## 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, 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.

## Math is Uplifting

Last week as teachers returned to school for faculty in-service, the school where I teach (Regents School of Austin) offered several talks/presentations that were broadly labeled as “Classical Christian Development.” There was a talk on western civilization, a talk on the importance of story, a talk on the centrality of theology, and a talk on math. I was asked to give the talk on math and this post is acting as a summary/recap. You can click the image above to download the slides I used in the presentation.

The title of my talk was “Math is _________ .” In introducing the talk I let that title just linger there for a while, asking the audience to consider what words or phrases come to their mind for filling in the blank. As the speaker I also enjoy soaking in the facial reactions of each member of the audience when it is announced that this next 45 minute talk will be about math. I contend that 100% of people (and as a statistics teacher, it means something when I say 100%) have a memorable, visceral experience from a math class. There are no neutral expressions on the faces of audience members. The sad thing from my perspective as a math teacher is that the majority of those memorable experiences are negative. My hope in giving this talk was to encourage people to consider some new words for the blank that they maybe had not thought of before.

I start by offering some familiar suggestions for the blank (familiar at least to our Classical Christian context where we teach). Here is the mission statement of Regents School of Austin:

The mission of Regents School is to provide a classical and Christian education, founded upon and informed by a Christian worldview, that equips students to know, love and practice that which is true, good and beautiful, and challenges them to strive for excellence as they live purposefully and intelligently in the service of God and man.

The bold emphasis is mine to point out a few words that might fit in the blank.

Math is TRUE.  This isn’t something I need to sell people on. I mean, 2 + 2 = 4 every time, amirite? To take it a step further though, I encouraged people to consider some ideas I put forward in another post: God, Math, and Order.

“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

When discussing mathematics from a Christian perspective, one statement that always seem to bubble to the top of the conversation is that mathematics reveals God as a God of order. This is true. This is also way underselling the connection between God and math.

Does God use mathematics because He is a God of order or does math have order because God uses it? I would argue that order is not a characteristic God displays but a quality that He defines by His nature and math gives us a glimpse into that nature. “Our God is a God of order” – By this claim we shouldn’t merely mean that God acts in an orderly fashion. We should mean God defines what an orderly fashion is. Order is not a quality God decided to portray, rather order flows from His nature.

If this can become our perspective, then when we speak of mathematics portraying God as a God of order, that description will carry so much more meaning. Instead of just correlating our mathematical results with some quality that God displays, we can realize those results are better understood as a manifestation of God’s nature. In a way we are communing with Him in our work as mathematicians, gaining deeper insight into His character.

Math is BEAUTIFUL. This is another category that I don’t have to do much convincing on. So many people have put together so many amazing presentations on the beauty of mathematics that any rational person could be convinced of math being beautiful after a quick Google search. Here is one my favorite videos in this regard and a few quotes.

“The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.”

~ G.H. Hardy, A Mathematician’s Apology

“The mathematical sciences particularly exhibit order, symmetry, and limitation; these are the greatest forms of the beautiful.”

~ Aristotle

Math is GOOD

Here is where the sell gets a little harder. As I mentioned above, a lot of people associate very negative things with math class. When asked to complete the phrase “Math is _________” they may think of words like “stressful,” “confusing,” “too abstract,” “not applicable to me,” or “the exact opposite of all that is good and holy.” Here is where I focused the remainder of my talk in an attempt to get anyone who fell into this boat to start seeing math in a different way.

Whenever I am presenting at conferences I like to do the following exercise: I ask people what the number one question asked in math class is, and without fail I always hear back “when am I ever going to use this?” The reality is that this is not a question, it is a statement. It is a statement of confusion and frustration. In other words the answer to “when am I ever going to use this?” has already formed in the student’s mind as “I am never going to use this” and then they withdraw from the mental activity at hand.

I would argue that what a student is really asking is “why should I value this?” It is not a question of finding application but of finding meaning. Maybe another rephrasing would be “why is this worth learning?” As Christian educators this deep longing should be familiar. If we believe Augustine in the Confessions that “Thou hast made us for thyself, O Lord, and our heart is restless until it finds its rest in thee,” then that doesn’t stop when students walk into math class. The most fundamental thing that is happening in math class is that students are seeking value (something we as teachers need to address in our curriculum) and are seeking to be valued (something we as teachers need to address in our pedagogy). In other words, the foundational issue of math class is an affective one as opposed to a cognitive one.

Affective issues are just present for students but for teachers as well. Another exercise I do at conference presentations: I ask people to close their eyes an imagine their best/ideal teaching moments (the O Captain My Captain moments). I then ask volunteers to share a word or phrase that describes that moment. Not once in all my years of doing this has a teacher mentioned anything about content. The language that is used is always affective – “engaging,” “curious,” “joyful.” Don’t get me wrong – I know the content was still there in the lesson and probably operating at a high level to produce those affective moments. The point of the exercise is simply to illuminate how central issues of affect are to the math classroom.

This is not just an anecdotal observation, but it is also affirmed in educational research:

When teachers talk about their mathematics classes, they seem just as likely to mention their students’ enthusiasm or hostility toward mathematics as to report their cognitive achievements.

Similarly, inquiries of students are just as likely to produce affective as cognitive responses, comments about liking (or hating) mathematics are as common as reports of instructional activities.

Affective issues play a central role in mathematics learning and instruction.

~ Douglas McLeod in Handbook of research on mathematics teaching and learning (1992)

It is also affirmed in national policy documents on math education (even though those documents never really develop how to go about obtaining these stated results – hence the motivation for my dissertation).

“Being mathematically literate includes having an appreciation of the value and beauty of mathematics as well as being able and inclined to appraise and use quantitative information.”

~ NCTM Standards for Teaching Mathematics

“Mathematical proficiency has five strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. Productive disposition is the habitual inclination to see mathematics as sensible, useful, and worthwhile.”

~ Adding it Up: Helping Children Learn Mathematics (National Research Council)

OK – so students and teachers both would admit that affect plays an important role in math education, this is supported by research, and it is affirmed in national policy documents and recommendations. With all of this motivation how are we (math teachers) doing?

As it stands our current methods of teaching mathematics are producing untold numbers of students who see mathematics more about natural ability rather than effort, who are willing to accept poor performance in mathematics, who often openly proclaim their ignorance of math without embarrassment, and who treat their lack of accomplishment in mathematics as permanent state over which they have little control.

~ McLeod (1992)

This quote may seem a little dated as far as research goes but I think it perfectly sums up the situation. No matter how dated the quote is, I know this is still true today because… well, I’m a math teacher. Plus I introduce myself to people in social situations. Other math teachers will quickly confirm this: whenever you meet someone and they ask “what do you do?” and you respond “I teach math” the next response will typically be something like “I was never any good at math.”

Math teachers are probably second only to priests in terms of the number of confessions we take from people.

Also, through these conversations it gets revealed that what these people really didn’t like about math were more factors of the math classroom schooling environment than the discipline of math itself. To me though, these actions are very foreign to the actual discipline of mathematics. For instance, people might say “I hated memorizing all of those formulas.” No mathematician would describe math as memorizing formulas. In essence what these people are doing is gossiping about math.

It as if they are saying “My friend’s cousin’s roommate’s teacher said that math is a jerk. He saw math behind the bleachers making out with history behind science’s back. No thank you – I want no part of math.”

To which I’d have to say “First, math is everywhere so math is probably making out with all of the subjects. Second, have you actually met math? Maybe you should talk to math face to face to sort this out.”

I think people have this false perception of what mathematics is because their experience of the math classroom was through forced, awkward, artificial “relevance” of math topics. For example:

The previous day another teacher had shared a story of how he actually jumped a cow in his car and miraculously survived – so I turned it into a word problem. Like most word problems it successfully takes and interesting event/story and kills it dead by now making it a chore for students to slog through. The artificial relevance of this problem makes it seem as if Mr. Williams was in the car doing this:

(NOTE: prior to this example coming up, I had planned to share an example I saw in a textbook of calculating parabolic motion on Steph Curry’s jump shot – this is clearly mathematical but we have to be careful not to oversell relevance as if Steph Curry makes his shots because he knows the math).

So how do we avoid this artificial relevance? How do we teach math differently at Regents? Because we teach at a Christian school, does that mean 2 + 2 = Jesus now? Here I had to share some thoughts from another previous post: 2+2=Jesus? Ultimately this type of question fails to see math as anything more than calculations. Math has calculations, but it is more than that. The question also sees Christianity as simply a new way of thinking (Jesus is now the answer to everything). As a pastor of mine would always say:

Christianity is always more than thinking, but never less.

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

A better understanding of how Regents approaches math differently can be summed up in the following quote:

“If you want to build a ship, don’t drum up people to collect wood and don’t assign them tasks and work, but rather teach them to long for the endless immensity of the sea.”

~ Antoine de Saint-Exupery

I pulled this from the first page of A Mathematician’s Lament, by Paul Lockhart, and it hits on the deeply affective aspect of what we do as teachers. I also proceed to read the opening imagery of Lockhart’s initial paper to communicate to everyone in the audience how we as math teachers feel about what often is described as math teaching.

So how do we go about cultivating mathematical affections? Well, I’ve written a lot about that here, but to quickly summarize:

Education is not primarily a heady project concerned with providing information; rather, education is most fundamentally a matter of formation, a task of shaping and creating a certain kind of people…. What makes them a distinctive kind of people is what they love or desire or value….. An education, then, is a constellation of practices, rituals, and routines that inculcates a particular vision of the good life by inscribing or infusing that vision into the heart (the gut) by means of material, embodied practices…. There is no neutral, nonformative education.

~ James K.A. Smith, Desiring the Kingdom (2009)

And also:

Mathematics educators who set out to modify existing, strongly-held belief structures of their students are not likely to be successful addressing only the content of their students’ beliefs…it will be important to provide experiences that are sufficiently rich, varied, and powerful in their emotional content.

~ G.A. Goldin in Beliefs: A hidden variable in mathematics education? (2002)

In other words, it is the practices of the math classrooms that shape mathematical affections. So I challenged the teachers in the audience to consider:

• Students want to know your story…
• What are the touchstone moments you can recall from a math classroom?
• What would you say are the “thick” practices/routines/liturgies of a math classroom?
• How has your experience of those practices shaped your perspective of mathematics?
• In light of our own experience of mathematics how do we work to shape our students’ experience of mathematics? How do we cultivate their mathematical affections?

To help answer these questions, I closed by offering three new words to fill in the blank of “Math is ____________ .”

Math is INVITING

Here I got to share about my role as an ambassador for the Global Math Project. First, an introductory video:

I challenged people to tell me how this math problem was different than the cow-jumping math problem above. A couple of different responses: this one makes you curious – you want to solve it. This problem has no words only images. This problem makes you ask questions rather than asking them for you.

Often our invitation into mathematics is already excluding some students. The words or terminology we use to introduce the problem may already shut people down. I’m not saying we shouldn’t use proper terminology, I am just asking to consider if it is always necessary. For instance, this example is teaching binary numbers but that term is never used. We don’t start by telling students “Ok, let’s learn the properties of binary numbers.” We have this interesting video instead. I would rather students understand the underlying concepts and connections (which this video very clearly portrays) than parrot terminology without understanding.

Math is ANALOGY.

I have written elsewhere about Flatland: A project of many dimensions. I love Flatland, and occasionally you’ll even see it’s title referenced as a parable. I love taking students through the though process of what would it look like for a sphere to enter into the 2D world of Flatland. In Flatland the inhabitants would only see the sphere as a circular cross section, completely unaware of the concept of a 3D sphere outside their literal plane of existence.

In this way the concept of dimension in mathematics offers a great analogy for issues of faith. How can Jesus be both fully God and fully man? Well, maybe it is kinda like the sphere still being a sphere but also a circle. I also love the illustration put forward in the chapter on dimension in Mathematics Through the Eyes of Faith about what if a hand entered Flatland. As your fingers went into Flatland the inhabitants would see cross sectional circles, none of which are connected. But if they could zoom out to 3D space then they could see fingers all connected to one hand. Maybe this can help us understand how the church is composed of many separate members but is still referred to as one body.

Finally there is Salvador Dali’s Crucifixion (Corpus Hypercubus). Just as a 3D cube can unfold into a 2D net that appears like a cross, a 4D hypercube can unfold into the 3D net seen above. Dali is speaking of the mystery of the crucifixion – of something that originated in a higher plane of existence unfolding itself into our world. These moments of insight for our students are made richer by the use of math as an analogy for faith.

Math is SERVING.

I have written a lot here about service-learning in mathematics. I won’t expand here, I’ll just summarize why I think teacher’s should consider service-learning:

• Affective learning objective is primary
• Cognitive learning objective is still present and operating at a high level
• Opportunity to communicate the value of affective learning outcomes through assessment
• “It is through our assessment that we communicate most clearly to students which activities and learning outcomes we value.”
• David J. Clarke, NCTM Assessment Standards for School Mathematics
• Reflection is key
• Moves toward inculcating a servant’s heart

Here is a quote from a student who would adamantly describe themselves as “not a math person” at the end of the year after going through a service learning project. When asked if they think their attitude towards math has become more positive:

“Yeah, definitely, much more positive. It was hard, don’t get me wrong and I’m not saying the ‘I’m no good at math thing’ didn’t change, but I do think … I am sure that I can learn it, because I am sure I can learn it. It just will take longer and when you don’t feel so completely discouraged about it … When you do feel that you do have shot to understand it and learn it, for me at least it really raises my attitude towards it. It doesn’t feel like it’s this hopeless thing that I just have to suffer through. It is kind-of just a hill you climb, right?”

I like this quote because it is honest. The point of cultivating mathematical affection is not to have every student now love math and have it be their favorite subject. The hope is that students who once saw math as this hopeless thing to be endured now see value in working hard at it. They start to see why they should value math.

Finally, returning to the Global Math Project as the inspiration for my talk in the first place:

Math is UPLIFTING.

I love that the motivation behind the Global Math Project is to change people’s experience of mathematics. I would love to see the students who would say that math is “confusing” and “stressful” now start to use words like “uplifting” to describe mathematics.

Returning to the Regents mission statement:

The mission of Regents School is to provide a classical and Christian education, founded upon and informed by a Christian worldview, that equips students to know, love and practice that which is true, good and beautiful, and challenges them to strive for excellence (inviting) as they live purposefully and intelligently in the service of God and man.

Notice the new points of emphasis. Sometimes focusing on math as true, good, and beautiful can still be an abstract exercise. Let’s start looking for math after the comma. Let’s look at the experience students are having of mathematics. Let’s care about the practices and liturgies of the math classroom so we can impact the mathematical affections of students.

If you are interested in starting the conversation with math (and leaving gossiping about math behind) I ended but sharing two great talks given by Francis Su, former president of the Mathematical Association of America:

The Lesson of Grace in Teaching

Math for Human Flourishing

## Where does math come from?

Here is a link to a talk given by Dusty Wilson of Highline College. Dusty gave a great presentation at the recent ACMS Conference on “A Triune Philosophy of Mathematics” (that I hope to cajole him into sharing here). This is a longer version of that talk given to a secular audience. Description of the talk:

Presented by Dusty Wilson. What is mathematics and is it discovered or invented? The Humanist, Platonist, and Foundationalist each provide answers. But are the options within the philosophy of mathematics so limited? This talk will provide a historical/philosophical overview, introduce an inclusive framework, and perhaps connect it to our critical work as community college educators.