Proving Math

From Stand to Reason

“What worldview makes the best sense of things like math? Certainly not materialism. Our worldview though, a worldview that entails both physical things and immaterial things, immaterial things that can be grounded in an intelligent, rational, reasonable God, that can make sense of things like math.”

I agree with his basic argument that math makes more sense in a theistic worldview. I also believe that there is actually a stronger argument that he can draw from rather than his apples example. There was an entire age of mathematics where mathematicians sought to eliminate any postulate (assumption) about mathematics and prove everything logically. Then Gödel came along and blew that ship out of the water (in part due to his Christian convictions). I talk about this significantly during the first unit of Geometry – mathematics is grounded on faith in something, it can’t stand by itself.

Snippet from perrymarshall.com:

Gödel’s Incompleteness Theorem:

The #1 Mathematical Discovery of the 20th Century

In 1931, the young mathematician Kurt Gödel made a landmark discovery, as powerful as anything Albert Einstein developed. Gödel’s discovery not only applied to mathematics but literally all branches of science, logic and human knowledge. It has truly earth-shattering implications. Oddly, few people know anything about it. Allow me to tell you the story.

Mathematicians love proofs. They were hot and bothered for centuries, because they were unable to PROVE some of the things they knew were true. So for example if you studied high school Geometry, you’ve done the exercises where you prove all kinds of things about triangles based on a list of theorems. That high school geometry book is built on Euclid’s five postulates. Everyone knows the postulates are true, but in 2500 years nobody’s figured out a way to prove them. Yes, it does seem perfectly reasonable that a line can be extended infinitely in both directions, but no one has been able to PROVE that. We can only demonstrate that they are a reasonable, and in fact necessary, set of 5 assumptions.

Towering mathematical geniuses were frustrated for 2000+ years because they couldn’t prove all their theorems. There were many things that were “obviously” true but nobody could figure out a way to prove them.

In the early 1900′s, however, a tremendous sense of optimism began to grow in mathematical circles. The most brilliant mathematicians in the world (like Bertrand Russell, David Hilbert and Ludwig Wittgenstein) were convinced that they were rapidly closing in on a final synthesis. A unifying “Theory of Everything” that would finally nail down all the loose ends. Mathematics would be complete, bulletproof, airtight, triumphant.

In 1931 this young Austrian mathematician, Kurt Gödel, published a paper that once and for all PROVED that a single Theory Of Everything is actually impossible.

Gödel’s discovery was called “The Incompleteness Theorem.”

 Gödel’s Incompleteness Theorem says:

“Anything you can draw a circle around cannot explain itself without referring to something outside the circle – something you have to assume but cannot prove.”

You can draw a circle around all of the concepts in your high school geometry book. But they’re all built on Euclid’s 5 postulates which are clearly true but cannot be proven. Those 5 postulates are outside the book, outside the circle. You can draw a circle around a bicycle but the existence of that bicycle relies on a factory that is outside that circle. The bicycle cannot explain itself.

Gödel proved that there are ALWAYS more things that are true than you can prove. Any system of logic or numbers that mathematicians ever came up with will always rest on at least a few unprovable assumptions.

Gödel’s Incompleteness Theorem applies not just to math, but to everything that is subject to the laws of logic. Incompleteness is true in math; it’s equally true in science or language or philosophy. And: If the universe is mathematical and logical, Incompleteness also applies to the universe.

Gödel created his proof by starting with “The Liar’s Paradox” — which is the statement

“I am lying.”

“I am lying” is self-contradictory, since if it’s true, I’m not a liar, and it’s false; and if it’s false, I am a liar, so it’s true. So Gödel, in one of the most ingenious moves in the history of math, converted the Liar’s Paradox into a mathematical formula. He proved that any statement requires an external observer. No statement alone can completely prove itself true.

His Incompleteness Theorem was a devastating blow to the “positivism” of the time. Gödel proved his theorem in black and white and nobody could argue with his logic.

Yet some of his fellow mathematicians went to their graves in denial, believing that somehow or another Gödel must surely be wrong.

He wasn’t wrong. It was really true. There are more things than are true than you can prove. A “theory of everything” – whether in math, or physics, or philosophy – will never be found because it is impossible.

So what does this really mean? Why is this important?

Faith and Reason are not enemies.

In fact, the exact opposite is true! One is absolutely necessary for the other to exist. All reasoning ultimately traces back to faith in something that you cannot prove.

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Proceedings from the 21st ACMS Conference

The Association of Christians in the Mathematical Sciences (ACMS) is pleased to announce the publication of Volume 21 of the ACMS Proceedings which can be found at the www.acmsonline.org webpage or at https://acmsonline.org/conferences/ (or click the image below).

The 21st biennial conference for the Association of Christians in the Mathematical Sciences was held at Charleston Southern University in May 2017.

 This is the first time the ACMS Proceedings have been refereed. The 22nd biennial conference will be held at Indiana Wesleyan University May 29-June 1, 2019. A call for papers for the 2019 conference will be announced in May 2018.

(My own contribution can be found at the end of the Proceedings“Cultivating Mathematical Affections through Engagement in Service Learning.” You can find my presentation here).

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John Roe (1959-2018)

At the end of March I was saddened to learn of the passing of John Roe, a professor of mathematics at Penn State University whom I had gotten to know through the Association of Christians in the Mathematical SciencesJohn moved from England to the United States in 1998 to join the math faculty at Penn State University.  Even while he was still in England, John was a ACMS member, but after his move to the US, he became increasingly active, attending the summer conferences, serving as a Board member, and speaking at one of the joint meetings receptions.
From the opening lines of his obituary:
John Roe — mathematician, teacher, rock climber, theologian, activist, and follower of Jesus — has departed from family and friends as well as the pain of cancer and has begun “a more focused time of peace and joy” with his Lord.
I felt blessed every time I interacted with John. Below is an excerpt from a post on the 20th ACMS Conference:
20th ACMS Conference Day 2

The day began with another excellent devotional from John Roe (who has graciously contributed his thoughts on GodandMath.com in the past). Personally, I feel blessed after every time I hear John Roe speak – he just has a way about him that seems infused with grace and deep spiritual understanding. John led us through Ephesians 3:14-19 with particular focus on the four dimensional analogy used by Paul:

For this reason I bow my knees before the Father, from whom every family in heaven and on earth is named, that according to the riches of his glory he may grant you to be strengthened with power through his Spirit in your inner being, so that Christ may dwell in your hearts through faith—that you, being rooted and grounded in love, may have strength to comprehend with all the saints what is the breadth and length and height and depth, and to know the love of Christ that surpasses knowledge, that you may be filled with all the fullness of God.

Some of John’s points:

  • When thinking of the love of God, don’t think in abstractions. Think of the concrete. Think of the cross.
  • Wideness – if you fold your arms across your chest this is the typical position of religion; inclusive and safe. If you stretch your arms wide open this is the position of Christ on the cross.
  • Longness – (a dimension of time perhaps) God’s patience and love are endless. God’s love wins because it endures more than we do.
  • Highness – The son of Man was lifted up. Christ does not shrink from being on display in that shameful place; He doesn’t hide.
  • Deepness – How deep Christ went – down to earth, down to the grave. How deep in our own hearts are the places that He can reach. He went there and He proclaimed freedom there.

To me, John Roe was a concrete example of the love of Christ. He will be missed here on earth but we rejoice in knowing that he is in the presence of his savior.

All to the glory of God
Succeed at home first
Communicate every day
Seek the heart of worship
Move out of the comfort zone
Teach from the heart
Prepare the ground for insight
Start with what matters most
Love alone endures

 

You can read John’s post as a guest contributor to GodandMath regarding his interest in the mathematics of sustainability: “Creation Care as a Focus for a General Mathematics Course.”

Here he is live and in person in a TEDx talk.

Mathematics for Sustainability will be published by Springer in May 2018.