Apart from Iran, it is in crowded, hot, dusty, Muslim Pakistan that Islam seems to be asserting itself most in public these days. In Karachi's biggest commercial bank, for instance, there are no posters extolling the latest interest rates on checking or savings accounts. Instead there is a string of cloth signs, brightly painted in Urdu script, proclaiming rather different messages:

''Whoever earns honestly, he is the friend of Allah [God],'' says one, quoting the Prophet Muhammad in yellow letters on a purple background above the heads of busy tellers.

''He has no religion who fails to fulfill his promises,'' warns pink script on a blue background as a man deposits a pile of rupee notes beneath.

''Do justice to others,'' flutters a line from the Koran.

And at the counters, all bank interest is rapidly being phased out to comply with Islamic law. Pakistan is the first Muslim country to ban interest by law.

Its effort is being echoed in a dozen countries as part of a rising campaign to express Islam in banking and finance as well as in faith, social affairs, government, and politics.

In turn, the campaigns are one element in what Muslims are convinced is a general revival of interest in Islam as a counter to, and protest against, the secularism of the West and the atheism of the East.

This final article in the Monitor's series on Islam looks not only at the financial challenge, but also at wider issues, still to be fully answered:

How much is Islam itself actually growing in the world?

How much should its developing sense of identity be seen as a threat to the West?

Where does Islam go from here?

How compatible is it with the modern, Western world?

The heart of the financial challenge, still only about 10 years old, is not in the signs above the bank counters but in the methods used to avoid the conventional Western system of interest rates.

Most Muslim scholars interpret the Koran as banning fixed interest (though not other forms of return and profit). Interest is held to be an unfair and exploitative use of money, which Islam sees not as a commodity in itself but as God-given wealth to be used to help those in need and to invest to make economies grow in accordance with Islamic law.

''The modern system of economics, based on Adam Smith, grew as religious faith declined,'' maintains a Muslim economist in London. ''It substitutes materiality and accumulation for faith. Interest rates themselves increase inflation and help redistribute money from the less well off, who deposit in banks, to those in business....''

So institutions such as the Habib Bank in Pakistan resemble Western investment and equity banks and finance houses.

They take depositors' money into profit and loss (PLS) accounts, invest it, and share out profits every three, six, or 12 months. They rely on most individual investments turning a profit to offset those that lose.

Rates of return are not fixed in advance. Until last year they were generally above conventional ones. Habib Bank in Karachi today has been paying 8.5 percent a year, against 8 percent on interest accounts. Term accounts range from 11.5 percent for six months to 15.25 percent for five years.

Money is also loaned to businessmen in various ways, adding up to forms of partnership which divide profits or losses at the end of the day.

About 30 such institutions have sprung up in Pakistan, Jordan, Egypt, Sudan, Kuwait, Bahrain, the Gulf states, Indonesia, and elsewhere, controlling assets of more than $9 billion.

Holding companies operate in Saudi Arabia, Geneva, and Luxembourg. The UBAF Arab American Bank is based in New York. No Islamic bank has yet won a license in the City of London, where the Bank of England recognizes only banks that base their operations on interest.

The biggest network - presided over by Prince Faisal of Saudi Arabia - is the Dar al-Maal al-Islami Trust, which claimed net assets in 1983 of $286 million.

Like the broader revival of interest in Islam as a whole, however, the new system faces a number of obstacles.

Western bankers and World Bank officials have their doubts. ''It's interest without being called interest,'' a businessman says.

''Arabs can experiment because they have a lot of oil money, but I prefer the guaranteed return of Western banks myself,'' says a successful Pakistani businessman in London. He is a Muslim.

Rates of return were pushed down by recession last year. The Faisal Bank in Egypt, which claims to be the biggest private bank in the Islamic world, with profits of more than $100 million in 1983, earned 11 percent in 1982. But it dropped below ordinary interest rates to 10.2 percent last year.

The Dar al-Maal earned a profit of $7 million in 1982 but lost $27 million last year and paid no dividend.

So far, only Pakistan has ordered its entire bankng system to turn to interest-free banking (by October of next year). Iran has published but not yet fully carried out a new banking law that bans interest.

In Saudi Arabia, interest is forbidden officially, but Pakistani bankers say Riyadh does not want to scare off foreign investment and so interest-based transactions do take place. ''Anyone challenging them in Islamic courts (under Islamic law) might win the case, but find himself without further finance,'' says a Muslim banker drily.

Problems include finding enough trained staff in underdeveloped countries as well as the costs of closely monitoring investment and partnership deals. Business borrowers tend to inflate losses and understate profits.

Yet bank heads interviewed for this series were enthusiastic, even though they conceded that final results were still not in.

''I call it the 'divine banking system,' '' says the president of the Habib Bank, Abdul Jabbar Khan, in Karachi. ''It's interest-free, but certainly not return-free. We will retain interest-bearing accounts for foreign capital and loans.''

Keeping these interest-bearing accounts is an acknowledgement by Pakistan that Islamic banking is in fact out of step with the way the rest of the world does business. Islamic banks must also cope with inflation - and they do so by imposing what seem to the West to be high ''service charges'' on their transactions.

''We Muslims will reach our goals whether the West helps us or not,'' comments the governor of the private Faisal Bank in Cairo, Mohammed Fouad el-Sarraf, in his office overlooking the Nile River. ''No one has the right to tell us that we can be Muslims when we pray but that we have to deal with interest in our banks.''

As for loans from the World Bank, ''two 'windows' of the bank are also free from interest,'' argues Dr. Zia Uddin Ahmed, director-general of the new International Institute of Islamic Economics in Islamabad. ''They are the International Finance Corporation (equity finance) and the International Development Agency, which works on a service charge....''

The West is largely uninformed about Islamic history and beliefs, but reads constant headlines about Iran, the Gulf war with Iraq, threats to oil supply lines, and turmoil in Lebanon and in North and West Africa.

The tendency is to see ''Islam'' as a monolithic, threatening, almost abstract force.

In fact, Western and Muslim scholars agree, Islam is many things: a religious faith that includes detailed rules for government, social life, economics, and even eating. It is a political culture offering an alternative focus, language, and commitment to both capitalism and communism; a sense of national identity for millions who don't pray or fast or give alms but who call themselves Muslims nonetheless.

Contrary to some Western belief, however, Islam does not seem to be spreading in the world, except among peoples already partly Muslim.

''There are some 800,000 Muslims living in Britain, between 5 and 6 million in Western Europe,'' says David Kerr, director of the Center for the Study of Islam in Birmingham, England.

''Yet most of these are immigrants from the Muslim world - Pakistan, India, Turkey, and so on,'' Dr. Kerr says.

''I myself personally know five or six Christian converts to Islam. They tend to praise what they see as a clear sense of moral guidance and were unhappy with the answers of their own clergy.''

Jorgen Nielsen of the same center agrees: ''Figures are very hard to pin down ,'' he observes. ''The Muslim world headlines prominent converts, like the pop singer Cat Stevens and the French intellectual Roger Garaudy.''

In Denmark Muslims have claimed two conversions a week. Many of these were Danish women marrying Muslim men: Whether the conversions were to satisfy their in-laws, or were genuine, is not known.

Dr. Nielsen estimates that the number of mosques in Britain alone (including those in ordinary houses), has risen to 450, but he says that the reasons are immigration, not conversion.

Charles le Gai Eaton, a former British diplomat, a convert to Islam, and a spokesman for the Islamic Center in London, estimates some 25 million Muslims live between the Atlantic and the Ural mountains. They include 3.5 million to 4 million in Yugoslavia (they have their own Islamic theology college in Sarajevo) , and more in Albania, Bulgaria, Romania, the Caucasus, and Turkey.

''There's a steady trickle of converts, but not all that many,'' Mr. Eaton says.

The area where Islam has made new gains in the last 50 years is in sub-Sahara Africa, from Senegal and Nigeria in the west, to Uganda and Kenya in the east.

In the US, Muslims are estimated to number between 2 million and 3 million, mostly immigrants, and many of them in the Detroit area, according to the National Council of Churches. These figures include followers of the orthodox American Muslim Mission led by Imam Warith Dean Muhammad, son of Elijah Mohammed , and a small splinter group under Louis Farrakhan. (See following stories on Islam in America.)

So the Islamic crescent lies across the globe in about the same places as it has for a century - stretching from Morocco in far western Africa across the top half of Africa, through Turkey and the Middle East, on through Iran, Afghanistan , and Pakistan, and down through northwest China and India to Malaysia and Indonesia.

Within the Islamic world, however, most Muslims contacted for this series agreed that signs of an awakening, or a revival, do exist.

Dr. Saad Eddin Ibrahim of the American University in Cairo says the growth is not just among young militants, but in ''establishment'' Islam: ''Religious broadcasting, and Islamic newspapers and books have at least tripled in the last decade,'' he explains. He also sees growth in Sufi orders and brotherhoods - ''members gathering for annual feasts and celebrations have quadrupled in 15 years.'' Finally he includes ''folkloric'' and points to a growing number of ''grass-roots gatherings called Zar in each village, a mix of religion and superstition in Egypt and East and North Africa....''

Arabs remain locked in conflict with Israel, but is there much dialogue between Islam and Christianity?

The initiative comes from the Christian side - the National Council of Churches in the US and groups such as the Center for the Study of Islam in Britain.

Progress is limited. The Second Vatican Council agreed that God and Allah were identical, and made overtures. Discussions on ecumenism continue in Europe and the US, yet to a Muslim, his own faith is the ultimate revelation of God to mankind. If he is devout, he believes that Jews and Christians were offered the ''true faith'' and misunderstood it, making Muhammad's reciting of the Koran inevitable.

''Dialogue is well worthwhile,'' says Dr. Kerr of the Center for the Study of Islam.

''Fear is always of the unknown.''

''Christians and Muslims coexist in Tanzania, for instance, and in India. In Britain, there's dialogue in individual communities, by Christian ministers who find more and more Muslims moving into their areas,'' Dr. Kerr says.

What happens now to Islam?

It confronts a Western world superior in science and technology, stronger economically, but largely secular in outlook and unattractive to many Muslims as a model of society.

Modern Muslim thinkers welcome scientific progress and the Western values of equality and the rule of law. More orthodox scholars stress the need to incorporate these within a strong ethical framework based on the Koran and the words and deeds of the Prophet Muhammad.

Thus the debate, the struggle, is between Westernizers and traditionalists.

''Yes, you in the West have the rule of law, and technology, but your society centers itself on man rather than on God,'' says Indonesian Cabinet minister Emil Salim in Jakarta. ''You live for this life. You don't worry much anymore about what happens when you die.''

Meanwhile, another struggle of ideas takes place: Between the now-ruling Shia Muslims in Iran, with their passionate preaching of traditionalist protest and reform, and the Sunni Arabs of wealthy Saudi Arabia, who spend millions of dollars a year promulgating their own Islamic orthodoxy abroad.

Muslims in general think Christians are too unaware of Islam's achievements in the past: its contributions to mathematics, science, and medicine a thousand years ago, and its domination of much of the world until 200 years ago.

The West remembers mainly the medieval crusades against Islam, the decadence of the Ottoman Empire, and the Turkish effort at genocide against the Christian Armenians at the turn of this century. It sees Iran and its war with Iraq as a constant threat to one-sixth of the world's oil supplies.

Western critics of Islam also say parts of it are out of touch with the modern age.

The month of daylight fasting during Ramadan (June this year) costs the Muslim world an estimated 12 percent of its economic production, according to some estimates (and has been banned in Muslim Tunisia as a result).

The treatment of women is thought to be paternal and conservative. Interest-free banking is held to be incompatible with today, as are some of the punishments of Islamic law. Islamic government is seen as leading to one-man rule.

Muslims themselves remain deep in transition. Many areas - Nigeria, Sudan, the Middle East, Iran, Lebanon - are in turmoil. Traditional values clash with modern economics and third-world poverty, hunger, and overcrowding.

To Dr. Muhammad Zaki Badawi, founder of the Muslim College in London, both Western and Islamic worlds need to define their identities in the face of modern technology and science.

Dr. Badawi sees three main experiments today to establish a separate Islamic way: in Iran, in Saudi Arabia, and in Libya, ''where, despite all the headlines, most of the people still support the Qaddafi government.''

''For militant Islam today the real field of battle is the schoolroom,'' Godfrey Jansen, the British journalist and author, has written. The struggle, he says, is to reclaim national history and tradition from decades of Western influence which either suppressed or diluted them.

''Islam has an identity crisis,'' observes Jivad an-Sari, general editor of the weekly Arabia news magazine in London. ''Does it accept the West, as the Chinese and Japanese have done?'' he asks. ''Or does it have something different to contribute about religion, life, and death - a religious approach to the world?''

Dr. Badawi adds: ''Should Islam absorb and assimilate the modern civilization , the result would be a civilization with power restrained by ethical values, and wealth justly distributed by religious law, and pleasure circumscribed by moral standards.''

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