What does 'times' even mean ?

We were having a great conversation on Christmas day with all the kids in the family. One of the side conversations turned into a quiz of multiplication—questions like “what is 2 times 2?” and “what is 4 times 6?” My 8-year-old niece was answering questions, thinking out loud. When asked “what is 2 times 3?” she got it wrong. She said 5.

I wanted her to think through the right answer by herself, so I asked, “Why is it 5?” She responded with a question that puzzled me: “What does ‘times’ even mean?”

I had to pause and think about it. I was trying to teach her using some M&Ms on a coaster, but it took me a while to get there. I don’t think I even understood clearly what “times” meant up until that point but I was able to give a pretty good explanation and I thought I’d write this down.

But before we get to that, let me explain what mathematics is. I think we need to understand that before understanding a concept of mathematics, in this case its ‘times’ ot ‘multiplication’

What is Mathematics?

There are many definitions of what mathematics is, but here is my attempt: Mathematics is a representation of reality. It is not reality itself. It’s a human-made concept to represent something real. So the number 2 doesn’t really mean much if we don’t say “2 what?” It could be 2 cars, 2 apples, or 2 of something real.

Once humans invented math, it allowed them to find things that cannot be seen or felt with any human senses. For example, we can mathematically describe wavelengths of light that are outside the visible spectrum—like infrared or ultraviolet—even though we cannot see them directly. Or we can see three dimensions, but there could be many other dimensions that we cannot see.

Humans sometimes make the error of believing something that is mathematically possible but not possible in reality. Reality is not certain. And when mathematics refers to reality, it is not certain either. But some people think math is always certain, and similarly, some leaders believe their metrics are certain. Here is a quote from Einstein about mathematics:

“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

This idea extends beyond pure mathematics. When we use data and metrics to understand reality, we face the same challenge. This reminds me of a quote from Jeff Bezos:

“When the data and the anecdotes disagree, the anecdotes are usually right.”

What Does “Times” Mean?

The technical term for “times” is multiplication. What it really is: a quick way of counting the total number of things if we know that they are grouped in a certain way. These things are real-world things, like the M&Ms.

Let’s say we take a group of 2 M&Ms. There are 2 M&Ms in total. If we have 2 groups of 2 M&Ms, we have 4 M&Ms in total. If we have 3 groups of 2 M&Ms, we have 6 M&Ms in total.

The idea that things can be counted easily if they are grouped in a certain way is very useful and saves time in calculation. But in order for it to be useful in reality, we need two pieces of information:

How many groups are there?
How many things are in each group?

Both pieces of information must be correct.

We use the word “times” linguistically in plain English to communicate this. Instead of saying “if we have 2 groups of M&Ms with 3 in each group, how many M&Ms do we have in total?” we say “2 times 3 is 6.” In some places, we don’t say “times”—we say “2, 3’s are 6,” which would be a better way of representing it linguistically. It is also written as 2 X 3 = 6, which is easier to read.

The Same Total, Different Grouping

Now, if you expand this to other examples: if we have 3 groups with 2 M&Ms in each group, that would lead to the same total of 6. In other words, “2 X 3 = 6” and so is “3 X 2”. It’s 6 because it’s just a different way of grouping the same M&Ms.

In Short

Mathematics is a representation of reality, and “times” or multiplication is a quick way to find the total number of things if they are grouped.