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The Importance Of Maths

GNC said:
Bobby Darin disagrees:
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Then algebra, trigonometry, differentiation and integration. I dare not go past there into that fearful zone of complex calculus. I once did but no longer do, nor likely ever will or can.
 
This Turing machine should run forever unless maths is wrong

By Jacob Aron

One hundred and fifty years of mathematics will be proved wrong if a new computer program stops running. Thankfully, it’s unlikely to happen, but the code behind it is testing the limits of the mathematical realm.

The program is a simulated Turing machine, a mathematical model of computation created by codebreaker Alan Turing. In 1936, he showed that the actions of any computer algorithm can be mimicked by a simple machine that reads and writes 0s and 1s on an infinitely long tape by working through a set of states, or instructions. The more complex the algorithm, the more states the machine requires.

Now Scott Aaronson and Adam Yedidia of the Massachusetts Institute of Technology have created three Turing machines with behaviour that is entwined in deep questions of mathematics. This includes the proof of the 150-year-old Riemann hypothesis – thought to govern the patterns of prime numbers.

Turing’s machines have long been used to probe such questions. Their origins lie in a series of philosophical revelations that rocked the mathematical world in the 1930s. First, Kurt Gödel proved that some mathematical statements can never be proved true or false – they are undecidable. He essentially created a mathematical version of the sentence “This sentence is false”: a logical brain-twister that contradicts itself.

Gödel’s assertion has a get-out clause. If you change the base assumptions on which proofs are built – the axioms – you can render a problem decidable. But that will still leave other problems that are undecidable. That means there are no axioms that let you prove everything.

Following Gödel’s arguments, Turing proved that there must be some Turing machines whose behaviour cannot be predicted under the standard axioms – known as Zermelo-Fraenkel set theory with the axiom of choice (C), or more snappily, ZFC – underpinning most of mathematics. But we didn’t know how complex they would have to be.

Now, Yedidia and Aaronson have created a Turing machine with 7918 states that has this property. And they’ve named it “Z”.
“We tried to make it concrete, and say how many states does it take before you get into this abyss of unprovability?” says Aaronson.

The pair simulated Z on a computer, but it is small enough that it could theoretically be built as a physical device, says Terence Tao of the University of California, Los Angeles. “If one were then to turn such a physical machine on, what we believe would happen would be that it would run indefinitely,” he says, assuming you ignore physical wear and tear or energy requirements.

etc...

https://www.newscientist.com/articl...1905-newGLOBAL&utm_medium=NLC&utm_source=NSNS
 
rynner2 said:
This Turing machine should run forever unless maths is wrong

By Jacob Aron

One hundred and fifty years of mathematics will be proved wrong if a new computer program stops running. Thankfully, it’s unlikely to happen, but the code behind it is testing the limits of the mathematical realm.

The program is a simulated Turing machine, a mathematical model of computation created by codebreaker Alan Turing. In 1936, he showed that the actions of any computer algorithm can be mimicked by a simple machine that reads and writes 0s and 1s on an infinitely long tape by working through a set of states, or instructions. The more complex the algorithm, the more states the machine requires.

Now Scott Aaronson and Adam Yedidia of the Massachusetts Institute of Technology have created three Turing machines with behaviour that is entwined in deep questions of mathematics. This includes the proof of the 150-year-old Riemann hypothesis – thought to govern the patterns of prime numbers.

Turing’s machines have long been used to probe such questions. Their origins lie in a series of philosophical revelations that rocked the mathematical world in the 1930s. First, Kurt Gödel proved that some mathematical statements can never be proved true or false – they are undecidable. He essentially created a mathematical version of the sentence “This sentence is false”: a logical brain-twister that contradicts itself.

Gödel’s assertion has a get-out clause. If you change the base assumptions on which proofs are built – the axioms – you can render a problem decidable. But that will still leave other problems that are undecidable. That means there are no axioms that let you prove everything.

Following Gödel’s arguments, Turing proved that there must be some Turing machines whose behaviour cannot be predicted under the standard axioms – known as Zermelo-Fraenkel set theory with the axiom of choice (C), or more snappily, ZFC – underpinning most of mathematics. But we didn’t know how complex they would have to be.

Now, Yedidia and Aaronson have created a Turing machine with 7918 states that has this property. And they’ve named it “Z”.
“We tried to make it concrete, and say how many states does it take before you get into this abyss of unprovability?” says Aaronson.

The pair simulated Z on a computer, but it is small enough that it could theoretically be built as a physical device, says Terence Tao of the University of California, Los Angeles. “If one were then to turn such a physical machine on, what we believe would happen would be that it would run indefinitely,” he says, assuming you ignore physical wear and tear or energy requirements.

etc...

https://www.newscientist.com/article/2087845-this-turing-machine-should-run-forever-unless-maths-is-wrong/?cmpid=NLC|NSNS|2016-1905-newGLOBAL&utm_medium=NLC&utm_source=NSNS[URL='https://www.newscientist.com/article/2087845-this-turing-machine-should-run-forever-unless-maths-is-wrong/?cmpid=NLC|NSNS|2016-1905-newGLOBAL&utm_medium=NLC&utm_source=NSNS[/QUOTE'][/QUOTE[/URL]]

Any engineering program basically computes high level math. Math that would be virtually impossibly to determine long hand. But the math is nothing wrong or new and allows us to do things ( simulate things) we otherwise could never accomplish. Such as to display the actual near and far fields of radiating devices (not just hypothesize or connect the dots via measurement). Thus allowing us to greatly expand our horizon and capabilities. However the math is still the same algebra, linear algebra, trig, calculus, complex calculus developed by mathematicians long before the computer era. I don't belief anything will change this fundamental truth, however that's me.
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Hi Jim! You've managed to insert your comments into the quote of my post, which must be confusing for some readers!

(Put your comments after the [/QUOTE] tag, and then all will become clear. :))
 
My apologies rynner2, not intentional! about mixing up the response to your Math post.
 
You guys are quick! This is the original source, that I discovered today:

http://www.scottaaronson.com/blog/?p=2725

Adam has also constructed a 4,888-state Turing machine that halts iff there’s a counterexample to Goldbach’s Conjecture, and a 5,372-state machine that halts iff there’s a counterexample to the Riemann Hypothesis.
 
Something for all you fans (like me) of red-headed female mathematicians:
The Joy of Data

A witty and mind-expanding exploration of data, with mathematician Dr Hannah Fry. This high-tech romp reveals what data is and how it is captured, stored, shared and made sense of. Fry tells the story of the engineers of the data age, people most of us have never heard of despite the fact they brought about a technological and philosophical revolution.

For Hannah, the joy of data is all about spotting patterns. Hannah sees data as the essential bridge between two universes - the tangible, messy world that we see and the clean, ordered world of maths, where everything can be captured beautifully with equations.

The film reveals the connection between Scrabble scores and online movie streaming, explains why a herd of dairy cows are wearing pedometers, and uncovers the network map of Wikipedia. What's the mystery link between marmalade and One Direction?

The film hails the contribution of Claude Shannon, the mathematician and electrical engineer who, in an attempt to solve the problem of noisy telephone lines, devised a way to digitise all information. Shannon singlehandedly launched the 'information age'. Meanwhile, Britain's National Physical Laboratory hosts a race between its young apprentices in order to demonstrate how and why data moves quickly around modern data networks. It's all thanks to the brilliant technique first invented there in the 1960s by Welshman Donald Davies - packet switching.

But what of the future? Should we be worried by the pace of change and what our own data could be used for? Ultimately, Fry concludes, data has empowered all of us. We must have machines at our side if we're to find patterns in the modern-day data deluge. But, Fry believes, regardless of AI and machine learning, it will always take us to find the meaning in them.

http://www.bbc.co.uk/iplayer/episode/b07lk6tj/the-joy-of-data
 
rynner2 said:
The film hails the contribution of Claude Shannon, the mathematician and electrical engineer who, in an attempt to solve the problem of noisy telephone lines, devised a way to digitise all information. Shannon singlehandedly launched the 'information age'.
Click to expand...
...and now using OFDM and FEC, we can transmit data that is effectively under the noise floor. Incredible stuff.
 
And so we have Shannon's limit, for which he is very famous for. The rest just slowly evolved as the digital age came about such OFDM and DSSS, as for low noise transmission methods. BTW DSSS can be effectivity transmitted under the AWGN spectrum thru a auto correlation method.
Shannon's limit is defined as: the maximum capacity of a channel in bits/second for the given channel, but simple.
 
Jim said:
And so we have Shannon's limit, for which he is very famous for. The rest just slowly evolved as the digital age came about such OFDM and DSSS, as for low noise transmission methods. BTW DSSS can be effectivity transmitted under the AWGN spectrum thru a auto correlation method.
Shannon's limit is defined as: the maximum capacity of a channel in bits/second for the given channel, but simple.
Click to expand...
Hey Jim, lay off the Alphabetti Spaghetti! Or your readership here won't know what you're on about, and will give up on you. (I don't, and I'm fairly well up with technology, etc!)
 
rynner2 said:
Hey Jim, lay off the Alphabetti Spaghetti! Or your readership here won't know what you're on about, and will give up on you. (I don't, and I'm fairly well up with technology, etc!)
Click to expand...
Sorry about that
 
Jim said:
And so we have Shannon's limit, for which he is very famous for. The rest just slowly evolved as the digital age came about such OFDM and DSSS, as for low noise transmission methods. BTW DSSS can be effectivity transmitted under the AWGN spectrum thru a auto correlation method.
Click to expand...

Yes but the dialytheum crystals will never take it.
 
Ye cannae change the laws of physics, cap'n.
 
Jim said:
And so we have Shannon's limit, for which he is very famous for. The rest just slowly evolved as the digital age came about such OFDM and DSSS, as for low noise transmission methods. BTW DSSS can be effectivity transmitted under the AWGN spectrum thru a auto correlation method.
Shannon's limit is defined as: the maximum capacity of a channel in bits/second for the given channel, but simple.
Click to expand...
So CDMA then :D
 
rynner2 said:
That's an amazing piece of maths I've never encountered before!

The whole universe seems more and more to be a mathematical simulation, although we may never know which mind or machine is running the simulation... :cool:
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Careful - you'll have Fudgetusk stamping all over this thread with talk like that........
 
Maths becomes biology's magic number
By Tom Feilden Science correspondent, Today programme

"If you want a career in medicine these days you're better off studying mathematics or computing than biology."
This pithy aside was delivered by Sir Rory Collins, the head of clinical trials at Oxford University, in the middle of a discussion about the pros and cons of statins.

It is a nice one-liner, but I didn't think much more about it until a few days later, when I found myself sitting in a press conference to mark the launch of a new initiative on cancer.

Rubbing shoulders on the panel with the director of the Institute of Cancer Research, Professor Paul Workman, was a scientist I didn't recognise, but it soon became clear this was exactly what Sir Rory had had in mind.

Dr Andrea Sottoriva is an astrophysicist. He has spent much of his career searching for Neutrinos - the elusive sub-atomic particles created by the fusion of elements in stars like our sun - at the bottom of the ocean, and analysing the results of atom smashing experiments with the Large Hadron Collider at Cern in Geneva.
"My background is in computer science, particularly as it applies to particle physics," he told me when we met at the ICR's laboratories in Sutton

So why cancer? The answer can be summed up in two words: big data. What Dr Sottoriva brings to the fight against cancer is the expertise in mathematical modelling needed to mine the vast treasure trove of data the information revolution has brought to medicine.

"The exciting thing is that we can apply all the new analytical techniques we've developed in physics to biology," he says.
"So we have all these new quantitative technologies that allow us to process an enormous amount of data, and all of a sudden we can start to apply that to implement the paradigm of physics in biology."

Of course, applying maths to solve biological problems is not entirely new. But it is only now, according to Sir Rory Collins, that the big data revolution is transforming medical science and ushering in a new era of bioinformatics.
"The big data era we're in provides extraordinary opportunities to understand the determinants of a range of different health conditions," says Sir Rory.

"The availability of data is unsurpassed, the ways of manipulating that data are also unsurpassed and so are the opportunities to work out what's going on and how to avoid disease."

But there's a problem. The vast data sets that give bioinformatics its power are also its Achilles heel.
The Professor of Science and Society at Arizona State University, Daniel Sarewitz, warns of "datageddon" - over-enthusiastic researchers risking being set adrift on a sea of irrelevant information.
"If mouse models are like looking for your keys under the streetlamp, big data is like looking for your keys all over the world just because you can," says Professor Sarewitz.

The epidemiologist Professor Liam Smeeth agrees. If researchers aren't very disciplined about what they're looking for, he argues, they can quickly disappear down rabbit holes and blind alleys.

"The analogy is like someone firing an arrow at a wall," he says. "They fire at a big blank wall and then go up and draw a target around the arrow and say we've hit bullseye.
"What you need is to be doing is precise science and to be firing at a pre-specified target."

The answer, according to Dr Sottoriva, may be to approach big data like a grandmaster approaches chess.
To use mathematical modelling to understand and decode the rules of the game cancer is playing.

"What grandmasters do is to predict the moves of the opponent," he says. "If we can decode the complexity and make predictions about what cancer will do three, four moves ahead, then we can develop really effective treatments based on a solid mathematical framework."

http://www.bbc.co.uk/news/science-environment-37630414
 
How maths can get you locked up
By Simon Maybin BBC News

Criminals in the US can be given computer-generated "risk scores" that may affect their sentences. But are the secret algorithms behind them really making justice fairer?

If you've seen the hit Netflix documentary series Making A Murderer, you'll know the US state of Wisconsin has had its problems delivering fair justice.
Now there's another Wisconsin case that's raised questions about how the US justice system works.
In the early hours of Monday 11 February 2013, two shots were fired at a house in La Crosse, a small city in the state.

A witness said the shots came from a car, which police tracked down and chased through the streets of La Crosse until it ended up in a snow bank. The two people inside ran off on foot, but were found and arrested.
One of them was Eric Loomis, who admitted to driving the car but denied involvement in the shooting.
In court, he was sentenced to six years in prison - and this is where the maths comes in.

In deciding to lock Loomis up, the court noted that he had been identified as an "individual who is at high risk to the community" by something called a Compas assessment.

The acronym - which stands for Correctional Offender Management Profiling for Alternative Sanctions - is very familiar to Julia Angwin of ProPublica, an independent investigative journalism organisation in the US.
"Compas is basically a questionnaire that is given to criminals when they're arrested," she says. "And they ask a bunch of questions and come up with an assessment of whether you're likely to commit a future crime. And that assessment is given in a score of one to 10."

Angwin says the questions include things like: "Your criminal history, and whether anyone in your family has ever been arrested; whether you live in a crime-ridden neighbourhood; if you have friends who are in a gang; what your work history is; your school history. And then some questions about what is called criminal thinking, so if you agree or disagree with statements like 'it's okay for a hungry person to steal'."

A risk score might be used to decide if someone can be given bail, if they should be sent to prison or given some other kind of sentence, or - once they're in prison - if they should be given parole.

Compas and software like it is used across the US. The thinking is that if you use an algorithm that draws on lots of information about the defendant it will help make decisions less subjective - less liable to human error and bias or racism. For example, the questionnaire doesn't ask about the defendant's race, so that in theory means no decisions influenced by racism. [But read the full article to see how this is actually works in practise.]

But how the algorithm gets from the answers to the score out of 10 is kept secret.
"We don't really know how the score is created out of those questions because the algorithm itself is proprietary," says Angwin. "It's a trade secret. The company doesn't share it."

And she says that makes it difficult for a defendant to dispute their risk score: "How do you go in and say I'm really an eight or I'm a seven when you can't really tell how it was calculated?"
It was partly on that basis that Loomis challenged the use of the Compas risk score in his sentencing. But in July the Wisconsin Supreme Court ruled that if Compas is used properly, it doesn't violate a defendant's rights.

Bad news for Eric Loomis. However, the court also ruled that future use of the risk score should come with a health warning explaining its limitations - for example, that the algorithm behind it is kept secret and that the risk score is based on how people with certain traits tend to behave in general, rather than on the particular person being considered.

etc...

http://www.bbc.co.uk/news/magazine-37658374

This is bad science. Science should be open to peer review. "The algorithm itself is proprietary.." is just a fig-leaf to hide potentially unfair assessments from examination. It's not the maths that gets you locked up, but the people who devised the algorithm. :mad:
 
rynner2 said:
http://www.bbc.co.uk/news/magazine-37658374

This is bad science. Science should be open to peer review. "The algorithm itself is proprietary.." is just a fig-leaf to hide potentially unfair assessments from examination. It's not the maths that gets you locked up, but the people who devised the algorithm. :mad:
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Exactly the case. Trait measurement is not an exact science in any way shape or form even if done openly and fairly. Psychology as a whole has enough problems with openness and repeatably without this kind of rubbish masquerading as science.

In any event, "If someone is hiding something, they've something to hide", so I've even graver reservations about the hiding of the 'algorithm'.

It feels only a short step away from eugenics.
 
The few paragraphs I understand are fascinating:

http://inference-review.com/article/fukugen

The natural numbers are infinite but discrete. The real numbers form a linear continuum. In the late nineteenth century, Georg Cantor demonstrated that there is no one-to-one mapping between the natural and the real numbers. This is still a shocking result. Fundamental changes in mathematics have arisen from the interplay between these structures, and many mathematicians suspect that, at a profound level, the mathematical structures behind the natural numbers are continuous. The history of physics and chemistry suggests as much. The periodic table appears to list a series of discrete atoms, but from the much deeper perspective of quantum field theory, these stable and discrete—indeed, isolated—structures are the adaptive masks worn by continuous fields.
 
Well, it greatly surprises me to present a Film Review in this thread! :evil:
Man versus machine: how computers replaced humans in the Space Race
3 February 2017 • 11:45am
Kerry Kolbe

Hidden Figures tells the story of three remarkable female mathematicians working for Nasa during the Space Race – a time when human calculations were more trusted than computers.

When the term “computer” was first used in the 17th century it referred to a human who performed mathematical calculations following fixed rules. Three hundred years later, in 1969 – at the height of the Space Race – electronic computers were in their infancy and Nasa still relied on “human computers” to perform advanced calculations because the new gadgets were simply not trusted to come up with the right answers.

The obvious need for reliability and safety on space missions meant that Nasa preferred to use proven methods and techniques. They relied on their pool of women mathematicians, who transcribed raw data from celluloid film and oscillograph paper and then used slide rules and electric calculators to turn it into standard engineering units. New film Hidden Figures follows the true story of three of these women and the incredible impact they had on American space exploration.

On manned flights, one wrong figure could cost lives, not to mention millions of dollars and political embarrassment on a global scale. For this reason leading-edge computing technology was slowly and carefully introduced – at first used only in ground systems, then occasionally on unmanned flights that allowed for greater chances to be taken.

When John Glenn was preparing to become the first American to orbit Earth in 1962, the electronic computer plotted his coordinates – but he asked for “the girl” – meaning Katherine G Johnson – to check the numbers. The fact he was more trusting of a woman than a machine was particularly notable, because in the same year he helped to shut down a privately-funded Women in Space programme by arguing that “the fact women are not in this field is a fact of our social order”. It was clearly a case of baby steps rather than giant leaps when it came to gender and space.

As computing progressed through the 1960s, Nasa began to trust these machines more and widely [], encouraging the creation of new technology for its ground systems and finding new ways to adapt proven equipment for use on unmanned space flights.

The biggest challenge they faced was creating a computer system that could survive the stress of a rocket launch and was able to operate in space.

In 1969, the Apollo’s Guidance Computer (AGC) was no more powerful than a pocket calculator, yet it successfully navigated Neil Armstrong’s crew across 223,693 miles of space. Although the astronauts would have preferred to fly the craft manually, only the computerised system could provide the level of accuracy and control needed to succeed.

Video: Watch | Journalist Elena Poniatowska on the inspirational Katherine G Johnson

The programme demonstrated that computers could be entrusted with human lives, and today both manned and unmanned spacecraft contain many computers that they could not function without.

Hidden Figures – Genius has no race. Strength has no gender
To celebrate the release of new film Hidden Figures, the Telegraph has created a wealth of fascinating articles about Nasa, the Space Race, the Cold War, and society in the sixties as well as the incredible individuals who fought against the mores of the time challenging boundaries around race and gender. Check out to tgr.ph/hiddenfigures now

Hidden Figures tells the incredible real-life story of three black female mathematicians who fought against segregation, discrimination and sexism to work and excel at Nasa during the Space Race – making and changing history in the process.

Starring an incredible cast including Taraji P Henderson, Octavia Spencer, Janelle Monae, Kevin Costner and Kirsten Dunst, Hidden Figures is in cinemas from February 17. To find out more and go behind-the-scenes of the film, go to hiddenfigurestickets.co.uk.

etc...

http://www.telegraph.co.uk/films/hi...tm_campaign=tmgspk_listfour_1633_AnRs3YHBTx9k





 
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