Is Math truly Logical?
You’ve probably been told that math is a logical subject. Perhaps, you know that to succeed in math, you should think logically. Each step should accurately, precisely, and logically follow from the previous step, based on a rigorous set of unforgiving assumptions. It is a boring, old-school subject restricting our creativity, requiring us to follow strict, unchanging rules, allowing only one right answer, and pushing us to think inside the box.
If you thought about math this way, then congratulations! You’re… well, partially correct. Why is that?
Let’s take a look at the definition of logic:
The first definition is what we’re looking for:
Logic is defined as reasoning conducted or assessed according to strict principles of validity.
First and foremost, yes, math requires logic. If you’ve done math at the upper elementary/middle school level, you might know that each expression must be logically connected to the previous and the next. Each step implies the next, meaning that it needs to be written with clear lines of reasoning and complete validity. In fact, the process of induction is seldom used in mathematical proof-writing, just to maintain the complete logical validity of mathematical statements (ironically, mathematical induction uses deductive reasoning; check that out for yourselves).
But, notice that the above process only happens when we write mathematics. It is one of the inherent traits of mathematics, and of most sciences, that your ideas must be expressed clearly and logically. The question is, does logical communication correspond to logical thinking?
Not really. It is sometimes true, and that may be the reason why many teachers link mathematics to logical thinking. However, even then, logical reasoning in mathematics is very different from what we typically consider logical reasoning, say, in a logic puzzle.
Let’s see an example of a typical example of logical reasoning used in a logic puzzle.
One day a mad scientist lined up Andy, Brandy, Candy and Dandy in a row, so that each of them could see the ones in front of them but not behind. Andy was able to see everyone else while Dandy couldn’t see anyone. Then the mad scientist declared,
“There is a red hat, a blue hat, a white hat, and another hat that is either red, blue or white. I will place them on your heads, so that you can’t see the color of your own hat. However, you can see the hat color of anyone in front of you.”
Starting from the back (Andy first), he asked them each in turn what the color of their hat was. To his surprise, they all were able to correctly deduce the color of their hat based on the responses that they heard.
Which 2 people had the same color hats?
Here’s a solution:
- If Andy had seen hats of 3 different colors, then he would not have been able to deduce his own hat color. Thus, he saw 2 hats of the same color and 1 hat of a different color.
- If Bandy had seen 2 hats of different colors, then he would not have been able to determine his own hat color. Thus, he must have seen 2 hats of the same color, and then called out the remaining color.
Thus, Candy and Dandy had hats of the same color.
This follows a very clear line of deductive reasoning, which goes by casework (exhaustively listing all cases), then continuing casework for each subcase until a contradiction is reached, eliminating the sub-cases. If you think about it, most logic puzzles have very strong implicit assumptions; in some sense, this makes them mere processes of guess-and-check, and a strict, final answer can always be reached through a limited number of such steps.
If you’ve ever gotten stuck on a math problem, you’ll realize that this isn’t the case. Let’s say you have to solve a simple quadratic equation:
Assuming you have never seen this before, how are you going to guess and check? It definitely wouldn’t be a great idea if you tested all real values; if you were to use logical reasoning, and assuming you have some mathematical knowledge, you may want to somehow simplify this into a product of linear expressions.
The problem is, nothing is guaranteed to work. And, if you’re unlucky enough, you may test infinitely methods that appear logical and get nowhere. In fact, the odds of this happening increase as the problem gets more complex.
Logic puzzles do not correspond well to math problems, but more sophisticated, real-life decision-making may be a bit closer.
Let’s say you’re working for a country’s ministry of education. You see that there has been an increase in depression amongst middle school students, and you’re asked to analyze the topic.
Such a problem requires thorough use of logical reasoning, but you can’t proceed by elimination. At least, it would be very difficult, as there are theoretically infinitely many situations. Here, you would be making implicit assumptions to simplify the problem as you go, either based on your personal experiences or (more likely) objective data.
This comes a lot closer to mathematical reasoning and is the core of reasoning in most natural sciences. Interestingly, using very strict logical deduction may not get you anywhere, and there is no guarantee that you can resolve the problem in a finite number of attempts.
Now, the difference between such a reasoning process and mathematical reasoning mainly lies in abstraction. Not only acknowledge that some letter corresponds to an unknown value and know the definitions of each abstract symbol, but you have to understand everything logically and think in terms of abstract concepts.
In the education example, one knows that middle school students are likely experiencing puberty, enjoy playing, and don’t like to do homework. They may also know the special characteristics of middle school students in different geographical locations, and all of this information motivates them to choose an assumption and continue a certain line of reasoning. Similarly, each abstract concept in math provides its own motivation.
For instance, in the quadratic equation example, a motivation could be that squared terms increase much faster than linear terms (albeit, this isn’t very useful). However, compared to decision-making, it is harder for humans to conceive of such motivation in an abstract, mathematical context. After all, most of the motivations used in decision-making come from our observations in everyday life, and it’s much more tedious for our brains to observe the details in every abstract mathematical expression.
With the above examples, you probably realized that even when logical reasoning is used in a mathematical context, it is quite different from the logic and reasoning we’re typically used to. And, it is one of the reasons why despite subjects like biology, chemistry, economics, etc. all requiring the usage of extensive logical reasoning, people proficient in those areas may not be proficient in mathematics.
Now, there are also situations where logic and reasoning play a minimal role during the mathematical problem-solving process. It proves that math isn’t as boring and inside-the-box as you may think, having just as much room for creativity and imagination as most subjects. That’s a much more sophisticated topic, and I’ll probably discuss that at another time.
In the meantime, have you had any peculiar or difficult experiences with mathematics? If so, please feel free to share! Math may seem difficult for some, but it can be really fun once you get the gist of it. It’s saddening how schools make math such a painful experience. So, instead of relying on others, why not try it out for yourself?
