severoon's Journal: Indirect Thinking

The unexamined life is not worth living. --Socrates, Apology, 38

I found myself recently wondering if a question exists that can be answered without directly addressing the question, but instead by considering the nature of the question itself. To give an example of what I mean by "directly addressing" a question, consider the following scenario.

I ask you: is 56*19 = 1040? To directly approach the answer to this question, you could do a number of things. You could calculate 56*19 and compare the result to 1040 to see if they are the same. You could do something algebraically equivalent as well, such as dividing 1040 by 56 to see if the result is 19. My use of the word direct in describing this approach has to do with the fact that these approaches depend upon executing exactly the operations (or the algebraic equivalent) present in the problem statement.

This is how we humans are taught to solve problems (at least, how this human was taught). This approach solves the problem, but not in a way that relies on creativity or intuition...it instead requires the solver to apply elementary knowledge in a simple and straightforward manner to find the answer. The solution is found through an application of knowledge and could more or less be carried out by someone that isn't necessarily intelligent but, rather, has memorized a sequence of steps without any real insight into the problem. Even so, using this approach tends to give people a feeling of satisfaction because it answers more than what the question explicitly asks (which could be addressed with a simple yes or no) and goes a step further in answering the unasked question: What is 56*19?

Humans have the habit of assuming the presence of these implicit questions. If you ask this question of your friends and relatives, I'm fairly confident that a good number of them will not simply respond with a simple yes or no, exactly enough to answer the question. Instead, they're likely to reply with a number.

So we've covered how to answer this question using a direct approach. What would be an indirect approach? Well, here's one example. I might notice that, whenever one multiplies two numbers together, the last digit of the product is always the same as the last digit of the product of the last two digits of the arguments. That was a mouthful, so if you don't feel like parsing it, fear not and read on. The rightmost digits of 56 and 19 are 6 and 9, respectively. The product of these two digits is 54, the rightmost digit of which is 4. So without actually multiplying 56 and 19 together, I can say with certainty that the final digit of the product will be 4. In the problem statement, the suggested product is 1040, which does not have a final digit of 4, and therefore cannot be the correct product.

This approach, while a bit more convoluted, is a bit more appealing to me because it seeks to provide an easy means of answering not just this particular question, but an entire class of similar questions. Furthermore, it requires a minimal amount of information from the problem statement while still providing the answer.

Consider, for instance, a slightly different problem. I ask you: is 734...211 (a 1000-digit number, of which I am only telling you the first and last three digits) times 349...214 (another 1000-digit number) equal to 936...001 (a 2000-digit number)? Using the direct approach of multiplying and comparing your result, you might conclude that this question is unanswerable because you can't carry out the multiplication; you don't even know what the actual numbers are.

Using the indirect approach, however, one can immediately see that not only is the answer to the question obtainable, but easily so! Even grade schoolers can do the required calculation in their heads. On the other hand, using the direct approach, even if I acquiesced and provided you all of the digits of the three numbers, finding the answer would require quite a long bit of calculating.

There is one little hang-up with the indirect approach, though. Perhaps you noticed it already...the class of problems that this approach applies to must be carefully considered. For instance, one might think this approach can be used to quickly answer any such question, such as: Is 21*21 = 421? You might be tempted, based upon the indirect approach, to say yes, it appears so. But do the calculation, and you'll discover that it is not so, it's merely that the actual product (441) happens to have the same final digit as the suggested, but incorrect, product. So this indirect approach is only useful in answering this question when it contradicts the suggested product, but outside the class of problems that fall into this cateogory, this approach has little to contribute.

Note I say it has little to contribute, but not nothing. If we assume that incorrect suggested products sport a last digit that is evenly distributed over all ten of the possible values, then the suggested approach will answer the question 9 times out of 10 (when the final digit of the suggested product does not match the actual product's final digit). That means when the problem doesn't belong to this method's problem space, while we cannot say the proposed product is correct, we can say we have a slightly higher degree of confidence that it is correct than before we applied this method. This alone is negligible, but consider if we had a long list of different methods such as this one to apply to the problem. If all of them failed to show the suggested product is incorrect, and there were enough of them, we might be able to have a very high degree of confidence that the proposed product is correct. It is still not a sure thing, depending on the completeness of the methods we've assembled in combination, but it's better than nothing and in some cases still far less work than the direct approach.

So we might conclude from this that using several indirect approaches in concert can be a very effective way to deal with a seemingly intractable problem. For the sake of example, let's consider a second indirect approach to the same kind of problem: is 22*47 = 1,048,224 true? Can a 2-digit number multiplied with a 2-digit number result in a 7-digit product? If you study this a bit, you can conclude that any m-digit number multiplied by any n-digit number will result in a product not larger than an (m+n)-digit number. (This is also known as checking by "order-of-magnitude.") Since the proposed product above is not 4 digits long, it cannot be correct. So here is a case where the first indirect method could not be applied, but the second one works. These two approaches, taken together, effectively expands the problem space that we can answer conclusively. If the problem at hand involved two 1000-digit numbers, one could develop quite a long checklist of these techniques and still find that the total amount of work involved in performing each and every check is still far less work than the direct approach.

This is all well and good for mathematics, but can this kind of indirect reasoning be applied to knotty "real life" questions, such as those that might arise in a discipline such as philosophy? I'm glad you asked, because this is exactly the thought I found myself considering a few days ago. And, I think I've found an example question that is related to the quote at the top of this essay.

The philosophical question is: is it useful to examine one's own beliefs? To directly answer this question, one might choose a particular position ("yes, examination of one's own beliefs is useful") and then set out to support this position. Then, one might choose the opposite position ("no, self-examination is futile and a waste of valuable time") and set about supporting that viewpoint. Once one is satisfied that both positions are bolstered to the best of one's abilities, one could compare the two arguments and see which one is more complete, more self-consistent, etc, and ultimately buy into one or the other. (It is worth pointing out, however, that addressing such questions in this way have been known to occupy great minds for years and even lifetimes without being resolved to the person's satisfaction.)

Now, let us consider the indirect approach to such a question. For the sake of this argument, let's assume that there exists support in favor of not examining one's beliefs, and this support trumps that of the opposing position. (That is not to say such support actually exists--only that we are assuming it as a basis to move forward along the indirect path.) If such an argument exists against self-examination and we were to discover it, we would without a doubt consider this newfound knowledge useful--why else would we put in the time and effort to uncover such a conclusion if the result of that work was not interesting to us?

But, if we were to achieve this state of affairs, we would immediately find ourselves in a quandary. The argument we have discovered is itself an example of the self-examination of one's beliefs yielding a result we find useful! It seems that the mere act of finding a good argument for this position disproves such an argument.

Now, let's consider our original assumption, which is: the argument against self-examination is better than the argument for it. All of the reasoning above is based upon this assumption, but what if this assumption is incorrect? If this assumption is wrong, then it means the opposite is true--that the argument for self-examination is better. So, even if our original assumption is invalid, we arrive at the same conclusion. Any way we cut it, the only logical stance is that self-examination of one's beliefs is useful, which is the answer to our question. We were able to arrive at this conclusion without even attempting to answer the question directly...instead, simply by considering the nature of the question itself, we were able to arrive at the correct answer.

Such methods of indirect reasoning are the hallmark of great thinking in many fields. Mathematics even sports formal methods of reasoning based upon indirect thinking...to name a couple: reductio ad absurdum and mathematical induction. Both have at their cores indirect thinking. ("Reductio ad absurdum" literally means "reduce to absurdity"...it is a method of answering a mathematical question by taking a position and showing that it irrevocably leads to impossibilities. Mathematical induction proves that a thing is generally true by proving that, if that thing is true in a single case, then it also must be true in each successive case. All that remains once this is done is to show any single example.)

What I have termed indirect thinking in this essay is, I think, an essential attribute of all truly deep understanding. It is an example of the synergy that can arise in the mind when that mind not only knows how to find an answer through brute force means, but also intimately understands the nature of the question and the essence of what allows the brute force method to work.

Problems with indirect thinking

[tootlemonde]

Any way we cut it, the only logical stance is that self-examination of one's beliefs is useful...

I will try to demonstrate that self-examination is not necessarily useful and that it does not by itself make life worth living.

The problem with applying methods that return either true or false in matters of ethics is that there is often a large area of ambiguity in what is right or wrong.

Socrates' principle offers a good example.

If we assume that the unexamined life is not worth living, does it follow tha