Sunday, January 27, 2013

Math Literacy: So Vital in Day-to-Day Business

A close relative to science literacy is math literacy - since good science often relies on doing the math right.

In the Richard Feynman collection of essays "Surely, You're Joking, Mr. Feynman!" is the chapter "Lucky Numbers" (the story is also available here "Feynman vs. The Abacus")

Feynman tells the story of his encounter with a man trying to sell abacuses at a restaurant and challenging customers to competitions of computation speed.  The waiters suggest the salesman challenge Feynman with his mathematical problems, which he does and Feynman accepts. The first problems are simple, and the salesman wins handily.  But as the problems become more complex, the salesman gets slower.  Eventually, Feynman wins on a more complex problem, computing the cube root of a number.  As Feynman explains - the salesman knew his tool (the abacus) but Feynman knew numbers.  A variant of this story appears in the Feynman biopic, "Infinity" (Wikipedia).

I've had a number of interesting experiences over my professional life which further illustrate the importance of math literacy, or math competence.

One of my early jobs was doing bookkeeping at a large tomato packing firm.  I was employed there my last years of high school and the job funded my first round of college education.  This was back in the day when small business computers were just coming on the scene and personal computers were looming on the horizon.  One of the company's accountants was trying to estimate the total amount paid by the company in social security wages (i.e. wages subject to social security tax).  The numbers readily available from the accounting system of the day were the total amount paid in wages,  (which included wages with and without social security tax deducted), the total amount paid in social security taxes, and the tax rate.

For those readers outside the U.S. or unfamiliar with the Social Security system in the United States (Wikipedia), Social Security taxes are deducted from paychecks at a fixed percentage, say SSrate, of a persons gross salary, Salary, up to some limiting amount, SalaryLimit, above which Social Security tax is not deducted.  Back in the 1970s,  employer contribution was equal to the employee contribution.

Now at the company I worked for, most of the employees paid Social Security taxes on their entire paycheck.   Only a few employees had annual pay that accumulated above the limit, SalaryLimit, to be exempt from the tax.  The accountant wanted to avoid computing the quantity of interest by adding up all the weekly employee paychecks for the year.

Now to me, the solution was obvious.  Since

SStax = SSrate * Salary

The total amount paid in Social Security is just the sum of SStax for each employee.  The tricky part comes in including employees who had part of their wages above the Social Security tax limit, SalaryLimit.

For anyone familiar with algebra, the obvious solution was that the total amount paid in SS wages was

 total amount paid in SS wages = (total paid in SS taxes)/(SSrate)

I suggested this to the accountant and he looked at me surprised. Had this obvious (to me) solution actually not occurred to him?  He ran his numbers through my suggestion and seemed satisfied.

Now this dollar amount computed this way will probably not be exact to the penny.  Because each individual tax computation is rounded to the nearest penny, adding them up for the total amount of the payroll propagates the rounding errors into the total.  Ideally, for an organization with a large number of employees with different pay rates, you round up about as many times as you round down, so the total should be close, but odds are it will not be exact.  This gets to Feynman's next point of his story - the power of approximation methods.

Why Does It Matter?
The point is that mathematical skills are valuable.  While many of the cranks whose science I discuss on this blog often try to dismiss the mathematics, it nonetheless can accurately describe how things work in the real world when the understanding of the fundamentals is correct.

In addition, scientific and mathematical literacy have applications far beyond the ability to do research.  As I demonstrate above, it has applications in day-to-day business.  It has applications in business budgets and inventory control, which requires us to understand flows of material and money to produce and distribute a successful product.  I have known a lot of people in businesses who fail not because they don't have a good product, or can't sell their product, but because they can't keep track of these fundamentals.

Exercise for Readers:
  • Assuming the accounting software did proper rounding, how would you calculate the largest possible error?  
  • How about the most probable error?

Additional Resources:
  • Wikipedia: Approximation
  • Enrico Fermi had a reputation for estimating quantities to determine if they were important or relevant to experiments or even everyday life (Wikipedia).  A collection of these type of "Fermi Problems" is maintained at the University of Maryland

Sunday, January 20, 2013

Explorations around the Web

Skeptoid Podcast
This is a collection of 10-15 minute podcasts on a wide variety of skeptical topics.  I found it useful for a quick introduction to a variety of topics which I've heard about over the years but never really researched.  There are many segments on the various flavors of creationism, scientific thinking, climate change, etc.  Periodically he posts a corrections and responses to comments from listeners.

The Scale of the Universe
The original Powers of 10 (released in 1977, YouTube) video illustrated the broad range of size scales in the universe, ranging from the sub-atomic realm to the edge of the universe.  An interactive version of Powers of 10,  going from our everyday sizes to larger and smaller is now available.

Heavenly Errors: Misconceptions about the real nature of the universe
by Neil F. Comins
A collection of questions and answers on misconceptions in astronomy

Hubble Sees Tribe of Baby Galaxies 13+ Billion Light Years Away
The Hubble Space Telescope takes yet another look at the ultra-deep field.

Is there a rising tide of irrationality?
Irrational behavior hasn't changed that much over the history of the human race, and odds are it will not change much in the future. 

The Flame Challenge
Ever wonder about the science behind something that seems so simple as a burning flame?  This was a very good explanation and I learned about some details of the physics I'd not heard before.

Wired: 8 Videogames to Get Your Kid Into Engineering 
I've tinkered with the "World of Goo" demo and the Kerbal Space Program looks interesting but there's not enough hours in the day...

Sunday, January 13, 2013

Is Big Bang Cosmology a 'Creationist' Model?

I've recently received a couple of comments of late, falling on both sides of the claim that the Big Bang is an explicitly "Creationist" model.   (Commenter1) claims this is a good thing. (Commenter2) regards it as evidence that Big Bang cosmology must be wrong.  It seemed like it was time to create a full post on the topic.  I will take the position that Big Bang cosmology is not a 'Creationist' model that depends on the religious views of the developers.

Both sides claim that the issue of cosmology is a question of Worldview (Wikipedia), and that somehow the model of the universe will be different if the writer has a different 'worldview'.  Most Young-Earth creationists, such as Barry Setterfield, oppose Big Bang cosmology, but some Old-Earth Creationists (such as those at Reasons To Believe) accept Big Bang cosmology as compatible with their theological beliefs.  Others from different philosophical camps hold a variety of views.

This is an hilarious quote from Commenter2:
"Lemaitre introduced a creation event into the equations of General Relativity and hence infused physics with the notion of God and His creation of the Universe."
This is a bizarre notion, as the equations for the FLRW metric (Wikipedia) are derived mathematically, starting from the assumptions of homogeneity (the universe, on large scales, has the same density everywhere) and isotropy (the universe should look the same in all directions from whatever point an observers looks).  The most general form includes a scale factor that can vary with time.  These are actually more general versions of the Copernican principle (Wikipedia). 

Kepler was an astrologer and was inspired by concepts of platonic solids.  Did his examination of planetary motion infuse physics with astrological concepts?  Does that invalidate Kepler's Laws?  What about the impact of Isaac Newton's supernatural beliefs (AIP) on his physics?  According to Commentor2's thinking, it must.  So does this person also believe spaceflight is a fraud, since it depends so much on these laws and their related phenomena?

How can mathematical operations be dependent on the religious beliefs of the writer?  Derivations of the FLRW metric from first principles are available in a variety of references.  Which of the two input assumptions carry the religious belief?  Which mathematical operation: addition, subtraction, multiplication, or division, is faith-based? Which terms in the equations represent this religious belief? 

You could change the input assumptions, such as examining metrics with non-uniform densities or which are not isotropic (have a preferred location).  Metrics with other characteristics have been explored, and found wanting (Wikipedia: Alternative Metric Cosmologies).  Beyond that, you could design a metric with the characteristics needed for Biblical Geocentrism and various flavors of relativity denial (see Global Positioning System).

Is Commentor2 saying the cosmological metric equations would be different if a non-religious person wrote them?  If so, then perhaps the Commentor2 should present a derivation of the the appropriate metric.

Many people were inspired by things that lead to correct results but the source of the inspiration was a faulty analogy.  William Harvey was inspired by ideas of planetary orbits when discovering circulation of blood. (Scientific Myth-Conceptions, Douglas Allchin, 2003).  The analogy was incorrect, but it inspired a correct result. 

The solutions to the Einstein field equations exist, regardless of the religious belief of the advocate.  Note that the metric was independently discovered by four people in time frame of the late 1920s to early 1930s (Wikipedia: Friedmann–Lemaître–Robertson–Walker metric).

Were all of them motivated by religious beliefs?  I find no conclusive evidence that any of the others expressed religious motivations for developing this metric.
Lemaitre's View

Lemaitre opposed the Church's attempts to claim that Big Bang cosmology validated Biblical Genesis.
  • "A Universe From Nothing", Lawrence M. Krauss (pp 5-6)  I write about this in my review of this book.
  • "The Day We Found the Universe", Marcia Bartusiak (pp 257-258)
  • "The Very First Light", John C. Mather & John Boslough,  footnote pg 40
  • "Cosmology & Controversy", Helge Kragh (pp 59-60)  'Lemaitre on Religion and Science'
Also consider this quote from Lemaitre from 1933:
"Hundreds of professional and amateur scientists actually believe the Bible pretends to teach science.  This is a good deal like assuming that there must be authentic religious dogma in the binomial theorem.  Nevertheless, a lot of otherwise intelligent and well-educated men do go on believing or at least acting on such a belief.  When they find the Bible's scientific references wrong, as they often are, they repudiate it utterly.  Should a priest reject relativity because it contains no authoritative exposition of the doctrine of the Trinity?"
This is pretty secular scientific thinking for a priest!

The bottom line is that there is loads of public evidence that Lemaitre opposed religious interpretations of Big Bang cosmology.  Lemaitre went up against the Pope to defend this position  - not a trivial thing for a priest to do!

If there is this much on the public record on Lemaitre's position, then what are we to make of the Alfven quote from Commenter2 which is apparently cut-n-pasted from Tony Peratt's web site?
Here is what Alfvén reported: “I was there when Abbe Georges Lemaitre first proposed this theory," he recalled. Lemaitre was, at the time, both a member of the Catholic hierarchy and an accomplished scientist. He said in private that this theory was a way to reconcile science with St. Thomas Aquinas' theological dictum of creatio ex nihilo or creation out of nothing.
Now the similar quote here does not contain the 'He said in private...' part.  And finally compare a similar reference in Eric Lerner's "The Big Bang Never Happened", pg 214:
"I felt at the time that the motivation for his theory was Lemaitre's need to reconcile his physics with the Church's doctrine of creation ex nihilo."
So Alfven's description goes from "I felt" this was what Lemaitre was doing, to "He said in private"?  This is suspicious on several fronts:
  1. Did Alfven misremember, turning a personal impression of what he thought Lemaitre believed into an actual conversation?  A similar question has been raised before about Alfven's recollections  concerning his contributions to the cosmic ray acceleration mechanism known today as Fermi acceleration (Wikipedia, see "Reading: Cosmical Electrodynamics")
  2. Why would Lemaitre make such a statement as attributed to him, apparently by Alfven, to individuals with possibly broad range of religious beliefs?  
  3. The differences with other references on this topic suggest the claim of what Lemaitre said in private was second- or third-hand, since it is outside quotation marks and could perhaps be outside of what Alfven actually said.
I suspect the most likely answer is a Plasma Cosmology supporter did a little historical embellishment in an attempt to bolster their case.

Einstein's View

Another snippet from Commentor2:
Einstein stood up applauded, and said, “This is the most beautiful and satisfactory explanation of creation to which I have ever listened.” So evidently Einstein was actually a creationist, revealing thereby that he too was actually theological in his real disposition, despite his often overt cryptic claims that he was not.
Einstein was an agnostic and this fact is far from 'cryptic'. 
Einstein even made public statements about religion that infuriated many in America to the point that they wanted him deported.  To say after all that that Einsten was really a closet 'Creationist' takes a real stretch of the imagination.

That the Commenter2 seeks to make their case by ignoring Einstein's well-documented religious views suggests Commenter2 may be blinded by their own dogmatic views rather than the evidence.

Notice the Einstein quote has 'creation', not 'Creation'.  What would you call an origin point for the cosmos if not the moment of 'creation'?  In quantum field theory (QFT), we speak of the creation and annihilation of particles.  Does that make QFT a 'Creationist' model?  The religious connotations are more likely in the mind of the reader. 

The notion that Einstein was a christian seems to be pushed by a lot of religious types, in spite of facts to the contrary (Snopes: Religion/Einstein).

Considering how many people wanted to run off einstein for his religious views, claiming that he was some kind of closet Christian "Creationist" (with a capital 'C') looks a little strange


I summarize with the quote from to Alfven from Commenter2:
“There is no rational reason to doubt that the universe has existed indefinitely, for an infinite time,"
This is in itself a human prejudice based more on a philosophy that Nature must conform to human worldviews - a position more religious than scientific.

Historically, Nature has demonstrated it is not constrained by human worldviews.  Quantum mechanics, relativity, etc. all flew in the face of human preconceived notions of how Nature actually worked.  In many cases, great problems in science were solved by a person with a compatible prejudice at the right time who had the right insight to solve it.

Commentor1 is incorrect because the FLRW metric is a mathematical consequence which could just as easily been discovered by someone with an atheistic 'worldview'.  Even Lemaitre supported this view.  Others derived this metric independently.

Commenter2 is incorrect because not only does it require that we reject the fact that the FLRW metric is a consequence of certain physical and mathematical assumptions, it also requires we reject all other evidence of Lemaitre & Einstein's public position (some taken at great professional and personal risk) and believe they were actually supporting some other secret agenda.  Commenter2's approach is not to prove their position, but to discredit an opposing position.  Commenter2 seems obsessed with conspiratorial thinking (RationalWiki).

Finally, to really make any kind of objective evidence in this type of claim, one must be able to answer the question: Where is the 'religious component' in the FLRW metric?  How would this metric

be different if the 'religious components' were removed?  If you can't answer that basic question, then the question becomes pointless and simply a matter personal prejudices.

To bump the silly meter up a notch, I'll point out that a Biblical Geocentrist who comments on this blog has some comments pending claiming that the FLRW metric is false because it denies the Earth as the 'Center of the Universe' contrary to their religious interpretation!

"Everyone is entitled to his own opinion, but not to his own facts."
Daniel Patrick Moynihan

Sunday, January 6, 2013

My 2012 Reading list

I thought I'd post a quick summary of my 2012 reading.  Some of it has been discussed on this blog already, others are in the planning

Programming in Python 3 : a complete introduction to the Python language
Mark Summerfield
ISBN: 0-321-68056-1
Publisher: Addison-Wesley
I'll need to upgrade my Python installation at some point. Fortunately, it looks like the changes are not too severe.

A Universe from Nothing: Why There Is Something Rather than Nothing
Lawrence M. Krauss, Richard Dawkins
ISBN: 145162445X
Publisher: Free Press
I've written a review of this book here.

A Princess of Mars
Edgar Rice Burroughs
Publisher: Nelson Doubleday, Inc.
I re-read this book due to the John Carter movie coming out (Wikipedia). I've still not seen the movie!

Henry Norris Russell
David H. DeVorkin
ISBN: 0691049181
Publisher: Princeton University Press
A very good history of astronomy in the United States in the early 1900s, centered around the astronomer Henry Russell (wikipedia) whose discoveries established much of our modern observational understanding about stars.

Science Friction: Where the Known Meets the Unknown
Michael Shermer
ISBN: 0805077081
Publisher: Times Books

Story of Mathematics 
Ian Stewart
ISBN: 1847240178
Publisher: Quercus Books
A general audience book describing the development of mathematics and its importance in our modern society.

Steve Jobs 
Walter Isaacson
ISBN: 1451648537
Publisher: Simon & Schuster

Space Chronicles: Facing the Ultimate Frontier
Neil deGrasse Tyson, Avis Lang
ISBN: 0393082105
Publisher: W. W. Norton Company
This book is a collection of essays by Dr. Tyson covering many topics related to space and space exploration.  Dr. Tyson is about my age, so many of his recollections about the early history of manned space flight brought back a lot of memories.

Seeing Red: Redshifts, Cosmology and Academic Science
Halton C. Arp
ISBN: 0968368905
Publisher: C. Roy Keys Inc 
I've been exploring this book pretty heavily during my hiatus.  I expect to post a review in the near future, and a series of articles about the things Dr. Arp DOES NOT TELL READERS.

Flatland: A Romance of Many Dimensions
A. Square (Edwin A. Abbott)
I've never read this before but it was an interesting read (Wikipedia article)