Tuesday, June 30, 2009
Not that modernism was well-received by Christians, either. Modernism, I am told, began with the skepticism of folks like Descartes, who decided to question everything he knew and build up a whole philosophy grounded in only those "self-evident" truths like, "I think, therefore I am."
(I'm glad I got to finally throw in that quote on this blog.)
This kind of skepticism seeks the "facts." Once you have the "facts," which are presumably unquestionable once firmly grasped, then it's a matter of interpreting the facts to gain sufficient theoretical knowledge of the universe.
Modern science derives from this, but so do a lot of other things. Skepticism about Christianity, for one. Modernism made humans the ultimate judge of truth. If something wasn't "self-evident" to us--either from just thinking about it or checking empirically--then it wasn't a fact, and if it wasn't a fact, then it bore a huge burden of proof. Christianity no longer sat so comfortably on the shoulders of faith.
Postmodernism, it's true, hasn't done much to bring people back to faith. What it has done, however, is call into question this whole notion of "self-evident" truths, pointing out that the way we see the world is largely shaped by our culture and personal experiences.
Modernism is still hanging on in a lot of circles, but postmodernism has pretty much made its way everywhere. One of my fellow math grads actually said that "science is just one perspective among many." Not even science, which has depended so heavily on modernism, can escape from the doubts of postmodernists.
But the problem with postmodernism is that it's not much more than a set of doubts about modernism, rather than a philosophy in itself. What could possibly fill the void?
Many evangelicals around me suggest that the only thing to fill the void is to go back to the revealed Truth of God found in the Bible. The problem I see with this is that the suspicions of both modernism and postmodernism really are quite penetrating. Like it or not, modernism brought with it a lot of scientific discoveries and historical criticism that raise huge doubts about the perfect accuracy of the Bible.
Perhaps the more crushing blow comes from the sheer number of different biblical interpretations there are. As hard as modern evangelicals try, we just can't help but read the Bible through layers upon layers of traditional interpretation that has been handed down to us.
But what, then, to take the place of modernism? How do we respond to postmodernism with something helpful?
It seems to me that the whole problem comes from a plain, one-dimensional approach to truth. The idea of obtaining revealed truth straight from the Bible is one-dimensional: truth goes from God to the Bible to you in a linear fashion.
Modernism, for all its skepticism, is no different. Humans observe nature, and "facts" come directly from nature to humans. By noticing patterns in these facts, we can ascertain truth.
In meditating on the Christian concept of the Trinity, I was struck by the idea that if God is not merely One, but also Three, why should we expect that truth would be merely one-dimensional?
After all, postmodernists have a point, don't they? Our thinking is always shaped by our culture. The way we perceive reality is shaped by our own experiences. There is, in fact, a trio of characters that interact in our pursuit of knowledge.
To be concrete, why don't I talk about the study of the Bible? The Bible is a collection of writings which exist independent of me. I cannot simply will the letters on its pages to change their form.
But when I read it, I am not simply being spoon fed "truth." The message I get will be shaped by my own mind. I may be prepared--whether through experiences or contemplation--to understand its words differently from someone else.
That's not the whole story. I am not on my own when I read the Bible; my culture reads with me. Sermons I hear, books I read, and conversations I have all shape what I am prepared to see in the text. Of course, the culture itself is shaped by the Bible, as people re-examine it. But the culture also shapes the Bible, in the sense that is shapes the message I receive from it.
That's not all, either. I shape the culture in which I live, and the culture shapes me. All of these different interactions are natural and inseparable. It is a trinity, if you will; three in one, and one in three. The Bible and I and the culture around me are ever three and ever one, and the search for meaning can only proceed with all three parts involved.
Each failed attempt at obtaining truth can be critiqued according to this model. Pure revelation cannot work, because it denies the role of self and culture. Attempts to evade self and culture are doomed from the start--they are inescapable, just as much as God the Father cannot be without the Son and the Spirit.
Modernism does not work because it denies the role of culture, and it seeks to trump up the self as impartial and objective (provided one is enlightened through skepticism). Postmodernism provides a critique of this, and I accept this critique.
In the place of Modernism, however, I would suggest "trinitarianism," the idea that truth is not a set of propositions revealed directly, nor a set of ideas derived from an objective study of the "facts," but that truth is the continually harvested fruit of a right relationship between the self, the thing studied, and that ever-present third party (e.g. one's culture, though one might think of something, or someone, else).
What, then, of doctrines? Doctrines are useful insofar as they promote this harmony. C. S. Lewis talked about how doctrine is not the destination, but it is a map toward the destination. Many people are too quick to line doctrines up beside one another, find them mutually exclusive, and therefore try to throw away one or the other; but I think this is not always the best approach. Ideas gradually evolve in their meaning as this "triune" process unfolds.
When I evaluate an idea about scripture (the doctrine of the Trinity, incidentally, is a perfect example), I must first see if it causes me to look for something I had not seen before. Then I must go and read the scriptures and analyze how this idea may have enlightened or conflicted with certain parts of my reading. Then I should share with other people what that experience was like, and let them share their own experience with me. The result of this will not be an artifact that we can put in a box labeled "truth." The result will be a better harmony between the three parts of my search for truth, and it is that harmony itself that is worth seeking.
Is this a shocking idea to Christians? Why are we so bent on having a tangible thing, like a statement, confession, or book, that we can call "Truth"? How do we come to know God the Father except through God the Son, by the power of God the Spirit? It is the same with all things. We don't know anything except through a three-fold harmony, and knowledge itself can be equated with this harmony.
Perhaps the doctrine of the Trinity can point us toward a vision of truth that is more attainable, a vision that appears to be needed in this time of uncertainty. I'd like to think so, anyway.
Monday, June 29, 2009
I gotta say, as frustrated as I am with Barack Obama, being the pro-lifer that I am, I try to give him the benefit of the doubt. But as fate would have it, I'm committed to another cause that is also getting frustrated with Obama.
Anyone remember Darfur in Sudan? Now 2.7 million people have been forced out of their homes, and hundreds of thousands have been killed at the hands of Sudanese President Omar al-Bashir. (Click here to learn more.)
I was hopeful back during election season that whoever became president would actually do something about this. This is one issue where I was happy with Obama (as well as John McCain and Hillary Clinton, since they all agreed). As you can see in the video, Obama certainly gave us the right words.
Edit: I didn't like hearing that video come on every time I loaded my blog, so I took it off. You can view the video here.
But where's the action? Now I'm getting letters from Jerry Fowler, President of the Save Darfur Coalition, telling me that "no long-term strategy for easing the suffering of the people of Darfur" has been made.
So I'm signing a letter that the Save Darfur Coalition has drafted to President Obama, saying that the US needs to send a clear message to Sudan, giving them two simple options:
"One: If Sudan permits immediate and unimpeded humanitarian access, secures peace in Darfur and South Sudan by fully implementing the Comprehensive Peace Agreement, and removes its President indicted for war crimes, a clear process toward normalization of relations and rejoicing the community of nations can begin.
Two: If President Bashir and his party are defiant and continue to undermine peace, a series of escalating costs will ensue, including diplomatic isolation, targeted economic sanctions, an effective and expanded arms embargo and, if necessary to stop massive loss of civilian life, eventual targeted military action."
I think Fowler's plan is pretty sound. Now all I'm wondering is whether Obama has the dedication to this issue to push this plan through.
Could the president's overall attitude towards foreign relations be stifling his actions on this issue? Maybe his propensity to shy away from anything that remotely smacks of George W. Bush style interventionism is keeping him from stopping this genocide.
I don't know what the deal is. But I just can't sit around and not say anything when hundreds of thousands have died, and millions are in dire straits.
I hate sounding like a public service announcement, but, if you care about this like I do, go to www.savedarfur.org to learn what you can do.
Thursday, June 18, 2009
The essential conclusion was nice and pragmatic. Livio thinks that the question of whether mathematics is invented or discovered is an ill-posed question, because it is both.
My feeling is that to most modern mathematicians this feels like common sense. We invent the mathematical concepts that we're going to study, and then we discover all the properties of, and relationships between, those concepts.
It's a little deeper than just inventing concepts at random, of course. These concepts typically find both motivation and application in the real world, whether we're talking about physics, economics, or whatever.
So that's great, but then, where did this question come from in the first place? One thing that's nice about Livio's book is how he traces the history of how this question was answered. There have been some really interesting answers.
The Platonists would say that math studies forms or objects that exist in some sort of reality beyond the world of the senses--a Platonic heaven, you might say. This idea has largely died out in the modern era, as people have discovered just how culturally unique certain mathematical concepts are.
For instance, the Babylonians, the Egyptians, and far Eastern cultures developed sophisticated mathematics, but only the Greeks thought of the concept of prime numbers. The kinds of questions you ask shape the kinds of concepts you formulate. There are other reasons why Platonism has lost credibility, but this is a pretty good one.
Then there are the formalists, who say that basically math is all just a construction of the human mind that happens to be self-consistent, nothing more. Gödel's incompleteness theorems created some problems with that idea, but I won't get into the details.
My own view accords well with the kind of realism that Livio seems to espouse in the last chapter of his book--the actual definitions and axioms with which we create mathematics are technically invented, but nevertheless there is objective content to mathematical theorems.
However, I don't really like the rabbit holes he goes down to try to explain this. He cites many different people who give pseudo-scientific answers that somehow get linked to the theory of evolution. They try to make the mathematics we now have out to be some "chosen species" weeded out from a number of potential candidates.
But evolution works on very long time scales, whereas the time it took for modern mathematics to develop was, comparatively, not that long. Besides, often the development of mathematical theories has a lot more to do with beauty than with fitness.
A "beautiful" mathematical concept is a purely subjective idea, but it's no less real because of that. What's so surprising to me is how beautiful mathematics can be--and still truly describe the world we live in. Some will say, "Oh, well, that's just because the world we live in happened to have these certain symmetries." Well, then, why not marvel at how beautiful the world is!
As I see it, mathematics has to develop as something of a human invention--but always, of course, seeking something that is objectively real. In fact, all knowledge must progress this way. We can gain no knowledge of the world in prepackaged form--it all comes in the form of changes in ourselves.
I have every theological reason to believe this as I contemplate the Trinity. Truth is not found in one person but in three. Just the same, truth is not found in nature alone, but it is found in nature, in ourselves, and in the interaction between the two. It is all about the interplay.
Consider an ancient mathematician charged with developing some accounting scheme for commerce. He starts by observing people trade their goods. He realizes that he can quantify these transactions using whole numbers. This is because the things traded always behave according to the rules of his number system. I have two cows and three sheep; you trade me a cow for my sheep; now I have 2 + 1 = 3 cows. Whether its cows or sheep or whatever, my goods always seem to follow these basic properties of arithmetic.
So this mathematician has immersed himself in the nature of the world around him, and he has come up with a way to describe it, the whole numbers. These are defined by simple rules: basically, that each number can be obtained by adding 1 to itself enough times, and given any number, adding 1 to that number gets you another number. But from here it is possible to define the operations of subtraction, multiplication, and division, where applicable, and it is possible to start proving all sorts of theorems in arithmetic.
This description of nature has a life of its own, apart from nature, but always intimately linked to nature. So long as the goods in this mathematician's world continue to behave like cows and sheep (which add together just like whole numbers) all the theorems about whole numbers will also be applicable to trading real goods.
But suppose the ancient mathematician's task is different. Perhaps it is to record the yearly cycles of water levels of the local river. If he has never done any arithmetic beforehand, who is to say that he will start by developing the whole numbers? Maybe he will not choose to create a system of definite quantities at all, but merely record comparisons throughout the year. His mathematics will be all inequalities, rather than definite quantities.
So we see that there is always a third party involved, one to whom we are accountable to and who influences how we see the world.
In this way the road to truth is not a linear motion, but rather a dance between three parties, each of whom must influence and be influenced by the other two. Just as God is one in three, so each of us must learn to be one of three, and we must learn that knowledge only comes from making these three one.
I am a bit mystified by this myself, but there's something compelling about it.
What I mean really is that I don't think a mathematician should not be discouraged by the fact that mathematical concepts are an invention. As part of this triad consisting of human, nature, and whatever this third party may be (perhaps "context" will suffice), the mathematician must simply work on his relationship with the other two. He must both establish his own real identity and work to seek the other two with sincerity. His own identity comes from his invention of mathematical systems. His seeking the other two comes from testing whether these systems help him describe something real in nature, and in seeing whether they are relevant in the context he is given.
It would be an unhealthy relationship if one part of this triad was favored over the others, so a mathematician should feel free to be creative, but never so free that he is not grounded in some reality. While this may be common sense to many, I like how there is this possibility of real theological grounding for it. If God is Trinity, why should that not affect all that we do?
Is God a mathematician? If by mathematician you mean one who studies the mathematical constructions that he invents, then probably the answer is "no." (Does God need to invent such things and study them?)
But on the other hand, I do believe that the study of mathematics is so linked with His character that I would have to say that God seems to have constructed this universe with mathematicians in mind. Somehow, God loves a quantifiable universe.
In any case, I know this will all sound like a lot of esoteric nonsense to most people out there, but after all my purpose in all this is written at the top of the page. I thought, therefore I blogged.
Tuesday, June 16, 2009
As much as I love the passage I quoted yesterday, equally do I dislike the following passage (from the Institutes, Book IV, Chapter 14, Section 3):
From the definition which we have given, we perceive that there never is a sacrament without an antecedent promise, the sacrament being added as a kind of appendix, with the view of confirming and sealing the promise, and giving a better attestation, or rather, in a manner, confirming it. In this way God provides first for our ignorance and sluggishness and, secondly, for our infirmity; and yet, properly speaking, it does not so much confirm his word as establish us in the faith of it. For the truth of God is in itself sufficiently stable and certain, and cannot receive a better confirmation from any other quarter than from itself. But as our faith is slender and weak, so if it be not propped up on every side, and supported by all kinds of means, it is forthwith shaken and tossed to and fro, wavers, and even falls. And here, indeed, our merciful Lord, with boundless condescension, so accommodates himself to our capacity, that seeing how from our animal nature we are always creeping on the ground, and cleaving to the flesh, having no thought of what is spiritual, and not even forming an idea of it, he declines not by means of these earthly elements to lead us to himself, and even in the flesh to exhibit a mirror of spiritual blessings. For, as Chrysostom says, (Hom. 60, ad Popul.) "Were we incorporeal, he would give us these things in a naked and incorporeal form. Now because our souls are implanted in bodies, he delivers spiritual things under things visible. Not that the qualities which are set before us in the sacraments are inherent in the nature of the things, but God gives them this signification."With all due respect to both Calvin and Chrysostom (pictured to the right), I disagree with the words I've italicized in a fundamental way.
I can't deny that this is the classic Christian position, which most Christians today would probably agree with--"our souls are implanted in bodies." But I have two fundamental problems with this. First, I see no scientific reason to agree with this statement. Second, I see no theological reason to agree with it.
Scientifically, we are constantly discovering more about the brain, which explains in deeper and deeper ways how it is that we have the capacity to do all those things which seem to distinguish us as "human."
This is an exciting thing, and it hasn't been until probably the last century that we've understood so much about the brain, intelligence, emotion, and consciousness. We should be celebrating this progress of science, but instead I constantly hear (and even used to make) abstract philosophical arguments designed to block classic Christian doctrines from being affected by science.
As I said yesterday, science can have theological consequences; why wouldn't it? But those theological consequences don't have to be frightening to Christians. Indeed, I see no theological reason to believe in the "soul in the body" picture.
The brilliant Reformed pastor and scholar Greg Bahnsen wrote a great article on the mind/body problem from a Christian perspective, in which he defends the idea that "man is a substantival monism, a material body which is special for reason of its capabilities (not its added substantival ingredient)." In other words, a person isn't a soul in a body, but that doesn't make him or her any less important in a theological sense.
I agree with Bahnsen, and moreover I think it's sad to see Calvin reason that God "accommodates himself to our capacity, that seeing how from our animal nature we are always creeping on the ground, and cleaving to the flesh, having no thought of what is spiritual," as if being physical beings distracts us from being spiritual.
In Genesis, God created the physical universe, including human beings, and there is no reason to believe he didn't create us as physical creatures. Our physical nature was no result of the fall. Therefore, what business does any theologian have in saying that God "condescends" to give us physical signs of His promises? Would He not rather rejoice to give us physical signs? Does He not love created things?
I think there's this false notion of spirituality that slips so easily into Christianity. It says that through the practice of religious activities (whether it's worship, prayer, or even good works) we can somehow transcend the physical universe and find some deeper truth beyond the created world. Many Christians, of course, emphatically believe this is true, but I disagree.
The picture of true spirituality I get from the Bible and from reason is that we ought to help care for the created world around us in such a way that there is harmony--that is, so that all things fulfill their purpose in God's sight.
In the Old Testament, sin polluted the land, not just our souls (as in Num 35:33). In the New Testament, it is the creation that longs for God's salvation, not just our souls. (Rom 8:19)
I think these ideas matter. Classic "soul in the body" Christianity will constantly seek to improve people's souls. But this could be damaging, especially if it's nothing more than an illusion. If humans are, after all, simply physical beings (which is not to say unspiritual) then it is frankly a waste of time to try to think of ways to improve one's soul as if it is distinct from the body.
On the other hand, if human beings are physical creatures created by God for a physical purpose, namely to care for His creation and to know Him more intimately through His creation, then we ought to be thinking of ways to do this.
In religious life, this may have something to do with the use of the sacraments, but I'm thinking in broader terms here. I'm thinking of a Christian spirituality that focuses in large part on the environment, living modestly (not consuming too many of the earth's resources), creating art and music, and thinking seriously about economic questions (as economics is the study of how to allocate the earth's resources).
The separation of these seemingly "secular" issues from "spiritual" issues is, for me, a contradiction, because these are all spiritual issues. They deal with the nature of things and the purpose for which they exist, and as such they are inherently spiritual.
I think this could have all sorts of practical consequences, but this is just one post, and I have to continue thinking about the details much more than I have. But I also think there is something just inherently more beautiful about not despising the body.
God created us as physical beings, and it is a good thing. Every tasty meal, every beautiful song, every breathtaking experience in nature, and every feeling of accomplishment after a hard day's work acts as a word from God's mouth, just as I feel that also mathematics and science are windows into His mind. If nothing else, this philosophy of life just makes life more satisfying.
Dualism probably appeals to most Christians in that it appears to separate us humans as unique. But I think the uniqueness of human beings lies in our purpose, and not in our substance, and I truly believe that if humans stop looking inward at an imagined soul in the body, we will be delighted to look outward at the world we truly live in, and discover in it a purpose for our lives that is beautiful, dignified, and even sanctified by the God who created this world.
Monday, June 15, 2009
Indeed, this post might be on the more esoteric side, but hey, anyone who reads my blog knows my interests are a little out there. Math, science, theology, and politics are usual fare here, so what can you expect?
I was reading through John Calvin's chapter on the sacraments in Institutes of the Christian Religion, originally because I was having a heated discussion about infant baptism. Being the good Presbyterian that I am, I decided to look to Calvin for some wisdom.
But then I got interested in Calvin's general idea of sacraments, and I couldn't help but look at it from the point of view of a mathematician interested in the relationship between God and nature.
Here's a rather long excerpt from the Institutes, Chapter 14, section 18:
The term "sacrament", in the view we have hitherto taken of it, includes, generally, all the signs which God ever commanded men to use, that he might make them sure and confident of the truth of his promises. These he was pleased sometimes to place in natural objects ...The emphases in the above quotation are mine. They point to an idea that is near and dear to my heart, the idea that there are two dimensions of creation that are firmly intertwined. The first dimension is physical existence, and the second is what I usually think of as meaning, though perhaps one could even call it spiritual existence.
...[W]e have an example, in his giving the tree of life to Adam and Eve, as an earnest of immortality, that they might feel confident of the promise as often as they ate of the fruit. Another example was, when he gave the bow in the cloud to Noah and his posterity, as a memorial that he would not again destroy the earth by a flood. These were to Adam and Noah as sacraments: not that the tree could give Adam and Eve the immortality which it could not give to itself; or the bow (which is only a reflection of the solar rays on the opposite clouds) could have the effect of confining the waters; but they had a mark engraven on them by the word of God, to be proofs and seals of his covenant. The tree was previously a tree, and the bow a bow; but when they were inscribed with the word of God, a new form was given to them: they began to be what they previously were not. Lest any one suppose that these things were said in vain, the bow is even in the present day a witness to us of the covenant which God made with Noah, (Gen. 9: 6.) As often as we look upon it, we read this promise from God, that the earth will never be destroyed by a flood. Wherefore, if any philosophizer, to deride the simplicity of our faith, shall contend that the variety of colours arises naturally from the rays reflected by the opposite cloud, let us admit the fact; but, at the same time, deride his stupidity in not recognising God as the Lord and governor of nature, who, at his pleasure, makes all the elements subservient to his glory. If he had impressed memorials of this description on the sun, the star, the earth, and stones, they would all have been to us as sacraments. For why is the shapeless and the coined silver not of the same value, seeing they are the same metal? Just because the former has nothing but its own nature, whereas the latter, impressed with the public stamp, becomes money, and receives a new value. And shall the Lord not be able to stamp his creatures with his word, that things which were formerly bare elements may become sacraments?
I think these two dimensions are sufficiently intertwined that one cannot exist without the other. For instance, in the Lord's supper, Christians always use bread and wine (or grape juice), even though some might say the meaning of the Lord's supper can still be conveyed through other elements. Christians still insist that using bread and wine is important, not least because it's just intuitively wrong to use, say, chips and coke, as if such things could ever represent as faithfully the body and blood of Jesus. It would be as if a man offered his fiancee a necklace instead of a ring to signify their engagement--the meaning of the thing and the physical nature of the thing cannot be separated so easily.
The reason all this is important to me is that we live in a scientific age, and I fancy myself something of a scientist. As a Christian, I have to grapple with the question, does science have any theological consequences? If so, do those consequences hurt or help Christianity?
I think the idea I've just described implies that yes, science must have some theological consequences. Learning about God surely cannot be separated from learning about His purposes in the world, and if what I've said above is correct, then learning about purpose cannot be separated from learning about the physical reality of nature. So science should have theological consequences.
But also this Christian way of understanding the natural world should give a powerful integrating framework for science, so that scientists need not find themselves studying the world for no particular reason. With a "sacramental" view of nature, we have a theological motivation for doing good science, and doing it carefully under sufficient ethical constraints.
This ability to integrate science into a broader framework also acts as a natural argument for the existence of God. I have read plenty of authors talk about how the Judeo-Christian view of nature laid the foundations for doing modern science, and I whole-heartedly agree. Is this not a testament to the strength of a belief, that it can open up greater understanding of the world we live in?
But it seems Christianity has been so obsessed with the "spiritual" as a separate category from the physical that it has lost ground in the scientific world. Most scientists today are not Christians, and I think this is because Christians have simply thrown away the philosophical fruit of their own worldview: creation is good, and its physical nature and purpose fit together beautifully in God's world.
This failure on the part of Christians is evident in our lack of clear moral leadership on crucial environmental issues (climate change, sustainability, etc.) And on bioethical issues, where I believe Christians do show impressive moral leadership, the world is not willing to listen, because we have sacrificed credibility on more general scientific questions.
Personally, I think understanding the natural world deepens my understanding of God. And not only that, but just experiencing the natural world deepens my relationship with Him. Music, for example, is fundamentally just a collection of sounds; we hear sounds occuring randomly every day, but when they are placed together in rhythm and harmony, these sounds become a means of conveying meaning. The physical nature of the sounds and the meaning they convey are inseparable. Music to me has always been an experiential proof of God's existence. I think most religious people can attest that they, too, have something in the natural world that "proves" God's existence to them.
In the ongoing battle between science and religion, I find it necessary to call Christians to simply embrace their own theological heritage and incorporate scientific knowledge, by which I simply mean greater familiarity with the natural world, into an ever-expanding view of who God is and what He is doing in the world. We should not be fearful that a greater familiarity with nature will distract us from knowledge of God, because I think just the opposite is true--understanding nature will deepen our knowledge of God.
This is too big an idea to iron out in one blog post, so I'll let it sit for now, and maybe come back to it tomorrow.
Friday, June 5, 2009
A reggae cover of Radiohead's "Karma Police" (from one of the best albums of all time). I heard it this evening on 106.1 fm, on the Corner Lounge. Apparently this group is in Charlottesville tonight. Who knew?
Monday, June 1, 2009
Basically this picture represents a result of my senior thesis. Here's more or less what I did:
Take a positive integer m. Make a table with m columns and infinitely many rows. Label the columns 0, 1, ... , up through m - 1. Now place a ball in the 0 column of some row, say row x. Then define a sequence of rotation remainders as follows:
- When the ball is in row x, you move it x spaces to the right, and if there aren't that many spaces left in the row, just wrap around to the next row (just like reading a page left to right, top to bottom)
- After each move, write down the column position the ball is in
So you might ask how the above picture represents this sequence at all. In fact, each row of pixels in the picture represents an integer (1, 2, 3, 4, 5, ...) and the pixels in a given row represent the numbers in the rotation remainder sequence corresponding to x. In this picture, m = 63.
Isn't it pretty? I promise there's a lot more complicated stuff behind it, too. I'm sure you'd be very impressed.