripts of his in 1872 and has since bought more. The online publication has started with Newton’s mathematical works of the 1660s and more papers will become available over coming months.

A philosopher of science at Flinders University, George Couvalis, said that Newton’s gravitational experiments – which largely corrected ancient observations of gravity – were sparked by his interest in magic and magnetism. “The idea that things might naturally attract one another is an idea that he got from magical ideas. He adapted it across to mathematical theory because it was a mystical theory,” Dr Couvalis said

It was important to remember that scientists of Newton’s era did not have what we would consider a modern sceptical outlook and – with the exception of the “exceptional” Galileo Galilei – instead held a fusion of views that we would consider deeply irrational, Dr Couvalis said.

“It was certainly far more common in the 17th and 18th centuries for scientists to be interested in magical beliefs and alchemical beliefs and religious beliefs. Johannes Kepler, for example, had all kinds of strange views about the music of the spheres, Copernicus had strange views about the sacredness of the sun, and Newton famously had views about the mysterious numerical meanings of Biblical passages and about alchemical material,” Dr Couvalis said.

Scientists of the period saw their work touching on many illogical and occult fields of interest, including Robert Boyle, a founder of modern chemistry, who had “an interest in doing experimental research on magical mirrors, which to us would sound bizarre but at the time it was thought to be a possibility,” said Dr Couvalis, who added that Boyle pulled back from some experiments for religious reasons. “He thought it might get him in touch with demons.”

Demonology may have fallen out of favour amongst scientists, but “the view that we’re getting everything right would be a serious mistake,” Dr Couvalis said. “To some degree science is always in the sway of the time it’s in; this is now the standard view of philosophers and historians.”

“Newton’s mechanics is in certain respects pretty much right, but in other respects it was shown by Einstein and others to be wildly wrong. By about 1900 we had people saying to their graduate students ‘You should give up physics because it’s all been done,’ but Einstein managed to show that it was wildly wrong in certain respects,” Dr Couvalis said.

The ideal of the scientific method is never met, and our beliefs and discoveries will likely on day be seen as flawed but perhaps useful stepping stones in the continuum of science, Dr Couvalis said. “People make mistakes, people have a lot of trouble leaving assumptions behind, and our tests are never rigorous enough to be absolutely certain that we’re getting things right. Future experimental studies and the sheer empirical facts will show us to be wrong in many ways that we can’t anticipate.”

“We work with what we have because we just don’t know anything better at the moment. It might turn out that Einstein’s special and general theories of relativity are wrong in some deep-seated way. It might turn out that some of our theories of the universe are wrong. It’s starting to look in biology as if neo-Darwinism isn’t completely right, so where will that go – I don’t know. Research will determine the direction. That doesn’t mean that we’re going to go back to being creationists – that view has been thoroughly debunked. Imre Lakatos wrote in the 1970s there are no good scientific theories, there’s only the best rotten theory we have.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Matthew Thompson, The Conversation

 

Mirrors of a magical scientist: Andromeda photographed through a Newtonian telescope.

The notebooks of Sir Isaac Newton, who was famously reported to have suffered a (scientifically) earth-shaking blow to the head from an apple, are being scanned and published online by the University of Cambridge.

Newton, a Biblical numerologist when he wasn’t developing calculus or building the first reflecting telescope, founded classical mechanics with Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), which was first published in 1687. In the book that made his name, Newton set out his three laws of motion, and his theory of universal gravitation (prompted by pondering what force plummeted the fruit straight down onto his head, or so goes the apocryphal tale).

Newton studied and later held the Lucasian Chair of Mathematics at Cambridge, which was given numerous manuscripts of his in 1872 and has since bought more. The online publication has started with Newton’s mathematical works of the 1660s and more papers will become available over coming months.

Striking a light for science.

A philosopher of science at Flinders University, George Couvalis, said that Newton’s gravitational experiments – which largely corrected ancient observations of gravity – were sparked by his interest in magic and magnetism. “The idea that things might naturally attract one another is an idea that he got from magical ideas. He adapted it across to mathematical theory because it was a mystical theory,” Dr Couvalis said.

It was important to remember that scientists of Newton’s era did not have what we would consider a modern sceptical outlook and – with the exception of the “exceptional” Galileo Galilei – instead held a fusion of views that we would consider deeply irrational, Dr Couvalis said.

“It was certainly far more common in the 17th and 18th centuries for scientists to be interested in magical beliefs and alchemical beliefs and religious beliefs. Johannes Kepler, for example, had all kinds of strange views about the music of the spheres, Copernicus had strange views about the sacredness of the sun, and Newton famously had views about the mysterious numerical meanings of Biblical passages and about alchemical material, ” Dr Couvalis said.

Scientists of the period saw their work touching on many illogical and occult fields of interest, including Robert Boyle, a founder of modern chemistry, who had “an interest in doing experimental research on magical mirrors, which to us would sound bizarre but at the time it was thought to be a possibility,” said Dr Couvalis, who added that Boyle pulled back from some experiments for religious reasons. “He thought it might get him in touch with demons.”

Demonology may have fallen out of favour amongst scientists, but “the view that we’re getting everything right would be a serious mistake,” Dr Couvalis said. “To some degree science is always in the sway of the time it’s in; this is now the standard view of philosophers and historians.”

“Newton’s mechanics is in certain respects pretty much right, but in other respects it was shown by Einstein and others to be wildly wrong. By about 1900 we had people saying to their graduate students ‘You should give up physics because it’s all been done,’ but Einstein managed to show that it was wildly wrong in certain respects,” Dr Couvalis said.

The ideal of the scientific method is never met, and our beliefs and discoveries will likely on day be seen as flawed but perhaps useful stepping stones in the continuum of science, Dr Couvalis said. “People make mistakes, people have a lot of trouble leaving assumptions behind, and our tests are never rigorous enough to be absolutely certain that we’re getting things right. Future experimental studies and the sheer empirical facts will show us to be wrong in many ways that we can’t anticipate.”

“We work with what we have because we just don’t know anything better at the moment. It might turn out that Einstein’s special and general theories of relativity are wrong in some deep-seated way. It might turn out that some of our theories of the universe are wrong. It’s starting to look in biology as if neo-Darwinism isn’t completely right, so where will that go – I don’t know. Research will determine the direction. That doesn’t mean that we’re going to go back to being creationists – that view has been thoroughly debunked. Imre Lakatos wrote in the 1970s there are no good scientific theories, there’s only the best rotten theory we have.”

For more such insights, log into www.international-maths-challenge.com.

*Credit For article given to Matthew Thompson*

 

In a project that has long been overdue, Cambridge University, thanks to a hefty gift from the Polonsky Foundation (supporter of education and arts) and a grant from Britain’s Joint Information Services Committee (JISC), has put some of Isaac Newton’s original papers online for any and all to see. Of particular interest to most will be Newton’s own annotated copy of Philosophiae Naturalis Principia Mathematica, considered by many to be one of the greatest published works by any scientist ever. For those looking for a little behind the scenes work, the University has also published Newton’s so-called “Waste Book,” a diary of sorts that Newton inherited from his step-father which he took along with him and used for jotting notes about such things as his ideas on calculus while away from school due to the Great Plague in 1665.

In viewing the material, which can be paged through in a PDF type format, by clicking arrows, it’s easy to see that the digitization of Newton’s papers have come none too soon, as many of the pages are tattered, smeared and even burned-looking in some places. Thus, not only has putting the papers online made them accessible to anyone with a computer and an Internet connection, it has also caused them to be saved for posterity in an electronic form that will ensure they will be accessible to all those who may wish to view them in the future as well.

It was in Principia Mathematica that Newton laid out his theories on the laws of motion and universal gravitation which some suggest laid the groundwork for Einstein’s theories on relativity. And if that weren’t enough, Newton is also widely credited with “inventing” calculus, a mathematical science without which the modern world would simply not exist.

In all there are more than 4,000 pages of Newton’s work displayed on the site, which took a team of photo copyists the better part of this past summer to capture, though it’s obvious in looking at the results that there were many slow-downs as pages had to have some restorative efforts made in order to present them. Those working on the project are to be commended as the results show great care and dedication to a single purpose; namely showcasing one of history’s brightest minds.

It’s intriguing to see the notes Newton himself made on the first edition of Principia Mathematica, in preparing for the second, and happily, the University has announced that they will be adding translations for all of the text and notes as early as next year.

The University has also announced plans to make the works of other famous scientists available as the future unfolds and hopefully will continue to add more of the Newton library too, as thus far only about 20% of their collection has been made available online.

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Credit of the article given to Bob Yirka , Phys.org

A major study of recent international data on school mathematics performance casts doubt on some common assumptions about gender and math achievement — in particular, the idea that girls and women have less ability due to a difference in biology.

“We tested some recently proposed hypotheses that try to explain a supposed gender gap in math performance and found the data did not support them,” says Janet Mertz, senior author of the study and a professor of oncology at the University of Wisconsin-Madison.

Instead, the Wisconsin researchers linked differences in math performance to social and cultural factors.

The new study, by Mertz and Jonathan Kane, a professor of mathematical and computer sciences at the University of Wisconsin-Whitewater, was published in Dec 2011 in Notices of the American Mathematical Society. The study looked at data from 86 countries, which the authors used to test the “greater male variability hypothesis” famously expounded in 2005 by Lawrence Summers, then president of Harvard, as the primary reason for the scarcity of outstanding women mathematicians.

That hypothesis holds that males diverge more from the mean at both ends of the spectrum and, hence, are more represented in the highest-performing sector. But, using the international data, the Wisconsin authors observed that greater male variation in math achievement is not present in some countries, and is mostly due to boys with low scores in some other countries, indicating that it relates much more to culture than to biology.

The new study relied on data from the 2007 Trends in International Mathematics and Science Study and the 2009 Programme in International Student Assessment.

“People have looked at international data sets for many years”, Mertz says. “What has changed is that many more non-Western countries are now participating in these studies, enabling much better cross-cultural analysis.”

The Wisconsin study also debunked the idea proposed by Steven Levitt of “Freakonomics” fame that gender inequity does not hamper girls’ math performance in Muslim countries, where most students attend single-sex schools. Levitt claimed to have disproved a prior conclusion of others that gender inequity limits girls’ mathematics performance. He suggested, instead, that Muslim culture or single-sex classrooms benefit girls’ ability to learn mathematics.

By examining the data in detail, the Wisconsin authors noted other factors at work. “The girls living in some Middle Eastern countries, such as Bahrain and Oman, had, in fact, not scored very well, but their boys had scored even worse, a result found to be unrelated to either Muslim culture or schooling in single-gender classrooms,” says Kane.

He suggests that Bahraini boys may have low average math scores because some attend religious schools whose curricula include little mathematics. Also, some low-performing girls drop out of school, making the tested sample of eighth graders unrepresentative of the whole population.

“For these reasons, we believe it is much more reasonable to attribute differences in math performance primarily to country-specific social factors,” Kane says.

To measure the status of females relative to males within each country, the authors relied on a gender-gap index, which compares the genders in terms of income, education, health and political participation. Relating these indices to math scores, they concluded that math achievement at the low, average and high end for both boys and girls tends to be higher in countries where gender equity is better. In addition, in wealthier countries, women’s participation and salary in the paid labor force was the main factor linked to higher math scores for both genders.

“We found that boys — as well as girls — tend to do better in math when raised in countries where females have better equality, and that’s new and important,” says Kane. “It makes sense that when women are well-educated and earn a good income, the math scores of their children of both genders benefit.”

Mertz adds, “Many folks believe gender equity is a win-lose zero-sum game: If females are given more, males end up with less. Our results indicate that, at least for math achievement, gender equity is a win-win situation.”

U.S. students ranked only 31st on the 2009 Programme in International Student Assessment, below most Western and East-Asian countries. One proposed solution, creating single-sex classrooms, is not supported by the data. Instead, Mertz and Kane recommend increasing the number of math-certified teachers in middle and high schools, decreasing the number of children living in poverty and ensuring gender equality.

“These changes would help give all children an optimal chance to succeed,” says Mertz. “This is not a matter of biology: None of our findings suggest that an innate biological difference between the sexes is the primary reason for a gender gap in math performance at any level. Rather, these major international studies strongly suggest that the math-gender gap, where it occurs, is due to sociocultural factors that differ among countries, and that these factors can be changed.”

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Credit of the article given to University of Wisconsin-Madison

There’s a contradiction between classical and quantum theories.

One of the outstanding discoveries made in the early part of the last century was that of the quantum behaviour of the physical world. At very short distances, such as the size of an atom and smaller, the world behaves very differently to the “classical” world we are used to.

Typical of the quantum world is so-called wave-particle duality: particles such as electrons behave sometimes as if they are point particles with a definite position, and sometimes as if they are spread out like waves.

This strange behaviour is not just of theoretical interest, since it is underpins much of our modern technology. It is fundamental to the behaviour of semiconductors in all our electronic devices, the behaviour of nano-materials, and the current rise of quantum computing.

Quantum theory is fundamental. It must govern not just the very small but also the classical realm. That means physicists and mathematicians have had to develop methods not just for understanding new quantum phenomena, but also for replacing classical theories by their quantum analogues.

This is the process of [quantization.](http://en.wikipedia.org/wiki/Quantization_(physics) When we have a finite number of degrees of freedom, such as for a finite collection of particles, although the quantum behaviour is often counter-intuitive, we have a well-developed mathematical machinery to handle this quantization called quantum mechanics.

This is well understood physically and mathematically. But when we move to study the electric and magnetic fields where we have an infinite number of degrees of freedom, the situation is much more complicated. With the development of so-called quantum field theory, a quantum theory for fields, physics has made progress that mathematically we do not completely understand.

What’s the problem?

Many field theories fall into a class called gauge field theories, where a particular collection of symmetries, called the gauge group, acts on the fields and particles. In the case that these symmetries all commute, so-called abelian gauge theories, we have a reasonable understanding of the quantization.

This includes the case of the electromagnetic field, quantum electrodynamics, for which the theory makes impressively accurate predictions.

The first example of a non-abelian theory that arose historically is the theory of the electro-weak interaction, which requires a mechanism to make the predicted particles massive as we observe them in nature. This involves the so-called Higgs boson, which is currently being searched for with the Large Hadron Collider (LHC) at CERN.

The notable feature of this theory for our present discussion is that the Higgs mechanism is classical and carries over to the quantum theory under the quantization process.

The case of interest in the Millennium Problem “Yang-Mills theory and Mass-Gap” is Yang-Mills gauge theory, a non-abelian theory which we expect to describe quarks and the strong force that binds the nucleus and powers the sun. Here we encounter a contradiction between the classical and quantum theories.

The classical theory predicts massless particles and long-range forces. The quantum theory has to match the real world with short-range forces and massive particles. Physicists expect various mathematical properties such as the “mass gap” and “asymptotic freedom” to explain the non-existence of massless particles in observations of the strong interactions.

As these properties are not visible in the classical theory and arise only in the quantum theory, understanding them means we need a rigorous approach to “quantum Yang-Mills theory”. Currently we do not have the mathematics to do this, although various approximations and simplifications can be done which suggest the quantum theory has the required properties.

The Millennium Problem seeks to establish by rigorous mathematics the existence of the “mass gap” – that is, the non-existence of massless particles in Yang-Mills theory. The solution of the problem would involve an approach to quantum field theory in four dimensions that is sophisticated enough to explain at least this feature of quantum non-abelian Yang-Mills gauge theory.

Doing the maths

Clearly this is of interest to physicists, but why is it of importance to mathematicians? It has become apparent in the last few decades that the tools that physicists have developed for doing quantum field theory, in particular path integrals, make precise predictions about geometry and topology, particularly in low dimensions.

But we don’t know mathematically what a path integral is, except in very simple cases. It is as if we are in a pre-Newtonian world – certain calculations can be done with certain tricks but Newton hasn’t developed calculus for us yet.

Analogously, there are calculations in geometry and topology that can be done non-rigorously using methods developed by physicists in quantum field theory which give the right answers. This suggests that there is a set of powerful techniques waiting to be discovered.

A solution to this Millennium Problem would shed light on what these new techniques are.

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*Credit for article given to Michael Murray*

 

Using a new mathematical methodology, researchers at MIT have created a scientifically rigorous analogy that shows the similarities between the physical structure of spider silk and the sonic structure of a melody, proving that the structure of each relates to its function in an equivalent way.

The step-by-step comparison begins with the primary building blocks of each item — an amino acid and a sound wave — and moves up to the level of a beta sheet nanocomposite (the secondary structure of a protein consisting of repeated hierarchical patterns) and a musical riff (a repeated pattern of notes or chords). The study explains that structural patterns are directly related to the functional properties of lightweight strength in the spider silk and, in the riff, sonic tension that creates an emotional response in the listener.

While likening spider silk to musical composition may appear to be more novelty than breakthrough, the methodology behind it represents a new approach to comparing research findings from disparate scientific fields. Such analogies could help engineers develop materials that make use of the repeating patterns of simple building blocks found in many biological materials that, like spider silk, are lightweight yet extremely failure-resistant. The work also suggests that engineers may be able to gain new insights into biological systems through the study of the structure-function relationships found in music and other art forms.

The MIT researchers — David Spivak, a postdoc in the Department of Mathematics, Associate Professor Markus Buehler of the Department of Civil and Environmental Engineering (CEE) and CEE graduate student Tristan Giesa — published their findings in the December issue of BioNanoScience.

They created the analogy using ontology logs, or “ologs,” a concept introduced about a year ago by Spivak, who specializes in a branch of mathematics called category theory. Ologs provide an abstract means for categorizing the general properties of a system — be it a material, mathematical concept or phenomenon — and showing inherent relationships between function and structure.

To build the ologs, the researchers used information from Buehler’s previous studies of the nanostructure of spider silk and other biological materials.

“There is mounting evidence that similar patterns of material features at the nanoscale, such as clusters of hydrogen bonds or hierarchical structures, govern the behaviour of materials in the natural environment, yet we couldn’t mathematically show the analogy between different materials,” Buehler says. “The olog lets us compile information about how materials function in a mathematically rigorous way and identify those patterns that are universal to a very broad class of materials. Its potential for engineering the built environment — in the design of new materials, structures or infrastructure — is immense.”

“This work is very exciting because it brings forth an approach founded on category theory to bridge music (and potentially other aspects of the fine arts) to a new field of materiomics,” says Associate Professor of Biomedical Engineering Joyce Wong of Boston University, a biomaterials scientist and engineer, as well as a musician. “This approach is particularly appropriate for the hierarchical design of proteins, as they show in the silk example. What is particularly exciting is the opportunity to reveal new relationships between seemingly disparate fields with the aim of improving materials engineering and design.”

At first glance, an olog may look deceptively simple, much like a corporate organizational chart that shows reporting relationships using directional arrows. But ologs demand scientific rigor to break a system down into its most basic structural building blocks, define the functional properties of the building blocks with respect to one another, show how function emerges through the building blocks’ interactions, and do this in a self-consistent manner. With this structure, two or more systems can be formally compared.

“The fact that a spider’s thread is robust enough to avoid catastrophic failure even when a defect is present can be explained by the very distinct material makeup of spider-silk fibers,” Giesa says. “It’s exciting to see that music theoreticians observed the same phenomenon in their field, probably without any knowledge of the concept of damage tolerance in materials. Deleting single chords from a harmonic sequence often has only a minor effect on the harmonic quality of the whole sequence.”

“The seemingly incredible gap between spider silk and music is no wider than the gap between the two disparate mathematical fields of geometry — think of triangles and spheres — and algebra, which uses variables and equations,” Spivak says. “Yet category theory’s first success, in the 1940s, was to express a rigorous mathematical analogy between these two domains and use it to prove new theorems about complex geometric shapes by importing existing theorems from algebra. It remains to be seen whether our olog will yield such striking results; however, the foundation for such an inquiry is now in place.”

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Credit of the article given to Denise Brehm, Massachusetts Institute of Technology