On Informed Ignorance

I was reading the other day and came across this name: Nicholas of Cusa. Nicholas was a cardinal and bishop in the mid 1400’s who made some significant contributions to the field of mathematics – most notably his influence on Johannes Kepler (another man of faith who demonstrated that planets move in an elliptical orbit around the sun).

I am always encouraged when I come across people of faith from history who also devoted serious study to mathematics and recognized how the two can be integrated. So while my knowledge of Nicholas is very limited (how well do you someone you just met?), I thought I would pass along what I found for anyone who is interested in examining his works further.

Nicholas is most known for his work De Docta Ignorantia, which is roughly translated “On Informed/Learned Ignorance.” Or perhaps another way to phrase it: “Recognizing the Limitations of Knowledge.” Nicholas used mathematical analogies to show that truth can be approached, but never fully reached (or comprehended).

While I believe Truth can be reached (as long as He reaches out to us), I also believe it is healthy to recognize our inability to fully understand it. I wrote in a previous post about the need for Christian humility in our mathematical scholarship.

The following are some quotes from De Docta Ignorantia:

If we achieve this, we shall have attained to a state of informed ignorance. For even he who is most greedy for knowledge can achieve no greater perfection than to be thoroughly aware of his own ignorance in his particular field. The more be known, the more aware he will be of his ignorance. It is for that reason that I have taken the trouble to write a little about informed ignorance. …

Thus wise men have been right in taking examples of things which can be investigated with the mind from the field of mathematics, and not one of the Ancients who is considered of real importance approached a difficult problem except by way of the mathematical analogy. That is why Boethius, the greatest scholar among the Romans, said that for a man entirely unversed in mathematics, knowledge of the Divine was unattainable. …

The finite mind can therefore not attain to the full truth about things through similarity. For the truth is neither more nor less, but rather indivisible. What is itself not true can no more measure the truth than what is not a circle can measure a circle; whose being is indivisible. Hence reason, which is not the truth, can never grasp the truth so exactly that it could not be grasped infinitely more accurately. Reason stands in the same relation to truth as the polygon to the circle; the more vertices a polygon has, the more it resembles a circle, yet even when the number of vertices grows infinite, the polygon never becomes equal to a circle, unless it becomes a circle in its true nature.

The real nature of what exists, which constitutes its truth, is therefore never entirely attainable. It has been sought by all the philosophers, but never really found. The further we penetrate into informed ignorance, the closer we come to the truth itself. …

Here are some links for more on Nicholas of Cusa:

De Docta Ignorantia

Translator’s Introduction

Book 1: Maximum Absolutum (God)

Book 2: Maximum Contractum (the universe)

Book 3: Maximum Simul Contractum et Absolutum (Christ)

Other works of Cusa

The Enduring Uniqueness of Mathematics

Why is mathematics different (in a good way) from every other subject you learned in school?

Two words: Pythagorean Theorem.

Let me explain. The Pythagorean Theorem in itself isn’t really the reason math is unique; it is merely an example I wish to use to illustrate my point. I chose this Theorem for an example because it has been my experience that it is one of the few things everyone remembers from math class, regardless of how much they enjoyed math or how well they did in the course. But just in case the P.T. slipped your mind, here is a recap:

For any right triangle, the square of the hypotenuse (side opposite the right (90 degree) angle), is equal to the sum of the square of the other two sides.

This result is attributed to the Greek mathematician and philosopher Pythagoras (hence the creative name for the theorem). Pythagoras lived between the 5th and 6th century B.C. and while he is ultimately the one credited with proving the theorem, there is evidence that the result of the theorem was known to the Babylonians 1000 years before Pythagoras was born. Notice this old tablet:

Wow, that is old. Here you can read more about the Babylonians and the Pythagorean Theorem.

My point is that in what other class are you performing the same operations as people were performing 3000 years ago? Certainly in history class you learn about earlier civilizations, but you are not being taught how to do history in the same manner as those civilizations. The precision that modern history requires was largely unknown to those ancient people. Perhaps in literature you read Homer’s Iliad and Odyssey, but again, you aren’t being taught to write in the same style of epic poetry.

So then why is it that in math class, while advancements have been made and technology certainly has come a long way, we still find it beneficial to perform calculations the way they were performed thousands of years ago?

My answer: there is nothing to perfect, nothing ot improve upon, when you come across truth. Real truth.

To all of us who hold the Christian belief that God is truth, anything that is true is a fact about God, and mathematics is a branch of theology.

~Hilda Phoebe Hudson

Pascal vs. Paulos, The Final Round: All Bets Are Off

Previous Entries:

Pascal vs. Paulos, Setting the Stage

Pascal vs. Paulos, Round 1: Pascal’s Wager

Pascal vs. Paulos, Round 2: Paulos Ups the Ante

Here is a concise summary of the objections raised by Paulos that I plan on responding to (For a more thorough treatment of the objections Paulos raises against Pascal’s Wager, please see the previous post):

  1. The wager argument can be used to justify heinous acts by people who judge human penalties to be inconsequential in comparison to their heavenly rewards.
  2. The concept of “betting on God” is ambiguous and can be used by people of various faiths.
  3. The phrase “the probability of God’s existence” is nonsensical because the universe is a singularity.

I would also like to respond to Paulos’ comments on ethics. To be fair, Paulos notes that his discussion of ethics is being given as an aside and it is not the primary focus of the chapter. However, there is another chapter in the book in which his discussion does center around morality. So then, I will save the discussion of ethics, morality and mathematics for a future posting.

In regards to objection 1: I believe this comment ultimately stems from a misunderstanding of Pascal’s Wager. Pascal’s argument is given as a response to the question: “Supposing the evidence for God’s existence is inconclusive, then is it rational to remain in unbelief?” The premise of this question is crucial to understanding the argument. The Wager argument is only useful for a person who isn’t sure one way or the other that God exists. Pascal would openly say that if the evidence is indisputably in favor for God, then believe in God. If the evidence is indisputably in favor against God, then don’t believe in God (though Pascal himself would personally say that the evidence indeed favors God’s existence).

People who would use a line of reasoning similar to Pascal’s Wager in order to justify horrible acts of violence are certainly not people who are in doubt about God’s existence. They are people who are already 100% sold on his existence. They are also people who are 100% misinformed about the nature of God and the desires He has for His creation.

It is not my desire to go into great prose here in addressing the atrocities that evil people commit in the name of religion. Just because evil people perform evil acts in the name of some religion, it does not justify their actions nor does it make that religion inherently evil (since it seems to me that people in this category have largely warped the religion they claim). Similarly, it should be stated, just because evil people perform evil acts in the name of non-religion (atheists), it does not justify their actions nor do their actions alone make atheism inherently evil. All this line of argument does is show that people are evil. ALL people. All people, when left to their own devices, are at their core depraved. We are all in need of redemption and the grace of God.

In regards to objection 2: It is certainly true that with Pascal’s Wager alone you cannot arrive at an argument for the God of Christianity. But the fact remains, if the argument can succeed in persuading a person to simply theistic belief over atheism, then the leap to specific Christian belief is not nearly as great.

There also arises the question of what does it mean to “bet on God.” By betting on God, Pascal does not mean that you can make yourself believe. Salvation remains a work of the Spirit and not the will. Perhaps it is instructive to again quote here Pascal’s own understanding of any argument for God.

That is why those to whom God has given religious faith by moving their hearts are very fortunate, and feel quite legitimately convinced, but to those who do not have it we can only give such faith through reasoning, until God gives it by moving their heart, without which faith is only human and useless for salvation. (Pensees, 282)

What Pascal means by betting on God is that you can put yourself in situations that are more conducive to belief. For instance you can attend church, you can pray, and you can leave behind your present lifestyle. Putting yourself in those situations places you in a seemingly better position to be open to the working of the Spirit, though it is no guarantee of salvation.

And without faith it is impossible to please Him, for he who comes to God must believe that He is and that He is a rewarder of those who seek Him. (Hebrews 11:6, Italics mine)

We can be comforted in knowing that God rewards those who seek Him.

In regards to objection 3: While I certainly do not claim to have a complete understanding of statistics or the use of religious language claims, I see no reason why the phrase “the probability of God’s existence” should fail to make since simply because we cannot count the number of universes there are and how many of those universes have gods, or something along those lines (even if we could count the total number of universes in existences, the claim is still that there exists one God who is over all of them).

One of the more popular examples that statistics students bring to me is the infamous sock drawer. Questions read along the lines of: “You are getting ready for school and your sock drawer has four argyle socks, two polka-dotted socks, and one blue-striped sock (since you lost one). What is the probability that you select two socks that match?”

If there was only sock in the drawer, would it be nonsensical to ask “what is the probability of one sock existing in the drawer?” I hardly think so. The probability is either 1 (meaning there is absolutely one sock in the drawer), or it is 0 (meaning there is absolutely not one sock in the drawer).

To extend the metaphor, it would seem that Paulos believes “the probability of God” does not make sense because we cannot open the drawer (on his view) to see if God is really there. Either God exists or He doesn’t (but He does), so the probability is either 1 or 0 (but it’s 1). So then, the phrase “the probability of God’s existence” seems to me to make perfect sense.

In Summary: While this post has certainly not been exhaustive in the treatment of the topic at hand (though those of you are still reading at this point may disagree), my critique of Paulos essentially boils down to this: Paulos’ critiques of Pascal’s Wager stem from a restatement of the argument that Pascal never originally intended. Pascal simply intended to demonstrate through sound mathematical reasoning that if you aren’t sure whether God exists or He doesn’t, it makes more sense for you to believe in Him than to not so believe.

I don’t fault Paulos for this approach. I simply offer it up as evidence that the presuppositions we bring to a subject (in this case Paulos’ atheism) always affect the way we reason through an argument.