Udayana also known as 'Udayanācārya' (Udyanacharya, or "Master Udayana) lived in the 10th century, in Kariyan village in Mithila, near present day Darbhanga, Bihar state, India. Udayana was a very important Hindu logician who attempted to reconcile the views held by the two major schools of logic (Nyaya and Vaisheshika). This became the root of the Navya-Nyāya school of the 13th century, established by Gangesha Upadhyaya (“New Nyāya”) school of “right” reasoning, which is still recognized and followed in some regions of India.
Udayana wrote a sub-gloss on Vachaspati's work called the Nyaya-vaartika-taatparya-tiikaa-parishuddhi. He wrote several other works such as the Kusumanjali, Atma-tattva-viveka, Kiranaavali and Nyaya-parishishhta (also called Bodha siddhi or Bodha shuddhi).
UdayanAchArya (a "tArkika-n") and Kumarilabhatta (a "mImAmsaka-n"). To the "mImAmsakA-s" the Buddhist's summary rejection of Vedic ritualism was the proverbial red rag waved under the nose of a raging bull !Kumarilabhatta, it can be seen, has written copiously criticising the Buddhist's distaste for Vedic ritualism। He and UdayanAchArya were chiefly responsible for the failure of Buddhism to acquire a large following in the country(Scholars mention here the texts of "tarkapAdam" of Kumarilabhatta and the "bauddhadhikAram" by UdayanAchAryA).
Udayanaachazrya was a Hindu philosopher and Nyaya-vaartika scholar from Mithila (Now North Bihar). Little is known about his biography, but he is famous for many of his seminal theses on Nyaya-vaartik and debate on buddhism and Vedic rituals. His birthplace is a village Kariyan in the modern Samastipur district of Bihar.
Two schools of thought for logical proof for the existence of God in Hindu philosophy , 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. Living in a period of lively controversy with the Buddhists, Udayana defended his belief in a personal God 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 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.
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:
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. According to Kisor Kumar Chakrabarti:[1]
In the third part we have shown how the study of the so-called 'restrictive conditions for universals' in Navya-Nyaya logic anticipated some of the developments of modern set theory. [..] In this section the discussion will center around some of the 'restrictive conditions for universals (jatibadhaka) proposed by Udayana. [..] Another restrictive condition is anavastha or vicious infinite regress. According to this restrictive condition, no universal (jati) can be admitted to exist, the admission of which would lead to a vicious infinite regress. As an example Udayana says that there can be no universal of which every universal is a member; for if we had any such universal, then, by hypothesis, we have got a given totality of all universals that exist and all of them belong to this big universal. But this universal is itself a universal and hence (since it cannot be a member of itself, because in Udayana's view no universal can be a member of itself) this universal too along with other universals must belong to a bigger universal and so on ad infinitum. What Udayana says here has interesting analogues in modern set theory in which it is held that a set of all sets (i.e., a set to which every set belongs) does not exist.