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Though Hume wrote in the 18th century, his work is still uncommonly relevant in the philosophical disputes of today compared to that of his contemporaries. Here is a summary of some of Hume's most influential work in philosophy.
When one event causes another, most people think that we are aware of a connection between the two that makes the second event follow from the first. Hume challenged this belief, noting that whereas we do perceive the two events, we don't perceive any necessary connection between the two. And how else but perception could we gain knowledge of this mysterious connection? Hume denied that we could have any idea of causation other than the following: when we see that two events always occur together, we tend to form an expectation that when the first occurs, the second is soon to follow. This constant conjunction and the expectation thereof is all that we can know of causation, and all that our idea of causation can amount to. Such a lean conception robs causation of all its force and some later Humeans like Bertrand Russell have dismissed the notion of causation altogether as something akin to superstition. But this violates common sense, thereby creating the problem of causation–what justifies our belief in a causal connection and what kind of connection could we have knowledge of? – a problem which has no accepted solution. Hume's view seems to be that we (as well as other animals) have an instinct-like belief in causality based on the development of habits in our nervous system, a belief that we cannot eliminate but which we cannot prove to be true by any kind of argument, deductive or inductive, just as is the case with regard to our belief in the reality of the external world.
For relevant contemporary work, see Wesley Salmon 's Hume and the Problem of Causation and Causality and Explanation.
We all think that the past is a reliable guide to the future. For example, physicists' laws of planetary orbit work for describing past planetary behavior, so we presume that they'll work for describing future planetary behavior as well. But how can we justify this presumption – the principle of induction? Hume suggested two possible justifications and rejected them both. The first justification is that, as a matter of logical necessity, the future must resemble the past. But, Hume pointed out, we can conceive of a chaotic, erratic world where the future has nothing to do with the past – or, more tamely, a world just like ours right up until the present, at which point things change completely. So there is nothing logically necessary about the principle of induction. The second justification, more modestly, appeals only to the past reliability of induction – it's always worked before, so it will probably continue to work. But, Hume pointed out, this justification is using circular reasoning, justifying induction by an appeal that requires induction to gain any force. The problem of justifying induction is still with us. Hume's view seems to be that we (as well as other animals) have an instinct-like belief that the future will resemble the past based on the development of habits in our nervous system, a belief that we cannot eliminate but which we cannot prove to be true by any kind of argument, deductive or inductive, just as is the case with regard to our belief in the reality of the external world.
For relevant contemporary work, see Richard Swinburne's compilation The Justification of Induction.
We tend to think that we are the same person we were five years ago. Though we've changed in many respects, the same person is present as was present then. We might start thinking about which features can be changed without changing the underlying self. Hume, however, denies that there is a distinction between the various features of a person and the mysterious self that supposedly bears those features. After all, Hume pointed out, when you start introspecting, you notice a bunch of thoughts and feelings and perceptions and such, but you never perceive any substance you could call "the self." So as far as we can tell, Hume concludes, there is nothing to the self over and above a big, fleeting bundle of perceptions. Note in particular that, on Hume's view, there is nothing that these perceptions belong to. Rather, Hume compares the soul to a commonwealth, which retains its identity not by virtue of some enduring core substance, but by being composed of many different, related, and yet constantly changing elements. The question of personal identity then becomes a matter of characterizing the loose cohesion of one's personal experience. (Note that in the Appendix to the Treatise, Hume said mysteriously that he was dissatisfied with his account of the self and yet never returned to it!)
For relevant contemporary work, see Derek Parfit's Reasons and Persons.
Most of us think that some behavior is more reasonable than others. There seems to be something unreasonable about, say, eating aluminum foil. But Hume denied that reason has any important role in motivating or discouraging behavior. After all, reason is just a sort of calculator of concepts and experience. What ultimately matters, Hume said, is how we feel about the behavior. His work begot the doctrine of instrumentalism, which states that an action is reasonable if and only if it serves the agent's goals and desires, whatever they be. Reason can enter the picture only as a lackey, informing the agent of useful facts concerning which actions will serve his goals and desires, but never deigning to tell the agent which goals and desires he should have. So, if you want to eat aluminum foil, reason will tell you where to find the stuff, and there's nothing unreasonable about eating it or even wanting to do so. Instrumentalism went on to become the orthodox view of practical reason in economics, rational choice theory, and some other social sciences. But, some commentators argue, Hume actually went a step further to nihilism and said there's nothing unreasonable about deliberately frustrating your own goals and desires ("I want to eat aluminum foil, so let me wire my mouth shut"). Such behavior would surely be highly irregular, granting reason no role at all, but it would not be contrary to reason, which is impotent to make judgments in this domain.
For relevant contemporary work, see Jean Hampton 's The Authority of Reason and David Schmidtz 's Rational Choice and Moral Agency.