## History of Mathematics Conference 2011 - Abstracts

- Speaker: Dr. Fiacre Ó Cairbre

Title: Beauty in Mathematics from a historical perspective.

Abstract: This talk will discuss the fundamental role that Beauty has played in the history of mathematics. I will give reasons for why I consider Beauty to be the most important feature of mathematics. Some of the reasons are that the pursuit of beauty has often been the motivation for why the great mathematicians do research in mathematics, practical power is often an offspring of the quest for beauty in mathematics and thirdly mathematics, as we know it today, was essentially born out of a search for aesthetic pleasure and beauty by the Classical Greeks around 600 B.C. - Speaker: Professor Philip Boland

Title: W. S. Gosset - an industrial "Student".

Abstract: William Sealy Gosset (alias "Student") was an immensely talented scientist of many interests, but he is best known for his contributions to modern statistics. Born in 1876 in Canterbury, England, he studied chemistry and mathematics at Oxford. After graduating in 1899, he was hired by St James's Gate Brewery of Arthur Guinness, Son & Co. in Dublin, Ireland, to assist in the development and application of scientific methods in brewing. His practical work led to important theoretical discoveries, in particular to "Student's t-test of significance". In this somewhat rambling presentation I plan to discuss various aspects of his career both in the brewery and in the wider scientific domain. -
Speaker: Micheál Brennan

Title: The History of Maths in the History of Art in these Islands.

Abstract: Art does not need mathematics but the history of art does. A good example is provided by the history of the involvement of mathematicians, professional and otherwise, in art-historical and archaeological studies of Western European art from the period 400-1200AD. At regular intervals over the past one hundred and forty years in these islands, mathematically-minded individuals have got involved, tangentially, in analysing the underlying structure of the ornamental genre commonly known as 'Celtic art'. The main focus of their interest has been 'interlace' the projection of alternating knots and links onto two-dimensional surfaces. It is fair to say that the mathematician quickly finds more interesting aspects of interlace than either those that are found in the archaeological record or those that could possibly have occurred to the early medieval artist but have not survived. In short, the paths of the mathematician and art historian soon divide. This talk will reflect on the reason for this, and argue that a job of work has been left undone. The extent of this job will be sketched out, and an example of how mathematical reasoning can help in appreciating zoomorphic interlace will be illustrated with a segment of a page from the Lindisfarne Gospels. -
Speaker: Mary Kelly

Title: Mathematics in medieval Glendalough: British Library, Egerton 3323, fo. 18, a twelfth-century witness.

Abstract: A marginal note places this single folio in Glendalough in 1106. The text of fo. 18, indicates that this fragment belongs to a much larger corpus of mathematical texts. These were core texts in the study of the quadrivium - the disciplines of the Liberal Arts curriculum.

As well as tables of Roman fractions fo. 18 contains a condensed statement of numerical philosophy based on the works of Boethius, notably De Institutione Arithmetica and De Consolatione Philosophiae. Indeed it is only with reference to these works that the text of fo. 18 can be understood.

Construe marks demonstrate that fo. 18 was used as a teaching text. This paper will show that the curriculum in the monastic school at Glendalough was on a par with that of contemporary cathedral schools in continental Europe and Britain. - Speaker: Dr. Maurice O'Reilly

Title: History in Mathematics Education - The past decade.

Abstract: The initiative to set up the International Commission on Mathematical Instruction (ICMI) was led by Felix Klein at the International Congress of Mathematicians in Rome in 1908. One of its affiliated organizations is the International Study Group on the Relations between History and Pedagogy of Mathematics, or HPM for short, which got off the ground in 1984 at the International Congress on Mathematical Education (ICME) held in Adelaide. HPM has promoted dedicated research in investigating the role of the history of mathematics in mathematics education for over a quarter of a century. In this paper some of the main features of the work of HPM are outlined focusing on the past decade since the publication of the tenth ICMI Study by Fauvel and van Maanen in 2000, dedicated to history in mathematics education, to the working group at CERME 7 in Rzeszów, Poland in 2011. - Speaker: Dr. Cáit Ní Shúilleabháin

Teideal: Fuascailt Mhatamaiticiúla: Fís de Valera i dtaobh chaidreamh na hÉireann leis an gComhlathas.

Title: Mathematical Freedom: De Valera's vision of Ireland's relationship with the Commonwealth.

Abstract: Éamon de Valera, former Taoiseach and President of Ireland, initially trained as a mathematician. Throughout this talk I will aim to show how his interest in mathematics, both pure and applied, permeated his political and personal life, leading him to devise mathematical concepts as solutions for political situations. It is necessary to consider various aspects of de Valera's life that involved mathematics. These aspects include his primary and secondary education along with his third level instruction, speculation as to the type of work he may have worked on while studying for a master’s degree, the time he spent teaching mathematics and mathematical physics in second and third level colleges, and his particular interest in the algebra of quaternions. Additionally, I will look at de Valera's use of mathematics as a tool to combat the loneliness he felt in jail, his use of mathematics to escape from prison and the impact of his mathematical and logical training on certain political decisions.

Oileadh Éamon de Valera, iar-Thaoiseach is iar-Uachtarán na hÉireann, mar mhatamaiticeoir i dtús báire. Le linn na cainte seo, léireod an tslí gur bhain a shuim i gcúrsaí matamaitice, glanmhatamaitic agus matamaitic fheidhmeach, lena chinntí agus lena choincheapanna polaitiúla. Chun é seo a mheas beadsa ag scrúdú roinnt gnéithe de shaol de phearsantacht de Valera. I measc na ngnéithe seo, féachfadsa ar a chuid oideachais idir bhunscolaíocht agus mheánscolaíocht agus an saghas oibre a dhein sé ar an gcoláiste tríí léibhéail, go háirithe an spéis fé leith a chuir sé in obair mhatamaitice an Éireannaigh William Rowan Hamiliton, na ceathairníonacha. Ar aon dul le seo, scrúdód an leas a bhain de Valera as an matamaitic mar chosaint ar an uaigneas a bhraith sé is é i bpríosún i Sasana, agus an úsáid a bhain sé as an matamaitic mar uirlis éalaithe ón bpríosún i 1917. Tríd is tríd taispeánfad an tionchar a bhí ag a chuid oiliúna sa mhatamaitic ar chinntí polaitiúla. -
Speaker: Dr. Gerry Keane

Teideal: Paradacsaí agus Stair na Matamaitice.

Title: Paradoxes in the History of Mathematics.

Abstract: Paradoxes can succinctly uncover flaws in reasoning or highlight counterintuitive findings. They have a long tradition in the history of mathematics. Resolution is not always straightforward.

Zeno argued that the hare could never capture the tortoise for when the hare reached where the tortoise was, the tortoise had already moved. It took mathematicians 2,000 years to finally dispel the `impossibility of motion' and to formalise the notion of limits. St. Paul mentions (inadvertently) the liar paradox in the Bible and the intricacies of this paradox carried through to Kurt Gödel's work on the Incompleteness of Mathematics.

The late 1800's/early 1900's was a rich time for paradoxes as mathematics became more structured. In particular Guttlop Frege sought to establish the foundation of mathematics (the natural numbers) on logic but became unstuck when Bertrand Russell revealed a paradox about sets that include themselves as members. Georg Cantor encountered many paradoxes of set theory through his work on the infinite. Joseph Bertrand highlighted a difficulty with the classical definition of probability.

These and other paradoxes (Augustus de Morgan, Grandi, Galileo, Tarski, Burali-Forti) will be discussed.

Is féidir le paradacsaí deacrachtaí a léiriú go gonta sa réasúnaíocht nó nuair a ritheann conclúid i gcoinne ár n-imfhios. Ní bhíonn réiteach éasca ann i gconaí.

Dar le Zeno, níorbh fhéidir leis an ngiorria breith ar an toirtís mar aon uair a shroicheadh an giorria an áit ina raibh an toirtís, bheadh an toirtís bogtha ar aghaidh. Thóg sé 2,000 bliain dos na matamaiticeoirí teacht ar mhíniú sásúil - agus sainmhíniú a thabhairt do theorainn fheidhme. Déanann Naomh Pól tagairt (i ngan fhios dó féin) do pharadacsa an bhréagadóra sa Bhíobla Naofa agus leanann an deacracht leis an bparadacsa seo go croílár na matamaitice mar d'úsáid Kurt Gödel é nuair a léirigh sé go raibh matamaitic neamhiomlán.

B'am saibhir é do pharadacsaí thart ar chasadh an chéid 1800/1900, nuair a thugadh aghaidh ar bhunstructúr na matamaitice. Theip go tubaisteach ar Frege, a bhí ar tí a leabhar a fhoilsiú faoi conas bunchloch na matamaitice a leagadh ar loighic, nuair a léirigh Bertrand Russell paradacsa faoi fein-bhallraíocht i dtacair. Phlé Georg Cantor leis an éigríoch agus is iomaí paradacsa a tháinig chun solais trína thaighde. Thaispeáin Joseph Bertrand gur gá athbhreithniú a dhéanamh ar an sainmhíniú clasaiceach a bhí le dóchúlacht. Chomh maith leosan luaite thuas, pléifear leis le paradacsaí ó Augustus de Morgan, Grandi, Galileo, Tarski agus Burali-Forti. - Speaker: Stacy Carter

Title: The positive impact of using the History of Mathematics in teaching at second level.

Abstract: This talk will discuss some of the research from a Project that involved teaching some History of Mathematics to Transition Year students in 2010. We will discuss some of the resources created and used for the Project and will also show how exposure to the History of Mathematics changed the students' attitude towards Mathematics in a very positive way. - Speaker: Gordon Lessells

Title: Incorporating New Mathematics into a History of Mathematics course.

Abstract: Topics from the History of Mathematics often give rise to mathematical problems which are different from the problems encountered in a mainstream mathematics course. I will endeavour to point out a few areas where interesting mathematical ideas can be encountered arising from topics arising naturally in a History of Mathematics course. Among these topics will be the following: Egyptian Fractions, Difference Equations, Hamiltonian Cycles, Amicable numbers, Partitions of integers, Elliptic curves, and Sums of Squares and quaternions. - Speaker: Dr. Ciarán Mac an Bhaird

Title: Dealing with an unexpected mathematical outcome.

Abstract: We look at some interesting cases in the history and development of maths where unexpected results have arisen. We look at the reactions to these situations and how many led to significant developments and generalisations in mathematics. This has implications for current students who are often only familiar with maths through rote learning. When students encounter an unexpected answer or situation they often stop. This is the key point where students should be encouraged to investigate unexpected outcomes and in doing so they should gain a much greater understanding of mathematics. - Speaker: Professor Rod Gow

Title: William Rowan Hamilton and Octonions - a lost opportunity.

Abstract: As is well known, Hamilton discovered quaternions in October 1843, and by December 1843, his friend John Graves had generalized the construction to a 8-dimensional system of octaves (now usually called octonions). Hamilton pointed out fairly soon that octonions are not associative, and although he occasionally tried to work on algebraic systems which are not associative and some of his notebooks are full of calculations, he made no worthwhile contributions to the study of octonions. We will argue that Hamilton missed an opportunity to make his work more significant, since octonions underlie the foundations of many exotic structures, unlike quaternions, precisely because they are non-associative.