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The Mathematics of Self Reflection

Mathematicians, in developing their fundamental axioms, search for the most powerful and inclusive assumptions. They then look to simplify these without limiting their power. The aim is to derive as much mathematics as possible from the simplest axioms. This led to the idea that every property should define a set of objects that satisfy the property. Gottlob Frege developed a formalization of mathematics in which one could, in effect, define the set of all sets that do not contain themselves. Assuming such a set exists leads to a contradiction. If the set contains itself it cannot contain itself and vice versa. Such self referencing or impredicative properties are both extremely useful in constructing powerful mathematical axioms and a potential source of inconsistency.

Defining powerful consistent self reflecting structures is an open ended problem with no finite solution. That follows from Gödel's Incompleteness Theorems. Gödel proved that any system powerful enough to embed a general purpose ideal computer12 (or equivalently the primitive recursive functions) must be incomplete or inconsistent. He did this by first showing how such a system must contain, as a valid proposition, the question of its own consistency. He then proved that, if the system could prove this proposition was true, then the system must be inconsistent with a paradox similar to, but far more complex than, the one that plagued Frege's formalization of mathematics[12].

Gödel's results led to a hierarchy of mathematical truth involving ever more complex levels of abstraction and self reflection. No finite axiomization of mathematics can capture more than an infinitesimal fragment of this hierarchy.

Many mathematicians have embraced a Platonic philosophy of mathematical truth that contrasts with the open ended implications of Gödel's results. This philosophy postulates a hierarchy of infinite sets that embody absolute mathematical truth that cannot be decided by finite means. Today mathematicians think it unlikely that set theory is inconsistent, but many of them are having doubts about the objective validity of the hierarchy of infinities that seem to be far removed from anything that could conceivably exist in physical reality. It is a philosophy that has always been questioned by some mathematicians.

The intuition that justifies the set of all real numbers in contemporary set theory is, in some ways, similar to that used to justify the parallel postulate or the claim that spatial extension is fundamental and irreducible. These assumptions seem intuitively obvious to some as if human intuition is sometimes the ultimate arbiter of truth. Increasing skepticism about this philosophical approach is raising doubts about one of the most important open conjectures in mathematics, the Continuum Hypothesis.13 One example comes from Solomon Feferman, the editor of Gödel's collected works.

I am convinced that the Continuum Hypothesis is an inherently vague problem that no new axiom will settle in a convincingly definite way.14 Moreover, I think the Platonic philosophy of mathematics that is currently claimed to justify set theory and mathematics more generally is thoroughly unsatisfactory and that some other philosophy grounded in inter-subjective human conceptions will have to be sought to explain the apparent objectivity of mathematics[13].15

Regardless of what we will ultimately conclude about infinite sets, the lessons about finite self referencing structures from Gödel's results and subsequent mathematics are clear. Formalizations of mathematics are finite rules for enumerating theorems. The combinatorial self referencing structure of these rules limit the power of the mathematical system they define. It is reasonable to assume that related self referencing combinatorial structures in the brain limit, to some degree, the depth and subtlety of human thought, perception and consciousness.

There are practical consequences of these limitations. No finite physical process, can solve the computer halting problem16 for all computer programs. The halting problem is an example of asking whether some event will eventually occur. No finite physical system can, in general, solve this problem, even in a deterministic universe where initial conditions are known exactly. If the event does occur one can determine this eventually, but one needs an infinite amount of computation to determine that it will never occur.

There may seem to be a huge gap between predicting what an ideal computer (with unlimited memory) may do and practical problems of human survival. However more powerful mathematical systems can be used to solve more problems more efficiently than weaker systems because they allow work at higher levels of abstraction. The ability to solve more halting problems in the abstract translates to an ability to deal with practical problems with more depth and efficiency. The human brain does not work like a formal mathematical system, but it does have a structure capable of subtle self reflection and abstraction that made the human creation of mathematics possible. That capacity could only have evolved to solve practical problems of survival.

The human brain contain a great deal of biological machinery dedicated to making good decisions. This is especially important in a world inhabited by creatures with similarly complex brains. The message from mathematics about this arms race in mental capacity may not be what one first suspects.

Gödel's result was a shock to the mathematical community because it dashed forever the hope of coming up with a single formalization of all mathematical truth. Mathematics was seen by many as the one source of absolute truth in a mostly uncertain world. There is only one way around the limitations of Gödel's proof in a finite universe17 and is not one widely embraced by the mathematical community but, in a sense, it has been embraced by biological evolution.

Any single path approach to expanding mathematics is bound to run up against a Gödelian limit. Within such a limit, progress can be made forever, but the entire sequence of results obtained over an infinite time can be fully captured in a finite axiom which will never be discovered. It is only a divergent process that follows an ever increasing number of paths that can avoid a Gödelian limit. There are two boundary conditions essential for exploring all mathematical truth. The first is ever expanding diversity and the second is ever more resources devoted to each viable path.18 The immense diversity of evolution on this planet and the enormous concentration of resources in the human brain and nervous system are examples.

The dialectic between diversity and concentration of resources is a universal theme in creative processes. Carl Jung, in defining the modern usage of the psychological terms, intravert and extravert, observed that it applies to these psychological dispositions and to the fundamental strategies for reproductive success.

There are in nature two fundamentally different modes of adaptation which ensure the continued existence of the living organism. The one consists of a high rate of fertility, with low powers of defense and short duration of life for the single individual; the other consists in equipping the individual with numerous means of self-preservation plus a low fertility rate. This biological difference, it seems to me, is not merely analogous to, but the actual foundation of, our two psychological modes of adaptation [intraversion and extraversion][16, ¶559]

Jared Diamond in Guns, Germs and Steel[10] observed a similar creative dialectic between diversity and concentration of resources in cultural evolution. He investigated why certain cultures came to dominate the planet while others remained relatively stagnant. One needed an appropriate balance between diversity and concentration of resources for modern civilization to arise. A culture dominated by a single ruling elite, like China, inevitably failed to pursue possibilities essential to future development. In contrast, a region, like Africa, with so many small communities, could never marshal the resources needed for certain kinds of progress. Europe presented the ideal combination of diversity and concentration of resources.

There are many reasons why a good tradeoff between diversity and concentration of resources may be advantageous. Mathematics proves expansion of both is a fundamental logical requirement for unrestricted creativity. The Totality Axiom and the rules of consciousness suggest that the same tradeoff applies to the evolution of consciousness not just historically but into an indefinite future.


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