Indeed, Freek Wiedijk of Radboud University Nijmegen in the

Netherlands says a revolution is already occurring. He writes in the December

Notices of the American Mathematical Society that in the future, “most

mathematicians will not consider mathematics to be definitive unless it has

been fully formalized”.

The first proof-validation programs were created more than 20 years ago.

Until recently, though, they were so cumbersome that the only users were the

researchers who had created and were trying to improve them. Furthermore,

even those researchers were only tackling relatively simple theorems. In the

last five years, though, those users have finally been able to verify some

remarkably complex and difficult proofs. Before long, they say, ordinary

mathematicians will be using these tools as part of their everyday work.

Perhaps the most remarkable success so far came in 2004, when Georges

Gonthier, a computer scientist at Microsoft Research in Cambridge,

England, verified the proof of the four-color theorem by computer. The problem

dates back to 1852, when a college student noticed that only four colors

were needed to fill in a map of the counties in England such that no adjacent

counties shared a color. It took until 1976 to mathematically prove that four

colors were enough for any map. That proof was more than 500 pages long and

relied on computers to check nearly 2,000 special cases. Many mathematicians

objected to the proof because it was impossible to check by hand.

Gonthier used a proof-checking software package to formalize the

entire proof, reducing both the text and the software for the special cases to

an enormously long series of simple deductions.

Of course, if the proof-checking software itself has bugs, Gonthier’s

Verification of the four-color theorem itself could be invalid. To guard against

this possibility, the designers of the proof-checking software make the “kernel”

of code that implements the axioms and rules of inference as short and simple

as possible. One program, HOL Light, has fewer than 500 lines of code in its

kernel, few enough that humans can check it by hand. John Harrison, the creator

of HOL Light, has also checked the code using other formal proof-checkers!

The software may not be able to produce perfect certainty, but Thomas

Hales, a mathematician at the University of Pittsburgh, calls the four-color

theorem “one of the most meticulously verified proofs in history.”

Hales is one of the first working mathematicians to embrace the proofcheckers,

because he ran up against the limits of mathematical certainty

himself. He proved the Kepler Conjecture in 1998, which is another theorem

that is simple to state but remarkably hard to prove. The Kepler Conjecture

says that the pattern grocers use to stack oranges packs the most oranges

into the smallest space. As with the four-color theorem, Hales used a computer

to check many, many special cases, and the proof consisted of 300 pages of

text and 40,000 lines of computer code.

When he submitted his result for publication, he received only a qualified

acceptance. The letter from the editor explained that “the referees put a

level of energy into this that is, in my experience, unprecedented.” Nevertheless,

the referees ended up only 99 percent certain that the proof was correct.

The referees were unable to check the computer code at all.

20 21

Hales decided that 99 percent certainty wasn’t good enough for him.

He started the “Flyspeck” project (named from the acronym FPK, for Formal

Proof of the Kepler Conjecture) to formalize his entire proof. When he

began, he estimated that it would take 20 person-years to complete it (i.e., one

person working for 20 years, for example, or 10 people working for two

years). He says now that he is about halfway through, and his team has

indeed devoted about 10 person-years.

Hales and Gonthier are managing to do more than simply check those

particular proofs. In the process, they are creating a library of basic formalized

results other mathematicians can use to formalize new proofs. Since new

proofs always rely on many, many previous proofs, this library provides the

essential foundation mathematicians need to efficiently use the proofverification

software programs in their daily work.

Once that library is created, widespread use may not be so far off.

Gonthier has been surprised to find that with experience, coding a proof

takes little more effort than typesetting an ordinary mathematics article,

which mathematicians do regularly. “The actual coding of results seems to

go on pretty quickly,” Gonthier says.

The hard part is the early stages, he says, teaching the computer what

an early graduate student would know. Mathematicians use many, many

tiny results and methods that they never write down explicitly. “Most of

what you learn from a textbook is in the exercises,” he says. “An entire part

of the theory is something never described literally. If you want to formalize

a theory, you have to find a good description for these things.”

Gonthier believes that ordinary mathematicians may start formally

verifying their proofs within the decade. Cameron Freer of the Massachusetts

Institute of Technology is beginning a collaborative project called Vdash that

he hopes will inspire many mathematicians to pitch in and help build the

basic library of results. Hales warns, though, that this will be a formidable

task. “To undertake the formalization of just 100,000 pages of core mathematics

would be one of the most ambitious collaborative projects ever undertaken in

pure mathematics, the sequencing of a mathematical genome,” he writes.

(Julie Rehmeyer, http://www.sciencenews.org, November, 2008)

5.2. Summarize the article using the following phrases:

The article provides information on … .

The article puts forward the idea ........... .

A careful account is given to … .

The article describes ................. in detail..

The results of ......are presented.

PROGRESS TEST

TASK 1: Match the expressions in column A with their Russian

equivalents in column B.

A.

1. The paper provides information on…

2. It is pointed out…

3. A detailed description is given to…

4. Particular emphasis is placed on…

5. It is claimed that…

6. The paper is of great interest.

7. The paper covers such points as…

8. The paper suggests the problem…

9. A careful account is given to…

10. The paper puts forward the idea…

11. The paper deals with the problem

of…

12. It is assumed that…

13. The effect of …on…is discussed…

14. Much attention is given to

15. The paper touches upon…

В.

a) В статье затрагивается…

b) Много внимания уделено…

c) Статья выдвигает проблему…

d) Обсуждается влияние…

e) Предполагается, что…

f) В статье выдвигается идея…

g) Особый акцент сделан на…

h) Утверждается, что…

i) Подробно описывается…

j) Статья информирует о…

k) Подчеркивается, что…

l) Статья рассматривает проблему…

m) Тщательно рассматривается…

n) Статья представляет большой

интерес…

o) Статья охватывает такие воп-

росы как…

TASK 2: Read and review the article.

The Bionic Age Begins

Neural implants will treat tremors, paralysis, and even memory loss

Theodore Berger, a Professor of Engineering at the University of Southern

California, is ready for the era of the bionic brain. He has spent 30 years developing

computer chips that can link with neurons in an effort to compensate for memory

loss. The chips that can do it exist. Most of the software exists. The challenge is to

make a reliable, long-term connection between the hardware and the wetware –

22 23

one that is unaffected by a corrosion, scar tissue, or the shifting and dying of cells

in the brain.”That’s the big showstopper,” Berger says.

He is a part of a growing movement of researchers struggling to perfect

neural prostheses, devices that employ electrodes to receive signals from

and transmit them to the brain. Cyberkinetics, a company cofounded by

neuroscientist John Donoghue at Brown University, has begun clinical

trials on an implant that can transmit signals from a paralyzed person’s

motor cortex to a computer or to a prosthetic limb. Several groups, including

one led by Ali Rezai of the Cleveland Clinic Center for Neurological

Restoration, have tentatively shown that stimulation of the thalamus can

relieve chronic pain, obsessive-compulsive disorder, and depression. Similar

devices may be able to treat blindness, epilepsy, and Parkinson’s disease.

All these applications will depend on solving the connection problem.

Groups at the University of Arizona and elsewhere have crafted arrays

containing 500 or more electrodes, trying to maintain a good link through

sheer numbers. Other strategies include building electrodes out of conducting

polymers, which are more compatible with neural tissue than are silicon or

metal, or coating electrodes with molecules that adhere to brain cells. A

team at Emory University is embedding electrodes in glass cones filled

with nerve-growth factors that encourage brain cells to sprout more dendrites

and axons. Several paralyzed patients using the Emory device have learned

to control a computer with their thoughts. But the ideal fix would be an

electrode that constantly moves to maintain connections.

Joel Burdick, a mechanical engineer at Caltech, and his colleagues are

developing an electrode array to do just that. Each electrode determines the

direction from which the neutron’s signals are strongest. A tiny motor then

moves the contact in that direction. The electrodes will be programmed to

search for specific types of neural signals-for example, those corresponding

to a subject’s desire to move her hand rather than her foot.

The first prototype of this device, which was successfully tested in monkeys

by Burdick’s Caltech colleagues Richard Andersen, had only four electrodes.

The motors were mounted outside the skull, and electrodes passed through plugs

in the scalp. The Caltech team is now working on downsized versions that will

have as many as 100 electrodes and be small enough to be implanted inside the

skull, thereby reducing the risk of infection. A comparison set of miniature injectors

could administer compounds to inhibit the formation of scar tissue or to stimulate

the activity of surrounding neurons. Power will be supplied by an external source

that beams radio waves through the skin and skull.

Andersen is still preparing a second round of animal tests to prove the

electrode array works. But ethicists already worry about a day when

implants are so effective that even healthy people elect to upgrade, less

they fall behind like some obsolete computer.

(John Horgan, Discover magazine, October 2005).

TASK 3: Check if you can remember the key words from the articles you

have read.

to treat/cure a disease

a tumour

cancer

poison

poisonous/harmful

a dumpster/landfill

environmental hazards/risks

to release pollutants into the air/toxic fumes

to dismantle by hand/with one’s barehands

wastes disposal/trash/litter

easily dissolved in ...

to deposite ... onto a sheet of...

a flexible substrate

data leakage

to cause disruption

the fast Internet access

spyware or adware/malware infections

theft of confidential data

to deduce a theorem/a statement/an equation

a proof-validation programme

to tackle theorems

to verify complex proofs

to object to the proof

the rules of inference

TASK 4: Choose to read and review 5 papers from the journals on the

subject of your research. Summarise their content. As a class

hold a mini-conference and discuss the innovations, research

problems and scientific achievements you have learned about.