The life and mathematics of Norberto Cuesta (1907-1989)

              Rev. Acad. Canar. Cienc., XXVI, 85-125 (2014) (publicado en julio de 2015)
THE LIFE AND MATHEMATICS
OF NORBERTO CUESTA (1907-1989)
José M. Pacheco
Departmento de Matemáticas. Universidad de Las Palmas de Gran Canaria
Campus de Tafira Baja. 35017 Las Palmas, Spain
pacheco@dma.ulpgc.es
Research partly supported by CICyT Project HAR20l2-39260
Abstract. This biography describes the activity as a mathematician, historian, and com­promised
person of the Spanish Professor Norberto Cuesta Dutari (1907-1989) [Cuesta, or
the Professor, in the rest of the paper] under the social, política! and scientific environ­mental
conditions in Spain during General Franco's regime (1939-1975) and the first years
after the democratic restoration. First, it is shown that his deepest and most original con­tributions
to Mathematics were obtained in the darkest years after the Spanish civil war,
a remarkable achievement in the 1940s' social and política! context, when strong interna!
tensions among the various fractions of the regime supporters, generalised poverty, and
international isolation made life - not only scientifically speaking- a difficult endeavour in
Spain. Second, the isolation of Cuesta, resulting from a mixture of personal and environ­mental
conditions, was also a constant in his vital trajectory, though he tried to overcome
t his handicap by steadily writing and publishing, and in his last life years as the active
promoter of a study track in Mathematics at Salamanca.
The body of the paper is preceded by a general introduction considering those aspects of
the general Spanish panorama after t he civil war needed to contextualise the narrative,
followed by a succession of portraits addressing personal facts, Mathematics, contributions
to mathematical Education and to the History of Mathematics, ending with the role played
by Cuesta in the establishing of new Mathematics studies in Spain. Appendices presenting
a hopefully complete list of publications and other facts are also included, among them a
non-technical study of his main mathematical result of 1943 and its 1955 generalisation.
AMS 2010 Mathematics Su bject Classifications: 01A60, 01A70, 01A72, 01A73,
01A80, 97 A30.
Key words: Cuesta Dutari, Francoism, History of Mathematics, Interior exile, Salamanca,
Spain.
Part 1
Introduction
The Spanish civil war ravaged Spain in the years between 1936 and 1939, during which
the country was divided in two zones, the so-called zona nacional under the rule of the
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rebel General Franco, and the zona roja that remained more or less Joya! to the legally
established republican Government. By mid-1938, well before the end of the war and with
Franco's victory in perspective, the various political groups gathered around his autocratic
regime had already started to shape an organisation of entities to tackle future scientific,
technological and education policies according with the regime ideology - or rather, lack
thereof (see e.g. Malet 2008).
It is in this suffocating environment where the scientific part of the present biography
begins. Professor Cuesta can be considered an interior exile, not on the usual sense of
someone having suffered a long prosecution, sometimes including prison, or having been
forbidden to work, or travel, as is usually stressed in studies dealing with the post-war
situation in the Arts, Literature, Medicine, and the Sciences (Balcells and Pérez-Bowie
2001 , Gómez Bravo 2009). Rather, the pressure exerted on a large number of professionals
- among which Cuesta may be counted- was of a subtler nature, a result of a behaviour
imposed on individuals of ali social classes that changed personal relationships through
an ever-present fear of publicly appearing close to sorne people, who although politically
neutral or in sorne cases even sympathising with the regime, were not absolutely trusted, as
a rule by minor personal or familiar questions. Eventually, this shift in personal behaviours
led these people to a self-imposed distance from others who in normal circumstances would
have been close collaborators, simply helpful people or just friends. This state of affairs
slowly changed in the mid-1950s when the regime politics evolved as a result of the new
political relations with the US started in 1952.
This study is the result of processing information directly gathered from primary sources
including Cuesta's personal papers and publications, interviews and talks with people who
knew him and his work, a number of secondary sources, and reflections on a personal
friendship spanned over sorne twenty years. Partial results have already been presented in
congresses (Pacheco 2009, 2010), and it has been completed during 2012 and early 2013,
once the Cuesta archive was made available by the university of Salamanca after classifying
and ordering it.
In order to organise the complete story, this introduction is split into severa! sections
referring to the various background aspects.
1 Doing and living science in Spain after the civil war
In a series of talks to university students in late 1930, the Spanish writer and philosopher
José Ortega y Gasset (1883-1955) had pointed well before the civil war to the need of
separating research and educational activities, leaving these last ones as the main aim of
universities. Those talks were subsequently published under the name La misión de la
universidad (Ortega 1931). An attempt to carry into practice this viewpoint was made by
the Franco regime when the war ended and a comprehensive organisational structure for
scientific policy was created in 1939: The Consejo Superior de Investigaciones Científicas
(CSIC), that took over ali facilities of the previous institution Junta de Ampliación de
Estudios (JAE), active since 1907. The CSIC developed a complicated and ever-changing
network of institutes, laboratories, libraries, museums, etc. According to Ortega's idealised
viewpoint, universities did not appear in this scheme, but as the CSIC had no fixed po­sition
scientists during its first life years, most of its areas and posts were occupied by
university Professors linked with it through a mixture of honorific and lobbying activities,
indeed including among them the control of university appointments, thus immediately
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disregarding any application of Ortega's old proposal. For an interesting though rather
informal overview of the joint history of the JAE and the CSIC, see the commemoration
book on the centennial year of the JAE (Puig-Samper 2007). See also (Sanz and López
1997).
To a large extent, the political history of the CSIC between 1941 and 1966 is that of
its general Secretary José María Albareda (1902-1966) , who exerted a strict control on
funding and appointments. In the early 1940s, the regime political ideology was a rather
foggy matter in Spain, but being in good relations with the military, or with officials of the
only allowed political party, Falange Española, or with certain social sectors close to the
Catholic Church like the Opus Dei organisation - the catholic group created in the mid-
1920s around the priest José María Escrivá (1902-1975) that had become quite relevant
in the zona nacional during the civil war- , or with the older and influent Asociación
Católica Nacional de Propagandistas (ACNP) from which the Opus Dei had branched,
was a necessary and sometimes a sufficient condition for success in certain academic and
scientific affairs. Nevertheless, struggles between the various groups over the control of the
educational and scientific policy systems were marked features of post-war Spanish poli tics.
The Education Minister and ACNP follower José Ibáñez Martín (1896-1969), was in office
for 12 years between 1939 and 1951, and he and Albareda, who was one of the earliest Opus
Dei affiliates, battled on the control of oposiciones - the usual name for public contests in
Spain- for chairs and other appointments through the presence of adequate members in the
examination boards (González-Redondo 2002a and 2002b, López-Sánchez 2006, Morente
2005, Peralta 2006).
Moreover, the bulk of intermediate Education and substantial parts of the primary were
largely put by the Education Ministry in the hands of the Catholic Church through severa!
religious orders, as an overreaction to the supposed laicity of the pre-war republican years.
In this way, part of the conflict between catholic factions carne to an end when people close
to the Opus Dei obtained control of higher education and research through the CSIC, while
intermediate and primary education were left as the business of the Ministry, channeled
through those religious orders. This state of affairs !asted for many years. As an example,
the protagonist of this biography wrote in his personal notebook: "El Sr Estella, alcalde de
Salamanca, opina que la educación es cosa del estado, lo que quiere decir de la Iglesia" [Mr
Estella, the Major of Salamanca, thinks that Education is a State matter, thus meaning a
Church matter] (entry of December 30th, 1962).
2 The Spanish rnathernatical panorama
The mathematical section of the CSIC, the Instituto ' Jorge Juan' de Matemáticas, active
between 1939 and 1982, concentrated in ita small élite that for years after the war governed
all official Spanish mathematical activity. The few available positions were unattainable
for candidates from outside the kernel gathered in the Jorge Juan, and Francisco Navarro
(1905-1970), an architect, mathematician and academician who held the chair of Rational
Mechanics at Madrid and was a personal friend of Albareda and Escrivá, chaired most
contests from the early forties until the earlyl960s, as a search in the Boletín Oficial del
Estado, the Spanish Official Gazette, easily shows. In this way a number of chairs in
Mathematics were soon awarded to young people whose fidelity to the new rulers was
out of question. Political loyalty to the regime was assessed in the early post-war from
a number of circumstances: For those who had spent the war in the zona nacional or
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had been enrolled in Franco's army, it was more or less an automatic matter, and having
been imprisoned or prosecuted in the zona roja could be a positive added value. In any
case, civil servants of both zones who tried to recover their posts once the war was over
were scrutinised in a depuration process whose result depended on a delicate network of
friendships and affinities, especially in the case of people who had had any sort of official
responsibilities in the zona roja at certain levels, even if they had not been directly involved
in political decisions.
In the case of Mathematics and mathematicians, most members of the pre-war school
around Julio Rey Pastor (1888-1962) could continue their work from within the Jorge
Juan, even promoting new lines after the usual negotiation for acceptation and funding.
A most typical example is Sixto Ríos (1913-2008) , a former student of Rey Pastor in
mathematical analysis, who described the pre-war years as los años de los investigadores
[the epoch of researchers] - he obviously included himself-, and after the war switched to
Statistics and eventually became the founder of today's Spanish statistical school. Ríos
maintained alife-long contact with Cuesta and tried to help him in various instances: in a
letter (Ríos to Cuesta, February 2nd 1945) he writes to Cuesta asking him to write a paper
of sorne 100 pages long to be published by the Jorge Juan, adding that there would be a
payment for that work. On the other hand, mathematicians who were actually punished in
the depuration process were more or less tolerated. To show but an example, the geometer
Pedro Pineda (1891-1983), who will appear afterwards, recovered his chair in spite of
informations against him, but the most interesting case is that of Tomás R. Bachiller
(1899-1980), who acted as Rey Pastor's alter ego in the post war years. Theoretically, he
could not occupy managerial or confidence positions, but nevertheless he was immediately
confirmed in his Madrid chair and appointed to the Jorge Juan from its very beginning. An
explanation is that he could be considered a safe value, for he had been - mathematically
speaking- quite active in the pre- and war years and had a large agenda of international
contacts which he soon reactivated through intensive correspondence and travelling when
possible. Bachiller started his post-war work by advising three PhDs to young people
who had been active in the side of the war winners, and travelled to Germany, Italy and
Portugal during WWII and to Princeton in early 1947. Moreover, on application as well
of the pre-war concept of doctorates as passports to tenured chairs, his young post-war
students obtained theirs immediately. See (Ausejo and Hormigón 1999, 2002), and a more
detailed account in (Pacheco 2014).
On the political landscape of those years, the official importance of Mathematics was
relatively small, for autarchy favoured other Sciences of a more practica! or applied nature,
e.g. sorne branches of Chemistry, of Physics, Pharmacy and applied Physiology. It is
enough to observe that Chemistry degrees were offered by ali twelve Spanish Universities,
while Mathematics degrees could be obtained only at three, namely Madrid, Barcelona
and Saragossa. An exception were the attempts to build aircraft during WWII, to whose
purpose even an Escuela de Aeronáutica, where personnel from the Jorge Juan were hired,
was created in a reedition of the previous Escuela Superior de Aeronáutica opened in
1928 that had been dismantled in the meantime, and for sorne years - after WWII, not
during it- Junkers and Messerschmitts were built in Spain until the early 1950s. In 1953,
when the Franco regime signed a cooperation treaty with the USA and Spain entered
the United Nations, the political situation slowly evolved into that of a more tolerable
dictatorship. Over the years, in addition to population increase and ageing of faculty,
this fact contributed to sorne mathematicians not directly involved in the Jorge Juan
to be eventually appointed to chairs and to become officially respected members of the
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established mathematical community. Professor Cuesta fits in this group, but he was
involved in the Jorge Juan as a collaborator only for a very short time in 1956-1957, again
with Ríos' support.
P ublication of mathematical results in Spain in the immediate post-war years was cen­tralised
in the Revista Matemática Hispano Americana (RMHA, series IV), the continu­ation
of the homonymous journal (series I and II) of the Sociedad Matemática Española
which sorne enthusiasts had managed to keep alive even during the war (series III) in the
zona roja. Results on elementary Mathematics and mathematical Education appeared in
Matemática Elemental (also series IV), which had been refounded as well in 1941 after
being discontinued by the war, see the accompanying figure. Its name was changed into
Gaceta Matemática in 1949. All these journals were published by the CSIC. The Revista
de la Academia de Ciencias also offered a vehicle for mathematical papers, both befare and
after the war, and the university of Madrid started as well in 1941 a Revista de la Univer­sidad
de Madrid, whose section Ciencias also accepted mathematical papers. A few years
later Collectanea Mathematica appeared in Barcelona: Now it is the oldest mathematical
journal in Spain, published without interruption since 1948.
~ U tX ClllNCIA El JEFE DEL ESTADO
Al«1111•nun MAll••AflU. Eu ... tHIAI .. 1 ......... jMriocjod .... publico­ción,
d••puéod•lc1vlct0<iooo Cr111oda1<td•nlOrt1,rind1- cólldomi....10
d1 ,......,..,, .. odl>ulón olo llgllfo próc1rd.!C.neroll.01110 Fronco, q111
•i•nl• .IP<edil.cto onh1lo6e n.,.•tro•'-•ocióftcienrilko«H110p<i .. •t
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::d~;,::;~:::11~~1:. .~ :. .: ..: ;:;~:~·,:~::::: ~.;: ;:;"" ·-;
yelpen14Mlen1opu1.to•nE1poi\o ~ '
Francoism and Mathematics: Dedication page in t he first after-war volume of Matemática Elemental,
1941. The grand iloquent rhetoric of t hese words is difficult to render into English.
ln spite of the centralistic mood of the regime, the higher divulgation journal Euclides,
mostly aimed to students preparing entrance examinations for the Engineering Schools,
appeared in 1941 with a large amount of Mathematics in it. Among the founders were the
mathematician José Barinaga (1890-1958), expelled from his chair at Madrid University
but later rehabilitated in 1945, and José Gallego-Díaz (1913-1965) a former communist
and enfant terrible who after the war completed his studies as an engineer and mathemati­cian,
Euclides survived until the 1960s, and it is an interesting example because its mere
existence shows something close to a Spanish mathematical community in the post-war
years and may be considered as a measure of the regime tolerance.
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Francoism and Mathematics: Welcome page in the first volume of Euclides, 1941.
Though there has been sorne arguing about the quality of materials in these periodicals
focusing on the lack of peer-review previous to publication, the objection may be overcome
by splitting the production into two different groups. As a rule, those who had already
published before the war went on doing so both in Spain and abroad, while the production
of those who started after the war was essentially published in Spain. Arguments favouring
the thesis of the low quality can indeed be found, especially if the Mathematical Reviews
are consulted: Taking as an example the case of Francisco Botella (1915-1987) - one of
those three young after-war PhDs and professors- the reader will find the usual reviewer
actually losing his patience with those papers. It is remarkable that the Zentralblatt für
Mathematik, though discontinued in 1943 and restarted in 1947, made an effort to review
the Spanish mathematical production during those years, and the lag was soon overcome.
Cuesta was a current author in the RMHA and later in Gaceta Matemática, and the reviews
of his mathematical papers in both journals are always either neutral or openly favourable
and long ones. A number of his papers were also reviewed in the Soviet Referativnii Zhurnal
Matematika from 1953 onwards, but those revisions were not available in Spain.
Part 11
Personal and academic data of Professor
Cuesta
Professor Norberto Cuesta Dutari (1907-1989), whose academic career spanned the whole
duration of the Franco regime, provides a most remarkable and interesting example of an
outlier mathematician as a result of interior exile in the sense explained above. He was the
youngest of five children of a couple formed by a local father and a Basque-Argentinian
mother, was born in Salamanca, the oldest university city in Spain, and lived and worked
continuously at his native place from 1947 until he passed away in 1989. Data on his life
have been obtained from his autobiographic last academic lecture (Cuesta 1978), a study
of his comprehensive last will, written in 1984 and opened in 1989, as well as from his
personal notebooks and articles published in newspapers.
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His father, over twenty years older than his mother, died when NCD was only six: therefore
his education was deeply marked by this early death, the strong influence of his mother and
two of his older brothers who had become Jesuits, secondary school at a Jesuit institution
in Tudela (Navarra), and a pre-university year in Argentina during 1924. He kept single
all his life and his only close relatives, living in Argentina, are the descendants of a brother
of his that migrated to that country in the early 1920s. They have inherited all private
papers left by the Professor.
The academic record of Cuesta includes studies of Chemistry at Salamanca and Mathemat­ics
at Saragossa. Upon obtaining his Mathematics degree he worked at Granada between
1933 and 1937 asan university assistant to the mathematician Juan Tercedor (1870-1956) ,
and in 1936-1937 he had to switch to secondary school teaching because the government
of the zona nacional - where Granada was- ordered Universities to stop their activity, and
late in 1937 he was sent to a secondary school in the small city of Teruel, quite close to
the war front, where a few days after his arrival a massive attack hit the town, which
was subsequently occupied by the republican army. Cuesta, as 'representating' the rebel
government, was captured and spent forty days in prison, then set free in Valencia, but
while trying to travel to Barcelona he was again imprisoned and spent about one year in
jail until November 1938.
Professor Cuesta around 1970. Picture taken from the Salamanca newspaper El Adelanto, February 6th
1989.
Once released, he was enrolled by the republicans and served as a soldier in administrative
tasks until the end of the war in March 1939. Then he underwent the depuration process,
and it is interesting to cite his own words on how it was conducted: "Mi expediente de
depuración, tras la guerra, estuvo detenido porque la Guardia Civil de Teruel me hizo un
rojo furibundo ( ... ) El razonamiento de ellos debió ser ¿lo conoce alguno? No. Entonces
'será' rojo" [After the war, my depuration file was delayed because the Guardia Civil at
Teruel made of me a furious red ( ... ) Their reasoning might ha ve been: Does anyone know
him? No. Then he "must be" a rojo (notebook, entry of December 16th 1962). Rojo (red)
was the name usually given to the republicans, in a clearly pejorative - and often dangerous
in those times- allusion. The Guardia Civil is the police force in charge of non-urban areas
in Spain since the middle of the 19th Century.
He was back in Granada for the academic year 1939-1940, and that same year, in a round of
contests promoted by the new authorities, he obtained a fixed secondary school professorate
at Ávila, a mere 100 km away from Salamanca. The next year he concurred again to another
national contest and obtained a new post, this time in Granada, at the secondary school
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of that city. The board for this contest was chaired by Navarro and a congratulation note
from him to Cuesta can be read among the correspondence preserved at the university
library in Salamanca. He remained in Granada until 1946 - and this seems to have been a
rather happy period in his private life, according to the correspondence with friends over
the years- when he moved to the small city of Segovia and then back to Salamanca from
1947 onwards, where he taught both at the secondary school and the university until 1966.
During his year in Ávila he managed to prepare a memoir on order properties of the real
numbers and sorne generalisations thereof inspired on his previous readings of Sierpinski
and Hausdorff, and with the help of Ancochea - in Salamanca since 1936 as a Professor- who
acknowledged its quality and agreed to act as adviser, he was awarded his PhD in Madrid
on February 5th, 1943. Cuesta's aim was to occupy a university chair, for which the PhD
was a necessary requisite, but he had to wait a long time to see his wishes fulfilled in 1958.
Indeed, university chairs in Spain had been awarded before the civil war to very young
people shortly after obtaining their PhDs if they happened to belong to the right group:
e.g. Ancochea himself, PhD 1934 within the Rey Pastor group, had obtained in 1935 his
chair at the age of 27. It has been shown above that this custom was readily continued after
the war by the new rulers, but apparently the last condition was not satisfied by Cuesta,
for he made three unsuccessful attempts before 1950 in contests presided by Navarro. The
main reason for not being appointed can be assessed to his relative isolation from the Jorge
Juan interna! politics and somehow outlier formation. It has been shown above that Ríos
tried to help him before one of such contests in 1945 by somehow involving him in the
Jorge Juan, without success. Moreover, in the mid-1940s he was already in his thirties and
that did not help.
As tenuring was by no means an easy matter, in order to duly install himself he participated
once more in a contest in 1950 to definitely obtain the secondary school professorate he
was occupying provisionally in his native city. At the university, a fourth opportunity was
at hand when a chair in Mechanics with teaching duties in Mathematics was offered for
public contest late in 1949. The examination board - once more with Navarro as president­was
published by the official gazette in 1953, and exams took place in early 1954. Cuesta
concurred to the contest, but the appointee was Jesús Tharrats (1923-2001) , a physicist
and brother of the artist Joan Tharrats (1918-2001), who only spent one academic year
in Salamanca before moving to Barcelona, and afterwards to Puerto Rico. The life and
achievements of Tharrats have been studied in (Soler 2008). As a result, Cuesta abandoned
the university but nevertheless, when Tharrats left Salamanca he was again lecturing at
the university, and even received an offer from Rey Pastor about moving to Argentina
(Rey Pastor to Cuesta, August lst 1957). On familiar grounds - he was already 49 and
was in charge of his only sis ter, seven years older than him and also single- he decided
not to go. Eventually the opportunity appeared for him to occupy a chair in 1958 after a
short collaboration with the Jorge Juan, with Pedro Puig Adam (1900-1960) chairing the
examination board, and finally obtained it at the age of fifty, by far much older than it
had been the rule all those years, and he proudly pronounced his inaugural lesson on May
12th 1958, though he did not resign his secondary school duties until 1966. An abridged
version of the memoir on the teaching of Mathematical Analysis he had prepared for the
contest was published shortly afterwards at the instance of Puig Adam (Cuesta 1958c). He
retired in 1977 and was emerited by the university until 1983. He died on February 5th
1989, exactly forty-six years after obtaining his PhD.
On the personal side, Cuesta was a cultivated person, with a solid cultural background
he attributed to his Jesuit formation, and broad interests outside Mathematics. One
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example was his study and edition of El Criticón, the masterpiece by the Jesuit writer
Baltasar Gracián (1601-1658) of the baroque epoch (Cuesta 1955b). This paper on the
interpretation of the complexities of the book through an analysis of punctuation is still
considered a fundamental reference on the tapie. He read severa! languages, including
Latin and sorne classical Greek, but always wrote in a very personal Spanish where he
insisted in keeping track of ethymologies, often disdaining the current orthographical rules
and wi th man y, ver y precise, quotations in the original languages spread both in faotnotes
and within the text (Pacheco 2007).
He was a deeply religious person, as stated by himself in his last will, but trusted very little
the Catholic Church and its satellite organisations. evertheless, he was always in good
terms with faculty of the catholic Universidad Pontificia at Salamanca, and even part of
his estate went to this institution.
Though in his university life he was never in charge of management duties, in a sense his
promoting Mathematics curricula far first-year students was something like that: In those
years funding was proportional to the number of enrolled students and the tuition fees of
the first and populous general-purpose year - the so called Curso Selectivo de Ciencias­represented
the source from which books were bought and journal subscriptions paid.
Nevertheless, although he was a political skeptic, between 1961 and 1966 he served as a
representative of the University in the City Council, where he took over the educational
area and contributed to plan a city school map and to implement hygienic measures in
primary schools.
Cuesta had an extremely strict sense of Justice - even to the point of rigidity- and his
personal behaviour was guided by a few salid principies: Hard work, an ordered life,
ceaseless effort, a good culture, a deep mistrust of official organisations, and an amusing
curiosity, sometimes mixed with a surprising sense of humour. Thus he acquired a name
as a polemic person, incurring in 'political incorrectness' whenever he was convinced of
his being right on anything. As a member of PhDs or contest juries he always read
the memoirs very carefully, annotated them, and farmulated many questions, a custom
that was sometimes mistaken with unpoliteness. In 1969 he even managed a thesis to be
rejected in Madrid when he showed that parts of it were strict copies of a certain text: He
produced the book and proceeded to a comparison between it and the memoir. Although
most possibly the copied pages were there as supplementary or introductory material, his
extreme attitude eventually earned him the serious enmity of a sector of mathematicians
in Madrid, with sorne bitter consequences in the years to come.
This peculiar behaviour led him to occasional crashes with ali sorts of establishment.
When the university awarded a honorary doctorate to the dictator Franco in 1954, Cuesta
was among those opposing that measure (see e. g. Claret 2004) , a fact that undoubtedly
had something to do with his not obtaining the Mechanics chair, and later on, while still
serving in the City Council as a representative of the university he quarrelled against the
local Bishop Mauro Rubio (1919-2000) when he realised that the projected demolition of
an old church in the city centre could be a cover far sorne speculative moves in 1965. He
went on tria! in 1967 and was condemned to a fine of pts 5000 (about half the monthly
wages of a secondary school teacher at that time) and "one month and one day" in prison
- he did not enter it- and the church is still there. Partly as a consequence of this frontal
crash with the Church, he was elected a corresponding member of the Academia de Bellas
Artes, and the Academia de Ciencias also elected him as a corresponding member. It
is not known why he was not appointed as a ful! member in this second one, where he
pronounced an inaugural dissertation in 1971.
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A lecture room in the Mathematics building and a street near the new university campus
have been named after him in his native city. His personal library, with more than five
thousand volumes and 123 boxes of diaries, notebooks, back-of-the-envelope computations,
drafts and letters are preserved in the university library at Salamanca. A chronological
summary of Cuesta's life is offered in Appendix IV.
Part 111
The mathematician
It is not difficult to establish how and why did he decide to become a mathematician, for
he declares to have been, on his own, an early discoverer of (the beauty of) Mathematics
at the young age of twelve (Cuesta 1975a):
" Yo recuerdo haberme hecho matemático, al entender los dos teoremas en que se funda la
regla del cálculo de la raíz cuadrada. Era el curso 1919-1920" [I remember becoming a
mathematician when I understood the two theorems on which the computation of a square
root is based. It was the academic year 1919-1920].
3 Early inftuences
Most possibly on familiar grounds, he first read for a degree in Chemistry at his hometown
in the period 1925-1929. Only after that could he study Ciencias Exactas - the name given
to Mathematics in Spain in the wake of the 1857 Ley Moyana- at Saragossa, where he ob­tained
his degree in 1932. From his student years at Salamanca, Cuesta always remembered
Professor Guillermo Sáez (1879-1932) as an outstanding teacher and mathematician, after
whom a street is named in Salamanca with no mention of who he was. It is known that he
had written a dissertation on elliptic functions in 1903 (Ortiz 2002) and spent nearly one
year with Holder in Leipzig during 1909-1910, where it is also possible he had met Haus­dorff
there befare this last one moved to Bonn. Sáez also delivered the opening lecture of
the academic year 1914-1915 at the university on Las Matemáticas aplicadas al estudio de
los fenómenos económicos [On the application of Mathematics to economic phenomenaj.
For sure, his lectures provided a firmer support to the mathematical vocation of Cuesta,
who had already bought and studied advanced Mathematics books during his years as a
Chemistry student. In his last life years, he attempted unsuccessfully a biography of Sáez.
Cuesta acknowledged the mathematical formation acquired in his student years at Saragossa,
but otherwise considered himself a sort of autodidact. Though, later he declared himself a
disciple of Rey Pastor, as shown by his two contributions to the centennial commemora­tion
of this one. A long letter by Ricardo San Juan (1908-1969) to Cuesta (no exact date,
but 1963) on the occasion of Rey's death shows light on Cuesta's mathematical ancestry:
" ... ¿Qué somos, pues, V d., Sunyer-Balaguer, Gaeta, A bel lanas y sus sucesores, A ncochea,
queriendo o no, sino una escuela de discípulos de Rey Pastor?" [ ... what else are we, i.e.
you, Sunyer-Balaguer, Rodríguez-Salinas, Augé, Castro, Vida! Abascal, Gaeta, Abellanas
and his offspring, Ancochea, want it or not, but a school of disciples of Rey Pastor?].
But the Professor was essentially a book reader, and a passionate writer: in his library
there is a dearth of carefully studied and annotated Mathematics texts, ali of them with his
signature and date of purchase: many had been bought and read between 1925-1930 while
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N. Cuesta in 1934, age 27, shortly after obtaining his Mathematics degree. Place and
photographer unknown. Photo: Archivo Universitario, Salamanca.
still being a Chemistry student. While a student in Saragossa he prepared in 1932 a set of
beautifully handwritten lecture notes on Geometry in two volumes for self-study after the
lessons by Pedro Pineda. They are preserved and are cited in the last will as something
valuable that might deserve publication to honour the memory of Pineda. Though a rather
abundant correspondence with Ancochea, Ríos, San Juan and other Spanish mathemati­cians
has been preserved, no mathematical considerations of interest have been drawn after
it. But sorne glimpses of the life within the Mathematics community can be obtained. A
couple of examples: "No veo apenas a nadie. Únicamente a Flores y Gaeta" [I hardly see
anyone. Only Flores and Gaetaj (Ancochea to Cuesta, September 22nd 1948). " ... ningún
Cebollero audaz llegaría a la universidad." [ ... no valiant Cebollero would have ever entered
a university.] (Gaeta - from Rome- to Cuesta, December 26th 1950). Cebollero refers
to Pedro Abellanas Cebollero (1914-1999) , and here it is used caustically by exploiting
the phonetical similarity between caballero (gentleman) and Cebollero (onion grower), by
pointing to El caballero audaz, the main character in a homonymous series of popular novels
by the writer José M. Carretero (1890-1951). Actually, Gaeta is wondering how Abellanas
could have deserved working at any university. Federico Gaeta (1923-2007) and Antonio
Flores de Lemus (1912-1992) were two other enfants terribles: Flores - who had written
his thesis in Vienna with Menger before the civil war- was not allowed to officially teach,
and Gaeta, an outstanding geometer, was a prolific creator of many sarcastic wordplays,
Abellanas being one of his favourite victims. Cuesta also played this game: At least in one
occasion he spoke about a meeting organised by Abellanas as "horticultura racional" jok­ingly
referring to both family names of Abellanas (hazelnuts) and Cebollero. The allusion
is celebrated in (Gaeta to Cuesta, December 3rd 1950). The author of this paper heard
him many times referring to Abellanas by his second family name in a rather unpolite and
funny way of naming him.
4 Order Theory
On how the Professor undertook research in set-theoretical questions, he writes in his
autobiographical essay (Cuesta 1978):
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"Yo había frecuentado mucho, desde que salió el 1929, el bellísimo libro de Sierpinski Lé<;ons
sur les nombres transfinis" [I had often read, since it was published in 1929, that most
beautiful book by Sierpinski, Lér;ons sur les nombres transfinis].
Therefore he might have considered: first , that this topic could be worked out on an indi­vidual
basis with a few basic books and journals; and second, that his prior knowledge of
Sierpinski and especially of the groups of papers by Hausdorff, Untersuchungen über Ord­nungstypen,
published in the Berichte der Abhandlungen der Sachsischen Akademie der
Wiss enschaften zu Leipzig [Proceedings of the Saxon Academy of Sciences at Leipzig], for
short Leipziger Berichte, in 1906-1907 was a good starting point for doing mathematical
research. Although he had also read the classical Grundzüge der Mengenlehre (Hausdorff
1914) , he always preferred to refer to the Berichte papers. In any case, he did not ask
for a research topic from sorne would-be future thesis advisor, and he developed a life­long
independent mathematical production of a very personal and original character, even
managing to correspond with his respected Sierpinski severa! times during the 1950s.
According to Cuesta's mathematical conception, if all Mathematics were to be described
in a set-theoretical setting, techniques should be provided for building larger and larger
sets in terms of smaller ones, therefore including the necessary tools for counting, order­ing,
classifying, and naming the newly created objects in a systematic and thorough way.
Ordering and naming general sets were the problems Cuesta chose to study. He addressed
the following questions, on which he spent sorne twenty years, between 1940 and the early
1960s:
l. How to build all linear - also known as total- orders on arbitrarily large sets. The
problem was solved in an original way in his doctoral dissertation, where in addition
he compared the performance of his method with the classical lexicographic one by
Hausdorff. Cuesta's solution was rediscovered by Sierpinski in 1949 and by Chipman
in 1960.
2. How to name ordinal numbers in all generality, the so-called onomastic problem (see
e.g. Kleene 1938). This is still an unsolved problem.
3. Generalisation of n° 1 above to partial order relationships. He achieved it in 1955,
incidentally rediscovering the now famous Sierpinski triangle (1915) by offering a
complete construction.
Very elegant results were obtained by Cuesta on how to construct all total order types on
any set - in particular on the real line- by diadic constructions, and later, triadic construc­tions
for partial orderings, adding elements one at a time and comparing its performance
with lexicographic orders, a research that in those years was in the mainstream of Order
Theory (see Denjoy 1946, 1952, 1954a, 1954b, Cuesta 1954c and 1956b, Johnston 1956) ,
and is still cited today as fundamental work on the topic. Many years after the original
proof obtained for the first problem, that wonderful result was to find important appli­cations
in Decision Theory and Econometrics, and is a standard citation (Fishburn 1974,
Indurain 2002). The theorem, of which a sketch of proof is offered in Appendix 111, reads:
"Let (X,--<) be a totally ordered set anda the first ordinal number such that IXI = lal.
Then (X,--<) is order-isomorphic with sorne subset of the lexicographically ordered function
space ({0,1}ª ,--<1ex) ".
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N . CUl.STA OUTARl 1
~::.~~~
~-..Y
. NUMEROS REALES .
OE.NERALIZADOS
Cuesta's personal copy of his first publications, with a record of citations and reviews.
More than thirty years later, the American mathematician John Conway described a num­ber
domain generalising the real numbers (Conway 1976), which was popularised by the
mathematician and computer scientist Donald Knuth under the name 'surreal numbers'
in an homonymous 1974 mathematical novel (Spanish translation, Knuth 1979): They
are equivalent to one of the constructions by Cuesta, and Norman L. Alling, a specialist
on these topics, even visited him at Salamanca by Easter 1987. See the references and
Appendix V on citations.
5 Publishing efforts
In spite of sorne sporadic apparitions of his name as a problem solver in Matemática
Elemental before the civil war, Cuesta's publishing life as a mathematician started with
his PhD thesis, split into a series of articles in the first numbers of the RMHA, series IV
(Cuesta 1942, 1943a, 1943b). The thesis itself is the third reference, the other two being
complementary materials included at the suggestion of Ancochea (letter from this one to
Cuesta, May 26th, 1941). This suite was later completed with a number of further results
obtained until 1955, and the whole set was reworked and published (1958-1959) in a three­article
format (Cuesta 1958e, 1958f, 1959a) by the Revista de la Academia de Ciencias
after being awarded a prize by this Academy in 1957, most possibly at the suggestion of
Ríos as a recognition to those long research years without much academic success: indeed it
helped Cuesta in his finally obtainig the chair the next year. The complete set of papers was
published in book form - a short edition of a mere 100 copies- under the title Matemática del
Orden (Cuesta 1959c) by the same institution. A facsimilar edition was prepared in 1995 by
the University of Salamanca using one of those exemplars with sorne autograph annotations
by the author. This contribution to pure Mathematics was and still is a really outstanding
one, as the German mathematician Egbert Harzheim, who corresponded with Cuesta in the
sixties, duly acknowledged in his encyclopedic book Ordered Sets (Harzheim 2005) . The
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postal exchange between them began when Cuesta pointed to Harzheim that sorne results
appeared in (Harzheim 1965) had already been obtained by him years before. Matemática
del Orden was translated into French in the mid-1960s by the author himself and two
French-speaking people, but it was never published and the handwritten manuscript can
be found at the Salamanca university library. It seems that there was also an attempt to
prepare an English translation, but no written evidence of it has been found (Emiliano
Hernández, personal communication, 2012).
The problem of finding a mathematically sound rule for naming ordinal numbers was
addressed in severa! papers between 1949 and 1981, but a solution could not be found; in
the meantime he had offered the problem as a PhD topic to Carlos Durán (1937-2012),
who could not develop it. Years after, in 1982, Cuesta presented to the Academia de
Ciencias a communication by the Seville mathematician Juan Arias de Reyna containing an
impossibility proof under certain conditions (Arias de Reyna 1984). The interest accorded
by the Professor to this problem can be read in his encyclopedia entry (Cuesta 1991).
The archive also contain a small amount of papers classified by the Professor himself as
unpublished, but curiously enough a paper in German - which will be referred to below­does
not appear among them. It is difficult to assess to what extent they were left unpub­lished,
for the Professor's work-in-progress method and the continuous recycling of ideas,
fragments, and quotations show that even if those papers were not published as such, their
content is present in other printed materials, as a rule more than once.
6 Attempts to publish in German and in English
Ali published materials - but one item- by the Professor are in Spanish, although he made in
1953-54 an attempt to publish in German with the help of the then young German student
Rudolf Haller, who was spending one year in Salamanca on an interchange between the
university and the Maximilianeumsstiftung of Munich, by sending through him a long paper
to the Munich mathematician Georg Aumann (1906-1980) , for this one to consider it for
publication in a German journal. The paper, from which a carbon copy of the manuscript
and a number of drafts with handwritten notes by both Cuesta and Haller are preserved,
has the heading "Algebra der Ordnungen. Van N. Cuesta in Salamanca" , is sorne thirty
pages long and was translated by Haller, who moreover completed a number of details
at Aumann's suggestion (personal communication by Rudolf Haller, March 2013). The
introduction reads: In diesem Artikel führen wir notwendige Begriffe und Operationen ein
zur Inangriffnahme des Studiums der Partialordnungen, um so der Algebra der Ordnung
ein Fundament zu geben ... [In this paper we introduce the necessary concepts and operations
for a complete study of partía! orderings, with the aim of establishing the foundations of
an Order Algebra ... J.
A letter from Haller to Cuesta (July 23rd 1953) invites him to send the manuscript to
Munich, it was sent July 31st 1953 and then translated and passed to Aumann on Septem­ber
3rd 1953. After that, Haller asked Cuesta to provide him sorne technical explanations
on the paper, which were sent, and on January 16th 1954 Haller writes again asking for
more explanations and telling that according to Aumann, it would be possible that Duro
Kurepa (1907-1993) had a complete solution to the problem posed by Cuesta, for he had
listened in Salzburg to a talk where Kurepa had spoken on the generalisation of the facto­rial
to transfinite numbers larger than No , a problem already addressed in (Cuesta 1945)
and much later in (Cuesta 1977b). Upon receiving an answer, on March 25th 1954 Haller
writes that Aumann has already received that definitive version. There is no more corre-
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spondence, and therefore no evidence of a refusal letter from Aumann or any other editor
of the intended journal, but the fact is that the paper went never to print. In the 1980s
Cuesta sent a pack of his publications to Kurepa, who acknowledged receiving it and wrote
back that "we have several common points in our works" and promising that " certainly I
shall have the opportunity to quote you" (Kurepa to Cuesta, April 20th 1984) , as Harzheim
had already done twenty years before.
As a consequence of the failed paper in German, Cuesta and his French friend Odile Broucke
- who later was to collaborate in a translation of Matemática del Orden into French- wrote
a poor English version of the main parts of the paper, which was published in the Acta
Salmanticensia (Cuesta 1955d). Those Mathematics surely deserved a better fate than a
precarious attempt into English language. In the accompanying footnote a letter, sent in
1982 by Cuesta to the author of this paper is shown, regarding a possible new English
translation of that 1955 paper. The task could not be performed, basically because at that
time the would-be translator hardly understood the Mathematics in it.
In any case, the letter clearly shows how well Cuesta knew the value of his Mathematics:
In addition to knowing the reviews in Mathematical Reviews and Zentralblatt, this last
one by Kurepa himself, he shows his concern about t hat in the Referativnii Zhurnal and
with sorne humour he refers to his English as aljamiado, in a joking reference to that old
medieval Spanish written in Arabic characters, thus acknowledging that the paper was
actually written in Spanish but with English words. As in those years he corresponded
with Marshall Stone (1903-1989), who even visited him at Salamanca in 1957, the reader
learns that Stone had told him that the text was not in English. Most interesting is the
reflection on his 1943 dissertation, which he considers a better work, from which the 1955
paper would be a rather natural by-product, possibly deserving a joint publication of both
results.2
2Transcription of the letter from N. Cuesta to the author with sorne reflections on the 1955 paper. In
order to convey the typical phrasing of the professor, the author of this paper has not felt the need to
translate it.
Salamanca, a del del 1982 Sr. Don ... Mi querido amigo: Te devuelvo mi viejo artículo que pusimos en
un extraño idioma "aljamiado" de inglés entre Broucke y "moi". Es evidente que ganaría bastante escrito
en inglés, y encima por quien, como tú, se lo ha estudiado y puede quizá sacarle algunas extensiones útiles.
Yo ahora ya no puedo hacer esto último, pues quiero terminar el libro "de la invención e introducción del
análisis infinitesimal en España", sólo terminándolo, servirá para algo la labor informativa que efectué en
archivos, en viejas revistas, y viejos libros. He revisado las reseñas que de ese trabajo conozco. La del
Zentralblatt 65 {1956) 284-285 de 24 líneas la hizo K urepa (es croata, casi de mi edad) inventor de las
"ordenaciones ramificadas". Tiene un pequeño error, pues dice se pone I < F cuando algún i y algún f de
los respectivos verifican i < f. No tiene eso interés; se ha de verificar i < f cualquiera que sean i y f. La
del Math. Rev. 17 (1956) 950 de 21 renglones la hizo Bagemihl (Notre Dame, Indiana U.S.A.) que me
reseñó bastantes. No alude al "inglés" del artículo: Stone me dijo que "no era inglés". Desconozco la de
las "Re/eralas Mathematicas" de los rusos, que no sé si se podrán encontrar en Madrid, o que quizá tenga
Emiliano Aparicio en Bilbao. Quiero advertirte que yo lo incluí, pero con una redacción muy diferente,
en mi "Matemática del Orden" (Rev. Acad. Cien. Madrid Tº 52 y 53, 1959) A mí me satisface más mi
tesis (Rev. Mat. Hisp-Am 1943) allí titulada "Teoría decimal de los tipos de orden" Si hoy la publicara
la la titularía más expresivamente "Construcción diádica de todas las ordenaciones totales" En realidad
los dos trabajos podrían juntarse en uno, y si tienen algo que aún me satisface es que la construcción es
sistemática, introduciendo los elementos uno a uno y haciéndolo de todas las maneras posibles: por eso se
obtienen "todas" en un caso y en otro. Tú harás con él lo que te parezca más oportuno y yo te lo agradeceré
en todos los casos posibles. Un abrazo muy cordial, N. Cuesta.
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7 Metamathematics and more
While doing research on order theory, Cuesta became interested in Metamathematics, a
field which occupied him for sorne time: as an example, in the seminars he conducted on
Mondays the then recent 1970 Mathematics of Metamathematics by Rasiowa and Sikorski
was read and commented during one year. This interest on Metamathematics was essen­tially
motivated by his attempts to provide a sensible conception of mathematical rigour
based on first principles. For him, principles are established by a deep analysis of the very
origins of Mathematics in real-life problems through writing down the most precise defini­tions
of all concepts involved and working them out with a minimum of added typographic
and syntactic complexity. This seemingly intuitive way of understanding Mathematics
coexisted peacefully in his theoretical universe with the existence of the actual infinite
quantities he had freely used in his basic papers. Consequently, for him the paradigm
of axiomatisation was not the way to advance in Mathematics. To support this vew he
often cited sorne words by Paul Lorenzen (1915-1994): "Mit geeigneten Axiomen laflt sich
nun zwar alles beweisen, aber nichts begründen" [from adequate axioms anything may be
proved, though nothing founded] (Lorenzen 1951). A deeper analysis of this quotation
and its consequences led him to consider the difference between axioms and postulates,
establishing a time-like priority of axioms on postulates (Cuesta 1962a). This was his con­ception
of rigour, far away from those by Cauchy and Bourbaki. He wanted to build his
Mathematics on the solid ground of sorne Metaphysics-free world where words are univoca!
and ful! of meaning, as his admired Leibniz had also tried to do. More papers on these
topics are (Cuesta 1951a, 1954b, 1961, 1962c).
In the above considerations lie the roots of Cuesta' interest on the work of Euler, whose
traditionally supposed lack of rigour he tried to definitively refute by considering that the
long first part of the Intmductio in Analysin Infinitorum may be thought as a catalogue
of precise definitions needed to found the wonderful results in its second part, where the
main mathematically interesting questions appear. The conclusion drawn was that rigour
in Euler is an ab initio condition, which evolves according to the mastery of words: in
this treatise, words are precisely those formulas compiled in the first part. In this sense, if
Mathematics is not a language, it is something close to it (see Lorenzen 1956).
INTRODUCTIO
I N ..INA L TS I N
IN F INITORUM.
.IUCT O lt.E
LEONHARDO EULERO,
P,..ftffwt Rtfjo Bll.OLl!fUUI> d ,A<,w,,,¡41,,..
}'O'UJil~ Pan.OPOLITUIA
S«io.
LAUSANN.if. ,
.A,..t M .ucuN-M1c n 1olLIN 80111QJ11T 11r Socio..
LA SINFONIA
DEL INFINITO
Y Y• tn el ~rellltl ck 1!1'llu
N'ORBHRTO CUESTA DUTARl
Cover pages of Jntroductio in Analysin Infinitorum, by Leonhard Euler (1748), and La Sinfonía del
Infinito, by N. Cuesta(1981)
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As a result of his quest for rigour, in the mid-sixties Cuesta started to turn his attention
to sorne topics in the History of Mathematics and to the writing of a series of very original
books, as well as sorne incursions on didactical questions and on the practica! teaching of
Mathematics at various levels. 1969 marks the end of his prolific collaboration with the
RMHA - to which he had contributed thirty-nine papers since 1943- and his subsequent
production was disseminated mainly in Gaceta Matemática (thirteen papers), in talks at
the annual meetings of Spanish or Iberian mathematicians, and other small publications
of more reduced circulation, but above all in his books. His last article in a broad diffusion
mathematical Spanish journal was published in Collectanea Mathematica: It dealt with
the arithmetic of twin primes (Cuesta 1986a). He had also contributed to this journal with
two more papers during his years of interest in Metamathematics, among them the already
cited (Cuesta1962a).
The last original mathematical writings by Cuesta, also on arithmetical problems, were
three articles published in the ephemeral journal Maternas a group of Mathematics students
at Salamanca kept active between 1986 and 1988 (Cuesta 1986, 1987, 1988).
A few papers were published in Philosophy journals, like his refiection on the infinite in the
CSIC's Revista de Filosofía (Cuesta 1958a), and two papers (Cuesta 1980a, 1980b) in the
heterodox journal El Basilisco edited in Oviedo by the philosopher Gustavo Bueno, who
in addition to having been also a secondary school professor at Salamanca was among the
first editors of the philosophical journal Theoria founded in 1952 by the logician Miguel
Sánchez-Mazas (1925-1995), for which the paper ultimately published as (Cuesta 1958a)
was intended in 1953. It was rejected there because Rey Pastor thought it to be "too
mathematical" for that journal (letter from Sánchez Mazas to Cuesta). By the way, Theoria
(series II) is still published today.
In whole, Cuesta's production comprises over ninety papers and presentations in Mathe­matics
meetings, and five books with different printings. He never co-authored any item,
but that was the rule during his creative life. A hopefully complete list not including
newspaper collaborations may be found in Appendix I.
It is interesting to remark here that Cuesta proceeded with a typical pattern or epistemic
method, a combination of severa! years of work-in-progress anda final compilation or 'clo­sure
step', clearly inspired on Cantor's construction method of transfinite ordinal numbers
so clearly explained in Hausdorff's Grundzüge that had provided one of the threads of
Cuesta's original research interests. Matemática del Orden was the first of these episodes
in 1959.
There were two more closures. A second one has to do with his life-long involvement in
teaching, will be considered in sorne depth below and can be dated in 1981.
The third and last one happened in 1982 and is contained in his long plenary address intitled
Matemáticas finitas, matemáticas asintóticas, matemáticas infinitas, presented at the IX
Jamadas Hispano-Lusas de Matemáticas [the yearly meeting of Iberian mathematiciansj
that took place in Salamanca: It is a sort of uninhibited mathematical last will with deep
ideas on the general evolution of Mathematics and sorne particular areas thereof (Cuesta
1982b) .
8 Advising PhD theses ( or not)
A shocking and somehow contradictory point in the university life of Cuesta, who spent
sorne forty years in Academia and was quite involved in its development, is the very small
number of PhDs he advised. Various explanations may be offered, one of them is indeed
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related with isolation through the lack of Mathematics studies in Salamanca, where the only
people who could undertake a PhD work were not too young persons in the secondary school
area with little time or forces to devote themselves to such a task, but it should be added
that his conception of rigour in the mathematical enterprise did not help much to invite
possible candidates to work under his guide. To put but an example, Felicísimo Albarrán
was another secondary school teacher at Salamanca who also taught at the university, he
was a few years older than Cuesta, and there is a handwritten (undated, around 1963)
note by Cuesta to the Dean of the Facultad de Ciencias explaining that Albarrán felt
tired because he was about to become sixty and in whole his teaching duties amounted to
lecturing 34 and a half hours a week during that academic year.
Only two students, Pablo Carpintero and María Isidro, completed their PhDs under Cuesta,
and both of them were teaching at secondary schools (Carpintero 1970, Isidro 1977). The
first one, on infinite Boolean algebras, was an output of the Monday seminars and the whole
process was completed in less than one year: An idea originated in one of the sessions in
early 1970 was developed by Carpintero, and in June 19th 1970 he sent a manuscript to the
Professor, who readily considered it an acceptable job. The PhD was awarded in December
1970 anda number of papers stemming from it were produced afterwards (Carpintero 1971,
1972). On the other hand, Isidro's PhD was an expression of the interest on the rigour
in the summation techniques of Euler Cuesta had developed in the early seventies. It
was never independently published, and only a short abstract in booklet form was printed
by the university in a reduced edition, as it was customary at that time. More will be
commented in the section devoted to Cuesta as a historian.
On theses that did not succeed, the case of Carlos Durán and the naming of ordinal
numbers has been already cited. The story of Eulogio Hernández (1922-1997) and his
historical research in the early 1970s on the introduction of Infinitesimal Analysis in Spain
will be dealt with below more in extenso. Another would-be student, Jesús Gómez (1937-
200X), also a secondary school teacher, worked for sorne time on ideas of Cuesta, who never
seemed to be satisfied with his results, and finally managed to write a memoir on convexity
properties of polyhedra in normed spaces, which Cuesta recommended for presentation in
Madrid under the formal advising of his old acquaintance from the Saragossa years Pedro
Abellanas - the victim of the jokes by Gaeta et al.- who in the meantime had become an
influential figure in the Spanish mathematical landscape. It was read in Madrid in 1980,
but never published (Gómez 1981).
Two tesinas -an old rough equivalent of a Master's thesis- of mathematical content were
also advised by Cuesta: Influencia en la primitivación de las integrales racionales de la
inexactitud de las raíces del denominador [On the influence of errors in the denominator
roots in the computation of primitives of rational functions] by Emiliano Hernández (Sala­manca,
1968), latera physicist, and Estudio crítico sobre ordenaciones [A critica! study of
order properties] by Puri Galindo (Universidad Nacional de Educación a Distancia, 1981),
latera statistician None of them originated published papers.
Part IV
The professor and the teacher
On the teaching side, the university professor Cuesta always prepared his lectures very
carefully: After them, by studying the blackboard - he tried to employ it only once without
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erasing- the leading thread could be easily followed and often the whole contents retrieved.
He used to write a lot, not only formulas, but also text, made many drawings, inserted
historical data, and even those little comments usually left as casually spoken remarks
deserved sorne chalk. Exams were read in an orderly way, according to the place students
had occupied while writing them, and commented with a lot of red pencil. During the exam­reading
periods the dedication of the Professor to this task was complete: No research, no
reading, only exams at a steady pace of sorne 15 a day.
His textbooks were a natural outgrowth of many lectures, refiections on them, and inten­sive
written ellaboration of the topics expounded: A very fine one is Geometría Vectorial
(Cuesta 1968b), intended asan intuitive introduction to Linear Algebra and Analytic Ge­ometry
and sorne applications thereof, extending up to the basics of elementary differential
operators and Stokes' theorem for multiple integrals. The origin of the book can be tracked
back to the preparation of the Mechanics chair, for which Cuesta - a non specialist- had
written a complete set of lecture notes for self-study. It was very popular in Latin America
for many years, and the author inaugurated in it the custom of inserting long and polemic
forewords, where he expounded his personal ideas not only on mathematical matters, but
also on Philosophy, Logic, History, Religion, university policy, general Politics and many
more. There are evidences pointing to a possible second edition, but it never went further
than a handful of handwritten notes. Compared with his chef d'oeuvre, the Sinfonía del
Infinito, Geometría Vectorial is totally independent of any research interest. The Sinfonía
will be presented in the next section on the contributions to the History of Mathematics.
As a secondary school teacher, Cuesta's idea on mathematical Education at the non­university
levels was essentially that Mathematics must be presented and studied from
a global viewpoint, insisting on the close relationship with language and other general
abilities, in addition to the automatisation of basic computational and geometric skills. His
opinions and elaborations on how to teach Mathematics are spread in a number of papers,
most of them in a lesson format, published in Gaceta Matemática or as presentations in
meetings from 1960 onwards. For instance, (Cuesta 1963) contains a lesson on algebraic
structures in the domain of rational numbers, in (Cuesta 1965b) the topic of Dedekind
cuts is explained at an elementary leve!, (Cuesta 1966d) is a long lesson on analytic and
synthetic geometry, and the classical core topic in Linear Algebra, the Rouché-Frobenius
theorem, is addressed in (Cuesta 1966f). A sharp critique regarding the onomastic problem
can be read in his paper on the Von Neumann ordinal numeration (Cuesta 1968c) where
it is shown that Borges' principie is violated in that construction. A number of letters
sent to the Professor by secondary school ex-students over the years remember him as an
excellent teacher.
During the seven-year period between 1957 and 1964 he conducted an interesting experi­ment
in Salamanca by following the evolution of a group of young students along the whole
secondary school track, at that time, in ages between 10 and 17. The aim was to study
the mental evolution of those children, not only in the field of the Mathematics he taught
them during ali those years, but also from a broader cultural and personal perspective. Six
boxes of student notebooks and other related files are in the archive, but no conclusions
seem to have been drawn by Cuesta himself on the experiment . Quite recently, a study
has brought it to light with a detailed analysis of the notebooks content and including
personal interviews with sorne of those students (González-Astudillo 2012). It confirms
the importance given by the Professor to the global relationship between Mathematics and
other disciplines in the school curricula, especially language abilities. Whether such an
experiment could have taken place in our days is an open question, but the mere fact that
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it was actually performed on a personal basis, without a supporting team, is a mark of
how isolation and cultural context also imprinted Cuesta's behaviour as a teacher.
Part V
A historian of Mathematics
The interest on historical questions was a constant in his mathematical endeavour, and he
recognised the methodological differences and affinities between Mathematics and History,
being well aware of the techniques historians employ in their work and thoroughly applying
them by using only primary sources whenever it was possible. The first strictly historical
paper - where his mastery of the technique is clear- was the obituary of José Barinaga
(1890-1965), containing a most interesting chapter with a detailed analysis of the polemic
1932 contest for a chair on Differential Equations in Madrid, where Barinaga acted as
Secretary of the board that did not appoint the already influential scientist, engineer and
entrepreneur Esteban Terradas (1883-1950) (Cuesta 1966c). The story of Terradas and
Rey Pastor in the Spanish mathematical panorama after the civil war has been studied
in depth by severa! authors (see e.g. González-Redondo 2002b, Malet 2008 and references
therein) . Another historical paper is the short obituary of San Juan (Cuesta 1969b),
but the inaugural address of 1971 at the Academia de Ciencias, titled Filosofía natural y
pugna de fa cultades en la Universidad de Salamanca, a fine piece of historical research with
no Mathematics, can be considered the starting point of professionally written historical
publications. Actually, it is a sort of introduction to the encyclopaedic biography of Juan
Justo García (1752-1830), an ilustrado [enlightened], later idéologue and liberal politician,
and author of a number of Mathematics textbooks who taught infinitesimal analysis at
Salamanca for the first time in the history of that university (Cuesta 1974b). The reputed
arabist and historian Juan Vernet (1923-2011), wrote about this biographical essay: " ... me
ha apasionado más que una buena novela" [ .. .I have found it more exciting than a good
novel] (Vernet to Cuesta, April 4th 1975).
On the History of Mathematics, Cuesta basically dealt with two questions:
l. The invention of Infinitesimal Calculus and its introduction in Spain .
2. A thorough study of the methods and ideas of Euler on series summation.
In the first part of the book Historia de la Invención del Análisis Infinitesimal y de su
Introducción en España, the Professor claims to have established that the main tools of
the Newtonian approach to Infinitesimal Analysis were the 'first and ultimate ratio' method
anda skilful manipulation of series expansions starting with the binomial formula (Cuesta
1976). The same ideas have been described by Niccoló Guicciardini in his books on the
history of Analysis in Britain (Guicciardini 1989, 1999). On the other hand, he wrote at
length on the Leibnizian approach and the idea of actual infinitesimals, and determined
that the true merit of the Nova Methodus was the obtention and integration of a differential
equation for the first time in history. The Historia also shows that Infinitesimal Analysis
was brought into Spain by the Jesuits and taught for the first time at the University
of Salamanca by the cited Juan Justo García, to whom he had devoted the exhaustive
two-volume biography already referred to.
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The second part of the Historia had had its origin on that projected and failed PhD thesis
by Eulogio Hernández on the Introduction of Mathematical Analysis in Spain. Hernández
had found in the archives of the Real Academia de la Historia evidences of the first Spanish
incursions into Infinitesimal Analysis by the Jesuit Tomás Cerdá (1715-1791), but his
findings and posterior study of the papers were not enough for him to deserve a doctorate.
When he tried to present his work in Madrid with Cuesta's approval in 1971-72, the
1969 thesis affair showed its consequences: anything coming from Cuesta was put on
a serious quarantine, and the scientific excuse was the short sighted argument that the
History of Mathematics could not be paramount to real Mathematics 'because of its lack
of mathematical method'. Cuesta blamed the then Dean at Madrid, the analyst Baltasar
Rodríguez-Salinas (1925-2007) for having agreed with the decision made by the governing
committee at Madrid banning theses on historical topics (draft of a letter from Cuesta
to Miguel Artola, no date, but September 1976). Therefore Hernández only obtained an
acknowledgement for his contribution to the History of Mathematics in Spain in sorne
paragraphs of the Historia. To the contemporary reader, large parts of the book may seem
a thriller with the vivid descriptions of visits to the Spanish Historical National Archive in
the small town of Simancas, orto the Royal Palace in Madrid, where a fierce Cerberus-like
librarían did not even allow the Professor to consult the library catalog, as well as the joy
of having found a number of volumes of the Philosophical Transactions and other materials
at the Royal Observatory of San Fernando, in Andalusia. The Historia was printed twice,
in 1985 and 1994. The 1985 printing deserved a one-page long review in Studia Leibnitiana
(Knobloch 1986), and the review by Ubiratan D'Ambrosio in Mathematical Reviews was
commented in sorne scholia by Cuesta which were included, as well as sorne handwritten
corrections of his own, on the 1994 posthumous facsímile reprint of the original manuscript
On series summation, he aimed to establish that Euler 's approach to that problem was
as rigorous as any other ulterior development, a recurrent topic in his lectures where he
insisted on the ontological status of infinitesimals and infinities and on how to use them
in computations on a sound basis: He was very familiar with the works of Leibniz - or
Leibnitz, as he always thought was the right spelling- and those of Du Bois-Reymond on
the frontiers of convergence (Du Bois-Reymond 1871, 1876, 1877a, 1877b, 1882, Fisher
1981) especially Die Allgemeine Functionentheorie, and other classical studies on series
convergence and summability. He acknowledged an early interest on these topics to Ríos,
who as an analyst had written his PhD thesis in the 1930s on series summation, a field
Rey Pastor had introduced in Spain in the last 1920s and to which he made sorne original
contributions. Always according to his conception of rigour, the main results establishing
that sorne methods of Euler are actually rigorous ones were included as the long 16th
Chapter of the third and definitive edition of the surprising saga of re-elaborations that
led to that magnificent treatise on Mathematical Analysis - or rather, a true and very
personal history of Analysis- finally intitled La Sinfonía del Infinito, y ya en el paraíso
de Euler (Cuesta 198la), a clear allusion to the deep importance he accorded to Euler
in the development of modern Mathematics. A reference showing that the topic is still
interesting and alive is (Ferraro 2008). Actually, that chapter contains with very minor
changes the PhD memoir of María Isidro, who is acknowledged there for her achievement.
As the Sinfonía was not reviewed, neither by Mathematical Reviews nor by Zentralblatt für
Mathematik and no further papers followed it on the series problem, there are no citations
of Cuesta on this topic, e. g. sorne classical references (Dutka 1996, Ferraro 1998, Katz
1987, Kline 1983) do not contain any.
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Although the Sinfonía had had its origin in classroom notes and the first version appeared
under the modest title Cuarenta (40) lecciones de Análisis Matemático (Cuesta 1974a) , the
1975 enlarged edition already added the words La Sinfonía del Infinito in the title of the
larger Sesenta y siete (67) lecciones de Análisis Matemático (Cuesta 1975), and distilled a
decidedly historical fiavour , not only for its many footnotes and quotations, but for the ex­position
of most materials in a sequentially coordinated manner. The complete title of the
third revised and substantially enlarged version - which was to be the definitive one- was
even a longer one: La Sinfonía del Infinito, y ya en el paraíso de Euler, Noventa y nueve
(99) lecciones de Análisis Matemático. Its writing was a real tour-de-force, where many
new parts and comments already published elsewhere were included, e.g. his short auto­biographical
Lección Académica Final from 1977, as well as the ideas of the 1968 address
on Von Neumann's construction of natural numbers, and many others: The astonishing
final result is the second instance of his working pattern culminating in a closure step. A
review by Miguel Á. Quintanilla appeared in the newspaper El País on March 14th 1982
under the title Filosofía y matemáticas.
Cuesta's contributions to the History of Science published in journals were four papers in
Llull, where an obituary appeared after his death (Hormigón 1989). The most interesting
one (Cuesta 198lb) is a vindication of Euclid as a reaction to a book on the teaching of
'modern mathematics' which had a large impact in Spain (Piaget et al. 1981). In 1980,
Ríos communicated that the Academia was considering the possibility of inaugurating a
historical section at the proposal of Alberto Dou (1915-2009) , but it seems it was somehow
late for Cuesta (Ríos to Cuesta, May 29th 1980), for none of his posterior work was
published by the Academia. A number of articles and notes on historical questions and
mathematical curiosities were also published in various newspapers from 1982 onwards.
Part VI
Cuesta's dream: a cure against isolation?
Of foremost importance in Cuesta's academic life was his interest in establishing a study
track in Mathematics at Salamanca, to which he devoted plenty of time and personal
involvement. Indeed he wanted a mathematically sound centre that in his personal view
should be a bright Academia where Mathematics and mathematical Education would reign
over any other consideration, in an idealised picture of the Güttingen golden age he very
much admired and tried to imitate in the seminars he conducted on Monday evening. As
he used to work on his own with no funding other than the amount the chair obtained
proportional to the number of undergraduate students - all of it spent in books and journals
subscriptions- and his ordinary wages, he expected this behaviour to be the rule rather
than the exception. For instance, when gathering research materials in places other than
Salamanca he travelled there at his own expenses, installed himself in a cheap hotel, and
a notebook and a few pencils were all his needs: For him, scientific policy as an everyday
affair meant little. This spartan lifestyle, for which indeed his being single did help, could
not be easily imposed on other people and had much to do with the way things developed
afterwards.
The university at Salamanca (founded ca. 1250) had had historically up to three chairs
in Mathematics, but never offered a degree in Mathematics. In more recent times and
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for many years, the only available scientific career was a degree in Chemistry, the one
for which Cuesta had read in the 1920s, and the posts in higher Mathematics had been
reduced to two chairs, a first one in Mathematical Analysis and a second one on Analytic
Geometry. A 1923 regulation switched those denominations to General Mathematics and
Special Mathematics in universities where no Mathematics degree was offered, the idea
being that the second chair could be offered to public contest in a rotatory way, establishing
an order arrangement: Astronomy, Statistics and Mechanics, something that in a most
Spanish way of understanding official regulations was often not taken into account, though
it was revived on the occasion of the 1949-55 contest for a Mechanics chair commented
above. Sorne proposals were made by Cuesta himself around 1947 on this question, but
that chair was thought to belong to the Mechanics shift: the problem of naming things
is always an important one in official matters. It must be remembered here that just
before the civil war, two chairs were awarded in Salamanca to two mathematicians of the
Rey Pastor school without much ado about the official names of their posts: San Juan in
1935 - he was the follower of Sáez, died in 1932- and Ancochea in 1936. San Juan was at
Salamanca only for one year, and had moved to Madrid before the war began. After the
war he was again in Madrid and had an active though mostly unofficial role in the Jorge
Juan. Ancochea remained at Salamanca and eventually moved to Madrid in 1947, leaving
behind him a few shelves of a mathematical library which was to be the pride of Cuesta
years after, a number of journal subscriptions, and the first mathematical volumes of the
Acta Salmanticensia, a non-periodic series to which several internationally distinguished
mathematicians contributed. The Acta provided a sound basis for library interchange for
many years, and quite a number of journals were collected through it.
From late 1947 until 1958 ali Mathematics chairs remained vacant, i.e. without tenured
professors - with the exception of the year Tharrats spent in Salamanca- when eventu­ally
Cuesta was appointed to one of them as had been explained before, and this time
with the specific mention of "Mathematical Analysis, with teaching duties for Chemistry
students". This cautionary language was and still is common in the Spanish university
jargon, employed both as a deterrent and as a pointer to some specific candidate. He was
the only full professor until 1966, but never gave up his project of establishing a Math­ematics
study track at Salamanca: that was his self-prescribed cure against isolation, or
at least he considered it that way, as stated by the phrasing shown in the accompanying
figure: "Attention to students and the extension of teaching duties certainly deserve a larger
number of chairs, and I propase to your consideration that the same is true for organising
a continuous and durable mathematical research. This isolation is a suff ocating one. There
is a need of an adequate environment for research, of colleagues with whom to exchange
ideas, and of [PhD] students whose interest may bring a new atmosphere by stimulating
curiosity".
Y sl la atención debida al alumnado, aboga por eae. incremento de cát~­dras
y por la extensión de las ensei\anzas, que propongo al estudio de ustedes,
tamb~n Jo aconseja la organización de una investigación matemática con­tinua
y duradera. Uno se asfixia en esta .soledad. ~n necesario ambiente
para la investigación, compañeros con qwenes cambiar ideas, y disdpulos
que, son su inte~s, renueven la atmósfera, y sirvan de aci.cate a la curiosidad.
Final words of the inaugural address to the IV Reunión Anual de Matemáticos Españoles, Salamanca,
1963 (Cuesta 1966b). The first sentence acknowledges the need of more faculty d ueto population growth
and the quality of research, the second one being a desperate call to put an end to the (his) loneliness in
the isolation of provincial universities.
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On the wake of the evolution of the Franco regime, a series of so-called technocratic gov­ernments
were in office, as a counterpoint with the previous ones with many military, and
Ministers were chosen among economists, lawyers and engineers, in order to cope with the
growing needs of an ever-increasing population in Spain and the problems related with
it. Spanish universities also entered a developing stage, and the Facultad de Ciencias at
Salamanca presented in 1965 a new degree in Biology, enlarging an offer where only Chem­istry
and the general-purpose first year in Science and Engineering had been taught for
decades. According to the available records, Cuesta might have forced the introduction of
a Mathematics degree that same year as well, but being on his own, it seemed a difficult
task, and when Rafael Mallo! (1925-1988) was appointed to a second chair in Algebra in
1966, and the statistician José P. Vilaplana, a student of Ríos, was also hired, there was
a time during which the classical three chairs were back and Cuesta's dream of a degree
in Mathematics in Salamanca seemed quite close to materialise. But things were not so
easy: Vilaplana, who was not a tenured Professor, left Salamanca after one year and Mal­lol
returned to Barcelona in 1969. Nevertheless, in his personal battle against isolation
Cuesta did not give up, and eventually during the Rectorate (1968-1972) of Felipe Lucena
(1923-1976) , who was very receptive to the idea, thirty-four Mathematics students entered
the university in October 1971. A detailed account containing first-hand information on
dates, names, meeting minutes and student statistics can be found in (Cabezas 2001), as
well as in the papers by Cuesta preserved in the corresponding box at the archive. It is
interesting to note here that the Professor insisted - without success- during those conver­sations
and meetings on knowing the students' opinions, and it is even more informative to
know from his notebooks that as late as 1976 two motions by him proposing the debates in
the Claustro Universitario to be held with the presence of journalists and to be recorded
on tape, were disregarded. Most possibly the times were not ripe enough to accept such
proposals, but they would change almost immediately after Franco's death in 1977.
It is also worth pointing out that the generalised creation of faculties, polytechnics, and
new universities, though started a few years before, was boosted by the 1970 Ley General
The group of mathematicians at Salamanca gathered in 1969 on t he occasion of the farewell party in
honour of Rafael Mallo!, who was moving back to Barcelona. From left to right: Teresa Cañadas, Pablo
Carpintero, Leandro Guevara, Eulogio Hernández, Professor Cuesta, Rafael Mallo! with his wife and son,
unidentified lady, José Manso and J esús Gómez. The picture was taken by Carlos Durán, whose shadow
appears in the foreground.
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de Educación promoted by the Education Minister José Luis Villar Palasí, a law aiming to
a general modernisation of the educational system in Spain when, in addition to population
growth, it was clear that the Franco regime was close to an end. Without those general
environmental conditions the isolation of Cuesta and the difficulties for the study track
in Mathematics would not have been overcome, and though he could not openly agree,
he was aware of it. Indeed there were more people in other universities with interests
similar to Cuesta's in various areas, but this general picture is a reasonable one, see the
accompanying table.
Mathematical studies established in Spain 1965-1971
f Year f University f Promoter f Promoter = School?
1965 Santiago Vida! Yes
1967 Seville Castro Yes
1970 Valladolid Mz. Salas No
1970 La Laguna Cascante No
1971 Salamanca Cuesta No
A group of mathematicians led by the very gifted mathematician Juan Bautista Sancho
(1926-2011), arrived in Salamanca from Barcelona in two steps during 1971 and 1972 as
new faculty, and the study track started its development in a somehow diverging way
from the idealised picture Cuesta had in mind, where a certain contempt was present
regarding servitudes like struggles for funding, fierce battles around tenures, and the like.
Most probably his past experiences with his old acquaintance Navarro back in the 1940s
contests was responsible for it, and in addition to it, his sometimes peculiar behaviour
and the ideas he held on how to do Mathematics on a personal basis matched neither the
paradigm introduced by the sudden expansion of the higher educational system fostered by
the Ley General de Educación nor the interaction with the new faculty upon their arrival
from Barcelona. For instance, even as late as 1976 he considered that the publishing
production of that group was too scarce, and he did not understand that none of them
coud read German, which he thought to belong to th basic toolkit of any mathematician
(draft of the letter to Artola cited above). His PhD student Carpintero obtained a tenured
position in 1972, on which occasion the Professor published in the local newspapers an
anonymous celebration note which ended with a clear allusion - borrowing words from El
Quijote and The Gambler by Dostoievski- to what he considered unfair manoeuvers by
the Sancho group around the tenuring process (notebook, entry of November 20th 1972,
with additions one week later) . This incident, somehow reminiscent of the 1969 PhD
one in Madrid, helped very little in smoothing the relationships within the Salamanca
mathematical community and contributed to further isolation of the Professor. When a
few years after that Carpintero moved from Salamanca to Santiago de Compostela, the
init ial impulse of the Professor could not be followed by a school of academic descendants
of his own, and undoubtedly the shaping and performance of the Salamanca mathematical
school is to a great extent the result of the activity of those new faculty members and their
students. Decidedly, paraphrasing Bob Dylan's lyrics: the times they had a 'changed.
Cuesta taught Multivariate Calculus and Mathematical Analysis courses until his definitive
retirement in 1983, but he kept on working at the university library, publishing and giving
talks until a few months before his death, early in 1989.
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Part VII
Conclusions and a final comment
A first conclusion is that the isolation of Cuesta after the civil war had much to do with
his personal character and private environment, that forced him to live in small cities away
from the decision centres, and although he kept postal contact with a number of mathemati­cians
and colleagues in the official environment, no real advantages, either mathematical
or political, were obtained from these interchanges until quite late in the mid 1950s, when
the regime reluctantly started loosening its rigid control mechanisms on individuals and
society.
Second, the mathematical production of Cuesta, elaborated and published in Spain in the
first after-war years isolation under difficult conditions, was an up-to-date one, and he
obtained results whose quality is comparable with those of many mathematicians of his
time.
Third, his switch from pure Mathematics towards mathematical Education, book writing,
and historical research may be understood as the epistemological change that imprinted
the mood in which he has been remembered.
As a fourth conclusion, though he succeeded in creating a study track in Mathematics at
Salamanca, his original conception could not be developed, so when he somehow melan­cholically
realised that the creation of new Mathematics degrees in Spain was to a large
extent a contingent result of the regime evolution started in the mid-sixties and of popu­lation
growth, his final feelings were mixed with deception: Isolation finally won that last
battle, and most possibly only the beauty of his main mathematical achievements would
deserve being remembered, as will be explained in the following concluding comment.
It is interesting to remark that former students and even professionals with a mathematical
or at least scientific formation remember him more by his demeanour as a Professor and
public person, much influenced by the environmental conditions during 1940-1970, than by
the quality of his Mathematics. In addition to that, a topic much commented by people who
knew the origins of the Mathematics studies at Salamanca usually points to a comparison
between the promoter , and Sancho as the leader of the group that actually developed
them afterwards. Instead of such a comparison, even when their scientific initial conditions,
academic careers, and teaching habits were actually very different ones, their mathematical
common points should be stressed by writing that, with a really deep mathematical insight,
both held strikingly similar viewpoints when addressing the generality, simplicity and - why
not- beauty of their understanding of Mathematics. This is how Professor Cuesta finally
overcame his long-life isolation.
Part VIII
Acknowledgements
The late Carlos Durán provided the author with photographical material; in Salamanca,
Maite González, Asun Herranz, Emiliano Hernández, and Manuel López - who very pro­fessionally
typed ali writings by the Professor between 1972 and 1988- talked at length
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with the author on their relationship with the Professor. Prof. Egbert Harzheim (Cologne,
2007) and Rudolf Haller (Munich, 2013) offered comments on points of the mathematical
achievement of Cuesta; Albert Presas (Barcelona and Berlin) was a most receptive listener
and pointed out to the consideration of interesting details. The staff at the Archivo Univer­sitario
in Salamanca were extremely helpful and they surely deserve being thanked here for
their kind attention. The author is grateful by the hospitality of the Max-Planck-Institut
für Wissenschaftsg eschichte in Berlin during various periods between 2009 and 2013.
Part IX
Appendices
9 Appendix 1: Publications by N. Cuesta
This is a hopefully complete list of Cuesta's publications. Newspaper articles are not
included. RMHA means Revista Matemática Hispano Americana.
l. Cuesta N (1942) Números reales generalizados, RMHA (Ser. IV) 2, 5-12, 62-66,
104-109, 218-225.
2. Cuesta N (1943a) Construcción de un conjunto ordenado denso y no continuo cuyo
número ordinal es Wa , RMHA (Ser. IV) 3, 38-40.
3. Cuesta N (1943b) Teoría decimal de los tipos de orden, RMHA (Ser. IV) 3, 186-205,
242-268.
4. Cuesta N (1944a) Continuos asimétricos, RMHA (Ser. IV) 4, 16-23.
5. Cuesta N (1944b) Disimilitud de conjuntos decimales, RMHA (Ser. IV) 4, 45-47.
6. Cuesta N (1944c) Continuos de Suslin, RMHA (Ser. IV) 4, 175-187, 215-233.
7. Cuesta N (1945) Permutaciones continuas con los números reales, RMHA (Ser. IV)
5, 191-203.
8. Cuesta N (1946a) Número de tipos de orden, RMHA (Ser. IV) 6, 59-65.
9. Cuesta N (1946b) Número de tipos de ordenación, RMHA (Ser. IV) 7, 3-9.
10. Cuesta N (1947a) Notas sobre unos trabajos de Sierpinski, RMHA (Ser. IV) 7,
128-131.
11. Cuesta N (1947b) Sobre el concepto de línea, RMHA (Ser. IV) 7, 249-254.
12. Cuesta N (1948a) Estructuras y sus automorfismos, RMHA (Ser. IV) 8, 277-282.
13. Cuesta N (1948b) Ordenación densa perfectamente escalonada, RMHA (Ser. IV) 8,
57-71
14. Cuesta N (1949a) Ordenación de infinitésimos, RMHA (Ser. IV) 9, 131-140.
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15. Cuesta N (1949b) Sucesiones ascendentes de números ordinales, RMHA (Ser. IV) 9,
83-96.
16. Cuesta N (1950) Complementos al artículo "Ordenación de Infinitésimos", RMHA
(Ser. IV) 10, 2-4.
17. Cuesta N (195la) Estructuras y Programa de Erlangen, RMHA (Ser. IV) 11, 17-51.
18. Cuesta N (195lb) Problema equivalente al del continuo, RMHA (Ser. IV) 11, 240-
242.
19. Cuesta N (1952a) Estructuras proyectivas, RMHA (Ser. IV) 12, 107-128.
20. Cuesta N (1952b) Correlación involutiva compleja, RMHA (Ser. IV) 12, 330-339.
21. Cuesta N (1953) Modelos deductivos ordinales, RMHA (Ser. IV) 13, 211-223.
22. Cuesta N (1954a) Álgebra ordinal, Revista Acad. Ci. Madrid 48, 103-145.
23. Cuesta N (1954b) Estructuras deductivas, RMHA (Ser. IV) 14, 104-117.
24. Cuesta N (1954c) Escalonamiento ordinal, RMHA (Ser. IV) 14, 237-268.
25. Cuesta (1954c) "L'enumération transfinie" de Arnaud Denjoy, RMHA (Ser. IV)
14, 287-290.
26. Cuesta N (1955a) Injust a atribución a Zorn del principio maximal, Gaceta Matemática
7, 174-176.
27. Cuesta N (1955b) Para un texto más correcto del Criticón, Boletín de la Biblioteca
Menéndez y Pelayo XXXI, 19-50.
28. Cuesta N (1955c) Sobre la aritmetización del transfinito, Acta Salmanticensia, Ser.
Ciencias, Nueva Serie, Vol. 1(2), Universidad de Salamanca.
29. Cuesta N (1955d) Triadic construction of partially ordered sets, Acta Salmanticensia,
Ser. Ciencias, Nueva Serie, Vol. 1(4), Universidad de Salamanca.
30. Cuesta N (1956a) Una consecuencia de la hipótesis ~ 1 < 2No, RMHA (Ser. IV) 16,
11-14.
31. Cuesta N (1956b) Ordenatrices de Denjoy, RMHA (Ser. IV) 18, 179-192.
32. Cuesta N (1957a) Ciencia deductiva con bases irreducibles separables, Collectanea
Math. 8, 73-83.
33. Cuesta N (1957b) Nota sobre el producto de números complejos, Gaceta Matemática
8.
34. Cuesta N (1958a) El infinito aritmético desde Zenón y Eudoxio, hasta Galileo y
Cantor, Revista de Filosofía A 17(65-66) , 181-194.
35. Cuesta N (1958b) El saber matemático: Lección de incorporación al Claustro de la
Universidad de Salamanca, Imprenta Calatrava, Salamanca.
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36. Cuesta N (1958c) Sobre concepto, método, enseñanza y fuentes del análisis matemático,
RMHA (Ser. IV) 18, 87-98, 201-213.
37. Cuesta N (1958d) Sobre un artículo de Mac Neille, RMHA (Ser. IV) 18, 3-9.
38. Cuesta N (1958e) Matemática del Orden I, II, Rev. Acad. Ci. Madrid, 147-321.
39. Cuesta N (1958f) Matemática del Orden III, Rev. Acad. Ci. Madrid, 609-770.
40. Cuesta N (1959a) Matemática del Orden IV, Rev. Acad. Ci. Madrid, 33-19.
41. Cuesta N (1959b) La estructura topológico-deductiva de medición, RMHA (Ser. IV)
19, 213-221.
42. Cuesta N (1959c) Matemática del orden, Real Academia de Ciencias Exactas, Físicas
y Naturales, Madrid.
43. Cuesta N (1960a) Un enredo semántico-probabilístico, Gaceta Matemática 12(1-2).
44. Cuesta N (1960b) Un enredo semántico-probabilístico 2°, Gaceta Matemática 12(7-
8) .
45. Cuesta N (1961) Generadores matemáticos, Actas de la I reunión anual de matemáti­cos
españoles (Madrid) , 183-198.
46. Cuesta N (1962a) ¿Postulado = Axioma?, Collectanea Math. 14, 81-94.
47. Cuesta N (1962b) Diario de las lecciones, Gaceta Matemática 14(7-8).
48. Cuesta N (1962c) Estructuras de implicación, Acta Salmanticensia, Ser. Ciencias,
Nueva Serie, Vol. 6(1) , Universidad de Salamanca.
49. Cuesta N (1963) Estructuras algebraicas de las fracciones (lección para Bachillerato),
Gaceta Matemática 15 (3-4).
50. Cuesta N (1964) Lógica de las cuestiones dialécticas, Actas de la V reunión anual de
matemáticos españoles (Valencia).
51. Cuesta N (1965a) Estructuración lineal de conjuntos Actas de la IV reunión anual
de matemáticos españoles, Acta Salmanticensia 6(4) ,27-40.
52. Cuesta N (1965b) Los números reales como triparticiones eudoxo-euclídeas del con­junto
de fracciones, Gaceta Matemática 22 (5-6).
53. Cuesta N (1965c) Nota crítica sobre un teorema de series funcionales, Gaceta Matemáticc
17, (1-2).
54. Cuesta N (1966a) Algunos aspectos del pensamiento matemático, Opening address
for the academic year 1966-1967, Ed. Universidad de Salamanca.
55. Cuesta N ( 1966b) Clima humano necesario para crear una escuela española de matemáti­cos,
Actas de la IV reunión anual de matemáticos españoles, Acta Salmanticensia
6(4) , 164-168.
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56. Cuesta N (1966c) Don José Barinaga: In memoriam, Gaceta Matemática 18(3-4),
63-86.
57. Cuesta N (1966d) Estudio vectorial de las homotecias, Actas de la IV reunión anual
de matemáticos españoles, Acta Salmanticensia 6(4).
58. Cuesta N (1966e) Integración conjuntista, Acta Salmanticensia, Ser. Ciencias, Nueva
Serie, Vol. 28, Universidad de Salamanca.
59. Cuesta N (1966f) Una lección sobre el teorema de Rouché, Gaceta Matemática 18,
28-43.
60. Cuesta N (1967a) Probabilidades en conjuntos amorfos, Actas de la VIII reunión
anual de matemáticos españoles (Santiago), 112-115.
61. Cuesta N (1967b) Promoción positiva y negativa de vocaciones docentes, Actas VI
reunión anual de matemáticos españoles (Sevilla), 183-190.
62. Cuesta N (1968a) ¿La Aritmética ciencia física? Gaceta Matemática 19, 36-47.
63. Cuesta N (1968b) Geometría vectorial: Introducción intuitiva al Algebra Lineal, Ed.
Alhambra, Madrid.
64. Cuesta N (1968c) La numeración ordinal de Von Neumann, Actas de la IX reunión
anual de matemáticos españoles (Granada) , 41-49.
65. Cuesta N (1969a) Escolios a un artículo de Uuno Saarnio, RMHA (Ser. IV) 29, 8-12.
66. Cuesta N (1969b) La creación matemática del Profesor San Juan, Gaceta Matemática
(Ser. 1) 21(5-6), 115-121.
67. Cuesta N (1971) Filosofía natural y pugna de facultades enla Universidad de Sala­manca,
1779-1796. (Inaugural address as a corresponding academician of the Real
Academia de Ciencias), Imprenta Comercial Salmantina, Salamanca.
68. Cuesta N (1972) Representación gráfica de la tripartición eudoxo-euclídea, Gaceta
Matemática 24 (5-6).
69. Cuesta N (1973) El postulado asintótico, fundamento esencial del Análisis, Actas de
la XI reunión anual de matemáticos españoles (Murcia, 1970), 37-47.
70. Cuesta N (1974a) 40 lecciones de Análisis Matemático, Author's edition, Salamanca.
71. Cuesta N (1974b) El Maestro Juan Justo García: presbítero natural de Zafra (1752-
1830), segundo catedrático de álgebra de la Universidad de Salamanca desde 1774 y
creador de su Colegio de Filosofía en 1792, Ediciones Universidad de Salamanca.
72. Cuesta N (1975a) Conceptuación matemática, Actas de las IV Jornadas matemáticas
Luso-españolas, 15-38.
73. Cuesta N (1975b) La Sinfonía del Infinito (67 lecciones de Análisis Matemático) ,
Ediciones de la Universidad de Salamanca.
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74. Cuesta N (1976) Historia de la invención del Análisis Infinitesimal y de su introduc­ción
en España, Ediciones de la Universidad de Salamanca.
75. Cuesta N (1977a) Conjuntos de oscilación de L zi y rr (1 + z2i) . Actas de las
l ::=;j<w l ::=;j<w
I Jornadas Matemáticas Hispano-lusitanas (Madrid, 1973), 95-112.
76. Cuesta N (1977b) Sobre ordenaciones continuas, Actas de las IV Jornadas matemáti­cas
Luso-españolas (1), 81-85.
77. Cuesta N (1978) Lección académica final, Universidad de Salamanca.
78. Cuesta N (1979) Introducción en España del Análisis Infinitesimal, Llull 2(3), 47-49.
79. Cuesta N (1980a) Análisis metamatemático de los números reales, El Basilisco 10,
4-7.
80. Cuesta N (1980b) Análisis metamatemático de la axiomática de los números natu­rales,
El Basilisco 11, 24-26.
81. Cuesta N (1981a) La Sinfonía del Infinito: y ya en el paraíso de Euler (99 lecciones
de Análisis Matemático), Ediciones de la Universidad de Salamanca.
82. Cuesta N (1981b) Tres notabilísimos pasajes de Euclides, Llull 4(6-7), 35-42.
83. Cuesta N (1982a) Las Matemáticas y el conocimiento científico factual, in M. Quin­tanilla
(comp.) Seminario de Teoría de la Ciencia (1978-1979), Ediciones Universi­dad
de Salamanca.
84. Cuesta N (1982b) Matemáticas finitas, matemáticas asintóticas, matemáticas infini­tas,
Actas IX Jornadas Hispano-lusas de Matemáticas 1, 31-71.
85. Cuesta N (1984a) Las Matemáticas en Europa y en España en tiempos de Torres
Villarroel, Universidad de Salamanca.
86. Cuesta N (1984b) En el tricentenario de las ecuaciones diferenciales de Leibnitz, Llull
7(12), 91-92.
87. Cuesta N (1985a) Número de primos entre an y a(n+ 1) y desde 1 hasta n+ l. Actas
X Jornadas Hispano-lusas de Matemáticas, 300-305.
88. Cuesta N (1985b) Poliedros regulares infinitos, Actas X Jornadas Hispano-lusas de
Matemáticas, 296-299.
89. Cuesta N (1985c) Primos gemelos cribando las sucesiones de Cataldi, Actas del Hom­enaje
al Prof. Rodríguez Vidal, 117-134, Zaragoza.
90. Cuesta N (1985d) Escolios a unos comentarios de Pascual Llorente, Llull 6(10-11) ,
191-192.
91. Cuesta N (1986a) Aritmética de las sucesiones 6n - 1, 6n+ 1 y de los primos gemelos,
Collectanea Mathematica, 37, 211-227.
92. Cuesta N (1986b) Los primos desde an hasta a(n + 1), Maternas 1, 71-76.
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93. Cuesta N (1987) Los primos gemelos, Maternas 2, 51-67.
94. Cuesta N (1988) Tabla de divisores de los números naturales, Maternas 3, 6-13.
95. Cuesta N (1991) "Aritmética", entry in the Gran Enciclopedia Rialp, Ed. Rialp,
Madrid.
10 Appendix 11: Unpublished materials
l. Cuesta N (1943) El ultracontinuo de F. Bernstein. Dated Salamanca / Granada,
Christmas 1943.
2. Cuesta N (1944) Sobre los conjuntos finitos . Dated: Granada, 1944.
3. Cuesta N (1947) La disposición circular en las formas de primera categoría. Dated:
Salamanca, Dec 6th 1947.
4. Cuesta N (1952) El teorema fundamental de la cinemática de sistemas rígidos. Dated:
Salamanca, May lst 1952.
5. Cuesta N (1957) Estructura de los conjuntos cerrados lineales. Dated: Salamanca,
July 4th 1957.
6. Cuesta N (1956) El infinito aritmético desde Zenón y Eudoxio hasta Galileo y Cantor.
Talk at the Biblioteca Provincial in Cáceres, March 3rd 1956.
7. Cuesta N (1961) Proyectores de Mn sobre M. Dated: August-December 1961.
8. Cuesta N (196X) Mathématique de l'ordre (handwritten, unpublished) Translated by
Norberto Cuesta, Antoine Pinéde and Odile-Joseph Broucke. (569+ 15 p.)
11 Appendix 111: A guided mathematical tour
First part
On algorithms
Cuesta's main result in pure Mathematics, his 1943 systematic construction of all total
orders on any set, was achieved by firmly adhering to the believers in actual infinities and
through a thorough usage of the well-ordering principie and the techniques developed by
Ernst Steinitz in his 1910 Algebraische Theorie der Korper. An elementary example of
Steinitz's procedure of adding elements to a given structured set, one ata time, is common
fare in courses for beginners in Linear Algebra.
In his papers Cuesta did not use the word algorithm, though he was plainly aware of
the possibility of automation for many processes he employed. A word of his invention is
maquinización. Moreover, he used the adjective 'chronological' referring to a previous well
ordering needed in order to develop his theory: It is interesting to note here time as an
essential ingredient. Here the rationale behind Cuesta's method is succinctly explained:
Let A be any set, and choose sorne element aEA. Now, proceed to well-order A, in such a
way that a be its first element: This order will play the role of time. Now, figure out that a
is placed in sorne position on an infinitely long ideal supporting line. Then a automatically
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creates two empty places: one in its left side, the other on the right one, yielding the
pattern -a-. According to the induced order, Jet b be the first element in A-{a} and use
it to occupy one of the empty places, say the one on the right. The result will be - a - b-,
i.e. the empty place at the right of a is turned into - b- . Note carefully that there is a
decision, choice, or branching, right vs. left. Now, Jet the first element ct:A- {a, b} occupy
one of the three empty places by changing it into the corresponding pattern. Repeat the
process.
The choice regarding which empty place is to be filled with the pattern -x- when intro­ducing
any element x constructs the decision tree, whereby ali possible total orderings of
the initial set are represented.
Once the process (ideally speaking) is over, the problem immediately arises of naming
the arder just constructed. Indeed, the arder itself could be used as its name, but it
would be largely impractical by severa! reasons: First, it would be a violation of 'the
Borges principie', where this name refers to a short story by the Argentinian writer Jorge
L. Borges (1899-1986) in his 1960 book El hacedor on an emperor who wanted such a
detailed map of his empire that eventually his cartographers presented him an obviously
useless 1:1 reproduction. Second, and worse, isomorphic orderings would receive different
names. Cuesta wanted an onomastic process for isomorphism classes, or arder types, not
for particular ones.
Although the onomastic problem is still an open one, there are partial solutions for de­numerable
ordinal numbers, and Cuesta criticised the proposal made by Arnaud Denjoy
where names for denumerable well orderings were established via napierian decimal expan­sions
of real numbers in the unit interval, and devised a procedure of his own according to
the following scheme:
Let D be a denumerable ordinal number, and nD the name accorded to it, written with the
symbols of sorne finite alphabet. The naming procedure must fulfil the condition that nD
be itself an ordinal number complying with Borges' principie, i.e. nD < D. Exceptionally,
Ordinal Na me Ordinal Na me Ordinal Name Ordinal Name
o o 1 1 CD+1 21 CDx2+1 201
2 10 CD+2 210 CD x2+2 2010
3 11 CD+3 211 rox2+3 2011
4 100 CD+4 2100 rox2+4 20100
5 101 ro+5 2101 rox2+5 20101
... ... ... ... ... ...
co 2{new) CDx2 20 CDx3 200
Class N1 Class N2
Boldface: ineffable ordinals. (new) = an alphabet extension
Denumerable ordinal naming according to Cuesta, worked out by t he aut hor of this paper.
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this principie may be violated in sorne cases: For example, the empty set 0 yields an
ordinal O whose name clearly <loes not satisfy it, thus being an example of the so-called
ineffable ordinals J for which nJ ;::: J , and though needed at sorne stages, they must be
kept to a mínimum either by occasional enlarging of the initial alphabet or by recurrence
on already existent names (Schütte 1954). A table below shows the first steps of Cuesta's
method for naming denumerable ordinal numbers starting with the initial binary alphabet
Inverting the viewpoint
One of the most fruitful techniques in the mathematical experience - whether it be creation
or discovery- is that of inverting, or at least changing, the usual viewpoint. In his attempt
to construct total orderings, Cuesta freely used the intuitive idea of 'filling an empty place'
with a given pattern in order to create the order relation. Here the viewpoint inversion
amounts to start with a pre-existent order on sorne set, to consider the addition of elements
one at a time, and to watch things happen:
Let (M, -<) be a totally ordered set. A gap in it is a disjoint pair of subsets (A, B ) of
M such that two conditions hold: A -< B and A U B = M. Now, !et p be sorne object
or element which will be used to fil! the gap, i.e. a new set MU {p}is built under the
condition thatA-< p -< B . The insertion of p creates two new gaps in MU {p}descending
from (A, B), viz. (A U {p} , B) and (A, B U {p} ). If O is a name given to (A, B ), then the
two descending gaps could be named, respectively Oi , O f meaning that the new element
was added either to the initial (i) section A or to the final (f) one, B of the gap.
Now, start with 0 and consider the gap (0,0). Then, adding elements one by one will
provide a construction of binary names like Oif fifiiif ... With the necessary elaboration,
this procedure yields the proof obtained by Cuesta of his theorem.
Second part
The generalisation to the case of part ial orders of the 1943 algorithm for total orderings
may be obtained by shadowing that binary technique through extending it to pairs of
gaps (Cuesta 1955): Let (M, -<) be a totally ordered set. A 'tripartition' of M is a triple
(A, B , C) of pairwise disjoint subsets of M such that the relationships A -< B -< C and
A U B U C = M hold. One such tripartition may be named again O. Now, !et p be sorne
element which will be used it to fil! one of the two gaps (A, B ) or (B, C) . The insertion of
p can now be done in any of the forms (AU {p} , B , C), (A, BU {p} , C), (A, B , Cu {p}) , so
again a branching process is generated according to the selected option. The descending
tripartitions will be named, respectively Oi, On or O f , meaning that the new element was
added to the initial (i) section A, to the central one B (n) or to the final (f) one, C.
Now, start with M = 0 and consider the tripartition (0,0,0) . Then, adding elements
will yield ternary names like Oniifnfin ... The procedure rediscovered the now famous
Sierpinski triangle, as shown in the illustrations of the original paper by Cuesta: Any
denumerable partial order can be described by a path joining the cent res of smaller and
smaller triangles according to the chosen sequence of i's, n's and f's.
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T•llLllt.. Cúli~'flll'C11 ...... UI l'AMTIAl.t.l' 0111n..au1 ... 'f. l:t
Th~ first fi~u1-e inilinUlS u prOCes.!>, with which thc HClhallvM1ca1
lrce, corre poodin~ to the triadícs of finite length, would be asymp-
1.0ticatly delinented on the cuclhleu n plwh:
The Sierpinski triangle as rediscovered in Cuesta, 1955.
12 Appendix IV: Succint biographical data of N. Cuesta
• 1907 Born at Salamanca.
• 1925-1929 Chemistry degree, Salamanca.
• 1930-1932 Mathematics degree, Saragossa.
• 1933-1937 Assistant Professor at Granada.
• 1935 Attempt (failed) to obtain a high school Professorate.
• 1937-1939 civil war and prison.
• 1940 -1966 High school teacher (Ávila, Granada, Segovia, Salamanca).
• 1942 First mathematical papers.
• 1943 Doctoral dissertation (Madrid).
• 1947 University (non-tenured) Professor at Salamanca.
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• 1954 Opposition to the Honoris Causa doctorate to General Franco.
• 1955 Failed attempt to tenure. Leaves university.
• 1957 Rey Pastor offers him a position in Argentina. Short collaboration with the
official agency CSIC. Academy prize.
• 1958 University tenured professorate. Book: Matemática del Orden.
• 1962 Crash with the local Bishop.
• 1968: Book: Geometría Vectorial.
• 1970 Corresponding Academician of two Royal Academies.
• 1971 Study track in Mathematics (Salamanca)
• 1976 Book: Historia de la Invención del Análisis Infinitesimal . ..
• 1977 Retired / Emerited
• 1981 Book: La Sinfonía del Infinito
• 1983 Retired
• 1988 Last mathematical paper
• 1989 Passed away at his native Salamanca
13 Appendix V: Sorne papers where citations of N . Cuesta
may be found
l. Acharyya S (2004) A generic method to construct P-spaces through ordered fields,
Southeast Asian Bull. M ath. 28, 783-790.
2. Alling N, Ehrlich P (1986) An abstract characterization of a full class of surreal
numbers, Compt. R. Acad, Sci. Soc. R. Ganada 8, 303-308.
3. Alling N, Swartz, T (1990) Monomorphisms of inductive number systems, Portugaliae
Mathematica 47(3) , 293-308.
4. Bourdeau M (2003) La critique de la Théorie des Ensembles dans la Dissertation de
Brouwer, Math. & Sci. Humaines 164, 29-43.
5. Candeal J , Indurain E (1999) Lexicographic behaviour of chains, Archiv der Mathe­matik
72 , 145-152.
6. Ehrlich P (1988) An alternative construction of Conway's ordered field No, Algebra
Universalis 25, 7-16.
7. Ehrlich P (2012) The absolute arithmetic continuum and the unification of numbers
great and small, Bulletin of Symbolic Logic 18(1) , 1-45.
120
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8. González-Redondo F (2003) Un modelo para la delimitación teórica, estructuración
histórica y organización docente de las disciplinas científicas: El caso de la Matemática.
Academia de Ciencias e Ingenierías de Lanzarote, Arrecife de Lanzarote.
9. Maz A (2005) Los números negativos en España en los siglos XVIII y XIX, PhD
Thesis, Universidad de Granada.
10. Pacheco J , Pérez-Fernández J , Suárez C (2007) On the Reception and Spread of Meta­physical
Explanations of Imaginary Numbers in Spain, Folia Canariensis Academiae
Scientiarum XVIII (1-2) , 101-132.
11. Pacheco J , Pérez-Fernández J , Suárez C (2013) Infinitesimals in Spain: Portuondo's
'Ensayo', Revista Brasileira de História da Matemática, to appear.
12. Poveda R, Morales Y (1999) El uso del software 'scheme' como modelo de lenguaje
matemático, Actas IV CIEMAC, San José de Costa Rica.
13. Prieto A (2008) Emular la Naturaleza, además de observarla, Academia de Ciencias
Matemáticas, Físico-Químicas y Naturales de Granada, Granada.
14. Purisch S (1996) A history of results on orderability and suborderability, Topology
Atlas, at.yorku.ca/ t / a/ i/ c/ 18.htm
15. Suárez C (2007) La introducción del rigor en las Matemáticas en España en el siglo
XIX, PhD thesis, Universidad de Cádiz.
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