The first time I entered the math library at Lebanon Valley College, I was struck by what I saw on top of the bookcases – a giant slide rule! Though I had never used one, I remembered my dad telling me about how he had to use a slide rule in his math classes in college. This iconic piece of mathematical technology owes its existence to the mathematical development that is celebrating its 400th birthday this year – the invention of logarithms.
I recently came across a link to a Science News article by Tom Siegfried entitled Logarithms celebrate their 400th birthday. The article discusses how John Napier revolutionized mathematics in 1614 with his invention of logarithms. Though many considered the logarithm a godsend, not everyone was convinced.
Napier’s mathematical wizardry wasn’t universally appreciated, though, as rumors swirled that he was actually a dark wizard, à la Lord Voldemort. For one thing, Napier’s grass seemed to be greener than other landowners. And he allegedly trained a magical black rooster to identify thieves among his workers.
The article also discusses how the invention of logarithms led to the development of the slide rule.
It wasn’t long until others figured out how to put the logarithms to use in mechanical calculations using sticks. Inscribing numbers on the sticks at intervals proportional to their logarithms made it possible to multiply numbers by proper positioning of the sticks. Edmund Gunter, a London clergyman and friend of Briggs, had the germ of the idea in 1620. But the honor of first to slide the sticks is usually accorded to William Oughtred, an Episcopal minister, who also devised a circular version of the slide rule in the 1620s.
The invention of logarithms introduced a new way for quickly performing complicated calculations, with one mathematician claiming that “logarithms effectively doubled a mathematician’s useful lifetime.” Without the tools that logarithms offer the sciences today, our understanding of the world would be greatly limited.
]]>Is Mathematics an art or a science? Calvin has a different perspective. Hmm…Calvin & I might need to converse. #ijs pic.twitter.com/99zx8nB6op
— Karen Morgan Ivy (@Afrikanbeat) March 7, 2014
(Transcription below by http://blog.onbeing.org/post/250746172/calvin-and-hobbes-math-is-a-religion)
First frame
Calvin: You know, I don’t think math is a science. I think it’s a religion.
Hobbes: A religion?
Second frame
Calvin: Yeah. All these equations are like miracles. You take two numbers and when you add them, they magically become one new number! No one can say how it happens. You either believe it or you don’t.
Third frame
Calvin: This whole book is full of things that have to be accepted on faith! It’s a religion!
Fourth frame
Hobbes: And in the public schools no less. Call a lawyer.
Calvin: As a math atheist, I should be excused from this.
Related to this, I recently saw a link on my Facebook page to an article on NYTimes.com by Elizabeth Green entitled Why Do Americans Stink at Math? The article discusses a Japanese teacher who has tried revolutionizing mathematics pedagogy.
Instead of having students memorize and then practice endless lists of equations — which Takahashi remembered from his own days in school — Matsuyama taught his college students to encourage passionate discussions among children so they would come to uncover math’s procedures, properties and proofs for themselves.
The article also talks about how the American mathematics teaching practices have seen several failed reform attempts in the past.
It wasn’t the first time that Americans had dreamed up a better way to teach math and then failed to implement it. The same pattern played out in the 1960s, when schools gripped by a post-Sputnik inferiority complex unveiled an ambitious “new math,” only to find, a few years later, that nothing actually changed. In fact, efforts to introduce a better way of teaching math stretch back to the 1800s. The story is the same every time: a big, excited push, followed by mass confusion and then a return to conventional practices.
How can we, as mathematicians, work to ensure new reform (we are in the midst of the Common Core) actually works?
]]>This weekend, I was helping paint flats for a play when an interesting problem arose – we wanted to use three colors of paint to create rectangles of different sizes on a rectangular flat, with the stipulation that no two adjacent rectangles were the same color. Prior to painting, we labeled the rectangles with their colors just to make sure everything worked out. We didn’t run into any problems, but one person did ask the question – was it possible to come up with a configuration of rectangles that was impossible to paint with our condition? Since we were only using three colors of paint, the answer, of course, was yes! In fact, when making up sample designs the night before, our director ran into the issue of three colors not being enough.
In mathematics, the Four Color Theorem (or Map Coloring Problem) states that, given any separation of a plane into contiguous regions (producing a figure we will call a map), no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. For this problem, two regions are considered adjacent if they share a boundary that is not just a corner. It seems like a simple problem – given a map, how many colors do you need so that no two adjacent regions are the same color? However, this problem, so simply stated in 1852, was tantalizingly difficult to prove, leading to many false proofs and false counterexamples. It was not until 1976 that, with the aid of a computer, a proof was obtained!
The Four Color Problem was first formulated by Francis Guthrie in 1852 when he was coloring a map of the counties of England and noticed that four colors sufficed. He asked his brother Frederick if it was possible to color any map using only four colors while also requiring that adjacent regions (i.e. those sharing a boarder) be of different colors. Frederick, who at this time was a student of Augustus De Morgan, posed the problem to his advisor. De Morgan, in turn, communicated the proposition to his friend William Rowan Hamilton, stating:
“A student of mine asked me to day to give him a reason for a fact which I did not know was a fact — and do not yet. He says that if a figure be any how divided and the compartments differently colored so that figures with any portion of common boundary line are differently colored — four colors may be wanted but not more — the following is his case in which four colors are wanted. Query cannot a necessity for five or more be invented… “
There were several early attempts to prove this theorem, though all of them failed. The first widely accepted proof was given by Alfred Kempe in 1879, with another proof given independently by Peter Guthrie Tait in 1880. It wasn’t until 11 years later that Kempe’s “proof” was shown to be incorrect by Percy Heawood. In 1890, in addition to exposing the flaw in Kempe’s proof, Heawood proved the Five Color Theorem and generalized the Four Color Conjecture to surfaces of arbitrary genus. In 1891, Tait’s proof was shown to be incorrect by Julius Petersen. Though both early proofs turned out to be fallacious, they were not without value. Kempe discovered what became known as Kempe chains, while Tait found an equivalent formulation of the Four Color Theorem in terms of 3-edge colorings.
During the 1960s and 1970s, German mathematician Heinrich Heesch developed methods using computers to search for a proof. Unfortunately, he was unable to procure the necessary supercomputer time to continue his work. However, others adapted his methods and computer-assisted approach.
The Four Color Theorem was finally proven in 1976 by Kenneth Appel and Wolfgang Haken, with some assistance from John A. Koch on the algorithmic work. This was the first time that a computer was used to aid in the proof of a major theorem. The Appel-Haken proof began as a proof by contradiction. If the Four Color Theorem was false, there would have to be at least one map with the smallest possible number of regions that requires five colors. Two technical concepts were used in the proof:
1. An unavoidable set is a set of configurations such that every map that satisfies some necessary conditions for being a minimal non-4-colorable triangulation (such as having minimum degree 5) must have at least one configuration from this set.
2. A reducible configuration is an arrangement of countries that cannot occur in a minimal counterexample. That is, if a map contains a reducible configuration, then the map can be reduced to a smaller map (using fewer countries). If this smaller map can be colored using only four colors, then the original (larger) map can also be colored using only four colors. This implies that if the original (larger) map cannot be colored with only four colors, then the smaller map cannot be colored with only four colors either. Thus, the original map could not be a minimal counterexample.
Using mathematical rules and procedures based on the properties of reducible configurations, Appel and Haken found an unavoidable set of reducible configurations, thus proving that a minimal counterexample to the Four Color Theorem could not exist. Instead of examining all possible map configurations (an impossible task), the Appel-Haken proof was able to reduce the problem to looking at a particular set of 1,936 maps. Each of these maps, it was shown, could not be part of a smallest-sized counterexample to the Four Color Theorem. Since checking all of these maps by hand would be tedious and time consuming, Appel and Haken used a special-purpose computer program to confirm that each of the maps had the desired properties. Checking the 1,936 maps one by one took over one thousand computer hours. The reducibility part of this work was independently checked by different programs and different computers. However, the unavoidability portion of the proof needed to be verified by hand, leading to hundreds of pages of analysis. The Appel-Haken proof concluded that no smallest counterexample exists because it must contain, yet cannot contain, one of the specially chosen 1,936 maps. This contradiction shows that there cannot be any counterexamples and therefore the Four Color Theorem must be true.
Initially, this long-awaited proof of the Four Color Theorem was met with trepidation and concern. It was the first major theorem to be proven with extensive computer assistance. In addition, even the human-verifiable portions of the proof were highly complex and prone to error. These two facts caused many mathematicians to reject the proof and led to considerable controversy. In the years that followed the original publication of the proof, it has become more widely accepted, although doubts still remain.
To this day, many mathematicians and philosophers debate whether or not the Appel-Haken proof is legitimate. Some claim that “proofs” should only be accepted if they are proved by people, not machines, while others question the reliability of both the algorithms used and the ability of machines to carry them out without error. Others argue that even proofs written by people can be found to be faulty and so the reliability argument is meaningless. No matter what your opinion on the matter is, the Appel-Haken proof of the Four Color Theorem has led to a serious discussion about the nature of proof which continues today and will continue into the future as we make use of computers that are growing ever more powerful.
Sources:
Appel, Kenneth; Haken, Wolfgang (1989), Every Planar Map is Four-Colorable, Providence, RI: American Mathematical Society, ISBN 0-8218-5103-9
Wilson, Robin (2002), Four Colors Suffice, London: Penguin Books, ISBN 0-691-11533-8
]]>
To apply and see all qualifications, visit https://www.mathprograms.org/db/programs/280. The deadline for applying is July 18, 2014 at 11:59 p.m. Eastern Time.
]]>
Assuming you’re able to secure a quorum, summer is the perfect time to begin a new student organization as many due dates occur at the beginning of the academic year. You will need a faculty advisor and five students, some willing to serve as officers. The petition process consists of two basic steps: establishing rules of procedure and writing an annual budget. More thorough instructions for the petition are detailed by the AMS. You should also check with your university about establishing an official student organization. At UCSB, clubs must register with the Office of Student Life at the beginning of the year. This provides your chapter with a bank account and procedures for disbursement of funds.
Where to find initial members? Instead of starting our SIAM chapter from scratch, we took the existing graduate student seminar in applied mathematics and brought it under the SIAM umbrella, giving us an instant pool of members. We found this helpful to build momentum for our first year. Additionally, if included in your budget proposal, then you will be able to enjoy coffee and cookies at your usual seminar! Which brings us to the next topic.
To write an annual budget you need activities. If you followed the previous advice, you already have one item to include in your budget. You may also consider:
The AMS provides student chapters up to $500 per academic year, and an additional $500 for “large groups with extensive activities”.
What do I mean by a themed colloquium? Here are two examples. First, think of all the interesting facts about the sphere! Why not host a graduate student colloquium focused on S^n? As someone who studies nonlinear partial differential equations, I would talk about Fourier analysis on the sphere and its relation to representation theory, or how spherical geometry is used when studying hyperbolic PDEs with finite speed of propagation. Each week one graduate student would present a topic in their focus area. The second example comes from the UCSB student chapter of SIAM. We are planning a lecture series this fall centered around the Top 10 Algorithms of the 20th Century. We’re currently in the process of finding graduate students from around campus to speak about their favorite algorithm at an advanced undergraduate level.
When writing your proposal, target more than the AMS budget document. Are there other funding sources on your campus? At UCSB, the Graduate Student Association offers clubs up to $600 per year directed towards events which provide “a means for graduate students to enrich their educational experience”. When you find a new source, make note of their deadlines and guidelines. Finally, after you have secured funding, you will need advertising copy for flyers, emails and the chapter website. I believe it is best to write a proposal which attempts to address all of these goals and later specialize it for each task.
I will close with a few tips for advertising your chapter events.
Do you already run a student chapter of AMS or another mathematical society? Please share your suggestions for events and advice for beginning a new chapter.
]]>Most folks have heard about Netflix, an internet video streaming website for users to be able to watch movies, TV shows, documentaries, etc. What about reading books, though. We have also heard about the Nook or Amazon Fire. But what if you want to have a service like Netflix for your books? (I’ve never used Amazon Fire, so I will stick with the Nook here.) The Nook has free books you can download, but often, I find them to be not very well written. The solution I found was Oyster. This is an app for i-Devices (iPhone, iPad, iPod). The fee for unlimited books is $9.99/month. It has more than 500,000 titles including many best-sellers to choose from including Stephen King, Dan Brown, John Irving, etc.
I am currently reading Stephen King’s 11/22/63, but I have several more books on my reading list. I have been able to find several math-related books. Here is a list (in no particular order) of the ones I have added to my reading list:
If you are an Oyster user, what math books have you found? If you are not yet an Oyster user, you can try a free trial. Let me know what you think about it and what you find!
]]>It is with great pleasure that I introduce Avery Carr as the new Senior Editor. Avery is a graduate student at Emporia State University. He completed his BS in Mathematics at the University of Memphis in 2010. Please join me in welcoming Avery to his new position.
]]>Between the light and darkness of mathematical knowledge exists an ever-extending boundary that pushes the limits of abstraction into the framework of tangible existence. What can and cannot be known converges with each symbol in a collection of coherent logic. Step after step of pure reasoning reveals an underlying component connecting complexity with simplicity. Oftentimes, this sought after link is not easily discovered. This is immediately true for the ever-growing list of proposed problems that span the entire spectrum of mathematics.
This notion also holds true for problems and puzzles that pose a more recreational nature. I was introduced to one such puzzle during a talk given by the English mathematician Oliver Riordan during my undergrad years at the University of Memphis. Riordan was presenting the methods he and his colleague, mathematician Alex Selby, used to solve the first version of the conundrum known as the Eternity Puzzle. In late 2000, with a combination of intuition, combinatorial reasoning, and statistical inference, Riordan and Selby were able to place all 209 irregularly fashioned pieces of the puzzle in a dodecagon shaped board and claimed the £1 million prize from the puzzle’s inventor, Lord Christopher Monckton.
This was somewhat of a shock since the Eternity Puzzle has an estimated combinations and it would take anyone using brute force computation longer than the entire age of the Universe to compute all possible combinations in hopes of finding a solution. However, mathematics is an employable tool often used to supplant computational complexity. Such was the case with the solution to the Eternity Puzzle.
With the success of selling over 500,000 copies of the first puzzle, Monckton recruited Riordan and Selby to help invent a second version of the puzzle known as the Eternity II Puzzle. Unlike the first version, the Eternity II is very much unsolved. The only people that know of a solution are its creators. Also, unlike the first version, there are 256 square pieces with designs in four directions such that any solution on the 16 x 16 gridded board must preserve a pairwise edge-matching in every direction (see above picture). From its release in 2007, a $2 million prize was offered for its solution that has since expired in 2010.
When I graduated from the University of Memphis, some administrators in the math department gave me one of the copies of the puzzle that Riordan brought with him for the talk. On several occasions I have attempted to solve it with no success. Maybe one day a solution will emerge from the depths of complexity into the realm of simplicity.
]]>Over the Memorial Day weekend, while visiting the Toledo Botanical Garden in Toledo, OH I came across this “Knot Garden” in the herb section of the park. I was pleasantly surprised to find this mathematical influence and it prompted me to wonder: what mathematical objects have you come across in your everyday life?
]]>The axioms of Presburger arithmetic are essentially a subset of the Peano axioms, excluding those pertaining to multiplication. For our purposes, it suffices to know that Presburger arithmetic deals with the nonnegative integers with standard definitions for equality, succession, and addition. Most of the usual properties of integer arithmetic can be built up from those definitions—for example, “less than” is a definable relation in Presburger arithmetic. It is also permissible to define constant multiplication for a given integer—that is, we are allowed to define the operation 3 * x because it is essentially shorthand for adding the term x to itself a specified number of times. However, we may not define x * y, because both terms are variables and this introduces a problematic ambiguity.
Now that we know what the ground rules are, what makes Presburger arithmetic interesting? Compared to other models of arithmetic, it may seem restrictive and boring, but its restrictiveness is precisely what makes it important. Gödel’s Incompleteness Theorems, which assert the existence of undecidable propositions, only apply to formal theories of a certain strength. By excluding generalized multiplication, Presburger arithmetic remains insufficiently strong for the Incompleteness Theorems to apply, and hence all its propositions are decidable. This makes Presburger arithmetic especially appealing for automated theorem proving—since you can find a finite decision algorithm for any proposition, you can program a computer to prove or disprove theorems in the language. The Coq proof assistant and the theorem prover Princess are both examples of automated theorem provers that base some or all of their functionality on Presburger arithmetic. These theorem provers can have applications beyond model theory—Princess has been used by such programs as Seneschal, a ranking function generator for bitvector relations, and Joogie, which detects infeasible code in the Java programming language. That’s a broad scope of applications for a model that can’t even define multiplication!
If you’re interested in reading more about Presburger arithmetic or its applications, Wikipedia has a great overview, and the official webpage for the Princess theorem prover provides a lot of documentation.
]]>