The primary purpose of this paper is to attempt a restatement of the logicist claim that mathematics is reducible to logic, and to produce some arguments in support of that claim.
A second purpose is to provide some motivation for this position from the applications of mathematics in systems engineering.
Motivation has greater importance in the establishment of doctrines which depend on some more or less arbitrary (even if carefully considered) choice in the meaning of technical terms. In this case it is primarily the scope of `logic' which is at issue, and in particular, whether the truths of logic are deemed to include matters of abstract ontology. We argue on pragmatic grounds and on the basis of current practise that abstract ontology should be considered to be part of logic, while conceding that statements with ontological content present greater difficulties both in their justification and in their claim to absolute rather than merely conventional truths.
The philosophy of Mathematics has, to judge by even quite recent accounts, been caught in a time warp since not long after the start of this century, when three mathematicians, namely Hilbert, Brouwer and Russell set forward their solutions to the paradoxes which had been causing some concern in the newly forming foundations of mathematics. The three schools associated with these men have been known as formalism, Intuitionism and Logicism. These are usually presented as exclusive approaches to mathematics without any careful examination of whether the tenets of each school are genuinely incompatible. In typical accounts, developments in mathematical logic since the formation of these schools are represented as having conclusively shown that the formalist program is unachievable, and that the logicist doctrine is insupportable. Intuitionism is often represented as the only one of these schools to have escaped lethal criticism. Nonetheless, the main body of working mathematicians do not work within the constraints imposed by intuitionist doctrine, but have continued throughout this century along lines most readily interpreted through the logicist position.
While Hilbert's formalist programme, containing in its original form important objectives which have since been shown to be unachievable, is probably beyond repair even though many of its methods have been shown to be of enduring utility, Logicism has suffered mainly from poor advocacy. Intuitionism has probably benefited from a combination of opacity and aggressiveness. The serious debilitation which intuitionism promised for mathematics was so horrifying to most mathematicians that they looked no further for flaws in the system or its justification. As time passed by the scope of intuitionistic mathematics has been shown to be much greater than it had been feared, and intuitionism as apostle of constructive mathematics has presented itself as the natural and most appropriate approach for computer science and information systems engineering. I believe that intuitionism deserves more cogent criticism than it has so far received.
The possibility of reducing mathematics to logic was independently perceived by Frege and Russell towards the end of the 19th century. Frege's magnum opus, in which he sought to effect the reduction, was with the publishers when Russell communicated to Frege the paradox which now bears his name. Though this was more a technical problem than a fundamental conceptual failure, Frege was devastated by the revelation and never repaired the defect. Russell however, after great difficulty and anguish mainly flowing from philosophical problems rather than technical ones, ultimately put forward his logical theory of types. Though through modern eyes this system is cumbersome it nevertheless was adequate for the task. The main defects in this system are philosophical. The first is the need for the axiom of reducibility and the lack of any convincing justification for this axiom. The second is the ontological content of the axioms of choice and of infinity.
In collaboration with A.N.Whitehead, Russell used his theory of types to develop a substantial part of mathematics as it then was, thereby demonstrating the reducibility of this mathematics to the theory of types. Principia Mathematica became the definitive example of how mathematics was to be reduced to Logic, and to this day criticisms of the Logicist programme are usually simply criticisms of Russell's theory of types, with little consideration of more modern systems, and few arguments to show that defects in Russell's system are inescapable.
From a more modern perspective, after more than half a century of further development in mathematical logic we can find many flaws in Russell's theory of types, its presentation in Principia Mathematica and its use in the derivation of mathematics. Most of thee detailed criticisms do not however bear upon the validity of then logicist thesis. Principia Mathematica is generally held to have failed to establish that thesis on the ground that three of the axioms used in Principia cannot easily be shown to be principles of logic.
The disputed axioms are:
All of these axioms are essentially ontological in that their effect is simply to admit proof of the existence of entities which could not otherwise be shown to exist.
The first of these, the axiom of reducibility, is the least defensible, but fortunately the one with which we can most readily dispense. It must surely be the one element of the system which gave Russell most heartache, for he found that he could not derive the mathematical results he required without it, but was completely unable to put forward any reason why we should believe it to be true.
Fortunately we can now see that the need for this axiom arose from the ramified type theory which Russell put in place to prevent the `vicious circularity' which he thought responsible for contradictions in Frege's system. Modern mathematics is content with less draconian constraints which suffice to avoid contradiction without giving rise to the need for an axiom of reducibility.