Strictly speaking, "pushing the envelope" is testing a system to find out where its limits actually are.
Test pilots routinely push the envelope
(in fact as far as I know the term is aviation slang) so that a set of normal limits can be defined for a new aircraft, but even a test pilot doesn't spend much time outside the envelope, because outside the envelope is where the plane doesn't fly, and if you don't get back inside the envelope real quick, you're gonna auger in.

The term “pushing the envelope” has been used in the field of mathematics for quite some time and according to the website I went to the phrase became popular in general/civilian use after it showed up in the Tom Wolfe novel The Right Stuff in 1979. Here’s the passage that is referenced.

"One of the phrases that kept running through the conversation was ‘pushing the outside of the envelope’... That seemed to be the great challenge and satisfaction of flight test."

I’ll stray off course here and interject a personal note. If you have a couple of days or maybe even a week or so to kill I’d highly recommend the book. If you have four or five hours to spare, I also highly recommend the movie.

Now, back to the topic at hand.

What exactly is the “envelope”?

Let me state right off that I’m no mathematician or aeronautical engineer. When it comes to that sorta stuff my claim to fame is being able to balance my checking account on a monthly basis or get to the airport on time. However, thanks to the miracle of the internet, here’s what I found out.

When it comes to math the envelope is defined as “the enclosing boundary of a set or family of curves that is touched by every curve in the system.” For a more scientific explanation of the term please go to the write ups posted here.

When it comes to aeronautics the envelope is defined as “the outer boundary of all the curves that describe the performance of the aircraft.” That takes into account the basic things such as speed, thrust, altitude and weather conditions as well as a host of other variables that test pilots must take into account on deciding whether on aircraft is safe or not and how much the aircraft will endure before it fails and either breaks into pieces or comes plummeting down to earth.

Just another personal insight. Both the men and women who test aircraft for a living have my utmost respect and admiration. From where I stand you can do all the simulation on computer models as much as you want but until some human being actually gets their hands on the aircraft you never know what might happen. They truly do have “balls the size of church bells.”

In closing, the math definition seems to set the boundaries that are considered “safe” to operate in while the aeronautical term strives to “push” those boundaries to their utmost.

I hope that adds some tidbits of information regarding the term itself. If it doesn’t maybe I’ve pushed my own personal envelope too far when it comes to math and science related issues.Source(s)

http://www.worldwidewords.org/qa/qa-pus1.htm
http://www.phrases.org.uk/meanings/push-the-envelope.html

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