How do we know something is true?

Chapter from a book (in progress) to help students transition from school to college mathematics.

When you read or hear something, how do you know it’s true? Something about the world is being stated, or reported. “The atom is mostly empty space.” “Africa, Europe and the Americas used to be a single continent.” “There was an earthquake in Indonesia yesterday.” What you’re hearing or reading isn’t part of your direct experience and isn’t something you already knew. But you have a lot of ways to assess truth. What is the source? Does it fit plausibly with other facts? Is there accompanying evidence being presented, such as a photograph? If you’re skeptical, you might push back – ask further questions, cross-check other sources, seek to understand why it might be true even if it’s not an obvious truth. If it seems that someone came to those conclusions based on their direct experience, and the chain of reporting (perhaps spanning generations) that brought it to you is reliable, you’ll believe it.

Now let’s talk about mathematical truth. Suppose you are looking at a statement about a mathematical object. Mathematical objects are numbers, functions, graphs, matrices. . . all the things you find in math textbooks. These objects don’t exist in the world in the same way as atoms, continents or earthquakes do; you can’t see them, you can’t come to any conclusions about them by examining any evidence that you can apprehend with your physical senses. They exist in – and are brought into existence by – the human imagination. Somebody once found it useful to create these abstract entities (in most cases while trying to make sense of the physical world), and as others found them useful too, the ideas spread. One by one, each one of these objects became a shared notion that belongs to all of us.

The most basic example of a notion we all share is that of the natural numbers. We all know what we mean by “3 chairs” or “3 children” – but we’ve gone beyond that; it is part of our human intellectual inheritance that we are able to think of 3 itself as a thing, as an entity in its own right. And we know how it is related to, say 2, and to 9, and to a whole bunch of other natural numbers. These natural numbers belong to all of us, and we all get to think about them and say things about them.

Here are some examples of things we can say about 3:

  • “3 is greater than 1”
  • “3 is greater than 4”
  • “3 is prime”
  • “3 squared is 9”
  • “The square root of 3 is 9”
  • “The square root of 3 is another natural number”

Now some of these statements probably jump out at you as being false: no, we can’t say that “3 is greater than 4”! Or that “The square root of 3 is 9”. Well, we just did, haha – but the thing is, just because you can say something meaningful doesn’t mean it is true. Now in this list of statements here you just know which ones are true. How do you know? Because

a) you are acquainted with the objects mentioned (1, 3, 4, 9),

b) you know what the terms used mean (greater than, prime, square root), and

c) you know how these objects are related to each other (your prior knowledge.)

If I say “3 squared is 9,” you know what is being said, and you also know that it is true. If I say “3 is greater than 4,” you know what is being said, and you also know that it is not true.

Let’s look at more things one can say about numbers. “Numbers” itself can refer to a lot of different things – for example √ 2 is a number – but for now we want to talk only about the natural numbers. So let’s agree that for now “number” is to mean “natural number”, by which we mean the numbers 0, 1, 2, 3, 4 and so on. (Yes, we’re including 0, which is a common mathematical practice.)

Here’s another statement:

  • The number 12 can be expressed as the sum of two prime numbers.

Is this true? It might take a moment’s thought, but it’s easy to settle. Here’s another statement:

  • The number 125 can be expressed as the sum of two prime numbers.

Is it true? In this case we can’t know immediately, but we do know what it means for it to be true or not. The number 125 is big, but we could make a list of all the primes less than it. Either some two of them add up to 125 , making the statement true, or else no two of them do, in which case the statement is false. So even if we can’t be bothered to do the actual computations, we do know what it means for the statement to be true, and know how to settle whether the statement is in fact true.

  • If you take any odd number n, the number n2 is odd.

Is this true or not? Let’s recall what even and odd numbers are – of course you know, but let’s go ahead and articulate it. That helps. Here goes. Even numbers are the ones that are multiples of 2. An even number is 2 times something, and anything that is 2 times something is even. An odd number is one that isn’t a multiple of 2, which means when you divide it by 2 you get a remainder of 1. An odd number is 2 times something plus 1, and anything that is 2 times something plus 1 is odd. Okay, so we know what the odd and even numbers are, and we have a description to refer to.

Now let’s examine the statement: “If you take any odd number n, the number n2 is odd.” The first few odd numbers are 1, 3, 5, 7, 9, 11, 13, and their squares are 1, 9, 25, 49, 121, 169 – they are all in fact odd. I can even check more. But I can never check ALL, because there is no last odd number. The list never ends. So if checking all is not possible, I need something else. I need an argument that applies to all odd numbers, without my having to do computations on one odd number at a time. Here’s one way to go about it:

Let’s take an odd number, say n. (Look at what we are doing here. We aren’t saying, “Let’s take an odd number, say 21.” We don’t want to get stuck with 21 – we want our argument to cover all odd numbers at the same time. So we don’t commit – we just say “n” – and this n could be 21 but it could also be any other odd number. This is the way we get to talk about all odd numbers simultaneously.)

Okay, now our n is odd – what does that mean? It means n is an even number plus 1, or in other words, n has the form 2k + 1 for some k. So now n2 is (2k + 1)(2k + 1), which we can multiply out to get 4k2 + 2k + 1. This last expression shows us that we get an even number plus 1. (Right? Because 4k2 + 2k + 1 is the same as 2(2k2 + k) + 1.) So yes, n2 is odd.

What did we use here? We used what it means for any number n to be odd, and we used the rules of basic arithmetic. These were our ingredients, and using these ingredients we cooked up an argument.

Now you try one.

  • If n is even, n2 is also even.

Now this may feel like you already know this is true, but the point here is, can you say WHY it is true? Can you figure out and write down why this must always be true? Try it, right here. Your ingredients could be what it means for a number to be even, and the rules of basic arithmetic.

Try writing down proofs of these:

  • The sum of two even numbers is always even.
  • The sum of two odd numbers is always even.

At this point we have established four things:

  1. If n is odd, n2 is also odd.
  2. If n is even, n2 is also even.
  3. Even + even is even.
  4. Odd + odd is even.

And the ingredients that got us there were the definitions of evenness and oddness and the basic rules of arithmetic.

Now let’s use these four things as our ingredients to cook a dish that’s a bit more complex. It’s exactly like using basic ingredients to make garam masala out of individual spices, or imli ki chutney out of tamarind and dates and water and spices, or yoghurt out of milk, and then using these to make more complex dishes.

So here’s our next-level dish.

  • If n is any number, then n2 + n is always even.

Try it!

You have now tasted how the whole of mathematics proceeds. We start with some basic facts, and use logical arguments to establish more complex facts. Anything that has already been established, by you or by other people, can become a new ingredient in the next argument. Going backwards, if there is a mathematical statement that is true, you can always ask, why is that true, and why is that true, and in turn why is that true, until at the very bottom of the pyramid there are some basic simple facts that we all agree on. In what we did here, we took the basic algebra of the natural numbers to be our starting point, or the base of our pyramid.

When we do any mathematics, the base of the pyramid is always specified. In the courses you take, basic facts about set theory and basic algebra of numbers (including the natural numbers, and importantly, the principle of induction) are generally part of that base. In addition, the base will include other facts, or axioms, that will be specific to the particular area of study, whether it is group theory or linear algebra or analysis. For example, in your analysis textbooks, you will find a list of axioms of the real numbers: a list of basic facts about the real numbers that we take as the starting point for our further investigations. For example, these axioms say that when you add two real numbers, say a and b, then the output a+b will always be another real number, and also that the order doesn’t matter, so the output when you perform a + b will be the same as the output of b + a. There are other axioms that tell us about the other behaviours of the real numbers. Taken all together, these will be your starting point from which you will prove, or deduce, all other truths about the real number system, all the way up to the theorems of calculus and beyond.