Chapter 2 in Yearning.
Prelude
Warmup What is $-5\times -7$? Why is that so?
What kinds of numbers do you know about? What distinguishes them from one another? Do they exist? What warrants their being called numbers?
Blog post
Existence – Mathematical and Otherwise
In class, we talked about existence of certain numbers. We noticed that with $-3, \sqrt 2, i$ they all seem to lie in different places for people in a spectrum of reality. We also talked about how something everyone agreed was unreal, unicorns, was nonetheless not-nonsense in a very strong way. The way in which we can agree on what a unicorn is even though we’ve never seen one is called inter-subjective agreement. In some ways this is similar to an objective fact, but we know that the reliability of this concept comes from our agreement to agree about what it is.
There’s a brief chapter about this idea, “Mathematical Objects and Structures: Existence” in The Mathematical Experience. It’s where I got the idea for the unicorn.
As we continue to discuss existence let’s try to keep in mind a few things. The first is that this should be a playful space for everyone in the room. We have already seen that different things feel to different people. So let’s not push so hard on how we feel about someone who doesn’t believe what we do. Besides leading to antagonism, which is our chief enemy in trying to talk math with one another, it misses the main point I want everyone to see about existence – it is not a simple dichotomy. Existence of one thing or idea can often hinge on others, or on hidden assumptions, value systems, or epistemologies (assumptions about how we can know stuff). These are rarely articulated openly, but by asking the right questions we can see how existence of certain things depend on them.
Math is a fun place to talk about existence if we give it a chance. Part of this is because the ideas of the things in question are synonymous with the things themselves (or are they?) – we don’t really need the messy world to intervene, and yet these ideas are not exactly fiction either. Although unicorns and real numbers might seem to be real in a similar way, zombies are an example from fiction that doesn’t feel like math: in every movie or book, zombies have variable properties. Are they fast or slow? Do you get infected from bites or blood? Do they hop like bunnies? Can they act collectively? We can say that real numbers do not exist, but if we grant them existence we can’t choose for them to be countable.
Even though I claim that existence is not a light switch, either on or off, I do think we all have a gut feeling about these things that does take sides and is worth noting before we go further. Sometimes an ability to acknowledge other perspectives is to first acknowledge your own as a perspective. Then we can see where these perspectives come from and how they can change. Even with the negative numbers, which everyone felt to be real on a gut level, there was a time in each of your lives that this was not true. And as we’ll see from Chapter 2 in Stillwell, it was not too long ago that this was the case for mathematicians.
One way in which we are often willing to grant reality to something is when it has “practical consequences” or a “real world application”. This is an important source of reality, but we should be careful; it is not self explanatory. In many ways, these reasons are their own questions. For example:
- Commerce and the concept of owing someone money has existed for millenia. It has been around for so long, no one really knows when it first came about. Today, we think about it as an obvious model and application for negative numbers. If it alone was what made the negative numbers real, shouldn’t we have had them much sooner?
- We would probably agree that zombies are not real, yet they have many “real world applications”. What were the box office receipts for World War Z last year?
By interrogating this concept, we may arrive at a more general substitute — the reality of a thing is described by the value it holds or the work it does within a social system. Within the entertainment business, the reality of zombies is guaranteed, while we still do not expect the CDC to be on watch for a zombitus outbreak. For a mathematician, $i$ is a simple, concrete number. Every bit as real as 2. For the random person we stop on the street, we would not expect them to feel the same way, partly because they likely don’t have much care about the solution of polynomial equations or unifying concepts of analysis.
Finally, the point of opening things up a bit isn’t a bland relativism – everybody thinks something different and that’s okay! These social realms are not typically isolated from one another or rigidly defined in terms of their members or prinicples. Ideas become stronger when they have use in more systems, and new uses are invented all the time. We learn more when we consider ideas from different perspectives, value systems, uses, and epistemologies. Just because that random person walking down the street doesn’t think of $i$ as something real doesn’t mean there’s nothing to gain.
So let’s keep an eye on $-3, i, \sqrt 2, \aleph_0$ and what about $\pi, e,$, or even $3$? Soon enough we’ll have new numbers too, probably no one in class has heard of at least some of them: the hyperreals, p-adic numbers, quaternions.
Warmup
we saw why the numbers $1, 2,3,\ldots$ , used for counting and basic arithmetic, do not meet all the needs of geometry. To measure line segments we need irrational numbers such as $\sqrt 2$, and indeed a continuous sequence of numbers that fill the spaces between $1, 2,3,\ldots$ with something like a line itself. We get a full line, called the real number line $\Bbb R$,
by extending the continuous sequence of numbers backward through $0, -1,-2, -3,\ldots$, thus balancing each positive number $x$ with its negative, $-x$.
In the above passage, what is the warrant?
On p. 27 Stillwell gives an example of how people might not actually use the distributive property in practice. This brings up a question about the differences between how things are supposed to work and how they actually work. I am reminded of grammar rules. Until I took a Linguistics course in college, I always thought of language like English teachers do. There’s the correct way and the wrong way. If you’re not saying something correctly, you’re making a mistake. Linguists have a different perspective though. English to them isn’t what’s in Strunk and White – a manual of style – but rather a natural phenomenon. To understand its grammar, they look at how its actually spoken, not how it should be. Similarly, Jean Lave thought to actually take a look at how people really do math (basic arithmetic) in the world.
Research Read some part of Cognition in Practice. How does it change how you think about how people do math?
Stillwell gives us 1585 as the first use he knows of the multiplication of negative numbers, but not until 1830 was this act common. He also says that negative numbers are familiar to anyone who has handled money, presumedly dating the existence of negative numbers far back into history.
Recall What is the relevance of the problem $(8-5)(9-7)$ for the development of negative numbers?
Reflect Find some of the details to this story. How does it contradict our usual sense of applying math to the world? Make an explicit statement about how you imagine math is usually developed in the world.
Reflect Unlike some other mathematical achievements in history, negative numbers seem almost obvious. There was no genius who saw farther than the rest of humanity. Why did it take so long for arithmetic with negative numbers to really happen?
Warmup Solve $ax^2+bx+c=0$ by completing the square visually and by symbol manipulation.
