Equations are profoundly simple in concept - two calculations (one of them may reduce to something trivial like a statement) that may seem quite unrelated that have the same meaning. The "=" sign makes the connection. Oddly enough, even though mathematics is very old, the symbol has been around for less than 500 years.
To avoide the deiouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or gemowe lines of one lengthe: =, bicause noe .2. thynges, can be moare equalle.
Robert Recorde coined it as a shorthand for "is equal to" when he wrote The Whetstone of Witte in 1557. The OED says gemowe means something like twin. It proved useful and stuck.
In math the meaning of the equal sign is usually rigorous. In science and many other disciplines it is often more nuanced whether we appreciate it or not.
One of the profoundly curious things about Nature is that so much of her can be approximately described using the language of mathematics. A theory is often stated as a formula and extensions are sometimes made using rigorous mathematical extensions that turn out to describe something previously unseen. At the very least these formulae give a powerful mechanism for playing with Nature and trying to understand a bit more deeply.
It is dangerous to become too enamored with these relationships as they can fall apart when you are in new territory or if you are in a bit past your own understanding. In fact this can be precisely where life gets interesting. You think your measurement of something will give a certain result, but a careful experiment gives a slightly different result. No one has ever made a measurement quite this carefully. You suspect something is wrong with your experimental apparatus or the steps you used to process the calculation and painstakingly eliminate any potential mistake. At some point you become convinced that your result is correct and submit your result for peer review. Ultimately the piece is published and the world takes notice and attempts to replicate your result. When they do it is time to recognize we need a richer description. Theorists may have constructions that immediately work or there may be nothing for years. You have learned something about Nature at a more fundamental level than anyone else. It can be an electric moment - the profound discoveries in science are not "aha!" moments, but rather usually begin with "that's odd..."
Imagine the discoveries you would make if your measurement tool had been a black and white camera and you suddenly were able to see the world in color.
This is why improved tools that allow you to measure Nature in previously unexplored areas and at better accuracy are important. Our senses, which turn out to be much more powerful than those of most animals, turn out to have dramatic limits. Hearing only gives access to a tiny portion of the world of acoustic sound, our vision only explores a tiny section of the electromagnetic spectrum
But the old equation still works for any measurement less accurate than yours and it still may be very useful to people. You may only be thinking at the leading edge preferring the delightful ignorance you have just uncovered, but most of the rest of the world may be built on the earlier theory.
Many of us build models of one type or another. It is centrally important to construct an appropriate model. It may be simple or complex, but it should be appropriate. There are times when what you are interested in may be too difficult to measure, but a simple model may allow you to easily set an upper and/or lower limit that gives you insight. And you have to avoid bias and not wed yourself to a model or the conclusion that an inappropriate model or calculation might give.1 Unless you are doing math it is critical to remember that the sweet little notation that Mr Recorde gave us may only give an approximation and you really need to calculate a confidence level to make your result strictly accurate.
Our models usually don't need to be perfect, but they are an important mechanism to find answers and sometimes a bit of insight. But remember that a they are never perfect representations and learn where they break down. So much of what we do is really grey rather than black and white.
A Danish friend who happens to be a great Lego fan pointed out this fascinating Marvin Minsky interview where he speculates that mechanical engineers who grew up with legos may be more limited than those who grew up with Tinker Toys.
But Minsky's point is valid even if he said it in jest. Some models provide a richer insight into how the world works than others. Playing with a diverse set of models can be enormously important.
1 Biases are very common. Recently there was a lot of speculation on the cause for a decrease in miles driven in the US. Many suggested it was caused by a change in how people between the ages of 18 and 30 relate to cars. It turns out a more careful analysis shows this is only a small component, but so many people wanted to believe that youth are falling out of love with the automobile that they seized upon it. In science there have been dramatic examples like the cold fusion debacle. People wanted to believe it - so they did for awhile. Some still cling to the belief more than twenty years later.
The photo is an Instagram photo by silverstar22b