Aftermath

Teaching History of Math this past semester gave me an excuse to read carefully two Dover Publications books that I have owned since high school, but only skimmed then. Imagine my delight to discover that if you are given a theorem that is hard to prove beforehand, you can prove that is transcendental in just a couple of lines. The hard theorem gives many other corollaries too, corollaries that I’ve known in my gut but never had a handle on how to prove.

Here are the details, from p. 76 of Felix Klein’s book Famous Problems of Elementary Geometry. You can read it on-line at Google Books.

Theorem (Lindemann): Over the complex field, in the equation not all of the
and can be algebraic, assuming that at least one

Corollary 1:
is transcendental.

Proof:
and 1 is trivially algebraic, so is transcendental, so then also is

Corollary 2: In the equation if x is algebraic and then y is transcendental. If x is a non-zero rational multiple of then y is algebraic.

Proof: Note first that by a power series argument, for example, Therefore, If x is algebraic then, Lindemann’s theorem applied to (*) shows that
is transcendental. If x is a rational multiple of then x is transcendental in a simple proof by contradiction from Corollary 1, hence y cannot be transcendental, again by Lindemann’s theorem.

Similarly, we can show that in only one of x or y can be algebraic (excluding the y = 0 case, as Lindemann’s theorem does, because ). Similarly for all of the rest of the trig functions and inverse trig function, which can also be expressed as rational functions of exponentials.

And finally, what is the heart of the proof of Lindemann’s theorem in the first place? It rests on the fact that as a power series has factorials in denominators, making any attempt to make it satisfy an algebraic equation a failure.

There is a universal distinction in Indo-European languages between knowing about a fact and knowing personally, between

1 wissen (saber Spanish, savoir French, scire Latin) and

2 kennen (conocer, connaître, cognoscere).

“Science” from scire “to know” is analytic. The root meaning of “science” is to “cut up,” so it is about knowledge by analysis, not synthesis. In fact the word “shit” in English comes from that same root, because excrement is “cut off” from the body.

Enlightenment rationalists used experiment and reason, both of them in an impersonal non-commital sort of way, being suspicious of the mystical– divorcing as it were “experiment” from “experience.” They had important axioms: Truth should be able to be interpersonally verified, and the universe is knowable because God is good and wouldn’t deceive our senses and reason. But truth without Truth led to deism and onward down to atheism.

“I cut open the body and I can’t find the soul.”

“Did it ever occur to you that the very cutting destroyed the soul?”

Micah Tillman reminded me again of this distinction in his blog today, where he comments on the familiar fact that pisteuo in Koine Greek means both faith and faithfulness.

1. faith, of the mind, analytic, held at arm’s length intellectually;

2. faithfulness, of the deeds, synthetic, embraced personally. “Cognoscere” is to about “personal knowledge,” to use Michael Polanyi’s term, and personal commitment.

Contrast Jesus before Pilate:

1 truth, academic, according to Pilate;

2. Jesus the Truth, personal.

Hear C. S. Lewis in his essay “Is Theology Poetry?” in Weight of Glory: “I believe in Christianity as I believe that the sun has risen:”

1. “not only because I see it” (analysis and experimental observation), but

2. “because by it I see everything else” (commitment).

Michael Ward has a great new book, Planet Narnia, about C. S. Lewis’s childrens’ stories. (A great review by N. T. Wright is here. ) In Planet Narnia Ward tells us that C. S. Lewis distinguished between

1. looking at a beam of light; and

2. looking along a beam of light.

(The C. S. Lewis quote is here in context.) Don’t just analyze the light; allow the light to serve its telos.

I was already prompted to think about this over the past weekend even before Micah’s blog entry. My first prompt was reading a fresh accurate translation of Exodus 20:3Exodus 20:3
English: World English Bible - WEB
3?You shall have no other gods before me.

“Thou shalt have no other gods between your face and Mine.”

The Hebrew for “before me” also means “in my face.” The traditional translation imagines (analytically) gods lined up along a line. Elohim is first; all the others rank distant seconds. This new more personalistic translation (thank you Thom Gardner) suggests that even in the Ten Words, God intended personal relationship not just legalistic obedience.

My second prompt was yesterday while I was listening to the Easter sermon at my church–a sermon on Romans 10:9Romans 10:9
English: World English Bible - WEB
9?that if you will confess with your mouth that Jesus is Lord, and believe in your heart that God raised him from the dead, you will be saved.

1 believe in your heart that Jesus resurrected (faith), and

2 confess with your mouth (faithfulness).

In my Ph.D. dissertation in 1979, I reviewed the artificial intelligence (AI) literature to find out how proponents of “strong AI” would ever bridge the gap between “knowing about” and “knowing personally.” The best that they could come up with– the best that they still can come up with 30 years later–is that personal knowledge is “precompiled” knowledge. But precompiling “X causes Y to die” into “X kills Y” does not bootstrap us nearly far enough up the “information pyramid” from mere data to wisdom:

As St. Agustine says, “believe that you may know.”

Get Wisdom. You can only get it by personally being committed to Him, for Wisdom is a person, not a proposition.

Thanks to my son John, my wife Emily has been dragged into the 21st century with her own blog. John used the free Joomla content management system to redesign her site.

Visit Emily at www.emilychase.com.

Emily still doesn’t have time for social networking, so if you want to network with us, you’ll have to friend Gene Barry Chase on Facebook.

I was hoping to be able to include mathematics in this blog, using LaTeX.  My son John found what I think is the easiest way to do that.  Visit the site http://www.codecogs.com/components/equationeditor/equationeditor.php

For example, if I type the following as the src field when inserting an image in the Wordpress editor

“http://latex.codecogs.com/gif.latex?\sin^2 x + \cos^2 x = 1″

then appears on the screen.

The site allows typing directly in LaTeX, watching the image to see that it’s right, and then cutting and pasting the URL for the image.  The site prompts for rarely-used LaTeX features that I’m likely to forget, now that I’m retired.  When I do mathematics here at Aftermath, I’ll put it on a separate page so that if you came here to read the promised philosophy, religion, sexuality, and (occasionally) politics, you won’t be distracted.  I’ll have to practice separate pages next.  Always something new to learn!

Our Christmas letter for 2009 is here. [PDF]

Below are more details. You can jump immediately to photos or videos if you’d like.

Emily’s humor articles at TheChristianPulse.com
• This is a perma-link to all of them including future ones.

The mathematics theses that Gene reviewed in 2009:

• Jonathan Dent, Geneva College, master’s thesis in Education, “Non-Trivial Natural Inclusion of Faith in Mathematics.”

• Richard Rast, Dickinson College, senior honors thesis in Computer Science, “Automated Interpretation of Arithmetic in First-Order Theories.”

Photos:

• Josiah & Anderson DeRosa, Summer 2009, Ocean City, MD. [Gene's favorite photo of the brothers so far.]

• Emily and I help to decorate Simeon’s nursery walls, May 2009, Frisco, TX.

• Emily & Simeon Chase, August 2009, Frisco, TX.

• Anderson DeRosa plays miniature golf, September 2009, Carlisle, PA.

• John & Kelly Chase and John & Eleanor Parke and Gene & Emily Chase, Thanksgiving 2009, Springfield, MA.

• Josiah and Anderson DeRosa visit cousin Caleb Masshardt, New Cumberland, PA.

• Josiah and Anderson DeRosa visit cousin Caleb (second shot).

•
Josiah DeRosa celebrates fourth birthday party, at several places, album.

• Anderson Ross DeRosa, Columbia, MD.

• Gene’s elementary school class when he was in fifth grade, Riverhead, NY. Beginning of a nostalgia trip.

Some of John’s juggling videos

• Juggling fire with his friend Matthew Wright, during an inner-city Philadelphia outreach. Video.

• Juggling clubs with a guy named Peter. Video.

• Juggling fire and other things at Boiling Springs High School, for the annual Labor Day fair, September 2009. This is also a permalink to Gene’s YouTube channel for future videos.

Videos of grandsons:

• Josiah DeRosa’s first soccer practice, Columbia, MD.

• Josiah DeRosa hitting a line drive in T-ball, Columbia, MD.

• Josiah DeRosa playing the drums, Columbia, MD.

• Simeon Chase dancing, Frisco, TX.

Photos of our family getting our houses in shape

• John and Kelly refinish their basement.

• Prisca and Anthony’s new house, Woodbridge, MD.

• Gene & Emily put new carpet down, Mechanicsburg, PA.

John’s paper at an NCTM math conference in Texas

“Increasing textual and spatial literacy through the examination and application of reading strategies utilizing the language of geometry”

It’s now possible to export all one’s Facebook photos. Here supposedly is my link to my photos. It appears to be a link to my whole Facebook site. I’ll try looking at this from another computer before I say that for sure.

Gene Barry Chase |

Here are three stories about the faith development of my grandsons, Anderson, 2, and Josiah, just now 4.

In March, Anderson and Prisca–his mom–visited his great-grandparents. Anderson was named after his great-grandmother, the former Eleanor Anderson. We call her “Grandy.” On Grandy’s wall is a crucifix made of palm leaves. Because it’s rather abstract, Anderson can be forgiven for not having figured out what it represents. He asked Grandy. She said it pictures Jesus on the cross with his arms outstretched.

Anderson, who knows more about baseball than he does about Jesus, said, “Safe!”

Anderson expressed perfectly the meaning of the cross …

… even if the metaphor itself doesn’t work perfectly, since Jesus is the Runner not the Umpire. Still, I will never again look at a cross without being thankful that I am “safe!” at home.

—–

In church, Josiah learned that God is everywhere. So when he got outside the building, he shouted into the sky at the top of his voice, “Hi, God!” Prisca pointed out that because God is everywhere, He’s not only in the sky, but in us as well, so that we can talk to God quietly, too. Josiah must have understood, because on several occasions during the following week Josiah was heard to whisper quietly, “Hi, God.”

—–

While Anderson was visiting Grandy, Josiah was visiting us. On his first morning, I asked Josiah how his sleep was. He said, “I had a dream about Jesus and God. I love them.”

On another day I caught him singing the words to the hymn “What a Friend We Have in Jesus.” I asked him how he knew the words. After all, some of them are big words: “griefs,” “privilege,” and “forfeit” to name just three. Josiah replied, “One of Anderson’s books has the song.”

—–

Psalm 8:2Psalm 8:2
English: World English Bible - WEB
2?From the lips of babes and infants you have established strength, Because of your adversaries, that you might silence the enemy and the avenger.

toddlers shout the songs
That silence atheist babble.

We are safe. God is near. And I love Him. What more do I need to know?

And so I whisper, “Hi, God!”

My niece’s husband Dan Masshardt inspired me to prepare this entry. He listed the above themes as separate topics and created them as timeless lists of talking points. It fits my style better to have a chronological arrangement and not to separate the two themes, since the reasons came to me in a clear order, but it has never been clear to me how to disentangle the two themes. I also couldn’t separate my very personal reasons from general theological speaking.

The list is cumulative. For example, those earliest stirrings of sin, love, beauty, and answered prayers continue unabated to this day. Besides those reasons, the Bible spoke to me as a unique book and the logic of the Christian worldview grounded me by the time that I finished college.

1. Elementary school
a. Sin. I was all too painfully aware that I was a thief and nosy, although I had “good reasons”—my family was poor and I was insecure about who I was. But my conscience worked, so I knew that I was doing wrong things. I wanted to stop, but I didn’t know how.

b. Love. Alice Huntington, my second and third grade teacher, and to a lesser extent Elizabeth Terry, my Sunday School teacher, showed me love which I did not get at home. They claimed that it came from God through Christ. I had no reason to disbelieve them.

c. Beauty. Alternate yellow and red tulips every Spring filled with beauty the hexagonal star in the center of the traffic circle in front of my apartment. The sleek lines of the 1956 red and white twotone Chevy Impala, were a graceful contrast with our black family clunker. A black and orange Monarch butterfly emerged from its jeweled green chrysalis. The Dona Diana Overture captivated me. (It was the theme song for the radio program Sergeant Preston of the Yukon. We get our beauty where we can.) These are just a few of the beautiful things that caught my attention as a young boy.

d. Answered prayer. Two of my big prayers were answered in elementary school. In 1954, I prayed to become a Christian, sensing a need to be forgiven and a need for love. I was flooded with a deep sense of a permanent relationship to God. Second, I prayed that my mother would be able to live away from the mental hospital where she was locked up for seven years. That happened in 1957. Two smaller prayers were answered as well, when I was near death breaking through the ice into a deep pond and when I was near death being rushed to the hospital with a burst appendix.

2. Junior high school
a. Sin. Add to the above sins indulging in sexual sin. My guilt increased by the fact that it more clearly involved others and by the fact that I sensed how much it displeased Jesus, Who was now without a doubt living in my heart.

b. Love. As I got to know my relatives, I saw a stark contrast between those on the one hand like my grandmother Emma Chase, and my aunts Flo Cavanaro, Florence Glidewell, and Ruth Finch —who were Christians—, and those on the other hand like Marjory Bacorn, George Cavanaro, Carl Chase, and Dad who were not.
     I grant you that there were exceptions. My neighbor Alice Jeneski did not seem to profess to be a Christian but she loved me; Uncle Art did profess to be a Christian but he had a bad temper anyway.

c. Beauty. I consumed every mathematics book that I could find, whether from libraries, from Christmas gifts that I asked for, or from my small allowance. Parabolas were beautiful; algebra was beautiful. I bought One, Two, Three, … Infinity by George Gamow, and read it at least five times.

d. Bible. For me in elementary school, the Bible was a book of heroism—of David, Gideon, Deborah, and Sampson. I did memorize Psalms 23 and 100 then. But thanks to Wednesday afternoon Bible memory work at church offered during “release time” from public school, I began to memorize a lot of verses of the Bible. I began to see the Bible’s themes. The Bible was like “fire in my bones” (Jeremiah 20:9Jeremiah 20:9
English: World English Bible - WEB
9?If I say, I will not make mention of him, nor speak any more in his name, then there is in my heart as it were a burning fire shut up in my bones, and I am weary with forbearing, and I can?t contain.

3. High school
a. Love. Rita Youngs, my high school youth group leader, showed me love, both directly to me, and to many foster kids she raised.

b. Beauty. High school teachers exposed me to the beauty of literature and of language. Although I messed around with a chemistry set since sixth grade, a high school science teacher showed me the beauty of Chemistry. Aunt Ruth’s gift of National Geographic Magazine showed me the beauty of far-off places. The book God’s, Graves, and Scholars showed me the beauty of far-off times. The Rev. Bill Blackwood’s Sunday School class on systematic theology that used George Pardington’s Outline Studies of Christian Doctrine showed me the logical beauty of the Christian faith.

c. Bible. Two summer camps continued my Bible memory, deepening my sense that this was no mere collection of human writings.

d. Prayer. It was in high school that I learned to “pray without ceasing.” Unanswered prayer didn’t bother me very much, because I was grounded in prayer as a relationship instead of prayer as a slot machine. I did pray that Dad would become a Christian and that I would be able to stay sexually pure. Neither of those happened fast. The first answer waited for six decades; the second answer, for one decade.

4. College
a. Love. Pastor George & Gladys Decker treated me like their third son.

b. Logic. I read C. S. Lewis’s defense of Christianity in Mere Christianity, and a defense of Jesus’ resurrection by Frank Morrison in Who Moved the Stone? Since I wasn’t argued into the Kingdom, it’s not likely that I’ll be argued out. I did not see apologetics as a weapon, but as a thing of beauty. The logic of my faith was like the logic of Mathematics—a powerful, coherent story.

c. Answered prayer. I saw the lives of others changed as I prayed that they too would discover the joy of following Jesus: for example, Rick Lueth, Tommy Wandless, Anne Macomber and her brother David.

c. Beauty. Before college I was exposed to Protestant hymns on piano, polkas on accordion, and country music on guitar. For some reason, the classical music taught in my public school didn’t “get to me”; like poetry, it was dry as a mere object of study. But at MIT, Bach concerts on the Kresge Auditorium organ were intensely emotional experiences for me.
     In college I discovered the beauty of worship itself—through Lutheran vespers in the chapel on Wednesday nights, and through Anglo-Catholic services around Christmas and Easter at a nearby church.
     You might see my appreciating Bach as merely becoming cultured, but both Bach and formal liturgy increased my faith by keeping it from becoming mere sterile logic. My faith continues to be emotional as well as logical. That is why I answer the title questions with my story rather than with propositions. The man with an experience is never at the mercy of a man with an argument, I reasoned.

5. After college
I now draw on revelation, on reason, on tradition, and on experience to ground my faith, aware that no one of them by itself is without its challenges. For me one test of a faith that conforms to reality is this: Is it simple enough that a child can have it, but profound enough to engage the wisest of adults? Jesus said both about faith in Him.
     Where does a sense of sin, of love, and of beauty come from? I reasoned with the faith of a child that they came from the same place where answered prayers come from. Now as an adult I do not see how atheism adequately accounts for even one of those themes.

For Christmas — just as a joke — my brother-in-law Richard, who is politically to the right of me, regifted a T-shirt that he had been given, also as a joke. It said, “I Obama.”

I told Richard, “But I do love Obama. I pray for him. I wish him well.” I have been meaning to follow up on my long post supporting John McCain before the November 2008 election. This T-shirt reminds me to do that. I promise a shorter post this time.

My comments are primarily about the US economy. I will wait to see what kinds of executive orders Obama produces in other areas. I’m already pleased with his keeping his promise to draw on a broad base of leadership in selecting his cabinet.

I’m pleased with Obama’s plan to provide public works projects that will simultaneously address unemployment and shore up our country’s ageing infrastructure.

I’m saddened by all the corruption already evident in how bailout money is being spent. But the lack of accountability falls both on Congress with a (slight) Democratic majority and the Republican administration, and Obama is neither senator (any more) nor administration (yet).

I am disappointed that Obama is not going on record as planning to slash Federal funding for things that I think are discretionary, controversial, and ineffective. I would disband the US Department of Education, eliminate Federal funding for abortions, and until the economy is back on track suspend spurious concerns about man-made global warming — as the president of the Czech Republic Vaclav Klaus explained so well at a conference on global warming in 2008.

But that’s what happens when you love someone. You don’t necessarily agree with them.

“Psychotic” is defined as “being out of touch with reality.” But whose reality?

The Hebrew Bible considers atheism psychotic. Twice the Psalms say: “The fool has said in his heart ‘There is no God.’ ” On the other hand, the New Atheists consider theism to be psychotic.

Psychotic_Theism and Psychotic_Atheism are quite different.

Before the American Psychiatric Association (APA) removed references to psychoanalytic jargon in its Diagnostic and Statistical Manual (DSM), Freudians used a broader term — “neurosis.” Neuroses included not only distressing breaks with reality but also thoughts that seemed to the analyst to be breaks with reality — whether or not they were ego-dystonic, which is to say personally distressing. Consider William Stekel’s 1922 book Homosexual Neurosis for a view of homosexual ideation as a psychosis contrary to (heteronormative) reality. But in 1973 the APA removed ego-syntonic homosexuality from its DSM.

Ronald Bayer documents in his 1987 Princeton University book
Homosexuality and American Psychiatry that the APA’s motive for the change was political, based on its (pansexual) view of reality.

I’m glad that psychoanalysis is out of favor with psychologists. Freudian analysis is pretentious and overly influenced by the analyst’s world view. Freud’s views are too easily politicized. Not a surprise! Freud is an intellectual child of Nietzsche, who — having abandoned God — was left with merely a “will to power” in making moral decisions.

Take the gulags of Marxist Russia for example, where prisoners were interred for their mental illness, to wit holding views contrary to the prevailing world view of the State. Marx is also an intellectual child of Nietzsche. Robert Heinlein’s 1941 science fiction short story, “They,” is clever allegory for this misuse of psychoanalysis.

Why am I thinking about these things today? Two sources came together for me this week.

I am finishing Arthur Goldberg’s 2008 book Light in the Closet (Torah, Homosexuality and the Power to Change). It argues that a conservative Jew cannot embrace the “gay is good” mantra.

Psychotic_GayIsBad and Psychotic_GayIsGood are quite different.

So are the anthropologists right after all? We must subscript all our terms with our world view.

Almost.

This week I also listened to Pastor Tim Keller’s masterful talk at Google headquarters (yes!) condensing the beginning of his book The Reason for God: Belief in an Age of Skepticism. Keller quotes Bishop Leslie Newbigin’s caution about the familiar Feeling the Elephant analogy, here (starting 21 minutes into the talk, and extending for 2 minutes):

We might say that anthropologists are

Right_Anthropology

after all. Humility is required.