Introduction

Udayana also known as 'Udayanācārya' (Udyanacharya, or "Master Udayana) lived in the 10th century, in Kariyan

Kariyan

Kariyan is village and a panchayat in Samastipur district in the Indian state of Bihar.- Location :About 16-km northeast of the Rosera ghat railway station of Northeastern Railways the modern village of Karian is situated on a mound, 20-ft high from the surrounding ground level and about 96 acres...

 village in Mithila

Mithila

Mithila was a city in Ancient India, the capital of the Videha Kingdom. The name Mithila is also commonly used to refer to the Videha Kingdom itself, as well as to the modern-day territories that fall within the ancient boundaries of Videha...

, near present day Darbhanga

Darbhanga

Darbhangā is a twin city and a municipal corporation and the capital city of the Darbhanga district and Darbhanga Division in the state of Bihar, India. It is one of the most important districts of North Bihar situated in the very heart of Mithilanchal. According to the latest 2011 census, the...

, Bihar

Bihar

Bihar is a state in eastern India. It is the 12th largest state in terms of geographical size at and 3rd largest by population. Almost 58% of Biharis are below the age of 25, which is the highest proportion in India....

 state, India

India

India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...

. Udayana was a very important Hindu logician

Indian logic

The development of Indian logic dates back to the anviksiki of Medhatithi Gautama the Sanskrit grammar rules of Pāṇini ; the Vaisheshika school's analysis of atomism ; the analysis of inference by Gotama , founder of the Nyaya school of Hindu philosophy; and the tetralemma of Nagarjuna...

 who attempted to reconcile the views held by the two major schools of logic

Logic

In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

  (Nyaya

Nyaya

' is the name given to one of the six orthodox or astika schools of Hindu philosophy—specifically the school of logic...

 and Vaisheshika

Vaisheshika

Vaisheshika or ' is one of the six Hindu schools of philosophy of India. Historically, it has been closely associated with the Hindu school of logic, Nyaya....

). This became the root of the Navya-Nyāya

Navya-Nyaya

The Navya-Nyāya or Neo-Logical darśana of Indian logic and Indian philosophy was founded in the 13th century CE by the philosopher Gangeśa Upādhyāya of Mithila. It was a development of the classical Nyāya darśana. Other influences on Navya-Nyāya were the work of earlier philosophers Vācaspati...

 school of the 13th century, established by Gangesha Upadhyaya

Gangesha Upadhyaya

Gangesha Upadhyaya was an Indian mathematician and philosopher from the kingdom of Mithila. He established the Navya-Nyāya school. His Tattvacintāmaṇi also known as Pramāṇacintāmaṇi is the basic text for all later developments...

 (“New Nyāya

Nyaya

' is the name given to one of the six orthodox or astika schools of Hindu philosophy—specifically the school of logic...

”) school of “right” reasoning, which is still recognized and followed in some regions of India.

Philosophy

Two schools of thought for logical proof for the existence of God in Hindu philosophy

Hindu philosophy

Hindu philosophy is divided into six schools of thought, or , which accept the Vedas as supreme revealed scriptures. Three other schools do not accept the Vedas as authoritative...

 , the old Nyaya system was concerned with the critical examination of the objects of knowledge by means of logical proof, while the earlier Vaiseshika system dealt with particulars—objects that can be thought of and named. Udayana assumed, with the Vaiseshika, that the world was formed by atoms, from which physical bodies also derived. But he was equally concerned with the mind and its right apprehension of objects in nature. His vigorous thinking was set forth in the Nyāya-Kusumānjali and the Bauddhadhikkāra, the latter an attack on the atheistic thesis of Buddhism

Buddhism

Buddhism is a religion and philosophy encompassing a variety of traditions, beliefs and practices, largely based on teachings attributed to Siddhartha Gautama, commonly known as the Buddha . The Buddha lived and taught in the northeastern Indian subcontinent some time between the 6th and 4th...

. Living in a period of lively controversy with the Buddhists, Udayana defended his belief in a personal God

God

God is the English name given to a singular being in theistic and deistic religions who is either the sole deity in monotheism, or a single deity in polytheism....

 by resorting to the two natures of the world: cause and effect. The presence of the world is an effect that cannot be explained by the activity of atoms alone. A supreme being

Supreme Being

The term Supreme Being is often defined simply as "God", and it is used with this meaning by theologians of many religious faiths, including, but not limited to, Christianity, Islam, Hinduism, and Deism. However, the term can also refer to more complex or philosophical interpretations of the...

 had to cause the effect and regulate the activity of the atoms; hence, according to Udayana, God exists.
In a debate with Buddhists in India he was the final victor. After him no Buddhist philosopher undertook again a debate with Nyāya. Thus the nine-centuries long debate ended.

Nyayakusumanjali and Set theory

• Udyan tried to prove the existence of God on the basis of inference.

They made this question a challenge to their own existence.Udayana's
Nyayakusumanjali gave the following nine arguments to prove the existence of creative God

1) Kāryāt (lit. "from effect"): An effect is produced by a cause,
and similarly,the universe must also have a cause.
Causes (according to Naiyayikas) are of three kinds:
Samavayi (in case of the universe, the atoms),
Asamavayi (the association of atoms)
and Nimitta (which is Ishvara).
The active cause of the world must have an absolute knowledge of all the
material of creation, and hence it must be God. Hence from the creation,
the existence of the Creator is proved.
2) Āyojanāt (lit., from combination): Atoms are inactive and properties are unphysical.
So it must be God who creates the world with his will by causing the atoms to join.
Self-combination of inanimate and lifeless things is not possible,
otherwise atoms would only combine at random, creating chaos.
There is to be seen the hand of a wise organizer behind the systematic grouping of the
ultimate atoms into
dyads and molecules. That final organizer is God.
3)Dhŗtyādéḥ(lit., from support): Just as a material thing falls off without a support,
similarly, God is the supporter and bearer of this world,
without which the world would not have remained integrated.
This universe is hence superintended within God, which proves his existence.
4)Padāt (lit., from word): Every word has the capability to represent a certain object.
It is the will of God that a thing should be represented by a certain word. Similarly,
no knowledge can come to us of the different things here unless there is a source of this knowledge.
The origin of all knowledge should be omniscient and, consequently, omnipotent.
Such a being is not to be seen in this universe, and so it must be outside it. This being is God.
5) Pratyayataḥ (lit, from faith): the Hindu holy scriptures, the Vedas,
are regarded as the source of eternal knowledge.
Their knowledge is free from fallacies and are widely believed as a source of proof.
Their authors cannot be human beings because human knowledge is limited.
They cannot obtain knowledge of past, present, and future, and in depth knowledge of mind.
Hence, only God can be the creator of the Vedas. Hence, his existence is proved from his being the
author of the Vedas,which he revealed to various sages over a period of time.
6) Shrutéḥ (lit., from scriptures): The Shrutis, e.g., the Vedas extol God and talk about his existence.
"He is the lord of all subjects, omniscient, and knower of one's internal feelings;
He is the creator, cause and destroyer of the world", say the Shrutis. The Shrutis are regarded as a source of proofs by
Naiyanikas. Hence, the existence of God is proved.
7)Vākyāt (lit., from precepts): World is governed by moral laws that are objective and universal.
These are again manifested by Shrutis. Hence there exists God, the promulgator of these laws.
8)Samkhyāviśeşāt (lit., from the specialty of numbers):According to the Nyaya, the magnitude of a dyad is produced by the
number of two atoms. The number "one" is directly perceived but other numbers are created by perceptions,
which is related to the mind of the perceiver. Since at the time of creation, souls, atoms, Adŗşţa (Unseen Power), space,
timeand minds are all unconscious, hence it depends on divine consciousness. So God must exist.
9)Adŗşţāt (lit., from the unforeseen): Everybody reaps the fruits of his own actions.
merits and demerits accrue from his own actions and the stock of merit and demerit is known as Adŗşţa, the Unseen Power.
But since this unseen power is unintelligent, it needs the guidance from a supremely intelligent god.

Udayana also developed theories on "restrictive conditions for universals" and "infinite regress" that anticipated aspects of modern set theory

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

. According to Kisor Kumar Chakrabarti: