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In addressing our current fiscal and economic woes, too often we neglect a key ingredient of our nation's economic future—the human capital produced by our K-12 school system. An improved education system would lead to a dramatically different future for the U.S., because educational outcomes strongly affect economic growth and the distribution of income.
Over the past half century, countries with higher math and science skills have grown faster than those with lower-skilled populations.
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fiscal and monetary policy are inextricably linked, and their research reflects the broad shift in economics from words to numbers — toward a level of empirical analysis that few outside the profession can readily grasp. But it contains a kernel of skepticism appropriate for these troubled times. In a world of uncertainty and constraint, cause and effect may not be what they seem. As a result, we must test and retest our assumptions — and try to prepare for the unexpected.
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“The most impressive thing about them as scholars,” says David Easley, an economist at Cornell University, “is that in recent years they have questioned the assumptions of the models they helped to create, and they have been at the vanguard of the efforts to go beyond them.”
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Submitted by mf344 on October 4, 2011
How many universes are there? What has made us into who we are? Is there absolute truth?
These are difficult questions, but mathematics has something to say about each of them. It can probe the physical reality that surrounds us, shed light on human interaction and psychology, and it answers, as well as raises, many of the philosophical questions our minds have allowed us to dream up.
On this page we bring together articles and podcasts that examine what mathematics can say about the nature of the reality we live in. They look at physical reality, the mind, consciousness, the emergence of life, philosophy and mathematics itself. The page will be continually updated with new relevant articles, so keep looking and get reading!
With countless biological details emerging from cancer experiments, there is a growing need for minimal mathematical models which simultaneously advance our understanding of single tumors and metastasis, provide patient-personalized predictions, whilst avoiding excessive hard-to-measure input parameters which complicate simulation, analysis and interpretation. Here we present a model built around a co-evolving resource network and cell population, yielding good agreement with primary tumors in a murine mammary cell line EMT6-HER2 model in BALB/c mice and with clinical metastasis data. Seeding data about the tumor and its vasculature from in vivo images, our model predicts corridors of future tumor growth behavior and intervention response. A scaling relation enables the estimation of a tumor's most likely evolution and pinpoints specific target sites to control growth. Our findings suggest that the clinically separate phenomena of individual tumor growth and metastasis can be viewed as mathematical copies of each other differentiated only by network structure.
One of the most difficult things about treating cancer is that each case is a very individual process; tumors recruit blood vessels in order to grow, yet also send out new vessels of their own. This harnessing of the body’s resources is what eventually allows a tumor to metastasize — be carried into other parts of the body where growth continues. On the other hand, sometimes tumors don’t grow at all. Predicting each patient’s unique response to cancer and its progression is a large part of the battle, as an accurate estimate is required to begin appropriate treatment. The field of Oncology is always seeing exciting developments, and this latest one is no different — its benefits touch future and existing cancer patients, alike. Knowing that hindsight is 20/20, it would be a lot easier if there was a “fast-forward button” with which doctors could view each unique case before it develops.
Physicist Sehyo Choe and colleagues at the University of Heidelberg, Germany have developed such a button in the form of a mathematical model. By inputting data about the tumor and its current location of blood vessels, the model allows doctors and researchers to see how the tumor will grow and move — if at all — giving them a very accurate helping hand when it comes to prompt and accurate treatment. Tested on mice, the model accurately predicted the progression of all cancer-stricken subjects, giving researchers that amazing “fast-forward” capability.
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Predicting each patient’s unique response to cancer and its progression is a large part of the battle, as an accurate estimate is required to begin appropriate treatment. The field of Oncology is always seeing exciting developments, and this latest one is no different — its benefits touch future and existing cancer patients, alike. Knowing that hindsight is 20/20, it would be a lot easier if there was a “fast-forward button” with which doctors could view each unique case before it develops.
Physicist Sehyo Choe and colleagues at the University of Heidelberg, Germany have developed such a button in the form of a mathematical model. By inputting data about the tumor and its current location of blood vessels, the model allows doctors and researchers to see how the tumor will grow and move — if at all — giving them a very accurate helping hand when it comes to prompt and accurate treatment. Tested on mice, the model accurately predicted the progression of all cancer-stricken subjects, giving researchers that amazing “fast-forward” capability.
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Co-author Neil Johnson from the University of Miami says that “in the future, treatments will no longer have to be based on population averages. People will get individual treatment based on the predictions of our model.”
a group of mathematicians from the British Society for the History of Mathematics have collected some examples of the unplanned impact of maths, which are reported in the 14th July issue of the journal Nature. Peter Rowlett, who coordinated the collection, said, "Although most mathematicians know that mathematics has this surprising nature, many that I have spoken to aren't aware of more than one or two specific examples. I thought the British Society for the History of Mathematics could help by searching through history for examples that are less well known. We hope this collection will only be the start and that more mathematicians will send their favourite stories to us."
Risk is a very complex topic, since it's all about things we can't predict, which just about includes everything. Many aspects of risk are studied by researchers all over the world. In the Big Risk Test, which is now live as part of BBC Lab UK, we want to find out how people deal with risk, particularly to try and understand what makes people have such different opinions and feelings about life's chances.
How would it feel to look in a mirror and see not your own reflection but instead how you would look as the opposite sex? You can explore this strange alternate reality at this year's Royal Society Summer Science Exhibition where Peter McOwan and his colleagues from Queen Mary, University of London and University College London will use mathematical wizardry to produce gender reversed images of faces.
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Despite the crisp clear details and blended colours, a photo is just a series of dots, called pixels, of different colours. Any photo, including one of your own lovely visage, is represented in a computer as a long string of numbers, each representing the colour at a particular pixel. Just as a string of three numbers marks a point in three-dimensional space (it gives its coordinates), so a string of N numbers sits in what mathematicians think of as N-dimensional space: so mathematically you can think of a photo as a point in an N-dimensional space, where N is the number of pixels in the photo.
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First of all the researchers calculate the average face by simply averaging the values at each pixel over all the photos in the set. This new string of numbers represents the average face and the position of the points representing the other faces show how they differ from the average face.
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The good Christian should beware the mathematician and all those who make empty prophecies. The danger already exists that the mathematician have made a covenent with the devil to darken the spirit and to confine man in the bonds of hell. – SAINT Augustine.
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Math isn’t real: “As far as the law of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” - Albert Einstein.
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The hot hand is a term used in basketball to describe a player making a series of successful throws at the basket, i.e., getting “hot”. But it seems that we, as humans, are too fast at jumping at the conclusion that someone is “hot” just because we saw some impressive streak of successes in a series of trials.
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I regularly play in my introductory statistics classes, geared at answering how long a run of consecutive successes should be in order to make us suspicious that something “hot” is going on. I tell my students that after I leave the classroom (a dangerous thing to do, but things usually work out), some of them should take out a coin and toss it 25 times (50 or 100 times would be much better, but who has time for that?) to obtain a random sequence of heads and tails. Some other students should just imagine throwing a fair coin and make up a sequence of 25 tosses in their mind. Each student then writes their sequence somewhere on the blackboard.
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How much math do you really need in everyday life? Ask yourself that -- and also the next 10 people you meet, say, your plumber, your lawyer, your grocer, your mechanic, your physician or even a math teacher.
Unlike literature, history, politics and music, math has little relevance to everyday life. That courses such as "Quantitative Reasoning" improve critical thinking is an unsubstantiated myth. All the mathematics one needs in real life can be learned in early years without much fuss. Most adults have no contact with math at work, nor do they curl up with an algebra book for relaxation.
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Those who do love math and science have been doing very well. Our graduate schools are the best in the world. This "nation at risk" has produced about 140 Nobel laureates since 1983 (about as many as before 1983).
As for the rest, there is no obligation to love math any more than grammar, composition, curfew or washing up after dinner. Why create a need to make it palatable to all and spend taxpayers' money on pointless endeavors without demonstrable results or accountability?
Mathematical models and philosophical progress.
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"Hannes Leitgeib: Overall, and ultimately, mathematical methods are necessary for philosophical progress, yes. But of course there can be points in a philosophical argumentation at which there is no payoff applying such methods. And while I do not think that there is any area of philosophy that is ‘beyond mathematical methods’, in some areas they do not pay off as yet because these areas are not quite developed enough. Or that’s at least the diagnosis of a mathematical philosopher!"
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let's accept that philosophical progress by mathematical methods ought to be understood in terms of "clarity" (as Leitgeib seems to suggest in the interview). Ought we to accept that it is cost-free?
Here are some possible costs within philosophy (I created the list while thinking of the role of Bayesianism as an aid to understanding scientific practice in the fields I am familiar with):
1. Focus on tractable machinery and toy-examples (at expense of complexity)
2. Training in technical skill at expense of good judgment
3. Inflated expectations from technique rather than learning how to ask right questions (or the making of distinctions)
4. Focus on producing 'results' rather than insight
5. Focus on the model and not the messy world
Pigeons outperform humans at the Monty Hall Dilemma
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Imagine that you’re in a game show and your host shows you three doors. Behind one of them is a shiny car and behind the others are far less lustrous goats. You pick one of the doors and get whatever lies within. After making your choice, your host opens one of the other two doors, which inevitably reveals a goat. He then asks you if you want to stick with your original pick, or swap to the other remaining door. What do you do?
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Most people think that it doesn’t make a difference and they tend to stick with their first pick. With two doors left, you should have even odds of selecting the one with the car. If you agree with this reasoning, then you have just fallen foul of one of the most infamous of mathematical problems – the Monty Hall Dilemma. In reality, you should actually swap every time – doing so means double the odds of getting the car.
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April 4, 2010, 5:00 PM
Take It to the Limit
By STEVEN STROGATZ
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if you keep moving halfway to the wall, will you ever get there? Something about this one was deeply frustrating, the thought of getting closer and closer and yet never quite making it.
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Around 500 B.C., Zeno of Elea posed a set of paradoxes about infinity that puzzled generations of philosophers, and that may have been partly to blame for its banishment from mathematics for centuries to come. In Euclidean geometry, for example, the only constructions allowed were those that involved a finite number of steps. The infinite was considered too ineffable, too unfathomable, and too hard to make logically rigorous.
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Power Tools: In math, the function of functions is to transform.
A mathematician needs functions for the same reason that a builder needs hammers and drills. Tools transform things. So do functions. In fact, mathematicians often refer to them as “transformations” because of this. But instead of wood or steel, functions pound away on numbers and shapes and, sometimes, even on other functions.
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Take It To The Limit -- Archimedes recognized the power of the infinite, and in the process laid the groundwork for calculus.
The key to thinking mathematically about curved shapes is to pretend they’re made up of lots of little straight pieces. That’s not really true, but it works … as long as you take it to the limit and imagine infinitely many pieces, each infinitesimally small. That’s the crucial idea behind all of calculus.
Following the excellent tradition he has established in this series, Strogatz then goes on to provide an intuitive proof for the well known relation between a circle's area (A) and its radius (r): A = π r2.
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Weiye Loh on 2010-04-05What's the difference between Archimedes' infinitesimals and the logical extremes of post modernism?
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