What is a circle?
A round thing, you might say. If you were my student, I would press for a better definition. What makes a circle different from an oval? Both are round. If you are intelligent, or if you simply remember the definition... a circle is the collection of points that are equidistant from a given center. It involves the idea of a fixed radius.
Have you ever seen a circle?
Yes, you may say. Duh, you may say.
Pardon me, but I think you have not. No circle you have ever drawn is perfect. Close maybe. Perhaps very close. But they are still approximations.
Well, if I can't, the computer can, you might say. Not so fast. What the computer projects onto the screen is a grid of pixels. It is, and someone might be able to correct me on this, something like a square grid. At least, as I press my face towards the screen, that seems to roughly be the case. The computer probably gives a good approximation, better than we can do by hand, yet it is still an approximation in the end.
How many points comprise a circle? Infinitely many. Supposing the computer could draw a true circle, how long does it take to calculate the location of infinitely many points? If it could locate and draw a billion, billion, billion, trillion points in a second... it would still take an infinite amount of time. Dividing infinity by a finite number still leaves you in a lurch.
Supposing you had a perfectly steady hand and a mind for drawing perfect geometric figures, would you even be able to see the circle you had drawn? A circle is a curved line comprised of points. How wide is a point? It has no width, or height, or depth, or breadth. A point is simply a location. No thickness, weight, nothing. But in order to draw a point on a piece of paper, you draw a dot. But for you to see the dot, the dot has to be extended in three dimensions. The dot itself is like a tiny, colored-in circle, which we would recognize as a two-dimensional figure. But it actually needs a little bit of height, too, a third dimension, however thin, for the ink or graphite to pop off the paper, to be distinct for our eyes to see it. The dot, then, is not a point. It is a way that we have decided is convenient to graphically represent a point.
What we would draw as a circle and recognize as a circle is not actually a circle. It is an approximate representation of a circle.
Are circles real?
I would say yes. We do have a definite idea of what a circle is. We didn't simply make it up. We recognized it. We recognize it pressing its way into reality, though the physical world can never quite give it to us without some slight bit of stretching or pulling, some slight bit of approximation, some slight bit of deviance from the ideal that we now have within our minds.
We could do the same bit of mental gyration for any number of things. Geometric objects especially. Take a straight line, for example, and start asking questions about it.
We wouldn't need to stick merely to geometric figures. What about negative numbers? What about formulas like the quadratic formula? What about...?
Lots of math is being done now that is beyond my current ability to comprehend. A lot of what is being done has no obvious correspondence to something in the real world. Complex analysis, for example deals with imaginary numbers. What is an imaginary number? I cannot hold an imaginary number of things in my hand. (Certainly, on occasion, it turns out that advances in pure mathematics will turn out to have physical applications in the real world. Though I can't hold an imaginary number of things, imaginary numbers are powerful tools in things like digital signals processing, which some of my friends from Tech can attest to.)
Is my idea of a circle a real thing? Is my idea of the number 8 a real thing? Is my conception of the quadratic formula a real thing?
This really is the type of question Plato was seeking to answer in his Platonic Forms.
The idea of a circle does seem to be a real thing. The mathematical structure behind the quadratic formula does seem to be a real thing. I say this because they do not seem to be things that we were free to make up as we wished. To pick a simpler example, 2+2=4. That is true whether or not I want it to be true. That was true before there were human minds around to comprehend its truth. The Chinese were not free to discover that 2+2=5... unless 5 were really a symbol that meant what we mean when we say 4, (assuming 2 and + and = still mean the same thing). Though different peoples might use different symbols, they are referring to the same underlying reality that is simply true.
I conclude, then, that there is an objective nature to mathematics. It is what it is. Mathematicians are discoverers. They are not writers of fiction. They are not symbolic conspirators. They have gotten in touch with things that are really there. They have quite definitely stumbled upon this true idea of a circle that we have never actually met in our experience of the physical world.
Mathematicians have gotten in touch with something that is really there. But where? If not in the physical world, in our minds? Yes, these mathematical truths are in our minds. But they were there first. It seems that they were somewhere already, embedded in things, awaiting for our minds to pick up on them, to discover them. So they are not merely in our minds.
The existence of these ideas is not a physical existence. Again, you can't hold them in your hand. And they appear to be timelessly true. 2+2=4 was around before we were. And it is not going out of style tomorrow. And any aliens who show up on my doorstep tomorrow are bound to agree.
What a can of worms! What a thoroughly disconcerting notion for the pure naturalist, the devoted materialist!
In math we have found a multitude of non-physical, timeless things that have an objective reality. And they are things that the human mind is suspiciously well-suited to grasp and explore.
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I actually started writing this entry to explore the following analogy, and I got a little sidetracked by circles...
Consider pi. Pi is an irrational number. That means it cannot be written as a fraction of two integers. There are infinitely many such irrational numbers. Take any irrational number and add 1 to obtain another irrational number. And now you have a quick and easy way of producing infinitely many of them.
Not being able to write pi - or any other irrational - as a fraction means also that its decimal expansion will admit no repeatable pattern.
Pi derives its existence from the idea of a circle. Consider that a calculation of pi from any drawn or observed circle in the physical world is going to be an approximation... because those circles are not exact. But for an exact circle, pi is the ratio of the circumference to the diameter. This ratio holds for any size circle.
Pi, then, is firmly anchored in the idea of a circle, and it proves its usefulness in our world because of how often "circles" pop up. Pi is what it is, and I cannot wish it any different, and it will never go out of style. It will always be what it always has been. It is a non-physical, timeless constant woven into the fabric of reality.
There is a grandness to it. Because in it we approach the idea of infinity.
3.14159265358979323846264338327950288419716939937510582097494459230781640628620
8998628034825342117067982148086513282306647093844609550582231725359408128481117
4502841027019385211055596446229489549303819644288109756659334461284756482337867
8316527120190914564856692346034861045432664821339360726024914127372458700660631
558817488152092096282925409171536436789259036001133053054......
If you start walking this road... you will never finish.
Here is a reality that is non-physical, timeless, infinite, simple and yet complex, objectively there regardless of whether we think about it or choose to approach it, hidden within things that we must deal with every day, exerting an undeniable influence on things we must deal with every day, able to be grasped truly, though not exhaustively or comprehensively known.
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Staring down pi might hold fascination for some, like myself, who enjoy that sort of thing. But fractals are another, more geometric way, of staring down the infinite and experiencing awe at objectively eternal objects...
From some simple equations, Benoit Mandelbrot discovered a set that now bears his name. Fractals are interesting because you can zoom in on them. You will never exhaust them. They are not atomistic. There is not a lowest level of zooming. They are simple, yet incredibly complex... and in them we find a beauty.
Mandelbrot discovered this set. He did not create it. It was there from eternity past and will continue on. The following video gives just a taste of this complexity, this inexhaustible infinity, this beauty...
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In an earlier post, I did a thought experiment on analogies with time travel. I ended up talking a great deal about the cosmological arguments for the existence of God.
I pressed on the question of why there is something rather than nothing.
Now, I think the question of the universe itself is a different sort of question from the truths of mathematics. I can easily imagine the universe being different from what it is, and I can easily imagine the universe not existing. It is possible, I suppose, that it might just have never gotten started.
But not so, at least as far as I can tell, with the truths of mathematics. It seems like the eternality of the mathematical truths would make them true independent of the universe itself. They are necessarily true, which means that they would be true in any conceivable universe.
But is it off the cards to ask, why are the mathematical truths there at all? Why are they there instead of not being there? Why are they what they are, instead of something else? Why is pi not 3.16?
The Leibnizian Argument calls for an explanation of the existence of anything, either by a necessity of its own nature or in some external explanation. Why then do the mathematical truths exist? We are on the edge of mysteries that probably few in high school math care to think about...
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Let me give a brief account of mathematics from a Christian worldview.
There are truths in the world that appear to have necessity. They are not dependent on what we think or do for them to be true. Yet, the only necessary being is God himself. All other necessary truths are grounded in who God is and what he thinks.
Because God is eternal, immutable, unchangeable, we can expect his thoughts on reality to reflect this. Hence, the unchanging nature of math.
The language of mathematics is suited very well to describe our world. Dogs and cats don't do math, but people do. We can do math because we are rational beings created in the image of a rational God. The truths of mathematics are grasped by minds. Yet they are not dependent on any of our finite and passing minds. They were prior to us. But that does not mean that they are wholly independent of Mind or Rationality. They are the outworking of God's eternal Mind.
The infinite complexity of math is a clue that we cannot exhaust the riches of who God is. Because he is infinite, he alone can traverse every digit of pi and every crevice of Mandelbrot's fractal. We will never be able to do this. Though our existence may never end with God, he alone has existed infinitely in both directions of time. He is above and beyond time.
What do we do with all of this? We worship our Creator, and we recognize our creaturely dependence for our knowledge upon his superior knowledge and wisdom.
Though it is awe-inspiring, we dare not worship mathematics or science.
Math is in the same position we are... a creature who exists to point to Creator in our own way.
For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who by their unrighteousness suppress the truth. For what can be known about God is plain to them, because God has shown it to them. For his invisible attributes, namely, his eternal power and divine nature, have been clearly perceived, ever since the creation of the world, in the things that have been made. So they are without excuse. For although they knew God, they did not honor him as God or give thanks to him, but they became futile in their thinking, and their foolish hearts were darkened. Claiming to be wise, they became fools, and exchanged the glory of the immortal God for images resembling mortal man and birds and animals and creeping things.
Therefore God gave them up in the lusts of their hearts to impurity, to the dishonoring of their bodies among themselves, because they exchanged the truth about God for a lie and worshiped and served the creature rather than the Creator, who is blessed forever! Amen. (Romans 1:18-25, ESV)
