10.11.2008

New computer tools have the potential to revolutionize the practice of mathematics by providing far more-reliable proofs of mathematical results than have ever been possible in the history of humankind.

These computer tools, based on the notion of "formal proof", have in recent years been used to provide nearly infallible proofs of many important results in mathematics.

A ground-breaking collection of four articles by leading experts, published today in the Notices of the American Mathematical SocietyNotices of the American Mathematical Society (http://www.ams.org/notices), explores new developments in the use of formal proof in mathematics.

When mathematicians prove theorems in the traditional way, they present the argument in narrative form. They assume previous results, they gloss over details they think other experts will understand, they take shortcuts to make the presentation less tedious, they appeal to intuition, etc. The correctness of the arguments is determined by the scrutiny of other mathematicians, in informal discussions, in lectures, or in journals. It is sobering to realize that the means by which mathematical results are verified is essentially a social process and is thus fallible. When it comes to central, well known results, the proofs are especially well checked and errors are eventually found.

Nevertheless the history of mathematics has many stories about false results that went undetected for a long time. In addition, in some recent cases, important theorems have required such long and complicated proofs that very few people have the time, energy, and necessary background to check through them. And some proofs contain extensive computer code to, for example, check a lot of cases that would be infeasible to check by hand. How can mathematicians be sure that such proofs are reliable?

To get around these problems, computer scientists and mathematicians began to develop the field of formal proof. A formal proof is one in which every logical inference has been checked all the way back to the fundamental axioms of mathematics. Mathematicians do not usually write formal proofs because such proofs are so long and cumbersome that it would be impossible to have them checked by human mathematicians. But now one can get "computer proof assistants" to do the checking. In recent years, computer proof assistants have become powerful enough to handle difficult proofs.

Only in simple cases can one feed a statement to a computer proof assistant and expect it to hand over a proof. Rather, the mathematician has to know how to prove the statement; the proof then is greatly expanded into the special syntax of formal proof, with every step spelled out, and it is this formal proof that the computer checks. It is also possible to let computers loose to explore mathematics on their own, and in some cases they have come up with interesting conjectures that went unnoticed by mathematicians. We may be close to seeing how computers, rather than humans, would do mathematics.

The four Notices articles explore the current state of the art of formal proof and provide practical guidance for using computer proof assistants. If the use of these assistants becomes widespread, they could change deeply mathematics as it is currently practiced. One long-term dream is to have formal proofs of all of the central theorems in mathematics. Thomas Hales, one of the authors writing in the Notices, says that such a collection of proofs would be akin to "the sequencing of the mathematical genome".

The four articles are:

Formal Proof, by Thomas Hales, University of Pittsburgh

Formal Proof---Theory and Practice, by John Harrison, Intel Corporation

Formal proof---The Four Colour Theorem, by Georges Gonthier, MicrosoftResearch, Cambridge, England

Formal Proof---Getting Started, by Freek Wiedijk, Radboud University,

Nijmegen, Netherlands

The articles appear today in the December 2008 issue of the Notices

and are freely available at http://www.ams.org/notices.

Founded in 1888 to further mathematical research and scholarship, today the American Mathematical Society has more than 32,000 members. The Society fulfills its mission through programs and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and to everyday life.

Prof. Thomas Hales | Newswise Science News

Further information:

http://www.ams.org/notices

http://www.ams.org

**Further reports about:**
> Proof by Computer
> computer tools
> formal proof
> mathematical genome
> mathematics

Cutting edge research for the industries of tomorrow – DFKI and NICT expand cooperation

21.03.2017 | Deutsches Forschungszentrum für Künstliche Intelligenz GmbH, DFKI

Molecular motor-powered biocomputers

20.03.2017 | Technische Universität Dresden

Astronomers from Bonn and Tautenburg in Thuringia (Germany) used the 100-m radio telescope at Effelsberg to observe several galaxy clusters. At the edges of these large accumulations of dark matter, stellar systems (galaxies), hot gas, and charged particles, they found magnetic fields that are exceptionally ordered over distances of many million light years. This makes them the most extended magnetic fields in the universe known so far.

The results will be published on March 22 in the journal „Astronomy & Astrophysics“.

Galaxy clusters are the largest gravitationally bound structures in the universe. With a typical extent of about 10 million light years, i.e. 100 times the...

Researchers at the Goethe University Frankfurt, together with partners from the University of Tübingen in Germany and Queen Mary University as well as Francis Crick Institute from London (UK) have developed a novel technology to decipher the secret ubiquitin code.

Ubiquitin is a small protein that can be linked to other cellular proteins, thereby controlling and modulating their functions. The attachment occurs in many...

In the eternal search for next generation high-efficiency solar cells and LEDs, scientists at Los Alamos National Laboratory and their partners are creating...

Silicon nanosheets are thin, two-dimensional layers with exceptional optoelectronic properties very similar to those of graphene. Albeit, the nanosheets are less stable. Now researchers at the Technical University of Munich (TUM) have, for the first time ever, produced a composite material combining silicon nanosheets and a polymer that is both UV-resistant and easy to process. This brings the scientists a significant step closer to industrial applications like flexible displays and photosensors.

Silicon nanosheets are thin, two-dimensional layers with exceptional optoelectronic properties very similar to those of graphene. Albeit, the nanosheets are...

Enzymes behave differently in a test tube compared with the molecular scrum of a living cell. Chemists from the University of Basel have now been able to simulate these confined natural conditions in artificial vesicles for the first time. As reported in the academic journal Small, the results are offering better insight into the development of nanoreactors and artificial organelles.

Enzymes behave differently in a test tube compared with the molecular scrum of a living cell. Chemists from the University of Basel have now been able to...

Anzeige

Anzeige

Event News

International Land Use Symposium ILUS 2017: Call for Abstracts and Registration open

20.03.2017 | Event News

CONNECT 2017: International congress on connective tissue

14.03.2017 | Event News

ICTM Conference: Turbine Construction between Big Data and Additive Manufacturing

07.03.2017 | Event News

Latest News

When Air is in Short Supply - Shedding light on plant stress reactions when oxygen runs short

23.03.2017 | Life Sciences

Researchers use light to remotely control curvature of plastics

23.03.2017 | Power and Electrical Engineering

Sea ice extent sinks to record lows at both poles

23.03.2017 | Earth Sciences

VideoLinks

NASA | A Year in the Life of Earth's CO2

NASA Computer Model Provides a New Portrait of Carbon Dioxide

Black Holes Come to the Big Screen

The new movie "Interstellar" explores a longstanding fascination, but UA astrophysicists are using cutting-edge technology to go one better.

NASA's Swift Mission Observes Mega Flares from a Mini Star

NASA's Swift satellite detected the strongest, hottest, and longest-lasting sequence of stellar flares ever seen from a nearby red dwarf star.

NASA | Global Hawks Soar into Storms

NASA's airborne Hurricane and Severe Storm Sentinel or HS3 mission, will revisit the Atlantic Ocean for the third year in a row.

Baffin Island - Disappearing ice caps

Giff Miller, geologist and paleoclima-tologist, is walking the margins of melting glaciers on Baffin Island, Nunavut, Canada.

The Infrasound Network and how it works

The CTBTO uses infrasound stations to monitor the Earth mainly for atmospheric explosions.

B2B-VideoLinks

Efficient reduction of odour and grease with Heraeus UV solutions

Kitchen exhaust air cleaning with UV in gastronomy

Drying and curing of paints on glass and ceramics

Bright and brilliant paints on glass and ceramics require safe solutions for drying and curing.

JULABO World of Temperature

Explore the World of Temperature with JULABO - Superior Temperature Technology for a Better Life.

Acoustic Wave Separation: How It Works

In this animated video, see how Acoustic Wave Separation technology works in full detail.

Infrared Heat for printed electronics

Drying and sintering of printed electronics by specialty light sources from Heraeus

All about Data Logger, how to use

Wolfgang Rudolph explains: all information worth knowing about the data logger and the practical test by means of a drone