Symbolic Mathematics and tools for symbolic math and algorithms – teach computers who teach students or who help others
Flammable Maths: An Insane Approach! The Fresnel integrals: sin(x^2) and cos(x^2) without complex analysis at https://www.youtube.com/watch?v=iCwiDPL3EMI
Writing math by hand is a bad thing to do to new generations. Get a decent symbolic math editor/solver and at least make the symbols standard. Do you trust programmers who make canned math manipulation tools, or some old guy named Leibniz or someone else?
You cannot let yourself get excited, and start rushing. When you think you are close to a good results, that is the time to be more careful, not less.
These are routine methods and issues in time series analysis. The integrals themselves, for different signals can be valid or not, depending on whether the original assumptions when the methods and steps were invented, match the conditions in the real world. Particularly, your approach is more like a one or few independent signal situation, where assumptions about normally distributed errors from a simple model make sense. But throw in multiple independent signal sources and it will not add up properly.
Stop wasting your time and the time of anyone you might want to help or teach. Put what you learn into the computer, so your equations can be “compiled”, verified, traced step by step, audited, reproduced, compared, shared, and used in real calculations.
Bourbaki had a good idea, but they and you both fail to see that humans, with their finite memory, can never completely represent some solutions where the number or steps and the number of pieces is beyond human capability. And, groups of humans get stupider as there are more humans trying to communication, not smarter. For relatively small problems, there is a sweet spot where a group of humans can get more done then one alone. But that number is small. On the Internet most of the problems, issues and opportunities require tens of thousands, tens of millions or billions of humans to cooperate – losslessly, open, auditable – in order to achieve some kind of global optimization for the human species.
The language AIs are not good mathematicians. The space of things possible with their representation of knowledge is much too limited. In the same way, hand mathematics on a blackboard that keeps disappearing will only work on tiny problems. You might think you put in a lot of work and were doing wonderful things when you could match up some things you have memorized and combine things in new ways. But relatively simple algorithms can whip through those same explorations, and never make a mistake, in a few seconds of computer time.
When I started college I mentioned one day to my room mate that I had made a perfect score in my Chemistry SAT. From where I came from that was a good thing. But he said he had not only made perfect score on the main test, but on 4 of the specialized tests as well. In my second year, I shared a group with 5 other students (individual rooms and shared kitchen and living room). One of my mates had memorized every integral and could combine things he had memorize quickly and efficiently. But I knew the books he had read, and had memorized much of that. I vowed then that was a losing battle. At most I could remember a million such thing, exactly, and their derivations, and the history and context of their development. But, eventually, with life getting more complex, and most large problems facing the human species requiring the usual tens of thousands or tens of millions of humans and their supporting computer software and data – to “solve”.
I am not saying to give up. Mostly, video is linear. It is one dimensional, and will not efficiently emulate multidimensional problems in efficient notions and visualizations. It is possible to put complex problems into videos, but they only give clues to humans (and a very limited set of AIs). The proper tool for most problems now is the computer, not a blackboard. I have tried to encourage groups who say they want to make tools for symbolic mathematics to help the human species. They all start out saying “open” and then as soon as they get it working, they lock it down and charge what market will bear. They solve a problem or two and that just becomes marketing case studies. They help a few organizations as an “expert” or “consultant” and that just requires more advertising and marketing than real work. It is hard to judge your age. I would say I am about 50 years older than you. If you look at your own skills and potential and look ahead at your own future, knowing what you know and learning as fast as you can, think what you will be in 10 years or 5 decades. But you cannot take it with you, and all that will evaporate when you die one day.
Putting things on paper is a losing battle. If you fill hundreds of paper books (or PDF which is effectively paper), you have a horrible way to try to share what you have memorized. If you write the perfect book on some topic, and a million people read it, they all have to take that information into their heads, match it to anything in their memory that might fit, and then recreate all that you have written down on paper, by hand. They have to re-invent what you wrote, because the form = paper does not itself do anything. On the Internet one of the saddest things is Wikipedia and all those ARXIV preprint servers. Both have LOTS of equations, data, graphs, charts, tables visualizations, units constants dimensions properties — all only in form where humans are required to use it at all, and where only humans working on paper can verify, combine, compare and apply. That can change a little, but the real answer, at least a small step for the human species and its survival – is to put symbolic equations and relations into shareable and immediately computer usable form that humans can query and guide and interact with. Teach the computers who teach generations of students, not humans who will eventually forget and who always die eventually,
Richard Collins, The Internet Foundation