According to the Catholic Church, is there any supernatural (non-physical, non-mathematical, non-scientific) part of a human being? My previous understanding of the Catholic Church's teaching was that a human being is of two conjoined natures: physical and spiritual. Also, my previous reasoning was that things like consciousness, the ability to feel and especially free will are impossible in the natural/physical/mathematical world, where all that is possible are non-conscious, non-free, more or less sophisticated mechanisms. Therefore, I would say, free will must have supernatural origins and the claim that a human being has free will implies that there must be a supernaturnatural, spiritual, non-mathematical, non-physical, non-scientific part of the human nature, that cannot be researched nor explained by scientific means.

However, now I am being told that what I was thinking and saying were all heresies. I am being told that according to the teaching of the Catholic Church there is no supernatural or spiritual part nor dimension in the human being or human soul. The only fundamental difference between humans and animals, according to the Church, is that God has gifted humans with His grace to a greater extend than animals, and because of this God's favor humans are granted immortality - but otherwise, humans are really simply more intelligent animals and nothng more. Therefore free will, if existent, is absolutely within the scope of science and probably will be fully explained by science.

Some people I was talking to were telling me that was I was believing were non-Christian views of Plato, and that I was guilty of believing and spreading the "heresy of Platonic dualism", since the Church has rejected the views of Plato and instead adopted the views of Aristotle, as described above, through St. Thomas. Other people were telling me that while my views were not yet heretical, because these views, since adopted by St. Augustine (who had been inspired by Plato), did have their place in the Church, nevertheless they were unfavorable, since the majority of theologians were instead holding the views of St. Thomas, who had been inspired by Aristotle.

The view that free will is a purely natural phenomenon that does not and can not have supernatural origins, since there is no such part nor dimension in the human being is, I have to admit, absolutely counter-intuitive for me and is absolutely opposite to how I used to understand the teaching of the Church. Therefore, could you explain to me, according to the teaching of the Church:


Is there no non-natural, non-scientific, non-physical, non-mathematical part or dimension in the human nature?
If so, does free will therefore have purely natural origins that are not fundamentally prohibited from being researched and fully explained by science?


If I may ask, could you kindly back your answers by appropriate citations of the teaching of the Catholic Church?

Disclaimer: Just to keep my conscience clear: This is a summation of views 3 different persons, out of them I know one to have academic background in theology and philosophy and have reasons to suspect one more of such background; these three peoples' views did clash on certain subtleties, however I do not feel competent to state them; finally I am likely to have misunderstood and misinterpreted certain subtleties. The big picture stands, however, for these reasons I cannot 100% say that every word I wrote is an accurate representation of what I was told to be the teaching of the Church.
The answer can be found here in the Catechism of the Catholic Church; in particular, it says: 


  366 The Church teaches that every spiritual soul is created immediately by God - it is not "produced" by the parents - and also that it is immortal: it does not perish when it separates from the body at death, and it will be reunited with the body at the final Resurrection.


This paragraph makes pretty clear that the Catholic Church says there is a part of man that is not “tangible” or “mathematical”.  

The other paragraphs in this sections explain it a little better and I think can provide an answer to your other doubts.