The Mathematics of Cognition .Beyond the Network A primary attribute of our world from the perspective of immaterialism is its relational structure. Objects and events have no meaning or substance when taken out of context. The meaning of any part of the world depends on the meaning of many other aspects of the world. This is the problem with artificial intelligence. Unlike natural intelligence, it operates by dealing with only a small aspect of reality at a time, and so is easily confused by novelties that are easily handled by our natural intelligence.
Computer programming languages must be defined with complete precision. Beginning programmers become frustrated when computers follow instructions to the letter, seeming to ignore the intent of the programmer.
Dictionaries demonstrate the highly relational structure of natural languages. There is much circularity in the definitions of words. Also the meaning of any utterance depends on its context or on the frame of reference of the speaker. This is also known as the horizon problem. Although machine translations are often passable, meanings are frequently distorted.
Mathematical language, like programming language, is defined with precision, but like natural language it is relational. There is a great difference in the frame of reference between the computer and the mathematician. The computer performs simple mechanical operations. An adding machine can add numbers with precision all day long, and yet could in no way respond to questions about the meaning of numbers.
Most mathematicians spend considerable time with rote calculation that could conceivably be handled by a computer. But this is ancillary to their main function of discovering and exploring structures or patterns. Like most scientists their work involves the framing and testing of hypotheses about the patterns they observe. The hypotheses are couched in terms of symbols of great abstraction. A computer can be instructed to manipulate rudimentary symbols. It can test hypotheses that can be reduced to combinatorial procedures such as the map-coloring problem. But the only relation it can recognize is whether one number is bigger than another.
Let me restate the problem here in terms of pattern recognition. The work of mathematics is to recognize and explain meaningful mathematical patterns. Most mathematical objects derive from such relational structures. There are no formal procedures for recognizing patterns. Patterns may be computable, but only after the fact. The machine must be instructed precisely what to look for.
The salient point is that the cognitive challenge of mathematics is not particularly different from any other area of human activity, but no other discipline has been as concisely documented. Nowhere else has the depth and breadth of the human psyche been so exposed. If there were fundamental limits to the mathmaticizing of the cosmic and human mind, the knowledge of those limits would be among our most important pieces of wisdom. Otherwise, we cannot fail to continue in our journey to self-knowledge.
The fact that the neuroscience industry has granted an exclusive franchise to the computer model of the brain is an increasingly serious obstacle to progress on the much broader front suggest above. Once again, the most effective challenge to the hegemony of neuroscience is likely to come from developments in quantum computing.
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rev. 10/27/98