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Jolly Mathen: Incompleteness and Scientific The...

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Jolly Mathen is an independent philosophical researcher residing in San Fransisco. His paper On the Inherent Incompleteness of Scientific Theories is a fascinating look at the consequences of some important concepts in the philosophy of science and mathematics, which I was able to question him on recently.

- Interviewed by Paul Newall (2005)

PN: Can you explain the basic aim of your paper?

JM: Primarily to provide an argument for the incompleteness of scientific theories. Secondarily to show connections between scientific incompleteness, belief, arbitrariness, self-reference, and some ideas in the philosophy as science, such as the Quine-Duhem thesis, the underdetermination of theory, and the observational/theoretical distinction failure, and some ideas surrounding the concept of mathematical incompleteness, such as complexity, infinity and computational irreducibility.

PN: What do you mean by an "incomplete" scientific theory? In particular, you claim in your paper that "every field of scientific inquiry stands incomplete." Why?

JM: By an incomplete scientific theory, I mean that there will always be non-trivial questions left unanswered by the theory. In regards to my claim, "every field of scientific inquiry stands incomplete", it is just an observation of the present situation. Some scientists may believe that they may be close to a complete or final theory, but I don't think any of them would claim that their theories are presently complete; they are obviously still working on it. Can anyone today identify a complete scientific theory? Of course, in my paper I am arguing that, even in the future, scientific theories will remain incomplete.

PN: You point out that a problem for scientific theories is the "experimental dilemma"; that is, whether novel observations will ever cease and allow us to call a theory complete. Do you think this requirement can ever be met?

JM: Any scientific measurement is always accompanied by an error. That is, every measurement is taken with a certain amount of precision, which can always be improved. Because this is characteristic of every scientific measurement ever performed, I think we can take it to be a fundamental principle. This is just an inherent feature of scientific measurements and, more broadly, of our every day natural experience. We can always zoom in and get a more detailed picture of any natural phenomena. Some contend that this may come to a stop someday, for example at the ultra-subatomic level of the Planck scale; but I don't think so. In this sense, I don't think that novel observations will ever cease. Besides precision, I identify in my paper three other aspects of a measurement that can be tuned to give us a new look at the phenomenon under consideration: perspective, range and interaction.

The more important question though is whether the continued improvements in the precision of any measurement will reveal any surprises or not. If not, then the novel experience issue becomes a moot point. Presently no scientific theory, or meta-scientific theory, can guarantee that more precise measurements will continue supporting a theory. Unless such a guarantee is given, it will remain uncertain whether improved measurement precisions will result in surprises or not. Because of the arguments and results in my paper, specifically the theorem of undefinability of valid observations, I take it that no such guarantee can be given, and therefore novel experiences are a point of concern and not moot. Strictly speaking, however, the empirical evidence alone does not require such an interpretation. I believe that it is worthwhile to examine the issue further and see if there is a more clearer connection between novel experiences, the theorem of undefinability of valid observations and theoretical incompleteness.

PN: In your paper you compare the philosophical debate on the existence of God with the scientific debate on whether a Theory of Everything (TOE) is possible. Why are the two related?

JM: On a surface level these questions are related because the demonstration of a miracle (which would prove the existence of God) would depend on our having a complete TOE (or complete theory on any specific domain). For example, when Moses parted the Red sea, how do we know that some alien race whose knowledge of physical laws is far superior to ours was not orchestrating some complex technological drama for their curiosity and amusement? The only way we can know that this is not the case is that if we understand those laws completely. For then we can say whether the parting of the sea under the given circumstances is a physically possible event or really a miracle. (For those of you who are Star Trek fans, there was a Star Trek Next Generation episode where a female goddess was subjugating the people of a planet to all sorts of catastrophic "miracles" until the Enterprise discovered that the "goddess" had an invisible ship orbiting the planet.)

On a deeper level these questions are related because belief (religious or otherwise) and understanding are related. For instance, one might ask, "Can't I still believe in God in spite of having achieved a complete physical theory? What do my beliefs have to do with scientific theories?" The argument here is that our cognitive capacity to believe prevents our cognitive capacity of understanding from being complete. Theories fail to be complete because of our capacity to doubt them and believe in God (or some other theory); or, vice versa, belief is only possible because of the inherent incompleteness of our ideas about the world.

PN: As a result of this comparison, you conclude that the existence of God is "empirically undecidable". Can you explain what this means, how you argue for it and how this position differs from the traditional atheist/theist/agnostic spectrum?

JM: By "empirically undecidable" I mean that no matter what we observe or experience, as long as our scientific theories about those experiences remain incomplete, we cannot use those experiences to argue for or against God's existence. In other words, proving whether some observed phenomenon is a miracle or not is an impossible task in the light of scientific incompleteness. From the answer to the above question we can see how proving whether some observed phenomenon is a miracle is impossible as long as our understanding of physical laws remains incomplete. By similar reasoning, we can also see that as long as this understanding remains incomplete it is not possible to rule out the role that God plays: observations that the incomplete theory is thus far unable to explain maybe ultimately due to God; moreover, the theory being incomplete may then turn out to be actually incorrect, thus possibly ceding the explanation of all our observations ultimately to God.

Since I argue in my paper that scientific theories are incomplete for cognitive reasons, then because of the above implications I am also arguing that the question of God's existence is undecidable for cognitive reasons. Therefore, on reflection, I guess the argument supports an agnostic position.

PN: In your paper you relate the incompleteness of scientific theories to G del's work in mathematics. How closely are the two related?

JM: They are very much related, but not in the manner of the traditional argument stating that any math based physics is incomplete owing to G del's proof that arithmetic is incomplete. Scientific theories are incomplete regardless of whether their mathematical models exhibit undecidability or whether they even have mathematical models. Let me elaborate. Since Goedel's time, mathematicians have discovered many other systems that exhibit incompleteness and undecidability, for example, computers and cellular automata. By studying these systems, they have been able to identify two fundamental features necessary for incompleteness: infinity and complexity. Using arithmetic as an illustration, if we limit the natural numbers to some maximum, no matter how large, say, for example, one billion, then arithmetic can be given a complete description. The same holds true if we lessen the complexity of arithmetic by removing either the addition or multiplication operation (even if we allowed an infinite amount of numbers).

I suggest in my paper that the interaction between nature and our sensorial-cognitive system gives rise to processes that are also complex and infinite in character, thereby preventing our experiences and the theories of our experiences from ever being complete. We can easily see how infinity is involved. As pointed out above, we can always have novel experiences of natural phenomena, by either improving the precision of a measurement or by tuning its range, perspective or interaction. Thus we can continually count up new experiences as we can count up new numbers. How complexity is involved is more difficult to assess at this point. Besides the complex processes going on within our sensory-cognitive system, I would also guess that languages used to describe our natural experience must have a minimal complexity. The complexity question is certainly an area for future research.

Besides infinity and complexity, there are a couple of other features that we can associate with incomplete systems: self-reference and arbitrariness. Self-reference is a central feature in both Gödel's mathematical proof and in my demonstration of the undefinability of valid scientific observations. In both instances, it is employed in a manner similar to the construction of the liar paradox, "This statement is false". Incomplete systems are also characterized by the fact that they lead to multiple arbitrary formulations, without singling out any one true formulation. For example, there are multiple formulations of arithmetic, set theory and geometry. In science, arbitrariness was recognized early on in the 20th century by philosophers as the underdetermination of theory, which states that it is always possible to have multiple theories on a given domain of phenomenon. The relationship between incompleteness and arbitrariness is this: because our language cum theories cannot completely capture our experiences (or mathematical ideas), we must allow for flexibility and mutability in its usage. In summary, we see that mathematical and scientific incompleteness share many features in common: infinity, complexity, self-reference and arbitrariness.

PN: What is the relationship between incompleteness, theory-ladenness and underdetermination, in your view?

JM: The underdetermination of theory is a philosophical position that states that an observation(s) does not determine a unique theory, but allows for multiple competing theories. Deservingly so, it has been a point of much contention and confusion. How is it possible that two or more mutually inconsistent theories can possibly describe the same observational data? I argue that it is only so because the observational data is incomplete; if the observational data on some domain could be complete, then only one unique (class of) theory can be supported. Let me make an analogy. If you were given a low resolution image of a photograph and asked to guess what it represents, you may entertain several possibilities. But as the resolution of the image improves, the possibilities that you're willing to entertain becomes less and less, until finally the resolution matches that of the human eye and you can see exactly what the image represents. I suggest that in science we are always looking at a low resolution "image" of our experiences, which we are continually trying to improve the resolution of (for example, by making our experimental measurements more and more precise), but never reaching perfection. Due to the cyclical nature of scientific progress, there are times when the observational data on some domain appears almost complete and times when it is found wanting. During the prior times, scientists will settle on one theory (assuming, of course, that any technical hurdles in candidate theories are resolved) and any claims of underdetermination would only fall on deaf ears, whereas during the later times, scientists are willing to entertain multiple theories.

Incompleteness also makes clearer the much discussed connection between underdetermination and theory-ladenness, or holistic models of science. If an observation is incomplete, then the description of the observation is also incomplete. The observation and its description, like the low resolution photograph, are both fuzzy around the edges. This as a result undermines our ability to assign a unique observational term to the observation. At the same time, it allows us some play in the description. We can ply and mold it along its fuzzy edges to fit different theories. On a higher level, the descriptive incompleteness of multiple observations gets translated into an integrated theoretical incompleteness. All the language terms in the theory, being incomplete, are now laden with one another--observational and theoretical terms are inter-laden with the likes of both--resulting in a holistic web of inter-laden terms, in which there is a massive and multidimensional pliability along the fuzzy edges of the new theoretical superstructure. A pliable holistic model as required by underdetermination is therefore granted by the inherent incompleteness of the observational and theoretical terms occurring in the language of any theory.

As a note, I would like to add that theory-ladenness is more fundamental than incompleteness and under-determination, and is the cause of the later two. The theory-ladenness of observational terms, or the observational/theoretical distinction failure, is the scientific manifestation of a cognitive symptom: the inseparability of sensory and thought processes, an issue that, like the observational/theoretical distinction failure, is much debated. I think that it is a worthwhile program of future research to determine if this inseparability in fact exists and why, and how it leads to incompleteness.

PN: Can you explain briefly what you call "the problem of the undefinability of valid observations" and its relationship with other critiques of completeness?

JM: Our scientific theories are supported or refuted by our observations about the world. Therefore we need to have a clear idea when a certain observation has taken place. At first, one may say what is the big deal, isn't it obvious? Some philosophers and cognitive scientists have argued it isn't; our theories and background knowledge affect what we see with our naked eye and how we interpret our scientific instruments--the issue of theory-ladenness. Whether this is so and the extent of it is much debated. Second, even if we can distinguish observations independently of our theories, can we clearly distinguish among different observations? For example, is some large bush perhaps really a tree? Can we offer distinguishing criteria, perhaps based on sub-definitions of the tree's components, that will be sufficient? Finally, can we give definitions of observations that are thorough enough so that we can't be fooled by the best technological imitations or even some virtual reality simulation? (After all, we can't have some imitation observation dupe us into thinking that some scientific theory is true or false.) The above, taken as a whole, is the problem of determining valid observations.

Many of the issues we're talking about here, such as completeness, under-determination, theory-ladenness, and the identification of miracles, come to a head on the ability to determine valid empirical observations. For example, if we could determine valid observations, then we could assign definite, theory-free observational terms to them. In my paper, I produce a self-reference argument to show that there can exist no scientific procedure to determine valid empirical observations. The novel experience problem, or observational incompleteness, also lends some credit to this conclusion: because observations are always incomplete, or fuzzy around the edges, we can never make a clear determination of their occurrence.

PN: You give several critiques of the notion of completeness. Which do you consider the strongest - and why? Is your argument cumulative or does it follow from any of the objections to completeness?

JM: Presently, the two strongest reasons are the theorem of undefinability of valid observations, which is based on a self-reference argument (as just mentioned previously), and the novel experience problem, which is based on the observation that the precision of all scientific measurements can always be improved. The argument does not have to be cumulative, but a cumulative argument, as given in my paper, can serve to flesh out the connections between the many different critiques.

PN: Your paper explores the philosophical pedigree of your thinking, concentrating in particular on the Duhem-Quine thesis. Can you explain this thesis and why it was important to your argument?

JM: The Quine-Duhem thesis is a generalization of the under-determination of theory and, like it, presupposes that all the observational and theoretical terms, concepts and laws of a scientific theory are interconnected in a holistic web, and that by suitably modifying aspects of this web, any theory can be accommodated to any observation. This curiously, and alarmingly, makes science a somewhat arbitrary affair, and has therefore become a hotly debated issue. The Quine-Duhem thesis is important to my paper on two points. As mentioned above, scientific incompleteness, like mathematical incompleteness, requires theories to have a certain amount of arbitrariness to them. The Quine-Duhem thesis fulfills this requirement. In fact, the pliability implied by the Quine-Duhem thesis, like in the case of the underdetermination of theory, is due to the incompleteness of the observational and theoretical terms occurring in a theory.

The Quine-Duhem thesis also brings to the surface the role played by belief in science. As I mentioned earlier, scientific incompleteness and belief are joined at the hip. But this is not restricted to religious belief, but any kind of belief, even a scientific one. In fact, the two components of belief, faith and doubt, find exact parallels within the Quine-Duhem thesis. The Quine-Duhem thesis itself has two components. The first component is the underdetermination of theory, which again states that an observation(s) does not determine a unique theory, but allows for multiple competing theories. The second component is the underdetermination of observation, which states that a theory can accommodate multiple incompatible observations. As you may guess now, the underdetermination of theory plays the role of doubt: we can doubt some theory in favor of some other theory. The underdetermination of observation plays the role of faith: we can always hold onto some given theory no matter what the observational evidence.

In light of this parallel, we can draw another parallel between the God existence debate and the Quine-Duhem thesis. The inability to prove that God does exist can be likened to the inability to prove whether any given theory on a given domain of phenomena is the correct one. The failure is due to doubt or the underdetermination of theory. Second, the inability to prove that God doesn t exist can be likened to the inability to prove whether any given theory on a given domain of phenomena isn t the correct one. This failure is due to faith or the underdetermination of observation.

PN: What consequences do you see for your paper and the critique of completeness?

JM: Some are: (1) The study of meta-science can tell us something about our ability to believe. (2) That no matter what we may observe, "miracle" or "scientific fact", we can never prove that God exists or doesn't exist. (3) That we will never achieve a theory of everything in physics nor a complete theory of any domain of phenomena, i.e., of chemical interactions, genetics, cancer, star formation, evolution, etc. (4) That our understanding (theories) of all natural phenomena will continue to evolve. (5) That we can't rule out the development of seemingly impossible technological advances, such as faster than light travel and anti-gravity devices.

But more importantly, what do the above consequences tell us about sentient entities like ourselves and how they sense and understand the world around them? About living organisms and how they interact with their environment? It seems to me that there is something peculiar and deep going on here. I think there are bigger questions ahead.

PN: What are your next projects? Will you be continuing to work on the ideas in this paper?

JM: This question dovetails on the end of my last response. So let me expand on that. For starters, how is it that our experiences can never be complete, that is, our scientific measurements can never be 100 percent precise? Is this purely an extraneous feature or something that arises out of our interaction with the external world, and, if the latter, how does it arise? Another question is whether cognitive processes are necessarily wrapped up with sensory process and, if so, why? I think that cognitive science would be an excellent avenue of pursuit for these questions.

Also, the study of language and formal systems may be able to tell us something about the complexity requirements, if any, of scientific (and natural) languages. (As I mentioned earlier in the interview, a minimal level of complexity is a requirement for incompleteness.)

Last, I think that quantum physics may be able to shed light on scientific incompleteness. Quantum physics has taken science to the point where the role of the observer has become an integral part of the theory. It doesn't merely tell us about the world, but about our knowledge of the world. Further research may then tell us whether this knowledge can be complete or not. (In this spirit, the uncertainty principle and the novel experience problem may appear related, but, non withstanding further investigation, I can only see a superficial connection between the two.)

PN: Why are you interested in the philosophy of science? What prompted you to consider this issue at all?

JM: The philosophy of science, like epistemology and cognitive science, is interesting to me because it addresses the nature of knowledge and understanding itself, one of the great mysteries. I didn't pursue this interest seriously in the past because I thought philosophy to be too speculative and, at the same time, I was already taken by the mysteries of physics. Then about four years ago I decided to finally find out what was all the fuss concerning G del's incompleteness theorem in mathematics, a result that I had only heard about here and there but never really understood. What G del had discovered stunned me. How could such a formal and logically tight system as mathematics be eternally incomplete? To me, this said something about the nature of knowledge. It harked back to some ideas I had during my philosophy and history of science courses as an undergraduate. The new-found mystery and the rekindling of old ideas prompted me to investigate whether incompleteness was a more widespread phenomena, and what were its causes. My paper represents a momentary culmination in this on-going investigation.

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