2. Let

Xbe a topological space. Then the empty set andXare sets that areboth open and closed!

Open and Closed Sets have got nothing to do with the day-to-day open and closed.

]]>Contradictory is perhaps not a good word to use in the title of this thread. Oxymoronic might be better, I believe. It is not contradiction that I wish to explore, but rather oxymoron: http://z8.invisionfree.com/DYK/index.php?showtopic=69

If I said that someone was an **unpleasant kind** of person, I would be uttering an oxymoron.

MathIsFun: Gimme a day or two to get a Vector Space thread together. I'm not sure I'm the right person to make a website page on it, but if I do start a thread you will all be free to make use of it for that purpose.

]]>Ricky has a few pages on this site (example: Introduction to Sets) and Ganesh has made a few pages, too.

It does take a bit of effort, though, with drafts, redrafts, graphics, etc. But worthwhile if it opens up the subject to people.

]]>Rather, we use a probabilistic thing such as a wave function to set off a trigger.

Well actually, it's the absolute square of the wave function psi, which is itself an eigenfunction. For, wave functions have, by definition, codomain [-1,1], which makes no sense probabilistically.

Ha! I am tempted to start a thread on linear vector spaces, where all this might be explained. But, judging from the responses to my previous "lectures" I am not confident it would be well received.

Anybody have any thoughts (it is a cool subject, and only involves simple arithmetic)?

]]>I think the point is that the "cat in a box" is NOT reasonable, because something the scale of a cat is not probabilistic.

The cat isn't what the probabilistic parts of quantum mechanics act on. Rather, we use a probabilistic thing such as a wave function to set off a trigger. When this trigger goes off, it does something (releases gas I believe is the most common) to kill the cat.

The trigger is small enough. However, it directly effects the macro world.

]]>Poor guy indeed, now Wikipedia uses it as the theme picture, see http://en.wikipedia.org/wiki/Quantum_mechanics

I think the point is that the "cat in a box" is NOT reasonable, because something the scale of a cat is not probabilistic.

]]>Then I could add to the list in my first post:

3. Letxbe a phenomenon (e.g. Schrödingers cat ) satisfying both conditionsAand conditionsB. Thenxisboth alive and dead!

OK Jane, nice post. Just for fun, note this. I am not a physicist, but was recently told that Schroedinger devised the "cat-in-a-box" thought experiment to illustrate what he regarded as the illogicality of quantum mechanics (or do I mean the Copenhagen interpretation? Not sure)

And now the cat in its box is in all the texts as a *perfect* illustration of how QM works. Poor old Schroedinger!

And there is nothing to apologize. I did not make it clear in the first place that it was not meant to be taken too seriously, so you were entitled to have your own views about it.

]]>I should point out at this stage that whole point of this thread is to take a light-hearted look at the way mathematical language is used in particular, at how certain mathematical terms differ in meaning from the way those terms are used in ordinary language. This thread is

notmeant to be a serious critique of mathetical language itself it is only a light-hearted comparison of mathematical and non-mathematical language.

In that case I apologize (I'm always the last to get a joke in a crowd).

You are right. In fact, when I first started in science, it really pissed me off at what I thought was the hijacking or "ordinary" words for technical purposes. I now see it as inevitable - as I said, ordinary usage is so imprecise as to be virtually useless.

Good, we agree.

]]>Ben, what you say is entirely correct. I agree with you. What I mean by contradictory is contradictory (or apparently so) only in the everyday-language sense, not in the mathematical sense. My point is that mathematics sometimes defines different concepts using terms which may seem contradictory in terms of everyday language. Thus, closed and open sets are defined in mathematics in not the same way as how one would ordinarily think of, say, a box which is why a set can be both open and closed in not the same way as a box can be. There is nothing wrong with the mathematical definitions themselves, and I am not saying that there is.

Lets take a hypothetical example. Suppose I am developing a brand-new branch of mathematics to study some newly discovered phenomena in the universe. I find that some phenemena satisfy some set of conditions, say *A*, and some phenomena satisfy some other set of conditions, say *B*. Now I wish to make some definitions. If a phenomenon satisfies conditions *A*, I define it as alive. If it satisfies conditions *B*, I call it dead. Now what if a phenomenon satisfies both sets of conditions *A* and *B*? Then I am perfectly entitled to call it both alive and dead!

Then I could add to the list in my first post:

3. Let *x* be a phenomenon (e.g. Schrödingers cat ) satisfying both conditions *A* and conditions *B*. Then *x* is **both alive and dead**!

This is just a hypothetical example. But as long as the terms alive/dead are interpreted in the hypothetical technical context, not as in their ordinary-language sense, there is no problem with the underlying semantics at all.

Contradictory is perhaps not a good word to use in the title of this thread. Oxymoronic might be better, I believe. It is not contradiction that I wish to explore, but rather oxymoron: http://z8.invisionfree.com/DYK/index.php?showtopic=69

]]>