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Alice, these are all good questions that will ultimately generate much discussion, I think. That said, before diving into some of these, I'd like to further clarify my thoughts on demarcation, some of which might seem a bit odd, if not inconsistent...though ultimately I don't think that they are. So why demarcation, and is it art?

Well, the demarcation problem is a core problem for science because, largely, of the goals of science...mainly, to understand things, and I'll go a step further to say, to construct things. And science is useful to most folks, largely, because of the latter. Pick your favorite piece of technology, your favorite drug, your favorite mode of transportation. They work because of the consistency of the processes that govern their action. Imagine the issues that inconsistencies could cause with regard to any one of your choices above. Basically, what this means, is that for science to progress, the patterns of nature that it uncovers need to be consistent (or predictably inconsistent); otherwise, science would cease to be useful.

Demarcation amongst science, non-science, and pseudo-science, then, is a real societal problem from a practical standpoint. Because time and resources are limited, where do we apply shared wealth to tackle technological problems (say in the areas of energy, disease, etc.)? Probably toward approaches that are likely to come to solutions efficiently and consistently...scientific approaches seek, and tend, to do this. The problem is, and here is where the art comes in, when confronted with the task of distinguishing science from the other categories... the lines can become blurry quickly. What tests can distinguish these types, and do the same rules apply consistently and objectively? This is a much longer discussion to have.

All of this being said, here is where I imagine folks will think I'm stating an inconsistency (and maybe this idea will need flushed out a bit)...formal mathematical systems do not have much to say about the scientific process of discovery (through induction). Math and science are different. Math is a tool that can be leveraged by scientists to form (or in some cases validate) hypotheses, but mathematical statements themselves are merely a statements of logic with well defined symbols and grammars. Certain statements can be proven, and in some circumstances, statements can be generated that reveal inconsistencies in the system (and thus our discussions of Godel). To equate these statements with science is a little problematic I think. What we are doing is making an assumption that we have captured the behavior of some scientific observation through a particular formulation of a mathematical statement...we are making an analogy. Maybe the analogy holds maybe it does not, but to equate the process of science with the field of math is a bit dangerous.

Science makes progress through successive feedbacks between induction and deduction. Take a set of observations, attempt to generalize consistent patterns from them, apply these generalizations in new ways to make predictions that otherwise could not have been made. Math does not say much about this process. Mathematical statements do help make some hypothetical statements tractable, and from a deductive standpoint, help to formulate testable hypotheses. But we shouldn't assume that the math is real. Which gets me to Godel...the limitations of formal arithmetical systems are not fatal for science or put bounds on what we can learn from science. It may say something about what might be computed in a scientific context, but that is a different problem entirely.

For example, I was reading an article last night in the NY Times discussing some recent discoveries in physics having to do with the question, why do we exist? The 'why' that science addresses is fairly unique here. If the mathematics were correct, the Big Bang should have created equal parts of matter and anti-matter, and these would cancel each other out...yet here we are...and there is matter in the universe (oh yeah, that kind of 'why'). Science as a process (of observation and experimentation) is discovering that certain types of physical reactions, which should produce this balance between matter and anti-matter, tend to leave a little extra matter than expected...good for us, but this is an example of where the math is used as a means to codify thought in a tractable manner. The arithmetic behind these physical calculations is surely consistent, probably correct, and likely provable within the system to which it has been applied. But the math is only a representation of the science...an analogy....not reality (whatever that is). To equate the two, I think starts to travel along the borderlands of science and pseudoscience. Further, making the same analogies of science to other disciplines is performing the same operation, and can be quite dangerous (e.g., intelligent design, social Darwinism, or astrology come to mind).

This all said, with respect to other disciplines, I have no problem with (and in fact want) inconsistencies. Ask me about Coltrane or Miles (or more modernly Rudresh or Vijay) and I'll be glad for the inconsistencies of the world, both external and internal to their individual beings. For whatever reason, my nervous system is stimulated in ways that I experience as good when I hear how they are different than others, and how they tend to evolve ( or mutate) with respect to themselves. Outside of the sciences, inconsistencies (or our perception thereof) have much to offer from the perspective of aesthetics. That said, I do like Paul's suggestion of how these questions could be posed in such a way that science can make statements about them...but these might be really boring conversations for some.