Another small, not insignificant point about a bill (official currency printed by a government), and a $10 IOU paper napkin. What would a bank teller say, if you asked for a $10 bill and said you can give them an IOU for it? After they stopped laughing?
We do that all the time. Ever borrowed hard cash on promise of repayment ? An IOU? (a piece of paper with your signature (a contract). The only difference is that you are using the pre-printed bank's paper stationery, instead of a restaurant paper napkin. Now, if I waited 30 minutes in line and then asking the teller change for a nickel, they will call other tellers over to witness the transaction, laughing......Please Register or Log in to view the hidden image! ....Please Register or Log in to view the hidden image!
Yes, from a friend, not a bank. The promise was verbal. I've pawned items too, promising to repay the loan and "unpawn" the items. Banks are another story, usually you ask for a line of credit, an overdraft "facility", etc. You can arrange to repay a bank with "future earnings", which is where being gainfully employed comes in handy. Banks don't usually accept IOUs written on paper napkins in my country.
This idea, possibly somewhat ill-conceived or one we don't bother with, the "place" that numbers exist, is quite possibly just us pulling them out of thin air, literally, when we think about physical patterns. We have "good" number systems and "theories of numbers" because we're good at seeing patterns. Music is patterns we can hear, a kind of abstract tiling of an even more abstract surface. Both though, are real physical objects. Mathematics can't object to the existence of real physical values, but doesn't need them. So for whatever reason, we struggle with this apparent separation, we can mathematically 'peel' a flat object from a plane and turn it into a cylinder, even a graph of some kind, without stretching anything, this is just a proof of how many dimensions you need to curve a two-dimensional surface in at least one of them. The unsurprising answer is 2 + 1. Physically, a cylindrical roller with an image on it that prints the image onto a flat sheet of paper, is a model of the mathematical operation which can have as large a bag of theories attached to it as you like, as long as it stays mathematical and doesn't contain errors. Like, you can buy a glass Klein bottle; this is a physical model of the mathematical theorem that proves the bottle embeds in more than three dimensions without self-intersection. The glass bottle doesn't say that though, it says this is what happens in three dimensions (duh) when you connect the ends of a cylinder with one end inverted.
Correct! Please Register or Log in to view the hidden image! A cylinder rolled across a two-dimensional surface produces a rectangle. How are we to know which is which??
The cylinder is the one doing the rolling and the rectangule is the result of the rolling Hint, if you cut the side of the cylinder, from one open end to the other you can make it a rectangular Or take the rectangle and join two sides you make a cylinder Please Register or Log in to view the hidden image! PS I suspect I have who you are addressing your post to on Iggy so my post might be well out of line Oh well Please Register or Log in to view the hidden image!
Please Register or Log in to view the hidden image! That's okay Michael345. I was replying to igged person. So the answer is, it is BOTH a cylinder AND a rectangle. Please Register or Log in to view the hidden image!
The Euclidean plane is isomorphic to a cylinder. It's also topologically a sphere with one point removed, which is then stretched open.
A rectangle is a two-dimensional plane. If drawn by a measuring wheel, can we say that Pi is a hidden aspect embedded in the rectangle?