Step 1: Understanding the Concept:
The question is based on the account of the Chinese Buddhist monk and traveler Xuanzang (Hsüan-tsang), who visited India in the 7th century CE. It asks to identify which of the given options is NOT one of the "five sciences" or "five vidyās" that formed the curriculum of Indian education at the time.
Step 2: Detailed Explanation:
Xuanzang described the system of education in India, which was based on the mastery of five broad fields of knowledge, known as the Pañca-vidyā. These five were:
1. Śabda-vidyā: The science of sound/words, i.e., grammar and lexicography. This matches option (1).
2. Śilpasthāna-vidyā: The science of arts, crafts, mechanics, and architecture. This matches option (2).
3. Cikitsā-vidyā: The science of medicine.
4. Hetu-vidyā: The science of cause/reasoning, i.e., logic and epistemology.
5. Adhyātma-vidyā: The 'inner science', i.e., philosophy, metaphysics, and spiritual knowledge. This matches option (4).
Now let's examine the option that is not on this list:
3. Ahetuvidyā: The prefix 'a-' in Sanskrit means 'not' or 'without'. So, \textit{Ahetu} means 'without a valid reason' or 'fallacy'. Ahetuvidyā would thus mean the 'science of illogical reasoning' or 'sophistry'. This is the opposite of Hetu-vidyā (the science of logic) and was certainly not part of the formal curriculum.
Step 3: Final Answer:
Ahetuvidyā is not one of the five vidyās described by Xuanzang. Therefore, option (3) is the correct answer.