Well, in pure set theory a and b are sets too, because it's sets all the way down. — litewave

Well, trivially, yes, since pure set theory is about nothing but sets of sets and the empty set.

a and b are sets too? — litewave

No, but {a} and {b} are.

Chairs are collections too. — litewave

We'd have to look into Wittgenstien's analysis of simples here, and ask if the chair or the leg or the table set is the individual.

A step too far, I think, for this thread.

"abstract" objects — litewave

I spoke a bit about how we might define "abstract" here - that we have a and b and then add the abstract item {a,b} without adding anything to the domain - it still contains just a and b, but we can talk as if there were an abstract thing {a,b}.

I was responding to your post in which you used the phrase "reified metaphysical entities". I understood them simply as real entities. — litewave

Cool. Too many words, too many crossed discussions. The aim might be to be clear about what the individuals we are talking about are.

So we can talk about a and b. And we can change the game a bit and talk about {a,b}. And in one way we have added a new thing to the conversation, yet in another way we are still just talking about a and b. We may be tempted to ask which way is real, but perhaps that question is irrelevant provided we talk clearly.

Chairs are collections too. — litewave

There's a whole new barrel of fish.

Yep.

I interact with collections of objects all the time. — litewave

Sure. Just not in the way you interact with chairs. Different domains.

I have always treated sets as real metaphysical entities. So if properties were sets, then properties would be real too. If properties are not sets, I am not sure if properties are real, but I tend to think they are. — litewave

Do you want me to go on?

What is a real metaphysical entity as contrasted with a real entity? What does the word "metaphysical" do here?

What is a real metaphysical entity as contrasted with a metaphysical entity? What does the word "real" do here?

So what more do we have then "I have always treated sets as entities", which seems quite agreeable.

Just leave aside the baggage of "real" and "Metaphysical".

So if properties were sets, then properties would be real too. — litewave

So instead think about these issues in terms of sets, with all the clarity of the formal apparatus that invokes, and just drop the use of "property", or use it as an anachronistic approximation.

If properties are not sets, I am not sure if properties are real, but I tend to think they are. — litewave

What does it mean to say they are real? What more can we do with real properties that we can't do just with properties? Or much better, with talk of sets or predicates?

This stuff:

What's curious here is how "the property of..." serves to confuse things. The very grammar of "the property of..." encourages us to think we're talking about entities when we're really just manipulating linguistic constructions. — Banno

This is the legacy of syllogistic logic. Since it can only deal in terms of "All S are P", "Some S are P", and so on, it obliges the user to think in terms of substances having properties. It squeezes the world in to an ontology of things and properties. Scholastic metaphysics elaborated on this logical limitation by inventing essences, accidents, substance and so on.

We now have better logical tools for dealing with all of this stuff. The answer on offer to ↪litewave is not to identify properties with sets but to drop talk of properties for talk of sets and predication and extension. Indeed, that is probably the intuition behind the OP.
a day ago — Banno

Odd.

You think she should throw it out on the basis of of her imaginings?

Yes, because my attempt to treat the set and the property as one and the same object seems to have failed. — litewave

But I hope you see that your intuition - that having the property of being red and being a member of the set of red things say much the same thing - remains valid?

That sets are objects in the ontology of set theory. — litewave

And so long as you do not expect to bump in to them as you walk down the street, that's fine, isn't it? What is needed is to keep track of which domain is which.

I think the intuition in the OP is quite right, and in a rough line with Quine and indeed pretty much all of more recent logic.

Cheers. Respect.

Mary Tiles (a philosopher of math) says she can imagine mathematicians ditching set theory someday. — frank

And I hope has the sense not to ditch it yet?

Well, they are, a bit. :wink: — bongo fury

:lol:

Which is why these threads are neverending.

I wanted to say that the set is the common property of its elements. — litewave

Thanks for the clarity.

To my eye, this reifies the property, making it a thing alongside the elements of the set.

That is, you now have the set and the property, separately, and are apparently defining the set in terms of the property.

But of course we could then stipulate a set with no common properties.

I had taken you as proposing to eliminate properties in favour of sets. I would agree with that. But it seems you have something else in mind.

And I'm not at all sure what.

I thought this was what Banno was pointing out to you 4 years ago? — bongo fury

Me, too.

But in set theory, sets do add to ontology. — litewave

What does this mean?

Here's one way to look at it. We have the domain <a,b>. The only items in that domain are a and b. Constructing the set {a,b} does not add to the domain. It does not add {a,b} to the domain.

Are you eliminating properties in favour of sets (which I would support), or making sets into reified metaphysical entities that ground properties?

One of the problems here was a classic for Tones - the move from formal exposition to philosophical exposition. The OP makes that move, by equating the formality of set theory with the informality of properties. So we are a bit stuck with it.

That's the next logic textbook cuz of you :D — Moliere

Sorry.

There’s no formal problem in set theory with counting sets as different from their elements. The “problem” arises only if one has an intuition that collections shouldn’t add to ontology—that a set should “just be” its members. In that case, the proliferation looks like an unnecessary or suspicious multiplication of entities.

https://st.openlogicproject.org/settheory-screen.pdf

Yeah, I deleted that response becasue it doesn't make the teaching point I would like. It does depend on what we mean by individual, just as it depends on what we mean by identical or by completely determined.

:yawn: .

Redacted. I can't help you here.

Yep. As explained earlier, "identical" has a very specific definition here. The set is not extensionally identical to it's elements.

Try "Is a set completely determined by its elements?"

Cool.

Tone's last few posts expressed frustration with a particularly recalcitrant contributor. :worry:

AH!

I missed it.

Yep. Tones can tell me, rightly, how all the stuff I've said here is a gross oversimplification.

@TonesInDeepFreeze?

Hasn't been seen for eight months.

Tones — jgill

Types?

:smile:

...statistical laws... — Ludwig V

I wasn't so much thinking of statistical laws as the basic equations of physics.

Some folk tend to think of F=ma as setting out how the force causes the acceleration. But what's actually happening here is that Newton defined the very notion of force as change in velocity times mass. F=ma is not a description of a cause and effect so much as the presentation of a new way of talking about motion. He didn't find a hidden cause - force - he set up a way to calculate changes in motion and mass. He found a new way to talk about the stuff around us, not a new thing in that stuff. It was a change in semantics, not in ontology.

But I can see that it is a very different model from the Aristotelian model. — Ludwig V

Yep.

That underdetermination stuff is a feature, not a problem. It's about being unhappy with a determinate causal answer such as "God willed it" and looking for more, doing the experiments, using your imagination, seeing what happens when you do this or that...

If we followed Tim we would still be in the monasteries.

I'd hoped that it was pretty clear there were two things - a and b. Kinda what I stipulated. You can say "the domain consists of a and b" to avoid boxes if you like.

Yep, and are talking at cross purposes.

Supose our domain of discourse - what we are talking about - contains only the letters "a" and "b". How many things are in that domain?

If we listen to Frank, then we have a, and we have b, of course; two things. But we also have the set {a,b}. So there are three things: a, b and {a,b}. But then we also have {a,b,{a,b}} - so there are four things in our domain - a, b, {a,b}, and {a,b,{ab}} - and off we go. I hope folk see the problem inherent in counting a set as a different thing to it's elements.

No, a set is no more than the things it contains.

There's also the problem that if a set consists in a criteria rather than it's members, then we can construct the criteria "the set of sets that do not contain themselves", and upset Bertrand Russell. No, set consists in its elements, not in a description of those elements.

That's part of the axiom of extensionality. We can have innumerably many descriptions of some set - the set of odd number, the set of all numbers one less than an even number, the set of numbers that are two more than the last number in the set, starting at one. These are not three different sets, but three different descriptions of the very same set. It's the elements that count (pun intended), not the description.

What this does is to define what we mean when we say that a set is an abstract object - the set {a ,b} is not something else in addition to it's elements, but a different way of talking about a and b. A bit of extra language, not a bit of extra ontology. We talk as if the set were a new thing, but it isn't one of the things in the domain.

Similarly, when we talk of the red of the sports car and the red of the sunset, we haven't thereby added a new thing to the word - the property of redness. Redness is just a new way of talking about the car and the sunset.

Properties dissolved by analysis. Tim will love it. Not.

That doesn't seem too far from the treatment of properties as things we actively atribute to individuals, as I suggested teasingly to Tim.

It also serves to bring out something of the intensional character of properties that might be considered to be missing from the extensional account of properties in terms of sets.

I don't know what a "potential for redness" might be, though, and might resist the idea that such an entity somehow inheres in the apple...

A curious approach.

...whores usually don't go in for a lot of batting practice — BC

I wouldn't know. There are other more obvious parallels - an enthusiastic amateur league, for one. And I am given to understand that there is a ready market for watching sex workers doing what it is they do, even as for cricket and athletics and swimming. Perhaps the missing ingredient is nothing more than finding suitable sponsorship deals.

Aristotle — noAxioms

- that odd idea that properties are "more real" than relations.

Really! Who knew? — BC

I don't see anything that debars sex workers from playing football, should they so choose. The question seems more to be as to whether doing so constitutes a change in profession, or not.

Ok. The move to considering the issue as methodological is worthwhile. "Every event has a cause" is one of Watkins' "haunted universes" doctrines, neither provable nor disprovable. I think the idea of claiming such principles as methodological rather than as ontological truths comes from Watkins. That's certainly were I learned it.

Causation, generally, is perhaps another idea that has hung around well past it's use by date. It made sense to Aristotle but suffered badly under Russell and continues to be difficult to characterise.

Science doesn't look to causes so much as to predictability. It's not about event A causing event B but about the relation between A's and B's, especially when that relation is expressed in an equation.

Those hierarchies are how we keep set theory consistent, underpinning Zermelo set theory, and hence ZFC, the accepted foundation of mathematics.

Not unimportant. And again, speaking in rough outline.

This is the legacy of syllogistic logic. Since it can only deal in terms of "All S are P", "Some S are P", and so on, it obliges the user to think in terms of substances having properties. It squeezes the world in to an ontology of things and properties. Scholastic metaphysics elaborated on this logical limitation by inventing essences, accidents, substance and so on.

We now have better logical tools for dealing with all of this stuff. The answer on offer to is not to identify properties with sets but to drop talk of properties for talk of sets and predication and extension. Indeed, that is probably the intuition behind the OP.

What's curious here is how "the property of..." serves to confuse things. The very grammar of "the property of..." encourages us to think we're talking about entities when we're really just manipulating linguistic constructions.

I didn't know there could be the property of having a property, so I learned something. — frank

This leads pretty quickly to Russell's paradox. Consider "the property of being a property that doesn't apply to itself."

Hence logicians and mathematicians introduced hierarchies. Individuals, then sets of individuals, then sets of sets of individuals, and so on, without intermingling.

For example, let's take property red or redness (X = red): The property of "being in set red" is the same as the property of "having property red", which is the same as the property of "being red", which is the same as property red. So, the property of "being in set red" and property red are one and the same property. — litewave

My advice would be to drop "...the property of..." from all of this. Then "being a member of the set of red things" is the same as "being red".

This kinda cuts to the heart of the issue.

I'll point out again the discourtesy of removing the automatic links when quoting.

See the summary I provided above for Moliere. — Count Timothy von Icarus

Where?

You appear here to have gone to great lengths to explain what your argument is not, without explaining what it is. Your reply is in such broad terms as to say very little.

Added: links and citations are conducive to clarity. It might be helpful if you did not remove them.

I hope not.

It's like because its trite, immature and ignores the specific criteria that causes prostitution to obtain. — AmadeusD

Love your sense of irony. Is professional sport that bad?