Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

340 jeremy avigadwon’t help in general. The choice of facts we derive must somehow be drivenby the context and the goal.A final observation is that often inferences involving inequalities are verifiedby combining methods from more restricted domains in which it is clear howone should proceed. For example, the additive theory of the reals is wellunderstood, and the multiplicative fragment is not much more complicated;more complex inferences are often obtained simply by combining these modesof reasoning. There is a body of methods stemming from a seminal paperby Nelson and Oppen (1979) that work by combining such ‘local’ decisionprocedures in a principled way.These three strategies—backward-driven search, forward-driven search, andcombining methods for handling more tractable problems—are fundamentalto automated reasoning. When it comes to real-valued inequalities, an analysisof the problem in these terms can be found in Avigad and Friedman (2006).But what we really need is a general theory of how these strategies can becombined successfully to capture common mathematical inferences. To someextent, it will be impossible to avoid the objection that the methods wedevise are merely ‘ad hoc and heuristic’; in real life, we use a considerableamount of mucking around to get by. But in so far as structural featuresof the mathematics that we care about make it possible to proceed inprincipled and effective ways, we should identify these structural features. Inparticular, they are an important part of how we understand proofs involvinginequalities.12.7 Understanding algebraic reasoningThe second type of inference I would like to consider is that which makesuse of algebraic concepts. The use of such concepts is a hallmark of modernmathematics, and the following pattern of development is typical. Initially,systems of objects arising in various mathematical domains of interest are seento share a common structure. Abstracting from the particular examples, onethen focuses on this common structure, and determines the properties thathold of all systems that instantiate it. This infrastructure is then applied innew situations. First, one ‘recognizes’ an algebraic structure in a domain ofinterest, and then one instantiates facts, procedures, and methods that havebeen developed in the general setting to the case at hand. Thus algebraicreasoning involves complementary processes of abstraction and instantiation.We will consider an early and important example of such reasoning, namely