But an empty set is nevertheless a concrete object? — Wayfarer

By a concrete object I mean any collection, so an empty collection would be the simplest concrete object.

I know that sets (collections) are often regarded as abstract objects, but in those cases a set is meant as a concept/property (which is instantiated in concrete sets). A concept/property is an object that is not a collection but it has instances in collections.

I understand, I just meant to point out that if all possible (logically consistent/coherent) universes are equally real as the one we live in, correspondence theory of truth becomes identical to coherence theory of truth.

I understand, I just meant to point out that if all possible (logically consistent/coherent) universes are equally real as the one we live in, correspondence theory of truth becomes identical to coherence theory of truth — litewave

How?

Every consistent description of a world corresponds to a real world.

I had the idea it was with land title claims and the tallying of agricultural output in Sumeria and Egypt. Land holdings had to be calculated across very irregular shapes, There was a recent discovery about this https://cosmosmagazine.com/science/mathematics/babylonian-tablet-trigonometry-pythagorean-triplets/ — Wayfarer

To simplify the discussion, let’s use a right-angled triangle with shorter perpendicular side s, longer perpendicular side l, and diagonal d, such that s² + l² = d² .

Columns two and three of Plimpton 322 simply contain values for s and d respectively for the series of Pythagorean triples. Column four is just a list of the numbers 1 to 15, so we can remember which row we’re up to. But column one represents the ratio d² / l², and since we’re given the value of d in column three, we can calculate l, and voila … a complete Pythagorean triple (s,l,d) is revealed! — cosmosmagazine.com

3, 4, and 5 are a pythagorean triple: 3² + 4² = 5²

The yield of the land corresponds to its area.

Imagine you had 9 square units of land (3²).

Suppose now, that new land is cultivated and the ratio of the total yield to that of the new land cultivated is 1.5625 (roughly one and a half times). This is basically the ratio of the total area of land that's cultivated to the area of the new land cultivated = d²/l².

We can now calculate l = sqrt(d²/1.5625) = 4 units. The new land cultivated can be thought of as a square 16 square units with sides 4 units. The area of land you began with (3²) + the area of the new land cultivated (4²) = Total land cultivated (5²). From how the yield scaled up (× 1.5625), we could determine the area and dimension of the new land cultivated. I dunno!

Every consistent description of a world corresponds to a real world. — litewave

Isn't that begging the question? By the way if a world has to be qualified with real as you do in "...a real world", it suggests that worlds can be unreal. Care to expand and elaborate.

Isn't that begging the question? By the way if a world has to be qualified with real as you do in "...a real world", it suggests that worlds can be unreal. Care to expand and elaborate. — TheMadFool

As I said, I think that all possible worlds are just as real as our world because I don't see any ontological difference between possible and real worlds.

The maths of the correspondence theory of truth is called model theory.

Commitment to the correspondence theory means commitment to a model's actual existence: properties, relations and all. (Platonism.)

The same commitment isn't required in order to do model theory, because models, like all mathematical entities, might be fictions, like Santa Claus.

Neither is it required in order to do nominalism (reference theory), and examine the correspondences (albeit conventional or pretended) between words and things, or other words. In order to take, that is to say, a mathematical or literary or pictorial story and examine its pretended connections to existing things or events (e.g. world war II) or, that perhaps not being an option, to other words and pictures (numerals, number lines, Santa pics, real old man pics, etc).

If what matters most according to the correspondence theory of truth, is the accurate portrayal of a particular or general 'state of affairs' - through language - of reality, — Shawn

Yes, I think so...

and therefore what can be platonically described as the mind's eye — Shawn

No idea what you mean, although actual existence of properties etc is what an anti-platonist can't handle. Is my understanding of 'platonic'. So if 'minds eye' means imaginary... No I can't parse it.

From a retired mathematician who still dabbles with it, — jgill

Hi there from an ignoramus.

That sequence of electronic dots has a kind of "physical" existence but is still in a way non-physical. How does this fit into the current discussion? — jgill

I would offer: the non-physical aspect is the pretended or conventional reference (by the dots, in sequence). That leaves it open to analyse the reference as fictive, like a Santa story, or factual, like a history. Either way, there is no need to infer reference to non-physical entities. If you don't want to... Do you?

PS why the scare quotes?

As I said, I think that all possible worlds are just as real as our world because I don't see any ontological difference between possible and real worlds. — litewave

Modal Realism! I quite like it that there are more worlds out there populated by unicorns, fairies, angels but what scares me are vampires, ghosts, werewolves, zombies.

Yeah.

Yeah — litewave

What's impossible to you?

What's impossible to you? — TheMadFool

Logically inconsistently defined objects. Objects that are not what they are. Objects that have properties they don't have.

It may not be obvious whether an object is consistently defined, because its definition, its properties include all its relations to all other objects in reality, so it must be defined consistently in relation to everything else. But at least when you interact with an object, you can know that it is consistently defined without having to check consistency of its relations to everything else, because if you interact with it it must exist and inconsistent objects cannot exist.

Perhaps surprisingly, whatever you are doing at this moment, it is impossible for you not to be doing it, at this moment. Simply because it would be a contradiction, an inconsistently defined event, if you were not doing what you are doing. For a copy of you in a different possible world it might be possible not to be doing what you are doing but not for you.

So, whence does logic end and mathematics begin, could be a short way of asking another person... — Shawn

Hmm. Thought I replied to this yesterday, but must not have clicked "Post"...

If physics is applied mathematics, then mathematics is applied logic...?

You appear to still be using "simples" - so you assume there is a "lowest level", and speak of "smallest parts".

But what is to count as a simple, as the atom from which you derive the world? Whatever you choose will be arbitrary - we might choose otherwise.

Go back to your first post:

Mathematics corresponds to the structure of reality (and omits the qualities that fill the structure). — litewave

...Can you provide an indubitable account of what that "correspondence" consists in?

That's the core problem for correspondence.

And what is the ontological (existential) difference between a possible universe and a "real" universe? I think none, so all possible universes exist and descriptions of all possible universes correspond to reality. — litewave

Seriously?

If this were so, then since in some possible world you didn't write that post; and since all possible universes exist and descriptions of all possible universes correspond to reality, you really didn't wright that post.

How will you avoid such inconsistency?

You have set the scope of "...exists" across all possible world instead of within the scope of each possible world, and that results in inconsistency.

When P matches R, there's a correspondence and then we can claim P is true. — TheMadFool

That already goes too far; you would next be asked to explain the nature of that "match" of the "correspondence". That's the fatal flaw of correspondence theory.

...so an empty collection would be the simplest concrete object. — litewave

As @Wayfarer impied, what is concrete about an empty set?

A concept/property is an object that is not a collection but it has instances in collections. — litewave

This is at odds with extensional logic, in which a property is a collection of objects; so "...is red" is the collection of red things.

Every consistent description of a world corresponds to a real world. — litewave

But then all you have done is claim that anything could be true.

The point is surely to sort out the way things actually are from the way things might be. If every possible world corresponds to the real world, you have no way to do this.

You appear to still be using "simples" - so you assume there is a "lowest level", and speak of "smallest parts".

But what is to count as a simple, as the atom from which you derive the world? Whatever you choose will be arbitrary - we might choose otherwise. — Banno

The smallest parts, empty sets, are obviously "simples" in the sense that they have no parts. But any collection is also a "simple" in the sense that the collection as a whole is an indivisible/unstructured thing that stands in parthood relations to other things that are its parts.

...Can you provide an indubitable account of what that "correspondence" consists in?

That's the core problem for correspondence. — Banno

A proposition describes an object by affirming that the object has certain properties. If the object really has those properties then the proposition is true - that's how a proposition corresponds to reality.

Example: Proposition "Planet Earth has approximately a spherical shape with a radius of 6,370 km" is true and thus corresponds to reality iff planet Earth has approximately a spherical shape with a radius of 6,370 km. And the properties attributed to Earth in this proposition are relational (geometric/quantitative) and thus mathematical.

The smallest parts, empty sets, are obviously "simples" in the sense that they have no parts. — litewave

...and yet to understand empty sets one needs all the paraphernalia of set theory. SO if they are to form the "simples" of a logical system, it is only by presuposing set theory. Which is not all that simple.

...any collection is also a "simple" in the sense that the collection as a whole is an indivisible/unstructured thing that stands in parthood relations to other things that are its parts. — litewave

Yes! Anything can count as a simple.

A proposition describes an object by affirming that the object has certain properties. If the object really has those properties then the proposition is true - that's how a proposition corresponds to reality. — litewave

..and? That does not explain the "correspondence" in the correspondence theory of truth. Indeed, while correspondence is about what is the case, you've moved to affirmation, which is distinct, and quite different. One can after all affirm things that are not true.

So I don't see this approach as getting very far.

If this were so, then since in some possible world you didn't write that post; and since all possible universes exist and descriptions of all possible universes correspond to reality, you really didn't wright that post.

How will you avoid such inconsistency?

You have set the scope of "...exists" across all possible world instead of within the scope of each possible world, and that results in inconsistency. — Banno

Assuming that it is really "me" who lives in different possible worlds (which I don't think is a correct definition of "me", since my consciousness is clearly limited to only one of those worlds), I can say that I wrote the post in this world but not in some other worlds and I can also say that I wrote the post in reality as a whole, which is the collection of all possible worlds. Similar to saying that I watched Citizen Kane in Germany but I did not watch Citizen Kane in France, and I watched Citizen Kane in reality as a whole. This way inconsistency is avoided.

We agree that it is true in this possible world that you wrote the post; it is not true in some other possible world?

Even though many descriptions of a universe by mathematicians don't correspond to our universe, they correspond to other possible universes. — litewave

So do you think mathematical statements are true in this possible world because they are true in some possible world?

Then if it is true that in some possible world you dd not write that post, wouldn't it be true in this possible world, too?

@litewave

Arn't mathematical statements true in all possible worlds?

...so an empty collection would be the simplest concrete object. — litewave

As Wayfarer impied, what is concrete about an empty set? — Banno

That it is a collection, rather than a property.

A concept/property is an object that is not a collection but it has instances in collections. — litewave

This is at odds with extensional logic, in which a property is a collection of objects; so "...is red" is the collection of red things. — Banno

I don't think that a property is a collection. Redness is not the collection of all red things but something that is had by all red things. The red things are the extension of the collection of red things as its parts, but they can also be said to be the extension the property redness as its instances. If you think that properties are collections then reality consists only of collections, which are concrete things, because properties as abstract things that have instances don't exist.

But then all you have done is claim that anything could be true. — Banno

Anything that is consistently defined and thus identical to itself.

The point is surely to sort out the way things actually are from the way things might be. — Banno

It is useful to sort out the way things are in our world from the way things are in other possible worlds.

That it is a collection, rather than a property. — litewave

That's the very definition of a property in first-order logic. First order logic is extensional by design.

So you using a non-standard interpretation?

Anything that is consistently defined and thus identical to itself. — litewave

SO for any proposition P you have:

P is true IFF it is consistent and identical with itself

Consistent with what? "Lightwave wrote this post" is consistent, but not true - I wrote this post.

It is useful to sort out the way things are in our world from the way things are in other possible worlds. — litewave

...appears incompatible with...

...all possible universes exist and descriptions of all possible universes correspond to reality. — litewave

My reading of the correspondence theory of truth requires two essential components:

1. An actual reality. Call this R
2. A proposition about that actual reality. Call this P

When P matches R, there's a correspondence and then we can claim P is true. — TheMadFool

That two things correspond is a judgement. Correspondence is never anything more than a judgement. So there's really no such thing as "when P matches R", just the judgement, and the claim.

...and yet to understand empty sets one needs all the paraphernalia of set theory. SO if they are to form the "simples" of a logical system, it is only by presuposing set theory. Which is not all that simple. — Banno

Set theory is based on the simple and self-evident fact that objects constitute a collection. A collection can be defined by listing all its parts or by specifying their common property. The problem with defining a collection by the common property of its parts is that such a definition may be inconsistent, so this kind of definition has been narrowed by certain axioms that select certain kinds of collections. It doesn't mean that some axiomatizations of set theory are correct and others wrong; they just select different kinds of collections.

..and? That does not explain the "correspondence" in the correspondence theory of truth. Indeed, while correspondence is about what is the case, you've moved to affirmation, which is distinct, and quite different. One can after all affirm things that are not true. — Banno

Affirmation is in the nature of propositions. Proposition is a tool of communication, which by affirming something provides information. If a proposition affirms that a certain object has a certain property and the object in reality does not have the property, then the proposition does not correspond to reality and thus is not true.

We agree that it is true in this possible world that you wrote the post; it is not true in some other possible world? — Banno

It is not true that in some other possible world I wrote the post.

So do you think mathematical statements are true in this possible world because they are true in some possible world? — Banno

Only if one of those possible worlds is this world.

Then if it is true that in some possible world you dd not write that post, wouldn't it be true in this possible world, too? — Banno

It would be true in this world that I did not write the post in a different world in which I did not write it.