The experience of mathematical beauty and its neural correlates

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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.


“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.”

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Who wants to live forever?

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Who wants to live forever?

We are living longer. How can we live better?

Your mother’s diet, your immune system and air pollution are among the many factors affecting how long you live and whether you develop Alzheimer’s or cancer. ERC researchers are unravelling the secrets of longevity, exploring ways of adding ‘life to years’ as well as ‘years to life’. (Hint: It helps to be a monk.)

Story by Gary Finnegan
Video by Sabina Brennan
Design by Ben Newton.

Let’s start with the good news: most Europeans born today will live long lives – on average, 78 years for men and 83 years for women.

Now the bad news: although they will live longer, they will not live better. Men born today will spend 17 of their years in poor health, and women will be ill 22 years. Hence the adage: ‘women are sicker but men die quicker’.

Nor is this un-healthy ageing problem a European issue only. In rapidly developing nations such as China and India, life expectancy is also on the up – but so too are chronic conditions, including cancer, heart disease, diabetes and dementia. It seems we have added quantity of life without making much progress on quality.

Top scorers in Mathematics- PISA Global Student Assessment

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Top 3 mean 2015 PISA score in Mathematics

  • Singapore 564
  • Hong Kong (China) 548
  • Macao (China) 544

Mean: 490

Most of the top scorers in Mathematics are countries from eastern Asia, followed by European countries.

For comparison

  • US 470
  • Greece 454
  • Cyprus 437

Bottom 3 mean 2015 PISA scores in Mathematics

  • Kosovo 362
  • Algeria 360
  • Dominican Republic 328

Some findings from the PISA global education survey

  • One in four boys and girls reported that they expect to work in a science-related occupation but opt for very different ones: girls mostly seek positions in the health sector and boys in becoming ICT professionals, scientists or engineers.
  • Poorer students are 3 times more likely to be low performers than wealthier students, and immigrant students are more than twice as likely as non-immigrants to be low achievers.
  • How much time students spend learning and how science is taught are even more strongly associated with science performance and the expectations of pursuing a science-related career than how well-equipped and staffed the science department is and science teachers’ qualifications.

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Self-learning AI emulates the human brain

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Self-learning AI emulates the human brain


European researchers have designed brain-like artificial neural networks capable of numerical and spatial cognition and written language processing without any explicit training or pre-programming. Their work, based on the machine-learning approach of generative models, significantly advances the development of self-learning artificial intelligence, while also deepening understanding of human cognition.

Research picture: © Ivilin Stoianov, Marco Zorzi

The research was led by Marco Zorzi at the University of Padova and funded with a starting grant from the European Research Centre (ERC). The project – GENMOD – demonstrated that it is possible to build an artificial neural network that observes the world and generates its own internal representation based on sensory data. For example, the network was able by itself to develop approximate number sense, the ability to determine basic numerical qualities, such as greater or lesser, without actually understanding the numbers themselves, just like human babies and some animals.

“We have shown that generative learning in a probabilistic framework can be a crucial step forward for developing more plausible neural network models of human cognition,” Zorzi says.

Tests on visual numerosity show the network’s capabilities, and offer insight into how the ability to judge the amount of objects in a set emerges in humans and animals without any pre-existing knowledge of numbers or arithmetic.

Much as babies develop approximate number sense without first being taught how to count, or fish can naturally tell which shoal is bigger and therefore safer to join, the GENMOD network developed the ability to discriminate between the number of objects with an accuracy matching that of skilled adults, even though it was never taught the difference between 1 and 2, programmed to count or even told what its task was.

The model was implemented in a stochastic recurrent neural network, known as a Restricted Boltzmann Machine, which simulates a basic retina-like structure that ‘observes’ the images and deeper hierarchical layers of neural nodes that sort and analyse the sensory input (what it ‘sees’).

Zorzi and his colleagues fed the self-revising network tens of thousands of images, each containing between 2and 32 randomly-arranged objects of variable sizes, and found that sensitivity to numerosity emerged in the deep neural network following unsupervised learning. In response to each image, the network strengthened or weakened connections between neurons so that its numerical acuity – or accuracy – was refined by the pattern it had just observed, independent of the total surface area of the objects, establishing that the neurons were indeed detecting numbers.

In effect, the network began to generate its own rules and learning process for estimating the number of objects in an image, following a pattern of neuronal activity that has been observed in the parietal cortex of monkeys. This is the region of the brain involved in knowledge of numbers and arithmetic, suggesting that the GENMOD model probably closely reflects how real brains work.

Learning number acuity like a child
“A six-month-old child has relatively weak approximate number sense: for example, it can tell the difference between 8 dots and 16 dots but not 8 dots and 12 dots. Discrimination ability improves throughout childhood. Our network showed similar progress in number acuity, with its ability to determine the number of objects improving over time as it observed more images,” according to Zorzi, who plans to discuss his research at the EuroScience Open Forum 2016 on 26 July in a session entitled ‘Can we simulate the human brain?’

The project’s work on numerical cognition could have important implications for neuroscience and education, such as understanding the possible causes of impaired number sense in children with dyscalculia, the effect of ageing on number skills and enhancing research into pathologies caused by brain damage.

GENMOD’s impact could be even more far-reaching in other fields, with applications in machine vision, neuroinformatics and artificial intelligence.

“Much of the previous work on modelling human cognition with artificial neural networks has been based on a supervised learning algorithm. Apart from being biologically implausible, this algorithm requires that an external teaching signal is available at each learning event and implies the dubious assumption that learning is largely discriminative,” Zorzi explains. “In contrast, generative models learn internal representations of the sensory data without any supervision or reward. That is, the sensory patterns, for example images of objects, do not need to be labelled to tell the network what has been presented as input or how it should react to it.”

A breakthrough in modelling human perception
The GENMOD team has also used deep neural networks to develop the first full-blown, realistic computational model of letter perception that learned from thousands of images of letters in a variety of fonts, styles and sizes in a completely unsupervised way. By inputting random images of natural scenes beforehand, the network learned over time to define lines, shapes and patterns. When it was subsequently given written text to observe, it applied the same processes to differentiate the letters and eventually words.

“This supports the hypothesis about how humans developed written language. There is no part of the brain evolved for reading, so therefore we use the same cognitive processes as we do for identifying objects,” Zorzi says. “The generative model approach is a major breakthrough for modelling human perception and cognition, consistent with neurobiological theories that emphasise the mixing of bottom-up and top-down interactions in the brain.”

Unsupervised learning neural networks could also be put to use for a wide variety of applications where data is uncategorised and unlabelled. For example, the network could be used to identify features of human brain activity from functional magnetic resonance imaging that would be impossible for other technology or human observers to explore. It could even be used to make smartphones truly smart, imbuing mobile devices with cognitive abilities such as intelligent observation, learning and decision-making to overcome the growing problem of network overload.

“Our findings demonstrate that generative models represent a crucial step forward. We expect our work to influence the broader cognitive modelling community and inspire other researchers to embrace the framework in future lines of research,” Zorzi says.

  • Project details:

    Researcher (PI):
    Marco Zorzi

    Host institution:
    Universita Degli Studi Di Padova, Italy

    Generative Models of Human Cognition, (GENMOD)

    ERC call:
    Starting Grant , ERC-2007-StG, panel SH3

    Max ERC funding:
    492,200 €

    Start date: 2008-06-01, End date: 2013-05-31


Marco Zorzi is a Full Professor of Cognitive Psychology and Artificial Intelligence at the University of Padua. He leads an interdisciplinary research group, the Computational Cognitive Neuroscience Lab, that explores the computational bases of cognitive functions such as numeracy, spatial recognition, visuospatial processing, reading and writing.

Inclusion across the Nation of Communities of Learners of Underrepresented Discoverers in Engineering and Science

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Inclusion across the Nation of Communities of Learners of Underrepresented Discoverers in Engineering and Science

Program Solicitation
NSF 16-544

National Science Foundation

Directorate for Biological Sciences

Directorate for Computer & Information Science & Engineering

Directorate for Education & Human Resources
Division of Human Resource Development

Directorate for Engineering

Directorate for Geosciences

Directorate for Mathematical & Physical Sciences

Directorate for Social, Behavioral & Economic Sciences

Office of Integrative Activities

Preliminary Proposal Due Date(s) (required) (due by 5 p.m. proposer’s local time):

April 15, 2016

Design and Development Launch Pilots

Full Proposal Deadline(s) (due by 5 p.m. proposer’s local time):

June 24, 2016

Design and Development Launch Pilots


Preliminary proposals and full proposals. Submission of a preliminary proposal is required for Design and Development Launch Pilots. Full Design and Development Launch Pilot proposals may be submitted by invitation only after the review of the preliminary proposal is completed.

Any proposal submitted in response to this solicitation should be submitted in accordance with the revised NSF Proposal & Award Policies & Procedures Guide (PAPPG) (NSF 16-1), which is effective for proposals submitted, or due, on or after January 25, 2016.


General Information

Program Title:

Inclusion across the Nation of Communities of Learners of Underrepresented Discoverers in Engineering and Science (NSF INCLUDES)

Synopsis of Program:

Inclusion across the Nation of Communities of Learners of Underrepresented Discoverers in Engineering and Science (NSF INCLUDES) is a comprehensive national initiative designed to enhance U.S. leadership in science, technology, engineering and mathematics (STEM) discoveries and innovations focused on NSF’s commitment to diversity, inclusion, and broadening participation in these fields. NSF INCLUDES supports efforts to develop talent from all sectors of society to build the STEM workforce. The initiative aims to improve the preparation, increase the participation, and ensure the contributions of individuals from groups that have traditionally been underrepresented and underserved in the STEM enterprise, including women, members of racial and ethnic groups, persons with disabilities, and persons with low socio-economic status. Significant advancement of these groups will result in a new generation of promising STEM talent and leadership to secure our nation’s future in science and technology.

The grand challenge of broadening participation in STEM is to transform the STEM enterprise at all levels in order to fully engage the nation’s talent for the ultimate improvement of the STEM enterprise. As a comprehensive national initiative, NSF INCLUDES aims to address the various complex equity and inclusion-related challenges and opportunities that characterize the nation’s cultural and linguistic diversity, with a specific emphasis on the aforementioned groups. The goal is to achieve national level impact and progress toward STEM inclusion. Viewing this challenge as a social innovation problem, NSF is particularly interested in using approaches to scaling and growth such as collective impact, networked communities and strategic partnerships. The objective is to develop networks that involve representative organizations and consortia from different sectors that are committed to a common agenda to solve a specific STEM inclusion problem at scale. The long-term goal of NSF INCLUDES is to support, over the next ten years, innovative models, networks, partnerships, and research that enable the U.S. science and engineering workforce to thrive by ensuring that women, blacks, Hispanics, and people with disabilities are represented in percentages comparable to their representation in the U.S. population.

In FY 2016, NSF seeks proposals for Design and Development Launch Pilots to catalyze the formation of NSF INCLUDES Alliances.

Cognizant Program Officer(s):

Please note that the following information is current at the time of publishing. See program website for any updates to the points of contact.

  • Bernice T. Anderson, telephone: (703) 292-5151, email: banderso
  • Janice Cuny, telephone: (703) 292-8900, email: jcuny
  • Tasha R. Inniss, telephone: (703) 292-4684, email: tinniss
  • Mark H. Leddy, telephone: (703) 292-4655, email: mleddy
  • Julio E. Lopez-Ferrao, telephone: (703) 292- 5183, email: jlopezfe
  • James L. Moore, telephone: (703) 292-7082, email: jamoore

Applicable Catalog of Federal Domestic Assistance (CFDA) Number(s):

  • 47.041 — Engineering
  • 47.049 — Mathematical and Physical Sciences
  • 47.050 — Geosciences
  • 47.070 — Computer and Information Science and Engineering
  • 47.074 — Biological Sciences
  • 47.075 — Social Behavioral and Economic Sciences
  • 47.076 — Education and Human Resources
  • 47.083 — Office of Integrative Activities (OIA)

Award Information

Anticipated Type of Award: Standard Grant

Estimated Number of Awards: 30 to 40

In FY 2016, 30 – 40 NSF INCLUDES two-year Design and Development Launch Pilot Projects awards will be made.

Anticipated Funding Amount: $12,500,000

In FY 2016, approximately $12.5 million is available to fund 30 – 40 NSF INCLUDES two-year Design and Development Launch Pilot Projects at levels up to $300,000 each.

Eligibility Information

Who May Submit Proposals:

The categories of proposers eligible to submit proposals to the National Science Foundation are identified in the Grant Proposal Guide, Chapter I, Section E.

Who May Serve as PI:

The PI must hold a permanent position at the lead institution. The PI must have experience in leading distributed teams and organizations. Collaboration for impact in STEM relevant activities is desirable but not required.

Limit on Number of Proposals per Organization:

An organization may serve as the lead institution on only one Design and Development Launch Pilot proposal.

Limit on Number of Proposals per PI or Co-PI: 1

An individual may serve as a PI on only one (1) Design and Development Launch Pilot proposal. An individual may serve as the Co-PI on up to three (3) Design and Development Launch Pilot proposals.

Proposal Preparation and Submission Instructions

A. Proposal Preparation Instructions

  • Letters of Intent: Not required
  • Preliminary Proposals: Submission of Preliminary Proposals is required. Please see the full text of this solicitation for further information.
  • Full Proposals:
    • Full Proposals submitted via FastLane: NSF Proposal and Award Policies and Procedures Guide, Part I: Grant Proposal Guide (GPG) Guidelines apply. The complete text of the GPG is available electronically on the NSF website at:
    • Full Proposals submitted via NSF Application Guide: A Guide for the Preparation and Submission of NSF Applications via Guidelines apply (Note: The NSF Application Guide is available on the website and on the NSF website at:

B. Budgetary Information

  • Cost Sharing Requirements: Inclusion of voluntary committed cost sharing is prohibited.
  • Indirect Cost (F&A) Limitations: Not Applicable
  • Other Budgetary Limitations: Not Applicable

C. Due Dates

  • Preliminary Proposal Due Date(s) (required) (due by 5 p.m. proposer’s local time): April 15, 2016

    Design and Development Launch Pilots

  • Full Proposal Deadline(s) (due by 5 p.m. proposer’s local time): June 24, 2016

    Design and Development Launch Pilots

Proposal Review Information Criteria

Merit Review Criteria:

National Science Board approved criteria apply.

Award Administration Information

Award Conditions:

Standard NSF award conditions apply.

Reporting Requirements:

Standard NSF reporting requirements apply.


Summary of Program Requirements

  1. Introduction
  2. Program Description
  3. Award Information
  4. Eligibility Information
  5. Proposal Preparation and Submission Instructions
    1. Proposal Preparation Instructions
    2. Budgetary Information
    3. Due Dates
    4. FastLane/ Requirements
  6. NSF Proposal Processing and Review Procedures
    1. Merit Review Principles and Criteria
    2. Review and Selection Process
  7. Award Administration Information
    1. Notification of the Award
    2. Award Conditions
    3. Reporting Requirements
  8. Agency Contacts
  9. Other Information


Diversity – of thought, perspective, and experience – is essential for excellence in research and innovation in science and engineering.1 Full participation of all of America’s STEM talent is critical to the advancement of science and engineering for national security, health, and prosperity. America’s STEM talent pool has a competitive advantage when it is enriched by diversity of perspectives and approaches, which in turn enriches knowledge across STEM. African Americans, Hispanics, Native Americans, women, persons with disabilities, and persons with low socio-economic status are underrepresented in various fields of science and engineering across all levels – from K-12 to long-term workforce participation.2 Inclusion of talent from all these sectors of American society is necessary for the health and vitality of the science and engineering community and its societal relevance.

NSF Inclusion across the Nation of Communities of Learners of Underrepresented Discoverers in Engineering and Science (NSF INCLUDES) is a comprehensive initiative to enhance U.S. leadership in science and engineering discovery and innovation by proactively seeking and effectively developing STEM talent from all sectors and groups in our society.

The overarching goal of NSF INCLUDES is to create a sustainable collaborative process for the inclusion in STEM of women, members of racial and ethnic groups that have been underrepresented in STEM, persons with low socio-economic status and people with disabilities. NSF INCLUDES will improve the preparation, increase the participation, and ensure the contributions of individuals from groups that traditionally have been underrepresented in the STEM enterprise.

NSF INCLUDES aims to mobilize communities concerned with STEM opportunities to bring renewed focus and effective collaboration to solving broadening participation challenges at scale. Collective commitment to specific objectives for inclusion is necessary for impact at scale in STEM. This initiative will leverage investments from NSF programs and projects focused on broadening participation, building on lessons learned, best practices, and proven mechanisms for achieving success.3 4

Collaborative alliances spanning both education levels and public and private sectors, and including new partners, will need to be developed, expanded, organized and built by leveraging state-of-the-art knowledge on scaling of social innovations. For example, the collective impact approaches that incorporate key success determinants of common agenda, shared measurements, mutually reinforcing activities, continuous communications, and backbone support organizations have the potential to yield large-scale progress towards NSF INCLUDES’ goals. While the latest knowledge from the science of broadening participation provides a strong foundation, novel systems approaches and designs for achieving scale are critical for advancing diversity and inclusion in STEM.5 6 7

NSF INCLUDES will fund new research, models, networks, and partnerships that lead to measureable progress in diversity and inclusion in STEM, and have the ability to scale to the national level. The multi-year goals of NSF INCLUDES are to:

  1. Synthesize and build the research base for broadening participation and foster the spread and adaptation of proven effective practices.
  2. Support the identification, development and attainment of a set of shared goals and objectives developed by stakeholders, including those from specific STEM disciplines, which are essential for achieving inclusion in the nation’s scientific workforce and in high quality STEM learning opportunities.
  3. Support local/regional and discipline-specific or crosscutting multi-stakeholder partnerships and networks (NSF INCLUDES Alliances) and support an NSF INCLUDES National Network.

With Passage of Every Student Succeeds Act, Life After NCLB Begins


With Passage of Every Student Succeeds Act, Life After NCLB Begins

By Tim Walker

On December 10, President Obama, with a stroke of a pen, made it official: the No Child Left Behind era is over. Obama signed into law the Every Student Succeeds Act (ESSA), one day after it was passed by an overwhelmingly bipartisan vote in the U.S. Senate, which followed broad passage in the House last week.

The Every Student Succeeds Act is the seventh reauthorization of the landmark Elementary and Secondary Education Act, first passed in 1965, and the first since 2002 when NCLB became law. This reauthorization has been years in the making and suffered through several false starts, but it picked up steam in 2015 as opposition to the rigid and punitive “test and punish” regimen imposed by NCLB intensified and several education groups, including the NEA, lobbied Congress to get the job done.

“Students can’t afford to live another year under No Child Left Behind,” NEA President Lily Eskelsen García said repeatedly this year. Major progress was made over the summer with the passage of two separate reauthorization bills – the Every Child Achieves Act in the Senate and the Student Success Act in the House. In November, leaders from both chambers met and hammered out the compromise final bill – the Every Student Succeeds Act.