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Filtering by Tag: history

Without =

All I want to say here is how interesting I think it is to consider:

For thousands of years people did Mathematics without an equal-sign! The “=” symbol, if memory of a history of Math text serves me correctly, made its first appears due to … Fibonacci? Perhaps I’m getting my mental wires crossed with the fact that he imported Arabic numerals. Maybe it was Recorde? If only there were a way to determine its source without leaving my computer … Well I guess we’ll never know.

But before the 13th century much of mathematics had no symbol for equality, and people merely used the word “equals” or just a blank space where an equality symbol might go. In some ways we might do well to return to that, at least for introductory students, because they often fail to realize that the equal-symbol means the same that the English word “equals” means: That the thing on the left is the same as the thing on the right. (That’s not actually entirely true, equality can be used for the definition of a function, which isn’t precisely, logically, the same thing.)

However, ancient Egyptians had a symbol for it similar to what we now use for proportionality. Certain later Greek speakers had their own symbol although it did not persist through history. Ancient Indians used pha which was a contraction in their language. Today’s equal-sign, “=”, derives from the signification of two parallel lines. This choice is due to the fact that parallelness behaves very similarly to equality. Some of the features of equality that are most useful are:

  1. Reflexivity: Every line is parallel to itself just as every object is equal to itself.

  2. Symmetry: If line m is parallel to line n then line n is also parallel to line m. A similar statement holds for equality.

  3. Transitivity: If m is parallel to n, and n is parallel to p, then m must be parallel to p. Again, the same sort of thing is true for equality.

Any relation satisfying the three properties above is called an equivalence relation. Equality and parallelness are both equivalence relations, and polygon similarity is also an equivalence relation. It is worth noting, though, that equality is stronger than a mere equivalence relation. The distinguishing feature of equality which separates it from all other equivalence relation is that of substitutability. If two quantities are equal, all the same sentences must be true about those numbers. If 11 is prime then, since 11 = 9+2, we must also be justified in saying 9+2 is prime. Substitutability does not hold for parallel lines. Line m may be parallel to n but a distance of 10 units away from it. If substitutability held we would be able to infer that line n is also 10 units away from n. Since that’s ridiculous, we see that parallelness does not permit of substitution.

But the use of a symbol for equality was not practiced even by startlingly modern European mathematicians such as Fermat. In some way I find this inspiring—Math really just is what we could express in many more words in regular English. Math is not categorically different reasoning from other kinds of reasoning, the symbols merely compact a very large amount of information and abstraction.

The End

I read a lot of history, and lately that’s been a lot of prehistory. You know, stuff that existed before writing, since academics officially regard history as analysis of text. What I’m now reading isn’t exactly that, it’s The Knowledge. It’s a literal manual for rebuilding the world after a futuristic collapse. In a way it’s the opposite direction to what I’ve been studying: Off in the future. But just like every post-apocalyptic story, part of the fun and the point is to see regress to our primitive state. The past and the future intersect.

At least on the face of it the book is a literal manual for how to rebuild civilization after some great collapse, be it meteor strike, nuclear war, or epidemic. But I don’t think the author believes in an impending collapse and isn’t a so-called “Prepper” who prepares for the end times. At least he seems to think it’s not very likely in the very near term. How likely should we think a global catastrophe is?

Here’s a very not-data-based answer that is correspondingly not very precise. It depends on the time frame and the size of the class of events we call “global catastrophe”. If we say 200 years is the time frame and catastrophe can include most of the world (but not, say, Africa and Australia) being in a nuclear shoot-out, or the avian flu depopulating the world by 10%, then … I’d say the odds of such an event are “good”. Especially given that global warming seems inevitable and disastrous, I think we can lump that in with a slow-burning global catastrophe, making some degree of catastrophe pretty near certain. But if we call the catastrophe, by definition, something that happens suddenly one day in the next few decades, and leaves no corner of the Earth unaffected (and definitely destroys New York because it’s not yet a global disaster if you don’t get a half-sunked Statue of LIberty) … I’d put the odds near zero.

Back to the point, although the book on its face is about The Fall and rebuilding, I think that’s largely just a fun and interesting conduit to re-conceive of modern technology. It gives an excuse to remember that the Serbian army besieged the city of Gorazde. The people inside built little turbines and placed them in a river, fixed to a bridge, to generate electricity. It reminds us that, if the survivors of The Fall did not take care to collect the seeds of domesticated crops, they would quickly lose in the competition with weeds and die out—and humanity would have to spend an extra several thousand years re-domesticating new or old species. It is a reminder of how much technology was necessary to build the modern world. So much of it is so deep in the background that we need it threatened or removed before we can see it again.