Are there any arguments provided by Sankhya school to hold Prakriti (Mula Prakriti or Pradhana), and not other things like atoms or consciousness, as the root cause of the universe? Maybe somewhere in their debates with other schools such as the Vaisesikas?
No. Avyakta (unmanifest) is root cause of all in Samkhya Philosophy, which itself has no cause (अहेतुमत् Ahetumat). This is explained well in Samkhya karika of Iswarakrishna.
Karika 10 says (translated by Har Dutt Sharma):
हैतुमदनित्यमव्यापि सक्तियमनकमाश्रितं लिङ्गम् ।
सावयवं परतन्त्र' व्यक्त', विपरीतमव्यक्तम् ॥ १० ॥
haitumadanityamavyāpi saktiyamanakamāśritaṃ liṅgam ।
sāvayavaṃ paratantra' vyakta', viparītamavyaktam ॥ 10 ॥
The Vyakta (Manifest) is caused, non-eternal, non-pervading, active, manifold, dependent, mergent, conjunct and subordinate. The Avyakta (Unmanifest) is just the reverse.
Karika 15 explains reasons why Avyakta is root cause of all in much more detail.
भेदान परिमाणात् समन्वयात् शक्तितः प्रवृत्तेश्च ।।
कारणकार्यविभागादविभागाई वरूप्यस्य ॥ १५ ॥
bhedāna parimāṇāt samanvayāt śaktitaḥ pravṛtteśca ।।
kāraṇakāryavibhāgādavibhāgāī varūpyasya ॥ 15 ॥
(The Unmanifest cause does exist), because of the finiteness of the specific objects, because of natural sequence, because of activity depending upon efficiency, because of distinction between cause and effect and because of the merging of this diverse (evolved).
Commentary of Gaudapadacharya:
The Unmanifest cause exists— this is the relation of government of subject and predicate in the sentence. Because of the finiteness of the specific objects. In this world, wherever, we find an agent, we see a finiteness of his. For example, a potter makes only finite jars from finite lumps of clay; so does Mahat also. The mergent Mahat and • the rest are finite and are the specific effects of the Nature. Intellect is one, ego is one, the subtle elements are five, the organs are eleven and the gross elements are five. Thus, on account of the finiteness of the specific objects, there is Nature as the cause which produces the If there were no Nature, then even this finite Manifest.
Manifest would have been infinite. And so, on account of the finiteness of the specific objects, there is Nature, wherefrom this Manifest has sprung up. Again, because of natural sequence. This is quite well known in this world that when one sees a boy engaged in performing sacred ri les, one infers that his parents are naturally brahmins. Similarly, seeing this mergent (i.e. the evolved), we arrive at a thing which must be its cause. Thus, by natural sequence there is Nature.
Again, because of activity depending upon efficiency. Here, a man does that for which he is efficient. For example, a potter who is efficient to produce a jar, produces only a jar and not cloth or chariot.
Again, there is Nature as the cause. How? — Because of distinction between cause and effect. Karana is that which produces: kärya is that which is produced. (There is) a distinction (of functions) of cause and effect. For example, as a jar is competent to hold curds, honey, water and milk, so is not a lump of clay (competent). Or, a lump of clay produces a jar, but a jar does not pi-oduce lump of clay. Thus, seeing the mergent Mahat and the rest, it is inferred that there is a separate cause from which this Manifest has separated ( i.e., evolved) itself.
And again, because of the merging of the diverse (evolved). Visva means universe; its rüpa is manifestation. The abstraction of visvarüpa is vais'varüpya (i.e., manifested or evolved ); on account of its merging there is Nature; because there is no mutual distinction between the three worlds anthe five gross e., the three worlds are included in the five gross elements. At the time of dissolution, the five gross elements, viz., earth, water, fire, air and ether, merge into the modified five subtle elements in the order of creation; the five subtle elements and the eleven organs (merge) into ego; ego (merges) into intellect; intellect (merges) into Nature. Thus, the three worlds merge into Nature at the time of dissolution. From such merging of the Manifest and the Unmanifest, like that of milk and curds, there does exist the Unmanifest as the cause.
I suggest you to read Karikas from 10 (or even from beginning) to understand it in detail.
Translation of Sammkhya karika by Radhanath Phukan is also good.
What @The Destroyed has said in his answer is true but what you have said is also true.
Mula Prakriti is the root cause in SAmkhya. And. it is the highest cause of everything else, no other higher causes exist. And, this Mula Prakriti is also known by the names of Avyakta and PradhAna.
To understand this we need to know what the 25 Tattvas are in SAmkhya and also how creation is described to occur in that doctrine.
I am quoting all verses from the SAmkhya KArikA.
25 Tattvas of SAmkhya Darshana
The Tattvas are described in KArikA 3:
Mulaprakritir vikriti mahadAdyAh prakritivikritayah sapta |
Shodashakantu vikAro na prakirtina vikritih purushah || 3
Mula Prakti (which is also called the PradhAna) is Avikirti (the only non-distorted or non-transformed Tattva besides Purusha); The seven [Tattvas], starting from Mahat, are Prakriti-Vikriti ( i.e a mixture of non-distorted and distorted both); The 11 Indriyas and the 5 MahAbhutas (the 5 elements; Prithvi etc)- these 16 are Vikriti; And Purusha is neither Prakriti nor Vikriti.
So, we have:
Mula Prakriti or PradhAna is not the effect (KArya or action) of another Tattva so it is Avikriti (non-transformed). It is the cause itself.
Mahat, AhamkAra, 5 Tanmatras ( smell, taste etc)- these 7 are Vikriti and Avikriti both. They are Avikriti ecause they are the cause of the other lower Tatvas but they are Vikriti too as they themselves are also the effect of some cause. For example, from Mahat, AhamkAra is born, so Mahat is Avikriti here, but since PradhAna (or Mula Prakriti) is the cause of Mahat so it is Vikriti also. Similarly for the other 6.
- Vikriti only
5 GyAna Indriyas (eyes, ears etc), 5 Karma Indriyas (hands, feet etc), mind, the 5 MahAbhutas- these 16 are all Vikritis. Because they are all effects of some other cause but they themselves are not the cause of any other Tattvas.
Now, if you count the number of Tattvas it is 25 (including the Purusha) and of which the highest is the Mula Prakriti. So, it is the root cause.
Process of creation in SAmkhya:
PrakritermahAmstatohahamkArastasmadganashcha shodashakah |
TasmAdapi shodashakAt panchabhyah panchabhutAni || 22
From Prakriti is born Mahat Tattva; from Mahat Tattva, Ahamkara is born; From AhamkAra are born the sixteen Ganas or the sixteen Vikritis (distortions or transformations); and from five among that sixteen the Pancha MahAbhutas are born.
This is the process of creation in SAmkhya. So, Prakriti or Mula Prakriti (which is also known as Avyakta or PradhAna) is the root cause in SAmkhya.
Now, coming to your question- "What reasons they give?"
They don't give any reasons actually. That is what (or how) the doctrine is.
In this doctrine the Purusha is inactive i.e non-doer. This is established in KArikA 19:
TasmAccha viparyasAt siddham sAkshitvamasya purushasya |
Kaivalyam mAdhashyam drashtritvam akatribhAvashcha ||
Due to Purusha's opposite-ness (as described in KArikA11), the Purusha's following attributes are established- SAkshitva, Kaivalya, MAdhyastha, Drashtritva and Akatritva.
SAkshitva- The Purusha in SAmkhya is not the doer, he is the witness of the play by Prakriti. He is just the onlooker (or DrashtA).
MAdhyastha- The Purusha disinterested.
Drashtritva- Already mentioned above.
Akatritva- Here Purusha does not act. He does not hold doership. Here Prakriti does everything selflessly for the Purusha.
But the creation manifests not only by Prakriti. It comes to existence only when Purusha unites with Prakriti. This is mentioned in KArikA 21:
Purushasya darsanartham kaivalyArtham tatha pradhAnasya |
pangandhavadubhayorapi samyogastatkritah sargah ||
For the Kaivalya (or liberation) of the Purusha and for the Darshana (or Bhoga or enjoyment) of the Prakriti, and from the coming together of (or the union of) the Purusha and the Prakriti, like that between the lame (which is the Purusha) and the blind (which is Prakriti), the whole creation comes into existence.
@SMJoe If your question is "Does it say why Prakriti is the root cause?" then no answers given.
If your question is "Why Prakriti is the cause and not any other like say Purusha?" then we can get the clue from the verse where Purusha's "Akatritva" is established.
But, still, if you note the 21st KArikA, then Pursha's involvement is also necessary for creation to come into existence.
They have compared Purusha with a person who can't walk and Prakriti with a person who can't see. Imagine both these persons trying to cross a road on their own--then neither of them will succeed. But they can do so if they help each other.
So, although, Purusha is mostly disinterested and inactive, his involvement is also necessary.
Now, if your question is "Did they give any argument how they concluded that a root cause (which is PradhAna or Avykata Prakriti) must exist?" then they have given an answer to it in one KArikA.
So, in short this is the kind of question where the answer is a NO.