The evolution of math over the last 50 years?
Understanding of math never moves backwards, only forwards. In that respect, it is unique amongst the sciences. Once a math theorem is proven, it is never disproven.
Not so with science. Newton's laws of physics stood for 250 years until Einstein came along and proved them incorrect. So scientific theory changed and so it will go.
Math is different. The Pythagorean theorem will be true a million years from now. It will never be disproven.
We will move forward, sometimes slowly, sometimes more rapidly, but we will never move backwards in math.
The most significant thing that changes with time is the way we teach or learn math. Technology has made math much easier to learn. When I went to college, there were no calculators, only slide rules. It wasn't easy to see graphical depictions of functions. Without the ability to see functions graphically, it was much harder to learn calculus, 3-D vectors, etc. Now 3-D vectors are taught to seniors in high school as opposed to graduate school in college back in 1970. That's largely due to the technology, not any fundamental changes in math. Also, the internet has brought us online teaching and tutoring, with online classes from MIT and Harvard professors, not to mention Khan Academy's 40 thousand educational videos.
While the upsides of technology are huge, there are some downsides to the tech revolution. Virtually everyone that grew up in the post-calculator generation, (post 1970), got a different elementary math education than those pre-1970. The emphasis switched from arithmetic skills to more complex math concepts, but the result today, is that there is a severe shortage of arithmetic skills in today's high school math students. This almost certainly is due to the ubiquitousness of the calculator, and it allows many students to get to high school without arithmetic skills.
As long as we have calculators available, this might seem acceptable, but, I believe it is the #1 fundamental cause of poor performance in algebra, geometry, trig, and pre-calculus.
The evolution of math over the last 50 years?
Understanding of math never moves backwards, only forwards. In that respect, it is unique amongst the sciences. Once a math theorem is proven, it is never disproven.
Not so with science. Newton's laws of physics stood for 250 years until Einstein came along and proved them incorrect. So scientific theory changed and so it will go.
Math is different. The Pythagorean theorem will be true a million years from now. It will never be disproven.
We will move forward, sometimes slowly, sometimes more rapidly, but we will never move backwards in math.
The most significant thing that changes with time is the way we teach or learn math. Technology has made math much easier to learn. When I went to college, there were no calculators, only slide rules. It wasn't easy to see graphical depictions of functions. Without the ability to see functions graphically, it was much harder to learn calculus, 3-D vectors, etc. Now 3-D vectors are taught to seniors in high school as opposed to graduate school in college back in 1970. That's largely due to the technology, not any fundamental changes in math. Also, the internet has brought us online teaching and tutoring, with online classes from MIT and Harvard professors, not to mention Khan Academy's 40 thousand educational videos.
While the upsides of technology are huge, there are some downsides to the tech revolution. Virtually everyone that grew up in the post-calculator generation, (post 1970), got a different elementary math education than those pre-1970. The emphasis switched from arithmetic skills to more complex math concepts, but the result today, is that there is a severe shortage of arithmetic skills in today's high school math students. This almost certainly is due to the ubiquitousness of the calculator, and it allows many students to get to high school without arithmetic skills.
As long as we have calculators available, this might seem acceptable, but, I believe it is the #1 fundamental cause of poor performance in algebra, geometry, trig, and pre-calculus.
Is Math Discovered or Invented?
Or a little bit of both. That is the question. Their are arguements for both. Ultimately, the reality of our universe will provide the answer.
We continually find that math predicts future discoveries about the universe. The Higg's Boson was such. Higgs used mathematics to predict its existence 50 years before its discovery.
At the same time, math is not able to define or predict really complex systems, like long range weather forecasting, quantum physics, or even the stock market.
If you were to assume that life is just an advanced, elaborate video game, where everything is described mathematically with code, then it is a discovery. Everything would be math, and humankind would be constantly just discovering different pieces of it.
But could it be perhaps just a human invention, essentially only inside our individual brains.
Most of the evidence points towards math defining, and perhaps being everything to or about our universe.
To me, that makes it a discovery.
History of Calculus
Calculus is called by most math experts the most important discovery/development in the history of math or possibly even mankind.
Who discovered it? The story gets even more interesting.
Sir Isaac Newton, whom almost everyone agrees is the greatest mathematician of all time, actually discovered differential calculus around 1660. The problem was that it involved division by quantities that approached zero, and as every mathematician knows, you can't divide by zero. So, Newton doesn't publish his discovery because he doesn't quite trust it enough.
Ironically, twenty years later, in 1680, he does publish the 'Principia Mathematica', which to this day is considered to be the greatest treatise on mathematics ever published, (even without calculus).
About the same time Newton is publishing 'Principia Mathematica', Gottfried Leibniz has discovered integral calculus, and published. Now it wasn't realized at first that the two math disciplines, diff calc, and int calc, were intrinsically entertwined. One was actually the inverse of the other. 2 sides of the same coin.
Eventually, both Newton and Leibniz realized this and both are credited equally with the discovery of calculus.
