Serious and Trivial Problems

In empiricism on 15/06/2011 at 3:09 pm

A chess problem is genuine mathematics, but it is in some way ‘trivial’ mathematics. However ingenious and intricate, however original and surprising the moves, there is something essential lacking. Chess problems are unimportant. The best mathematics is serious as well as beautiful — ‘important’ if you like, but the word is very ambiguous, and ‘serious’ expresses what I mean much better.

I am not thinking of the ‘practical’ consequences of mathematics. … The ‘seriousness’ of a mathematical problem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is ‘significant’ if it can be connected, in the natural and illuminating way, with a large complex of other mathematical ideas. (G.H. Hardy, A Mathematician’s Apology, p. 88-9)

This insight into ‘serious’ and ‘trival’ problems in mathematics is, I think, analogous to science: the depth and breadth of scientific theories is of a different kind than the kind of puzzles that occupy our day-to-day lives. “Where did I park the car?” is a practical problem for me, but it is an unimportant problem for science.

While there is an intuitive understanding of the differences between ‘trivial’ and ‘serious’ problems, there is a rigorous concept: a ‘trivial’ problem or puzzle does not question one’s overarching theories. Puzzles are difficulties in games that, when resolved, do not tell us something new outside the game. It is working on a crossword puzzle or playing Go. Serious problems occur when our understanding of the world collides with, as far as we can determine, the world as it is, not as we would want it to be. A posited explanation, a conjectured ontology, is thrown off-balance by running up against a wall.

I need a serious answer to my serious problem. When my understanding of the world is flipped on its head, I don’t answer with “I parked my car on the south side of the fourth floor of the parking garage”. Instead, we make something comparable to the radical shift from geocentrism to heliocentrism.

The first ‘serious’ problem was, I think, Zeno’s paradox of motion, which produced in response the novel metaphysical (at the time) theory of atomism. It would be an understatement to claim that this answer was somewhat productive. Our understanding of the world changed: the universe is understood now to be mostly empty space. As a bonus, the theory gracefully unified inorganic and organic chemistry.

This concept of ‘seriousness’ can be relativized: terrestrial mechanics is ‘trivial’ when it is compared to a theory that unifies terrestrial and celestial mechanics. By increasing the content of the theory, I make my ‘serious’ answer even more ‘serious’. To paraphrase Hardy, a scientific idea is ‘significant’ if it can be connected, in a natural and illuminating way, with a large complex of other ideas. To claim that the orbits of the stars and the falling of an apple follow the same guiding principle is to unify two worlds. In a single instant, the way we understand the world changes in light of our new theory. There is, though not in the psychological sense, a Gestalt switch, for the answer to the problem has dramatically changed.

If a scientific theory is replaced by a superseding theory, one that answers all the past problems, along with the problem that eventually overturned the previous theory, and also makes novel predictions, then it can be said that science progresses — not in the sense that science accumulates observations, but in the sense that the content of scientific theories increases. Our superseding theories explain more about the world, which in turn produces even deeper problems when they fail to deliver on their predictions.

Yet, our problems become even more remote, less ‘practical’ while at the same time becoming more ‘serious’ to our fragile ontology. These new answers are in turn more testable (read: falsifiable) than their ancestors, since the new way of viewing the world rules out more states of affairs.


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