Augustine’s Philosophy of
Mathematics
Jim Bradley
Nov. 3, 2006
Why am I interested in
Augustine’s philosophy of
mathematics?
modernism
post-modernism
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 preEnlightenment Christian thinker
Why is Augustine interested in
the philosophy of mathematics?
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
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
“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.”
Such a thing does exist –
Number (Latin numerus)
x
…
y
z
Human reason
Inner sense
Sight
Hearing
Taste
...
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
My main thesis:
Augustine addresses all of these questions in a
way that provides a viable starting point for a
Christian philosophy of mathematics.
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.
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.
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.
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?”
Mathematical truths are instances of truths and
hence are:
- necessary
- immutable
- eternal
- and they transcend human minds
Ontology
Some widely held positions:
•Platonism (realism)
•Nominalism
•Kant’s anti-realism
•Constructivism
To Augustine, numbers are ideas in the mind
of God.
Menzel: Is mathematics created or
uncreated?
Both sides are problematic:
Created:
coherence
freedom
Uncreated:
uniqueness
sovereignty
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
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?
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.
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?
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?
One more
Should Calvin merge the mathematics and
religion departments?
2 Timothy 2:13:
“If we are faithless, he will
remain faithful, for he
cannot disown himself.”
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?”