On the applicability of mathematics : philosophical and historical perspectives

The present thesis is a contribution to the debate on the applicability of mathematics; it examines the interplay between mathematics and the world, using historical case studies. The first part of the thesis consists of four small case studies. In chapter 1, I criticize "ante rem structuralism", proposed by Stewart Shapiro, by showing that his so-called "finite cardinal structures" are in conflict with mathematical practice. In chapter 2, I discuss Leonhard Euler's solution to the Königsberg bridges problem. I propose interpreting Euler's solution both as an explanation within mathematics and as a scientific explanation. I put the insights from the historical case to work against recent philosophical accounts of the Königsberg case. In chapter 3, I analyze the predator-prey model, proposed by Lotka and Volterra. I extract some interesting philosophical lessons from Volterra's original account of the model, such as: Volterra's remarks on mathematical methodology; the relation between mathematics and idealization in the construction of the model; some relevant details in the derivation of the Third Law, and; notions of intervention that are motivated by one of Volterra's main mathematical tools, phase spaces. In chapter 4, I discuss scientific and mathematical attempts to explain the structure of the bee's honeycomb. In the first part, I discuss a candidate explanation, based on the mathematical Honeycomb Conjecture, presented in Lyon and Colyvan (2008). I argue that this explanation is not scientifically adequate. In the second part, I discuss other mathematical, physical and biological studies that could contribute to an explanation of the bee's honeycomb. The upshot is that most of the relevant mathematics is not yet sufficiently understood, and there is also an ongoing debate as to the biological details of the construction of the bee's honeycomb. The second part of the thesis is a bigger case study from physics: the genesis of GR. Chapter 5 is a short introduction to the history, physics and mathematics that is relevant to the genesis of general relativity (GR). Chapter 6 discusses the historical question as to what Marcel Grossmann contributed to the genesis of GR. I will examine the so-called "Entwurf" paper, an important joint publication by Einstein and Grossmann, containing the first tensorial formulation of GR. By comparing Grossmann's part with the mathematical theories he used, we can gain a better understanding of what is involved in the first steps of assimilating a mathematical theory to a physical question. In chapter 7, I introduce, and discuss, a recent account of the applicability of mathematics to the world, the Inferential Conception (IC), proposed by Bueno and Colyvan (2011). I give a short exposition of the IC, offer some critical remarks on the account, discuss potential philosophical objections, and I propose some extensions of the IC. In chapter 8, I put the Inferential Conception (IC) to work in the historical case study: the genesis of GR. I analyze three historical episodes, using the conceptual apparatus provided by the IC. In episode one, I investigate how the starting point of the application process, the "assumed structure", is chosen. Then I analyze two small application cycles that led to revisions of the initial assumed structure. In episode two, I examine how the application of "new" mathematics - the application of the Absolute Differential Calculus (ADC) to gravitational theory - meshes with the IC. In episode three, I take a closer look at two of Einstein's failed attempts to find a suitable differential operator for the field equations, and apply the conceptual tools provided by the IC so as to better understand why he erroneously rejected both the Ricci tensor and the November tensor in the Zurich Notebook.

Investigation of Techniques for Accelerating the Construction of Bridge Deck Overlays, 2016

Use of bridge deck overlays is important in maximizing bridge service life. Overlays can replace the deteriorated part of the deck, thus extending the bridge life. Even though overlay construction avoids the construction of a whole new bridge deck, construction still takes significant time in re-opening the bridge to traffic. Current processes and practices are time-consuming and multiple opportunities may exist to reduce overall construction time by modifying construction requirements and/or materials utilized. Reducing the construction time could have an effect on reducing the socioeconomic costs associated with bridge deck rehabilitation and the inconvenience caused to travelers. This work included three major tasks with literature review, field investigation, and laboratory testing. Overlay concrete mix used for present construction takes long curing hours and therefore an investigation was carried out to find fast-curing concrete mixes that could reduce construction time. Several fast-cuing concrete mixes were found and suggested for further evaluation. An on-going overlay construction project was observed and documented. Through these observations, several opportunities were suggested where small modifications in the process could lead to significant time savings. With current standards of the removal depth of substrate concrete in Iowa, it takes long hours for the removal process. Four different laboratory tests were performed with different loading conditions to determine the necessary substrate concrete removal depth for a proper bond between the substrate concrete and the new overlay concrete. Several parameters, such as failure load, bond stress, and stiffness, were compared for four different concrete removal depths. Through the results and observations of this investigation several conclusions were made which could reduce bridge deck overlay construction time.

Enclosed Bridge

One of the so called enclosed bridges connecting modules of the Mackenzie Chown Complex.

Lounge Area

One of the lounge areas situated on the enclosed bridges.

Armourstone revetments : will standard design criteria prevent failure? /

Bank stabilization structures are used to prevent the loss of valuable land within the urban environment and the decision for the type of structure used depends on the properties of the stream. In the urban areas of Southern Ontario there is a preference for the use of armourstone blocks as bank stabilization. The armourstone revetment is a free standing stone structure with large blocks of stone layered vertically and offset from one another. During fieldwork at Forty Mile Creek in Grimsby, Ontario armourstone failure was identified by the removal of two stones within one column from the wall. Since the footer stones were still in place, toe scour was eliminated as a cause of failure. Through theoretical, field, and experimental work the process of suction has been identified as a mode of failure for the armourstone wall and the process of suction works similarly to quarrying large blocks of rock off bedrock streambeds. The theory of lateral suction has previously not been taken into consideration for the design of these walls. The physical and hydraulic evidence found in the field and studied during experimental work indicate that the armourstone wall is vulnerable to the process of suction. The forces exerted by the flow and the resistance of the block determine the stability of the armourstone block within the wall. The design of the armourstone wall, high surface velocities, and short pulses of faster flowing water within the profile could contribute to armourstone failure by providing the forces needed for suction to occur, therefore adjustments to the design of the wall should be made in order to limit the effect.

A journey into narrative inquiry : one teacher's lived experience with eating disorders /

In "A Journey Into Narrative Inquiry: One Teacher's Lived Experience With Eating Disorders," an elementary teacher searches for answers regarding how education can help prevent eating disorders by journeying into her own experience of having had such a disorder. This qualitative study is a personal narrative based on an individual's experience, a method appropriate to the sharing of personal voices and stories told in education research. It is an attempt to address the gap found in the research on this topic by offering a subjective and unique perspective of what it is like to live within the nightmare of an eating disorder and by sharing the wisdom gained from having survived such an experience. This narrative inquiry explains how a teacher found herself at a stage where she was willing and ready to share her experience for the sake of research. The story of having had an eating disorder, consisting of both anorexia and bulimia, for over a decade is shared in a genuine, reflective manner. The researcher then shares the analysis of her own story, unpacking the themes of journeying toward voice, self-esteem, self-acceptance, and self and the completion of an M.Ed. degree. Bridges are made which connect these themes to the personal and professional life of the researcher, to the schools in terms of both curriculum and climate, to research directions, and to the larger culture. Suggestions are made for possible changes in educational settings that may help teachers in providing students with some tools and strategies to prevent turning to eating disorders as coping mechanisms. A literature review of eating disorders is included as well, as a guide for others to use when undertaking such qualitative studies.

Office of the City Clerk, St. Catharines Records, 1928-1974, n.d.

The site of present-day St. Catharines was settled by 3000 United Empire Loyalists at the end of the 18th century. From 1790, the settlement (then known as "The Twelve") grew as an agricultural community. St. Catharines was once referred to Shipman's Corners after Paul Shipman, owner of a tavern that was an important stagecoach transfer point. In 1815, leading businessman William Hamilton Merritt abandoned his wharf at Queenston and set up another at Shipman's Corners. He became involved in the construction and operation of several lumber and gristmills along Twelve Mile Creek. Shipman's Corners soon became the principal milling site of the eastern Niagara Peninsula. At about the same time, Merritt began to develop the salt springs that were discovered along the river which subsequently gave the village a reputation as a health resort. By this time St. Catharines was the official name of the village; the origin of the name remains obscure, but is thought to be named after Catharine Askin Robertson Hamilton, wife of the Hon. Robert Hamilton, a prominent businessman. Merritt devised a canal scheme from Lake Erie to Lake Ontario that would provide a more reliable water supply for the mills while at the same time function as a canal. He formed the Welland Canal Company, and construction took place from 1824 to 1829. The canal and the mills made St. Catharines the most important industrial centre in Niagara. By 1845, St. Catharines was incorporated as a town, with the town limits extending in 1854. Administrative and political functions were added to St. Catharines in 1862 when it became the county seat of Lincoln. In 1871, construction began on the third Welland Canal, which attracted additional population to the town. As a consequence of continual growth, the town limits were again extended. St. Catharines attained city status in 1876 with its larger population and area. Manufacturing became increasingly important in St. Catharines in the early 1900s with the abundance of hydro-electric power, and its location on important land and water routes. The large increase in population after the 1900s was mainly due to the continued industrialization and urbanization of the northern part of the city and the related expansion of business activity. The fourth Welland Canal was opened in 1932 as the third canal could no longer accommodate the larger ships. The post war years and the automobile brought great change to the urban form of St. Catharines. St. Catharines began to spread its boundaries in all directions with land being added five times during the 1950s. The Town of Merritton, Village of Port Dalhousie and Grantham Township were all incorporated as part of St. Catharines in 1961. In 1970 the Province of Ontario implemented a regional approach to deal with such issues as planning, pollution, transportation and services. As a result, Louth Township on the west side of the city was amalgamated, extending the city's boundary to Fifteen Mile Creek. With its current population of 131,989, St. Catharines has become the dominant centre of the Niagara region. Source: City of St. Catharines website http://www.stcatharines.ca/en/governin/HistoryOfTheCity.asp (January 27, 2011)

Second Welland Canal - Book 1, Survey Map 3 - Old Lock 2 in Grantham Township

Survey map of the Second Welland Canal created by the Welland Canal Company showing a portion of the Grantham Township near Port Dalhousie. Identified structures associated with the Canal include the new towing path. The surveyors' measurements and notes can be seen in red and black ink and pencil. Features of the First Welland Canal are noted in red ink and include the old Lock 2, old towing path and the original bed of the Twelve Mile Creek. Local area landmarks are also identified and include streets and roads (ex. Side Line and Old Road), four unnamed bridges, and a tree stump along the old towing path. A New Road to Port Dalhousie is featured in red ink. Properties and property owners of note are: Concession 3 Lots 21, 22 and 23, Concession 4 Lots 21, 22 and 23, Jabez Johnson, Adam Gould, Peter Weaver and Samuel Wood.

Second Welland Canal - Book 1, Survey Map 5 - Through Grantham Township

Survey map of the Second Welland Canal created by the Welland Canal Company showing a portion of the Grantham Township near Port Dalhousie. Identified structures associated with the Canal include the tow path and floating tow path. The surveyors' measurements and notes can be seen in red and black ink and pencil. Features of the First Welland Canal are noted in red ink and include the old towing path, the Old Canal, two unnamed bridges traversing the Old Canal, and the bed of the Twelve Mile Creek. Local area landmarks are also identified and include streets and roads (ex. Old Road, Side Line, Road to Port Dalhousie), Samuel Wood's house, Peter Weaver's house, J. R. Tenbroeck's house, a poplar and walnut tree along the east shore, an oak and hickory tree along the west shore, and a walnut tree stump along the west shore. Properties and property owners of note are: Concession 3 Lots 21, 22, and 23, Concession 4 Lots 21, 22, and 23, Adam Gould, Samuel Wood, Peter Weaver, and John R. Tenbroeck.

Second Welland Canal - Book 1, Survey Map 12 - St. Catharines and Lock 4

Survey map of the Second Welland Canal created by the Welland Canal Company showing the Town of St. Catharines. Identified structures associated with the Canal include Lock 4, Hydraulic Race, floating tow path, and the Canal waterway itself. The surveyors' measurements and notes can be seen in red and black ink and pencil. Local area landmarks are also identified and include streets and roads (ex. Geneva Street, St. Paul Street, Queenston Street, Niagara Street and Mill Street), C. Phelp's Grist Mill and Store House, Stinson's Distillery, and several unnamed bridges. Properties and property owners of note are: Concession 6 Lot 16, J. R. Benson, Calvin Phelps, J. Hudson, David Gray, A. Roberts, Mrs. McDonell, J. S. McDonell, T. B. Wragg, J. Donaldson, W. Barr Jr., C. L. Hall, G. Ward, Ridout Bros and Co., and the St. Catharines Grass Co.

Second Welland Canal - Book 1, Survey Map 14 - St. Catharines and Lock 5

Survey map of the Second Welland Canal created by the Welland Canal Company showing the Canal along the eastern edge of the Town of St. Catharines. Identified structures associated with the Canal include Lock 5 and the towing path. The surveyors' measurements and notes can be seen in red and black ink and pencil. Local area landmarks are also identified and include bridges, streets, and roads (ex. Queenston Street, St. Catharines Macdamized Road and Suspension Bridge), a Pond, a number of unnamed bridges, Stinson's Distillery, and R. Collier's Saw Mill. Properties and property owners of note are: Concession 6 Lots 14 and 15, R. Collier, W. Gillespie, Orson Phelps, W. Chase, M. Bryant, John Soper, Winsor Chace, John Berryman, John Boyle, J. Madigan, B. F. Reynolds, W. Smaill, F. Stinson, G. Ward and Mrs. Soper.

Second Welland Canal - Book 1, Survey Map 15 - Hydraulic Race and Aqueduct in Grantham

Survey map of the Second Welland Canal created by the Welland Canal Company showing the Canal along the eastern edge of the Town of St. Catharines. Identified structures associated with the Canal include Lock 7, Lock House Lot, and the towing path. The surveyors' measurements and notes can be seen in red and black ink and pencil. Local area landmarks are also identified and include bridges, streets, and roads (ex. Queenston Road, St. Catharines Macdamized Road and Suspension Bridge), a hydraulic race, and the Hydraulic Aqueduct. Properties and property owners of note are: Concession 7 Lots 12, 13, and 14, M. Bryant, Mrs. Soper, J. Capner, O. Phelps, P. Marren, Mrs. Parnell, J. Carty, Mrs. Ward, and J. Goodenew.

Second Welland Canal - Book 2, Survey Map 3 - Locks 8, 9 and 10 in Grantham

Survey map of the Second Welland Canal created by the Welland Canal Company showing the Grantham Township between the Town of St. Catharines and Merritton. Identified structures associated with the Canal include Locks 8, 9, and 10, waste weirs, the towing path, and several floating bridges. The surveyors' measurements and notes can be seen in red and black ink and pencil. Several stones and tree stumps likely used in the measurements are identified on the map. Local area landmarks are also identified and include streets and roads(ex. Macadamized Road to Thorold), J. Hamilton's Hotel, a school house, McCoy's Farm House, Bradley's House, O. Phelps Saw Mill, Disher and Hait's Woolen Mill, Centreville Mills, a bridge, several barns, and a number of structures (possibly houses, cabins, or shops) belonging to: P. McCoy, E. McLachlan, T. Wilson, W. Wilson, M. Bradley, S. Bradley, P. Boyle, J. Bradley, E. Grant, and W. Church. Lock 12 and 15 of the original canal are also identified. Properties and property owners of note are: Concession 8 Lots 12, 13 and 14, O. J. Phelps, P. McCoy, A. Bradley, C. Bradley, T. Reed, O. Clifford, J. Bradley, W. C. Loan Company, Duffin, and T. Towers Mill Lot.

Second Welland Canal - Book 2, Survey Map 8 - Little Deep Cut in Thorold

Survey map of the Second Welland Canal created by the Welland Canal Company showing the canal in Thorold South. Identified structures associated with the Canal include the Little Deep Cut and the towing path. The surveyors' measurements and notes can be seen in red and black ink and pencil. Local area landmarks are also identified and include streets and roads (ex. Road to Beaverdams and Road to Allanburgh), two unnamed bridges, the Spoil Bank, a pond, and the Back Water. Properties and property owners of note are: Lots 29 and 30, Jacob Keefer, John Brown, William Bouck, C. Gisso, and a property reserved for Bridge Tender.

Second Welland Canal - Book 2, Survey Map 17 - Deep Cut and Port Robinson

Survey map of the Second Welland Canal created by the Welland Canal Company showing the canal at Port Robinson. Identified structures and features associated with the Canal include the Deep Cut, Old Channel of Canal, and the towing path. The surveyors' measurements and notes can be seen in red and black ink and pencil. Local area landmarks are also identified and include streets and roads (ex. Road to Port Allanburg), the Spoil Bank, an island, several bridges, and a church. Several unidentified structures are present but not labeled. Properties and property owners of note are: Lots 202, 203, and 204. Lot 203 is divided into several properties labeled A - J. Owners of these properties include James McCoppen, John Coulter, James Griffith, John C. Jordan, W. Hendershot, John Greer, Charles Richards, C. Stuart, and S. D. Woodruff. Other property owners include D. McFarland.