Hi, I love your website and I have enjoyed reading the articles. Please could you help me with a question? I would like to ask the question regarding 'negative numbers'. Can there be such a thing as a negative? Please allow me to explain. My daughter recently brought home some Math homework that asked what -20 + -10 =. So this had me thinking, -20 (or-10) does not exist. There is no difference between having no apples to having minus a million apples both equal me having no apples. I don't think this is the same as debt as the amount in question (as in financial debt) does physically exist, even if you owe it. My daughters teacher explained that you have to see it as a scale. But I do not feel this explains the question either. For example if a car travels one direction on a scale (say North) at 100mph, if the scale is reversed the car is not travelling minus 100mph, it is now simply travelling South at 100mph. Scale I feel is inaccurate, surly its a measure of direction along an axis i.e. left or right. Another person said we have minuses in temperature as in minus 5 degrees Celsius. But again inaccurate as temperature is mealy a scale measured at the temperature of water freezing. There is no temperature lower than absolute zero (−273.15°), all temperature starts at -273.15°. As you may have noticed I am not the greatest at explaining things, but hopefully you will understand where I am coming from. I believe there is no such thing as a negative as it does not exist. Even to ask the question what is -20 + -10 does not make sense, as neither number exists the answer must always be zero? Thank you for you time Regards Philip

If I understand you correctly, there's a plausible point behind you question: things either exist or they don't. There's no such thing as what I'll call "negative existence" for shorthand, if that means a state that's somehow less than plain non-existence. And while there's no view so strange that some philosopher won't defend it, I'm betting most philosophers will agree: "negative existence" is a confused idea.

I'm pretty sure mathematicians will be equally willing to go along with that. And since the point about negative existence seems so uncontroversial, that suggests we need to ask: when people use negative numbers, do they really mean to suggest that there's something "below" non-existence?

I don't think so. Start with numbers themselves. There's a long-standing debate abut whether numbers of whatever sort exist, but we can sidestep that. There's a consistent, useful and highly successful enterprise called mathematics. From it, we learn all sorts of interesting and surprising things. Furthermore, we can use mathematics to do science, and to make exquisitely successful predictions that we find born out in the real world. Using math for this purpose inevitably includes using negative numbers. When we do that, or when we math for more ordinary purposes, are we really somehow committing ourselves to the weird metaphysics of something beyond (or below) existence and non-existence?

It's hard to see why. Think about my net worth. Suppose I have $500 in the bank, that the goods I owe could be sold for $1000, and that I owe $2000 to my credit card company. Then my net worth is -$500. That doesn't mean that there's some kind of metaphysically shady stuff called negative money. It just means that, for example, if I cleaned out my bank account, sold my goods and gave it all to the credit card company, I'd still owe them $500. The first point is that there's nothing mysterious there. The second point if that this is pretty much all we mean when we say that someone's net worth is a negative number. And the third point is that being able to use negative numbers for this sort of bookkeeping is extraordinarily useful. We could avoid the word "negative." We could say that I owe more than I own. But using negative numbers in this context is precisely a way of keeping track of the "owed" versus "owned" distinction.

Or think about your car example. Suppose I drive 100 miles due north, then turn around and drive 200 miles due south. Now 100 miles - 200 miles is -100 miles. You point out quite correctly that all this is really doing is telling me where I am along an axis. In this case, I'm at a point 100 miles south of my original starting point. That's absolutely right. And using negative numbers is an extremely useful way to keep track of that. The negative numbers don't carry a metaphysical commitment. Indeed, we can label points north of my starting point with negative numbers, or we could use the negative numbers for points south.

This suggests that what's really at stake here is a worry over a word. If we take the word "negative" in a certain way, it seems to point to metaphysical bogeymen. But when we look at how so-called negative numbers are actually used, we see that they don't carry any such implication.

A bit more abstractly, a mathematician might say this. We can add numbers. There's a special number, called "zero" in English, and usually written "0," that has a special property: when we add it to another number, nothing changes. And when we consider the full number system, each of the non-zero numbers has what's called an additive inverse. When we perform the operation we call addition on a number and its additive inverse, the result is the special number called 0. When I subtract the number p from the number q, I am adding the additive inverse of the number p to the number q. All of this is consistent, useful, and apart from worries about what numbers are in general (worries that aren't the same as the one you're raising), metaphysically innocuous.

So to sum up: you're quite right that we don't want to commit ourselves to fantastical states of sub-existence. But when we look at mathematics both as an internal, abstract enterprise and as it functions when we apply it, we see that talk of negative numbers doesn't bring the fantastical implications with it.

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