Ruined your life? Well, almost. But now that you’re intrigued with my wild claim, let me explain.
The science of mathematics has a very long heritage. The language of mathematics, as we recognize it today, is a bit younger – widely credited to François Viète, who introduced the first systematic algebraic notation in the last half of the 16th Century.
Along the way, over the course of many centuries, the power of mathematics increased steadily, through the contributions of many great men.
Between 800 and 1500 Islamic mathematicians from Persia, North Africa, and the Middle East made many contributions, including the words “algorithm” and “algebra”, as well as the first known proof by mathematical induction. Beginning around 1400 many important advances began in Europe. Today we all recognize the names and the contributions of Fibonacci, Bradwardine, Descartes, Galileo, Brahe, Kepler, Newton, Leibnitz, Fermat, Pascal, Euler, Riemann, Cantor, Hilbert and many others.
Records of prehistoric counting and recognition of geometric patterns go back as far as 70,000 years. The first known mathematical text was written in Babylonia by Plimpton in 1900 BCE. Indian mathematicians calculated the value of , explored irrational numbers, prime numbers and cube roots between 800 and 500 BCE. Between 500 and 300 BCE Greek mathematicians including Pythagoras, Euclid, Aristotle and Archimedes made major contributions. Chinese mathematicians made important advances beginning about 500 BCE, including the first known mathematical formula for what we now know as Gaussian elimination.
The power and the ease of use of mathematics were, and still are, closely tied. More sophisticated tools enabled mathematicians to tackle more challenging problems, and more challenging problems led to the development of more sophisticated tools. One of our mottos at Maplesoft is, “People can do great things if you give them great tools.” I think that’s been proven over a few millennia.
Then something dramatic happened: The modern computer was born.
Credit for the invention of the computer is hotly debated, and I’m not going to join that debate here. Some things are clear, though. Charles Babbage laid the foundation for the modern computer in the mid-19th Century. The Enigma machine, widely known as the German cipher machine of World War II, was actually introduced commercially in the early 1920’s. The British began Operation ULTRA in 1939 which resulted in 1943 in the release of Colossus, one of the earliest digital electronic computers. In order to accomplish real work using these new computers, Alan Turing applied the concept of algorithms to digital computers and programming languages were born.
So what’s the problem?
It’s quite simple. Computers are rather stupid things. They can only do a few things (add, shift, compare, branch), but they can do those things incredibly fast. Those early computers couldn’t do the sophisticated things that mathematicians and engineers needed, and up to that point did by hand (i.e. calculus), so mathematicians and engineers had to become programmers to instruct their computers how to calculate. The language of mathematics was replaced by the language of the computer. Power went way up (thanks to speed), but ease of use went way, way down.
It wouldn’t be for another two generations before advanced computer languages that could solve calculus problems would become available, so in the mean time the field of “numerical methods” was born – basically, more and more clever ways to solve analytic problems numerically. Finding the area under a curve (integrating) is an obvious example. Numerical analysis does not seek exact answers, because exact answers are impossible to obtain in practice. Instead, numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors. The comparative advantages and disadvantages of numerical analysis versus symbolic (analytical) analysis is not the purpose of this blog, though it’s comforting to know that Maple offers world-class capabilities in both areas, and in fact each area is enhanced by the presence of the other.
And finally, the language of mathematics is back – it’s alive and well, married to the modern computer. No longer are engineers and mathematicians encumbered by difficult to learn and use programming languages. No longer does the language of the computer obfuscate the language of mathematics. (Obfuscation is the concealment of meaning, making something confusing and harder to interpret.)
And thankfully, with modern systems such as Maple the ease of use of doing mathematics has once more caught up with the power of mathematics.
Coming back, then, to my opening line… Although the computer has “ruined the life” of two generations of engineers and mathematicians, thankfully it doesn’t have to ruin yours.