Monday, August 21, 2023

The Historical Acceptance of Negative Numbers

Mathematician tries to trace the acceptance of negative numbers, in 2010 essay:
I was flabbergasted when I first read Augustus De Morgan’s writings about negative numbers1. For example, in the Penny Cyclopedia of 1843, to which he contributed many articles, he wrote in the article Negative and Impossible Quantities:
It is not our intention to follow the earlier algebraists through their different uses of negative numbers. These creations of algebra retained their existence, in the face of the obvious deficiency of rational explanation which characterized every attempt at their theory.
In fact, he spent much of his life, first showing how equations with these meaningless negative numbers could be reworked so as to assert honest facts involving only positive numbers and, later, working slowly towards a definition of abstract rings and fields, the ideas which he felt were the only way to build a fully satisfactory theory of negative numbers. On the other hand, every school child today is taught in fourth and fifth grade about negative numbers and how to do arithm
Wallis and Newton had fully accepted negative numbers by 1685.

Closely related is the discovery of zero.

It is repeated everywhere that the Indians invented zero and place notation and that the Arabs learned it from them and later transmitted this to Europe. It’s bizarre that such a misunderstanding should be widespread but in fact, the Babylonians invented place notation (albeit using base 60) and their arithmetic was used by many Greeks, e.g. Ptolemy. I hope I have made the case that the most substantial arithmetic discovery of the Indians – and independently the Chinese – was not merely that of zero but the discovery of negative numbers. Sadly this discovery was not absorbed in any but a superficial way by the Arabs.
His essay has examples of famous mathematicians being leery about negative numbers. Also imaginary numbers, infinities, and other constructs.

I am not sure how well these are accepted today, outside of Mathematics. If you read the business section of the newspaper, a company's loss is just a negative profit, but the articles hardly every express it that way.

1 comment:

  1. Roger,
    Seriously man, there are no 'negative profits', or 'positive losses' for that matter. Semantic bullshit is nothing but 'creative accounting', it's always misleading and usually harmful.

    'Numbers' are a somewhat deceptive word or term in the mouth of a mathematician.
    Historically, people were quite naturally skeptical of that which isn't even remotely actual. This isn't stupidity, but the plain old observation of many intelligent people smarter than us who weren't idiots and who didn't like being fooled.

    Negative numbers are in fact a different kind of thing than counting numbers, which are closely related to quantity. You can encounter an apple, see it on a plate. You can eat an apple. While you can easily say 'I have no apples', you can't encounter, see, or eat, or interact with a negative apple, much much less 'i' apples. Same goes for time, which is why answers that produce negative time quantities are usually discarded altogether as useless.

    Things like negative numbers, and especially 'imaginary' numbers are entirely Cartesian in nature, they exist only as a product of graphing on a grid, which is not the same context, visualization, or conceptualization of numbers and how they were used until that point. Once again, This isn't a failing of people, this is people saying 'you are conflating different kinds of things and calling them the same thing'...which mathematicians are historically notorious at doing, often dishonestly eliding between entirely different concepts in sloppy and convoluted ways...and frequently used to bamboozle people in everything from accounting to card games. Counting isn't terribly abstract as it pertains to the actual world, but calling a coordinate place holder like (10, -3) or 'i' is. While a coordinate can be said to be composed of numbers, like 40°42'51.4'' N 74°0.358' W, or an address like 123 Mainstreet, 22141, VA, it isn't a number, it's a geographical location that uses numbers within a framework outside of quantity, which is how people at the time primarily used numbers.

    A tail is not a leg, even if you call it one, and especially if you then try to walk on it. Calling infinity a number is just sloppy nomenclature, as it is not a number at all, it has no fixed value either known or unknown. It's like saying 'go North', or 'keep on trucking', left, right, straight up, down, or forwards is a number. Well, it ain't. You can't add, multiply, divide, or subtract 'forwards' or 'that-a-way'.

    Infinity has also been misused excessively in physics, to the point we have entire branches of word-game mickey-mouse reasoning from calculations dependent upon quantities that can never be observed or measured. A great example of this is space time, where physicists like to mix and match impossibles, 'I place my point of zero volume on my magic graph and magically apply mass to it (without any physical extension), and Voila, magic black hole of infinite density! Aren't I clever?!'