AN Augustinian Perspective on MATHEMATICS

1. AN Augustinian Perspective on MATHEMATICS

2. A dilemma Could God have created a world in which 2 + 2  4 without changing the meanings of 2, +, =, and 4?

3. Augustine • Born: 356 AD • Died: 427 AD • Lived in N. Africa, spoke Latin • Probably the most influential Christian thinker outside the biblical writers • Much influenced by Plato; the “gold of the Egyptians” • Wrote On Free Choice of the Will, c. 395 AD

4. Philosophy of Mathematics • Four basic themes: • 1. Ontology • 2. Epistemology • 3a. Meaning of truth • 3b. How do we account for the certainty of mathematical truth? • 4. Effectiveness

5. Ontology For Augustine, mathematical objects (well, at least the natural numbers) are ideas in the mind of God and have been such from eternity.

6. Epistemology Elementary truths of mathematics are present to all who think - neither deduced nor induced but perceived. Not perceived by bodily senses - our understanding of infinity is enough to prove that Are more foundational than bodily senses. Accessible to anyone who uses reason.

7. Concept of truth • Def’n: A truth is a necessary and therefore immutable proposition. • Distinctive characteristics of all truths: • necessity • immutability • eternity • common to all minds that contemplate them • Some items of rational knowledge are truths.

8. Examples of truths One ought to live justly. Like should be compared with like. Everyone should be given what is rightly his. The uncorrupted is better than the corrupt, the eternal than the temporal, the invulnerable than the vulnerable. A life that cannot be swayed by any adversity from its fixed and upright resolve is better than one that is easily weakened and overthrown by transitory misfortunes.

9. Mathematical truths • Mathematical truths are instances of truths and hence are: • - necessary • - immutable • - eternal • - and they transcend human minds

10. Effectiveness “Every material object, however mean, has its number.” Augustine says things have form because they have number - take away their number and they cease to be. Math is effective because the number of things existed in the mind of God at creation and because we are created in the image of God.

11. Reason The ability to form and operate upon abstract concepts For Augustine, it’s how we learn anything and includes deductive reasoning, i.e., logic

12. Benedict …the faith of the Church has always insisted that between God and us, between his eternal Creator Spirit and our created reason there exists a real analogy, in which - as the Fourth Lateran Council in 1215 stated - unlikeness remains infinitely greater than likeness, yet not to the point of abolishing analogy and its language.

13. Logos • John 1 – “In the beginning was the logos…” • The Greek concept of “logos” includes reason and order. • Colossians 1 – all things were made by and for him • So all mathematics finds its origin and its meaning in Christ – that is, he is the alpha and omega of mathematics.

14. Kepler I was merely thinking God's thoughts after him. Since we astronomers are priests of the highest God in regard to the book of nature, it benefits us to be thoughtful, not of the glory of our minds, but rather, above all else, of the glory of God.

15. Galileo Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these one wanders about in a dark labyrinth.

16. BUT

17. Descartes …the essence of the Enlightenment was the belief that the world could now be seen in mechanical terms and defined in mathematical language…all reality…acts by natural laws , if we had eyes to see them. The world of human affairs, in sum, is the same as the natural world because the same laws that govern each govern all. The task for Enlightenment thinkers, then was to ascertain those general laws that governed reality and then apply them to the various cases that came up, whether political, economic, social, or religious. Ronald Wells, History Through the Eyes of Faith

18. 19th Century England • George Boole – called the “father of pure mathematics” by Bertrand Russell • Mary Boole – “Mathematics had never had more than a secondary interest for him; and even logic he cared for chiefly as a means of clearing the ground of doctrines imagined to be proved, by showing that the evidence on which they were supposed to give rest had no tendency to prove them. But he had been endeavoring to give a more active and positive help than this to the cause of what he deemed pure religion.”

19. The secularization of mathematics • Poor apologetics • Events within mathematics

20. Apologetics • Oliver Byrne (1839) The Creed of Saint Athanasius Proved by a Mathematical Parallel  +  +  = 

21. Within mathematics • Rigorization • Professionalization • Non-Euclidean geometry • Logicism – Russell • Formalism - Hilbert

22. Russell “It has gradually appeared, by the increase of non-Euclidean systems, that Geometry throws no more light on the nature of space than Arithmetic throws on the population of the United States…Whether Euclid’s axioms are true, is a question as to which the pure mathematician is indifferent…The [modern] geometer takes any set of axioms that seem interesting and deduces their consequences.”

23. Hilbert “The fundamental idea of my proof theory is none other than to describe the activity of our understanding, to make a protocol of the rules by which our understanding actually proceeds … Already at this time, I would like to assert what the final outcome will be: mathematics is a presuppositionless science. To found it, I do not need God or a special faculty of our understanding…”

24. What are we to make of this effort to establish the autonomy of mathematics?

25. If we grant that reason is rooted in the nature of God, the secular position on mathematics has fundamental intellectual and spiritual problems.

26. 1. Autonomy • The assertion of human autonomy from God is the essence of sin. • “The gold of the Egyptians”

27. 2. It starts from a false premise Science / religion debate. Atheistic perspective starts with the premise: If science can explain x, then we don’t need God. Russell and Hilbert: If mathematics can be cut loose from moorings in anything external to itself, then it doesn’t need God or anything external to human thought.

28. Reason • The saying is sure: If we have died with him, we shall also live with him; If we endure we shall also reign with him; If we deny him, he will also deny us; If we are faithless, he remains faithful– for he cannot deny himself.2Tim 2: 11-13 RSV

29. 3. It doesn’t account for: • The fact that we have these incredible capabilities • The certitude mathematics provides • Effectiveness of mathematics in describing physical reality

30. 4. Fractures human thought • Enlightenment defines reason as empiricism and mathematics; Russell sees his view as glorifying human thought. • But these views exclude • Ethics • Culture • Religion

31. Some implications Augustinian view of mathematics: • Is inspiring – the capacity to do math is a gift of God, its content originates in God, a mathematical career is a calling to think about God’s wonders, it leads to service of his kingdom, and to worship • Leads us to work for a more holistic view of mathematics – one that includes history, philosophy, ethics, and culture

32. Our calling • Restoration – to reestablish the vision of mathematics as rooted in the mind of God, given to us that we might worship God and build God’s kingdom • Transformation – to counter the turn away from reason in popular culture by helping our students value, trust, and use reason