When people say that they put money into the stock market, it's just a figure of speech. The stock market is not a big tank that people throw money into, it's just a meeting place where people who want to buy and sell stock can get together. It's a little more complicated than that because there are middle men called "brokers" involved, but let's ignore them for this example in the interest of keeping it simple.
Trader A sells a share of stock to Trader B for $100. The next day, Trader B sells the same share of stock to Trader C for $50. The $50 that Trader B now has did not come from the original $100 that he gave to Trader A, it came from Trader C. The original $100 is now in the possession of Trader A, and will be until he spends it. If Trader A spends the $100 on something else other than stock, then the $100 is no longer in the stock market, but it's still in the economy. If Trader A doesn't spend the $100, but deposits it in his bank account, the $100 is still in the possession of Trader A, but it's no longer in the stock market. Assuming that Trader C didn't get his $50 from the sale of stock, then it represents new money coming into the stock market, but it's only new to the stock market, it's not new to the general economy. Subtract that $50 from the $100 that Trader A took out of the market, and the market is down $50, but the general economy is not. Since the general economy has now gone international, that $50 could end up anywhere in the whole wide world. It may have disappeared from the stock market, it may have disappeared from the United States, but it has not ceased to exist, it's still out there somewhere.
Okay, let's talk about existence. You're right that money has no material existence, neither does language, but both money and language can be used to represent things that do have material existence. Space-time doesn't have any material existence either but, if space-time didn't exist, then all the things that do have material existence would have nothing to exist in. It would seem, then, that some non-material things, like space-time, are essential to the existence of material things, while other non-material things, like money and language, are mere conveniences that make material existence easier for people to navigate.
Since money is a human invention, human beings can make it exist or cease to exist but, if everybody was allowed do that, the economy would be a lot more chaotic than it already is. The U.S. Constitution reserves the power to "coin money and regulate it's value" to Congress, which has since delegated it to the Federal Reserve. Somewhere along the line the power to "coin money" was expanded to include printing paper money and creating money on paper. I said yesterday that the Fed can bring money into existence, but I wasn't sure if it could make it cease to exist. After thinking about it, I concluded that the Fed can certainly take money out of circulation which, for all practical purposes, means the same thing, so I will agree that money can cease to exist, but the stock market cannot cause that to happen. There is at least one other way that money can be taken out of circulation, and that is if it is lost or destroyed, say in a fire. Maybe that's one reason the Fed keeps cranking out new money, to replace money that has been lost in this manner. Another reason might be that the money supply needs to continually expand to keep up with the growing human population and the increase of economic activity as the developing countries come on line. I once read on Wiki that one reason we came off the gold standard is that there is not enough gold in the world to cover all the economic activity that is going on. Another way to look at it is that gold has become to valuable to use as money anymore. If the central banks were not creating money like they do, it would have to come from somewhere else, or we would be back to the barter system, which would be damn inconvenient.
You would know more about than I do, but I think the way scientists "know" things like the conservation of matter and/or energy is that somebody comes up with a theory that seems to make sense. In the absence of any empirical evidence to the contrary, the theory will stand until somebody comes up with another theory that makes even more sense. If they waited till they could prove everything beyond a shadow of a doubt, they wouldn't get anything else done. Who's going to pay them for that?
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