Dear Great sophist Marcus,
The Buddha knows all things science, but science doesn't know all knowledge of the Buddha. Like a satellite in space can zoom into any landscape of the Earth, but a traveler tracking the Himalaya can only observe the surrounding with his eyes, or his drone with monitor; or by checking the compass in reference to the map to confirm his location. Science is in discovering new unknowns everyday. Buddha has known all, is the all, science is a set within the all. This is important to bear in mind. Aside from this, your questions are answerable. Here is my perspective, I don't expect all to agree, nor do I get the best verbal explanation for the ineffable.
But how about the soul?
In Buddhism, the soul should be understood as consciousness. It is consciousness that animates matter. It is consciousness that gives rise the notion of self. However, in contrast, Buddhism discerns the soul - consciousness, impermanent. Hence it cannot be the self. Therefore, different from Vedas, there is not any eternal, unchanging self, nor an undying soul existed.
What is the proof that consciousness is impermanent? Or,
What proof does Buddha have for soul?
If consciousness is permanent: a) you will never forget anything; b) you will not at one moment feel happy then the next moment unhappy; c) information procured by the eyes, ears and nose etc. shouldn't be in different format. Etc.
Therefore, consciousness - the soul, is impermanent. It is a momentarily continuum, like a river. If a thing is impermanent and cannot be controlled, it is not the self. Like you can't stop forgetting, or stop hearing.
What about reincarnation? What are the proofs for it?
Reincarnation is an autonomous, self-induced birth. The entity reincarnated is a clone of the self, in Vedas called avatar. But in Buddhism, birth is not autonomous, self-induced. Often it translates as rebirth. A rebirth takes place when the matter the consciousness animated deteriorated, i.e., the death of a physical body. As the deteriorated matter no longer fits for animation, the consciousness will animate an associated matter. Like the electricity powers the microwave, when broken, you can plunge in an oven to replace it.
Rebirth will have to take place because consciousness cannot be destroyed, just like electricity cannot be destroyed. You can destroy the circuit, but cannot destroy electricity.
What about the Maitri?
Yes, compassion is an "energy" field. Thought, which is conscious consciousness, is energy. The proof is that there is headgear invented (the best and lightest is a Taiwanese made) which amplifies the signal generated by the mind to switch on/off lights (certain brain-wave), without physical contact. I don't think this the perfect solution with proper understanding of how the mind works, but a good try. Normal human's consciousness is scattered, it jumps here and there incessantly. An accomplished meditator can have more condensed, focused consciousness. That's why the Buddhist training on Dhyana. Such training also exists in ancient India and China, though not exact the same as the Buddha's method.
Also what about Karma?
Karma will have to take place too. Because the mind doesn't bind by space-time, but space-time is created by the mind. This is answered in a sublime and profound teaching of the Buddha, that preserved only in the Chinese Tripitaka:
「...隨眾生心, 應所知量, 循業發現。」 ── 《楞嚴經》
...yields to the mind of the sentient, responding to its capability of quantification, manifesting according to its Karma. -- Surangama Sutra
Quantification - quantum, long ago has been mentioned in Buddha Sutra.
This good and bad is cached in the mind.
From my limited and amateur knowledge of quantum particles, a) that could exist in multiple places at once, b) and they correlate with each other instantly despite astronomical distance. These quantum attributes are recognized based on assuming space-time a constant and primary factor. But in Buddhism, the Chinese Mahayana understanding, space-time is a structure we projected to the phenomenal world. It is also a framework where the tangible consciousness operates. However, in the intangible, consciousness never arises nor ceases, space-time doesn't exist, it is one.
When a bad deed is carried out, there is the offender and victim. In order to fulfill the roles, the intangible consciousness materialized into the offender and victim in space-time. Here we have sentient A and B, like the quantum particles appear in different places at once, or correlating. In the ultimate, both the experiences of the offender and victim are lived. However, in space-time, our world, these experiences will be lived consecutively. Hence the old idiom says, "you reap what you sow", a wisdom procured by observation and intuition.
What about Amitabha?
Amitabha is the Light. Consciousness its material manifestation is light. An accomplished meditator can directly see it in his meditation. (That's all I should say, for an esoteric knowledge.)
Because there's no scientific/cognitive proofs of the aforementioned
elements given our sensorimotor apparatus.
I don't think consciousness can be measured by instrumental apparatus. We can measure brain-wave, but brain-wave is a manifestation of consciousness. Like space cannot be measured by sensorimotor apparatus, can scientist deny the existence of space? In the same way, accepting the existence of space by the scientist then is anything but blind belief, which should be dropped too.
Or suggest me some books?
If you want to have the purest understanding of Buddhist doctrines, you should only read the Sutras, as the beginning.
Unfortunately its unlikely you can read Chinese Sutras, which were translated by the most extraordinary and accomplished Indian and Chinese historical Buddhist masters who proficient in both Sanskrit and Chinese. If you read Pali Suttas, which have approx. 1/10 of the complete teaching of the Buddha, the best and most sincere translator is Bhikkhu Thanissaro. However, from my reading, probably Pali Suttas can't satisfactorily satisfy the mind of a scientist and philosopher.
Of import, I would advise to get away from those:
- books written by modern Buddhist-scholars on introduction to Buddhism, full of misinterpretations; also avoid
- books written by those Rinpoches/Geshes who are not scientists neither philosophers nor psychologists but trying to interpret with a distorted scientific, philosophic or psychologist's tone; keep from
- books written by scholars on Nagarjuna, Madhyamaka, Two Truths, Middle Way, Conventional vs Ultimate Truth, or (Vedanta) non-duality. These topics can only be correctly understood by a Buddhist master who proficient in Buddhist doctrines, with direct insight acquired in Samadhi. Scholars without such are just doing the mind-games or playing with words. It only adds pollution.
Mine is an unconventional approach, but closest to the Buddhist hereditary. May it delight you.