1. Augustine’s Philosophy of Mathematics Jim Bradley Nov. 3, 2006

2. Why am I interested in Augustine’s philosophy of mathematics? modernism post-modernism

3. Neither is satisfactory Modernist approach is a form of idolatry. Post-modern approach can’t account for mathematics’ sense of transcendence. Augustine a (the?) preeminent pre- Enlightenment Christian thinker

4. Why is Augustine interested in the philosophy of mathematics?

5. Augustine’s argument for the existence of God Faith seeking understanding a sure starting point - our own existence levels of being - existence, life, understanding we have senses, an inner sense, and reason Is there anything in human nature more exalted than reason? Among all things greater than reason, if such exist, the greatest is God

6. Inner sense - means by which we perceive our senses - animals have it - is not reason but is an agent of reason Reason (Latin ratio) - (Gilson) “Reason is the movement whereby the mind passes from one of its knowledges to another to associate or disassociate them.” - is how we attain knowledge - grasps itself by itself

7. “The bodily senses perceive material objects. No bodily sense can perceive itself. The inner sense, however, perceives material objects through the bodily senses and also perceives the bodily senses themselves. And by reason all of these things, as well as reason itself, become known and are part of knowledge.”

8. Such a thing does exist – Number (Latin numerus)

9. Human reason Inner sense SightHearingTaste... x y z …

10. What is the 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

11. My main thesis: Augustine addresses all of these questions in a way that provides a viable starting point for a Christian philosophy of mathematics.

12. 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.

13. Examples of truths One ought to live justly. Inferior things should be subjected to superior things. 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.

14. Truth Is a kind of light - it’s possessed by all who perceive the same truths at a given moment but it’s not changed by any of them. Cannot come from any individual reason as it is common to every reason. Rules the mind therefore is independent of it - transcends our minds. Cannot be gained from sensible objects. E.g., the idea of unity must precede my perception of it.

15. In seeing any truth, the mind perceives something above itself and immutable, therefore perceives aspects of God. “Is God greater than truth or is the truth God?”

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

17. Ontology Some widely held positions: Platonism (realism) Nominalism Kant’s anti-realism Constructivism To Augustine, numbers are ideas in the mind of God.

18. Menzel: Is mathematics created or uncreated? Both sides are problematic: Created: coherence freedom Uncreated: uniqueness sovereignty

19. Menzel’s answer: created but... Continuous creation So numbers and truths about them are - thoughts that God necessarily thinks - distinct from individual minds, superior to them - complete, immutable

20. Epistemology 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. How do we account for this presence?

21. 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.

22. Questions this opens up If we ground number in God’s thoughts, do we ground logic there also? Menzel’s continuous creation only solves half the freedom problem. What about more sophisticated mathematical structures such as groups or fields? Do we want to ground these in God’s nature also?

23. More questions Is all mathematics discovery of God’s thoughts? Is there room for human creativity in this approach? Augustine wants the structure of mathematics to precede creation. He also wants time created “In the beginning…”. He recognizes that “before time” is self-contradictory. Thus he says that mathematics precedes creation in a causative sense but not in a temporal sense. What does causation mean apart from time?

24. One more Should Calvin merge the mathematics and religion departments?

25. 2 Timothy 2:13: “If we are faithless, he will remain faithful, for he cannot disown himself.”

26. Aside Faith, Reason and the university: memories and reflections Benedict XVI at the University of Regensburg, September 12, 2006 “Is the conviction that acting unreasonably contradicts God’s nature merely a Greek idea, or is it always and inextricably true?”