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.