Tag Archives: mathematics

Unanswered Questions

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Two recent experiences provoked me to reflect on the disparity between what I want to know and how much I actually know. To live at peace with this disparity, I’ve had to develop strategies for dealing with my ignorance without falling into skepticism or dogmatism. I share three of those lessons below.

Two Humbling Experiences

First. My newspaper never arrives before I finish breakfast! For this reason, I keep my Oxford Dictionary of the Christian Church on the dining table. This amazing resource contains 1798 double-column pages and many thousands of entries of relevance to church history. It begins with “Aaron” the brother of Moses and ends with “Zwingli” the Swiss reformer. I learn something new, interesting, and useful every morning. But I am also stung with something I’d rather not think about. Almost every time I read from that huge book I am reminded of my ignorance and my insignificance: the millions of lives lived, experiences had, institutions founded, and systems created.

Second. Recently, a long-time friend with whom I have not spoken in years called to catch up. During the course of our conversation, he asked what I thought of the state of the church in the United States. Perhaps he thought that because I’ve taught theology for 35 years to thousands of students from two generations I would have a grasp on what things had been, how they have changed, and where they stand. After a few lame attempts to say something intelligent, I think I said, “I really don’t know.”

You must understand that I’ve always had a passion to know, to understand my world, the flow of human history and my place in it. As a young person I had questions to which I needed answers. How can you act intelligently in a world you do not understand? So, I read and read. I read the Bible, theology, the history of nations, church history, and philosophy. Hundreds of books and tens of thousands of pages! I found satisfactory answers to some of my questions. But many more remain, and new ones arise almost every day.

The first lesson: know your ignorance. Perhaps I should say “feel” your ignorance, because you cannot know the extent of what you do not know. Speak and act with humility and caution.

Big Picture Theories

Of course, there is no shortage of big picture theories. Theories of society and politics, metaphysical theories, theories of human nature, climate futurology, economic theories, and theological visions. Comprehensive theories give us an illusory feeling of omniscience, of knowing the essential truth of all things. Perhaps that is why we like them. But there is no consensus about which theories are true. For abstract theories paint only the vaguest general outlines of their subject matter. They cannot be verified, for they do not create transparent understanding of why things in all their intricate interrelations are exactly as they are.

Our minds long for simplicity and clarity. Simplicity allows us to see the whole thing at one instant, and clarity gives us confidence that we see things as they truly are. Mathematical knowledge is the paradigm case of simplicity and clarity. Physics is the most mathematical of the natural sciences. But what you gain in certainty and clarity of knowledge you lose in descriptive power. For there is more to nature than quantity, much more! Knowing the basic physical/mathematical laws of matter does not give you the power to describe the actual state of the vast array of different kinds of things we experience in the universe. Such non-mathematical theories as we find in psychology, sociology, economics, and politics must be abstract, simple, and general for us to understand them. Like theories in physics, theories that deal with human behavior, insofar as they are abstract, simple, general, offer little help in understanding why things are the way they are in all their actuality.

The Second lesson: The wise person will avoid mistaking theory for actuality or the model for the thing. Theory never exhaustively explains actuality. Let the model direct your attention to the thing.

First-Hand Experience

We all know the limits of first-hand experience. No individual human being’s experience extends to every place and time. The very purpose of education is to make available to each individual the experiences and insights of countless other individuals from other places and times. It is the distinctive glory of humanity that we are not limited in knowledge and wisdom to our natural instincts or to what we can learn from our own experiences. Still, each individual must integrate information received from diverse sources into a unified whole centered in themselves. We are limited to the information we receive and to our powers of integration; we cannot leap outside of ourselves to get a God’s-eye point of view. Nor can we know how well our limited vision of things corresponds to a universal consciousness.

Though we must acknowledge the limits of our knowledge, few of us can believe that one individual’s vision of the world bears no resemblance to those of other individuals or to that of a universal consciousness. For this belief would render futile all attempts to learn from each other, to understand each other, or to achieve consensus. And why strive for a common vision apart from the conviction that this common vision bears some resemblance to reality? For then we would be limited to exploring the internal powers and possible objects of the human mind without reference to the way things truly are.

Now let’s revisit my two humbling experiences mentioned above. Admittedly, I cannot come to know and understand the experiences, thoughts and deeds of every human being who has ever lived. However, if I assume that all human beings possess the same powers, possibilities and weaknesses, I can learn more and more about my own humanity by studying the history of the human spirit in the lives of past individuals. (Reading entries from the Oxford Dictionary of the Christian Church makes sense after all!) Making the same assumption—that all human beings possess the same powers, possibilities and weaknesses—and thinking in the reverse direction, I can learn better to understand the experiences of other people by examining carefully my own first-hand experiences.

As I stumbled to answer my friend’s question about the state of the church in the United States, I ended up saying something like this: “I can’t speak from extensive, detailed knowledge of the big picture, but I can tell you how it looks from where I stand.” This humble and cautious approach seemed to satisfy my friend. He could compare his limited perspective with my limited perspective in a process that promised to clarify and supplement his understanding. Combined together many limited perspectives may move us all closer to the goal of comprehensive understanding.

The Third Lesson: Avoid both arrogance and despair (or dogmatism and skepticism) in your quest for true understanding. Listen carefully and dialogue respectfully to all voices and allow them to clarify and purify your first-hand understanding of the matter at issue.

God is Not Just (Or Even Primarily) A Mathematician!

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Last week we explored the consequences of the early modern shift from the organic model of the world to the mechanical model. When we imagine the world and every process within it as working like a machine we place ourselves outside of everything and every process in world. Our point of view is always that of an external observer of how the surfaces of things relate to each other. The external viewpoint is maintained even if we modify the metaphor to include waves, fields and strings. And the interrelationships of particles, waves or fields can be described scientifically only in mathematical terms.

Today let’s think a bit more about what science is, what it does, and what it can and cannot do; and I will apply this to the question of Big Bang Cosmology and divine creation. As I pointed out above mathematics is the native language of modern science. All other languages are at most pre-scientific; yet as we will see, science cannot rid itself of all pre-scientific concepts. As a most conspicuous example take the concept of a “thing.” Things are designated by names and properties. Its name designates a thing as a whole in its difference from other things: This is a dog, not a cat or a chair or a star. Its properties describe the distinct intelligible aspects that make the thing what it is and identify it as such: Cats meow, dogs bark and rivers flow. One of the properties of things is quantity. It seems to me that the only things that have only the property of quantity are numbers; and I even have doubts about this: can a thing have only one property? In general, the meaning of a thing cannot be fully described by expressing its quantity.

Pure mathematicians use only numbers and quantitative operations in their science. But as soon as you attempt to understand the empirical world in mathematical terms like physicists do you leave pure mathematics and begin to speak of things. And empirical things are more than numbers and can appear to us only through their non-quantitative qualities. The importance of this transition cannot be overstated; for it means that even physics, the most mathematical of all the sciences that study the real world, cannot escape the language of things and qualities into the clarity of pure mathematics. In order to increase our understanding of the world we experience through the senses physicists must tell us what things they are measuring. What is the mass of an electron? What is the electromagnetic charge of a proton. Physicists must relate their mathematical formulae to something we experience through the senses or their work illuminates nothing. And what is a proton or an electron or energy…or any of the other things physicists name? The answer cannot be a mere quantity! That would be a number. It must be a quantity of something. And things have qualities!

Physics and other natural sciences pride themselves on being empirical, that is, their goal is to explain theoretically the world we experience through our five senses. A scientific theory should be able to predict the occurrence of some event in the empirical world and the measure of success is whether or not its predictions turn out to be correct. Hence natural science begins with empirical experience and ends with empirical experience. Between the beginning and end of the scientific process scientists abstract from these empirical experiences aspects that are amenable to theoretical generalization, ideally in mathematical language. But when scientists abstract only the quantitative information from empirical experience what do they lose? What is the status of this ignored information?

On the purely empirical level, before the operation of our minds in reading the sense data, we receive only physical impacts that cause physical and chemical changes in our sense organs. But unless we are completely skeptical about our ability to know the external world, we understand that raw sense data encode information that are decoded by the mind. The information communicated by the senses to the mind includes the quantitative properties of things, but it also includes the other properties as well. Galileo dismissed all properties other than the quantitative as secondary. He considered such qualities as color, heat and cold, and smell to be mental reconstructions of more primitive mechanical qualities and these reconstructions do not tell us the truth about the external world.

But Galileo’s dismissal of qualities as secondary is blind to a huge fact: the sense data that the mind decodes and experiences as qualities is itself information. And information is created only by minds and is understood only by minds. Galileo and modern science in general are so focused on quantitative information that they relegate other types of information—esthetic and moral and religious—to the subjective realm. But why privilege quantitative information over qualitative information? If we read novels like physicists read nature—strictly as physicists, not as human beings who happen to be physicists—we would examine the quantitatively measurable properties of the paper and ink but completely miss the story. If however we read nature like we read novels we would find ourselves united with another Mind for an inside view of that Mind, its beauty, goodness and power. Why shouldn’t the Creator use the physical properties of the world to impact the senses, which the created mind decodes into various qualities, which in turn makes meaningful esthetic, moral and religious experience possible?

Next Week: I did not get to the Big Bang today. Next week, I promise! Hint: All physical theories of cosmology relate one empirical state of the cosmos to another state by way of theoretical explanation of the transition from a previous to the present state. Big Bang Cosmology is no different. It is not a theory of creation. It is a theoretical account of the development of the present cosmos from a previous state. On a theoretical level it speaks the language of mathematics. That is the secret of its explanatory power but also of its poverty. It sees numbers where we see color and hear music and feel the cool of the evening air! It can’t read the messages of running water, a singing bird, a sunset, the smell of a rose, the touch of a loved one. But these too speak truth! God is not merely a mathematician! God is also a composer, an architect, a lover, an author and a painter. Perhaps math is merely the medium whereas love, life, goodness and beauty are the messages!