Axioms

A couple of young fresh-faced smiling missionaries knocked on my door the other day, wanting to know if I read the bible. I turned them away – but tried to be nice about it. I’m not sure why, since I actually enjoy having conversations about fundamental issues. Superficially I suppose I thought I was too busy. But I really wasn’t. I was recuperating from a vacation and taking a day off to get over my jet lag. A more plausible reason is that I knew a discussion wouldn’t get us anywhere. Having thought about this sort of stuff (whatever it is) for decades, I’m pretty confident of my basic principles and I suspect so are they – and our agreement on basic assumptions would be pretty small. On the other hand, by having discussions with persons we disagree with, we get the opportunity to probe and refine our own thinking in ways we can’t do well alone, even if nobody’s mind gets changed about anything.

When I talk to such folks I sometimes find that they claim that they take the bible literally. As a theoretical physicist, I have some sympathy for the idea that our thinking can be much improved by not treating life as just a collection of snippets of knowledge we have come to know; that placing some “stakes in the ground” – some organizing principles that we choose to trust under a wide variety of circumstances – helps us to “get things right” more often, not to be led astray by little bits of knowledge that might be more complex and less straightforward than we realize.

But what should we choose as our holdfasts? The principles of physics I was implicitly thinking of in the above paragraph organize limited realms of knowledge – Newton’s laws, the Schrödinger equation, quantum field theory. Further, I’m used to thinking of those principles as both really, really good, but also temporary and local. Newton’s laws of classical physics are not superceded by the laws of quantum physics or relativity, but learning quantum physics and relativity helps us understand the boundaries of applicability of Newton’s laws. We have both a stability in knowing the places where the laws work and to what accuracy, and a flexibility in knowing that new places might be found where new laws have to be generated. These are not our basic principle – our axioms. Axioms are starting points for reasoning: principles we take as true because they are “self-evident” to everybody. Well, we now know better. Even some of Euclid’s axioms about geometry are now known to be (useful) approximations to the physical world rather than exact. Even in the abstract unphysical world of mathematics we know we have a choice of axioms for geometry.

Identifying axioms may help us understand the extent that people have irretrievable philosophical differences and where they might be able to seek out common truths in systems that appear at first to be fundamentally at odds. Let’s explore the nature of our axioms and in particular, let’s contrast the difference in the approach to axioms in the religious and scientific communities.

Religious fundamentalists want to take as an axiom that the bible is literally true – to be taken as the direct word of god. This seems strange and untenable to me. I’ve read at least the Judeo-Christian bible and have even read much of Genesis in the original Hebrew. My reaction is, “that can’t be right.” The first problem occurs right away. We’re all supposed to be descended from Adam and Eve. Cain and Abel didn’t make it to be ancestors, so we all get to descend from Seth, Adam and Eve’s other son. But who was Seth’s wife? If you start from only a single pair there has to be incest to get you started – Eve as mother for a few generations and sisters marrying brothers. That’s creepy and you don’t hear it talked about much by the fundamentalists. But even worse, that assumption makes a firm prediction – one the authors of the bible clearly did not know they were making. If all men are directly descended from Adam we should all have the same Y chromosome. If all women are directly descended from Eve they should all have the same X chromosome – and we should all have identical mitochondria. If we don’t (and we don’t) then there has to have been enough time from Adam and Eve for those genetic patterns to drift a lot – and that would imply a much too rapid rate of evolutionary change. Given what we now know about the human genome, the slow rate of evolutionary change is actually far more problematic for young-earth-bible-literalists than it is for old-earth-scientific-evolutionalists.

The only way I see of getting around this is by inventing lots of creation stuff that is not in the bible. Once you accept the incompleteness (or inconsistency) of the bible as literal truth in one place, it seems to me very hard to identify it as a basic principle.

So if I’m not willing to accept either the bible or Newton’s laws as “basic principles that underlie all knowledge”, what do I have as my epistemological axioms, the things that I accept as being fundamental to deciding what I, as a scientific rationalist, think I know?

Let’s see if I can “open the hood” of my thinking and clarify what goes on.
I suppose my first and simplest axiom is:

Axiom 1: There is a real world that exists independent of human observation.

This one seems to me a very good bet. Especially as we learn more and more about the size and extent of the universe around us. We are a very, very small part of it. It seems to me incredibly arrogant to assume that we are the whole point of the universe. Of course, there could be another universe that has some other point and we could be living in “The Matrix” – a universe constructed just for us – but this seems to me not to be a useful assumption, at least right now. Some scientists think that the laws of quantum physics require “an observer” and therefore we are essential to reality; but given my axiom 1, I find this very hard to take seriously.

So I answer the question, “if a tree falls in the forest does it make a sound?” by “yes it does” without any hesitation. Of course you have to decide what you mean by “sound”; if you mean something that is heard by an observer, then the answer is no. But if you mean something that produces the physical oscillations that we take to be “sound” then I say yes. Why would you tie an observer to sound more tightly than you do to vision? This is like asking, “if no one is there to see a rock is it invisible?” The both seem to me to be silly questions if you accept Axiom 1. (Of course, in philosophy a lot of why you might want to discuss such a question is to question the validity of axiom 1.)

Axiom 2: We each live in our own virtual reality, which is an approximation to the real world, not identical with it.

This one also seems incontrovertible. Each of us only has a limited set of information about the real world that we collect through our sensory apparatus – eyes, ears, skin. We assemble that data to create a model of the real world that we live in.

One bit of evidence for this is that through the process of science we have discovered many, many things that are undetectable to us directly through our senses. These can have great importance to our lives and kill us or save us – things ranging from bacteria and asteroids, from UV-light and x-rays to atoms and chemistry. A second bit of evidence is that psychological research has demonstrated very clearly how the brain assembles an internal version the real world – and have created strikingly powerful illusions show how we often get it wrong. Two of my favorites are Ed Adelson’s checkershadow, and Daniel Simon’s Invisible Gorilla.

So Axiom 1 says reality exists, axiom 2 says we can’t know it very well by direct observation. Axiom 3 holds out the hope that we can know it better by working together and trying to figure it out using the best tools we’ve got.

Axiom 3: Science is possible.

What I mean here by “Science” is a community activity addressed at finding out more about reality through observation and attempting to build coherent descriptions of what we see. The nature of this activity is complex and not easily described by a “scientific method” of a few steps. The critical point is that although we may use a theoretical framework to guide our observations and predictions, at base science is fundamentally empirical: built on our observations of what actually happens and our ability to weave an increasingly consistent story about how things are, behave, and work from fewer and fewer principles. I’ll write again later about my view about the nature of scientific activity and the lemmas that tell me when I define something as “science” and when not.

These aren’t so radical. Reality exists; we don’t see it directly; but we can figure it out if we’re careful. Even at this simple level I butt heads against some of the fundamentalists who want to say, “this is not the real world – what’s real is what we will experience after death.” Others will accept my 3 axioms but not the way I think I should apply them. But there is another fundamental difference between me and many religiously-based thinkers. We both believe that we need to develop a broadly accepted code of human behavior. How are we to know right from wrong? Where is the “good”? For them, it must arise from axioms. For me, the axioms are about finding out what “is”. What “should be” must be consistent with what is. IMHO, separating these two is an essential first step

Psychological symmetry or the cold really creeps into your bones

Last weekend, my wife and I attended the play at the Arena Stage in Washington. It was a one-man show, a monologue called “R. Buckminster Fuller: The History (and Mystery) of the Universe.” Now I’m not usually a great fan of one-man monologues, although as an academic I’m typically perfectly happy sitting all day in lectures at a conference if the topics are interesting. But I thought that Fuller was a thinker and his musings might be interesting. Well, I was right. The text was basically taken straight from Fuller’s writings, but it was implemented in a very dynamic way using videocameras and projection in interesting ways – showing different angles of what we were watching and supplementing it with images so it was theatrically fun as well. And the character (actor Rick Foucheux) even used some interactive pedagogy that I have been trying to use in my classroom. He sometimes turned up the lights and asked the audience questions, insisting on responses. I quite enjoyed it.

Interestingly enough, I found myself agreeing with many aspects of Fuller’s philosophy, much more than I expected. But there was one place where I strongly disagreed with what he said. At one point he said something like, “The wind doesn’t blow, different places on the earth suck!” This got a good laugh from the audience, but it wrinkled my brow. That didn’t feel right to me and I’ve been thinking about why I had that reaction.

When we describe phenomena in physics (or in any science) we typically choose a level at which to make our description. If we’re talking about the wind, we talk about the density of the air and its velocity – macroscopic concepts that we can see, measure and feel. If we’re talking about hot and cold we talk about temperature and heat flow. If we’re talking about electric currents, we talk about the electric potential difference and the current – volts and amps. We can’t see or feel those (Well – actually we can, in terms of the hair on our skin rising or feeling shocks, but we don’t have a lot of experience with that and we don’t tend to interpret those feelings quantitatively the way we do wind speed or heat and cold.) but we can measure them directly with devices we can hold in our hand.

It’s a question of ontology. Wikipedia current defines that as the philosophical study of the nature of being and the categories of being. I like to think of ontology as answering the question, “What kinds of things are there?” – or perhaps better, “What kind of concepts should I use to describe a particular phenomenon?” If we are discussing the motion of the air, the thing we are talking about is “air” – a substance that we consider has the properties that it is continuous (no gaps), is everywhere, has a variable mass density (any volume of it has mass but the same volume can have different masses in different places) and it can move. If we want to describe the causes of its motion, it is natural to try to apply Newton’s laws, which give the general principles that successfully describe the motion of essentially all classical objects. This description relies on our being able to identify forces – pushes and pulls.

Fuller’s statement about the wind makes clear that, if we are talking only at the macroscopic ontological level, there is a possible symmetry in our description: we can say that there are forces in the air that push it from place to place, or there are forces in the air that pull it from place to place. We can describe the motion in terms of repulsive (pushing) forces or attractive (pulling) ones. Newton’s synthesis of the laws of motion tells us that forces on objects are caused by other objects, so we have to identify who is doing the pulling or pushing, but that’s not a big deal. It could be the air acting on itself or it could be bits of the earth pulling or pushing on the air (Fuller’s “some places suck!”).

Another example of this kind of symmetry is heat and cold. When I am teaching about thermodynamics I ask my students to place their hand first on the cloth part of their chair and then on the metal part of their chair. They all respond that the metal feels colder than the cloth, even though we have just had a discussion that concluded when things are left to stand together for a long time they tend to come to the same temperature. We resolve this by deciding that it is the rate at which they come to the same temperature that matters in what we feel. The temperature of your hand is higher than the temperature of the chair in my classroom, so when you touch either the cloth or the metal heat energy will tend to flow out of your hand into the chair. It flows into the metal faster so it feels colder.

In temperate climates, we tend to see the “hot” as the active agent that moves. But I suspect that if we lived in a climate that was extremely cold most of the time we would see the “cold” as the active agent. When you go out in a temperature of 40 below, you might feel that “the cold is sucking the heat right out of you” or that “the cold just seeps right into your bones.” This is an ontological symmetry: we could describe things equally well in terms of the motion of “heat” (better: “heat energy”) or of “cold”. We would just reverse the direction of the flow if we switched our description from heat flow to cold flow. Everything else would look the same and it would work fine. In my classes some student often makes this suggestion.

In physics, symmetries are of great importance. If we decide that we are going to create a theory in which the placement of the origin of our coordinates doesn’t matter, then the theory we create will necessarily conserve momentum. If we decide that our theory should not depend on the orientation of our coordinate system, then the theory we create will necessarily conserve angular momentum.

But the symmetries I’m discussing here are different. They’re not really physical symmetries; rather, they’re psychological symmetries. It’s a question of how we look at the system we’re describing. Whereas physical symmetries are about what we think we are able to explain with our theories. [Aristotle made no attempt to create a theory of gravity. The result of the earth’s gravity was imposed on the theory without physical explanation. The center of the earth was the center of the universe where everything tended to go. There was a “special point” in the theory, so theory was not independent of coordinate system and momentum did not emerge as a relevant concept.]

Another nice psychological symmetry is in the construction of Newton’s laws and contact forces. For a physicist or an engineer looking at a bowling ball being hit by a hammer it seems natural to look at the changing motion of the ball as responding to the forces it feels. But for a biologist looking at an active organism, it looks like it’s the intent of the organism that causes its motion to change. If you are riding a scooter and push your foot on the ground, your backward push is what sends you forward. Since Newton’s third law says that for every force that one object exerts on another, the other objects exerts an equal and opposite force backward on the first, we could write Newton’s second law to say that an object responds negatively to all the forces it exerts instead of positively to all the forces it feels. Since mostly in physics we are talking more about inanimate objects we tend to prefer the traditional choice. [This is discussed in detail in my paper with Rachel Scherr, Newton's zeroth law: Learning from listening to our students, The Physics Teacher, 43, pp. 41-45 (2005).]

So if these psychological symmetries are really symmetries and it doesn’t matter which way we look at it, why was I uncomfortable with Fuller’s statement?

The reason is that the current paradigm of science is to relate things across levels. Up until the 19th century, most of science was about learning to describe the regularities in the world as we saw it. But when, at the end of the 19th century, we began to understand the structure of matter in terms of atoms and molecules, another level became available. We could now describe the properties of matter we had observed phenomenologically in terms of the structure of matter and what is happening to its molecules. The flow of the air is naturally understood in terms of pressure – the air pushing on itself – and that pressure can now be interpreted in terms of the average momentum and the number density of the molecules moving around. The temperature of matter that controls heat flow can now be interpreted in terms of the average kinetic energy of a molecule. At the molecular level, the ideas of pressure and heat flow have natural and simple explanations; the concepts of “sucking” and “cold flow” do not. It’s not always useful or necessary to think down to the molecular level. An electrician can be perfectly competent thinking about volts and amps without ever considering electrons. But crossing levels changes the ontology of how we think about matter and enriches our view of what’s happening. And it provides a “psychological symmetry breaking” that chooses which of our symmetrical descriptions are more appropriate.

Now when I’m teaching these physical concepts I think not just about the entire package that I have learned through my many years of studying physics. I also think about what I can expect my students to know and what might appear natural to them that appears bizarre to me. For many of my university science students, although they know perfectly well about atoms and molecules, they don’t necessarily have the idea that their observations at the everyday level should in fact be consistent with what we know about the structure of matter. This has lots of implications that I will discuss in other posts.