{"id":2084,"date":"2005-09-16T11:25:34","date_gmt":"2005-09-16T15:25:34","guid":{"rendered":"https:\/\/engineering.jhu.edu\/magazine-archive\/?p=2084"},"modified":"2014-12-16T11:26:50","modified_gmt":"2014-12-16T16:26:50","slug":"proof-truth-numbers","status":"publish","type":"post","link":"https:\/\/engineering.jhu.edu\/magazine-archive\/2005\/09\/proof-truth-numbers\/","title":{"rendered":"The Proof of Truth in Numbers"},"content":{"rendered":"

Node by node in networks, mathematician Edward R. Scheinerman models the motion along direct routes.<\/strong><\/p>\n

\"Edward<\/a>
To get from I-495 to I-695, from blood cell to brain cell, from e-mail to server, the pathway is a network. In our digital era, the specific applications for discrete mathematics provide Edward R. Scheinerman with many routes for research.<\/strong><\/figcaption><\/figure>\n

The Internet…interstate highways…interpersonal relations… e-mails…cell phones…cellular growth…Google. Welcome to a world composed entirely of networks. Edward R. Scheinerman thinks that\u2019s just fine. A professor of mathematics in the Whiting School of Engineering\u2019s Department of Applied Mathematics and Statistics, Scheinerman loves to model and analyze networks\u2014and he uses advanced mathematics to do just that across many engineering disciplines.<\/p>\n

\u201cNetworks are everywhere,\u201d Scheinerman explains, \u201cfrom looking inside of a cell at how genes interact to finding your way from Baltimore to Colorado. My field is discrete mathematics, especially graph theory, partially ordered sets, random graphs, and combinatorics, with applications to robotics and networks. The explosion of networks in the world has also led to an explosion in discrete mathematics.<\/p>\n

\u201cMapQuest is a perfect example,\u201d Scheinerman observes. \u201cYou want to get from point A to point B.\u201d Instead of simply seeking the shortest distance, however, \u201cyou have to follow a network. When you come to a stoplight, you have to make a decision. There is no, \u2018Well, maybe I\u2019ll take a half left,\u2019 because there is no road there.\u201d<\/p>\n

This fall, Scheinerman returned to campus after a year\u2019s sabbatical, during which he completed the second edition of his popular textbook, Mathematics, a Discrete Introduction (now available from Brooks Cole). He also conducted applied mathematics research; began \u201cand mostly completed,\u201d he says, a new book on computer programming for mathematicians; and was a visitor at the Center for Computing Sciences in Bowie, Maryland.<\/p>\n

The mathematician has served the Whiting School as chair of his department and recently as interim associate dean for academic affairs. He received his master\u2019s (1981) and PhD (1984) degrees in mathematics from Princeton University, where he was first introduced to the charms of discrete mathematics and graph theory.<\/p>\n

In his textbooks, Scheinerman provides a better understanding of discrete mathematics, graph theory, and ways that networks can explain human interaction systems. The first edition of his Mathematics: A Discrete Introduction was published in 2000. Invitation to Dynamical Systems (Prentice Hall) was released in 1996.<\/p>\n

\u201cWhen I started at Hopkins in the mid-\u201980s,\u201d Scheinerman recalls, \u201cthere was no introductory-level course in discrete mathematics, and I thought it was important that there be one.\u201d Much to his surprise, then-department chair Robert J. Serfling suggested he develop one. Out of this experience, Scheinerman produced Mathematics, A Discrete Introduction. \u201cThat\u2019s one of the reasons I love Hopkins,\u201d he continues. \u201cIt has afforded me the opportunity to make changes, to have some influence, to make a difference.\u201d<\/p>\n

Another area where the mathematician is making a difference is in helping computer science more directly address the needs of mathematicians. In fact, the book Scheinerman has almost completed seeks to raise the profile of the programming language C++ among this group. \u201cThere has long been a great need\u201d for such a resource, he notes. \u201cC++ is widely used in the computer engineering world, but it has not been widely embraced by the mathematics community.\u201d<\/p>\n

The mathematical problems that Scheinerman tackles are not merely abstract. \u201cSome of my papers would perhaps be of interest only to mathematicians,\u201d he says. \u201cOthers deal directly with specific application areas, such as robotics.\u201d What does tie them all together is that they all draw on the relatively new field of mathematics known as \u201cdiscrete.\u201d<\/p>\n

\u201cIn discrete mathematics, you\u2019re either A or B. You are not something muddled in between.\u201d Edward R. Scheinerman<\/cite><\/strong><\/p><\/blockquote>\n

What Makes This Math \u201cDiscrete\u201d? <\/strong><\/p>\n

\u201cMost people study continuous mathematics,\u201d Scheinerman explains. \u201cCalculus, the study of things in motion, is one of the crowning achievements of continuous mathematics.\u201d Calculus developed to answer such questions as, \u201cHow does the Moon move around the Earth, and why?\u201d For this reason, he says, \u201cfor a very long time, there was no clear distinction between mathematics and physics. To ask whether Newton was a mathematician or a physicist would not have made any sense. In his time, the two things were intermixed.\u201d<\/p>\n

Fast-forward to the 20th century, and a world of finite (as opposed to continuous) mathematical problems. \u201cWe have invented everything from computers to MapQuest, and Google to the Rubik\u2019s Cube. Not one of these is readily analyzed by continuous mathematics,\u201d Scheinerman says. \u201cLikewise with computers, everything is digital. In discrete mathematics, you\u2019re either A or B. You are not something muddled in between.\u201d<\/p>\n

Much of mathematics is application-driven, and the need for a mathematics that can deal effectively with motion through networks\u2014as opposed to smooth and continuous motion\u2014has grown up with the computer era. Using discrete mathematics to model and understand networks finds applications in many fields. \u201cOne of my colleagues at Hopkins, Carey Priebe [professor of Applied Mathematics and Statistics], analyzed the e-mail communications of Enron employees to better understand how the social networks differed from corporate networks,\u201d Scheinerman explains. \u201cNothing happens in engineering these days that doesn\u2019t have a mathematics connection.\u201d<\/p>\n

No network, it seems, can escape the discrete mathematician\u2019s ability to capture it in a graph that can be studied to better understand the connections between individuals\u2014human or otherwise.<\/p>\n

\u201cGraph\u201d is really somewhat a misnomer, the mathematician says, because the graphs he makes bear little resemblance to a typical pie or bar chart. Scheinerman\u2019s graphs model discrete connections between entities, or nodes. Depending on the network one hopes to graph, there can be hundreds\u2014or thousands\u2014of direct connections between various nodes. Consider for example, the e-mail communication paths within an organization, or the connections represented by the growth and changes inside of a cell. The graphs reveal patterns, and Scheinerman tests theorems against these patterns, looking for mathematical proofs that can be used to discover the \u201ctruths\u201d lying hidden in the complex interconnections.<\/p>\n

We have a Johns Hopkins mathematician to thank for this specific use of the word \u201cgraph.\u201d J. J. Sylvester, the legendary 19th-century British-born mathematician, coined the term \u201cgraph\u201d for the kind of representation used by Scheinerman. (Perhaps contributing to Sylvester\u2019s legendary status is the fact that in 1877, as a condition for accepting the position as the University\u2019s first mathematics professor, he stipulated that his annual salary of $5,000 be paid in gold.)<\/p>\n

Indeed, language and logic are exceedingly important to mathematics. As the chapter titles in Mathematics: A Discrete Introduction attest, mathematics seems more closely connected with philosophy than science, with which mathematics is often linked. Such chapter titles as \u201cWhy?\u201d \u201cThe Nature of Truth,\u201d and \u201cIf-then,\u201d would sound right at home on a shelf next to Kierkegaard, Bonhoeffer, and Kant.<\/p>\n

\u201cThe whole idea of \u2018definition, theorem, proof\u2019 is key,\u201d Scheinerman explains. \u201cThis is universal to all of mathematics, to a mathematical way of knowing.\u201d As it turns out, this is the exact opposite of the scientific method of discovering truth, which is to prove by empiricism. In science, \u201cwhen you make a statement that has a one-in-a-million exception, no one is going to criticize you,\u201d he says, \u201cespecially in a biological or social science.\u201d<\/p>\n

In mathematics, however, \u201ctruth means 100 percent. When we say a theorem is true, it is true without exceptions. Period. You cannot achieve that by empirical methods,\u201d Scheinerman says with the certainty of a man who speaks mathematical truth. In all probability, he can produce a graph to prove it.<\/p>\n

To learn more about Edward R. Scheinerman, visit www.mts.jhu.edu\/~ers . To see a graph of how the Applied Mathematics and Statistics faculty members are networked in their research, visit www.ams.jhu.edu\/ams\/ research\/general.html<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

Node by node in networks, mathematician Edward R. Scheinerman models the motion along direct routes. The Internet…interstate highways…interpersonal relations… e-mails…cell phones…cellular growth…Google. Welcome to a world composed entirely of networks. Edward R. Scheinerman thinks that\u2019s just fine. A professor of mathematics in the Whiting School of Engineering\u2019s Department of Applied Mathematics and Statistics, Scheinerman loves…<\/p>\n","protected":false},"author":4,"featured_media":2085,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[84],"tags":[],"class_list":["post-2084","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-high-perfomers","issue-fall-2005"],"acf":[],"yoast_head":"\nThe Proof of Truth in Numbers - JHU Engineering Magazine<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/engineering.jhu.edu\/magazine-archive\/2005\/09\/proof-truth-numbers\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Proof of Truth in Numbers - JHU Engineering Magazine\" \/>\n<meta property=\"og:description\" content=\"Node by node in networks, mathematician Edward R. 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