We frequently assign numbers to measure quantities in our lives. For example, we get assigned grades in school to measure our understanding of material.

We take these measurements for granted, but they have significant impact in our lives. Our GPA and SAT scores have influence on what colleges we get accepted to, what jobs we get, and whether or not we get scholarships.

The issue with these measurements is that they are 1-dimensional. That is, they are only one number. The nice thing about 1-dimensional measurements is that they can be easily compared. For example, a score of 85 on a test is better than a score of 77. However, the downside is that they do not contain very much information. So exactly how much information do these measurements contain?

There are many factors that go into determining who you are as a person. Your height, race, gender, favorite ice cream flavor, etc. Imagine every parameter that determines who you are as a person. It is not possible to answer exactly how many parameters, there are, but for our purposes let us suppose it is a very large number .

Let us call the space of all possible parameters that determines a person . Then the dimension of is . Suppose some 1-dimsnonal measurement on is taken. That is, we consider a map . In other words, for each person, we assign some number to that person depending on what we want to measure. Let us assume that is a smooth function.

Now, given some measurement , we would like to deduce something about the person who received that measurement. For example, if someone received a 90 in their calculus class, what can we deduce about that person? That is, if person received measurement , then , where , and we would like to know the dimension of .

It turns out that for general values of , the implicit function theorem tells us that the dimension of is . This is only one fewer dimension than the very large number that we started with. In other words, a 1-dimensional measurement only describes one parameter of information about a person.

If we wish to measure a complicated aspect of a person, such as their intelligence, then one number cannot provide us with a sufficient amount of information to make conclusions about that person. So why do we place such heavy emphasis on these quantities? One answer is that people tend to be lazy. If we can reduce the complicated problem of comparing two people to a much easier problem of comparing two numbers, then that saves us a lot of work. Another possible answer is that people do not realize how little information these measurements contain since it is difficult to intuit high-dimensional space.

It is no wonder why schooling and tests tend to be anxiety inducing experiences. When one is attempting to measure something complex, it can be extremely arbitrary which dimension actually ends up being measured.

While these 1-dimensional measurements still have some value, as they do provide some information, they do not contain nearly as much information as we tend to assume they do.