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Probably most chess players will have the intuitive notion of what kind of positions are not "tension-filled" and what kind of positions are. Usually these are middlegame positions.

Here's an idea of what kind of positions are "tension-filled": In the position there are many pieces that are either hanging or, many times, a piece of lower material value attacks a piece of higher material value (which may or not be defended). Perhaps it could be defined in such a way as "the total value of hanging pieces plus the total value of defended pieces which are attacked by pieces of inferior material value minus the material value of the pieces of inferior material value attacking the defended pieces", or something like this. This is just a "materialistic" attempt to define the concept, since it doesn't incorporate advantages or disadvantages due to positional factors. Perhaps one could consider Tal vs Hecht, 1962 Varna Olympiad at move 21 as an example of a complicated position.

Nonetheless, is there actually an "objective" or "mathematical" way to quantify this idea which can distinguish a position which is not so complicated from one which most chess players would intuitively agree is "complicated"? Has this been done before?

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    Before I opened your question, just from the title I was thinking roughly along the lines of what you already suggest: total number of attacks on pieces (including pawns) as a crude measure of tension in the position. Like you say, this is purely materialistic; one positional factor that could perhaps be added is this: reduced overall king safety (which is routinely quantified by engines these days) could be said to increase tension. Anyway, interesting question, and welcome to chess.SE! (By the way, my choice for maximum tension in Tal-Hecht would be the position after 18. ... b5.)
    – ETD
    Aug 20 '12 at 1:35
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    Thanks man, glad to know that this intuitive notion is quite universal amongst chess players, and remains so even when translated into some quantifed form.
    – user1539
    Aug 20 '12 at 2:26
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To me, a position looks "tension-filled" only when there are many threats on the board and (the important part!) - when these threats are mutual. A position that contains many threats but all of them being made by one player - that's rather realization of huge positional advantage.

I don't know very well the quality notation of moves in English chess books, but in Russian ones some moves get the '!?' quality notation, which means "this is a slightly risky move that can give you advantage if your opponent fails to find the right countermove (which is not trivial to find, that's the whole point of the idea), otherwise your position will get worse".

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    In English chess books, !? usually means "interesting", and ?! means "dubious". Surely a move that will worsen your position if the opponent finds the right countermove is ?!.
    – RemcoGerlich
    Apr 18 '13 at 20:28
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Professional players don't generally quantify this, but it is definitely pretty important. The most common, insider-y word for this kind of thing (in English) is 'sharp', which tends to mean that accuracy is exceptionally important. Sharpness usually lapses after a few moves. Games that remain sharp for longer are considered remarkable. Other adjectives can be applicable, depending on the situation.

As for an objective measure, an excellent starting point would be, IMO, to see how the top choices of an engine look -- for example, if the top 8 choices are all equivalent in evaluation, odds are the position is quiescent. On the other hand, if 2 choices are much better than the rest, and if humans would find those 2 choices non-obvious, the position is sharp. If you try to just isolate factors like material, king safety, etc., you are going down a path that logically ends with an intricate evaluation function, which is what engines do.

Also, I think (not 100% sure) that this is actually done by engines themselves -- they want to steer towards volatile positions, and the above is a handy method of identifying them.

Another word used is 'pressure', which has a different nuance of meaning than 'sharp'. Or 'tense' as you mentioned. 'Complicated' can apply to quiet/deep positions, though -- that's more about the number of reasonable options, or the breadth of the analysis tree. For example, Karpov had some very complicated Nimzo-Indians with 8 pawns per side and long, drawn-out maneuvering. And Carlsen has shown that positions we tended to think of as simple and safe can be quite complicated and dangerous. So I think 'complicated' may be different from 'tension-filled' and easier to quantify.

In general, there are lots of qualitative words that strong players use, and this is related to the fact that 'objective measures' of positional characteristics tend to be pretty hard to define, especially in a way that has practical value for human players.

tl;dr -- None that is commonly used, but a good start would be looking at the engine-based evals of move choices, and categorizing the patterns.

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