For what it's worth, I spent time arguing with you because the ideas you were proposing were interesting, but the problem with math is you have to make rigorous definitions and such. The ideas as stated cannot be defended, but as I said elsewhere, you can make something like infinitesimals work with some more effort, but they aren't the real numbers anymore.
"Define it that way, but show me the theorems" is an important organizational philosophy of math. It also helps remove ego from everything. It can be painful creating math without realizing this, and communicating the philosophy was the main point I was trying to make. (I also had some hopes you would show interesting consequences!)
It's not like what you were saying was obviously wrong. It wasn't until the mid-1800's that people really sorted out the real numbers. I myself spent some time thinking I "solved" the 1/0 problem and thought about "numbers" like 0.000..infinitelymany...01, but the nice thing about math is that performing experiments isn't too expensive.
(I didn't downvote you. Sorry for the full-contact lesson in math philosophy, and don't get the idea this is a "sore spot in mathematics," rather the non-existence of infinitesimals in the real numbers is easily defended.)
Edit: Beyond the algebraic way of making infinitesimals work, there is also the real analysis version: limits to 0. The idea of infinitesimal there is that no matter what positive real number you give me, I can give you a smaller one. This concept of infinitesimal isn't a number per se. Similarly, one of the many ways infinity shows up is that no matter what number you give me, I can give you a larger one.
The metaphor of infinity also shows up in: cardinality of sets, as the added point in a one-point compactification, the extended real line, the Riemann sphere, arbitrarily large numbers (limits), and that's all I can think of at the top of my head.
Proofs and explanations are not always the same. Proofs depend on logic and rigor, while explanations depend on the audience. Mathematics and pedagogy are not usually considered to be closely related areas of study. And yet universities make mathematicians teach mathematics.
Are they the best teachers? No. No, they are not. But they are the only ones that understand the subject matter well enough to do it. And that leads to the vicious cycle where you have to think like a mathematician in order to learn math from one, because they have difficulty explaining anything to any other type of person. A student that needs an explanation gets a proof, which is technically correct, but still fails to elucidate.
I think some kinds of math are fun and interesting, but proving the math is [currently] less than 1% of my job, and I have never had to worry about precision that would underflow a 64-bit floating point double.
Take another look at the whole thread tree, originating at https://news.ycombinator.com/item?id=15236430 , and look at the posts by "zelah". Realize that all the responses made them realize that they got something wrong somewhere, but it looks like they are still as confused as ever, and probably net negative karma from being wrong on the Internet and not knowing why.
My original goal was to help zelah understand, and I failed. My secondary goal was to play the game alluded to by lisper, who essentially said I cheated. There is nothing left for me to accomplish here. I wasn't trying to be pissy and storm out the door in a cloud of drama, but rereading, it seems like that's probably the simplest interpretation of my last post. So... sorry for that. I'm still not writing you any theorems.
"Define it that way, but show me the theorems" is an important organizational philosophy of math. It also helps remove ego from everything. It can be painful creating math without realizing this, and communicating the philosophy was the main point I was trying to make. (I also had some hopes you would show interesting consequences!)
It's not like what you were saying was obviously wrong. It wasn't until the mid-1800's that people really sorted out the real numbers. I myself spent some time thinking I "solved" the 1/0 problem and thought about "numbers" like 0.000..infinitelymany...01, but the nice thing about math is that performing experiments isn't too expensive.
(I didn't downvote you. Sorry for the full-contact lesson in math philosophy, and don't get the idea this is a "sore spot in mathematics," rather the non-existence of infinitesimals in the real numbers is easily defended.)
Edit: Beyond the algebraic way of making infinitesimals work, there is also the real analysis version: limits to 0. The idea of infinitesimal there is that no matter what positive real number you give me, I can give you a smaller one. This concept of infinitesimal isn't a number per se. Similarly, one of the many ways infinity shows up is that no matter what number you give me, I can give you a larger one.
The metaphor of infinity also shows up in: cardinality of sets, as the added point in a one-point compactification, the extended real line, the Riemann sphere, arbitrarily large numbers (limits), and that's all I can think of at the top of my head.