An example why Scott’s Representation Theorem does not preserve infinite conjunctions and disjunctions

 

Consider the Lebesgue measurable subsets of $[0,1]$ module the null sets. I.e. $A,B\subset [0,1]$ are equivalent is they agree
a.e. This is a $\sigma$-algebra. Consider the set of ultrafilters on this $\sigma$-algebra.

Also, consider the family of sets $\{[0,\frac{1}{n}]|n\in\omega\}$ (each set is identified with its equivalent class. E.g.
$[0,\frac{1}{2}]$ means the equivalent class of $[0,\frac{1}{2}]$). This collection satisfies the finite intersection property.
So, there is an ultrafilter $U$ that extends this collection. For every $n\in\omega$, let $X_n$ be the set of all
ultrafilters that contain $[0,\frac{1}{n}]$. By Scott’s Representation Theorem each $[0,\frac{1}{n}]$ maps to $X_n$.

Now, notice that the intersection $\bigcap_n [0,\frac{1}{n}]$ is the empty set.
If Scott’s Representation theorem was respecting countable conjunctions, the intersection $\bigcap_n [0,\frac{1}{n}]=\emptyset$
would map to the intersection $\bigcap_n X_n=\emptyset$. But the intersection of the $X_n$’s is not empty as it contains the
ultrafilter $U$. Contradiction.

Bottom Line: What goes wrong is that in the above example, the Boolean Algebra given by Scott’s Representation theorem is not
$\sigma$-complete.

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The experience of mathematical beauty and its neural correlates

See the full article at https://doi.org/10.3389/fnhum.2014.00068

Original Research ARTICLE

Front. Hum. Neurosci., 13 February 2014 |

The experience of mathematical beauty and its neural correlates

newprofile_default_profileimage_new.jpgSemir Zeki1*, newprofile_default_profileimage_new.jpgJohn Paul Romaya1, newprofile_default_profileimage_new.jpgDionigi M. T. Benincasa2 and Thumb_24.jpgMichael F. Atiyah3

  • 1Wellcome Laboratory of Neurobiology, University College London, London, UK
  • 2Department of Physics, Imperial College London, London, UK
  • 3School of Mathematics, University of Edinburgh, Edinburgh, UK

Many have written of the experience of mathematical beauty as being comparable to that derived from the greatest art. This makes it interesting to learn whether the experience of beauty derived from such a highly intellectual and abstract source as mathematics correlates with activity in the same part of the emotional brain as that derived from more sensory, perceptually based, sources. To determine this, we used functional magnetic resonance imaging (fMRI) to image the activity in the brains of 15 mathematicians when they viewed mathematical formulae which they had individually rated as beautiful, indifferent or ugly. Results showed that the experience of mathematical beauty correlates parametrically with activity in the same part of the emotional brain, namely field A1 of the medial orbito-frontal cortex (mOFC), as the experience of beauty derived from other sources.

Introduction

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty”

Bertrand Russell, Mysticism and Logic (1919).

The beauty of mathematical formulations lies in abstracting, in simple equations, truths that have universal validity. Many—among them the mathematicians Bertrand Russell (1919) and Hermann Weyl (Dyson, 1956; Atiyah, 2002), the physicist Paul Dirac (1939) and the art critic Clive Bell (1914)—have written of the importance of beauty in mathematical formulations and have compared the experience of mathematical beauty to that derived from the greatest art (Atiyah, 1973). Their descriptions suggest that the experience of mathematical beauty has much in common with that derived from other sources, even though mathematical beauty has a much deeper intellectual source than visual or musical beauty, which are more “sensible” and perceptually based. Past brain imaging studies exploring the neurobiology of beauty have shown that the experience of visual (Kawabata and Zeki, 2004), musical (Blood et al., 1999; Ishizu and Zeki, 2011), and moral (Tsukiura and Cabeza, 2011) beauty all correlate with activity in a specific part of the emotional brain, field A1 of the medial orbito-frontal cortex, which probably includes segments of Brodmann Areas (BA) 10, 12 and 32 (see Ishizu and Zeki, 2011 for a review). Our hypothesis in this study was that the experience of beauty derived from so abstract an intellectual source as mathematics will correlate with activity in the same part of the emotional brain as that of beauty derived from other sources.

Plato (1929) thought that “nothing without understanding would ever be more beauteous than with understanding,” making mathematical beauty, for him, the highest form of beauty. The premium thus placed on the faculty of understanding when experiencing beauty creates both a problem and an opportunity for studying the neurobiology of beauty. Unlike our previous studies of the neurobiology of musical or visual beauty, in which participating subjects were neither experts nor trained in these domains, in the present study we had, of necessity, to recruit subjects with a fairly advanced knowledge of mathematics and a comprehension of the formulae that they viewed and rated. It is relatively easy to separate out the faculty of understanding from the experience of beauty in mathematics, but much more difficult to do so for the experience of visual or musical beauty; hence a study of the neurobiology of mathematical beauty carried with it the promise of addressing a broader issue with implications for future studies of the neurobiology of beauty, namely the extent to which the experience of beauty is bound to that of “understanding.”

See the full article at https://doi.org/10.3389/fnhum.2014.00068

In Memoriam: Hartley Rogers, Jr.

From https://math.mit.edu/about/history/obituaries/rogers.php

Hartley Rogers, Jr., Professor Emeritus of Mathematics

Published: July 22, 2015

Hartley Rogers, Jr.

Photo courtesy of the MIT Museum.

Hartley Rogers, Jr., professor emeritus of mathematics at MIT, died at the Meadow Green Rehabilitation and Nursing Center in Waltham, Massachusetts, on Friday, July 17. He was 89.

Rogers joined the MIT mathematics faculty in 1956 as an assistant professor, following a year’s visit at MIT. He was promoted to full professor in 1964, and retired from MIT in 2009.

Rogers’ research interests were in mathematical logic, and he is credited as one of the main developers of recursion theory, and of the usefulness and validity of informal methods in this area. His 1959 paper “Computing Degrees of Unsolvability” obtained semantical completeness results for higher levels of arithmetical complexity, and underlies current methodology in studies of computable structures. Rogers authored the 1967 book “Theory of Recursive Functions and Effective Computability,” which has become a central and standard reference in the field, and remains in print.

Rogers served as vice president of the Association for Symbolic Logic, senior editor of the Journal of Symbolic Logic, senior editor of Annals of Mathematical Logic, and associate editor of the Journal of Computer and Systems Sciences. Among his distinctions, Rogers received the Lewis R. Ford Award of the Mathematical Association of America for his expository papers in 1965.

Rogers’ career at MIT included significant administrative service during the 1960s and 1970s. From 1962 to 1964, he was a member of the Committee on Curriculum Content Planning, whose report radically modified the General Institute requirements for undergraduate education. In 1968, he chaired the Panel on November Events and the MIT Community, whose findings further developed the judicial processes of the Institute. Rogers served as chair of the MIT faculty from 1971 to 1973, and as associate provost from 1974 to 1980. He chaired the editorial board of the MIT Press from 1974 to 1981, as the press became an arm of the Institute’s educational mission.

At Rogers’ suggestion in 1996, the Department of Mathematics initiated its Summer Program in Undergraduate Research (SPUR). Teams pair a graduate student mentor with an MIT undergraduate; each team then works intensively on a research problem over a six-week summer period, culminating with the undergraduate giving a presentation and submitting written materials to a group of math faculty. Under Rogers’ direction through 2006, SPUR became popular with students who saw its educational benefits.

In 2001, the Rogers family established the Hartley Rogers, Jr. Prize for the top SPUR teams selected by the faculty. The prize has not only boosted the competitive spirit of its participants, but has attracted participation by graduate students from Harvard University and exchange students from Cambridge University.

From 1993 to 2006, Rogers supervised the MIT mathematics section of the Research Summer Institute program for advanced high school students. From 1995 to 2008, he also helped develop the MIT problem-solving seminar into an important resource for students, especially freshmen, interested in participating in the William Lowell Putnam Mathematics Competition. (Each year, he invited all attendees to his home in Winchester, Massachusetts, for dinner prior to the competition.) During this period, MIT’s Putnam team placed among the top three teams 10 times, twice in first place.

Rogers was a popular and respected teacher, particularly with his development of course 18.022 (Multivariable Calculus with Theory). In 1993, he received the Teaching Prize for Undergraduate Education from the School of Science. Rogers’ graduate lectures in mathematical logic were known for their beauty and clarity, and he was known for assigning challenging problem sets. He produced 19 doctoral students at MIT, with 557 mathematical “descendants” in total.

“[Rogers] presented an innovative, intuitive approach to recursion theory (computability) in his lectures and classic text,” says Richard Shore, a professor of mathematics at Cornell University and former president of the Association for Symbolic Logic, who studied with Rogers as a PhD student under Gerald Sacks from 1968 to 1972. “His approach was a major influence on my development and on all other students of the subject for the past 50 years. He was both a gentleman and a scholar who was devoted to his students, university, and academic community. For me, personally, he was a model and mentor for professional conduct and service to the community for many years.”

Along with mathematics, Rogers maintained a love for English literature, the field of his undergraduate degree. In the 1960s, he took up rowing with a passion. He was a founding member of the Charles River All Star Has-Beens (CRASH-B) sprints, and served as its unofficial guru for three decades. He won numerous medals at the CRASH-B sprints as well as at World Rowing Masters competitions, and in the Head of the Charles Regatta. He was the president of Boston Rowing Center, which prepared many top athletes for the U.S. national team, in the 1980s and early 1990s.

Hartley Rogers, Jr., was born in Buffalo, New York, on July 6, 1926. He received his BA in English from Yale University in 1946. Following a year at Cambridge University as a Henry Fellow, he returned to Yale to complete his MS in physics in 1950. He continued his studies at Princeton University in mathematics, receiving his MA in 1951 and his PhD in 1952, with Alonzo Church was his thesis advisor. Rogers’ first academic appointment was as Benjamin Peirce Lecturer at Harvard from 1952 to 1955.

Rogers was a devoted father, fiercely proud of his children and their accomplishments. He is survived by his wife, Dr. Adrianne E. Rogers; by his three children, Hartley R. Rogers, Campbell D.K. Rogers, and Caroline R. Broderick; and by 10 grandchildren.

Gifts in Rogers’ memory may be made to the Hartley Rogers Jr. Fund in the Department of Mathematics.

Mathias and Set Theory

Full article address: http://onlinelibrary.wiley.com/doi/10.1002/malq.201500072/full

Abstract

On the occasion of his 70th birthday, the work of Adrian Mathias in set theory is surveyed in its full range and extent.

1 Introduction

Adrian Richard David Mathias (born 12 February 1944) has cut quite a figure on the “surrealist landscape” of set theory ever since it became a modern and sophisticated field of mathematics, and his 70th birthday occasions a commemorative account of his mathematical oeuvre. It is of particular worth to provide such an account, since Mathias is a set theorist distinctive in having both established a range of important combinatorial and consistency results as well as in carrying out definitive analyses of the axioms of set theory.

Setting out, Mathias secured his set-theoretic legacy with the Mathias real, now squarely in the pantheon of generic reals, and the eventual rich theory of happy families developed in its surround. He then built on and extended this work in new directions including those resonant with the Axiom of Determinacy, and moreover began to seriously take up social and cultural issues in mathematics. He reached his next height when he scrutinized how Nicolas Bourbaki and particularly Saunders Mac Lane attended to set theory from their mathematical perspectives, and in dialectical engagement investigated how their systems related to mainstream axiomatic set theory. Then in new specific research, Mathias made an incisive set-theoretic incursion into dynamics. Latterly, Mathias refined his detailed analysis of the axiomatics of set theory to weaker set theories and minimal axiomatic sufficiency for constructibility and forcing.

We discuss Mathias’s mathematical work and writings in roughly chronological order, bringing out their impact on set theory and its development. We describe below how, through his extensive travel and varied working contexts, Mathias has engaged with a range of stimulating issues. For accomplishing this, Mathias’s webpage proved to be a valuable source for articles and details about their contents and appearance. Also, discussions and communications with Mathias provided detailed information about chronology and events. The initial biographical sketch which follows forthwith is buttressed by this information.

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