What is the nature of a problem? Aristotle figured he could answer the question this way: "The difference between a problem and a proposition is a difference in the turn of the phrase" (Topics, Bk I, 4). Thus, to the proposition: 'Man is an animal that walks on two feet", there corresponds the problem: 'Is man an animal that walks on two feet?" - basically the proposition with a change in phrasing. For Aristotle then, there is a one-to-one correlation between problems and propositions, where every proposition has its corresponding problem. 

While this is a tidy way to understand the nature of problems, it is also an exceedingly crappy one. For one, it is entirely banal. That a problem consists of a mere 'turn of the phrase' means that a solution never gives us anything other than what was already implied in the problem; to 'solve a problem', on the Aristotelian model, means nothing other than to affirm or deny it, to nod or shake one's head in mute agreement or disagreement. But it is clear that the most interesting problems - and I want to say all truly philosophical problems - are not of this kind. The most interesting problems always demand more from their solutions than what can be already found in the problem to begin with. What does this mean?

The realm of math provides a concrete example. While basic math problems like "1+1=?" do abide by the Aristotelian model of problem-posing ('=2'), any step in the direction of higher-order math takes us far beyond such conceptions. Consider a more general equation, like "x+y=3", where 'x' and 'y' can take on multiple/different values. Or else consider the (even more) general equation of a circle, "(x – h)^2 + (y – k)^2 = r^2", which has a huge number of solutions (every circle in Euclidean space can be represented by the equation). And this is all to say nothing about group theory and other, further mathematical generalizations.

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These kinds of problems break with the Aristotelian model in two ways. First and most obviously, each problem correlates with not just one, but multiple solutions. Second and more importantly, every solution has 'more' in it than can be found in the problem itself: the problem forces one to think 'beyond' what can already be found there (which is why Kant - for those familiar with him - specified that mathematical propositions were 'synthetic', and not merely 'analytic' propositions). And of course, math is just one model on which to think about problems.

The fields of engineering and evolution provide further examples in which solutions always 'exceed' the problems from which they are born. Thus a wind turbine, a dam, and a coal powered fuel plant can be considered 'solutions' to the problem of generating energy. Or else in the realm of biological evolution, one can speak - in the manner of Gilles Deleuze and Henri Bergson - of the 'eye as a solution to the problem of light' posed to living beings; or in the way in which cold-bloodedness and warm-bloodedness are different 'solutions' to the problem of heat-regulation in animals. One can begin to see the outlines of what might be called a 'realism of problems' or 'problematic realism', where problems themselves have ontological standing. But that's just a side-thought - for now.

Thinking about problems in this way has a few advantages. For one, it 'dynamizes' problems, keeping them 'open' such that problems are not static artefacts to be solved once and for all, but instead force creative and ongoing engagement. The problem of energy generation remains an 'open' one, for instance. Second, it allows us to frame the nature of philosophical questions in a more interesting way. People often complain about philosophy as providing no definite answers; but such complaints are often motivated by treating the problems of philosophy in the anodyne Aristotelian manner outlined above. Problems - true problems - inspire and compel (not unlike a physical force); they take us beyond, into the new, the unexplored, the unfamiliar. And this is, at minimum, what philosophy does best.