Early in the Twentieth Century, Logical Positivism proposed the "verifiability principle" which states that the meaning of a proposition should be identified with the method used to verify it. From which it follows that any proposition that cannot be verified has no meaning. With one quick stroke, the Positivists could eliminate all of Metaphysics, Epistemology and Ethics. The effect of the principle was to propose that an intellectual pursuit only be treated as valuable if it could result in knowledge, and that real knowledge can only result from empirical verification.

The attraction for the thesis was felt by almost anyone who was frustrated by the attempt to understand statements in transcendental metaphysics like "The Absolute is beyond time". Even though the verifiability principle is easily repudiated by a simple embarrassing question, its attractiveness gave it the strength to be influential long after its time. The question? -- How does one verify the verifiability principle?

For those in the scientific and mathematical community who accepted the principle or some weaker version of it, it raises a haunting question. What if I discover that there are propositions in my field that cannot be verified or confirmed? It was as if this possible self doubt had ironically entered into the causal chain, for what followed were some of the major discoveries in the twentieth century concerning what would provably never be known in the field of science and mathematics.

Those who accepted the verification principle, which was designed to show that science is meaningful in ways in which other fields could never be, were about to discover that there are regions of science which are in the same metaphysical boat. Consider the following fields...

Quantum Physics. Historically, the well known Heisenberg principle of uncertainty is the first encounter with the unknowable. What Heisenberg proved was that one cannot know the precise position and momentum of an atomic particle at any given time. The process of observing or gathering information about the atomic, no matter by what method, is, by its very nature, an interruption that affects the quality of the data in a predictably unpredictable way.

If we could in some way learn about this system without changing it then there would be no problem. However in atomic physics, in order to study the system we inevitably affect it. In a sense we are a pervasive part of the system that we are examining and this recursiveness is the source of the unavoidable uncertainty. Even when seeking knowledge in the most remote area, we need to always be aware that we are the part of the universe that investigates itself.

Those subscribing to the positivist position needed a way to understand this result. Their conclusion: The electron does not have an exact position and momentum at the same time. It is not meaningful to say that it does. Instead, it has a probability cloud that represents its possible positions and the probability of being in those positions. In reality, the electron is not in any single place. Thus, we can again be assured that there is nothing meaningful that cannot be known. This imaginative remedy to correct what was unknowable and make everything knowable again has the unfortunate consequence that probability must be thought of as a feature of the world, not a relation between the observer and the world. I say 'unfortunate', because if probability is now part of objects in the world then it is only "probability" in a somewhat metaphorical sense and doesn't have much to do with the sense of probability that we have come to understand over the last few centuries.(1)

Relativity Physics. The theoretical possibility of a black hole was shown long before black holes were suspected of actually existing. Black holes are objects that are so massive that their gravity will not let light escape. If light cannot escape their boundaries then nothing can. So, there is in principle no way for us to get any information from inside the "event horizon" of a black hole and thus, by its very nature, there is no way to characterize any event from within. The positivist would conclude that there are no events inside a black hole. Others might say that we must accept that it is in principle impossible to know anything about such events.(2)

Cosmology. The big bang theory, the most widely accepted cosmological theory concerning the beginning of the universe, presents the view that there was a time when the whole universe was smaller than a pin point. It exploded and thereby became the universe as we now know it. Those who handle the mathematics have shown that within this model there is no way to characterize the first few instances of expansion. These moments and anything about them, in principle, cannot be known.(3)

Mathematics. Gödel's most famous theorem is that any rich consistent mathematics will contain true statements that cannot be proved. (Curiously enough, Gödel was one of the founders of Logical Positivism.) This theorem, since it is a theorem, is provable. The proof itself is complex but uses an interesting trick. Since mathematics is a recursive language, we are capable in mathematics of giving every possible statement a number. After this is done, we may construct a mathematical statement that is about all of the statements with a number of a certain kind. We could, for instance, create a mathematical statement about all of the even numbered statements.

This consequence seems most peculiar when we realize that the statement about all even numbered statements itself has a number. The number may even be an even number. Because it is possible for mathematical statements to represent anything in the language, including those statements that refer to mathematical statements, i.e., statements in the meta-language, it is thereby possible to construct statements that refer to themselves. It is common in mathematics, for instance, to make a statement about all provable statements, and this statement itself may or may not be a provable statement.

What Gödel was able to show is that a statement saying of itself "I am not provable" is constructible in mathematics. Let us consider what this implies. Suppose this statement is constructed. Thus, it is either true or false. If true, there is a true statement that cannot be proved, namely this one, and thus the system is not complete. If it is false, there is a statement that is provable but false, namely this one, and thus the system is not consistent. Therefore the system of mathematics is either incomplete or inconsistent.

The positivist is likely to have no qualms with this result, since mathematics was never thought to have any real content anyway. Nevertheless, there are mathematicians with positivist sentiments who feel uneasy with these results and thereby declare that if a statement cannot be proven as true and cannot be proven as false then it is neither true nor false. Since it has no truth value, its truth value is not unknown. Very convenient!

Chaos theory. Chaos is an interdisciplinary science that studies dynamic systems which are not linear. In a linear system, a small change produces a correspondingly small systematic change. However, a nonlinear system is extremely sensitive to initial conditions, thus small differences in initial conditions, measurable or not, can lead to consequences not only of different magnitude but also to outcomes that are different in kind.(4)

Imagine, if you will, two equally massive, perfectly spherical objects orbiting each other.(5) There is a point equidistant between them where a perfectly spherical object could be placed in precise suspension for an indefinite period of time. It would have no tendency to move one way or the other. What Chaos theory emphasizes is that, if we do not place it on the single mathematical point exactly between these two spheres, the object will soon fall into one of the spheres.

Of course, no method of placement could be that precise. If in placing the object we place it so close that we don't know which way we err, then it becomes impossible to predict which way the object will eventually fall. Since measurement is always with some error, no matter how small that amount, perfect prediction is not possible. Furthermore, even if we possessed a tool for precise measurement and placement, if successive calculations were necessary to predict future positions, then the calculator must be able to maintain in its memory an infinite number of places of accuracy or it too will soon be significantly wrong. The discouraging feature is that this result follows even in this most ideal of situations, with perfect spheres, a perfect universe and perfectly known equations.(6)

Fractal mathematics, considered to be the mathematics of Chaos theory, has many classes of mathematical objects that are infinitely complex. Objects of Fractal mathematics, called fractals, are created by analyzing what happens when one repeatedly and indefinitely takes the output of an equation and feeds it back into the equation as input. This cyclical process of recursion is the source of a fractal's interesting features. When we consider a partial section of an ordinary non-fractal figure and magnify it, we obtain a simpler structure. When one magnifies a polygon, for instance, one is in the world of straight lines. However, by magnifying a fractal object in the appropriate area, we obtain an object that is no less complex than the original. A fractal is by its nature infinitely complex.

Imagine a fractal to be multicolored. Because the mathematics of fractals is capable of characterizing infinite complexity, it can be shown that there are fractals where each ideal point has a single color, but any area, no matter how small that area is, contains more than one color.

There are physical systems that are considered to be best modeled by such fractals. Thus, you might imagine by using the analogy of the sphere example above that a multicolored fractal could have been colored so that objects that begin on red points fall into sphere A, and blue points fall into sphere B. However, with this fractal there is no small partial area within it that contains only red points or only blue points.

Thus, there are situations where it is in principle impossible to predict a result, even if you were to know the initial conditions with the accuracy of the best instruments imaginable and even if the equations of behavior were perfectly known. Furthermore, unlike the earlier example of orbiting spheres, predictability in this system is not just a local problem near a single point, but it is instead a problem everywhere in the physical system.

So Chaos theory implies that there is a necessary limit to what can be predicted even under the most ideal of circumstances, drawing another boundary around the knowable. The culprit is often the feedback of recursion, the iterative process that makes a relationship like y=x2 (x squared) look like y=x4 in one generation and y=x8 in the next. Small differences become large very fast. An error in the 100th decimal place soon becomes an error in the 10th and then the 1st.

Logic. Formal logic is arguably the purest discipline. Mathematics and Set Theory presuppose logic in their methodology, which are in turn presupposed by all of the sciences, and yet it is provable that an unknown resides in the midst of this purity. There is no problem when the logical objects studied are statements taken as indivisible units in what is called the sentential calculus. For this simple language, most often seen in the context of truth tables and sentence connectors like 'and', 'or', 'not' etc., there is a procedure that will decide for any argument whether or not it is valid. However, when we consider adding things called quantifiers, which make a logic just slightly more complex, clouds start rolling in.

In English, one can think of quantifiers as those logical modifiers that attempt to capture the meaning of "all" and "some", the same logic that Aristotle explored a bit. More formally, the universal quantifier is used to represent "for each and every thing in the universe", the existential quantifier would represent "There is at least one thing in the universe, such that: ". This is not an extremely high level language. It need not introduce notions of identity, set membership or properties of properties. Nevertheless, a problem of decidability arises.

Alonzo Church's undecidability theorem states that there exists no one procedure for deciding the validity of any argument in the system. What this means is that you could not program a computer so that every time you give it a formal argument, it responds with 'valid' or 'invalid'. You could program it with three options-- 'valid', 'invalid', 'don't know'. If this is well done then most of the time the answer would be 'valid' or 'invalid', but sometimes by necessity it would be 'don't know'.

The proof again involves a clever trick of recursion. If we assume that there is a procedure that decides validity, we can form a formal argument mirroring the procedure in certain key respects to guarantee that the procedure will not work on that argument. You might think that you can use a different procedure for that one, but as soon as you give me all the procedures you are willing to use, a new undecidable argument can be formed from the procedures taken as a whole. Conclusion: There is at least one argument whose validity will always be in question.

Computer Science. After the above example in logic, it should come as no big surprise that there would be similar problems in computer science. The interesting result in computer theory, called the halting problem, is concerned with the classification of computer programs into two categories: programs that eventually stop and programs that go on forever. Again by recursive techniques it has been shown that a program cannot be written that will tell us which programs will go on forever and which shall eventually halt. Again, the proof involves a tentative assumption that there is such a program and then constructs a variation on this program to provide the undecidable case.(7)

Economics. In 1995, the Nobel prize for Economics went to a Dr. Robert Lucas who showed a most interesting result. To put it succinctly, the expectations of an individual influence his/her economic behavior. This result seems obvious and harmless enough until one realizes that much of the making of economic policy is based upon a more static view of the consequences of its policy.

When the Federal Reserve Board, for instance, meets to decide whether to raise or lower interest rates, it does so to bring about some perceived desirable consequence. However, even before the board meets, news of the plan to meet, itself, often will bring about consequences in the economic markets. These consequences can easily be contrary to the aims that were the original reasons for the board to meet.

To take a simple example, suppose you have an economic theory that predicts that the market will sell off wildly tomorrow at 3:00 p.m. Eastern time, causing prices to fall. If your theory is correct, then the prediction will be true and you'll make money. However, suppose that you decide to go on national TV this afternoon and share your findings with the rest of the world. Suppose, also that you have good reasons for your theory and because of this everyone believes it.

This prediction will be self-defeating. Some will not be willing to wait until 3:00 p.m. They may worry that others who believe the theory might not wait. They will sell in the morning. It is not too difficult to imagine that the prediction itself brings about a market crash far in advance of 3:00 p.m., thereby invalidating the theory. The conclusion is clear, such a theory cannot be accurate if it is shared with the community whose behavior it is attempting to predict. However, if the theory has faulty predictions, then it should be rejected.

It might seem as if the difficulty was the fact of the announcement, but instead the problem is built into the heart of what a theory is. If someone created this theory and found it to be well confirmed then someone else could have discovered it also. Clearly, the theory cannot be an accurate general model, for it cannot handle successfully the situation where others also understand and believe the theory. Recursion, in the form of feedback is again the source of unknowability in the area of economics and tells us that in principle it is impossible for there to be an acceptable general economic theory.(8)

In contrast to the above prediction, let us consider another prediction that may have the complete opposite effect. Suppose our theory says that it is rational to buy Apple stock at $34 and it is rational to sell it at $35. This is great. There is money to be made. Anytime it swings to $34 or lower we buy and anytime it reaches $35 or higher we sell. But wait. Suppose everyone believes and has access to the theory. This implies that anyone would be irrational to sell their Apple at $34. As a result the opportunity to buy Apple at $34 will never arrive. So with the Stock prediction the result is true but useless, and for the market crash prediction the result is significant but false.

Comparing economic theory to earlier examples of the unknowable, a feedback loop with greater viciousness becomes apparent. Suppose there is a fairly good economic theory, one that is not perfectly general but is at least acceptable. If the phenomena that this economic theory is intended to predict and explain is the phenomena of human action and human action is largely based upon beliefs and one of those beliefs makes reference to which economic theory is true, then every prediction and explanation will involve recursion, a feedback that can potentially destroy the prediction/explanation. Thus, even a fairly good theory will be hard to come by, since it carries within it the seed of its own destruction.

1 It makes little sense to refer to the probability of something being in a certain spot without it being possible that it actually be there.
2 When we say "nothing can be known about those events", we are, of course excluding statements like this one, since this statement is itself something we know about events.
3 Same as footnote #2.
4 Actually, some non-linear systems suppress error while others expand it. Chaos is usually known for its research with expanding error.
5 Exclude from consideration any other entity in the Universe.
6 It has been proven that the motions of the nine planets and sun cannot be shown to be an indefinitely stable system, even when considered in isolation from the rest of the Universe.
7 Some take the undecidability theorem of Logic and the halting problem of Computer Science to be the same problem.
8 Curiously enough, but maybe not surprisingly, there are economists who label themselves as contrarians. They believe you should sell a stock when all of the experts say "buy" and buy when the experts say "sell". If most economists were contrarians, they would be out of a job.
9 And Sociology and Political Science.
10 A theory is like a computer program where the input is the initial conditions and this input is used by the theory to create the output (i.e., the prediction).
11 Some theories of psychology use the notion of "information states" instead of "beliefs", but the argument remains the same, since there is no limit on the complexity on information states and the output of a theory is able to change an information state.
12 Instrumentalists take prediction and control to be the aims of science. They focus upon questions of "how" instead of "why".
13 Also, the initial conditions should not have been inferred from the fact of the event being explained.
14 Logical Empiricism is a more mellow descendent of Logical Positivism
15 Our concept of "electron" may change over time but this does not change what actually exists
16 A science must give this kind of assurance.
17 Philosophy is recursive on a regular basis. For any subject in Philosophy, there is a context in Philosophy where it is appropriate to ask questions about the adequacy of the subject. It has been convincingly argued that there is no clear distinction between Philosophy and Meta-Philosophy.
18 My thoughts on this have been influenced by Douglas Hofstadter in an article that introduces the self-modifying Game :Nomic, in his book "Metamagical Themas"
19 Rules about how to change rules are a part of any institution that was established to be flexible enough to survive in a changing world.
20 In Political Science "Institutionalism" comes closest to imposing these limitations.
21 It has been argued that if Democracy does not meet any single one of these conditions that it loses some of its ethical appeal or moral authority.
22 Arrow regards the possibility of this situation as unacceptable in democracy or inconsistent with the idea of democracy.
23 Some would say "tricky".
24 Rationalism, a political theory, is such a theory and focuses upon maximizing social gain. It is a kind of political Utilitarianism.