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Current Essay Contest

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**Undecidability, Uncomputability, and Unpredictability Essay Contest**

*December 24, 2019 - April 24, 2020*

Contest Partners: Fetzer Franklin Fund, and The Peter and Patricia Gruber Foundation

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**What Is “Fundamental”**

*October 28, 2017 to January 22, 2018*

*Sponsored by the Fetzer Franklin Fund and The Peter & Patricia Gruber Foundation*

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**Wandering Towards a Goal**

How can mindless mathematical laws give rise to aims and intention?

*December 2, 2016 to March 3, 2017*

Contest Partner: The Peter and Patricia Gruber Fund.

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**Trick or Truth: The Mysterious Connection Between Physics and Mathematics**

*Contest Partners: Nanotronics Imaging, The Peter and Patricia Gruber Foundation, and The John Templeton Foundation*

Media Partner: Scientific American

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**How Should Humanity Steer the Future?**

*January 9, 2014 - August 31, 2014*

*Contest Partners: Jaan Tallinn, The Peter and Patricia Gruber Foundation, The John Templeton Foundation, and Scientific American*

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**It From Bit or Bit From It**

*March 25 - June 28, 2013*

*Contest Partners: The Gruber Foundation, J. Templeton Foundation, and Scientific American*

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**Questioning the Foundations**

Which of Our Basic Physical Assumptions Are Wrong?

*May 24 - August 31, 2012*

*Contest Partners: The Peter and Patricia Gruber Foundation, SubMeta, and Scientific American*

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**Is Reality Digital or Analog?**

*November 2010 - February 2011*

*Contest Partners: The Peter and Patricia Gruber Foundation and Scientific American*

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**What's Ultimately Possible in Physics?**

*May - October 2009*

*Contest Partners: Astrid and Bruce McWilliams*

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**The Nature of Time**

*August - December 2008*

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Current Essay Contest

Previous Contests

Contest Partners: Fetzer Franklin Fund, and The Peter and Patricia Gruber Foundation

read/discuss

read/discuss • winners

How can mindless mathematical laws give rise to aims and intention?

Contest Partner: The Peter and Patricia Gruber Fund.

read/discuss • winners

Media Partner: Scientific American

read/discuss • winners

read/discuss • winners

read/discuss • winners

Which of Our Basic Physical Assumptions Are Wrong?

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FQXi ESSAY CONTEST

July 16, 2020

2015

First Prize

Children of the Cosmos

Essay Abstract

Our mathematical models may appear unreasonably effective to us, but only if we forget to take into account who we are: we are the children of this Cosmos. We were born here and we know our way around the block, even if we do not always appreciate just how wonderful an achievement that is.

Authors Bio

Sylvia Wenmackers is a professor in the philosophy of science at KU Leuven (Belgium). She studied theoretical physics and obtained a Ph.D. in Physics (2008) as well as in Philosophy (2011). In her current project, she explores the foundations of physics, with a special interest in infinitesimals and probabilities.

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Our mathematical models may appear unreasonably effective to us, but only if we forget to take into account who we are: we are the children of this Cosmos. We were born here and we know our way around the block, even if we do not always appreciate just how wonderful an achievement that is.

Authors Bio

Sylvia Wenmackers is a professor in the philosophy of science at KU Leuven (Belgium). She studied theoretical physics and obtained a Ph.D. in Physics (2008) as well as in Philosophy (2011). In her current project, she explores the foundations of physics, with a special interest in infinitesimals and probabilities.

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Second Prizes

Mathematics is Physics

Essay Abstract

In this essay, I argue that mathematics is a natural science---just like physics, chemistry, or biology---and that this can explain the alleged "unreasonable" effectiveness of mathematics in the physical sciences. The main challenge for this view is to explain how mathematical theories can become increasingly abstract and develop their own internal structure, whilst still maintaining an appropriate empirical tether that can explain their later use in physics. In order to address this, I offer a theory of mathematical theory-building based on the idea that human knowledge has the structure of a scale-free network and that abstract mathematical theories arise from a repeated process of replacing strong analogies with new hubs in this network. This allows mathematics to be seen as the study of regularities, within regularities, within ..., within regularities of the natural world. Since mathematical theories are derived from the natural world, albeit at a much higher level of abstraction than most other scientific theories, it should come as no surprise that they so often show up in physics.

Authors Bio

Matt Leifer is a visiting researcher at Perimeter Institute for Theoretical Physics. His research interests include quantum foundations, quantum information, and particularly the intersection of the two. He is hoping to break the world record for the number of FQXi essay contests won by a single individual.

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In this essay, I argue that mathematics is a natural science---just like physics, chemistry, or biology---and that this can explain the alleged "unreasonable" effectiveness of mathematics in the physical sciences. The main challenge for this view is to explain how mathematical theories can become increasingly abstract and develop their own internal structure, whilst still maintaining an appropriate empirical tether that can explain their later use in physics. In order to address this, I offer a theory of mathematical theory-building based on the idea that human knowledge has the structure of a scale-free network and that abstract mathematical theories arise from a repeated process of replacing strong analogies with new hubs in this network. This allows mathematics to be seen as the study of regularities, within regularities, within ..., within regularities of the natural world. Since mathematical theories are derived from the natural world, albeit at a much higher level of abstraction than most other scientific theories, it should come as no surprise that they so often show up in physics.

Authors Bio

Matt Leifer is a visiting researcher at Perimeter Institute for Theoretical Physics. His research interests include quantum foundations, quantum information, and particularly the intersection of the two. He is hoping to break the world record for the number of FQXi essay contests won by a single individual.

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My God, It's Full of Clones: Living in a Mathematical Universe

Essay Abstract

Imagine there's only math: physics is nothing more than mathematics, we are self-aware mathematical substructures, and our physical universe is nothing more than a mathematical structure "seen from the inside". If that's the case, I will argue that it implies the existence of the Maxiverse, the largest imaginable multiverse, where every possible conscious observation is guaranteed to happen. I will attempt to explain why, of all the worlds in the infinite Maxiverse, we happen to live in one that can be understood by physical laws simple enough to be discovered (or, at least, approximated well enough for predictive and technological purposes). I will consider the question of personal identity in the context of a Maxiverse that contains an infinite number of exact clones of myself, and whether I should expect my future subjective experience to be unbounded. I will also consider the question of whether the Maxiverse hypothesis makes predictions that can be put to the test.

Authors Bio

Marc Séguin holds two master's degrees from Harvard University: one in Astronomy (under the supervision of David Layzer) and another in History of Science (under the supervision of Gerald Holton). He teaches physics and astrophysics at Collège de Maisonneuve, in Montréal, and is the author of several college-level textbooks in physics and astrophysics.

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Imagine there's only math: physics is nothing more than mathematics, we are self-aware mathematical substructures, and our physical universe is nothing more than a mathematical structure "seen from the inside". If that's the case, I will argue that it implies the existence of the Maxiverse, the largest imaginable multiverse, where every possible conscious observation is guaranteed to happen. I will attempt to explain why, of all the worlds in the infinite Maxiverse, we happen to live in one that can be understood by physical laws simple enough to be discovered (or, at least, approximated well enough for predictive and technological purposes). I will consider the question of personal identity in the context of a Maxiverse that contains an infinite number of exact clones of myself, and whether I should expect my future subjective experience to be unbounded. I will also consider the question of whether the Maxiverse hypothesis makes predictions that can be put to the test.

Authors Bio

Marc Séguin holds two master's degrees from Harvard University: one in Astronomy (under the supervision of David Layzer) and another in History of Science (under the supervision of Gerald Holton). He teaches physics and astrophysics at Collège de Maisonneuve, in Montréal, and is the author of several college-level textbooks in physics and astrophysics.

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Third Prizes

Most Creative Presentation

Let's consider two spherical chickens

Let's consider two spherical chickens

Essay Abstract

Confronted with a pythagorean jingle derived from simple ratios, a sequence of 23 moves from knot theory, and the interaction between a billiard-ball and a zero-gravity field, a young detective soon realizes that three crimes could have been avoided if math were not so unreasonably effective in describing our physical world. Why is this so? Asimov's fictional character Prof. Priss confirms to the detective that there is some truth in Tegmark's Mathematical Universe Hypothesis, and reveals him that all mathematical structures entailing self-aware substructures (SAS) are computable and isomorphic. The boss at the investigation agency is not convinced and proposes his own views on the question.

Authors Bio

Tommaso Bolognesi (Laurea in Physics, Univ. of Pavia, 1976; M.Sc. in CS, Univ. of Illinois at U-C, 1982), is senior researcher at ISTI, CNR, Pisa. His research areas have included stochastic processes in computer music composition, models of concurrency, process algebra and formal methods for software development, discrete and algorithmic models of spacetime. He has published on various international scientific journals several papers in all three areas. He obtained two 4th prizes at the FQXi Essay Contests of 2011 and 2014.

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Confronted with a pythagorean jingle derived from simple ratios, a sequence of 23 moves from knot theory, and the interaction between a billiard-ball and a zero-gravity field, a young detective soon realizes that three crimes could have been avoided if math were not so unreasonably effective in describing our physical world. Why is this so? Asimov's fictional character Prof. Priss confirms to the detective that there is some truth in Tegmark's Mathematical Universe Hypothesis, and reveals him that all mathematical structures entailing self-aware substructures (SAS) are computable and isomorphic. The boss at the investigation agency is not convinced and proposes his own views on the question.

Authors Bio

Tommaso Bolognesi (Laurea in Physics, Univ. of Pavia, 1976; M.Sc. in CS, Univ. of Illinois at U-C, 1982), is senior researcher at ISTI, CNR, Pisa. His research areas have included stochastic processes in computer music composition, models of concurrency, process algebra and formal methods for software development, discrete and algorithmic models of spacetime. He has published on various international scientific journals several papers in all three areas. He obtained two 4th prizes at the FQXi Essay Contests of 2011 and 2014.

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The Deeper Roles of Mathematics in Physical Laws

Essay Abstract

Many have wondered how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as exemplified by the laws of physics. In this essay, I claim that much of the utility of mathematics arises from our choice of description of the physical world coupled with our desire to quantify it. This will be demonstrated in a practical sense by considering one of the most fundamental concepts of mathematics: additivity. This example will be used to show how many physical laws can be derived as constraint equations enforcing relevant symmetries in a sense that is far more fundamental than commonly appreciated.

Authors Bio

Kevin Knuth is an Associate Professor in the Departments of Physics and Informatics at the University at Albany. He is Editor-in-Chief of the journal Entropy, and is the co-founder and President of the robotics company Autonomous Exploration Inc. He has 20 years of experience in applying Bayesian and maximum entropy methods to the design of machine learning algorithms for data analysis applied to the physical sciences. His current research interests include the foundations of physics, autonomous robotics, and the search for and characterization of extrasolar planets.

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Many have wondered how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as exemplified by the laws of physics. In this essay, I claim that much of the utility of mathematics arises from our choice of description of the physical world coupled with our desire to quantify it. This will be demonstrated in a practical sense by considering one of the most fundamental concepts of mathematics: additivity. This example will be used to show how many physical laws can be derived as constraint equations enforcing relevant symmetries in a sense that is far more fundamental than commonly appreciated.

Authors Bio

Kevin Knuth is an Associate Professor in the Departments of Physics and Informatics at the University at Albany. He is Editor-in-Chief of the journal Entropy, and is the co-founder and President of the robotics company Autonomous Exploration Inc. He has 20 years of experience in applying Bayesian and maximum entropy methods to the design of machine learning algorithms for data analysis applied to the physical sciences. His current research interests include the foundations of physics, autonomous robotics, and the search for and characterization of extrasolar planets.

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How Mathematics Meets the World

Essay Abstract

The most obvious explanation for the power of mathematics as the language of physics is that the physical world has the right sort of structure to be represented mathematically. But what this in turn means depends on the mathematical language being used. I first briefly review some of the physical characteristics required in order to unambiguously describe a physical situation using integers, and then take up the much more difficult question of what characteristics are required to describe a situation using geometrical concepts. In the case of geometry, and particularly for the most basic form of geometry— topology—this is not clear. I discuss a new mathematical language for describing geometrical structure called the Theory of Linear Structures. This mathematical language is founded on a different primitive concept than standard topology, on the line rather than the open set. I explain how some other geometrical concepts can be defined in terms of lines, and how in a Relativistic setting time can be understood as the feature of physical reality that generates all geometrical facts. Whereas it is often said that Relativity spatializes time, from the perspective of the Theory of Linear Structures we can see instead that Relativity temporalizes space: all of the geometry flows from temporal structure. The Theory of Linear Structures also provides a mathematical language in which the fact that time is a fundamentally directed structure can be easily represented.

Authors Bio

Tim Maudlin is Professor of Philosophy at NYU. He received his B.A. in physics and philosophy from Yale and his Ph.D. in History and Philosophy of Science from the University of Pittsburgh. His books include Quantum Non-Localtiy and Relativity (Blackwell), The Metaphysics Within Physics (Oxford), Philosophy of Physics: Space and Time (Princeton), and New Foundations for Physical Geometry: The Theory of Linear Structures (Oxford). He has been a Guggenheim Fellow.

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The most obvious explanation for the power of mathematics as the language of physics is that the physical world has the right sort of structure to be represented mathematically. But what this in turn means depends on the mathematical language being used. I first briefly review some of the physical characteristics required in order to unambiguously describe a physical situation using integers, and then take up the much more difficult question of what characteristics are required to describe a situation using geometrical concepts. In the case of geometry, and particularly for the most basic form of geometry— topology—this is not clear. I discuss a new mathematical language for describing geometrical structure called the Theory of Linear Structures. This mathematical language is founded on a different primitive concept than standard topology, on the line rather than the open set. I explain how some other geometrical concepts can be defined in terms of lines, and how in a Relativistic setting time can be understood as the feature of physical reality that generates all geometrical facts. Whereas it is often said that Relativity spatializes time, from the perspective of the Theory of Linear Structures we can see instead that Relativity temporalizes space: all of the geometry flows from temporal structure. The Theory of Linear Structures also provides a mathematical language in which the fact that time is a fundamentally directed structure can be easily represented.

Authors Bio

Tim Maudlin is Professor of Philosophy at NYU. He received his B.A. in physics and philosophy from Yale and his Ph.D. in History and Philosophy of Science from the University of Pittsburgh. His books include Quantum Non-Localtiy and Relativity (Blackwell), The Metaphysics Within Physics (Oxford), Philosophy of Physics: Space and Time (Princeton), and New Foundations for Physical Geometry: The Theory of Linear Structures (Oxford). He has been a Guggenheim Fellow.

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A naturalist account of the limited, and hence reasonable, effectiveness of mathematics in physics

Essay Abstract

My aim in this essay is to propose a conception of mathematics that is fully consonant with naturalism. By that I mean the hypothesis that everything that exists is part of the natural world, which makes up a unitary whole.

Authors Bio

Lee Smolin is founding and senior faculty member at Perimeter Institute for Theoretical Physics. He has contributed to quantum gravity, cosmology, quantum foundations through more than 180 research papers. He is the author of five semi-popular books on philosophical issues which illuminate the current crisis in physics and cosmology: Life of the Cosmos, Three Roads to Quantum Gravity, The Trouble with Physics, Time Reborn and, most recently The Singular Universe and the Reality of Time, written with Roberto Mangabeira Unger, from which this essay has been abstracted.

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My aim in this essay is to propose a conception of mathematics that is fully consonant with naturalism. By that I mean the hypothesis that everything that exists is part of the natural world, which makes up a unitary whole.

Authors Bio

Lee Smolin is founding and senior faculty member at Perimeter Institute for Theoretical Physics. He has contributed to quantum gravity, cosmology, quantum foundations through more than 180 research papers. He is the author of five semi-popular books on philosophical issues which illuminate the current crisis in physics and cosmology: Life of the Cosmos, Three Roads to Quantum Gravity, The Trouble with Physics, Time Reborn and, most recently The Singular Universe and the Reality of Time, written with Roberto Mangabeira Unger, from which this essay has been abstracted.

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And the math will set you free

Essay Abstract

Can mathematics help us find our way through all the wonders and mysteries of the universe? When physicists describe the laws governing the physical world, mathematics is always involved. Is this due to the fact that the universe is, at least in part, mathematical? Or rather mathematics is merely a tool used by physicists to model phenomena? Is mathematics just a language to tell the story of our universe, a story which could be told with the same or even more effectiveness using another language? Or quite the opposite, the universe is just a mathematical structure?

Authors Bio

Theoretical physicist. Research interests: foundations of physics, gauge theory, foundations of quantum mechanics, singularities in general relativity. Interested especially in the geometric aspects of the physical laws. ArXiv: http://arxiv.org/a/stoica_o_1 Blog: http://www.unitaryflow.com/ Scholar: https://scholar.google.com/citations?user=aleEOtsAAAAJ

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Can mathematics help us find our way through all the wonders and mysteries of the universe? When physicists describe the laws governing the physical world, mathematics is always involved. Is this due to the fact that the universe is, at least in part, mathematical? Or rather mathematics is merely a tool used by physicists to model phenomena? Is mathematics just a language to tell the story of our universe, a story which could be told with the same or even more effectiveness using another language? Or quite the opposite, the universe is just a mathematical structure?

Authors Bio

Theoretical physicist. Research interests: foundations of physics, gauge theory, foundations of quantum mechanics, singularities in general relativity. Interested especially in the geometric aspects of the physical laws. ArXiv: http://arxiv.org/a/stoica_o_1 Blog: http://www.unitaryflow.com/ Scholar: https://scholar.google.com/citations?user=aleEOtsAAAAJ

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Mathematics: Intuition's Consistency Check

Essay Abstract

There is a well-noted overlap between mathematics and physics, and in many cases the relevant mathematics was developed without any thought of the eventual physical application. This essay argues that this is not a coincidental mystery, but naturally follows from 1) a self-consistency requirement for physical models, and 2) physical intuitions that guide us in the wrong directions, slowing the development of physical models more so than the related mathematics. A detailed example (concerning the flow of time in physical theories) demonstrates key parts of this argument.

Authors Bio

Ken Wharton is a professor in the Department of Physics and Astronomy at San Jose State University. His research is in Quantum Foundations.

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There is a well-noted overlap between mathematics and physics, and in many cases the relevant mathematics was developed without any thought of the eventual physical application. This essay argues that this is not a coincidental mystery, but naturally follows from 1) a self-consistency requirement for physical models, and 2) physical intuitions that guide us in the wrong directions, slowing the development of physical models more so than the related mathematics. A detailed example (concerning the flow of time in physical theories) demonstrates key parts of this argument.

Authors Bio

Ken Wharton is a professor in the Department of Physics and Astronomy at San Jose State University. His research is in Quantum Foundations.

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How not to factor a miracle.

Essay Abstract

Wigner's famous and influential claim that mathematics is "unreasonably effective" in physics is founded on unreasonable assumptions about the nature of mathematics and its independence of physics. Here I argue that what is surprising is not the effectiveness of mathematics but the amenability of physics reductionist strategies. I also argue that while our luck may run out on the effectiveness of reduction, mathematics is still our best hope for surpassing this obstacle. While I agree that human understanding of the natural world in mathematical terms evinces a miracle, I see no way to factor out the human dimension of this miracle.

Authors Bio

Derek Wise is a mathematical physicist at the University of Erlangen, and has been working at the interface of mathematics and physics since undergraduate studies. He was a Visiting Assistant Professor of Mathematics at UC Davis under the NSF VIGRE program until 2010, and has since been a Postdoctoral Fellow at the University of Erlangen, where he has held positions both in the Institute for Quantum Gravity and in the Department of Mathematics. He is dedicated to getting mathematicians and physicists to work together, and promoting mutual understanding.

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Wigner's famous and influential claim that mathematics is "unreasonably effective" in physics is founded on unreasonable assumptions about the nature of mathematics and its independence of physics. Here I argue that what is surprising is not the effectiveness of mathematics but the amenability of physics reductionist strategies. I also argue that while our luck may run out on the effectiveness of reduction, mathematics is still our best hope for surpassing this obstacle. While I agree that human understanding of the natural world in mathematical terms evinces a miracle, I see no way to factor out the human dimension of this miracle.

Authors Bio

Derek Wise is a mathematical physicist at the University of Erlangen, and has been working at the interface of mathematics and physics since undergraduate studies. He was a Visiting Assistant Professor of Mathematics at UC Davis under the NSF VIGRE program until 2010, and has since been a Postdoctoral Fellow at the University of Erlangen, where he has held positions both in the Institute for Quantum Gravity and in the Department of Mathematics. He is dedicated to getting mathematicians and physicists to work together, and promoting mutual understanding.

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Fourth Prizes

GENESIS OF A PYTHAGOREAN UNIVERSE

Essay Abstract

Wide range, high precision and simplicity of the fundamental laws of nature rule out the possibility for them to be randomly generated or selected. Therefore purpose is present in their selecton.

Authors Bio

Alexey Burov, PhD, is a physicist at the Fermi National Accelerator Laboratory (USA). He has numerous publications in professional journals, and he also is an organizer of Fermi Society of Philosophy and the Russian Chicago Philosophy Forum. Lev Burov is an amateur philosopher and a software developer, focusing on work with start-up firms, currently, Scientific Humanities of San Fransisco, CA.

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Wide range, high precision and simplicity of the fundamental laws of nature rule out the possibility for them to be randomly generated or selected. Therefore purpose is present in their selecton.

Authors Bio

Alexey Burov, PhD, is a physicist at the Fermi National Accelerator Laboratory (USA). He has numerous publications in professional journals, and he also is an organizer of Fermi Society of Philosophy and the Russian Chicago Philosophy Forum. Lev Burov is an amateur philosopher and a software developer, focusing on work with start-up firms, currently, Scientific Humanities of San Fransisco, CA.

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Beyond Math

Essay Abstract

In this essay I reflect on the use and usefulness of mathematics from the perspective of a pragmatic physicist. I first classify the different ways we presently think of the relation between our observations and mathematics. Then I explain how we can do physics without using math -- that we are in fact already doing it. In the end the pragmatic reader will know why math is reasonably effective, why we are all models, and how to go beyond math.

Authors Bio

Sophia made a bachelors degree in mathematics before losing her way and ending up at the department of philosophy. She lives in Gothenburg with her partner, two sons, and three bunnies, and wishes she had studied physics.

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In this essay I reflect on the use and usefulness of mathematics from the perspective of a pragmatic physicist. I first classify the different ways we presently think of the relation between our observations and mathematics. Then I explain how we can do physics without using math -- that we are in fact already doing it. In the end the pragmatic reader will know why math is reasonably effective, why we are all models, and how to go beyond math.

Authors Bio

Sophia made a bachelors degree in mathematics before losing her way and ending up at the department of philosophy. She lives in Gothenburg with her partner, two sons, and three bunnies, and wishes she had studied physics.

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Why Mathematics Works So Well

Essay Abstract

A major question in philosophy of science involves the unreasonable effectiveness of mathematics in physics. Why should mathematics, created or discovered, with nothing empirical in mind be so perfectly suited to describe the laws of the physical universe? We review the well-known fact that the symmetries of the laws of physics are their defining properties. We show that there are similar symmetries of mathematical facts and that these symmetries are the defining properties of mathematics. By examining the symmetries of physics and mathematics, we show that the effectiveness is actually quite reasonable. In essence, we show that the regularities of physics are a subset of the regularities of mathematics.

Authors Bio

Noson S. Yanofsky has a PhD in mathematics (category theory). He is a professor of computer science in Brooklyn College. In addition to writing research papers he also co-authored “Quantum Computing for Computer Scientists”(Cambridge University Press, 2008) and “The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us” (MIT Press 2013). The second book is a popular science book that has been received very well both critically and popularly. He lives in Brooklyn with his wife and three children.

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A major question in philosophy of science involves the unreasonable effectiveness of mathematics in physics. Why should mathematics, created or discovered, with nothing empirical in mind be so perfectly suited to describe the laws of the physical universe? We review the well-known fact that the symmetries of the laws of physics are their defining properties. We show that there are similar symmetries of mathematical facts and that these symmetries are the defining properties of mathematics. By examining the symmetries of physics and mathematics, we show that the effectiveness is actually quite reasonable. In essence, we show that the regularities of physics are a subset of the regularities of mathematics.

Authors Bio

Noson S. Yanofsky has a PhD in mathematics (category theory). He is a professor of computer science in Brooklyn College. In addition to writing research papers he also co-authored “Quantum Computing for Computer Scientists”(Cambridge University Press, 2008) and “The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us” (MIT Press 2013). The second book is a popular science book that has been received very well both critically and popularly. He lives in Brooklyn with his wife and three children.

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Demystifying the Applicability of Mathematics

Essay Abstract

Essential tensions remain in our understanding of the reasons underlying the striking success achieved in science by applying mathematics. Wigner and many likeminded scientists and philosophers conclude that this success is a miracle, a ``wonderful gift which we neither deserve nor understand.'' This essay seeks to dissipate that aura of mystery and bring the factors underlying the success of applied mathematics into the fold of scientific rationality.

Authors Bio

Nic is an assistant professor of philosophy and member of the center for scientific computing at Simon Fraser University, Canada. His work focuses on philosophy of science, philosophy of mathematics, logic, and numerical analysis.

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Essential tensions remain in our understanding of the reasons underlying the striking success achieved in science by applying mathematics. Wigner and many likeminded scientists and philosophers conclude that this success is a miracle, a ``wonderful gift which we neither deserve nor understand.'' This essay seeks to dissipate that aura of mystery and bring the factors underlying the success of applied mathematics into the fold of scientific rationality.

Authors Bio

Nic is an assistant professor of philosophy and member of the center for scientific computing at Simon Fraser University, Canada. His work focuses on philosophy of science, philosophy of mathematics, logic, and numerical analysis.

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The Language of Nature

Essay Abstract

Galileo considered mathematics the language of nature. However, Wigner thought the effectiveness of mathematics in physics "miraculous" and noted that much of the mathematics needed for quantum mechanics had been previously developed by mathematicians for purposes having nothing to do with physics. I argue that Galileo's view is correct; but that the examples cited by Wigner in support of his view can be explained using two deep truths, one about mathematics and the other about physics. These truths are: (1) Since the advent of non-Euclidean geometry, new mathematics has been developed by abstracting and generalizing old mathematics. (2) New physical theories have old physical theories as limiting cases.

Authors Bio

David Garfinkle is a Professor of Physics at Oakland University, in Rochester, Michigan. He has a BA in physics (Summa cum laude) from Princeton University and a PhD in physics from The University of Chicago. His field of research is Einstein's general theory of relativity, especially the study of spacetime singularities. He is the author (along with his brother Richard Garfinkle) of "Three Steps to the Universe" a book for general readers on black holes and dark matter.

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Galileo considered mathematics the language of nature. However, Wigner thought the effectiveness of mathematics in physics "miraculous" and noted that much of the mathematics needed for quantum mechanics had been previously developed by mathematicians for purposes having nothing to do with physics. I argue that Galileo's view is correct; but that the examples cited by Wigner in support of his view can be explained using two deep truths, one about mathematics and the other about physics. These truths are: (1) Since the advent of non-Euclidean geometry, new mathematics has been developed by abstracting and generalizing old mathematics. (2) New physical theories have old physical theories as limiting cases.

Authors Bio

David Garfinkle is a Professor of Physics at Oakland University, in Rochester, Michigan. He has a BA in physics (Summa cum laude) from Princeton University and a PhD in physics from The University of Chicago. His field of research is Einstein's general theory of relativity, especially the study of spacetime singularities. He is the author (along with his brother Richard Garfinkle) of "Three Steps to the Universe" a book for general readers on black holes and dark matter.

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The ultimate tactics of self-referential systems

Essay Abstract

Mathematics is usually regarded as a kind of language. The essential behavior of physical phenomena can be expressed by mathematical laws, providing descriptions and predictions. In the present essay I argue that, although mathematics can be seen, in a first approach, as a language, it goes beyond this concept. I conjecture that mathematics presents two extreme features, denoted here by irreducibility and insaturation, representing delimiters for self-referentiality. These features are then related to physical laws by realizing that nature is a self-referential system obeying bounds similar to those respected by mathematics. Self-referential systems can only be autonomous entities by a kind of metabolism that provides and sustains such an autonomy. A rational mind, able of consciousness, is a manifestation of the self-referentiality of the Universe. Hence mathematics is here proposed to go beyond language by actually representing the most fundamental existence condition for self-referentiality. This idea is synthesized in the form of a principle, namely, that mathematics is the ultimate tactics of self-referential systems to mimic themselves. That is, well beyond an effective language to express the physical world, mathematics uncovers a deep manifestation of the autonomous nature of the Universe, wherein the human brain is but an instance.

Authors Bio

Christine C. Dantas has an undergraduate degree in Data Processing Technology (PUC-RJ/Brazil, 1991), BS in Astronomy (UFRJ/Brazil, 1993), MSc in Astrophysics (INPE/Brazil, 1996) and PhD in Astrophysics (INPE/Brazil, 2001). She is interested in all areas of science and philosophy. Scientific papers can be downloaded at http://arxiv.org/a/dantas_c_1 .

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Mathematics is usually regarded as a kind of language. The essential behavior of physical phenomena can be expressed by mathematical laws, providing descriptions and predictions. In the present essay I argue that, although mathematics can be seen, in a first approach, as a language, it goes beyond this concept. I conjecture that mathematics presents two extreme features, denoted here by irreducibility and insaturation, representing delimiters for self-referentiality. These features are then related to physical laws by realizing that nature is a self-referential system obeying bounds similar to those respected by mathematics. Self-referential systems can only be autonomous entities by a kind of metabolism that provides and sustains such an autonomy. A rational mind, able of consciousness, is a manifestation of the self-referentiality of the Universe. Hence mathematics is here proposed to go beyond language by actually representing the most fundamental existence condition for self-referentiality. This idea is synthesized in the form of a principle, namely, that mathematics is the ultimate tactics of self-referential systems to mimic themselves. That is, well beyond an effective language to express the physical world, mathematics uncovers a deep manifestation of the autonomous nature of the Universe, wherein the human brain is but an instance.

Authors Bio

Christine C. Dantas has an undergraduate degree in Data Processing Technology (PUC-RJ/Brazil, 1991), BS in Astronomy (UFRJ/Brazil, 1993), MSc in Astrophysics (INPE/Brazil, 1996) and PhD in Astrophysics (INPE/Brazil, 2001). She is interested in all areas of science and philosophy. Scientific papers can be downloaded at http://arxiv.org/a/dantas_c_1 .

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Special Prizes

Non-academic Prize

A Metaphorical Chart of Our Mathematical Ontology

A Metaphorical Chart of Our Mathematical Ontology

Essay Abstract

This is my vision of the relationship between mathematics and physics based on my observations on the nature of physical laws and mathematical structures. I use mainstream ideas from quantum gravity such as string theory, holography, the landscape and loop quantum gravity as a base for my reasoning. In addition I bring together more speculative ideas such as the Mathematical Universe Hypothesis, multiple quantisation, universality, iterated integration maps and complete symmetry. In my view these clues come together into a consistent whole where a structure from higher category theory is the central piece from which all else stems. The future of fundamental physics is going to be much more challenging than the past on both the experimental and theoretical sides and these meta-physical structures need to be understood to guide us towards the more specific physical laws which rule our universe. On a more philosophical level they can provide an explanation for why we exist and why the laws of physics are so steeped in mathematical abstraction.

Authors Bio

I hold a PhD in theoretical physics from the University of Glasgow. I have also published a number of papers on fundamental physics as an independent researcher. In addition I love problem solving in mathematics and have made modest contributions including progress on a problem in number theory proposed by Diophantus himself and recently a new improved solution to Lebesgue's universal covering problem. The philosophical links between physics and mathematics are something that I have given much thought to over many years.

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This is my vision of the relationship between mathematics and physics based on my observations on the nature of physical laws and mathematical structures. I use mainstream ideas from quantum gravity such as string theory, holography, the landscape and loop quantum gravity as a base for my reasoning. In addition I bring together more speculative ideas such as the Mathematical Universe Hypothesis, multiple quantisation, universality, iterated integration maps and complete symmetry. In my view these clues come together into a consistent whole where a structure from higher category theory is the central piece from which all else stems. The future of fundamental physics is going to be much more challenging than the past on both the experimental and theoretical sides and these meta-physical structures need to be understood to guide us towards the more specific physical laws which rule our universe. On a more philosophical level they can provide an explanation for why we exist and why the laws of physics are so steeped in mathematical abstraction.

Authors Bio

I hold a PhD in theoretical physics from the University of Glasgow. I have also published a number of papers on fundamental physics as an independent researcher. In addition I love problem solving in mathematics and have made modest contributions including progress on a problem in number theory proposed by Diophantus himself and recently a new improved solution to Lebesgue's universal covering problem. The philosophical links between physics and mathematics are something that I have given much thought to over many years.

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Entertainment Prize

The Raven and the Writing Desk

The Raven and the Writing Desk

Essay Abstract

In this essay, I use a dialogue between characters from Lewis Carroll’s Alice’s Adventures in Wonderland to discuss the relationship of mathematics to physical reality. In it, I propose that there are two realities: representational and tangible. Mathematics belongs to the former. We can reconcile the two by taking Eddington’s stance that the universe is nothing more than our description of it.

Authors Bio

Ian Durham is Professor of Physics at Saint Anselm College where he sometimes ventures — unharmed! — into the Mathematics Department. They even let him serve as Acting Chair of Mathematics once. He holds a PhD in mathematical physics from the University of St. Andrews (Scotland) where he and his wife once danced with Will & Kate. His alter ego, Cyrus Bohm, helped promote FQXi’s recent video contest.

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In this essay, I use a dialogue between characters from Lewis Carroll’s Alice’s Adventures in Wonderland to discuss the relationship of mathematics to physical reality. In it, I propose that there are two realities: representational and tangible. Mathematics belongs to the former. We can reconcile the two by taking Eddington’s stance that the universe is nothing more than our description of it.

Authors Bio

Ian Durham is Professor of Physics at Saint Anselm College where he sometimes ventures — unharmed! — into the Mathematics Department. They even let him serve as Acting Chair of Mathematics once. He holds a PhD in mathematical physics from the University of St. Andrews (Scotland) where he and his wife once danced with Will & Kate. His alter ego, Cyrus Bohm, helped promote FQXi’s recent video contest.

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Creative Thinking Prize

Cognitive Science and the Connection between Physics and Mathematics

Cognitive Science and the Connection between Physics and Mathematics

Essay Abstract

The human mind is endowed with innate primordial perceptions such as space, distance, motion, change, flow of time, matter. The field of cognitive science argues that the abstract concepts of mathematics are not Platonic, but are built in the brain from these primordial perceptions, using what are known as conceptual metaphors. Known cognitive mechanisms give rise to the extremely precise and logical language of mathematics. Thus all of the vastness of mathematics, with its beautiful theorems, is human mathematics. It resides in the mind, and is not `out there’. Physics is an experimental science in which results of experiments are described in terms of concrete concepts – these concepts are also built from our primordial perceptions. The goal of theoretical physics is to describe the experimentally observed regularity of the physical world in an unambiguous, precise and logical manner. To do so, the brain resorts to the well-defined abstract concepts which the mind has metaphored from our primordial perceptions. Since both the concrete and the abstract are derived from the primordial, the connection between physics and mathematics is not mysterious, but natural. This connection is established in the human brain, where a small subset of the vast human mathematics is cognitively fitted to describe the regularity of the universe. Theoretical physics should be thought of as a branch of mathematics, whose axioms are motivated by observations of the physical world. We use the example of quantum theory to demonstrate the all too human nature of the physics-mathematics connection: it is at times frail, and imperfect. Our resistance to take this imperfection sufficiently seriously [since no known experiment violates quantum theory] shows the fundamental importance of experiments in physics. This is unlike in mathematics, the goal there being to search for logical and elegant relations amongst abstract concepts which the mind creates.

Authors Bio

Anshu Gupta Mujumdar is a freelance researcher and visiting faculty in Mathematics/Physics for IB diploma program at the Fazlani L'Academie Globale, Mumbai. She holds a doctorate from the Physical Research Laboratory, Ahmedabad (1997) in the field of general relativity. She has held several postdoctoral positions and was a recipient of a post-doctoral fellowship award in Mathematical Physics (in memory of S. Chandrasekhar, 1998) and Peter Gruber post-doctoral fellowship in 2001. Her research interests are in inflationary cosmology, quantum effects in biological systems, and history of Mathematics. Tejinder Singh is Professor of Physics at the Tata Institute of Fundamental Research, Mumbai.

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The human mind is endowed with innate primordial perceptions such as space, distance, motion, change, flow of time, matter. The field of cognitive science argues that the abstract concepts of mathematics are not Platonic, but are built in the brain from these primordial perceptions, using what are known as conceptual metaphors. Known cognitive mechanisms give rise to the extremely precise and logical language of mathematics. Thus all of the vastness of mathematics, with its beautiful theorems, is human mathematics. It resides in the mind, and is not `out there’. Physics is an experimental science in which results of experiments are described in terms of concrete concepts – these concepts are also built from our primordial perceptions. The goal of theoretical physics is to describe the experimentally observed regularity of the physical world in an unambiguous, precise and logical manner. To do so, the brain resorts to the well-defined abstract concepts which the mind has metaphored from our primordial perceptions. Since both the concrete and the abstract are derived from the primordial, the connection between physics and mathematics is not mysterious, but natural. This connection is established in the human brain, where a small subset of the vast human mathematics is cognitively fitted to describe the regularity of the universe. Theoretical physics should be thought of as a branch of mathematics, whose axioms are motivated by observations of the physical world. We use the example of quantum theory to demonstrate the all too human nature of the physics-mathematics connection: it is at times frail, and imperfect. Our resistance to take this imperfection sufficiently seriously [since no known experiment violates quantum theory] shows the fundamental importance of experiments in physics. This is unlike in mathematics, the goal there being to search for logical and elegant relations amongst abstract concepts which the mind creates.

Authors Bio

Anshu Gupta Mujumdar is a freelance researcher and visiting faculty in Mathematics/Physics for IB diploma program at the Fazlani L'Academie Globale, Mumbai. She holds a doctorate from the Physical Research Laboratory, Ahmedabad (1997) in the field of general relativity. She has held several postdoctoral positions and was a recipient of a post-doctoral fellowship award in Mathematical Physics (in memory of S. Chandrasekhar, 1998) and Peter Gruber post-doctoral fellowship in 2001. Her research interests are in inflationary cosmology, quantum effects in biological systems, and history of Mathematics. Tejinder Singh is Professor of Physics at the Tata Institute of Fundamental Research, Mumbai.

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Out-of-the-Box Thinking Prize

The Descent of Math

The Descent of Math

Essay Abstract

A perplexing problem in understanding physical reality is why the universe seems comprehensible and correspondingly why there should exist physical systems capable of comprehending it. In this essay I explore the possibility that rather than being an odd coincidence arising due to our strange position as passive (and even more strangely, conscious) observers in the cosmos, these two problems might be intimately related and potentially explainable in terms of fundamental physics – that is, if we are willing to make the concession that information can physically influence the world. My premise begins by distinguishing physically possible states of the world, defined as everything permissible by the laws of physics, from physically accessible states of the world, defined as those that are achievable from a given initial state. For universes where the laws of physics are such that the number of accessible states is less than all that is possible, I argue that the most probable states among all possible states are those that include physical systems which contain information encodings – such as mathematics, language and art – because these are the most highly connected states in the state space of everything that is possible. Such physical systems include life and - of particular interest for the discussion of the place of math in physical reality - humans. Within this framework, the descent of math is a natural outcome of the evolution of the universe, which will tend toward states that are increasingly connected to other possible states of the universe, a process greatly facilitated if some physical systems know the rules of the game. I therefore conclude that our ability to use mathematics to describe, and more importantly manipulate, the natural world may not be an anomaly or trick, but instead a natural part of the structure of reality.

Authors Bio

Sara Imari Walker is a theoretical physicist and astrobiologist and Assistant Professor in the Beyond Center for Fundamental Concepts in Science and the School of Earth and Space Exploration at Arizona State University. She received her PhD in Physics from Dartmouth College and has held postdoctoral appointments in the Center for Chemical Evolution at the Georgia Institute of Technology and as a NASA Postdoctoral Program Fellow.

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A perplexing problem in understanding physical reality is why the universe seems comprehensible and correspondingly why there should exist physical systems capable of comprehending it. In this essay I explore the possibility that rather than being an odd coincidence arising due to our strange position as passive (and even more strangely, conscious) observers in the cosmos, these two problems might be intimately related and potentially explainable in terms of fundamental physics – that is, if we are willing to make the concession that information can physically influence the world. My premise begins by distinguishing physically possible states of the world, defined as everything permissible by the laws of physics, from physically accessible states of the world, defined as those that are achievable from a given initial state. For universes where the laws of physics are such that the number of accessible states is less than all that is possible, I argue that the most probable states among all possible states are those that include physical systems which contain information encodings – such as mathematics, language and art – because these are the most highly connected states in the state space of everything that is possible. Such physical systems include life and - of particular interest for the discussion of the place of math in physical reality - humans. Within this framework, the descent of math is a natural outcome of the evolution of the universe, which will tend toward states that are increasingly connected to other possible states of the universe, a process greatly facilitated if some physical systems know the rules of the game. I therefore conclude that our ability to use mathematics to describe, and more importantly manipulate, the natural world may not be an anomaly or trick, but instead a natural part of the structure of reality.

Authors Bio

Sara Imari Walker is a theoretical physicist and astrobiologist and Assistant Professor in the Beyond Center for Fundamental Concepts in Science and the School of Earth and Space Exploration at Arizona State University. She received her PhD in Physics from Dartmouth College and has held postdoctoral appointments in the Center for Chemical Evolution at the Georgia Institute of Technology and as a NASA Postdoctoral Program Fellow.

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