I've been trying to understand what is meant by words like reduction and reductionism in different contexts. Being somewhat scientifically minded, I enthusiastically embrace reduction as a strategy of explanation when it is done one level at a time. But I get lost when people talk about collapsing all levels down to one, as if that were an end in itself. So here is a loose analogy where things are much simpler. You have the rules of chess (and the objective of winning by checkmate). Most people would agree that there are strategies and principles of play that are objectively good in the sense that they don't depend on anyone's opinion of them. They just tend to work. While nothing is guaranteed, they tend to result in wins and avoid losses.

It's not like a gymnastics competition, where you have to impress judges to score points. If you happen upon a decent chess strategy, you'll tend to beat your opponents even if you don't completely understand why it works. That's assuming they haven't already discovered even better strategies. (I'm using strategy in an expanded sense that also includes the recognition of tactical opportunities, and general rules of thumb like "castle early". What I do not mean by strategy is what game theorists mean by it--i.e. an extensive form/tree that lists the reply to every possible move your opponent could make. EDIT: In other words, I'm referring to heuristics.)

Given the rules and objective of the game, all possible games of chess can, in principle, be enumerated. The rules determine the set all possible future positions, including checkmate positions. In other words, the rules determine all legal move sequences leading to a termination of the game.

Does that mean that good chess strategy can be reduced to the rules?

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    Winning strategies (in the sense of game theory) do "reduce" to the rules exactly along the lines you describe, see Zermelo's game theorem. But you emphatically do not wish to call them "strategies". Rules of thumb and other heuristics, that may be more practical/efficient with whatever computational hardware or software one has to use (including one's brain), obviously do not "reduce" to just rules, but also to features of the said ware. – Conifold Mar 27 '20 at 22:58
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    @Conifold Yes, "heuristic" is the word I should have used. That covers everything from "castle early" up to complicated "systems". I think there's a good reason for avoiding the game-theoretic meaning of strategy as applied to chess. For a perfect-information game, a game theoretic "strategy" simply solves the game, and then there's no game anymore. So using the ordinary language meaning of "strategy" seems more appropriate, since that's the only kind of strategy anyone uses. – Willie Betmore Mar 28 '20 at 0:57
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    This is a good philosophical question, because it is analogical reasoning about the nature of theories and explanation, the primary focus in the philosophy of science. But it seems to be a false analogy. Analogical eduction would be the act of taking a set of rules of one game, like chess, and converting them to another game to some extent, like checkers. That's game theoretic. Both games are played on the same board, but somehow some or all of the rules of chess would have to become equivalent to those of checkers. I'm going to reflect on this a bit before giving it a shot. – J D Mar 31 '20 at 16:42
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    Three points. First, "objective fact" is ambiguous, I may have a subjective hallucination, but it is an objective fact that I have it, or even what its content is. Second, "castle early" only makes sense in conjunction with the rules of chess. And third, it is likely not universal, there might be special positions where it is a bad idea, but identifying them is so expensive computationally (on human ware) that it is not worth it. Game theory disregards such imperfection. To summarize, it is an objective relational fact about playing chess by humans, not a fact about either one separately. – Conifold Apr 1 '20 at 4:27
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    I do not think this is so much about bounded resources as about the difference between what Kant called discursive and intuitive intellect. On this model, God does not employ his unbounded discursive resources, he has resource-cheap direct intuition instead. We encounter something similar, scaled down, with "mathematical intuition", photographic memory or fast mental calculation, that allows people to outperform common heuristics, which probably means some idiosyncratic "wiring" making them akin to "aliens". Not superior resources but differently configured resources. – Conifold Apr 2 '20 at 23:53

I wouldn't say that advantageous heuristics reduce to the underlying rules of the game, although you could do this if you can define "advantageous" strictly in terms of those rules. I think we can at least say that this is a case of supervenience where a given set of effective heuristics supervene over the rules or possibly a combination of the rules and player psychology. Reducibility of one set onto another entails a supervenient relation, but whether the inverse is true is a topic of debate.

  • I think I like where you are going with that. "Advantageous" here is a bit like "fit" in that it is hard to pin down a mathematically precise definition that is appropriate for all occasions. As used in my question, it means tending to result in wins or avoid quick losses. For example, when you are first learning you tend to lose very quickly. As your strategic awareness develops from "Ooh! I can move my horse here! Oh, darn it!" to something better, you still lose, but not as quickly. Of course, "win" and "loss" are defined strictly in terms of the rules and game objective. – Willie Betmore Apr 1 '20 at 2:02
  • @JD That wasn't bad. Why couldn't that have been your answer? I'm not joking. Could you modify your answer to say that? – Willie Betmore Apr 3 '20 at 23:14
  • @JD Also, please do not delete any more of your comments that contain something nasty or hateful. If you do, it will look like I've been talking to myself the whole time :) – Willie Betmore Apr 4 '20 at 1:25
  • @WillieBefore As you feel my comments are nasty or hateful, then the best course is to withdraw them! #ReactanceTheory Best of luck! – J D Apr 4 '20 at 21:30

The short answer is a resounding no, because rules do not encompass the values of the agents that use the rules.

A theory, as often conceived, can be abstracted to a set of set-theoretic, logical, and arithmetic principles represented by a syntax. For instance, in math, foundationalists claim they can reduce the axioms of arithmetic to set-theory expressed logically. Hence, the claim is that ideas expressed in the axioms of ZFC somehow can express anything that can be expressed in arithmetic or consequentially algebra. There are connections between topoi, categories, and logic whereby partial or full reduction can occur. But games are a fundamentally different creature because they require an analysis of agency.

Take a game of chess. The rules of chess are well-understood facts, but the psychological motivation for playing is a question of value. Is a game played by two narcissistic chess masters going to proceed fundamentally different than a friendly game between a mother and daughter? Of course. The more fully the theory accounts for the nature of the game, the more the theory matches the phenomenon. Nowhere is this more apparent in contemporary psychological circles than in behavioral economics. The work of Kahneman and Tversky and the death of homo economicus show the impact of game theory to understand how scientific theory is produced and reduced.

The key to understanding reductionism is that there are two factors at play. First, there is the logical connection of the propositional expressions of the theory, but secondly, there are value-laden decisions made in developing, adopting, defending, and reducing (or resisting reduction) of theory. Even the most stringent physical theories have normative aspects as they are the product of defeasible reasoning and subject to the normativity of games of abstraction.

Hence, a set of rules operates in conjunction with the values of the agent and the choices the agent makes, which in the case of humans can often be irrational and non-deterministic. A man who let s his nephew win constantly at chess because he is dying of leukemia will play a game that cannot be predicted without taking into account his love for his kin, and THAT is not in the rules.

  • "Is a game played by two narcissistic chess masters..." That is too real. You've clearly been to a chess meeting before :). That is a very broad answer with excellent points about the role values play in theory development. But for simplicity, let's just assume the motivation of the chess players is extremely simple and unchanging. They want to win and, if they can't, they'd rather at least make the game last longer. Do good heuristics reduce to the rules then? – Willie Betmore Apr 1 '20 at 2:40
  • Nobody is trying to outsmart anything or escape anything. And how do you know what I realize? What I tried to do is make the values involved simple enough that we could avoid what you're doing here. Is it really that hard to imagine the effect that wanting to win as often as possible, and to put up a good fight in losing situations, would have on someone's (or some thing's) approach to the game? It's pretty much how anyone who's ever learned the game approached it. "Don't get killed" is the most basic of all values. It is a rationale that supports itself. There doesn't ... – Willie Betmore Apr 2 '20 at 1:48
  • ...even need to be anyone consciously aware of it. Things that act according to that value stick around so you can see them. Things that don't, don't, so you can't. And yes, "is-a-heuristic" is a vague predicate. What else would one expect for a concept that operates at a higher level of description than the most basic one--the rules? – Willie Betmore Apr 2 '20 at 1:55
  • "...but how does one determine what is a heuristic and how does one determine what good is in terms of the rules which makes it true? Answer me in terms of the rule set of chess, and we might get somewhere" So you're asking me to reduce good heuristics to the rules before we can talk about whether good heuristics are reducible to the rules? Cool. – Willie Betmore Apr 2 '20 at 2:05
  • ." As a former 1st board who went to state, I can define "good heuristic" in my sleep...." Well, no need to keep us waiting. "Think about why the rules do not encompass fully tactics and strategy in beginning, middle, and end game." At long last. So the answer to my original question is "No". After all of that. Thank you. – Willie Betmore Apr 2 '20 at 23:07

When we talk about the rules of chess, we mean the rules that govern the movements of individual pieces. In other words, we can enumerate a short list:

  • All pieces capture opponents by occupying their space, except as noted for pawns
  • Pawns move one space forward, except:
    • When capturing, they move on the diagonal
    • Their first move, they may move two spaces, capturing pieces they pass (en passante) without occupying their space
  • Rooks move orthogonally any number of spaces, unless blocked by other pieces
  • Bishops move diagonally any number of spaces, unless blocked by other pieces

...and so on. The issue of reductionism revolves around the question of whether these singular (atomistic, independent) rules are both necessary and sufficient to describe the game of chess. The rules are clearly necessary: one cannot play chess without knowing the rules of the game. But it is far less clear whether the rules are sufficient to the task. Can a given strategy — even a simple one, like using a pawn to support a bishop — be explained solely in terms of the movements of the individual pieces?

Part of the reason that reductionism is so appealing is that it actually works well in many contexts. For instance, in ballistics people regularly ignore factors like the shape of the projectile, the distribution of mass within it, its interaction with the surrounding air through friction or rotation, etc. The projectile is reduced to a point mass subject to the simple independent factors of the force of gravity and the initial velocity vector, and for the most part it works well. The omitted factors produce small variations that are easily accounted for by statistical error handling, and no one generally cares whether their cannonball lands six inches off target in one direction or another, as long as it gets where it's going.

However, in other contexts this reduction to simple independent variables produces larger errors or incoherent results. We often see this in the social sciences, where something like (say) gender might only account for 35% of the variance, and we won't get decent results unless we consider gender and ethnicity and age all together at the same time. So then we have to ask: is there some other singular 'rule' or 'variable' that this complex situation can be reduced to, or do we need to take a non-reductionistic (holistic) approach to analyzing these contexts? That question remains to be solved.

Going back to the chess analogy, we can see the three cases:

  • Is strategy reducible to the simple (isolated) rules of individual pieces?
  • Is strategy reducible to some other simple (isolated) rule that we have not yet discovered?
  • Is strategy a holistic principle that emerges from (and is distinct from) the simple (isolated) rules of individual pieces?

We know the rules are necessary; we don't know whether they are sufficient. So pick your side...

  • I found this insightful and upvoted. I've addressed the issue of sufficiency in my own reply – J D Mar 31 '20 at 17:12
  • "Bob, I'll take door number 3". Lots of good points here. The reason I picked the chess example is that the rules are known. So you don't have to worry about whether your theory of how knights move is correct. Or whether there's some hidden rule that explains on a deeper level the way all pieces move. The rules are fixed. Once the rules are in place, a toy "world" is opened up with all sorts of opportunities to be exploited. So, reduction appears to be about sufficiency. But what counts as sufficient to a reductionist? – Willie Betmore Apr 1 '20 at 3:25

Reduction is, in its relevant sense, an invertible and explanatorily adequate simplification. So, if X is reduced to Y:

  • Y is simpler to understand, to handle, etc. in regard to the reductive purpose. It should not give the impression of obscurum per obscurius.

  • X and Y is so related that X can be recovered or derived from Y. Y shouldn't be an arbitrary or random splitting of X, or an outcome of an irreversible processing of X.

  • Y accounts for all whatever we know and would like to know about X. To wit, we should be able to understand X to our satisfaction by referring to Y.

For example, suppose that a biological phenomenon X is explained by a chemical phenomenon Y (notice that we talk about a translation from biological vocabulary to chemical vocabulary; although very interesting, it is far fetching to go into ramifications of this change). No doubt, the knowledge of Y enhances our understanding of X. But is this a reduction of a biological matter to a chemical one? We have to check further to affirm: Does Y enable us to make sense of X in its biological environment? Does Y allow us to predict sensibly about X, especially, for that environment?

This is not a question of computational resources - whether Y can actually replace X, or only potentially. It is a question of concepts, laws and representational systems. Even though we would accomplish to simulate X by Y in the most detailed manner, our understanding of X might still remain quite poor.

Many reducibility issues arise around how we ascribe simplicity and explanatory value to the reducing base, Y in our case. There may be borderline or degenerate cases in which differences between X and Y collapse as regards our assessment.

As for the chess example, does the brute force search reveal something significant to our understanding of the game? I assume that what is described is virtually a combinatorial exercise, so not a game anymore. Hence, it could only simulate game strategy, not represent a reduction of it. However, the brute force method should not be viewed as inferiority; it depends on the nature of the subject. It might well show us that a simplistic sequence of moves would suffice and there was no need to wrack our brains with chess strategy at all.

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