Proof in Mathematics

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This article is taken from the book 50 Mathematical ideas

Proof
Mathematicians attempt to justify their claims by proofs. The
quest for cast iron rational arguments is the driving force of
pure mathematics. Chains of Correct deduction from what is
known or assumed, lead the mathematician to a conclusion
which then enters the established mathematical storehouse.
Proofs are not arrived at easily – they often come at the end of a great
deal of exploration and false trails. The struggle to provide them occupies
the center ground of the mathematician’s life. A successful proof carries the
mathematician’s stamp of authenticity, separating the established theorem
from the conjecture, bright idea or first guess.
Qualities looked for in a proof are rigor, transparency and, not least, elegance.
To this add insight. A good proof is ‘one that makes us wiser‘- but it is also
better to have some proof than no proof at all. Progression on the basis of
unproven facts carries the danger that theories may be built on the
mathematical equivalent of sand.

Not that a proof lasts forever, for it may have to be revised in the light of
developments in the concepts it relates to.
What is a proof? When you read or hear about a mathematical result
do you believe it? What would make you believe it? One answer would be
a logically sound argument that progresses from ideas you accept to the
statement you are wondering about. That would be what mathematicians
call a proof, in its usual form a mixture of everyday language and strict logic.
Depending on the quality of the proof you are either convinced or remain
skeptical.
The main kinds of proof employed in mathematics are:

the method of the counterexample;

the direct method;

the indirect method;

and the method of mathematical induction.

<———-Foot Notes——————>
Euclid’s Elements provides the model for
mathematical proof  c.300 BC
Descartes promotes mathematical
rigor as a model in his Discourse
on Method   AD 1637

<——————————————->
The counterexample: Let’s start by being skeptical – this is a method of
proving a statement is incorrect. We’ll take a specific statement as an example.
Suppose you hear a claim that any number multiplied by itself results in an
even number. Do you believe this? Before jumping in with an answer we should
try a few examples. If we have a number, say 6, and multiply it by itself to get
6 x 6 = 36 we find that indeed 36 is an even number. But one swallow does nor
make a summer. The claim was for any number, and there are an infinity of
these. To get a feel for the problem we should try some more examples. Trying
9, say, we find that 9 x 9 = 81. But 81 is an odd number. This means that
the statement that all numbers when multiplied by themselves give an even
number is false. Such an example runs counter to the original claim and is
called a counterexample. A counterexample to the claim that ‘all swans are
white’, would be to see one black swan. Part of the fun of mathematics is
seeking out a counterexample to shoot down a would-be theorem.
If we fail to find a counterexample we might feel that the statement is correct.
Then the mathematician has to play a different game. A proof has to be
constructed and the most straightforward kind is the direct method of proof.
The direct method:  In the direct method we march forward with logical
argument from what is already established, or has been assumed, to the
conclusion. If we can do this we have a theorem. We cannot prove that
multiplying any number by itself results in an even number because we have
already disproved it. But we may be able to salvage something. The difference
between our first example, 6, and the counterexample, 9, is that the first
number is even and the counterexample is odd. Changing the hypothesis is
something we can do. Our new statement is: if we multiply an even number by
itself the result is an even number.
First we try some other numerical examples and we find this statement verified
every time and we just cannot find a counterexample. Changing tack we try
to prove it, but how can we start?’ We could begin with a general even number
n, but as this looks a bit abstract we’ll see how a proof might go by looking at
a concrete number, say 6. As you know, an even number is one which is a
multiple of 2, that is 6 — 2 x3. As 6 x 6 = 6 + 6 + 6 + 6 + 6 + 6 or, written
another way, 6 x 6 = 2 x 3 + 2 x 3 + 2 x 3 + 2 x 3 + 2 x 3 + 2 x 3 or,rewriting
using brackets,

<————-Foot Notes —————>
De Morgan introduces the term
‘mathematical induction’   1838
Bishop proves results
exclusively by constructive
methods  1967
lmre Lakatos publishes the
influential Proofs an d Refutations  1976
<———————————————–>

6×6=2x(3+3+3+3+3+3)
This means 6 x 6 is a multiple of 2 and, as such, is an even number. But in this
argument there is nothing which is particular to 6, and we could have started
with n = 2 x k to obtain
n x n = 2 x (k+k+…+k)
and conclude that n x n is even. Our proof is now complete’ In translating
Euclid’s Elements, latter-day mathematicians wrote ‘QED’ at the end of a proof
to say job done – it’s an abbreviation for the Latin quod erat demonstradum
(which was to be demonstrated). Nowadays they use a filled-in square l. This is
called a halmos after Paul Halmos who introduced it.
The indirect method:  In this method we pretend the conclusion is false
and by a logical argument demonstrate that this contradicts the hypothesis’
Let’s prove the previous result by this method.
Our hypothesis is that n is even and we’ll pretend n x n is odd. We can write
n x n : n + n + . . . + n and there are n of these. This means n cannot be even
(because if it were n x n would be even). Thus n is odd, which contradicts the
hypothesis.
This is actually a mild form of the indirect method. The full-strength indirect
method is known as the method o{ reductio ad absurdum

(reduction to the
absurd), and was much loved by the Greeks. In the academy in Athens,
Socrates and Plato loved to prove a debating point by wrapping up their
opponents in a mesh of contradiction and out of it would be the point they
were trying to prove. The classical proof that the square root of 2 is an irrational
number is one of this form where we start off by assuming the square root of 2 is
a rational number and deriving a contradiction to this assumption.
The method of mathematical induction  Mathematical
induction is powerful way of demonstrating that a sequence of statements P1,
P2, P3, .. . are all true. This was recognized by Augustus De Morgan in the
1830s who formalized what had been known for hundreds of years. This specific
technique (not to be confused with scientific induction) is widely used to prove
statements involving whole numbers. It is especially useful in graph theory,
number theory and computer science generally. As a practical example, think
of the problem of adding up the odd numbers. For instance, the addition of the
first three odd numbers 1 + 3 + 5 is 9 while the sum of first four I + 3 + 5 + 7 is
16.Now 9 is 3 x 3 = 3 squared and 16 is 4 x 4= 4 squared,

so could it be that the addition of
the first n odd numbers is equal to n squared?

If we try a randomly chosen value of n,

say n = 7, we indeed find that the sum of the first seven is 1 + 3 + 5 + 7 + 9 +
1 1 + 13 = 49 which is 7 squared. But is this pattern followed for all values of n?

How can we be sure? We have a problem, because we cannot hope to check an
infinite number of cases individually.
This is where mathematical induction steps in. Informally it is the domino
method of proof. This metaphor applies to a row of dominoes standing on their
ends. If one domino falls it will knock the next one down. This is clear. All we
need to make them all fall is the first one to fall. We can apply this thinking to
the odd numbers problem, The statement P’ says that the sum of the first n
odd numbers adds up to n squared.

Mathematical induction sets up a chain reaction
whereby P1, P2, P3,. . . will all be true. The statement P1 is trivially true
because1 = 1 squared. Next, P2 is true

because1 + 3 = l squared  +3 =2 squared, P3 is true because
1 + 3 + 5 = 2 squared + 5 = 3 squared and P4 is true

because I + 3 + 5 +7 = 3 squared + 7 = 4 squared
‘We use the result at one stage to hop to the next one. This process can be
formalized to frame the method of mathematical induction.
Difficulties with Proof: Proofs come in all sorts of styles and sizes.
Some are short and snappy, particularly those found in the text books. Some
others detailing the latest research have taken up the whole issue of journals
and amount to thousands of pages. Very few people will have a grasp of the
whole argument in these cases.
There are also foundational issues. For instance, a small number of
mathematicians are unhappy with the reductio ad absurdam method of
indirect proof ‘where it applies to existence. If the assumption that a solution
of an equation does not exist leads to a contradiction, is this enough to prove
that a solution does exist?  Opponents of this proof method would claim the
logic is merely sleight of hand and doesn’t tell us how to actually construct a
concrete solution. They are called ‘Constructivists’ (of varying shades) who
say the proof method fails to provide ‘numerical meaning’- They pour scorn on
the classical mathematician who regards the reductio method as an essential
weapon in the mathematical armory. On the other hand the more traditional
mathematician would say that outlawing this type of argument means working
with one hand tied behind your back and, furthermore, denying so many
results proved by this indirect method leaves the tapestry of mathematics
looking rather threadbare.
The condensed idea
 Signed and Sealed

INTRODUCTION TO THE PHILOSOPHY OF HOLISM

INTRODUCTION TO THE PHILOSOPHY OF HOLISM

 

“The whole is more than the sum of its parts.” — Aristotle

What is Holism?

The concept of “whole as greater than the sum of its parts” has ancient roots. But the term “holism” (more reasonably but less often spelled ‘wholism’) as fully developed rarely appears in anyone’s conversation except somewhat narrowly in that of the philosophers or sociologists. It is a scholarly word that originated from the Greek ‘holos’, meaning ‘whole’. In its present context, as defined by General Jan Christian Smuts (1870-1950), 4th Prime Minister of South Africa and a British Commonwealth military leader, statesman and philosopher conceived “holism” as “The tendency in nature to form wholes that are greater than the sum of the parts through creative evolution.” Smuts, arguing in the Holism and Evolution (1926) says: “This factor, called Holism in the sequel, underlies the synthetic tendency in the universe, and is the principle which makes for the origin and progress of ‘wholes’ in the universe. . . this whole-making or holistic tendency is fundamental in nature, that it has a well-marked ascertainable character, and that Evolution is nothing but the gradual development of progressive series of wholes, stretching from inorganic beginnings to the highest levels of spiritual creation.” (Smuts, page-V)

 

The holistic concept in ancient theological belief, per Heraclitus (c.535-475 BCE), was strongly reflected in the concept of Logos and Pantheism. The Chinese philosopher Zhuangzi (c. 369-286 BCE) was an exponent of the holistic philosophy of life, projecting a way of understanding that is uncommitted to a fixed system, a way that is fluid and flexible, and that maintains a pragmatic attitude towards the applicability of the “multiplicity of diverse modes” of realization among different creatures, cultures and philosophical outlooks. Philosophers and thinkers even before Socrates (c.469-399 BCE) have rationally as well as theologically speculated that wholes, both animate and inanimate, are real, while parts are abstract analytical distinctions, and that wholes are flexible patterns, not simply mechanical assemblages of self-sufficient elements. Implicit in this view is that, when individual components of a system are put together to produce a large functional unit, a holistic quality develops which is not predictable from the behavior of the components in their individual capacity.

Along genuine holistic paths, whether theistic or non-theistic and whatever they are called, there is a potential evolutionary movement in the consciousness of the human being. It is a movement from the ordinary level of being, doing, and having that most of us know in our daily lives to something more fulfilling. The ordinary level is one where exist many misunderstandings, frequent periods of frustration and stress, remittent moments of happiness and pleasure, a somewhat scattered attention, and for some an underlying sense that we are not living as fully as we might until it is too late. The holistic path—which is mystical path for a theistic and an evolutionary for a non-theistic—is intended to help us experience another level where life reveals a much deeper inner meaning, where our thoughts, feelings and actions are integrated by a clear intelligence and knowledge, and a feeling of intimacy and participation with something greater than our normal selves occurs. A theistic describes it as a level where a profound spiritual dimension appears. But Plato called it higher knowledge. Many great artists tell of mysterious creative moments. Speaking holistically, we might say that the ordinary daily level that most of us know is fragmented and partial, one where experiences are driven by one part or another, such as a strong desire, a thought, or a physical urge. The higher level is experienced as more whole, more free, where fragments of formerly disparate and conflicting physical, emotional, intellectual and spiritual energies are unified by a love-wisdom of the heart and a new sense of inner unity and oneness arises. The spirit is now filled with love and emerges as an active, creative, participating force in life. One must learn to distinguish the permanent transformation to the higher level from a temporary or gradual changing.

 

Holism Today

            The modern proposition of holism stems from an old idea that existed spontaneously in the ancient cultures of the Chinese, Babylonians, Egyptians, Indians, and Homeric Greeks. It viewed the human being as a compound of body and soul. With physical death, the soul was considered no longer an alert consciously living entity. In some cultures it would become a pathetic shade or ghost doomed to reside in a gloomy underworld. In other cultures it would reincarnate in another body, and for still others, on account of being no longer whole without a body, the soul would dissolve into nothingness. This theme of soul unable to function without physical body, still holding ground in the modern age, impregnated itself especially into most of the monotheistic faiths within the concept of an eschatological (religious belief of judgment and destiny) resurrection as a basic theological concern with death, destiny and day of judgment. Historically, these theological considerations, originating from Zoroastrianism, entered first into Judaism when the Jews, during their Babylonian exile, came into contact with the Zoroastrian culture. From Judaism this idea passed on to Christianity and Islam where it formed into a belief that a human being is a compound whole of body, mind and soul (or spirit), and that not one of these by itself is fully alive and whole without the other two.

 

Though the concept of holism was vividly and concisely reflected by Aristotle (384-322 BCE) in his Metaphysics (1045a10) that, “The whole is more than the sum of its parts” but holism in the mystical dimension of western philosophy and sociology emerged strongly when Baruch de Spinoza (1632-1677) developed a holistic philosophy in a way reminiscent of Parmenides (c. 515-450 BCE). Spinoza conceived that all the visible divisions and differences in the world are in fact aspects of an invisible single substance. He speculated that there is only one substance, “God, or Nature”, as nothing finite is self-subsistent. His holistic view proposed a pantheistic religious experience which was already being reflected in the mystical thinking of many religious traditions as “spiritualism.”

After Spinoza, George Wilhelm Friedrich Hegel (1770-1860), based on his holistic philosophy that nature consists of one timeless, rational and spiritual reality and state—reflected a mystical vision of the invisible unity underlying all visible objects. Hegel’s underlying invisible, unitive state is a quasi-mystical collectivism of an “invisible and higher reality.” The whole is identified by Hegel as the Absolute in a spiritual sense. All modern exponents of collectivism in the political and social sciences, including even Karl Marx (1818-1883), stress some higher collective reality—a unity, a whole, a group—though nearly always at the cost of minimizing the importance of the role of the part and the individual. Against individualism, they emphasize the social whole or social forces that somehow possess a character and a will which is greater than or over and above the characters and wills of the individual members summed up together. Thus, in the past hundred years, holism has tended to represent a collectivism and to sometimes be perceived as opposed to individualism.

 

In the second half of the twentieth century, the concept of holism began to inspire a broader thinking that the wholes, whether in biological organism, medicine, science, art, individual behavior, philosophy of language, cultures, etc., are much more than the sum of their parts. In the philosophy of history and social science, holism asserts that the objects of social inquiry are collectives rather than individual actions. In Gestalt psychology, it sets the focus on “Gestalt”—an organized whole that is perceived as more than the sum of its parts—not on isolated or separate elements. In philosophy of biology, holism opposes mechanism and vitalism, maintaining that life consists in the dynamic system of the organism. In the realm of physics, the holistic concept is reflected in the modern quantum field theory that describes all existence as an exhortation of the underlying quantum vacuum, as though all existing entities are like ripples on a universal pond—a very modern theory yet remarkably similar to a very ancient Indian theory that likens all entities to waves forming and un-forming on the surface of a vast and deep ocean.

 

Holism and Islam

            Hundreds of years before, Spinoza (1632-1677) developed a holistic philosophy or Hegel (1770-1831) conceived a mystical vision of the unity of all things, or modern thinkers like Karl Marx (1818-1883) could propound a sociopolitical collectivism, a sophisticated and remarkable holistic development had already occurred in the Islamic world of eighth century CE when holism emerged in the mystical branch of Islamic tradition, identifying a mystic of a certain high level of consciousness as a “perfect” Qutb, or a whole human being. It is important to add here that later on the famous philosopher of the eighteenth century, Emmanuel Kant (1724-1804)—though not directly from the mystics of Islam—defined this concept of mystical unity as a “transcendental unity of perception or self-consciousness.” Muslim thinkers believed that it is human spirit or soul whose windows can open to embrace all directions in unlimited dimension and their contents.  Subsequently, more than a few renowned mystics, men and women both, appeared in the culture of Islam believing and preaching pantheism or in the modern sense holism. Pantheism maintains that everything is divine, that God and Nature are identical. The Islamic mystics expressed pantheism within their belief of wahdat al-wujud, “the Unity of Being”, a concept tinged with metaphysics and a philosophical way of putting the same simple idea. The Arabic term describes the doctrine of pantheism the easy way, that all possible views about the Ultimate Reality can be termed as ‘pantheistic’ if they are focused exclusively on the Unity of Ultimate Reality, whatever its nature may be. According to Khalifa Abdul Hakim’s views in The Metaphysics of Rumi, “Even most of the evidently atheistic doctrines can be identified with it, to justify the witty remark of Schopenhauer that ‘Pantheism is the poetry of Atheism.’ Ethical Monism like that of Fichte or Panlogism like that of Hegel, the One Substance doctrine of Spinoza with a number of others in so far as they are monistic are pantheistic.” (Hakim, 2006, p. 148)

Muslim mystics were both experiencing and unfolding a mysterious and invisible factor (called later “holism”) that is enfolded but “hidden” within the fundamental synthetic tendency of the universe. Long sought by philosophers, mystics, and scientists, this factor brings an evolutionary leap in consciousness—a process with a phenomenal result we are describing as holistically human. Whereas a study of the ordinary outer mind, thought to be conscious, highlights reason and projects the power of a human being as an individual, the study of the inner or unconscious mind reflects the importance of genuine passion and reflects the power of a relationship between human beings, an invisible bond that yearns for and brings contact, connection, harmony, and wholeness. For the mystics, the “unconscious” is innate, emotional, and sensitive, is capable of perceiving and creating brilliantly. It is the unconscious mind that wants to reach out, aspires to love, and to commune with fellow human beings, emphasizing a feature of inner or unconscious mind that the learned are as one soul; in particular, the oneness of all the monotheistic prophets that cannot be broken up into fragments. If one disbelieves in one of the prophets, one’s faith in any other prophet is fractured.

Interpretation and development of a rational, Hellenistic-style philosophy in Islam had reached its highest point in the period between al-Kindi (801-873) and Averroes or Ibn Rushd (1126-1198). Now in the twelfth century and in reaction to Neoplatonism, the renowned Islamic theologians, most prominent amongst them al-Ghazali (1058-1111) interpreted that religion cannot be reconciled with philosophy. Mystic thought and life had also experienced a long and sustained tradition from the first known ascetic in Islam, Abu Hashim (d. 767 CE) of Syria to whom the word Sufi was applied, to Sanai (c.1044-1150) and Farid-ud-din Attar (c.1120-1193) of Persia and to Mohi-ud-Din ibn Arabi (1165-1240) of Arab Spain, a contemporary of Jalaluddin Rumi (1207-1273). Thus Rumi, appearing at a relative high point in the development and perfection of philosophical thought and religious experience in Islam, was able to inherit an exceptional intellectual and spiritual wealth. He had the theoretical influences of Greek philosophical interpretation, Jewish and Christian religious life, and Islamic jurisprudence on the one hand and the influences of Persian and Indian traditions on the other. Rumi, an orthodox Muslim, was guided and inspired by his mentor Shams-i-Tabriz who said: “The universe exists through the whole, not parts—and the whole universe is within one human being. When he knows himself, he knows all.” (Shams-i-Tabriz, Maqalat) Thus the concept of holistic humanism evolved to embrace all human beings as one “whole humankind.”

Rumi benefiting from his predecessors and then followed by many Muslim scholars and mystics, viewed the human being as the sum of several different, but interacting, energies within one body, namely, the physical, intellectual, emotional, and spiritual components. These entities or systems of energy mutually interact. The potential exists for both alignment and misalignment, for right or wrong relation among these energies. In ordinary daily activity, these energies are usually pulled in different directions. But, at moments, without notice, they can align—and there is a new level of awareness and consciousness that is substantially greater than the sum of its previously separated or conflicting parts. For as long as it lasts, this wholeness brings a much greater sense of unity and oneness, an awareness of itself with a new integrity, and a new relation to its inner and outer environments. An ordinary person, living with at least some conflicting thoughts, feelings, and impulses etc., inside him or her is unable to imagine how it will be when these are reconciled or resolved, what it will be like when “the conflicting desires and voices” (within) vanish and there is a wholeness—one intention, a unity reflecting as a whole. In many, but not in all cases, when this previously unimaginable wholeness occurs the individual may be aware of a spiritual component or presence and recognize it as such. This paradigm of holism within an individual (who experiences even a flash of holism) is a clue of what may be possible for the larger whole of humankind. The universal “holistic humanism” for the species appears dependent on the same paradigm as a single human being seen as a whole and bearing a common strand of spiritual harmony, unity and love.

 

Secular Humanism and Holism

Historically, humanism in its philosophical characteristic projected by the human being’s conscious mind or reason, has been associated with two main groups. One is the 15th century “Italian humanists” who were concerned with art and literature; the other is the present-day humanists who have a secular outlook. Neither group, at least until now, has defined humanism within the context of a human being’s inner or unconscious mind which is the source of an invisible bond between the people seeking harmony, connection, and love. Rather, the key attribute of secular humanism is to project the power of an individual human being pursuing worldly status, material gains or social contribution. Though secularism has a role in theist’s realm, it is not typically secularist. His holistic outlook is neither relative nor confined to one period’s art and literature nor to the present time’s secularist view. For him the core of human beings is the mind, soul or spirit that projects empowerment and self-actualization a, that in right relation produce harmony within an individual and within humanity. The ability of the human being to think and feel in all aspects, “material” and “spiritual”—to reason and intuit, to love, hate and fear, to receive and sense through “senses” and to perceive and believe beyond matter and form—is a vital element in the holistic essence of any humanism.

 

Present day humanists pose serious questions about the validity of religious traditions. Some reject religion outright, arguing that religions are intolerant of each other and thus create, conflict, violence and war. Yet history does not support the thesis that secularists have done less harm or damage to mankind with their conflicts, wars and killings. Religion has often espoused moral behavior by and toward those within the group and have been a source of motivation to individuals to bind together into a society. The theists endorse the faith instinct, believing that it is hardwired into human nature. The prophets and sages who introduced religions are like different windows through which one light enters, but is reflected differently to accommodate a diverse humankind. The consequences depend on the capacity of the receivers, how they accept this light and/or manipulate it.

 

After the two World Wars of the twentieth century, a new dimension of humanism among philosophers, theologians, and men of sciences began to emerge as “Renaissance of Holistic Humanism.” This view emphasizes that since human beings are holistically body, mind, and spirit, their appreciation must not be based only on matter and form. This form of humanism prioritizes our common human needs and seeks both rational and spiritual ways of solving out problems as physical, intellectual, emotional, and spiritual beings. Though many atheists and agnostics of the twentieth century humanism give primacy to humans in contrast to an ideology or a religion, they nevertheless do profess faith in the human beings’ capacity to evolve further in the realm of reason and love, and in their ability to grow toward whatever they potentially are. For the religious people, reason and science have their limitations while the human imagination, being naturally a religious imagination, is intrinsically drawn to spiritualism—irrespective of the faith or ideology one follows. For them, the spirit is the real substance and the phenomenal world of intellectual physical properties is a collection of its attributes.

Philosophically, an understanding of Al-Ghazali, Ibn Arabi and Rumi’s works, allows the premise that the holistic human view of these great thinkers may have been the forerunners of modern trends towards spiritual pluralism, voluntarism, and activism. The impact of Muslim philosophers’ translations and commentaries of Greek philosophers and on Latin philosophical thought was such that Western thought between the thirteenth and sixteenth centuries is inexplicable without considering the conceptual discourses of Muslim thinkers that there are different routes to the same truth. This served the modern foundation for theoretical openness, political freedom, and religious tolerance in Western thought. Muslim thinkers’ thought helped shape philosophy in the post-Kantian period of Goethe and Spinoza, and is related with the cognitions of Nietzsche, Schopenhauer, Bergson, Iqbal and many other thinkers of east and west. One salient aspect in especially Rumi’s work is his personal experiences in unraveling the religious problems that surround questions of free will, ego, and resurrection. His theory of emergent evolution and creative development and his emphasis on “intuition” and “love” (as opposed to barren intellectualism) converge into a supreme philosophy that an individual’s “self” is isolated, indeterminate, indistinct and featureless unless and until he can incorporate himself in the natural and social holism of humanity.

Mirza Iqbal Ashraf

February 17, 2012

 

A Ghazal by Ghalib – Shared by Mirza I Ashraf

GHAZAL OF GHALIB

جنوں تہمت کشے تسکیں نہ ہو گر شادمانی کی
نمک پاشِ خراشِ دل ہے لذت زندگانی کی

کشاکش ہائے ہستی سے کرے کیا سعیِ آزادی
ہوئی زنجیر موجِ آب کو فرصت روانی کی

پس از مُردن بھی دیوانہ زیارت گاہِ طفلاں ہے
شرارِ سنگ نے تربت پہ میری گُل فشانی کی

غالب