Minus numbers

Bad Blood by literary critic Lorna Sage is a memoir of her postwar childhood in a poor Welsh village. It’s full of memorable and quotable passages; this is just one of the paragraphs that struck me.

Lorna Sage Hanmer school left its mark on my mental life, though. For instance, one day in a grammar school maths lesson I got into a crying jag over the notion of minus numbers. Minus one threw out my universe, it couldn’t exist, I couldn’t understand it. This, I realised tearfully, under coaxing from an amused (and mildly amazed) teacher, was because I thought numbers were things. In fact, cabbages. We’d been taught in Miss Myra’s class to do addition and subtraction by imagining more cabbages and fewer cabbages. Every time I did mental arithmetic I was juggling ghostly vegetables in my head. And when I tried to think of minus one I was trying to imagine an anti-cabbage, an anti-matter cabbage, which was as hard as conceiving of an alternative universe.

The power of abstraction that allows us to contemplate negative numbers shouldn’t be taken for granted; it’s downright miraculous. And the “alternative universe” comparison is spot on — the difference between imagining the existence of negative numbers and imagining the existence of extra dimensions of space is one of degree, not of kind.

To indulge in some pop evolutionary psychology (a bad habit, I admit), I can’t help but wonder whether our faculty of abstract mathematical reasoning is somehow related to the development of grammar. One of the more intriguing parts of Steven Pinker’s The Language Instinct is where he suggests that the important difference between humans and other species resides in grammar, and in particular in the subjunctive mood. We can speak in counterfactuals, and make statements of the form “If X had been the case, Y would have happened instead of Z.” An incredibly useful skill, allowing human beings to contract with each other in arbitrarily complicated ways, and therefore opening up the possibility of laws and morality and all that.

Best of all, it allows for math. It doesn’t seem such a great leap from speaking about situations that are not the case to speaking about quantities that can’t exist, abstracting from a certain set of cabbages to the general notion of “numbers.” And once you’re there, it’s a short distance to negative numbers, and imaginary numbers aren’t far behind. Pretty soon you’re talking confidently about the Riemann hypothesis and category theory, and people know not to invite you to cocktail parties.

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14 Responses to Minus numbers

  1. Saucy Wench says:

    A physicist, a biologist, and a mathematician are sitting outside a coffee shop across the street from a house. They see two people go into the house, and then, a few minutes later, they see three people come out.

    The physicist says, “The original measurement must have been inaccurate.”

    The biologist says, “They must have reproduced!”

    The mathematician says, “If one person goes into the house, then it will be empty.”

  2. Risa says:

    “I got into a crying jag over the notion of minus numbers.”

    This is the best sentence I’ve read in a long time.

  3. Clifford says:

    That’s good!

    Actually, a particle physicist or condensed matter physicist might well be more likely to give the mathematician’s reply than the mathematician.


  4. JMC says:

    I appreciated the excerpt and would like to continue on the topic of “some pop evolutionary psychology.” Loma Sage was perplexed trying to use conventional rules to fathom negative numbers. What are the laws necessary to understand negative numbers in math or speaking about situations that are not the case in language?
    Extrapolating meaning from the logical form of language becomes more complex when you introduce indefinites or conditionals. Similarly with mathematics when meaning disconnects from common logical form many, like Ms. Sage, are brought to tears. Is there some great divide between truth and meaning? Anti-matter exists. But what does that mean??!!?? Something that’s physical but not matter…? Ah, I must stop before I cry.

  5. Kuas says:

    Michio Kaku says: ” Look, experimental proof of string theory!”

  6. Jennifer says:

    Sean I like this so much. I can relate to people having existential crises when they are required to jump from one state to another. Like here, the state of numbers=cabbages, which is wonderful, simple, and then this discrete step up to numbers= …well, abstract things, defined more by their properties than by their is-ness. I’ve always wondered what I would tell a child who asked me *why* 1+1=2, or, more specifically, what is 1? The only way I know how to describe it is with cabbages. According to Merriam-Webster online, a number is:
    the property that a mathematical set has in common with all sets that can be put in one-to-one correspondence with it

    Mm. I remember taking quantum mechanics and writing notes, the wave function is written like this, yes, hit it with another with another wavefunction, from a different space, squeeze a Hamiltonian in the middle if you must, square it and you get the probability that you’ll end up in the state represented by the second wavefunction. And this stuff is so amazing, but we were supposed to take notes and learn how to calculate probabilities. And that was cool, but at some point I would have loved the professor to just stop speaking and look at us. As if to say, can you believe this? Isn’t this wild? Because we were being asked to make a jump, to grow up in a way, and you need some space to realize what you are really doing. My sister the poet said Jennifer you guys need to have cots in the backs of your classrooms, so when this stuff hits you you can go lie down for a while and try to take it in. Because when I would explain what I was learning to her, she would make me stop so she could digest it – it makes her head whirl.

    I’d always thought that people who need to lie down when making a leap were somehow…maybe a bit more advanced, as far as being a human being and really living life, understanding the immensity of what they are trying to take in. But, maybe we’re just a bit more in touch with our earliest ancestors…that would be cro-magnon and his ilk, not the fig-leafed ones…

  7. JMC says:

    Jennifer, I like the idea of lying down for a bit to take things in. Definitely would love a professor who put cots in the back!

    Sean, you mentioned whether “our faculty of abstract mathematical reasoning is somehow related to the development of grammar.” I wonder what film has to do with it. In my personal experience its made me more flexible with alternate worlds and things. I can envision special effects and multi-dimensional images that make any abstract notion feasible.

  8. MobyDikc says:

    Sean wrote,

    >the difference between imagining the existence of negative numbers and imagining the existence of extra dimensions of space is one of degree, not of kind

    Care to explain or support this notion somehow?

  9. Arun says:

    I thought negative numbers were first used to denote debts.

  10. Sean says:

    Moby, nothing profound: just that the concept of an n-dimensional manifold is an abstract notion, exactly as is the concept of a negative number. Neither is something you can simply point to out there in the world.

  11. David says:

    But there are perfectly concrete ways to represent negative numbers. Instead of thinking about piles of cabbages, think about two piles of cabbages with the condition that two piles with i and j cabbages are considered the same as two piles with i+n and j+n cabbages for any number of cabbages n. Add two sets of cabbages by combining the first piles and then the second piles.

    What you would normally call n cabbages, call n and 0 cabbages and then negative n cabbages just becomes 0 and n cabbages. No pesky minus signs to remember, just order your cabbages piles! That’s how I still think about it – helps with K-theory.

  12. Doug says:

    A great passage to quote! It ought to be included in every textbook for elementary math teachers. I had a similar experience when I was introduced to imaginary numbers by an inept teacher who couldn’t, or didn’t bother, to adequately explain the idea behind the “number” ‘i’. What I have learned since is that, just because we can grasp an idea abstractly by divorcing it from physical concepts, it doesn’t necessarily mean that we understand the meaning underlying the concept, only that we can understand how to use it as a method.

    For instance, the idea of negative space is absurd, but, nevertheless, we can use it to great advantage. So what does this mean, that we are divine creatures? It took mankind centuries to take this giant leap for the first time, and it was not done without a lot of hand wringing and pain. Yet, we expect children to grasp it as a matter of course in between watching American Idol and playing Mortal Kombat.

    Fortunately, I’ve finally found that we can understand the meaning of the idea as well as how to use it abstractly, but only if we are willing to ferret it out by thinking on our own, definitely not by reading textbooks. The key is to understand that space doesn’t exist as stuff that has properties, like cabbages, or fabric, and that the early Greeks were wise in keeping the ideas of magnitude and numbers separate.

    It turns out that, if we introduced to children the difference in the ideas of operationally defined magnitudes as opposed to quantitatively defined magnitudes, and the proper use of real numbers in these respective definitions, then we could avoid a lot of grief, not only for naïve children, trying to learn mathematics, but also for the sophisticated adults they later turn into, who then try to formulate physical theories.

  13. D Fitch says:

    On the subject of pop psychology about abstract mathematical reasoning somehow being related to the development of grammar and symbolic thought, you may be interested in the works of Keith Devlin (The Math Gene.)

  14. JMC says:

    Post 12 was very well put. I must ponder for a while and also check out the book mentioned in post 13. It’s stuff like this that makes mathematics so worth the excruciating efforts to understand both its functionality and meaning!