Lot of posting in this thread but the explanation I left out. What is exactly being done and how does it differ from the way math has been done since Euclid?
We start with this quote:
Twenty-three centuries ago Euclid compiled the most influential book in all mathematics, The Elements. The elegance with which he proved the key discoveries of Greek geometry has entranced mathematicians ever since. He also established a simple paradigm: mathematics is what you can deduce through a series of logical steps. On the rock of that paradigm, mathematicians built the language of modern science. Now, 2000 years on, the first cracks in the paradigm are beginning to show. And they are fracturing the world of mathematics.
The cause of this disturbing turn of events is the computer. Invented by one generation of mathematicians and dismissed as a toy by the next, this handy algorithm processor has come back to haunt today's generation. By giving mathematicians the ability to do billions of complicated calculations on their own desks, the computer has spawned a whole new way of doing mathematics known as experimental maths. ...
Basically these are the uses:
Gaining insight and intuition.
Discovering new patterns and relationships.
Using graphical displays to suggest underlying mathematical principles.
Testing and especially falsifying conjectures.
Exploring a possible result to see if it is worth formal proof.
Suggesting approaches for formal proof.
Replacing lengthy hand derivations with computer-based derivations.
Confirming analytically derived results.
One thought I would like to add, what would geometry look like if Euclid and his pals had a CAS and geogebra instead of sand and stone tablets?
When Giotto di Bondone was asked to prove his worth as an artist. He drew a perfect circle, freehand. Supposing there were some race of beings that could estimate by eye angles and lengths to 100 digits of precison, even surpassing Giotto? Who could multiply 200 digit numbers in their head, add up thousands of numbers in seconds would they have invented a mathematics based on proof or one based on calculation?
Invented by one generation of mathematicians and dismissed as a toy by the next,
Would those creatures have agreed with this decision? At any rate, the fact that most mathematicians have turned their back on it is a unique opportunity for the amateurs to move right in.
In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.
The thing about Nature is it is NOT linear.
The things going on are NOT the linear superposition of n happenings each working out as it would do ALONE (in the absence of the other things going on).
All that stuff we were taught at school in wrong!
Linears things are far less than 1% of things actually observed and going on.
If the Greeks, Indians and Etruscans had owned a computer they would NOT have stuck us in the blind alley of "linear systems"
To talk of "non linear systems" as though they were the exception is like talking of Biology as "Non-elephant biology"!
My eyes were opened to this by a pupil. It was a 'low ability' class (not my choice of words) but what happened shows that at least one pupil disproved this label.
The topic was "The angle sum of a triangle is 180 degrees." Euclidean geometry was one of my strengths and I could 'prove' the result, but I decided to take an experimental approach.
I asked the class to draw any triangle, measure the three angles, and then add up the numbers.
A quick round_the_class survey showed results ranging from 178 to 182 degrees.
I asked what conclusion they might make from these results, expecting, indeed hoping, that someone would say "The angle sum of a triangle is 180 degrees."
This lad put his hand up and announced "The three angles in a triangle add up to a number that is approximately 180, sometimes a bit more, sometimes a bit less." Well, you cannot fault that, can you?
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
The human brain yearns for "simplicity", even over "truth"
Indeed the angles of no real (drawn on paper) triangle sum to 180 degrees
It was a mere dream (idealisation) someone had thousands of years ago!