PHOTO: TOMASZ WALENTA
9 COMMENTS
In trying to make sense of our complicated world, we naturally look for boundaries and cutoff points. They make it easier for us to navigate our lives and make practical decisions. But this need for clear lines often comes with confusing anomalies. Fortunately, math has tools for helping us to make better sense of gray areas.
Consider body-mass index, or BMI, which is one measure doctors use to assess how fat someone is. It is calculated as weight (in kilograms) divided by height (in meters) squared. If you weigh a lot for your height, your BMI will be higher. Medical guidelines say that adults with a BMI of 25 or more are “overweight.” But is there really an important difference in terms of health between someone just under 25 and just over 25?
Or consider the not-unrelated question (for me, anyway) of how much cake to eat. I’m never sure how to indulge my love for cake without becoming a glutton. If I eat just one bite, surely that’s not a problem. But where do I draw the line if each individual bite is not a problem?
The principle of mathematical induction explains why it can be so hard to draw a line. An argument by induction is like climbing a staircase: You only have to know how to climb one step, and then you can climb any staircase, one step at a time. This sort of thinking was a great help to me when I ran the New York City Marathon.
But we’d be in trouble if we applied this logic to every case. It would mean that I could eat any amount of cake, one bite at a time, or safely gain weight so long as it was just a few grams at time. Induction doesn’t allow us to pin down exactly which increment takes us too far over the line—and that can leave us stuffed with cake and unable to fit into our clothes.
A different piece of math, called the intermediate value theorem, gives us a way out. Like many math theorems, this one gives us an overall picture, not an answer to a specific problem. It holds that if a value changes continuously and there is a point where it is negative and a point where it is positive, it must be zero somewhere in between.
PHOTO: TOMASZ WALENTA
9 COMMENTS
MORE IN EVERYDAY MATH
- Lessons of the Möbius Bagel August 10, 2017
- The Mysteries of Means and MediansJuly 13, 2017
- How to Measure Diversity June 14, 2017
Mathematicians use the intermediate value theorem to show that it’s possible for an equation to have a solution in a given range, in cases where they don’t need to know exactly what the solution is. It allows them not to waste time and effort on producing a more precise answer.
In real-life cases, the theorem tells us that instead of trying to draw an exact line between “safe” and “dangerous,” we should envision a gray area where the line might be. For cake, we might tell ourselves that danger lurks somewhere between eating one slice of cake and eating three. If we want to keep the pounds off, the inductive logic of “one more bite makes no difference” will be a safe guide only if we stay out of the gray zone and have no more than one slice of cake. For BMI, a score of 25 defines a similar gray area: it isn’t good for us to be close to that mark, and going too far beyond it means real risks to our health.
This approach is often useful in making public policy. We consider 18 to be old enough to drive, because we need that bright line as a practical matter, but we also know there is a gray area for younger teens who are ready to drive, so we allow for various kinds of provisional or restricted licenses.
Math gives us tools to approach the same situation in various ways, and it can produce different interpretations that are all logically valid. But it can’t tell us which approach makes sense in practical situations. We have to figure that out ourselves.
Math has tools for making better sense of gray areas
PHOTO: TOMASZ WALENTA
In trying to make sense of our complicated world, we naturally look for boundaries and cutoff points. They make it easier for us to navigate our lives and make practical decisions. But this need for clear lines often comes with confusing anomalies. Fortunately, math has tools for helping us to make better sense of gray areas.
Consider body-mass index, or BMI, which is one measure doctors use to assess how fat someone is. It is calculated as weight (in kilograms) divided by height (in meters) squared. If you weigh a lot for your height, your BMI will be higher. Medical guidelines say that adults with a BMI of 25 or more are “overweight.” But is there really an important difference in terms of health between someone just under 25 and just over 25?
Or consider the not-unrelated question (for me, anyway) of how much cake to eat. I’m never sure how to indulge my love for cake without becoming a glutton. If I eat just one bite, surely that’s not a problem. But where do I draw the line if each individual bite is not a problem?
The principle of mathematical induction explains why it can be so hard to draw a line. An argument by induction is like climbing a staircase: You only have to know how to climb one step, and then you can climb any staircase, one step at a time. This sort of thinking was a great help to me when I ran the New York City Marathon.
But we’d be in trouble if we applied this logic to every case. It would mean that I could eat any amount of cake, one bite at a time, or safely gain weight so long as it was just a few grams at time. Induction doesn’t allow us to pin down exactly which increment takes us too far over the line—and that can leave us stuffed with cake and unable to fit into our clothes.
A different piece of math, called the intermediate value theorem, gives us a way out. Like many math theorems, this one gives us an overall picture, not an answer to a specific problem. It holds that if a value changes continuously and there is a point where it is negative and a point where it is positive, it must be zero somewhere in between.
MORE IN EVERYDAY MATH
- Lessons of the Möbius Bagel August 10, 2017
- The Mysteries of Means and MediansJuly 13, 2017
- How to Measure Diversity June 14, 2017
Mathematicians use the intermediate value theorem to show that it’s possible for an equation to have a solution in a given range, in cases where they don’t need to know exactly what the solution is. It allows them not to waste time and effort on producing a more precise answer.
In real-life cases, the theorem tells us that instead of trying to draw an exact line between “safe” and “dangerous,” we should envision a gray area where the line might be. For cake, we might tell ourselves that danger lurks somewhere between eating one slice of cake and eating three. If we want to keep the pounds off, the inductive logic of “one more bite makes no difference” will be a safe guide only if we stay out of the gray zone and have no more than one slice of cake. For BMI, a score of 25 defines a similar gray area: it isn’t good for us to be close to that mark, and going too far beyond it means real risks to our health.
This approach is often useful in making public policy. We consider 18 to be old enough to drive, because we need that bright line as a practical matter, but we also know there is a gray area for younger teens who are ready to drive, so we allow for various kinds of provisional or restricted licenses.
Math gives us tools to approach the same situation in various ways, and it can produce different interpretations that are all logically valid. But it can’t tell us which approach makes sense in practical situations. We have to figure that out ourselves.
