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This section is taken with very little change from my article on ``The Sources of Certainty in Computation and Formal Systems.''
The concept of formal systems was not designed by a committee to satisfy a research contract---it arose from centuries of work by mathematicians and philosophers who were addressing individual problems with formal components, rather than seeking to design a general formal method, and it was characterized rigorously only in the early twentieth century. Nonetheless, I propose that we understand formal systems as a concept that is adapted very precisely to meet sensible goals, even though its design was much more evolutionary than conscious. The value of formal systems can best be appreciated by considering how a fictitious conceptual engineer might have designed them.
The earliest useful formal system that I can identify is the system of nonnegative integers, also called the counting numbers. We are so familiar with the counting numbers that many people sense that they are real objective physical objects. I find that numbers are real and objective, but not physical. The physical view of numbers appears to be founded on experiences with collections of discrete objects, such as pebbles placed in a bowl. Arguably 3+1=4 is an observable physical fact, because the result of placing 3 pebbles in a bowl, then adding 1 more pebble, yields 4 pebbles in the bowl. But, the physical observations that support the concept of counting numbers must be filtered by some previous understanding of number. When we place 3 drops of water in a bowl, then add 1 more drop of water, we find 1 drop of water in the bowl. This experience does not lead us to question the validity of 3+1=4, nor even to view it as a mere approximation of the truth about numbers. Rather, it leads us to conclude that the number of distinguishable drops of water in a bowl does not follow the rules of addition of counting numbers.
If the counting numbers are not physical observables, are they merely psychological ephemera, or even illusions? I think not. The actual practice of arithmetic in the world suggests that we understand the counting numbers very well as essentially conceptual constructs in a formal system (Curry called them formal objects, or obs for short). A bit more carefully, the system of counting numbers represents the common properties of a large class of formal systems, each of which contains objects representing the numbers and derivations corresponding to arithmetic calculations with numbers. 3+1=4, then, is an observation about the qualities of the formal systems of counting numbers. We may convince ourselves very reliably that 3+1=4 by choosing a particular formal system, and carrying out a derivation similar to the one in the binary increment example. In fact, the practice of placing pebbles in a bowl can be understood as a presentation of a formal system for the counting numbers---perhaps as an approximate presentation, since the bowl has a limited capacity.
The usefulness of formal systems (and the effectiveness of mathematics) derives from the fact that many natural phenomena are exact or approximate presentations of formal systems. If we can identify such formal systems and discover their properties, we can characterize some of the properties of the corresponding natural phenomena. Since formal systems may use any precisely and unambiguously characterized notion of pattern, we have a chance to apply formal systems to all natural phenomena in which we recognize such patterns. Reasoning about formal systems is useful because it often tells us that the presence of one pattern that we have observed entails the presence of another pattern that we have not noticed. For example, the pattern of definition of numerical addition in terms of successor entails that the result of addition is independent of order. This famous pattern is the commutative property of addition, written x+y=y+x in algebraic texts.
Our understanding even of primitive propositions, such as 3+1=4, involves the recognition of patterns among formal systems, rather than the mere exercise of a single formal system. Nobody really endorses 3+1=4 just because of the specific derivation in the binary increment example. Rather, we recognize that a large class of formal systems exhibit certain qualities in common, which we regard as the form of numerical arithmetic, and that all of these systems share a pattern which we describe by 3+1=4. The derivation in the example is simultaneously an example of the 3+1=4 pattern in a particular formal system, and a formal presentation of the pattern of reasoning that we follow in recognizing the 3+1=4 pattern as an unavoidable consequence of the qualities shared by all of the formal presentations of numerical arithmetic.
If they are mental and abstract, rather than physical, how can formal systems be real and objective? They were evolved to be so, by eliminating from mental processes all the parts that we cannot agree on reliably. Suppose that we tasked our fictitious conceptual engineer to produce a method of reasoning, and communicating the results of reasoning, with the greatest possible capability for
If he were sufficiently brilliant, our engineer might consider that manipulation of symbols can be made physically very cheap, because we are always at liberty to substitute a lighter weight symbol for a heavier one, as long as all parties to a discussion recognize the same selection of symbols. Next, the rules for manipulating symbols should be based on the forms of arrays of those symbols, rather than a particular assigned meaning, in order to allow reasoners to maximize objective certainty. If the rules for manipulating symbols depend on their meanings, then our certainty about the correctness of manipulations is limited by our certainty about the behavior of the things that they refer to. We would like to use reasoning to illuminate qualities of things about which we are initially quite uncertain. So, we choose rules that refer only to the forms of arrays of symbols, and then we are at liberty to present the arrays and the rules in ways that we have discovered in practice to be thoroughly clear and unambiguous. It appears that our conceptual engineer has just designed for us the highly successful method of reasoning with formal systems.
Those with a taste for ontological studies may find my treatment of formal systems disturbingly circular. Instead of characterizing the essence of formal systems, and showing why that essence leads to certainty about the results of reasoning, I suggest that formal systems are the systems that we design for ourselves by refusing to deal with material about which we are uncertain. I concede the circularity, but I think that it represents the best way of understanding formal systems. They are social objects, designed for the purpose of communication. Because the design is so successful, they acquire an a posteriori air of necessity. In a later section, I trace the quest for certainty through the work of Descartes and Hilbert, and I think that this circular and somewhat negative view is at least highly consistent with their published ideas.
Another famously successful early example of a formal system is Euclidean geometry. Practitioners of geometry did not show clear understanding of the formal quality that characterized their work, but they had a strong intuitive sense that this work was reliable in a way quite different from other philosophical inquiries. Oddly, geometry was held for centuries to be an exact description of the layout of the physical universe, and the coexistence of formal certainty with physical factuality puzzled thinkers, such as Emmanuel Kant. Now we understand very well the sense in which geometry is a formal system and the resulting certainty in its conclusions, but we no longer believe that it describes the physical universe exactly.
The certainty derived from observations of derivations in formal systems appears to derive from our confidence in our ability to work out any ambiguity and achieve precise communication, as long as there is no fixed connection between what we are communicating and the physical world. By employing familiar forms of arrangement, such as linear sequences of symbols, and then answering whatever questions arise about which qualities of the physical presentations of those arrangements are and are not significant, mathematicians and other formalists create satisfyingly strong consensus about formal systems. The strength of that consensus origniates in our agreement about what is and is not a correct derivation according to the rules of a particular formal system. It carries over into observations about patterns in and among formal systems. The strength of this carry-over increases due to our ability to formalize the patterns themselves through reflective use of formal systems to study formalism.
A dark side of all consensus is that it consists partly of inducing voluntary agreement, and partly of excluding those who do not agree. Those excluded from useful consensus about formal systems are sometimes said to lack ``mathematical maturity.'' Those excluded from more basic and comprehensive consensuses are sometimes called ``insane.'' I don't have any particular ideas of my own about the exclusionary side of formalism, but I think it deserves a thorough study.
On one side, the participation of many people (even though they are a minority of all humanity) in the consensus about formal systems generates a sense of objective independent reality for the systems. Members of the consensus tend to think that, in principle, anyone willing to make sufficient mental effort can join. On the other side, the description of a formal derivation is in some sense a sort of mental choreography---instructions about the form of thoughts which must be executed in the mind of the reader. From this point of view, our feeling of certainty in formal derivations appears to be more like confidence in our abilities than like certainty of an external fact. I suspect that many other sorts of knowledge besides knowledge of formal systems suffer from a similar tension between fact and performance.
As a thought experiment, try to imagine a presentation of a formal system as a computational device in a box that spits out presentations of derivations. How is our sense of certainty about formal derivations connected to confidence in the workings of the box? Only partly. It's hard to imagine that formal derivations would generate a sense of certainty without highly reliable methods for generating them correctly (human methods are as good as truly mechanical ones). But, after building the best possible formal system presentation in a box, and observing that it fails, we don't immediately lose our confidence in the abstract for the formal system itself. Rather, we fix the box. Formal systems themselves are specifications rather than observations (mathematical ideas are then observations about these specifications). When a presentation of a formal system fails (e.g., when we put so many pebbles in the bowl that gravitational collapse merges them into one planet or star), we recognize an inaccuracy in the presentation, but do not blame the formal system. This sort of warranty as a substitute for physical reliability is very common in nonmathematical life, and it's rather amusing that it plays a fundamental role in the certainty of mathematics as well.
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