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Mathematical Metaphors


Theories in all areas of science tell us something about the world. They are images, or models, or representations of reality. Theories tell stories about the world and are often associated with stories about their discovery. Like the story (probably apocryphal) that Newton invented the theory of gravity after an apple fell on his head. Or the story (probably true) that Kekule discovered the cyclical structure of benzene after day-dreaming of a snake seizing its tail. Theories are metaphors that explain reality.

A theory is scientific if it is precise, quantitative, and amenable to being tested. A scientific theory is mathematical. Scientific theories are mathematical metaphors.

A metaphor uses a word or phrase to define or extend or focus the meaning of another word or phrase. For example, "The river of time" is a metaphor. We all know that rivers flow inevitably from high to low ground. The metaphor focuses the concept of time on its inevitable uni-directionality. Metaphors make sense because we understand what they mean. We all know that rivers are wet, but we understand that the metaphor does not mean to imply that time drips, because we understand the words and their context. But on the other hand, a metaphor - in the hands of a creative and imaginative person - might mean something unexpected, and we need to think carefully about what the metaphor does, or might, mean. Mathematical metaphors - scientific models - also focus attention in one direction rather than another, which gives them explanatory and predictive power. Mathematical metaphors can also be interpreted in different and surprising ways.

Some mathematical models are very accurate metaphors. For instance, when Galileo dropped a heavy object from the leaning tower of Pisa, the distance it fell increased in proportion to the square of the elapsed time. Mathematical equations sometimes represent reality quite accurately, but we understand the representation only when the meanings of the mathematical terms are given in words. The meaning of the equation tells us what aspect of reality the model focuses on. Many things happened when Galileo released the object - it rotated, air swirled, friction developed - while the equation focuses on one particular aspect: distance versus time. Likewise, the quadratic equation that relates distance to time can also be used to relate energy to the speed of light, or to relate population growth rate to population size. In Galileo's case the metaphor relates to freely falling objects.

Other models are only approximations. For example, a particular theory describes the build up of mechanical stress around a crack, causing damage in the material. While cracks often have rough or ragged shapes, this important and useful theory assumes the crack is smooth and elliptical. This mathematical metaphor is useful because it focuses the analysis on the radius of curvature of the crack that is critical in determining the concentration of stress.

Not all scientific models are approximations. Some models measure something. For example, in statistical mechanics, the temperature of a material is proportional to the average kinetic energy of the molecules in the material. The temperature, in degrees centigrade, is a global measure of random molecular motion. In economics, the gross domestic product is a measure of the degree of economic activity in the country.

Other models are not approximations or measures of anything, but rather graphical portrayals of a relationship. Consider, for example, the competition among three restaurants: Joe's Easy Diner, McDonald's, and Maxim's de Paris. All three restaurants compete with each other: if you're hungry, you've got to choose. Joe's and McDonald's are close competitors because they both specialize in hamburgers but also have other dishes. They both compete with Maxim's, a really swank and expensive boutique restaurant, but the competition is more remote. To model the competition we might draw a line representing "competition", with each restaurant as a dot on the line. Joe's and McDonald's are close together and far from Maxim's. This line is a mathematical metaphor, representing the proximity (and hence strength) of competition between the three restaurants. The distances between the dots are precise, but what the metaphor means, in terms of the real-world competition between Joe, McDonald, and Maxim, is not so clear. Why a line rather than a plane to refine the "axes" of competition (price and location for instance)? Or maybe a hill to reflect difficulty of access (Joe's is at one location in South Africa, Maxim's has restaurants in Paris, Peking, Tokyo and Shanghai, and McDonald's is just about everywhere). A metaphor emphasizes some aspects while ignoring others. Different mathematical metaphors of the same phenomenon can support very different interpretations or insights.

The scientist who constructs a mathematical metaphor - a model or theory - chooses to focus on some aspects of the phenomenon rather than others, and chooses to represent those aspects with one image rather than another. Scientific theories are fascinating and extraordinarily useful, but they are, after all, only metaphors.




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Yeast as a Metaphor: Élisabeth and Félix Leseur

By Sr. Joan L. Roccasalvo, C.S.J.

It’s a wonderful phenomenon—yeast.  It permeates lifeless flour and causes it to rise and expand.  The power of yeast effects the brewing of beer and the making of wine.   The yeast plant is a fungus that grows without limits to its borders.  Only if yeast is alive and active will it interact with the dough.

On her TV program, “Martha Bakes,” the talented Ms. Stewart cannot contain her delight when she makes yeast dough: “Look at the sheen—so soft and shiny! The aroma is “bee-you-tee-ful,” and the fragrance gratifies all the senses!” Follow these instructions: proof active yeast, blend it into the flour mixture, and let it rise to double the size.  From yeast dough come baked goods such as breads, sticky buns and sugar buns, and monkey bread.  “Soo pretty, soo delicious,” Ms. Stewart swoons over her culinary works of art.

Yeast as a Metaphor

In the Matthean parable (13:33), the reign of God is like yeast that a woman took and kneaded into three measures of flour.  Eventually the entire mass of dough began to rise.  The image of yeast was a favorite in the Early Church.  Everyone understood the inner power of yeast with its limitless ability to make things grow, even in small beginnings with “three measures of flour.”  They grasped the comparison.  The yeast referred to the Church as an unlimited and growing reality, “destined ultimately to be present everywhere and to affect everything, though by no means to convert everything into itself” (Walter J. Ong, “Yeast: A Parable for Catholic Higher Education,” America Magazine, April 7, 1990).  The Church is catholic because it has always been expanding into new and shiny ‘dough’ without limit. Katholicos, from kata or kath and holos, means “through-the-whole or “throughout-the-whole.”

The Laity: Worldly and Yet Unworldly    

The laity are catholic, yeast in business and finance, entertainment, nursing and medicine, arts and science, law and law enforcement, politics, and sports.  They are the inner power with its limitless ability to make things grow, even in small ways. The laity find their holiness in the world with its financial concerns and family responsibilities.  Those who marry and have children become not just a family but also the Domestic Church.

In 1987, the Catholic Church held a World Synod on the Laity, one of many, beginning with Vatican II in the 1960s.  According to the synod’s final document, the laity are equal with clergy and consecrated religious in the life and mission of the Church.  

The call to holiness of the laity differs from the vocation of consecrated religious.  The laity are to be in the world in an unworldly way.  They approach life with wisdom that teaches the limited and relative value of material things. This would seem to be a contradiction in terms.  How to be worldly and unworldly at the same time?   It cannot be easy, for at times, the challenges seem insurmountable.  Yet, it remains for the lay vocation to find a theology of being present in the world. It is a practical spirituality of the family and the workplace.  For the laity, this is where holiness resides.*  

Holiness of the Laity

The holiness of the laity began with Jesus himself.  He was a rabbi and teacher, as were his disciples. Peter was a married man, and for all we know, so were the other apostles, the exception being John, the Beloved Disciple.  

St. Paul addresses and refers to those he evangelized as ‘saints,’ meaning that they were on their way to becoming saints.  In the Early Church, there were no consecrated institutes of men and women.  All Christians grasped the importance of living as disciples and ambassadors of the Lord.

As increasing numbers of Christians came to view the world as wicked, they flocked to the desert to live alone. When the desert grew so overcrowded with these solitaries, they came together and formed religious communities.  Thus, the start of monastic orders of men and women.

Prayer

Consecrated men and women, and especially those who live in cloisters, spend several hours a day in prayer.

This is not the way of the laity. Their days focus almost entirely on family and the means of supporting it.  Their prayer is measured not in hours but in minutes—two here, five there, perhaps a Holy Hour or Retreat Day on rare occasions.

The conciliar document on the sacred liturgy encourages Catholic families to pray portions of the Liturgy of the Hours (#102-111).  The Hours are not private or devotional prayer but the prayer of the entire Church, the Church at prayer.  Praying the psalms nourishes Catholic family life whose welfare is daily beset with conflicting external forces. If prayer is the underlying power of strong family life, then parents can find ways to incorporate parts of the Hours into their daily schedule. In prayer, married couples derive the strength of God’s grace to live their married vocation.  

As children mature, they too must learn to travel the road to discipleship in the Lord.  Small children can be taught to pray a psalm or two at bed time. If this is not feasible during the week, then prayer on weekend is an alternate possibility.  

A minimal and external Christianity will not fortify today’s Domestic Church but only a vibrant Christianity in which Christ is a living reality.  It takes a few minutes to pray short sections of the Hours, even on public transit.  It is a consoling thought to recall that “in him, we live and move and have our being” (Acts 17:28).  

At Pre-Cana instructions, couples can learn the practice of making the Hours an integral part of their married life.

Can Yeast Corrupt?

The image of yeast is not always positive.  In First Corinthians 5:6-8, St. Paul mentions what all Jews understood.  At the Paschal festival time, they were to destroy all yeasted products because leaven was a metaphor for the corruptive influence of evil, for puffing up the self, leaving no room for God.   

Proofing the yeast in warm water will yield bubbles around the surface, and the yeast will become puffed up if it does not interact with the flour dough.  The puffed up yeast will die.  In this sense, neither the laity, nor any minister in the Church, can afford to be puffed up with pride.

Élisabeth Leseur (1866-1914) and Félix Leseur (1861-1950)

The story of Élisabeth Arrighi Leseur exemplifies the limitless power of marital love.  Élisabeth was born into a wealthy French Catholic family of Corsican descent.  As a child, she had contracted hepatitis, a disease from which she suffered all her life.  At twenty-one, she met Félix Leseur, a medical doctor, who also came from an affluent Catholic family.  Shortly before they were to be married, Élisabeth discovered that Félix was no longer a practicing Catholic.  Soon he became well known as the editor of an anti-clerical, atheistic newspaper.  

Despite the circumstances, the couple married, for Élisabeth was deeply in love with Félix.  They were unable to have children, a fact that made their marriage all the more difficult.  His attack on her religious devotion prompted an even more serious fidelity to the faith. She bore the brunt of his hatred of the Church with patient love.  At thirty-two years of age, Élisabeth experienced the grace to a deeper form of prayer.  She was convinced that her task now was to love her husband and pray for his conversion while remaining steadfast during his taunts against religion, and the Church in particular.

Homebound and Bed-Ridden

Élisabeth’s deteriorating health forced her to lead a sedentary life.  She received visitors and was able to conduct a vibrant apostolate from the confines of her home.  She became a devotee of St. Francis de Sales who wrote for the layperson in the seventeenth century. His Introduction to the Devout Life, perhaps the most famous spiritual guide of all time, is an offshoot of the Ignatian Exercises. During this period, Élisabeth kept a secret spiritual diary.  

When, at the age of forty-five, Élisabeth underwent surgery and radiation for the removal of a malignant tumor, she recovered and continued to receive visitors to her home. Three years later, she succumbed to cancer.  Her life has been recommended for sainthood. Why?  We turn the page to continue the narrative of her husband.

Dr. Félix Leseur

After Élisabeth’s death, Félix found a note addressed to him.  Not only did it predict his conversion, but he would also become a Dominican priest.  His hatred of the Church prompted him to expose her note as a fake, and he decided to do so at Lourdes, the famous Marian shrine in France.  There, something prevented him from carrying out his intended project—call it God’s intervening grace. As Élisabeth had predicted, he experienced a conversion and published her spiritual journal.  In 1919, Félix entered the Dominican Order, was ordained a priest four years later, and spent his remaining years speaking about his wife’s difficult yet remarkable life with him.  

In 1924, the future Archibishop Fulton J. Sheen made a retreat under Fr. Leseur’s direction.  It was at this time that he learned of Élisabeth’s life and her husband’s conversion.  In 1934, Fr. Leseur, O.P. worked to begin the cause for her canonization, and the Archbishop shared the story of this remarkable married couple in many presentations.  Élisabeth is currently a Servant of God, the first step in the cause for sainthood.

Élisabeth Leseur’s suffering was not wasted. On the contrary, her lifelong devotion to Félix was central to his conversion.  She became the yeast that permeated the lifeless soul of her husband.  It forever transformed his life so that he could affect change in the lives of others. Love begets love.

*The Ignatian “Prayer for Finding God in All Things” by Joan L. Roccasalvo, C.S.J. can help the busy person find God throughout the day.  Copies are available from the Institute of Jesuit Sources, Boston, MA.



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Human Metapneumovirus Infection in Hospitalized Children

BACKGROUND:Most children are exposed to human metapneumovirus (HMPV) by the age of 5 y. This study aimed to describe the morbidity associated with HMPV infections in a cohort of children in the Midwest of the United States.METHODS:This was a retrospective 2-center cohort study including children (0–17 y old) hospitalized with HMPV infections at 2 tertiary care pediatric hospitals from 2009 to 2013. Demographics, chronic medical conditions, viral coinfections, and hospitalization characteristics, including the need for respiratory support, high-flow nasal cannula, CPAP, bi-level positive airway pressure, invasive mechanical ventilation, pediatric ICU admission, acute kidney injury (AKI), use of extracorporeal membrane oxygenation, and length of stay, were collected.RESULTS:In total, 131 subjects were included. Those with one or more comorbidities were older than their otherwise healthy counterparts, with a median age of 2.8 y (interquartile range [IQR] 1.1–7.0) compared to 1.3 y (IQR 0.6–2.0, P < .001), respectively. Ninety-nine (75.6%) subjects required respiratory support; 72 (55.0%) subjects required nasal cannula, simple face mask, or tracheostomy mask as their maximum support. Additionally, 1 (0.8%) subject required high-flow nasal cannula, 1 (0.8%) subject required CPAP, 2 (1.5%) subjects required bi-level positive airway pressure, 15 (11.5%) subjects required invasive mechanical ventilation, 4 (3.1%) subjects required high-frequency oscillatory or jet ventilation, and 4 (3.1%) subjects required extracorporeal membrane oxygenation. Fifty-one (38.9%) subjects required pediatric ICU admission, and 16 (12.2%) subjects developed AKI. Subjects with AKI were significantly older than those without AKI at 5.4 y old (IQR 1.6–11.7) versus 1.9 y old (IQR 0.7–3.5, P = .003). After controlling for the presence of at least one comorbidity and cystic fibrosis, each year increase in age led to a 16% increase in the odds of AKI (P = .01). The median length of stay for the entire cohort was 4.0 d (IQR 2.7–7.0).CONCLUSIONS:Children hospitalized with HMPV may be at risk for AKI. Risk of HMPV-associated AKI appears to increase with age regardless of severity of respiratory illness or presence of comorbidities.




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