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.

Advertisements

2 + 2 = Jesus?

I think I understand the author's intentions... but that is just an awful title.
I think I understand the author’s intentions… but that is just an awful title.

Whenever I talk about my passion for the integration of Christian faith and the study of mathematics, the typical response is one of confusion. I can tell immediately that the listener is going to a place in their head where they envision the arithmetic lessons from elementary school somehow combined with the Bible stories from Sunday school – where the answer to every question (even 2+2) was always “Jesus.” When the listener can no longer contain the awkwardness of this mental image they eventually blurt out something along the lines of “How can 2 + 2 = 4 (or insert other trivial math problem) be Christian?”

Where the Question Originates

In a weird way this reasoning process and questioning makes sense to me. It makes sense because in my years as a math major/tutor/teacher/Ph.D. student whenever I might have occasion to interact with the general public on the issue of what it is I do, the one inquiring of me would typically respond by boasting in their ignorance – “You’re a math teacher? That’s great. I was never any good at math.” I’m quite certain no other profession receives that response. “You’re a dentist? That’s great. I never floss… You’re a lawyer? That’s great. I steal my neighbor’s newspaper everyday.” Um, I just met you, but thank you for that confession.

Though other professions may not get as blatant of a response as math teachers, I do think there is a social tendency of politeness to try and interact as best you can with whomever you may be speaking. This tendency causes people to revert back to their earliest point of connection with the subject at hand. So I may not be a dentist or a lawyer but I do have a shared experience of extra years of schooling for my profession (plus I also brush my teeth…and have seen a lot of “Law & Order.” A lot). For mathematics, the earliest shared experience that the average person feels comfortable reverting back to is arithmetic. Occasionally someone may have had calculus and remembers a bit even though they “don’t use it anymore” or someone may mention their experience of geometry (since that is one of the specific subjects that I teach). But in general, a common baseline across America is Math = Arithmetic.

I also believe, from my years of experience as a Christian, that the baseline across America is Christianity = something roughly akin to the rigid Sunday school classroom experience. Maybe this is because the number of those who leave the church as they grow older is heartbreakingly large, leading to a large population who only experienced Christianity as a child in Sunday school, I don’t know. Regardless I think there is a general impression that we Christians are those who read the Bible and believe that Jesus is the answer to every question, without question.  Who fed the 5,000? Jesus. Who saved the animals on the ark? Jesus. Who discovered America? Jesus. Perhaps this is an overly dramatized rendering of the general perception of Christians, but I do believe that by in large we aren’t viewed as very academically sound thinkers.

Put together the general societal experience with mathematics and the general societal view of Christianity and it isn’t hard to see where the question “How can 2 + 2 = 4 be Christian?” My ultimate response to this question is the following: it’s a bad question. You may have had a teacher that told you there are no bad questions. They were being polite. There are bad questions and there are two things specifically that make this a bad question: 1) it misunderstands the nature of mathematics as arithmetical calculations and 2) it misunderstands the gospel of Christianity as an intellectual endeavor.

Responding to the Question – Math is More than Calculations

With the dental/legal analogy above I have already hinted at the fallacy of associating an entire field of study with one basic component of that field. There is obviously more that goes into being a dentist than brushing teeth and there is obviously more that goes into being a lawyer than emulating Sam Waterston. I don’t think any honest person would believe that all there is to math is arithmetic. I do think that a lot of honest people believe that all there is to math is calculations – so like arithmetic, just more complicated.

This couldn’t be further from the truth. I am personally at a point as a student where I can’t remember the last math class I was in where they asked me to do calculations. And while I certainly do teach calculation methods to my high school students, I would be a horrible math teacher if my only concern for them was that they be able to memorize algorithms for completing calculations and solving equations. In fact, if that was the job description then I would be in a different profession (youngest general manager in Atlanta Braves history).

The vast majority of my time as a math teacher is spent trying to get students to think logically/rationally/creatively/independently – not algorithmically. I want them to be able to solve problems like sustainable energy, human trafficking or world hunger – problems whose solutions are not numbers that can be arrived at by way of a memorized formula or a graphing calculator. They need math to solve those problems and any other problem of importance that they can imagine. It is my job (and my passion) to get them to see that. Math is everywhere. Math is pervasively engrained in the both the physical and social structure of the world around us and it is equally as pervasive in rational processes of the human mind as we attempt to explore, understand, appreciate, and communicate knowledge of anything around us. Math is more than calculations.

Responding to the Question – Christianity is More than Thinking

Christianity is always more than thinking, but never less.

– Neil Tomba, Senior Pastor, Northwest Bible Church, Dallas, TX

Christianity is more than thinking; it is more than an intellectual endeavor. Christianity is more than learning new facts and being able to give new answers/responses to the questions of the world. The gospel is transformative of the whole person, not just of the intellect. Beyond that, the gospel of Jesus Christ is transformative of all of creation. When rightly understood, the gospel is a message about the redemption of something that is broken – broken people in a broken world – not just fixing our mental understanding to be correct. Sin is a horrible thing. It is much more than wrongful actions that we commit. Sin seeps down into our souls, perverting our intentions, decaying our physical body, and spreading through all humanity into the creation we were designed to oversee. Sin is not a thing that we do, sin is a thing that we are. Sin is pervasive.

“But where sin abounded, grace abounded all the more” (Romans 5:20). In other words, if sin is pervasive then grace is not only pervasive but also prevalent, permeating, extensive, all-inclusive, boundless, unrestricted, and inescapable. The gospel changes everything about us to the core of our being in more ways than we can even comprehend. To think that applying Christian faith to mathematics implies there is a “Christian” way of computing calculations and a “non-Christian” is to vastly underestimate the message of the gospel.

In sum, I believe my students need math to solve any meaningful problem that they will encounter in life. I also believe that the greatest problem that they will encounter is that of their own sin nature. It is only by experiencing the full grace of God that my students will ever have a proper perspective on themselves and the world around them. Through this lens, deep beyond the surface level of life, is where I hope my students will explore the integration of mathematics with their Christian faith.

I believe that this is how mathematics is done Christianly. Though it is admittedly a longer answer anyone I may be exchanging introductions with would be expecting. Maybe their response can now be “Um, I just met you, but thank you for that confession.”

UPDATE:

After posting I received a very insightful comment from Scott Eberle that I wanted to include within the body of the post in hopes that it would be seen by more people. My hope is that you can look forward to more contributions from Scott in the future. Enjoy.

The only part I wonder about is where you write “They need math to solve those problems.” Students certainly do need math to solve these very real and very Christian problems. I agree that it is right to have math courses center around these problems so that students never lose sight of the use of math in the real world. But I wonder if this does not also leave the impression that math is just a neutral tool for solving problems, that the Christian aspect resides in the use to which math is put rather than math itself.

I think I often separate in my mind pure math from applied math, though in practice they go hand in hand together. You give a beautiful description of how applied math is used to solve big issues that face us as Christians, but what about the math itself, the “pure” math that is actually used to solve “problems like sustainable energy, human trafficking or world hunger”? Does it have no intrinsic value until applied? Is it really neutral? I think this may be the thought left in many people’s minds even after they begin to see how math can be done from a Christian perspective—that math can be used Christianly, but that math itself is “non-Christian”.

Theologians from Augustine onward have affirmed that math comes directly from the mind of God. Mathematicians know that pure math is breathtakingly beautiful, amazingly logical, and unexpectedly useful in the real world. And we Christians know why. While we are teaching students to use math to fulfill God-given mandates, I think it would also be good to give students a glimpse of the divine origins, beauty, and nature of math itself.

Case Study of Faith-Academic Discipline Integration: Statistical Inference

(Hello world! I know it has been quite some time since I posted anything new here. This semester of Ph.D. coursework has kept me otherwise engaged, and that on top of my full-time teaching responsibilities. There are a number of projects that should come to completion this summer which will give me a whole new wave of exciting posts to share. In the meantime I was grateful to receive an email from guest contributor Andrew Hartley sharing the slides and his notes from a presentation he gave at Dordt College on the integration of Christian faith and statistics. Click on the image below to view the presentation or click here. Enjoy.)

by Andrew Hartley

Andrew Hartley is the author of Christian and Humanist Foundations for Statistical Inference; Religious Control of Statistical Paradigms. For more information on this work, please visit the Resource Book page. Guest author Steve Bishop posted an interview with Andrew as part of his series on Christian Mathematicians.

Hartley Lecture