Prabhupāda: Transcendental means it is not in my experience, but I get the experience from higher authority, paramparā.
Śyāmasundara: I think his definition of transcendental is slightly different.
Prabhupāda: Transcendental means beyond your sense experience. That is the real meaning. You can see the dictionary. Transcendental is that which transcends.
Śyāmasundara: "Transcendental: of an a priori character, not based on experience; intuitively accepted; innate in the mind; superrational; supernatural; consisting of or dealing in or inspired by abstractions.' The way he is using "transcendental" is simply he is trying to understand knowledge through abstraction, by abstracting.
Prabhupāda: Transcendental knowledge means knowledge received from a source which is beyond the reach of my material senses. That is transcendental. Just like we are reading Bhagavad-gītā. So we have no knowledge that there is a spiritual world, but Kṛṣṇa says that there is another nature, a spiritual nature, beyond this material nature. So we understand through the source of transcendental knowledge. We cannot experience. That is explained, ataḥ śrī-kṛṣṇa-nāmādi. God, His name, His qualities, His pastimes—nothing can be understood by these material senses. But if you engage yourself in service, they become revealed. That will become confirmed: "Yes, there is Vaikuṇṭha, there is Vṛndāvana, where Kṛṣṇa's pastimes are going on, and I am perceiving myself." These things become revealed gradually, not abruptly you can understand. Therefore common men cannot understand that they say " 'Going back to home, back to Godhead?' What nonsense they are saying?" They cannot understand, because it is transcendental, beyond the reach of these gross senses. But it is revealed: sevonmukhe. If you become submissive, if you engage yourself in the service of the Lord, guru-Kṛṣṇa, and the spiritual master, then these things become revealed. Now one who has got the knowledge by revelation, nobody can mislead him. Just like we believe in the transcendental abode, cintāmaṇi, Goloka Vṛndāvana. If somebody pays out millions of dollars and asks you to forget all these things, we cannot do that. If you give him hundreds and thousands of dollars, that "You believe in this," no, he will not believe. That is transcendental knowledge. So transcendental knowledge is not speculation. It is receiving from higher authority and gradually, by your service attitude, things become clear to you. That is transcendental.