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English in Mathematics

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By R.N.A. de Silva

 

“Which subject did you have most difficulties with, having switched the medium of instruction from Sinhala to English?” I posed this question to a Sri Lankan student who was following a pre-University course in an educational institution in Hong Kong, having completed studies up to the GCE Ordinary Level programme in the Sinhala medium in a leading girls’ school in Colombo. “It is definitely mathematics,” she replied. Having served as a teacher for a long period of time at this educational institution with students from over 80 countries, I realised the above-mentioned view was shared by other students, too, who had to change the medium of instruction to English. This does not seem to make sense as one would have expected mathematics to be the easiest subject to follow as it has its own symbolic language. Why then has this situation arisen?

I would like to separate these difficulties into two categories:

1. Hastiness due to mindset

2. Vocabulary issues

Sometimes hastiness can automatically occur due to the mindset that mathematics should be easy to follow even if you change the medium of instruction as you are dealing with symbols. This attitude can cause enormous problems as students may skip instructions or avoid reading the question fully and concentrate only on the symbolic part of the problem

As an example, consider the following question.

The graphs of lines 3y = 5x + 1 and 2y = 7 – 3x intersect at point P. Find the coordinates of P.

Seeing the word ‘graphs’ and the two equations, a student maybe tempted to draw the graphs of the two lines and thereby find the point of intersection, which is a time-consuming affair. If it was read properly, the student could have noticed that the solution can be obtained by solving the two equations algebraically, which is much more efficient.

To a fast reader, obtaining the correct answer to the following question can be a problem as it may end up with just finding the value of x.

If 2x+3 = 5x-3, find the value of 2x+3.

The students need to be trained to read the question fully and understand what is required to be done, before attempting it.

The time spent to grasp the aim of the question is not wasted time.

Many children consider mathematics as an alien language consisting of symbols and expressions. Most of the difficulties that students encounter is related to vocabulary. The mathematical interpretation of the meaning of a word may differ from the meaning given to it in the English language. The word ‘find’ in mathematics means to obtain an answer showing the working while in the English language, it refers to discover or search. The following sketch shows the funny side of this difference.

Two of the words that has caused much confusion are ‘or’ and ‘and’.

In general usage, A or B is considered as either A or B but not both, as shown in picture.

However, in mathematics ‘A or B’ means ‘it can belong to A or B including intersection’. This is shown in picture.

The above, in normal usage is interpreted as ‘A and B’. However, in mathematics A and B refers to only what is common to A and B as shown in picture.

Here are the mathematical meanings of some of the other words which can have a different meaning with the English language definitions.

Determine

– Obtain the only possible answer

Plot

– Mark the position of points on a diagram

Write down

– Obtain the answer (Working need not be shown)

Constant

– A number that does not change

Similar

– Having the same shape but not the same size

Deduce

– To show a result using known information

Operation

– A procedure such as addition, subtraction, multiplication, etc.

Element

– A member of a set

Volume

– The extent of space occupied by a solid

The following illustrate some of the difficulties that the difference of meanings brings:

How odd these odd numbers are? The even numbers are even stranger.

Don’t be mean and help me to find the mean of these numbers.

Is right angle the right answer? Let me write it on the board.

The polysemous nature of some of the mathematical terms make it confusing for the students in the understanding of mathematical concepts. Mathematical terms have precise definitions to describe numerical relationships. At times these definitions resemble the everyday usage meaning but there are instances where the definitions notably differ. Consider ‘in general’ as an example. In mathematics there can be no exceptions to a result if it is considered to hold in general. However, in everyday usage, if a claim is said to be true in general, it would mean that it is true most of the time, but exceptions are possible.

To add to the problem, there are some terms such as ‘degree’ that can have many different meanings within mathematics while having a different meaning in everyday use. In mathematics, degree can refer to the measurement of an angle, the complexity of an algebraic equation and a unit of temperature.

Although mathematics deals essentially with symbols, it is taught through the medium of language which is the major means of communication. Students build understanding as they process ideas through language. It is important for students to give emphasis to the familiarisation with the mathematical vocabulary and at the same time understand the difference of meanings of terms mathematically and everyday usage. Teachers have an important role to play here in highlighting such terms and using them in different contexts for comfortable acclimatization. As Marcus Quintilianus quoted, “One should not aim at being possible to understand, but at being impossible to misunderstand.”

(The author is a senior mathematics examiner of the International Baccalaureate Organization and a member of the faculty of the Overseas School of Colombo.)

 

 



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Govt. needs to deal with impunity

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By Jehan Perera

The drama over Prisons Minister Lohan Ratwatte could not have come at a worse time for the government. But it can also be the turning point. President Gotabaya Rajapaksa is due to address the UN General Assembly in New York this week.  The attention of the international human rights community has been focused on Sri Lanka during the past week due to the recently concluded sessions of the UN Human Rights Council in Geneva.  Sri Lanka was a country of interest due to its checkered human rights track record, especially in relation to the war, and subject to a special address by the UN High Commissioner for Human Rights, Michelle Bachelet.  Next week will see an EU delegation visiting Sri Lanka to assess the human rights situation in relation to the GSP Plus tariff privilege that the country obtained again in 2017 having lost it for seven years in 2010.

 The immediate challenge to the government’s plans for economic recovery hinge on its ability to keep the GSP Plus and not lose it.  This privilege, which permits the country’s exports entry to the EU market without the payment of import duties, has helped to sustain Sri Lankan business enterprises during the Covid-induced economic downturn.  With the exception of the tourism-related business sector, several export-oriented business sectors have been showing healthy profits due in large measure to the GSP Plus facility.

The GSP is given on condition that human rights are being protected in the country concerned, which has become a question mark for Sri Lanka. The questions have grown since the change of government. The attempt to withdraw from the commitments with regard to human rights given by the previous government when it co-sponsored UNHRC Resolution 30/1 in 2015 has not been well received by the international human rights community.

 The alleged offences committed by Minister Ratwatte, and the delayed response by the government undermines its presentation of facts and arguments that Foreign Minister Prof G L Peiris made during his presentation in Geneva last week intended to show that the Rule of Law, independence of state institutions and human rights are to be found in the country.  Minister Ratwatte is alleged to have entered two prison compounds, visited the gallows in one, along with his personal friends and threatened Tamil prisoners in another with his gun and made them kneel before him.  However, he has denied doing anything improper and his version has, according to media reports, been corroborated by prisons authorities who have claimed that no untoward incidents have occurred.  This has been amidst claims and counter claims about the availability of CCTV cameras in the high security prisons.

EXTERNAL INTERVENTION

The alleged act of unprovoked disregard for the human dignity of the two prisoners held as LTTE suspects under the Prevention of Terrorism Act, which itself violates international standards of human rights, has become a rallying call for unity amongst the Tamil community.  This justifies or adds credence to the view that the minorities have security issues in the country and their rights are inadequately protected. Rival Tamil political parties have been issuing statements of condemnation. They have also got onto common platforms in opposition to the Minister’s conduct. On the positive side, the protests have not only come from the Tamil community.

The mainstream media and opposition political parties have also joined in the condemnation and demanded that the government takes immediate action.  In addition, the ruling party, the Sri Lanka Podujana Peramuna (SLPP) Secretary Sagara Kariyawasam has said that “This type of action cannot be permitted in any civilised society and appropriate action should be taken after investigations.”

There is a need for governmental action that goes beyond the mere resignation of the accused Minister from his portfolio of prisons while retaining his other ministries. There is an urgent need for this. In June of this year the European Parliament passed a resolution that the GSP Plus should be withdrawn if Sri Lanka did not show signs of progress in its protection of human rights. The resolution expressed “deep concerns over Sri Lanka’s alarming path towards recurrence of grave human rights violations as described by the most recent UN report on the country, which lists among the early warning signals the accelerating militarisation of civilian governmental functions, reversal of important constitutional safeguards, political obstruction of accountability, exclusionary rhetoric, intimidation of civil society, and the use of anti-terrorism laws.”

 The government’s position in Geneva was that there is no need for international involvement in either monitoring or investigating human rights violations in Sri Lanka as the country has its own internal mechanisms to do so. Foreign Minister Prof GL Peiris speaking in Geneva said, “We reject the proposal for any external initiatives purportedly established by Resolution 46/1 while domestic processes are vigorously addressing the relevant matters. This will polarise our society, as we experienced with Resolution 30/1. The Council must adhere to its founding principles. External initiatives embarked upon without the cooperation of the country concerned cannot achieve their stated goals, and will be subject to politicization.”

 JUDICIAL ACTION

 The problem of impunity for state actors is long standing and precedes the present government. Successive governments have whitewashed the crimes of their members as taking punitive action is politically costly to them. Usually the culprits have political assets that outweigh their crimes in the eyes of the political parties and voters alike.  It is this tolerance of conduct with impunity and without accountability by those in government, who set themselves above the law, that has taken the country to the dock of international human rights opinion in Geneva. Unfortunately this type of conduct gained political acceptability during the period of the three decades long war and long periods of emergency rule.  However, the war is now over more than 12 years and the last election was fought on promises of a restoration of law and order and the practice of discipline as being national values.

 The key area of governance failure highlighted by the UN High Commissioner has been the area of impunity and lack of follow up in emblematic cases where human rights violations are alleged to have occurred.  It was on account of this failure of previous governments to ensure accountability, which continues, that the UNHRC Resolution 46/1 of March 2021 mandated the establishment of a special investigation unit at the UN level to monitor the human rights situation in the country and collect information.

According to the UN Human Rights Commissioner this “Office’s work to implement the accountability-related aspects of Resolution 46/1 has begun, pending recruitment of a start-up team.  We have developed an information and evidence repository with nearly 120,000 individual items already held by the UN, and we will initiate as much information-gathering as possible this year.”

 As Minister Ratwatte is a member of Parliament it would be appropriate that an independent inquiry be conducted with the investigators being appointed in a bipartisan manner by the Prime Minister and Opposition Leader to instill public and international confidence as this is absent at this time.

n the meantime the Minister should be persuaded to resign from all ministerial appointments pending investigation or be removed from such positions as this is an issue that goes to the heart of the issue of impunity. The failure of the Sri Lankan system of justice to act with transparency and integrity in the case regarding Minister Ratwatte will only serve to further justify the stance of the international community regarding the need for external intervention. It will undermine the efforts of the government to argue otherwise.  A credible investigation and judicial action is necessary if external intervention is to be kept at bay and the GSP Plus is to be retained.  This can be the turning point.

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Seeing the market through the spectre

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By Panduka Karunanayake

Leftist ideologues like to create an image of ‘the market’ as a terrible place, like how grown-ups frighten children with their ghost stories. The market is portrayed as a ruthless, heartless machine that thrives on unfairness and corruption, crushing the poor and fattening the rich. It must have become easier to create this image after the Soviet bloc fell in the 1990s and China emerged out of communism soon afterwards, because after that people quickly forgot that socialist and communist economies too have markets – even markets every bit as ruthless, heartless, unfair and corrupt as any capitalist market. Today, these ideologues can write as if the market and capitalism are synonyms. Everything ‘un-socialist’ can be easily ‘explained away’ by saying that it ‘promotes marketisation’.

No amount of argument or explanation would change these ideologues’ minds – after all, an ideologue is a person who pursues an ideology in an inflexible manner. But let me set out some related matters, for the benefit of the rest of us.

Emergence and evolution of the market

Markets have probably existed throughout human existence, because human beings are social animals that thrive on social interaction. There is an illuminating passage in Charles Darwin’s book The Voyage of the Beagle, where he described an encounter with the inhabitants of Tierra del Fuego in South America, who were ‘primitive’ hunter-gatherers and nomads with no previous encounter with civilisation:

“Some of the Feugians plainly showed that they had a fair notion of barter. I gave one man a large nail (a most valuable present) without making any signs for a return; but he immediately picked out two fish, and handed them up on the point of his spear. If any present was designed for one canoe, and it fell near another, it was invariably given to the right owner.”

This passage shows that they were well-versed in the moral principles – such as free choice, trust, fairness and reciprocity – associated with exchange of goods and services that form the basis of the functioning market.

The market became prominent after the emergence of agriculture about 10,000 year ago. Agriculture enabled farmers to produce a food surplus, which then enabled the rest of the villagers to work on other crafts – promoting division of labour and specialisation. This created an overall increase in the quality of the villagers’ lives, because they now had a wider variety of produce to consume. Villagers now depended on each other more, for the produce they wished to consume. As the village increased in size and complexity, a place where producers and consumers could meet, to exchange goods and services, became a necessity. In the physical realm this was the marketplace, and in the conceptual realm it was the market. Initially, exchange occurred through the barter system without a medium of exchange, but the invention of money made it easier.

But until industrialisation, this market was small and sluggish. It produced very little, compared to today. Most of what a village produced was consumed within it, and only a tiny proportion of it left the village, to be consumed by outsiders – the market was still not much more than the marketplace.

Reason for smallness

The reason for this smallness was not entirely because there were no industrial factories. The villagers could have produced more if they wanted to, but they didn’t, because they saw that any extra produce created problems. There were difficulties with storing it, protecting it, preserving it or transporting it elswhere, and in any event the lords could easily expropriate it under the feudalistic modes of production. Items that were considered luxury items were an exception; they were carried to distant destinations by camel, caravan or boat.

But starting in the eighteenth century, industrialisation changed all that. Factories produced large quantities of produce (or ‘commodities’) cheaply, and the market expanded to distribute a much larger variety and quantity of goods much more widely. Specialisation became the norm and a necessity. The crucial factor that made all this possible was probably the improvement in transport. Look at your lunch plate today, and try to figure out from where and how far each of the food items on it – not to mention the plate itself or the energy for the fire that cooked your lunch – have come from.

Today, production and consumption are almost totally separated from each other (with a few exceptions, like farmers who sell their produce by the roadside in front of their homes, and ‘factory outlets’). It is the market that enables this to happen. The market, which is no longer simply the marketplace, gives us access to a bewildering variety of goods and services, thereby enabling us – even the poorest amongst us – to enjoy a greater choice and higher quality of life, compared to pre-industrial times. The healthcare and education that even the poorest amongst us enjoy would not reach them if not for the market.

As an example, let us take soap. Until industrialisation this was a luxury item that only the elite enjoyed. In Roman times, even the elite cleaned themselves mostly by simply immersing themselves in their baths and rubbing off dirt; indeed, using soap would have made the bath too disgusting to get into. Soap was available only to those in the very highest echelons of society. The masses were ‘dirty’ and their skin was infested with scabies, pediculosis and lice, and they commonly suffered diseases like impetigo and erysipelas – they lacked even the water necessary to wash themselves (especially hot water in cold climates). But today, soap is so ubiquitous that we take it for granted – it was industrialisation that enabled its cheap mass production and the market that enabled its wide distribution.

Value of simple things

The real value of something as simple as soap was driven home powerfully to me in the aftermath of the 2004 tsunami, when people had lost everything and were accommodated in make-shift camps. What did they ask for, from the donors and volunteers who went to help them? First, they asked for food, water and certain medicines. A few days later, they began asking for soap, a change of clothing and sanitary pads: after three or four days, they were itching and suffering with fungal skin infections. That was an unfortunate re-enactment and reminder of the pre-industrial life of the masses. I remembered how a textbook of public health that I had read a few decades earlier had cleverly classified infectious diseases according to whether they were prevented by soap and water, clean drinking water, safe food, and so on. The post-tsunami experience showed me the sagacity of that – and the value of the ubiquitous soap, industrialisation and the market.

In his book The Third Wave, Alvin Toffler compared a modern-day market to an efficient telephone exchange or switchboard. A switchboard connects thousands of senders and recipients accurately and enables messages to be sent across to their intended destinations – like producers, consumers, and goods and services in the market. Like telephone messages, goods and services are produced, sent across and consumed according to need and availability: demand and supply.

Such a market cannot exist on its own. It needs inputs from important sectors in society, such as law and order (which upholds the right to private property, prevents or punishes theft, and arbitrates when there are contractual disagreements), education (which creates an educated and trained workforce, not only for manufacture but also for distribution), communication, energy, transport, ports, etc. Such external supports have existed not only in capitalist markets but also in pre-capitalist and socialist markets. These supports are provided because everybody realises that markets are useful to everyone, especially when the population expands and the demand for commodities increases.

So, to say that something should be abhorred because it promotes ‘marketisation’ is disingenuous.

Organising the market

Markets can be organised in various ways. It helps to think of these as lying on a spectrum ranging from capitalism to communism, which are the extreme forms at the two ends. In-between, there are lots of compromises, combinations or ‘middle ways’. For instance, the current Chineses model is sometimes called state capitalism – a good example of a middle way.

But the natural form of the market that emerged spontaneously was the free market: a market where no authority-imposed restrictions or controls, nor introduced any encouragements or inducements. The activities in the free market merely recognised the concepts of private property and voluntary exchange, and operated on demand and supply. That was all.

Opponents

Throughout this time, the free market has had many opponents who have tried to impose limits or controls to it. Division of labour and specialisation were resisted – by the cultural elite who tried to maintain the status quo in society, such as the caste system in ancient society and feudal-peasant relations in medieval times. During industrialisation when factories came up, that was resisted too – by guildsmen who felt that their business was threatened, and those like the Luddites who felt threatened by the new manufacturing technology. Karl Marx proposed that private property, including ‘the means of production’, should be taken over by the state and brought under its control.

Some of the concepts that they used against the growth of the free market were traditional values such as loyalty and caste-based duties (especially upheld by the cultural elite), simplicity in life and charity as well as opposition to ‘ursury’ and banks (especially the clergy), and equality. It is only now, after centuries of change, that words like ‘industry’ (which initially meant industriousness), ‘entrepreneurship’ and ‘individualism’ have emerged as ‘good words’, to create an environment conducive to a free market. Charles Dickens’ novel Martin Chuzzlewit nicely captured the mood of the era when industrialisation was struggling to emerge through the feudal society, by portraying the struggle of a typical ‘upstart’ who had to go to America to make a fresh start.

According to Stephen Fry, even today, the typical British comedy mostly parodies the upstart’s ineptness (“celebrate failure”), whereas the typical American comedy glorifies the industrious entrepreneur or smart-aleck (“life is improvable”). When we read comments about the market, we must take care to ‘read’ this subtext too.

Inequality

Ironically, inequality had previously been tolerated and even celebrated, as long as the only inequality was between the elite and the masses – the masses were ‘equal’ in their poverty and the elite were rich by birthright. But after industrialisation, the moment the masses gradually became enriched and a middle class emerged, inequality became a big social issue. When a part of the masses remained poor and another part became better-off, socialism was born. Different segments of the masses quickly became each other’s enemies – thanks to socialism. It was an example of applying the brakes even before the vehicle had started to move in earnest. They were ennobled by socialism’s new words, like ‘fraternity’ and ‘equality’ – which basically meant, ‘Those who are not poor like you are not one of you, and have no right to be rich if you too cannot be rich’.

But while the leaders of socialism may have harboured such beliefs, their proletariat comrades had simpler minds. In his book The Road to Wigan Pier written after World War One, George Orwell, himself a socialist, reflecting on the tensions and contradictions in English society as it grappled with this new-found inequality and ‘class struggle’, wrote:

“To the ordinary working man, the sort you would meet in any pub on Saturday night, Socialism does not mean much more than better wages and shorter hours and nobody bossing you about….[No] genuine working man grasps the deeper implications of Socialism. Often, in my opinion, he is a truer Socialist than the orthodox Marxist, because he does remember what the other so often forgets, that Socialism means justice and common decency….His vision of the Socialist future is a vision of present society with the worst abuses left out, and with interest centring round the same things as at present – family life, the pub, football, and local politics.”

Orwell’s main message was that any effort to ‘improve’ society, by whatever name, that had lost touch with the common man was bound to deteriorate into a fascism. The remainder of the twentieth century proved him right.

Today, Orwell’s England has come through quite nicely in spite of the disintegration of the British Empire soon afterwards, and shows none of the class struggle and poverty Orwell’s and Dickens’ books have recorded. So, should we champion an equality of the poor, or should we patiently work towards a gradually enriching society with a tolerable level of inequality?

Necessary controls

At the same time, it is also advisable to control some forms of exchange in the market. For instance, if a country considers that it is necessary to ensure food security through its own, local cultivation of important crops, it is wise to put in place some safeguards to protect local agriculture from the adverse effects of competition from imported foods, at least for important foods. Even advanced capitalist countries like USA and Japan do this (for wheat and rice, respectively).

Similarly, it would be prudent to protect certain crucial markets, such as the energy sector and ports. It is also important to ensure distribution of crucial goods & services (such as healthcare, basic education, basic housing, basic clothing, basic transport) for all members of society, with a view to protecting the poor who have limited purchasing power. This requires the institution of safety nets and price control. In this age of climate change, resource depletion and environmental degradation, nobody would argue against environmental protection, which naturally requires the imposition of certain restrictions on the free market. Finally, nobody would argue that sectors such as national defence and law & order should be floated in the free market. So markets do need judicious controls and regulation.

More than ‘free’

On the flip side, in some markets there are mechanisms created specifically to encourage a bigger flow of goods & services than what the natural, ‘free’ market would sustain. These include patent laws, laws restricting monopolies, bankruptcy laws, the financial and share market, and so on; some of them may be good, while others are not.

The market is then not merely a place of exchange; it is also a place to make massive profits, where the falling crumbs accumulate to produce huge volumes of ‘wealth’. There are those who would profit exactly from this, while such ‘wealth generation’ or ‘productivity’ brings no intrinsic value to society while needlessly destroying our environment and culture – this can then become the new status quo that these new elite wish to protect.

Conclusion

A free market would promote exchange of goods & services and increase the volume of exchange, and this in turn would increase employment, productivity, taxation and funds for welfare expenditure. Both restricting it and encouraging it, while sometimes necessary, must be done only cautiously. Naturally, therefore, both extremes – and their supportive ideologies – are not good. What we need to have is a ‘middle-way’ market that enables enough economic activity and protects the poor, while protecting the environment for future generations.

So, there is no need to fear the market. What we need to do is understand it, be able to predict its behaviour, and try to modify it so that it creates the benefits we need and avoids harm. The child must grow up, overcome the fear of ghosts and learn to deal with darkness. Disingenuous, sleight-of-hand arguments that promote the darkness are of no use, and their supportive ideologies can only lead to fascism – as the twentieth century amply taught us.

The writer teaches medicine in the University of Colombo (email:

panduka@clinmed.cmb.ac.lk). He acknowledges helpful comments from Professor Sirimal Abeyratne (Professor of Economics, University of Colombo) and Dr G. Usvatte-aratchi.

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PLANES OF EXISTENCE

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(from THE BUDDHA AND HIS TEACHINGS by Venerable Narada Mahathera)

“Not to be reached by going is world’s end.” ANGUTTARA NIKAYA

According to Buddhism the earth, an almost insignificant speck in the universe, is not the only habitable world, and humans are not the only living beings. Indefinite are world systems and so are living beings. Nor is “the impregnated ovum the only route to rebirth.” By traversing one cannot reach the end of the world, says the Buddha.

Births may take place in different spheres of existence. There are altogether thirty-one places in which beings manifest themselves according to their moral or immoral Kamma.

There are four states of unhappiness (Apaya) which are viewed both as mental states and as places.

They are:

1. Niraya (ni + aya = devoid of happiness) woeful states where beings atone for their evil Kamma. They are not eternal hells where beings are subject to endless suffering. Upon the exhaustion of the evil Kamma there is a possibility for beings born in such states to be reborn in blissful states as the result of their past good actions.

2. Tiracchana-yoni (tiro = across; acchana = going), the animal kingdom. Buddhist belief is that beings are born as animals on account of evil Kamma. There is, however, the possibility for animals to be born as human beings as a result of the good Kamma accumulated in the past. Strictly speaking, it should be more correct to state that Kamma which manifested itself in the form of a human being, may manifest itself in the form of an animal or vice versa, just as an electric current can be manifested in the forms of light, heat and motion successively — one not necessarily being evolved from the other.

It may be remarked that at times certain animals particularly dogs and cats, live a more comfortable life than even some human beings due to their past good Kamma.

It is one’s Kamma that determines the nature or one’s material form which varies according to the skilfulness or unskilfulness of one’s actions.

3. Peta-yoni (pa + ita) lit., departed beings, or those absolutely devoid of happiness. They are not disembodied spirits of ghosts. They possess deformed physical forms of varying magnitude, generally invisible to the naked eye. They have no planes of their own, but live in forests, dirty surroundings, etc. There is a special book, called Petavatthu, which exclusively deals with the stories of these unfortunate beings. Samyutta Nikaya also relates some interesting accounts of these Petas.

Describing the pathetic state of a Peta, the Venerable Moggallana says:

“Just now as I was descending Vultures’ Peak Hill, I saw a skeleton going through the air, and vultures, crows, and falcons kept flying after it, pecking at its ribs, pulling apart while it uttered cries of pain. To me, friend, came this thought: O but this is wonderful! O but this is marvellous that a person will come to have such a shape, that the individuality acquired will come to have such a shape.”

“This being,” the Buddha remarked, “was a cattle-butcher in his previous birth, and as the result of his past Kamma he was born in such a state. “

According to the Questions of Milinda there are four kinds of Petas — namely, the Vantasikas who feed on vomit, the Khuppipāsino who hunger and thirst, the Nijjhamatanhikaā, who are consumed by thirst, and the Paradattapajavino who live on the gifts of others.

As stated in the Tirokudda Sutta these last mentioned Petas share the merit performed by their living relatives in their names, and could thereby pass on to better states of happiness.

4. Asura-yoni — the place of the Asura-demons. Asura, literally, means those who do not shine or those who do not sport. They are also another class of unhappy beings similar to the Petas. They should be distinguished from the Asuras who are opposed to the Devas.

Next to these four unhappy states (Duggati) are the seven happy states (Sugati). They are:

1. Manussa The Realm of human beings.

The human realm is a mixture of both pain and happiness. Bodhisattas prefer the human realm as it is the best field to serve the world and perfect the requisites of Buddhahood. Buddhas are always born as human beings.

2. Catummaharajika — the lowest of the heavenly realms where the Guardian Deities of the four quarters of the firmament reside with their followers.

3. Tavatimsa — lit., thirty-three — the Celestial Realm of the thirty-three Devas where Deva Sakka is the King. The origin of the name is attributed to a story which states that thirty-three selfless volunteers led by Magha (another name for Sakka), having performed charitable deeds, were born in this heavenly realm. It was in this heaven that the Buddha taught the Abhidhamma to the Devas for three months.

4. Yama — “The Realm of the Yama Devas.” That which destroys pain is Yāma.

5. Tusita — lit., happy dwellers, is “The Realm of Delight.”

The Bodhisattas who have perfected the requisites of Buddhahood reside in this Plane until the opportune moment comes for them to appear in the human realm to attain Buddhahood. The Bodhisatta Metteyya, the future Buddha, is at present residing in this realm awaiting the right opportunity to be born as a human being and become a Buddha. The Bodhisatta’s mother, after death, was born in this realm as a Deva (god). From here he repaired to Tavatimsa Heaven to listen to the Abhidhamma taught by the Buddha.

6. Nimmanarati — “The Realm of the Devas who delight in the created mansions.”

7. Paranimmitavasavatti — “The Realm of the Devas who make others’ creation serve their own ends.”

The last six are the realms of the Devas whose physical forms are more subtle and refined than those of human beings and are imperceptible to the naked eye. These celestial beings too are subject to death as all mortals are. In some respects, such as their constitution, habitat, and food they excel humans, but do not as a rule transcend them in wisdom. They have spontaneous births, appearing like youths and maidens of fifteen or sixteen years of age.

These six Celestial Planes are temporary blissful abodes where beings are supposed to live enjoying fleeting pleasures of sense.

The four unhappy states (Duggati) and the seven happy states (Sugati) are collectively termed Kamaloka — Sentient Sphere.

Superior to these Sensuous Planes are the Brahma Realms or Rupaloka (Realms of Form) where beings delight in jhanic bliss, achieved by renouncing sense-desires.

Rupaloka

consists of sixteen realms according to the jhānas or ecstasies cultivated. They are as follows:

(a) T’he Plane of the First Jhana;

1. Brahma Parisajja –– The Realm of the Brahma‘s Retinue.

2. Brahma Purohita — The Realm of the Brahma’s Ministers.

3. Mahaā Brahma — The Realm of the Great Brahmas.

The highest of the first three is Mahaā Brahma. It is so called because the dwellers in this Realm excel others in happiness, beauty, and age-limit owing to the intrinsic merit of their mental development.

(b) The Plane of the Second Jhāna:

4. Parittābhā — The Realm of Minor Lustre,

5. Appamānābhā — The Realm of Infinite Lustre,

6. Ābhassarā —

The Realm of the Radiant Brahmas.

(c) The Plane of the Third Jhāna:

7. Parittasubha — The Realm of the Brahmas of Minor Aura.

8. Appamānasubha — The Realm of the Brahmas of Infinite Aura.

9. Subhakinhaā — The Realm of the Brahmas of Steady Aura.

(d) The Plane of the Fourth Jhana:

10. Vehapphala — The Realm of the Brahmas of Great Reward.

11. Asaatta — The Realm of Mindless Beings,

12. Suddhavasa — The Pure Abodes which are further subdivided into five, viz:

i. Aviha — The Durable Realm,
ii. Atappa — The Serene Realm,
iii. Sudassa — The Beautiful Realm,
iv. Sudassi — The Clear-Sighted Realm.
v. Akanittha — the Highest Realm.

Only those who have cultivated the Jhanas or Ecstasies are born on these higher planes. Those who have developed the First Jhana are born in the first Plane; those who have developed the Second and Third Jhanas are born in the second Plane; those who have developed the Fourth and Fifth Jhanas are born in the third and fourth Planes respectively.

The first grade of each plane is assigned to those who have developed the Jhanas to an ordinary degree, the second to those who have developed the Jhanas to a greater extent, and the third to those who have gained a complete mastery over the Jhanas.

In the eleventh plane, called the Asaatta, beings are born without a consciousness.

Here only a material flux exists. Mind is temporarily suspended while the force of the Jhāna lasts. Normally both mind and matter are inseparable. By the power of meditation, it is possible, at times, to separate matter from mind as in this particular case. When an Arahant attains the Nirodha Samāpatti, too, his consciousness ceases to exist temporarily. Such a state is almost inconceivable to us. But there may be inconceivable things which are actual facts.

The Suddhavasas or Pure Abodes are the exclusive Planes of Anagamis or Never-Returners. Ordinary beings are not born in these states. Those who attain Anāgāmi in other planes are reborn in these Pure Abodes. Later, they attain Arahantship and live in those planes until their life-term ends.

There are four other planes called Arupaloka which are totally devoid of matter or bodies. Buddhists maintain that there are realms where mind alone exists without matter. “Just as it is possible for an iron bar to be suspended in the air because it has been flung there, and it remains as long as it retains any unexpended momentum, even so the Formless being appears through being flung into that state by powerful mind-force, there it remains till that momentum is expended. This is a temporary separation of mind and matter, which normally co-exist. “

It should be mentioned that there is no sex distinction in the Rupaloka and the Arupaloka.

The Arupaloka is divided into four planes according to the four Arupa Jhanas.

They are:

1. The Sphere of the Conception of Infinite Space.

2. The Sphere of the Conception of Infinite Consciousness.

3. The Sphere of the Conception of Nothingness.

4. The Sphere of Neither Perception nor Non-Perception.

It should be remarked that the Buddha did not attempt to expound any cosmological theory.

The essence of the Buddha’s teaching is not affected by the existence or non-existence of these planes. No one is bound to believe anything if it does not appeal to his reason. Nor is it proper to reject anything because it cannot be conceived by one’s limited knowledge.

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