How are they any different?
Heuristic: a way people solve problems, using experience from the past.
Analogy: a way people solve problems, by remembering similar situations from the past.
In fact, I just encountered them in a psychological text on problem solving methods. Heuristic and analogy were introduced as two distinct methods. Two others are, you know, trial and error and algorithm.
There’s a classic book on heuristics called How to Solve It by George PÃ³lya. Back when I was teaching, I often recommended it to my students. You are correct that heuristics is the “science” of general problem solving, and it can be applied to anything from mathematics to psychology to management.
Within the discipline of heuristics, several specific problem solving methods are identified. Using analogy is just one specific method. Others are simplification, restatement, visualization, reductio ad absurdum, trial and error, etc. If I recall correctly, there’s about a dozen specific methods mentioned in his book.
So the answer to your question is: Analogy is a subset (or component) of heuristics.
The Mutilated Chessboard Problem is a great example of heuristics in action:
- a chessboard has 64 squares
- you have 31 dominoes, each of which covers exactly 2 squares of the chessboard
- you remove 2 diagonally opposite corner squares from the chessboard (leaving 62 squares)
- can the chessboard now be totally covered by those 31 dominoes, without cutting any dominoes?
Using the trial and error method, you could spend a huge amount of time looking for a solution (and wouldn’t find one). Removing two diagonally opposite corner squares from a chessboard removes two squares of the same color. Each rectangular domino must cover two squares of opposite colors. Thus, it is impossible to cover the 62-square chessboard with the 31 dominoes. PÃ³lya would cite this as an example of the visualization and restatement methods of problem solving.
You ask why “that” is heuristics. I assume by “that” you’re referring to the method of solution for the Mutilated Chessboard Problem?
First, note that the MCP is only asking for an answer in the form of either “Yes” or “No.”
Finding that answer is an example of heuristics because the method most people would try first (trial and error tiling of the board with dominoes) will fail to provide a solution. Success would immediately provide an answer of “Yes,” but that ain’t gonna happen. Failure would NOT provide an answer unless every possible tiling were tried (and that’s a LOT of possible tiling arrangements).
After several such failures, it might start to seem this is a problem requiring some kind of deep mathematical insight related to geometry or combinatorics. But it doesn’t, and that’s where heuristics comes in.
The more observant solver would notice all failed attempts leave two squares of the same color when the 31st domino is in-hand (which is a hint to the solution). That’s visualization, or perhaps pattern recognition.
So then we restate the problem as: Can I cover 32 white squares and 30 black squares with 31 dominoes? This would lead via basic arithmetic to the solution “No.” At least it would if combined with the insight that each domino must cover both a white and a black square (more pattern recognition).
PÃ³lya notes that we all use heuristics without realizing it. It’s just an intellectual process engaged during problem solving. What he did in his book was formalize that process by giving names to the specific methods used, and providing examples of each. Granted, the names are somewhat arbitrary, and there’s a lot of overlap between methods. But heuristics is an intellectual process, so things do get a bit fuzzy.
I’d like to suggest a different angle on the MCP which, I think, better illustrates a heuristic. I suspect a person would not spend a huge amount of time using trial and error. Instead, after one or two trials, one would use a heuristic: having experienced a couple of trials, it would be evident that there is no way to cover the two squares that have no mate of the opposite color or to cover a row having 7 squares when each domino must cover 2 squares. I think the key concept about heuristics is that such decisions are not backed up by rigorous analysis and are not necessarily sound logically or mathematically.
I’d like to address the original question, too, regarding how heuristics and analogy are different. Both horses and dogs are herd animals. If you had experience with horses, you know that where there are two or more horses, one must be the leader and the rest followers. Without any experience with dogs (no heuristic) you could understand (using analogy) the behavior they engage in to establish the leader. I suppose one could argue that the experience with horses provides a heuristic to understanding dog behavior but I’d argue that such a heuristic is only applicable due to the analogous herd behavior.
Nothing I said should be construed as a definition. I accepted the definitions in your original post and in the first reply from Heimhenge. I was attempting to help you answer your question, “How are they any different?” I did not infer that you needed a definition.
The part to which I believe you are aluding with your quote ‘ Inexact ‘ was what I think is a key concept which is that one can act without knowing everything there is to know; one only needs “enough” information. In other words, you decide what is enough for you and then act, thus avoiding “analysis paralysis.” How you decide is what (I think) heuristics is all about.
This riddle is an opportunity to apply the analogy principle (though really, just to share some weekend entertainment).
3 boxes, 1 containing 1M dollars, all yours if you guess right which. But must follow this 3-step rule:
1— You pick 1.
2— Of the other 2 boxes, the ‘manager’ (or ‘game show host,’ doesn’t matter) looks inside both to know the content (or non-content), then opens 1 of them to show you that it is empty.
3— Of the 2 unopened ones, you have the right to keep the one you picked, or switch to the other one.
The question of the riddle is which action gives you the better chance, keep or switch? And of course what is the chance numerically?
Some people can reason it out airtight (or so they think). For those who can’t (like me, even after hearing them and half-way convinced), there is help from this imagined analogy : Suppose you play the same game, but with 1,000 boxes: after you pick the first one, the ‘manager’ will open 998 boxes to show they are empty; what is your final decision then? Without a doubt you will switch to the one that he didn’t open.
The obvious implication of the analogy is that you should also switch in the case of just 3 boxes. But really? How valid can the lesson from 1,000 items be when you have just 3?
And then there is this to contemplate: does it make any difference whether the ‘manager’ looks in the boxes, or just plays by chance just like you?
I am feeling like a headache coming on just writing this. So much for entertainment!
Thinking again, that is not a good example of analogy, more a modification. But enjoy anyway.
No Robert, I agree with your first assertion. It is a type of analogy. From one online dictionary:
- A comparison between two things, typically on the basis of their structure and for the purpose of explanation or clarification.
- A correspondence or partial similarity.
Now PÃ³lya had a different name for this strategy. He called it “Think of a similar problem you can solve.” That almost always involves analogy, and one way to do that is to exaggerate one of the variables (as you did when you considered 1000 boxes). You were using a classic heuristic method.
On the puzzle itself, yours is one of many variations of what is called “The Monty Hall Problem.” It was first posed in the 70s, when the host of a popular TV show did exactly what the problem says (only with prizes hidden behind curtains). The answer is totally counter-intuitive, and I’ve heard mathematicians disagree on the solution. I will not present that solution, as I don’t want to be a spoiler. But the most lucid explanation I’ve read is here: http://montyhallproblem.com/
In fact, the author of that page first proves the answer rigorously, then reverts to an analogy with more boxes, as you did.
Interestingly, the answer does depend on whether Monty randomly chooses the door to eliminate, or whether he knows which door the prize is behind and chooses accordingly. It is usually assumed he does know, but that’s rarely stated in versions of the problem seen online. Your version did include that condition.
On ‘analogy’ thank you for agreeing, though because of that I must disagree- first to keep with my last point, but more on account of your most agreeable definition, which though you brought forward for the purpose to agree with me, in my view regrettably should be good cause to (rightly) disagree; and all that is because of how the definition sounds to me, that is, so agreeably I could not any more agree, which is why, paradoxically, and absolutely mindful of how I might come out the only party disagreeable in this discussion, a one not only all in all agreeable, but whose trajectory at this agreeable juncture (that is, just prior to this disagreeable wrench thrown in by me) appears to be nothing if not agreement bound, that is, if it’s not there already ( as anyone who might be reading would readily agree), I should disagree.
See, your definition mentions ‘two things.’ Accordingly I see these paired concepts as apt illustrations of analogies, because of the distinct entities:
Ant hauling bread crumb 30x own weight ==Man hauling large truck
Cadillac to cars == Sony to TVs
Grains of sand == Stars in cosmo
On the other hand these involve the same entity, only with changed parameters:
Vehicle racing through air at high velocity >< Vehicle in wind tunnel blasting at high velocity
Astronaut walking in space >< Astronaut training in free-fall airplane
Game played with 3 items >< Same game played with 1,000 items (which I rejected for analogy in last post)
It’s all subjective, but I like narrow definitions, because narrow definitions lend clarity as fine pixels give high-definition graphics.
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