Conclusion

First of all, we required that the theory be able to describe all of pure mathematics. This requirement pervaded the work described in this book, although its impact was rarely explicit. Had we been giving a theory suited to one or two domains, large portions of the book would have been unnecessary. For example, if we had only been concerned with real analysis, then all of our discussions of structures and relational types would have been unnecessary. If we had only been concerned with combinatorics, our extensive discussion of the number system could have been dispensed with. (And so on.) More importantly, if we had only been discussing individual domains, we would 250 9 Conclusion not have found many of the generalisations which we did; for example, it is unlikely that we would have discovered a general mechanism that could track both the dimensions of a matrix and the group in which a particular instance of group multiplication occurred (§8.3). Thus our requirement of generality exerted a quiet but continual pressure throughout the book, forcing us to discover deeper patterns rather than giving superficial analysis of the phenomena at hand.