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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Finding Faith in School

Here is another great article that I highly recommend. It is from the Christian Courier and is authored by David Smith of Calvin College and the Kuyers Institute (I’ve linked to the Kuyers Institute’s outstanding math resources in the past).

Personally, I am in the midst of reading Teaching and Christian Imagination by Smith (which I also highly recommend) and I always find his questions great points for reflection – on how I view teaching, on how I view my discipline, on how I view my students.

Here is a (long) quote from the article on how math might be used to seek justice.

School resources have not looked the same across the centuries and across cultures, but as far as your imagination is concerned, all mathematics classrooms are pretty similar, and they are like the ones you have experienced. They are just part of how the world works.

So what happens when someone decides the emperor does not have to keep wearing those particular clothes? Why would anyone start a mathematics textbook chapter in the Indian Ocean? Well, what if the chapter went on to explore the complex mathematics involved in describing the shape and acceleration of a wave? And what if it then pointed out that if we can use mathematics to do this, we can build early warning systems for tsunamis? What if it prompted some reflection along the way about what it might mean if people in poorer countries are more likely to die en masse when tsunamis happen? Is it just a natural disaster or might some human responsibility be involved? And suppose it then explored how mathematics is also involved in the aftermath. You need to drop food and water to people from helicopters – how would you figure out the best height from which to drop the crates so that you neither waste time and fuel descending too low (helping fewer people) nor damage the contents of the crates (helping fewer people)? The chapter from which I drew the example does in fact go on to explore these kinds of questions. It was designed by a group of Christian mathematics teachers and professors who wanted to explore how learning mathematics might be connected to matters such as seeking justice, enacting compassion and serving one’s neighbour.

Might students learn mathematics from such a chapter? Surely they could, if the problems are designed well. What else might they learn? How might time spent in this particular class help shape the way they imagine the world, their role in it, their future actions and responsibilities, or the reasons for being in school at all?

Smith’s point is that the way in which we present the material as teachers shapes the imagination of our students – how they see and interact with the world. Smith continues:

My point here is not to advocate for a blanket approach to mathematics, or even to claim that this is the best mathematics book chapter ever. My aim is simply to ask us to think about how the way we picture the world, our deep-down beliefs about how things work, might influence what happens in classrooms, whether or not religion is getting mentioned.

What happens to the shaping of our imagination as we pass through school if most of the examples in our mathematics textbooks are about shopping and sports? Or if there is never a mention of what is done with mathematics in the world? Or if mathematics is only related to science and technology? Or if it is, at least occasionally, shown to be possible that the knowledge and skills offered by mathematics might intersect in various ways with the effort to love God and neighbour?

Smith offers a very vivid portrayal of how the work we do as teachers can impact students at a level beyond their cognitive understanding of the material. We can use mathematics in service-learning, or in examining issues of sustainability or social justice. Not that every lesson will always address these issues, but if we are going to teach math Christianly then we should always be considering ways in which we might use mathematics to teach our students to love mercy, seek justice, and walk humbly with God (Micah 6:8).

Mathematics as Culture Making

I recently came across a great article from Redeemer University College (Canada) summarizing the work of Dr. Keven Vander Meulen that I thought would be worth sharing here. Here is a link to the article:

“Mathematics as Culture Making”

A few apt quotes:

The large variety of applications for matrix algebra illustrate that mathematics has cultural power, that it can be a tool for stewardship and culture-making. Mathematicians unfold the potential of creation and can stand in awe of our Creator as they discover the order and patterns within our world. Far from being a disembodied subject in an academic vacuum, mathematics impacts our broader culture.

And also:

Christians can reflect on mathematics as a culture-making activity. As noted by Andy Crouch, culture-making includes not only what we create, but also how we shape our understanding of the world around us… Mathematics is not neutral, Vander Meulen highlighted. He also defined mathematics as the naming of numerical and spatial aspects of creation. And so the study of mathematics complements rather than detracts from faith.

For those interested in this topic, here is a link to more information on Andy Crouch’s work on culture-making.

Kevin has spoken on this topic at past conferences of the ACMS. I strongly encourage readers of this site to check out his work.