Chess involves abstract entities different chess pieces, different squares on the board, etc. For example, one relation that a piece may have to a square is that the former is standing on the latter. Another relation that a piece may have to a square is that it's allowed to move there. There are many equivalent ways of describing these entities and relations, for example with a physical board, via verbal descriptions in English or Spanish, or using so-called algebraic chess notation.
So what is it that's left when you strip away all this baggage? What is it that's described by all these equivalent descriptions? The Mathematical Universe Hypothesis implies that we live in a relational reality, in the sense that the properties of the world around us stem not from properties of its ultimate building blocks, but from the relations between these building blocks.
The external physical reality is therefore more than the sum of its parts, in the sense that it can have many interesting properties while its parts have no intrinsic properties at all. This crazy-sounding belief of mine that our physical world not only is described by mathematics, but that it is mathematics, makes us self-aware parts of a giant mathematical object.
As I describe in the book, this ultimately demotes familiar notions such as randomness, complexity and even change to the status of illusions; it also implies a new and ultimate collection of parallel universes so vast and exotic that all the above-mentioned bizarreness pales in comparison, forcing us to relinquish many of our most deeply ingrained notions of reality. Indeed, we humans have had this experience before, over and over again discovering that what we thought was everything was merely a small part of a larger structure: our planet, our solar system, our Galaxy, our universe and perhaps a hierarchy of parallel universes, nested like Russian dolls.
However, I find this empowering as well, because we've repeatedly underestimated not only the size of our cosmos, but also the power of our human mind to understand it.
They'd been told beautiful myths and stories, but little did they realize that they had it in them to actually figure out the answers to these questions for themselves. And that the secret lay not in learning to fly into space to examine the celestial objects, but in letting their human minds fly.
When our human imagination first got off the ground and started deciphering the mysteries of space, it was done with mental power rather than rocket power. If you decide to read it, then it will be not only the quest of me and my fellow physicists, but our quest. Known as "Mad Max" for his unorthodox ideas and passion for adventure, Max Tegmark's scientific interests range from precision cosmology to the ultimate nature of reality, all explored in his new popular book, "Our Mathematical Universe.
His work with the SDSS collaboration on galaxy clustering shared the first prize in Science magazine's "Breakthrough of the Year: Already a subscriber? Sign in. Thanks for reading Scientific American. Create your free account or Sign in to continue. See Subscription Options. Discover World-Changing Science. Since the ball is made of elementary particles quarks and electrons , you could in principle describe its motion without making any reference to basketballs: Particle 1 moves in a parabola.
Image: Courtesy of Max Tegmark Life without baggage Above we described how we humans add baggage to our descriptions. Get smart. Sign up for our email newsletter.
Sign Up. Support science journalism. Knowledge awaits. See Subscription Options Already a subscriber? Create Account See Subscription Options. The applications use different forms of simulation, discrete optimization, Markov decision processes, dynamic programming, network modeling, and stochastic control. As health care practices move to electronic health care records, enormous amounts of data are becoming available and in need of analysis; new methods are needed because these data are not the result of controlled trials.
The new field of comparative effectiveness research, which relies a great deal on statistics, aims to build on data of that sort to characterize the effectiveness of various medical interventions and their value to particular classes of patients. Embedded in several places in this discussion is the observation that data volumes are exploding, placing commensurate demands on the mathematical sciences. This prospect has been mentioned in a large number of previous reports about the discipline, but it has become very real in the past 15 years or so.
What really matters is our ability to derive from them new insights, to recognize relationships, to make increasingly accurate predictions. Our ability, that is, to move from data, to knowledge, to action. Large, complex data sets and data streams play a significant role in stimulating new research applications across the mathematical sciences, and mathematical science advances are necessary to exploit the value in these data.
However, the role of the mathematical sciences in this area is not always recognized. Indeed, the stated goals for the OSTP initiative,. Multiple issues of fundamental methodology arise in the context of large data sets. Some arise from the basic issue of scalability—that techniques developed for small or moderate-sized data sets do not translate to modern massive data sets—or from problems of data streaming, where the data set is changing while the analysis goes on.
Data that are high-dimensional pose new challenges: New paradigms of statistical inference arise from the exploratory nature of understanding large complex data sets, and issues arise of how best to model the processes by which large, complex data sets are formed.
Not all data are numerical—some are categorical, some are qualitative, and so on—and mathematical scientists contribute perspectives and techniques for dealing with both numerical and non-numerical data, and with their uncertainties.
Noise in the data-gathering process needs to be modeled and then—where possible—minimized; a new algorithm can be as powerful an enhancement to resolution as a new instrument.
Often, the data that can be measured are not the data that one ultimately wants. This results in what is known as an inverse problem—the process of collecting data has imposed a very complicated transformation on the data one wants, and a computational algorithm is needed to invert the process.
The classic example is radar, where the shape of an object is reconstructed from how radio waves bounce off it. Simplifying the data so as to find its underlying structure is usually essential in large data sets.
The general goal of dimensionality reduction—taking data with a large number of measurements and finding which combinations of the measurements are. Various methods with their roots in linear algebra and statistics are used and continually being improved, and increasingly deep results from real analysis and probabilistic methods—such as random projections and diffusion geometry—are being brought to bear.
Statisticians contribute a long history of experience in dealing with the intricacies of real-world data—how to detect when something is going wrong with the data-gathering process, how to distinguish between outliers that are important and outliers that come from measurement error, how to design the data-gathering process so as to maximize the value of the data collected, how to cleanse the data of inevitable errors and gaps.
As data sets grow into the terabyte and petabyte range, existing statistical tools may no longer suffice, and continuing innovation is necessary. In the realm of massive data, long-standing paradigms can break—for example, false positives can become the norm rather than the exception—and more research endeavors need strong statistical expertise. For example, in a large portion of data-intensive problems, observations are abundant and the challenge is not so much how to avoid being deceived by a small sample size as to be able to detect relevant patterns.
In that approach, one uses a sample of the data to discover relationships between a quantity of interest and explanatory variables. Strong mathematical scientists who work in this area combine best practices in data modeling, uncertainty management, and statistics, with insight about the application area and the computing implementation.
These prediction problems arise everywhere: in finance and medicine, of course, but they are also crucial to the modern economy so much so that businesses like Netflix, Google, and Facebook rely on progress in this area.
A recent trend is that statistics graduate students who in the past often ended up in pharmaceutical companies, where they would design clinical trials, are increasingly also being recruited by companies focused on Internet commerce. Finding what one is looking for in a vast sea of data depends on search algorithms.
This is an expanding subject, because these algorithms need to search a database where the data may include words, numbers, images and video, sounds, answers to questionnaires, and other types of data, all linked. New York Times , August 5. New techniques of machine learning continue to be developed to address this need.
Another new consideration is that data often come in the form of a network; performing mathematical and statistical analyses on networks requires new methods. Statistical decision theory is the branch of statistics specifically devoted to using data to enable optimal decisions. What it adds to classical statistics beyond inference of probabilities is that it integrates into the decision information about costs and the value of various outcomes.
Ideas from statistics, theoretical computer science, and mathematics have provided a growing arsenal of methods for machine learning and statistical learning theory: principal component analysis, nearest neighbor techniques, support vector machines, Bayesian and sensor networks, regularized learning, reinforcement learning, sparse estimation, neural networks, kernel methods, tree-based methods, the bootstrap, boosting, association rules, hidden Markov models, and independent component analysis—and the list keeps growing.
This is a field where new ideas are introduced in rapid-fire succession, where the effectiveness of new methods often is markedly greater than existing ones, and where new classes of problems appear frequently. Large data sets require a high level of computational sophistication because operations that are easy at a small scale—such as moving data between machines or in and out of storage, visualizing the data, or displaying results—can all require substantial algorithmic ingenuity.
As a data set becomes increasingly massive, it may be infeasible to gather it in one place and analyze it as a whole. Thus, there may be a need for algorithms that operate in a distributed fashion, analyzing subsets of the data and aggregating those results to understand the complete set.
One aspect of this is the. This is essential when new waves of data continue to arrive, or subsets are analyzed in isolation of one another, and one aims to improve the model and inferences in an adaptive fashion—for example, with streaming algorithms.
The mathematical sciences contribute in important ways to the development of new algorithms and methods of analysis, as do other areas as well. Simplifying the data so as to find their underlying structure is usually essential in large data sets. The general goal of dimensionality taking data with a large number of measurements and finding which combinations of the measurements are sufficient to embody the essential features of the data set—is pervasive.
Various methods with their roots in linear algebra, statistics, and, increasingly, deep results from real analysis and probabilistic methods—such as random projections and diffusion geometry—are used in different circumstances, and improvements are still needed.
Related to search and also to dimensionality reduction is the issue of anomaly detection—detecting which changes in a large system are abnormal or dangerous, often characterized as the needle-in-a-haystack problem. The Defense Advanced Research Projects Agency DARPA has its Anomaly Detection at Multiple Scales program on anomaly-detection and characterization in massive data sets, with a particular focus on insider-threat detection, in which anomalous actions by an individual are detected against a background of routine network activity.
A wide range of statistical and machine learning techniques can be brought to bear on this, some growing out of statistical techniques originally used for quality control, others pioneered by mathematicians in detecting credit card fraud.
Two types of data that are extraordinarily important yet exceptionally subtle to analyze are words and images. The fields of text mining and natural language processing deal with finding and extracting information and knowledge from a variety of textual sources, and creating probabilistic models of how language and grammatical structures are generated. Image processing, machine vision, and image analysis attempt to restore noisy image data into a form that can be processed by the human eye, or to bypass the human eye altogether and understand and represent within a computer what is going on in an image without human intervention.
Related to image analysis is the problem of finding an appropriate language for describing shape. As part of this problem, methods are needed to describe small deformations of shapes, usually using some aspect of the geometry of the space of. Shape analysis also comes into play in virtual surgery, where surgical outcomes are simulated on the computer before being tried on a patient, and in remote surgery for the battlefield.
Here one needs to combine mathematical modeling techniques based on the differential equations describing tissue mechanics with shape description and visualization methods. As our society is learning somewhat painfully, data must be protected. The need for privacy and security has given rise to the areas of privacy-preserving data mining and encrypted computation, where one wishes to be able to analyze a data set without compromising privacy, and to be able to do computations on an encrypted data set while it remains encrypted.
The mathematical sciences have a long history of interaction with other fields of science and engineering. This interaction provides tools and insights to help those other fields advance; at the same time, the efforts of those fields to push research frontiers routinely raise new challenges for the mathematical sciences themselves.
One way of evaluating the state of the mathematical sciences is to examine the richness of this interplay. Some of the interactions between mathematics and physics are described in Chapter 2 , but the range extends well beyond physics. A number of such illustrations have been collected in Appendix D. The role of the mathematical sciences in industry has a long history, going back to the days when the Egyptians used the right triangle to restore boundaries of farms after the annual flooding of the Nile.
That said, the recent period is one of remarkable growth and diversification. Even in old-line industries, the role of the mathematical sciences has expanded. For example, whereas the aviation industry has long used mathematics in the design of airplane wings and statistics in ensuring quality control in production, now the mathematical sciences are also crucial to GPS and navigation systems, to simulating the structural soundness of a design, and to optimizing the flow of production.
Instead of being used just to streamline cars and model traffic flows, the mathematical sciences are also involved in the latest developments, such as design of automated vehicle detection and.
Whereas statistics has long been a key element of medical trials, now the mathematical sciences are involved in drug design and in modeling new ways for drugs to be delivered to tumors, and they will be essential in making inferences in circumstances that do not allow double-blind, randomized clinical trials.
The financial sector, which once relied on statistics to design portfolios that minimized risk for a given level of return, now makes use of statistics, machine learning, stochastic modeling, optimization, and the new science of networks in pricing and designing securities and in assessing risk.
What is most striking, however, is the number of new industries that the mathematical sciences are a part of, often as a key enabler. The encryption industry makes use of number theory to make Internet commerce possible. The social networking industry makes use of graph theory and machine learning. The animation and computer game industry makes use of techniques as diverse as differential geometry and partial differential equations.
The biotech industry heavily uses the mathematical sciences in modeling the action of drugs, searching genomes for genes relevant to human disease or relevant to bioengineered organisms, and discovering new drugs and understanding how they might act.
The imaging industry uses ideas from differential geometry and signal processing to procure minimally invasive medical and industrial images and, within medicine, adds methods from inverse problems to design targeted radiation therapies and is moving to incorporate the new field of computational anatomy to enable remote surgery.
The online advertising industry uses ideas from game theory and discrete mathematics to price and bid on online ads and methods from statistics and machine learning to decide how to target those ads. The marketing industry now employs sophisticated statistical and machine learning techniques to target customers and to choose locations for new stores.
The credit card industry uses a variety of methods to detect fraud and denial-of-service attacks. Political campaigns now make use of complex models of the electorate, and election-night predictions rely on integrating these models with exit polls. The semiconductor industry uses optimization to design computer chips and in simulating the manufacture and behavior of designer materials.
The mathematical sciences are now present in almost every industry, and the range of mathematical sciences being used would have been unimaginable a generation ago. This point is driven home by the following list of case studies assembled for the SIAM report Mathematics in Industry. The reader is directed to the SIAM report to see the details of these case studies, 14 which provide many examples of the significant and cost-effective impact of mathematical science expertise and research on innovation, economic competitiveness, and national security.
Another recent report on the mathematical sciences in industry came to the following conclusions:. It is evident that, in view of the ever-increasing complexity of real life applications, the ability to effectively use mathematical modelling, simulation, control and optimisation will be the foundation for the technological and economic development of Europe and the world.
Only [the mathematical sciences] can help industry to optimise more and more complex systems with more and more constraints. Strasbourg, France, p. The major effort concerned with the construction of reliable, robust and efficient virtual design environments is, however, not recognised.
As a result, mathematics is not usually considered a key technology in industry. The workaround for this problem usually consists of leaving the mathematics to specialised small companies that often build on mathematical and software solutions developed in academia. Unfortunately, the level of communication between these commercial vendors and their academic partners with industry is often at a very low level. The latter can only be addressed adequately if an effort is made to drastically improve the communication between industrial designers and mathematicians.
To this list, the European Science Foundation report adds optimization, noting p. This is of vital importance to industry. To support the maturation of these systems, the report goes on to identify 22 science and technology initiatives. In particular, the report recommends that due to the fundamental importance of complex systems, the Chinese government should provide sustained and steady support for research into such systems so as to achieve major accomplishments in this important field.
A number of these capabilities rely on research in modeling and simulation, control theory, and informatics:. Other areas for research include a general theory for adaptive systems that could be translated into manufacturing processes, systems, and the manufacturing enterprise; tools to optimize design choices to incorporate the most affordable manufacturing approaches; and systems research on the interaction between workers and manufacturing processes for the development of adaptive, flexible controls.
Virtual prototyping of manufacturing processes and systems will enable manufacturers to evaluate a range of choices for optimizing their enterprises. Promising areas for the application of modeling and simulation technology for reconfigurable systems include neural networks for optimizing reconfiguration approaches and artificial intelligence for decision making.
Processes that can be adapted or readily reconfigured will require flexible sensors and control algorithms that provide precision process control of a range of processes and environments. The 11 cross-cutting technology areas identified in that report p. National security is another area that relies heavily on the mathematical sciences.
The National Security Agency NSA , for example, employs roughly 1, mathematical scientists, although the number might be half that or twice that depending on how one defines such scientists. NSA hires some mathematicians per year, and it tries to keep that rate steady so that the mathematical sciences community.
The NSA is interested in maintaining a healthy mathematical sciences community in the United States, including a sufficient supply of well-trained U. For example, few mathematicians would have guessed decades ago that elliptic curves would be of vital interest to NSA, and now they are an important specialty underlying cryptology.
While cryptology is explicitly dependent on mathematics, many other links exist between the mathematical sciences and national security. One example is analysis of networks discussed in the next section , which is very important for national defense. Another is scientific computing.
Years later, with atmospheric and underground nuclear testing banned, the country relies once again on simulations, this time to maintain the readiness of its nuclear arsenal. Because national defense relies in part on design and manufacturing of cutting-edge equipment, it also relies on the mathematical sciences through their contributions to advanced engineering and manufacturing.
The level of sophistication of these tools has ratcheted steadily upward. The mathematical sciences are also essential to logistics, simulations used for training and testing, war-gaming, image and signal analysis, control of satellites and aircraft, and test and evaluation of new equipment. Figure , reproduced from Fueling Innovation and Discovery: The Mathematical Sciences in the 21st Century, 23 captures the broad range of ways in which the mathematical sciences contribute to national defense.
New devices, on and off the battlefield, have come on stream and furnish dizzying quantities of data, more than can currently be analyzed. Devising ways to automate the analysis of these data is a highly mathematical and statistical challenge. Can a computer be taught to make sense of a satellite image, detecting buildings and roads and noticing when there has been a major change in the image of a site that is not due to seasonal variation?
How can one make use of hyperspectral data, which measure light reflected in all frequencies of the spectrum, in order to detect the smoke plume from a chemical weapons factory? Can one identify enemy vehicles and ships in a cluttered environment? These questions and many others are inherently dependent on advances in the mathematical sciences.
A very serious threat that did not exist in earlier days is that crucial networks are constantly subject to sophisticated attacks by thieves, mischief-makers, and hackers of unknown origin. Adaptive techniques based on the mathematical sciences are essential for reliable detection and prevention of such attacks, which grow in sophistication to elude every new strategy for preventing them. The Department of Defense has adopted seven current priority areas for science and technology investment to benefit national security.
Data to decisions. Science and applications to reduce the cycle time and manpower requirements for analysis and use of large data sets. Engineered resilient systems. Engineering concepts, science, and design tools to protect against malicious compromise of weapon systems and to develop agile manufacturing for trusted and assured defense systems. Cyber science and technology. Science and technology for efficient, effective cyber capabilities across the spectrum of joint operations. New concepts and technology to protect systems and extend capabilities across the electro-magnetic spectrum.
Science and technology to achieve autonomous systems that reliably and safely accomplish complex tasks in all environments. Human systems. For a long time, no applications were found -- or were not even searched for -- for the theory until ten years ago, when it was understood that they would be useful in the efficient data transmission required by modern data networks.
The challenge was that, despite numerous attempts, the best possible codes described in the theory had not been found and it was therefore believed they did not even exist. Searching for them is a gigantic operation even if there is very high-level computational capacity available.
Therefore, in addition to algebraic techniques and computers, we also had to use our experience and guess where to start looking, and that way limit the scope of the search. The perseverance was rewarded when the group consisting of five researchers found the largest possible structure described by the theory.
The results were recently presented in the scientific publication Forum of Mathematics, Pi, which publishes only a dozen carefully selected articles per year. Although mathematical breakthroughs rarely become financial success stories immediately, many modern things we take for granted would not exist without them.
For example, Boolean algebra, which has played a key role in the creation of computers, has been developed since the 19th century. Our discovery will not become a product straight away, but it may gradually become part of the internet.
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