Mathematics in Engineering from Ancient Civilizations to Complex Artificial Intelligence

1. Introduction:
Man is a social animal that needs to understand how things came to be and why. He also solved some of the practical problems, built or engineered constructions to help him in his quest for deeper understanding of astronomy and the origins of his existence. This is shown in the ancient Greek philosophies and the ancient Egyptian civilization. Archimedes of Syracuse was a Greek mathematician, engineer, and astronomer. Pythagoras of Samos discovered the Pythagorian theorem in geometry (the square of the hypotenus in a right-angled triangle is equal to the sum of squares of the two other sides). See this YouTube presentation on the Pythagorian theorem here: https://www.youtube.com/watch?v=AA6RfgP-AHU
When I was 14 years-old I developed a little theorem in geometry - If you divide any two angles in any triangle, the angle that results from the two lines coming from the angles divided must be greater than a right angle.

The ancient Egyptians built the pyramids with stones whose sizes and the way they were put together still amazes archaeologists and engineers today. On YouTube a little presentation shows how the Great Pyramid in Giza shows mathematical symmetry and alleged "secrets": https://www.youtube.com/watch?v=N7o0rf6o-z0

Renewal of the Self and of the Church

Those who do not pass from the experience of the cross to the truth of the resurrection condemn themselves to despair! For we cannot encounter God without first crucifying our narrow notions of a god who reflects only our own understanding of omnipotence and power

Pope Francis

Indeed imitating Christ's life, teachings, and acceptance of his cross is the only way in which we will be raised to eternal life with him...

I think that the renewal of the Church must start with Rome. When St. Athanasius was exiled to Rome in the 4th century in his battle with the Arians and Semi-Arians, he wrote a book on St. Anthony the Great, and the Desert Fathers in Egypt. Following him, Pope Saint Leo the Great met Attila the Hun and persuaded him to turn back from his invasion of Italy. The Tome of Pope Leo to the Fathers of the Ecumenical Council in Chalcedon (451 AD) settled the dispute on the human nature and divine nature of Christ in the single Hypostatic union. Traditionally, it was Pope Saint Gregory the Great who started the reform that endured Christianity in Europe for centuries following his departure. St. Benedict inherited the Desert Fathers tradition and expanded it in the West.

The reform called by St. Francis of Assisi was acted upon by Pope Innocent III.  Thomism which is still taught today in seminaries and Catholic universities such as the University of St. Thomas in Texas and others is due to the reform started by St. Albert the Great and his disciple St. Thomas Aquinas at the great universities of Paris and Rome. Fifty years after Thomas' departure to God, he was proclaimed a saint by Pope John XXII in 1323. Proclaimed a Doctor of the Church by Pope Saint Pius V in 1567, St. Thomas most enduring thought recorded in his Summa theologiae was placed at the Ecumenical Council of Trent on the altar alongside the Bible and the Council's Decretals. His reform of philosophical thought introduced Aristotle's philosophy of nature where Thomas taught that the mind can discover God and the created beings from studying nature. This introduced the first scientific thought on nature in the West which was carried out by Galileo, Newton, and more recently Einstein and Bohr in quantum physics. In 1879, Pope Leo XIII ordered Catholic universities and seminaries to use Thomism as the basis of their theological and philosophical studies.

For more on Thomism see: http://www.aquinasonline.com/
Following the Franciscans, the Benedictines, and the Dominicans, a fourth renewal came through St. Ignatius of Loyola whose Spiritual Exercises are a great source of helping the faithful to have an insight in the Truth of Christ. 

I believe that renewal in contemporary Catholic thought and action was started by Pope Saint John Paul II in his New Evangelization. His philosophy stems from "Personalism" based on Thomistic thought in the 20th century as formulated by the Jesuit priest Fr. Joseph Maréchal.  We find it most clearly in Vatican II's Constitution where the Council's Fathers proclaim human dignity, freedom, and responsibility (see Gaudium et spes here). 

Pope Benedict XVI stressed inter-personalism in his encyclical Caritas in Veritate:

As a spiritual being, the human creature is defined through interpersonal relations. The more authentically he or she lives these relations, the more his or her own personal identity matures. It is not by isolation that man establishes his worth, but by placing himself in relation with others and with God.