What would set theory say - that all things have a complement?
Lynx ~ 'Lynx

It says that all sets have a compliment; it says nothing about 'things'.

But, even if it did, that would have ontological implications only for Platonists.

Most sets include things, except for the empty set, {} which includes nothing.
Thus the set of all things, U, would have a complement, U' ?

I find that peanut butter complements a banana nicely.

Lynx:

Sets include elements, which are well-defined mathematically; the word 'thing' is no use at all here.

Maybe so, but this has nothing to do with 'things', still less with the material world.

Sets can include anything, including fuzzy stuff.
Yes, but I believe Venn diagrams are a better tool for understanding certain relationships than words or numbers alone. For example, if I drew two circles with one intersection and labeled the circle on the left Communism, the one on the right Capitalism and the intersection Socialism, would that be an accurate enough representation for the average apolitical person?

Set theory and geometry were about the only times I had fun in math class

Lynx:

Now, this is where we enter controversial territory; this is a hotly contested subject in the philosophy of mathematics:

http://en.wikipedia.org/wiki/Intensional_definition

http://en.wikipedia.org/wiki/Extensional_definition

I personally think that mathematics went astray in the late 19th century when certain influential theorists accepted an extensional definition of a set (which would allow sets to contain anything you like; this transformed mathematics into a poor copy of empirical science, and led, among other things, to Godel's ridiculous theorem).

But, even if you (or this approach) were right, it must be possible to say which items are elements of a particular set and which are not.

In that case, it will have to be well-defined, as I said.

The background to my way of viewing mathematics and set theory can be found here:

http://plato.stanford.edu/entries/wi...n-mathematics/

No it doesnt...try that line of argument on science and see where it gets you,....probably Lakatos. Your taking short cuts Rosa.

Gil:

I like the way you settle things in logic -- ask a scientist!

But wouldn't "not-banana" imply all things that are not bananas? Opposites should be, as you say, "clear opposites". Thus, hot-cold are opposites, but hot and not-hot cannot be opposites as that would imply banana is the opposite of hot, as well as me and you, every poster in this thread, the computer you type on, and all other things that are not the entity we call "hot". "not-" is too vague to imply an opposite.

Escape Artist, you are right in much of what you say, but those who are lost in this mystical way of viewing things long ago gave up on good sense, and are quite happy to derive fundamental cosmic principles from a brief consideration of a few words (most notoriously, the word 'not').

Science on the cheap, or what?

One controversy is with regard to infinite sets and the other with rough sets?
'Communism' as a set would be finite and rough.

Lynx:

No, it's over how we define a set and its allowed elements.

There are no useful links I could find other than these; there are loads of books and articles if you are interested...

This is largely incomprehensible for me. Wittgenstein has a problem with infinite sets and set theory, that much I can tell. Why? If infinite sets don't exist in the practical sense then the sets we use are merely subsets or samplings. The purpose of using Venn diagrams in high school was to aid in understanding certain relationships, not to be comprehensive.

A finite set can be enumerated, so that ought to end the extensional controversy.

Not so. All that does is re-iterate the extensional point of view.

Intensionalists criticise this since it is unclear from this 'definition' what counts as a legitimate element of a set, and what does not. That cannot be left to chance in mathematics.

This book makes things a little clearer:

http://books.google.co.uk/books?id=0...mbnail#PPP7,M1

I would love to see a dialectician provide a coherent explanation of what the opposite of an "angry man" is.

Is it a "happy man", an "angry woman", or a "happy woman"?

Indeed, and what is the opposite of Chomsky's example: pretty little girls school?

You said it. Some contemporary philosophical concepts discussed (especially here on RL) entail extremely vague, rather mysticist posters.

That's an interesting thought, but I believe you're just confusing what is meant here by "not". I'm referring to the logical operator NOT, not the algebraic operator not-equal.