Some time ago — and I forget when, I’m embarrassed to say, and can’t seem to find it because the search tool doesn’t work on comments — I was asked about how negative numbers got to be accepted. That’s a great question, particularly since while it seems like the idea of positive numbers is probably lost in prehistory, negative numbers definitely progressed in the past thousand years or so from something people might wildly speculate about to being a reasonably comfortable part of daily mathematics.

While searching for background information I ran across a doctoral thesis, Making Sense Of Negative Numbers, which is uncredited in the PDF I just linked to but appears to be by Dr Cecilia Kilhamn, of the University of Gothenburg, Sweden. Dr Kilhamn’s particular interest (here) is in how people learn to use negative numbers, so most of the thesis is about the conceptual difficulties people have when facing the minus sign (not least because it serves two roles, of marking a number as negative and of marking the subtraction operation), but the first chapters describe the historical process of developing the concept of negative numbers.

Particularly, Kilhamn — here following a stages-of-development scheme outlined by Arcavi and Bruckheimer’s The Negative Numbers: A Source-Work Collection For In-Service and Pre-Service Teacher Courses which I haven’t read either — traces negative numbers in western mathematics along a path like this: around the 16th century they’d be simply not recognized. By the 17th century and the time of René Descartes, they’d be seen as things which come up in finding roots of equations and a computational convenience. By the end of the 17th and start of the 18th century, they’d be used, but with an uneasy feeling because they didn’t seem to be quite logical. Since they were useful, though, over the 18th century negative numbers would get used pretty freely with problems like what they actually mean avoided by the simple means of not saying what they actually mean. This was also a period when the philosophical opposition to negative numbers within the mathematics community rose. Come the 19th century and we start to get serious attempts at giving negative numbers a rigorous mathematical definition, which was found by the late 19th century, which feels surprisingly late to me.

Kilhamn points out a distinction in negative numbers that I hadn’t thought of, or at least hadn’t considered since I was learning negative numbers long enough ago to have forgot the details: while the metaphor of representing a negative number as a debt makes tolerably good sense for things like money or apples or gallons of milk that you might owe during word problems, it’s harder to apply to things like time and distance (although since those can be oriented we can save the metaphor), and pretty absurd to put to things like weight or heat. I shall have to consider this when I next teach an introduction to negative numbers.

Kilhamn also points out that the desire to avoid negative numbers, still unfamiliar and not quite worked out, not only influenced Daniel Gabriel Fahrenheit to set the zero point of his temperature scale at “colder than it gets in the Netherlands” (where he lived), but might have influenced Anders Celsius in his temperature scheme as well. One of the little footnotes that usually gets overlooked in the history of Celsius’s scale is that while, yes, he picked the range from freezing to boiling of water to be a neat 100 degrees, he originally set the boiling point of water at zero and the freezing at 100 degrees centigrade, the temperature reading decreasing as the thing measured got hotter. This does suggest a desire to measure things like the temperature in northern Europe without having to deal with negative numbers.

Since Kilhamn is more interested in teaching negative numbers, the thesis doesn’t go into detail about the opposition to the concept past pointing out that there weren’t satisfactory explanations for what they were or how to use them at the time, or how metaphors used to justify them didn’t fit well enough. I’m not yet aware of a reference that talks about the difficulties and false starts in understanding negative numbers. I also admit being fascinated by references I’ve seen to Immanuel Kant having a minor hand in the understanding of negative numbers, but this thesis doesn’t help me understand that better either.

Nevertheless, at least the first couple chapters of this thesis do provide an overview of what problems negative numbers presented and roughly when they were answered. There is some mathematical jargon involved, necessarily, but I don’t believe it gets to be dense or obscure enough to be confusing.

It was me. And thank you for a fascinating post.

(Reading the Comics, Feb 26)

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Oh, thank you so. I had a feeling it was one of the Reading the Comics threads but couldn’t pin down which.

I’m really quite interested in trying to understand different models people had for negative numbers. References I’ve seen to people hypothesizing that negative numbers were larger than positive ones are intriguing, particularly when I think about the statistical mechanics definition of temperature.

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I am surprised that there was opposition so recently in history. I wonder when imaginary numbers came to be accepted?

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Folklore in the mathematics department is that imaginary numbers still aren’t really accepted at least over in the electrical engineering department.

(Electrical engineers, the mathematics folks say, use ‘j’ rather than ‘i’ to denote imaginary numbers, which are a convenient way to represent properties of alternating currents. The lore is that this is because electrical engineers won’t put up with ‘imaginary’ numbers because they’re not real, although the notion that this is because ‘i’, the symbol, is already doing heavy enough work representing quantities like current is more compelling to me.)

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In the real world:

We count, which generates the need for numbers, the natural numbers (including zero).

We need to measure amounts of stuff, weight, volume, area etcetera, which requires a unit of measurement and fractional numbers.

We need to describe positions, levels and changes, temperature, voltage, height, which leads to signed numbers (positive and negative).

These are three very different types of activity, and the simple minded idea that each of these number systems is simply an extension of the previous is not helpful to the understanding of what is going on.

There is a big difference between 3 apples and 3 feet.

There is an even bigger difference between 3 feet and 3 volts.

Algebra assumes that we are working in the signed number system, although some of the quantities involved, when algebra is applied to the real world, may be amounts, or counts. (Diophantine equations excepted).

With operations difficulties can arise unless we are very careful.

The worst case is “subtraction”. In the counting numbers it means “take away”.

In the measuring of amounts it means “cut off” or “pour away”.

In the measurement of position or level it means “lower by”.

The sign of a signed number says “above” or “below” zero, and also it specifies the direction of a change.

Here is my extract from A N Whitehead’s “An Introduction to Mathematics” (1911). It’s a good read.

http://howardat58.files.wordpress.com/2014/08/whitehead-intro-to-math-negative-nos.doc

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That Whitehead extract is a very good read, yes. I like the precise outlining of the different ways we might mean signed numbers to be; it

isprobably a slipping of intuitive feeling between one model of negative numbers and the others that causes a lot of trouble working with them.LikeLike