Sherlock Holmes and Basic Problem Solving
*This is an adaptation of a post about Sherlock Holmes and
Biblical Interpretation taken from the gospel coalition website.1
I have kept most of author's section headings and generally
maintained his editing of the Holmes and Watson quotes, but I have
added a quote because of an insight by Dr. Doug Jackson. I have also
added different comments under each quote. I intend to use this for
mathematics and public speaking students. Conan Doyle's collection of stories is utterly marvelous to read. Holmes is great fun, most movies have made him insufferably boring, the Robert Downey Jr. iterations have bucked the trend. Anyhow, the novels not only point to some excellent principles for reasoning, but they have shown themselves to be fruitful for the field of expertise because of Doyle's apparent insight into how people with expert capacities in multiple fields of knowledge would behave.2
- Try to discern the true nature of the problem at hand and
gather appropriate data before finding (or suddenly claiming to
know) a solution, conclusion, answer, or thesis statement.
Holmes: “I have no data yet. It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.”
Similarly, “This is a very deep business,” Holmes said at last. “There are a thousand details which I should desire to know before I decide upon our course of action.”
And again, “I had,” said Holmes, “come to an entirely erroneous conclusion which shows, my dear Watson, how dangerous it always is to reason from insufficient data…I can only claim the merit that I instantly reconsidered my position.”
Holmes is right, during the course of solving any problem, it is necessary to gather the appropriate amount and kind of data. For instance, the wrong data can lead somebody to asking the wrong question to begin with, and from that wrong question, assuming the wrong answer before gather more data can lead to a conclusion that is off by many degrees and miles. But, if the appropriate questions are asked and the appropriate data is examined before the conclusions are made, then the makings of a good solution are in order.
- The kind of looking that solves mysteries.
Holmes: “You have frequently seen the steps which lead up from the hall to this room.”
Watson: “Hundreds of times.”
Holmes: “Then how many are there?”
Watson: “How many? I don’t know!”
Holmes: “Quite so! You have not observed. And yet you have seen. That is just my point. Now, I know that there are seventeen steps, because I have both seen and observed.”
One of the most important steps for gathering data is simply being observant. Seeing a triangle and observing that it is isosceles are two different things. Noticing that the door to your house is open when you arrive is less helpful than observing that it has rubber marks from being kicked open. When doing research (and math exercises) it is important to make observations about a topic or problem. If you try to make observations about the kind of arguments made toward a certain thesis, then you might gain an insight into what kind of evidence is actually available. If most of the arguments advanced against an author's idea are ad-hominem it does not follow that the idea is false, but it may call into question the motives of those attacking it.
- Know what to look for.
Watson: “You appeared to [see] what was quite invisible to me.”
Holmes: “Not invisible but unnoticed, Watson. You did not know where to look, and so you missed all that was important.”
In word problems involving mathematics, one must know what to look for. Key words are central here. Words like integer, consecutive, or mixture tell you a great deal about what tools to use to find a solution. If you have a problem with your car starting you need to know to look for data related directly to the problem at hand. Are you out of gas, is your battery dead, do you need a new alternator, is the correct key in the ignition? Look for data related to the problem at hand.
- Mundane details are important!
Watson: “I had expected to see Sherlock Holmes impatient under this rambling and inconsequential narrative, but, on the contrary, he had listened with the greatest concentration of attention.”
When doing humanities research, one must notice mundane details, like repeated words, repeated settings, and small allusions to prior works. These details are commonplace, but it is precisely their commonplace nature that the author may have been relying upon, a common technique to convey meaning. On a larger scale, somebody researching the role of the American government in marriage should start with something mundane: the Constitution and the documents related to understanding it.
- Use solutions to little mysteries to solve bigger ones.
Holmes: “The ideal reasoner would, when he had once been shown a single fact in all its bearings, deduce from it not only all the chain of events which led up to it but also all the results which would follow from it.”
Or elsewhere, Watson: “Holmes walked slowly round and examined each and all of [the pieces of evidence] with the keenest interest.”
Many mathematics problems involve finding dozens of small pieces of data first. The most common thing I hear from students is, “I don't know.” But that's the point, it is a problem, there is no knowing, only solving. If you can break a difficult problem into numerous smaller problems for which you have the data to solve, the many small solutions can lead you to a competent solution. This is especially important in Geometry wherein in proving some angle measure might unlock a whole series of possibilities (for instance same side interior angles being supplementary proves that two lines are parallel, at this point numerous things can be said about a transversal passing through parallel lines!). Similarly, if you were writing a Bible paper about the meaning of a controversial passage, the problem can be broken into numerous pieces: Greek syntax, intertextuality, ancient rhetoric, discourse analysis, history of interpretation, and its relation to the rest of Scripture. But each problem has some solution which can contribute to the over all problem, which when phrased as a question is, “What does this passage mean?”
- Simple solutions are okay if the problem is simple. Do not over complicate things.
Holmes: “The case has been an interesting one…because it serves to show very clearly how simple the explanation may be of an affair which at first sight seems to be almost inexplicable.”
If the solution is easy to discover using simple methods, then by all means, solve it simply.
- But, do not use simple thinking to solve complicated problems.
Holmes: “This matter really strikes very much deeper than either you or the police were at first inclined to think. It appeared to you to be a simple case; to me it seems exceedingly complex.”
But, if a problem is very complex, do not imagine that shortcuts will do. This is true in mathematics and humanities research. Just because some problems involve a nail, it does not mean that all problems require a hammer. Nuance is necessary in mathematics as well as in studying the big ideas.
- Learn to look for similarities between new and old
problems:
Holmes: "'Quite an interesting study, that maiden,' he observed. 'I found her more interesting than her little problem, which, by the way, is a rather trite one. You will find parallel cases, if you consult my index, in Andover in '77, and there was something of the sort at The Hague last year." And again, "I was able to refer him to two parallel cases, the one at Riga in 1857, and the other at St. Louis in 1871, which have suggested to him the true solution."
All right triangles have an hypotenuse. All quadratic equations with discriminates greater than zero have two roots. Most authors are trying to be understood, most biographies are trying to convey meaningful, accurate information, and most quotations or statistical studies should be fact checked.
