What produces philosophical originality? One answer is genius: from time to time a genius crops up and from his or her fertile brain originality flows. Then we have a golden age. No doubt the greatest philosophers were geniuses, so it is natural to suppose that this is what brings originality about. The trouble with this answer is that originality is too sporadic for this explanation to be plausible: geniuses will crop up in the population at a constant rate (assuming a genetic basis), but philosophical originality does not historically occur in this way—it comes in waves separated by arbitrary intervals of time. The best way to answer our question is to survey the history of philosophy (Western philosophy) and try to discern patterns and possible causes. Are there historical conditions that conduce to bursts of creativity?
There are two possible types of explanation: internal to philosophy and external to it. Internal explanations say that the causes are internal to the subject of philosophy; external explanations say that the causes are external to the subject of philosophy. Thus either something about philosophy itself leads to innovation or something outside it does—or possibly both. I have come to the conclusion that the causes are principally external, and indeed that one type of cause is typical (which is not to say necessary). Obviously these are large historical and psychological questions, inherently difficult to assess, but a broad picture seems to emerge when we examine the history of the subject. Not to keep the reader in suspense, it appears that the prime cause of original thought in philosophy has been advances in mathematics. (I will restrict myself here to the parts of philosophy that don’t include ethics, aesthetics, and political philosophy—metaphysics, epistemology, logic, and related fields.)
Plato must be counted as a great original, and it is well known that he was much influenced by Pythagoras and his school. Greek geometry, later assembled by Euclid, formed the intellectual environment in which Plato forged his philosophical ideas. Thus we have the idea of a changeless perfect world of forms to be contrasted with what the senses reveal, where truths about this world can be established by rigorous proof. Geometry can be described as the mathematics of space, so it was the mathematical treatment of space that acted as a trigger to Plato’s originality. The objects of geometry supply the ontology and the method of proof supplies the epistemology—this is what a serious subject looks like. Aristotle continues in the same vein (substance and form) but reacts against it to some degree: he is less mesmerized by mathematics than Plato—but it forms the background to his thought. An intellectual stimulus can have either a mimetic or an antipathetic response. One can be creatively against something. Aristotle was against Plato’s excessively mathematical outlook and shaped his philosophy accordingly.
There then followed a rather unoriginal period—the Middle Ages. During this time nothing comparable to Greek geometry occurred in mathematics and philosophy took no major steps forward (I am speaking broadly). Then we reach the Renaissance in which there was a great flowering: Descartes, Leibniz, Locke, Berkeley, Hume, and others. What happened? Physics is what happened—mathematical physics (Newton’s book is entitled Principia Mathematica). Calculus was invented and the mathematics of motion formulated. The physical world was conceived quantitatively, with mass, force, and motion mathematically measured. This new paradigm of knowledge led to a reinvigoration of philosophy—with adherents and dissenters (notably Berkeley). It provides a framework for metaphysics (matter in motion) and an epistemology (observation and calculation), as well as a model of what a real science should look like. The question of materialism took on new life now that physics was in the ascendant. Thus a good deal of original philosophy was stimulated by the new mathematical physics—not from the insights of philosophers working on their own internal problems (worthy as that may be). The agenda was set, the map laid out—by a development in mathematics. Just as the major influence on Plato was a non-philosopher (Pythagoras), so the major influence on the philosophers of the Renaissance was a non-philosopher (Newton—also Descartes in his capacity as physicist and mathematician).
Again, there followed a relatively static period in philosophy (though stirred somewhat by Darwin ) until the dawn of the twentieth century. Then we have the spectacular rise of mathematical logic—the application of mathematics to logical reasoning. Frege, Russell, and Wittgenstein were philosopher-mathematicians impressed by the power of symbolic logic, with its formulas, proofs, and theorems. Russell and Whitehead’s Principia Mathematica was a mathematical treatise on the subject of valid reasoning (among other things), and it formed the shiny new object onto which philosophers could latch. Some saw it as the bright future of philosophy, others as its death knell. Again there is adherence and reaction: analytical philosophy versus continental philosophy (roughly), or the Tractatus and the Investigations. Mathematical logic played the historical role previously played by geometry and mathematical physics—a model and inspiration, or a threat to all that is holy. It was not the achievement of a professional philosopher qua philosopher that caused this ferment, but the achievement of mathematicians; the trigger was external to philosophy.
This stimulus received a boost later in the century, particularly from Turing, with the idea of a formal computation. This idea led not only to the computer but also to developments such as cybernetics, automata theory, and mathematical information theory. A new branch of mathematics supplied new tools with which to think about the mind and knowledge. The doctrine known as “functionalism” arose from these developments—a kind of mathematical theory of the mind (mental processes as functions from inputs to outputs, formally implemented). We are still living with Turing’s contribution in today’s cognitive science (including linguistics). And once again, there are followers and rebels—some who think we now have the key to understanding the mind, others who think the mind is quite other. It is the mathematical conception that sets the agenda and captures the imagination. Philosophy responded to computation theory as it did to the rise of mathematical logic. Nothing else has had this kind of impact on the field—not chemistry, biology, psychology, history, or whatever. Philosophy seems uniquely susceptible to the charms of mathematics. Not its slave, to be sure, but its keen observer, its ardent pupil–or its stern critic. You either love mathematical philosophy or you hate it.
So now we have an interesting question: what is the next wave of mathematics that will drive the agenda of philosophy, shaking it up, reshaping the subject? We have had the mathematics of space, of motion, of logical reasoning, and of computation—what will it be next? I don’t think anything that now exists in mathematics can play the role played by these earlier innovations, so we need something new to get the ball rolling (whether we can achieve it or not). I suggest that what we need is a new mathematical theory of mind, especially of consciousness: we need a mathematical theory that does for the conscious mind what earlier mathematical theories did for space, motion, logical reasoning, and computation. I have no idea what such a theory might look like; my point is just that it would be likely to trigger a new wave of philosophical originality—perhaps greater than any seen heretofore. Think about it: a mathematical treatment of what lies at the center of human existence and human knowledge—what connects us to the world and to each other. Surely that would be an impressive body of mathematical thought with enormous implications. How would philosophy respond to it? What would it do to traditional philosophical problems? It would change the contours of the subject. Maybe we will have to wait a long time for a mathematical theory of consciousness to be constructed (look how long it took for the previous developments to come about), in which case we won’t see the degree of originality in philosophy that we saw in the earlier periods any time soon. Of course, I am speculating wildly and claim nothing more—it is an interesting idea to think about. There does seem to be an historical pattern here and a mathematical theory of consciousness would surely set the cat among the pigeons. It would set a standard of intelligibility and precision that isn’t even dreamed of today—a psychological Principia. The properties of consciousness would be as clear and exact as geometrical forms, motion through space, logical reasoning, and formal computation.
Mathematics crystallizes things, converts them into rigorous abstract patterns, and analyzes their structure, thus rendering them transparent to the intellect. This is why mathematical innovation impresses philosophers so much—it represents a distant ideal seldom if ever achieved in philosophy itself. We dearly wish that philosophy could achieve such clarity and precision—or we fear (some of us) that it would remove the charm of philosophical obscurity. Mathematics is like philosophy’s successful elder sibling, an inspiration and a rebuke. The affinity between mathematics and philosophy has often been remarked; it is no surprise, then, if philosophers keep a watchful eye on mathematics. When Spinoza wrote his Ethics in the style of Euclid’s Elements he was acknowledging the force of Euclid’s example. Empirical science can never exercise this kind of hold on the philosophical imagination because it is too caught up in the passing concrete empirical world; mathematics by contrast shares the abstract necessity of philosophy. Mathematics provides the kind of vision of things that philosophers (many of them) resonate to– so they are apt derive inspiration from it. Philosophers are mathematicians manqué. 
 Darwin’s theory has a mathematical aspect: a random process leads to the selection of organisms or traits that increase in frequency in a population. It is abstract and quantitative, a kind of algorithm; also statistical.
 Philosophers who model their subject more on literature or history (such as Collingwood) recoil from mathematical philosophy; they cannot, then, use mathematics as a source of new ideas. Their philosophical tradition will be independent of mathematical innovations. But they are in the minority.