Vedanta schools extensively deal with the concept of brahman. What do non-vedanta schools like nyaya-vaishesika, samkhya-yoga etc say about shruti statements about Brahman? How do they reconcile their atomism or dualism with shruti statements?
I know of two non-Vedantic Astika schools which seriously engage with the statements in the Upanishads about Brahman:
As I discuss in this question Mimamsakas believed that the Upanishads provide knowledge of the Jivatma, not Paramatma, and that this knowledge serves a useful purpose in Yagnas. See this excerpt of Shabara's commentary on Jaimini's Purva Mimamsa Sutras for details, but suffice it to say that when the Upanishads say things like "The Atman is neither the body, nor the mind, nor the senses.", Mimamsakas interpret that as a message to Yagna priests along the lines of "You know that there is a self within you which is neither the body, nor the mind, nor the senses. Similarly your king has a self within him which is neither the body, nor the mind, nor the senses. If you want to send the king's self to Swarga after he dies, then you should perform this Yagna for him."
But the Purva Mimamsa philosopher Kumarila Bhatta thought that Atmavidya or knowledge of one's own soul served three distinct purposes: it's useful for Yagnas (which I discuss above), it leads happiness in this life (which I discuss here), and it leads to Karma Yoga through which Moksha is attained (which I discuss here). Here's what he says in this excerpt from his Tantra Vartika:
As for the knowledge of Self; both by Conjunction and Disjunction it is found to help the sacrifice as well as the Person; because unless are knows his self (to be something other than the body that perishes) he would never undertake the sacrifices whose results are said to accrue to the man in another birth. And then again, such passages as - "the Self free from all evil...is to be sought after, &c.,” “one should worship the Self" -lay down the knowledge of Self as accomplished by a process accompanied by due reflection, &c.; and then from such knowledge, we find that there accrues to the agents both kinds of result - Happiness as well as final Deliverance, as is shown by the following passages:-"He obtains all worlds and all desires, passes beyond all sorrow, &c, &c.,"-which speaks of all the eight perfections of Yoga accruing to the person knowing the self; and the passage-“passing his life thus he, after death, reaches the regions of Brahma, and from there he never returns'--which points to the attainment of the Supreme Self (Final Deliverance) also as following from a due knowledge of the Self.
I should add that Kumarilaa Bhatta, unlike some other Mimamsakas, did believe in the existence of a supreme being. He didn't believe that this being was responsible for creating the world, which he saw as eternal and uncreated, or responsible for authoring the Vedas, which he saw as Apaurusheya or authorless, but he did see it as being the soul of the Vedas, as I discuss in my question my question here.
Kapila's Samkhya school believed that when the Upanishads speak of Brahman, they are really speaking about Prakriti, which the Samkhya school saw as the material cause of the Universe. Vyasa refutes the notion that Brahman refers to Prakriti in Adhyaya 1 Pada 1 of the Brahma Sutras (which you can read here):
- The Pradhana of the Samkhyas is not the cause of the universe, because it is not mentioned in the Upanishads, which fact is clear from the fact of seeing (or thinking).
- If it be argued that the seeing is in a secondary sense, we say, not so, owing to the use of the word Self.
- (Pradhana is not the meaning of the word “Self”), because liberation is promised for one who holds on to That.
- (Pradhana has not been spoken of even indirectly), because there is no subsequent mention of its rejection, and (because that militates against the assertion at the beginning).
- Because of the merger of the individual into his own Self.
- Because the knowledge (gathered from the various Upanishads) is the same (as regards Consciousness being the cause).
- And because (Brahman is) revealed (as such) in the Upanishads.
In any case, here is how Adi Shankaracharya summarizes the Samkhya school's view regarding this in this section of his Brahma Sutra Bhashya:
The Sânkhyas who opine that the non-intelligent pradhâna consisting of three constituent elements (guna) is the cause of the world argue as follows. The Vedânta-passages which you have declared to intimate that the all-knowing all-powerful Brahman is the cause of the world can be consistently interpreted also on the doctrine of the pradhâna being the general cause. Omnipotence (more literally: the possession of all powers) can be ascribed to the pradhâna in so far as it has all its effects for its objects. All-knowingness also can be ascribed to it, viz. in the following manner. What you think to be knowledge is in reality an attribute of the guna of Goodness 1, according to the Smriti passage 'from Goodness springs knowledge' (Bha. Gîtâ XIV, 17). By means of this attribute of Goodness, viz. knowledge, certain men endowed with organs which are effects (of the pradhâna) are known as all-knowing Yogins; for omniscience is acknowledged to be connected with the very highest degree of 'Goodness.' Now to the soul (purusha) which is isolated, destitute of effected organs, consisting of pure (undifferenced) intelligence it is quite impossible to ascribe either all-knowingness or limited knowledge; the pradhâna, on the other hand, because consisting of the three gunas, comprises also in its pradhâna state the element of Goodness which is the cause of all-knowingness. The Vedânta-passages therefore in a derived (figurative) sense ascribe all-knowingness to the pradhâna, although it is in itself non-intelligent. Moreover you (the Vedântin) also who assume an all-knowing Brahman can ascribe to it all-knowingness in so far only as that term means capacity for all knowledge. For Brahman cannot always be actually engaged in the cognition of everything; for from this there would follow the absolute permanency of his cognition, and this would involve a want of independence on Brahman's part with regard to the activity of knowing. And if you should propose to consider Brahman's cognition as non-permanent it would follow that with the cessation of the cognition Brahman itself would cease. Therefore all-knowingness is possible only in the sense of capacity for all knowledge. Moreover you assume that previously to the origination of the world Brahman is without any instruments of action. But without the body, the senses, &c. which are the instruments of knowledge, cognition cannot take place in any being. And further it must be noted that the pradhâna, as consisting of various elements, is capable of undergoing modifications, and may therefore act as a (material) cause like clay and other substances; while the uncompounded homogeneous Brahman is unable to do so.
And here is how Ramanujacharya summarizes it in this section of his Sri Bhashya:
The Pûrvapakshin maintains that the Pradhâna is meant. For he says, the Khândogya text quoted expresses the causal state of what is denoted by the word 'this', viz. the aggregate of things comprising manifold effects, such as ether. &c., consisting of the three elements of Goodness, Passion and Darkness, and forming the sphere of fruition of intelligent beings. By the 'effected' state we understand the assuming, on the part of the causal substance, of a different condition; whatever therefore constitutes the essential nature of a thing in its effected state the same constitutes its essential nature in the causal state also. Now the effect, in our case, is made up of the three elements Goodness, Passion and Darkness; hence the cause is the Pradhâna which consists in an equipoise of those three elements. And as in this Pradhâna all distinctions are merged, so that it is pure Being, the Khândogya text refers to it as 'Being, one only, without a second.' This establishes the non-difference of effect and cause, and in this way the promise that through the knowledge of one thing all things are to be known admits of being fulfilled. Otherwise, moreover, there would be no analogy between the instance of the lump of clay and the things made of it, and the matter to be illustrated thereby. The texts speaking of the origination of the world therefore intimate the Pradhâna taught by the great Sage Kapila. And as the Khândogya passage has, owing to the presence of an initial statement (pratigñâ) and a proving instance, the form of an inference, the term 'Being' means just that which rests on inference, viz. the Pradhâna.