Forum for Science, Industry and Business

Sponsored by:     3M 
Search our Site:

 

Proof by Computer

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, Microsoft
Research, 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

More articles from Information Technology:

nachricht Stable magnetic bit of three atoms
21.09.2017 | Sonderforschungsbereich 668

nachricht Drones can almost see in the dark
20.09.2017 | Universität Zürich

All articles from Information Technology >>>

The most recent press releases about innovation >>>

Die letzten 5 Focus-News des innovations-reports im Überblick:

Im Focus: LaserTAB: More efficient and precise contacts thanks to human-robot collaboration

At the productronica trade fair in Munich this November, the Fraunhofer Institute for Laser Technology ILT will be presenting Laser-Based Tape-Automated Bonding, LaserTAB for short. The experts from Aachen will be demonstrating how new battery cells and power electronics can be micro-welded more efficiently and precisely than ever before thanks to new optics and robot support.

Fraunhofer ILT from Aachen relies on a clever combination of robotics and a laser scanner with new optics as well as process monitoring, which it has developed...

Im Focus: The pyrenoid is a carbon-fixing liquid droplet

Plants and algae use the enzyme Rubisco to fix carbon dioxide, removing it from the atmosphere and converting it into biomass. Algae have figured out a way to increase the efficiency of carbon fixation. They gather most of their Rubisco into a ball-shaped microcompartment called the pyrenoid, which they flood with a high local concentration of carbon dioxide. A team of scientists at Princeton University, the Carnegie Institution for Science, Stanford University and the Max Plank Institute of Biochemistry have unravelled the mysteries of how the pyrenoid is assembled. These insights can help to engineer crops that remove more carbon dioxide from the atmosphere while producing more food.

A warming planet

Im Focus: Highly precise wiring in the Cerebral Cortex

Our brains house extremely complex neuronal circuits, whose detailed structures are still largely unknown. This is especially true for the so-called cerebral cortex of mammals, where among other things vision, thoughts or spatial orientation are being computed. Here the rules by which nerve cells are connected to each other are only partly understood. A team of scientists around Moritz Helmstaedter at the Frankfiurt Max Planck Institute for Brain Research and Helene Schmidt (Humboldt University in Berlin) have now discovered a surprisingly precise nerve cell connectivity pattern in the part of the cerebral cortex that is responsible for orienting the individual animal or human in space.

The researchers report online in Nature (Schmidt et al., 2017. Axonal synapse sorting in medial entorhinal cortex, DOI: 10.1038/nature24005) that synapses in...

Im Focus: Tiny lasers from a gallery of whispers

New technique promises tunable laser devices

Whispering gallery mode (WGM) resonators are used to make tiny micro-lasers, sensors, switches, routers and other devices. These tiny structures rely on a...

Im Focus: Ultrafast snapshots of relaxing electrons in solids

Using ultrafast flashes of laser and x-ray radiation, scientists at the Max Planck Institute of Quantum Optics (Garching, Germany) took snapshots of the briefest electron motion inside a solid material to date. The electron motion lasted only 750 billionths of the billionth of a second before it fainted, setting a new record of human capability to capture ultrafast processes inside solids!

When x-rays shine onto solid materials or large molecules, an electron is pushed away from its original place near the nucleus of the atom, leaving a hole...

All Focus news of the innovation-report >>>

Anzeige

Anzeige

Event News

“Lasers in Composites Symposium” in Aachen – from Science to Application

19.09.2017 | Event News

I-ESA 2018 – Call for Papers

12.09.2017 | Event News

EMBO at Basel Life, a new conference on current and emerging life science research

06.09.2017 | Event News

 
Latest News

Fraunhofer ISE Pushes World Record for Multicrystalline Silicon Solar Cells to 22.3 Percent

25.09.2017 | Power and Electrical Engineering

Usher syndrome: Gene therapy restores hearing and balance

25.09.2017 | Health and Medicine

An international team of physicists a coherent amplification effect in laser excited dielectrics

25.09.2017 | Physics and Astronomy

VideoLinks
B2B-VideoLinks
More VideoLinks >>>