About the origins of the equals sign(blog.plover.com)
blog.plover.com
About the origins of the equals sign
http://blog.plover.com/math/recorde.html
23 comments
I will second his recommendation to read original works.
Be aware, though, that this recommendation does have a flip side. I haven't tried to read Maxwell's original Treatise on Electricity and Magnetism, but I've heard it said that it's really, really difficult to get through. The fact that Maxwell's equations, the marvel of their era, are now a subject for second-year undergraduates is due to Oliver Heaviside's later work. From Wikipedia:
In 1884 [Heaviside] recast Maxwell's mathematical analysis from its original cumbersome form (they had already been recast as quaternions) to its modern vector terminology, thereby reducing the original twenty equations in twenty unknowns down to the four differential equations in two unknowns we now know as Maxwell's equations.
Just imagine Maxwell's equations as a system of 20 simultaneous equations. Now marvel at the fact that Maxwell was able to conclude anything at all while operating under such a heavy handicap.
Similarly, one of the reasons that intelligent nonphysicists today find quantum mechanics terribly confusing is that the original formulators of quantum theory -- guys like Bohr, Heisenberg, Schroedinger, and Einstein -- were so influential and so well-spoken that their writings and ideas have refused to go quietly into the dustbin of history, where many of them arguably belong. The mental models used by these folks -- the cat in the box, the atom as a miniature solar system, the notion of "god playing dice" -- are largely obsolete. They reflect the viewpoint of classically-trained physicists desperately scrambling for a grip on quantum theory. Now that quantum theory is much better understood, there are probably better ways to approach the subject than the analogies and metaphors used by its original pioneers.
Another example: While Darwin's Origin of Species is a great work, guys like Dennett and Dawkins and Gould have noted that one of Darwin's awesome literary talents is his ability to distract you from the fact that he didn't know about genetics. Without the notion of a gene it's rather hard to explain why the individual members of a species exhibit variation in the first place. Darwin, whose theory was already very difficult to sell, understood this problem but was careful not to call too much attention to it; it remained as a real hole in his theory until later work cemented it shut.
Be aware, though, that this recommendation does have a flip side. I haven't tried to read Maxwell's original Treatise on Electricity and Magnetism, but I've heard it said that it's really, really difficult to get through. The fact that Maxwell's equations, the marvel of their era, are now a subject for second-year undergraduates is due to Oliver Heaviside's later work. From Wikipedia:
In 1884 [Heaviside] recast Maxwell's mathematical analysis from its original cumbersome form (they had already been recast as quaternions) to its modern vector terminology, thereby reducing the original twenty equations in twenty unknowns down to the four differential equations in two unknowns we now know as Maxwell's equations.
Just imagine Maxwell's equations as a system of 20 simultaneous equations. Now marvel at the fact that Maxwell was able to conclude anything at all while operating under such a heavy handicap.
Similarly, one of the reasons that intelligent nonphysicists today find quantum mechanics terribly confusing is that the original formulators of quantum theory -- guys like Bohr, Heisenberg, Schroedinger, and Einstein -- were so influential and so well-spoken that their writings and ideas have refused to go quietly into the dustbin of history, where many of them arguably belong. The mental models used by these folks -- the cat in the box, the atom as a miniature solar system, the notion of "god playing dice" -- are largely obsolete. They reflect the viewpoint of classically-trained physicists desperately scrambling for a grip on quantum theory. Now that quantum theory is much better understood, there are probably better ways to approach the subject than the analogies and metaphors used by its original pioneers.
Another example: While Darwin's Origin of Species is a great work, guys like Dennett and Dawkins and Gould have noted that one of Darwin's awesome literary talents is his ability to distract you from the fact that he didn't know about genetics. Without the notion of a gene it's rather hard to explain why the individual members of a species exhibit variation in the first place. Darwin, whose theory was already very difficult to sell, understood this problem but was careful not to call too much attention to it; it remained as a real hole in his theory until later work cemented it shut.
I was happy with Dawkins book until being told, by a geneticist if I remember right, that he wasn't very serious, that Gould was serious reading but Dawkins was not completely so. I'd be glad to know if it's right or not.
Wow, you asked the WRONG geneticist. Gould is regarded as worthless and possibly dishonest by many/most professional evolutionary biologists (which is not quite the same thing as genetics).
John Maynard Smith, possibly the preeminent evolutionary theorist of our era:
"Gould occupies a rather curious position, particularly on his side of the Atlantic. Because of the excellence of his essays, he has come to be seen by non-biologists as the preeminent evolutionary theorist. In contrast, the evolutionary biologists with whom I have discussed his work tend to see him as a man whose ideas are so confused as to be hardly worth bothering with, but as one who should not be publicly criticized because he is at least on our side against the creationists. All this would not matter, were it not that he is giving non-biologists a largely false picture of the state of evolutionary theory."
http://www.overcomingbias.com/2007/11/beware-of-gould.html
John Maynard Smith, possibly the preeminent evolutionary theorist of our era:
"Gould occupies a rather curious position, particularly on his side of the Atlantic. Because of the excellence of his essays, he has come to be seen by non-biologists as the preeminent evolutionary theorist. In contrast, the evolutionary biologists with whom I have discussed his work tend to see him as a man whose ideas are so confused as to be hardly worth bothering with, but as one who should not be publicly criticized because he is at least on our side against the creationists. All this would not matter, were it not that he is giving non-biologists a largely false picture of the state of evolutionary theory."
http://www.overcomingbias.com/2007/11/beware-of-gould.html
Thanks. I was thinking about dropping stronger hints of this, actually -- I distinctly recall seeing various arguments of Gould's being torn to shreds by Dawkins and/or Dennett, and I have gradually developed the habit of taking my extensive collection of Gould's books (given to me as gifts), stacking them in a little pile on my shelf, and then actually reading something else -- but I am neither educated enough in evolutionary biology nor recent enough in my reading to pull a specific quote, and the last thing we need is more heat and less light from a total non-biologist like myself.
Also, I haven't read an essay that is quite so direct as this one that you link to. I'd point out that this link illustrates my point: When scientists get down to business and talk about each other's work, they do not mince words. It's kind of refreshing, actually.
Also, I haven't read an essay that is quite so direct as this one that you link to. I'd point out that this link illustrates my point: When scientists get down to business and talk about each other's work, they do not mince words. It's kind of refreshing, actually.
Thanks a lot, both of you for your replies. I hadn't imagined that Gould could be on that position. I might have asked the wrong geneticist indeed!
You're going to need to ask more specific questions. This is science. The closer to the rock face you get, the more criticism flies about.
The three guys I named have various disagreements on one thing or another, so it wouldn't surprise me to find that each has his partisans. Also note that science writing for the public is a completely different art than science, itself. Gould may well be taken more seriously as a practicing scientist, but that shouldn't necessarily matter to me as a reader of his popuLar works. Isaac Asimov was a great science writer, but his scientific career was brief and undistinguished.
The three guys I named have various disagreements on one thing or another, so it wouldn't surprise me to find that each has his partisans. Also note that science writing for the public is a completely different art than science, itself. Gould may well be taken more seriously as a practicing scientist, but that shouldn't necessarily matter to me as a reader of his popuLar works. Isaac Asimov was a great science writer, but his scientific career was brief and undistinguished.
I'm sorry I didn't understand what "closer to the rock face" means.
As far as I know, Dawkins' books give his version of evolution, that is somewhat non-standard and not yet proven. Gould used to write about evolution more as it was widely accepted by scientists on his time. I know they both have disagreements, but as far as I know Dawkins is faster in writing about not very accepted subjects as widely accepted science.
As far as I know, Dawkins' books give his version of evolution, that is somewhat non-standard and not yet proven. Gould used to write about evolution more as it was widely accepted by scientists on his time. I know they both have disagreements, but as far as I know Dawkins is faster in writing about not very accepted subjects as widely accepted science.
Hm, perhaps I am misusing that metaphor. Let me try a different metaphor: As you move from well-established, abstract, general principles to newly-proposed ideas that are still being tested against a bunch of specific data, you'll find scientists disagreeing with each other more and more. At the very edge of science, where new hypotheses get started, the put-downs can get pretty sharp. Just because Scientist A calls Scientist B a statistically-ignorant fool doesn't necessarily mean, in any objective sense, that Scientist B is a statistically-ignorant fool. They may just be embroiled in one of those academic debates.
As for your generalizations: I think they're way too sweeping. Gould and Dawkins have written hundreds of thousands of words. Some of those words are probably too daring, others not daring enough; some are words about cutting-edge subjects, some are about classic works in their fields. Some of their words have caught fire in the popular imagination to a much greater extend than their authors intended, i.e. Dawkins' "meme" concept was originally intended as a half-joking metaphor, and he himself doesn't take it too seriously.
As for your generalizations: I think they're way too sweeping. Gould and Dawkins have written hundreds of thousands of words. Some of those words are probably too daring, others not daring enough; some are words about cutting-edge subjects, some are about classic works in their fields. Some of their words have caught fire in the popular imagination to a much greater extend than their authors intended, i.e. Dawkins' "meme" concept was originally intended as a half-joking metaphor, and he himself doesn't take it too seriously.
Indeed, “meme” is a stupid name for what was completely understandable when simply called an “idea”. The analogy of ideas to genetics is not particularly insightful, and at this point quite worn through.
People congratulate themselves when they have an idea. A group doesn't get an idea. You get a meme from someone else, and it seems to survive on its own. Not all ideas are memes, and not all memes are ideas.
But finding a meme these days is like mentioning the long tail. It seems a little antiquated and cutesy.
But finding a meme these days is like mentioning the long tail. It seems a little antiquated and cutesy.
Not all memes are ideas? How is that so?
Getting a song stuck in your head (commonly called "earworm", but I think that's a terrible name for it. I think "memelody" makes more sense) when you hear the tune or someone says something that reminds you of it I think qualifies as a meme but not an idea, especially if there is some shared culture reference that makes multiple people think of the same tune from similar stimuli.
Thank you for your interesting replies.
After my (university) freshman physics class, we were all presented with a copy of Einstein's original papers on Brownian Motion. It's incredible how much he says things to the effect of, "well, I think this effect is the same thing that people are talking about, but I haven't actually seen it first hand, so I could be wrong".
There was a nice documentary about the birth of the calculus, that focused on Newton and Leibniz's notebooks. It was cool to see them making up notation on the fly.
(A quick check of YouTube finds it here: http://www.youtube.com/watch?v=dwZNg237x1M)
(A quick check of YouTube finds it here: http://www.youtube.com/watch?v=dwZNg237x1M)
So that's where the PHP guys got "===" from. Now it's obvious.
Now I can feel much more dignified every time I have to use identity comparison!
Robert Recorde wanted to make his math books clear and accessible. He was the first to write such books in English, rather than in the Latin or Greek of the educated elite.
Also, little more about this subject on the PBS site:
http://www.pbs.org/wgbh/nova/einstein/ance-equals.html
Also, little more about this subject on the PBS site:
http://www.pbs.org/wgbh/nova/einstein/ance-equals.html
"It doesn't seem that Recorde gained from his innovation, for it remained in bitter competition with the equally plausible "//" and even with the bizarre "[;" symbol, which the powerful German printing houses were trying to promote. But by Shakespeare's time a generation later Recorde's victory was finally certain."
I think that sounds exactly like people defending bizarre syntax rules in modern languages.
I think that sounds exactly like people defending bizarre syntax rules in modern languages.
It's very interesting to see the use of the tilde for nasalized vowels in an English text, the same as it is currently used in Portuguese today.
I will second his recommendation to read original works. They aren't always the best, but in many instances they can teach you a lot more than just the topic at hand. If you have your grammar school science textbooks at one end of the spectrum---nth hand knowledge echoed by an author with no understanding of the subject---then original works are at the opposite end---the idea in its first appearance, recorded by someone with the power necessary to understand and discover it for the first time. The advantage isn't only in that the author takes great pains to present his new discovery, it's in the countless unrepeatable ways he presents it dictated by the climate of his time. Just like there will never be another 17th century Italy to produce those particular violins, there will never be another Victorian Era to produce On the Origin of Species, in the way it was originally written. You can rewrite and distill the idea, but you can't recapture all the implications that originally flowed from it. Reinterpretations strip an idea of its historical context, and frequently these are as important as the idea itself.