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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A Short Post on Infinity

I was recently asked to write a few paragraphs on the mathematical concept of infinity for a school news letter. I have copied it below. It is indeed brief for the subject that it deals with. I encourage those interested to do additional reading. I especially encourage reading the chapter on infinity in Math through the Eyes of Faith.

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Infinity is a difficult concept to grasp. We often misuse the term “infinity” to mean “something really, REALLY, big.” When Buzz Lightyear exclaims, “To infinity, and beyond!” the implication is “Let’s go really, really far… and then past that… I guess.” However, when we say that we worship an infinite God, we must be saying more than simply, “God is really, REALLY, big.” So then what are we saying? We can find some insight in mathematics.

To get an idea of infinity in mathematics we have to first be clear on some basic terms and definitions. We can count the numbers in a set by comparing them to the natural numbers (whole, positive numbers). {3,5,7,9} has 4 numbers because we can match each number in this set to the natural numbers 1 through 4: {1->3,2->5,3->7,4->9}. We say this set has size 4. A finite set is any set that can match to the numbers {1,2,3,…,n} where n is some number. An infinite set is a set that is not finite. In other words, there is no stopping point. The natural numbers themselves are infinite: {1,2,3,…}. They just keep going.

Now for some fun.

Consider the even numbers {2,4,6,8,…}. Also infinite. Half as big as the natural numbers right? Wrong. The set of even numbers, which is the set of natural numbers minus the odd numbers, is actually the SAME SIZE as the natural numbers!!! This seems counterintuitive, but we can match each even to the natural numbers {1->2, 2->4, 3->6, 4->8…}. For any even number you can think of, I can give you the natural number it pairs with. We can also prove that the set of all fractions {1/1, 1/2, 1/3, 1/4, …, 2/1, 2/2, 2/3, ….} which intuitively seems much bigger than the set of natural numbers, is also the SAME SIZE as the natural numbers.

So is every infinite set the same size? Nope. The set of all real numbers (every decimal expansion) is infinite, but LARGER than the set of natural numbers. This was proven by George Cantor in the late 1800’s. In fact, it has been proven there are infinitely many different sizes of infinity! Try wrapping your brain around that. Cantor spent his whole life working with concepts of infinity… and he went insane… seriously.

So when we say that we worship an infinite God, what are we saying? From a math perspective there are some familiar aspects about infinity, but it is also wholly different than anything we have ever encountered. It seems to follow rules of logic, yet it is surprising and mysterious. A lot of the characteristics of God that may seem paradoxical on the surface (transcendent yet immanent, perfectly just and yet perfectly gracious, one and three) may not be so paradoxical when you are talking about the infinite.

I’m not proposing any answers to questions of faith based on mathematics. It is my hope that you will see how studying math may give us just as much opportunity to reflect on the wonder of God as does a beautiful painting, song, or piece of poetry.

Enjoy pondering the infinite!

[After my initial post, I received another great comment from Scott Eberle that I thought would be worth including in the post itself]

Infinity is such a great subject for exploring the impact of our faith on mathematics!

Yes, Cantor suffered from depression and had multiple mental breakdowns, partly because of the intense opposition to his ideas. But what is really interesting to me is the whole reason he pursued the study of infinity to begin with.

Up until Cantor’s time, Aristotle’s idea that “actual infinity” does not exist was generally accepted by everyone. This was Aristotle’s way of avoiding the seeming paradoxes associated with infinity. Aristotle taught that we could accept “potential infinity”—that we could always keep going out as far as we needed—but that a real, “actual infinity” does not exist; we can never “get there.” And because mathematicians could not figure out how to deal with infinite paradoxes (like there being as many even numbers as whole numbers), Aristotle’s ideas were accepted. A few mathematicians, like Bolzano and Galileo, toyed with attempts to study actual infinity, but without modern set theory, they did not get very far.

Cantor, on the other hand, was a devout believer. He knew that God was infinite and that “actual infinity” must really exist. And because of this deep-seated conviction, he passionately pursued the study of infinity and developed set theory to describe infinite sets in the face of much opposition, especially from Kronecker, one of his teachers. Cantor insisted that his pursuit of infinity was founded on the theological premise that infinity was an attribute of God and that it was right for us to study it. Studying infinity, for Cantor, was a call of God.

At the time, many mathematicians rejected Cantor’s work and there was quite a lot of opposition. Today, virtually all mathematicians accept it, and the set theory he developed is today considered the very foundation for all mathematics. A real story of faith.

God: One and Infinite

by Dr. Daniel Kiteck, Indiana Wesleyan University

The following is from a talk given by Dr. Kiteck at the ACMS Conference this past May. It is with his gracious permission that I am sharing it here. 

Abstract

The ontology of mathematical objects has been of interest for millennia. I focus on the ontology of the number one in relationship to the ontology of God.

1. Introduction

Let ONE represent “the essence of what ‘the number one’ is.” I focus on the cardinal nature of the number one as opposed to its ordinal nature. The primary question is “What is ONE?” First, consider:

“There is no physical entity that is the number 1. If there were, 1 would be in a place of honor in some great museum of science, and past it would file a steady stream of mathematicians gazing at 1 in wonder and awe.” [Fraleigh and Beauregard, Introduction]

Why would mathematicians be in wonder over the number one?

2. ONE in Mathematics

ONE is foundational to counting, which is foundational to much of mathematics. This is nicely shown by Doron Zeilberger in his Fundamental Theorem of Enumeration: [Gowers et al., 2010, p. 550]

number one

With some philosophical hand waving (which I admit deserves more attention, but that will be suppressed for now), ONE can be seen as foundational to much of the mathematical spaces that mathematicians work within. Just starting with ONE, we immediately have a unit {1}. From this we can imagine the existence of the unit and the non-existence of the unit: {0, 1}. But, imagining ONE twice, ONE three times, and so on brings us to N. Bringing in the concept of symmetry of an anti-ONE, an anti-ONE twice, etc., gives us Z. But then considering the set of all integers over non-zero integers, Q, and the set of all Dedekind cuts, R, (recalling that each Dedekind cut is an infinite set of rational numbers), we quickly imagine ordered pairs, triples, etc. to then have R 2 , R 3 , etc. So ONE is important to mathematics. But, is ONE important to Jesus?

3. ONE in Theology

3.1 Mark 12

In Mark chapter 12, someone asks Jesus “Of all the commandments, which is the most important?” Jesus famously responds with loving God and loving neighbor. But, the first part of Jesus’ response in verse 29 is

“The most important one,” answered Jesus, “is this: ‘Hear, O Israel The Lord our God, the Lord is one.’ ”

Jesus is quoting one of the most important pieces of scripture for the Jews, from Deuteronomy 6:4. My understanding is that this is commonly not seen as God being numerically one, so differing from ONE, but I still see it as reasonable to ask about how “One God” is related to ONE. But, first, where else in the Bible is it common to use the word “one?”

3.2 More than One?

In Genesis 2 we have two (husband and wife) becoming “one flesh.” But then in Ephesians 5 Paul informs us that this is a mystery actually referring to Christ and the Church. In traditional Christian Theology we have God as trinity [1] : “Three in One.” How are these connected to ONE?

4 The Ontology of ONE

4.1 Trinity

Does the concept of the trinity lead us to ONE and THREE as eternal concepts? Or does it lead us to conclude that we must leave the relationship between God and numbers a mystery? [2]

4.2 One God

Was there one God before God created the number one? I like asking this question to my students. Here are some responses attempting to honor God’s sovereignty: ONE is…

  • part of a “continuous creation,” so distinct from God [Howell and Bradley, 2001, p.71]
  • uncreated, so part of the nature of God in a mysterious way [Boyer and Huddell III, Spring 2015]
  • not “real” in a Platonic sense [Bradley and Howell, 2011, p. 230-235]
  • …wait!…Is “one” the same in “one God” and “number one?” [Ibid.]

Let’s explore this last question.

4.3 Necessity

Is ONE necessary in any universe God may create? I claim “yes” since one universe would be recognized by God.

But what if God had chosen not to create anything? Is ONE necessary then? [3] Perhaps another way to put it is: Must God’s eternal unique existence correspond to ONE? In light of the trinity, I am not confident in making a claim either way. And, in fact, in light of the oneness of Christ and the church, there seems to be some things concerning ONE that may be beyond our understanding. Or perhaps ONE should be seen as a rather distinct idea from many other ideas that are represented by the word “one.”

5 Conclusion

ONE is important to mathematics. “One God” is important to Jesus, but we must be careful with how this relates to ONE, especially in light of the trinity and the sovereignty of God. May our questions concerning ONE, and with mathematics in general, increase our wonder of the ultimate one, God.

I give a special thanks to my wife Karen for helping me sort my thoughts.

Bibliography

Steven D. Boyer and Walter B. Huddell III. Mathematical knowledge and divine mystery: Augustine and his contemporary challengers. Christian Scholar’s Review, XLIV(3):207–235, Spring 2015.

W James Bradley and Russell W Howell. Mathematics through the eyes of faith. HarperCollins Publishers, 2011.

J Fraleigh and R Beauregard. Linear algebra. 1990.

Timothy Gowers, June Barrow-Green, and Imre Leader. The Princeton companion to mathematics. Princeton University Press, 2010.

Russell W Howell and W James Bradley. Mathematics in a postmodern age: A Christian perspective. Wm. B. Eerdmans Publishing Co., 2001.

References

[1] I thank William Lindsey for pointing out the importance of including the trinity in this presentation

[2] I thank Kevin Vander Meulen for this question.

[3] A conversation with Jim Bradley sparked this question. Thank you Jim Bradley for the opportunity!