So in detail. I think we're construing the scope of behavioural concepts a bit differently still. I'm including statements like {"this is a duck"} and concepts/norms/behaviours like {what makes "this is a duck" correctly assertible} as part of the same idea. They're functions of a linguistic community and its environment {yes I am that debased, seeing language as functional}. — fdrake

How does this address such statements as:

1. The universe was created by a supremely powerful deity
2. Intelligent extra-terrestrial life has visited Earth in secret
3. I will get married next year
4. If Hitler hadn't killed himself then he would have been assassinated

If there's any truth to these statements it certainly doesn't seem to concern norms and behaviours.

Mathematical platonists distinguish themselves from non-platonists like nominalists. Each group seems to understand what the other means, hence their disagreement. I'm asking if infinitesimals exist in the sense that would satisfy mathematical platonism.

In short, it isn't obvious that mathematical platonism necessitates a commitment to only one construal (one use of ∃) of what it means to exist. — J

From here, "[p]latonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices."

I'm not sure how this can be further distinguished. Either some set is a mind-independent abstract object or it's not.

Different people may mean different things by "exist" but I don't think different people mean different things by "mathematical platonism", else there wouldn't be much sense in discussing which of platonism, formalism, intuitionism, fictionalism, nominalism, etc. are correct.

My claim is that it doesn't make sense to argue that both of these are true:

1. Quine atoms exist in the platonistic sense
2. Quine atoms don't exist in the platonistic sense

One of them is true and one of them is false. However, regardless of which of (1) and (2) is true, both of these are true:

3. Quine atoms exist according to New Foundations
4. Quine atoms don't exist according to ZFC

And notice the difference between saying that Quine atoms exist according to New Foundations and saying that New Foundations is correct to claim that Quine atoms exist.

Mathematical fictionalists claim that (1) is false and that (2), (3), and (4) are true.

Perhaps a more helpful phrasing of (3) and (4) is this:

3. There is a set that contains only itself according to New Foundations
4. There is not a set that contains only itself according to ZFC

I assume Harry is arguing for platonism and Sally is arguing against platonism.

Suppose I say, "x exists according to Harry." You say, "x does not exist according to Sally." What is the subject of the dispute between Harry and Sally? Are they in disagreement about x, or about what 'exists' means? — J

The question is ambiguous. Consider these two claims:

1. According to the Lord of the Rings canon, orcs exist
2. According to crazy folk, fairies exist

Mathematical fictionalists are saying something like (1), not (2).

I don't know what Harry and Sally are saying.

Repeating the same error does not correct it. — Banno

I haven't made an error. You have. I explained it quite clearly in my last post here which you opted not to address.

Firstly, you brought up chess as an analogy to propositions. My claim is that there is no chess in a barren world because there is nobody in that world playing chess and that there are no propositions in a barren world because there is nobody in that world using propositions.

Secondly, you conflate propositions about a world and propositions in a world. That we use propositions to talk about a world without language does not entail that there are propositions in a world without language.

I believe that there is a truth to logical laws that is not dependent on one or another philosophical doctrine — Wayfarer

Which logical laws, and why those? There's classical logic, intuitionistic logic, dialetheism, three-valued logic, fuzzy logic, free logic, and so on.

How does the issue of correctness arise? — J

If platonism is correct then I suppose a "correct" logic is one that includes these mind-independent logical facts and doesn't include any logical "fictions".

For example, if the law of excluded middle is a mind-independent fact then classical logic is more correct that intuitionistic logic, and if the law of noncontradiction is a mind-independent fact then classical logic is more correct than dialetheism, and if truth is mind-independently bivalent then classical logic is more correct than fuzzy logic.

But if platonism isn't correct then no logic is "correct". They can be consistent or useful, but nothing more substantial.

Couldn't both types of logic exist platonically -- awaiting discovery by sentient beings? To put it another way, if you believe that any abstracta can exist platonically, why draw the line at a single, putatively correct logic? — J

I addressed that in the OP with respect to incompatible mathematical entities, e.g. the Quine atom which is a set that contains itself. New Foundations allows for such a thing but ZFC doesn't.

If platonism is correct then either Quine atoms are mind-independent mathematical entities or they're not.

I don't think it makes any sense to say that they platonistically exist in New Foundations but don't platonistically exist in ZFC. We can only take the approach of mathematical fictionalism and say that they exist according to New Foundations but not according to ZFC.

The salient question I asked which you failed to address was 'if existence is mind-independent, is being prime likewise mind-independent?'. — Janus

I suppose that depends on whether or not numbers are mind-independent, which I discuss in a different topic.

But if we're discussing physical objects then I already stated several times over the past several weeks and pages that gold can exist in a world without minds.

My claim is only that a) truth and falsity are properties of truth-bearers, that b) truth-bearers are propositions, sentences, utterances, beliefs, etc., and that c) propositions, sentences, utterances, beliefs, etc. are not language-independent. This then entails that d) a world without language is a world without truth-bearers is a world without truths and falsehoods.

It's unclear to me which of a), b), c) , or d) you disagree with.

If you accept a), b), and c) but reject d) then you are clearly equivocating, introducing some new meaning to the terms "truth" and "falsehood" distinct from that referenced in a).

What about the laws of logic, like the law of the excluded middle? Does that cease to obtain in the absence of rational sentient beings? — Wayfarer

Classical logic uses the law of excluded middle but intuitionistic logic doesn't, allowing for sentences that are neither true nor false.

You seem to be suggesting that one of these logics is correct. Which may be so if platonism is correct, but not if it isn't.

It being prime and it being true that it is prime are exactly the same. — Janus

Which I addressed above. If "is is prime" and "it is true that it is prime" are being used interchangeably then the phrase "it is true that" is vacuous; "it" refers to nothing and so truth is not being predicating of anything.

But when "truth" and "falsehood" are being predicated of something – when truths and falsehoods are some thing – that thing is a proposition/sentence/utterance, and if platonism about propositions is incorrect then even if there are truths about a world without language there are no truths in a world without language.

for any number we write down it will be true or false that it is prime, even if we don't know the answer. — Janus

What is the word "it" referring to here?

Either it's referring to a proposition, as I have been arguing, or it's not referring to anything, in which case truth and falsity are being predicated of nothing, and so the phrases "it will be true that" and "it will be false that" are vacuous, as I have been arguing.

If all you are saying is that any number we write down is either prime or not, even if we don't know the answer, then I agree and have never claimed otherwise.

I'll try one last time If you won't address what I actually write, there is no point continuing. I haven't uttered any proposition; I've just nominated some extremely large number and asked the question about its primeness. Do you deny that the truth regarding the number's primness is prior to my proposing anything about it, and in fact prior to my even nominating it, or not. If not, why? — Janus

I went over this with the existence of gold, but I'll do it again with a number being prime:

1. "11 is prime" is true
2. It is true that 11 is prime
3. 11 is prime

(1) asserts that a proposition is true. (3) asserts that a number is prime (and says nothing about truth).

(2) either means the same thing as (1), in which case it asserts that a proposition is true, or it means the same thing as (3), in which case it asserts that a number is prime and the phrase "it is true that" is vacuous, being nothing more than grammatical fluff.

When you ask about the truth regarding a number's primeness it's unclear if you're asking me about the truth of the proposition "X is prime" or if you're asking me about the number's primeness, and this ambiguity is causing you to equivocate.

The unambiguous and correct answers to your question are:

1. If I say that the number 11 is prime then what I say is true
2. The number 11 is prime even if I don't say that it is
3. If I say that the number 12 is prime then what I say is false
4. The number 12 is not prime even if I don't say that it isn't

And to a different question:

1. If I say that gold exists then what I say is true
2. Gold exists even if I don't say that it does
3. If I say that vibranium exists then what I say is false
4. Vibranium does not exist even if I don't say that it doesn't

When you clear up the grammar of the questions and answers then it's clear that truth and falsity are properly properties of propositions/sentences/beliefs/utterances, not something that can be divorced from them by clever word play, and is why the SEP article on truth says:

We thus find the usual candidate truth-bearers linked in a tight circle: interpreted sentences, the propositions they express, the belief speakers might hold towards them, and the acts of assertion they might perform with them are all connected by providing something meaningful. This makes them reasonable bearers of truth.

So the relevant discussion concerns whether or not platonism about truthbearers is correct, or if we should adopt a non-platonistic interpretation that allows for a distinction between truths in a world and truths at a world, and I am firmly in favour of the latter.

If you open your mind and think about it you will see that my example of prime numbers throws that assumption into question — Janus

You seem to be saying that the proposition "X is a prime number" is true or false before it is uttered but denying that this is a case of a proposition being true or false.

For a start paintings do not enjoy pre-existence prioir to their being painted, and thought as pre-existents they are not determinate objects like prime numbers are. — Janus

You appear to be assuming mathematical platonism?

Also, it is an observable object—the painting—which will be assessed for accuracy once it exists. What exactly is it that will be assessed for primeness? — Janus

The proposition "X is a prime number" is assessed as accurate/true when uttered.

I don't have to propose anything I can simply present some number: say 579,836,642,549,743,762,649 and there is a truth about whether or not that number is prime. No proposition required. — Janus

But "a truth" means "a true proposition", and so you are saying "there is a true proposition about whether or not that number is prime; no proposition required".

Also, accuracy is not a precisely d;eterminable quality. — Janus

Then neither is the truth of the proposition "the painting is accurate".

Although I'm not a mathematical platonist, I do have some questions about the epistemological argument, specifically with premise (3):

3. If there exist any abstract mathematical objects, then human beings could not attain knowledge of them.

If knowledge is justified true belief then this can be rephrased as:

3. If there exist any abstract mathematical objects, then human beings could not attain justified true beliefs of them.

Is this saying that if mathematical objects are abstract then we cannot conceive of an equation, that we cannot believe that the equation is true, or that we cannot be justified in believing that the equation is true?

Or is there some other sense of knowledge, distinct from justified true belief, at play here?

I think the platonist response would be that premise 2 is false. — Count Timothy von Icarus

At least according to the SEP article here, (2) is platonism:

Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental.

Mathematical objects exist in spacetime. There is twoness everywhere there are two of something (e.g. in binary solar systems). — Count Timothy von Icarus

This is moderate/immanent realism:

Immanent Realism: Advocates of this view agree with platonists that there do exist such things as mathematical objects — or universals, or whatever category of alleged abstract objects we're talking about — and that these things are independent of us and our thinking; but immanent realists differ from platonists in holding that these objects exist in the physical world.

Isn't it easier then to accept that mathematics does not exist objectively, and is simply a very useful tool? — Tzeentch

I certainly believe so. Given my thoughts in the OP and Occam's razor, I think that mathematical platonism ought be rejected.

As far as my understanding of platonism goes, it argues that ultimate reality can be accessed (with great difficulty) via mystical experiences which go beyond the intellect, and are thus unintelligible?

So any platonic mathematics implies someone had a mystical experience and discovered math still exists 'beyond the veil'? — Tzeentch

It's certainly unclear, and is precisely what gives rise to the epistemological argument against platonism:

1. Human beings exist entirely within spacetime.
2. If there exist any abstract mathematical objects, then they do not exist in spacetime. Therefore, it seems very plausible that:
3. If there exist any abstract mathematical objects, then human beings could not attain knowledge of them. Therefore,
4. If mathematical platonism is correct, then human beings could not attain mathematical knowledge.
5. Human beings have mathematical knowledge. Therefore,
6. Mathematical platonism is not correct.

Why do you think realism is the prevailing view in Phil of math? Why is it found to be a valuable perspective in spite of its drawbacks? — frank

I don't know, I'm not a psychologist.

When you say 'exist in a platonic sense', what exactly do you mean? — Tzeentch

Platonism in the Philosophy of Mathematics

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented.

you just aren't interested in even looking into why propositions still hold a prominent spot in AP and phil of math. — frank

I don't have a problem with propositions. I have a problem with mind-independent propositions, à la platonism.

Even a mathematical platonist like Quine rejects mind-independent propositions, which seems to set up the interesting case where numbers are mind-independent abstract objects but that equations aren't.

And as for maths, I'm not a mathematical platonist, and I don't think that this is the discussion to discuss the merits of mathematical platonism.

It makes you wonder if accuracy is just something people say about the painting. — frank

Yes, and "true" and "false" are adjectives that we use to describe a sentence. I think Wittgenstein's account of language is more reasonable than any account that suggests that our utterances are somehow associated with mind-independent abstract objects and properties (related somewhat to this, this, and this).

I think that language is behavioral and psychological, not something more as platonism would seem to require.

If I utter a proposition that says that some X is a prime number the truth or falsity of that proposition is pre-determined. That seems to throw a spanner in the works for the idea that truth is exclsueively a property of uttered propositions. — Janus

Why?

If I paint a red ball accurately then is the accuracy of that painting "pre-determined", and so evidence that painting-accuracy is not exclusively a property of painted paintings?

When you're here, you admit that vibranium is an idea. When you're in wV, you say it's real. But you're never in wV. — frank

I know, and that is why you are equivocating.

In the real world, the film The Santa Claus is a fiction.
You then conclude that within the film The Santa Claus, Santa is a fiction.
Except that is not the case. Within the film The Santa Claus, Santa is a living, breathing person. That is the very premise of the film.

But I've already told you that this has nothing to do with what I have been arguing. I am only arguing that a world without language is a world without propositions is a world without true propositions (truths).

Unless you want to argue that propositions are language-independent (platonism) or that a world without language is incoherent/empty (strong anti-realism), there's nothing else to discuss.

The entire argument equivocates, as I explained earlier.

That we are using language to talk about a world without language does not entail that language exists in this world – by definition, it doesn't.

That we are using propositions to talk about a world without propositions does not entail that propositions exist in this world – by definition, they don't.

And a world without language is a world without propositions.

Conclusion: wV contains an idea.

Correct? — frank

No.

Again, there are fictional worlds in which Santa is an idea (e.g. Breaking Bad) and there are fictional worlds in which Santa is a living, breathing person (The Santa Claus).

But again, this is unrelated to anything I am saying.

My question is about the ontology of the world where magic exists and Santa is real. That whole thing is just a set of ideas, right? — frank

Yes, and completely unrelated to anything I am saying.

Here are two more fictions:

1. A world in which vibranium, people, and languages exist
2. A world in which vibranium exists but people and languages don't

And two more fictions:

3. A world in which vibranium, languages, and propositions exist
4. A world in which vibranium exists but languages and propositions don't

You claimed earlier that a world without propositions is an empty world, and you are wrong. (2) and (4) are worlds without propositions but they are not empty; they contain vibranium.

Real planets and stars? Or fictional ones? — frank

Do you understand the difference between these two fictions?

1. A world in which magic exists and Santa is a fiction
2. A world in which magic exists and Santa is real

Something can be real within a fiction without being real in the real world.

A world with planets and stars but no people is not an empty world; it's a world with planets and stars.

A world with planets and stars but no languages is not an empty world; it's a world with planets and stars.

A world with planets and stars but no propositions is not an empty world; it's a world with planets and stars.

The fact that these are imaginary worlds and that we are people using language and propositions to talk about them is irrelevant.

Because Frodo definitely isn't a physical object in spacetime. — frank

Again, you're equivocating.

When we talk about a fictional world in which there is gold but no people we are not talking about a fictional world in which there is imaginary gold but no people; we're talking about a fictional world in which there is actual, real, physical gold but no people.

Even if this fictional world is imaginary.

If World X is just a fiction, then it wouldn't be a set of physical objects in spacetime, would it? — frank

It's a fictional world in which planets and stars exist but people and propositions don't, just as the Lord of the Rings universe is a fictional world in which orcs exist but computers don't.

Like Banno you're equivocating. The fact that we use language and propositions to talk about a fictional world does not entail that there are languages and propositions in this fictional world.

A world without language is, by definition, a world without language and so a world without propositions.

If you want to claim that a world without propositions is incoherent/empty then you must claim that a world without language is incoherent/empty, but that's a strong form of anti-realism, and presumably not something that you (or Banno) are willing to endorse.

So what's the ontology of World X? Is it in another dimension? — frank

That depends on whether or not there is an (infinite) multiverse. If there is then there is likely some universe in which there is gold but no people playing chess or using language (and so no propositions). If there isn't then World X is just a fiction.

Doesn't that mean World X is empty? A world is basically a set of propositions. — frank

No, a world can be a set of physical objects situated in spacetime.

As I said many pages and weeks ago, the existence of gold does not depend on the existence of the proposition "gold exists".

In a world without wood, can there be no chess? — Banno

There might be something else that is used other than wood but so far you haven't offered any replacement for language that allows for propositions in that world.

At the moment your position is akin to saying that there is chess in a barren world, and I'm the one saying that there isn't – that there is chess in a world only if there are people (or computers) in that world playing chess.

It is clear that there are propositions, including those that set up the world in question. — Banno

We are using language and propositions to talk about that world, but there are no languages or propositions in that world.

You continue to equivocate.

Try reading the section on Truth in a World vs. Truth at a World again. As a very explicit example it offers:

A proposition like <there are no propositions> is true at certain possible worlds but true in none.

If "there are no propositions" is true at World X then there are no propositions in World X, just as if "there is no English language" is true at World X then there is no English language in World X.

There is a difference between an utterance and a proposition, hence there is a difference between a world in which there are no utterances and one in which there are no propositions. — Banno

And now you're back to contradicting what you said earlier when you said that propositions are constructed by us using words.

If propositions are constructed by language users using words then if there is no language use in a world then there are no propositions in that world.

You really can't make up your mind.

You want to say that there is no truth to there being gold in that world — Banno

No I don't.

I'm only saying what I am literally saying, which is that there is no language in that hypothetical world and so no propositions in that hypothetical world and so no true propositions (truths) in that hypothetical world.

I have repeatedly said that there is gold in that world.

If you are reading something into my words that isn't there then that's on you.

Sure, there is no English in that hypothetical world. But there is gold [in that hypothetical world]. — Banno

And I have never disagreed with this.

I have only ever claimed that because there is no language in that hypothetical world there are no propositions in that hypothetical world and so no true propositions (truths) in that hypothetical world.

The fact that we are using the English language and its propositions to truthfully talk about that hypothetical world is irrelevant.

Where in any of this are we not doing things with words? — Banno

We're asking about a hypothetical world in which there are no people doing things with words. This is where the distinction between "truth at" and "truth in" is important.

Obviously we are using the English language to describe this hypothetical world but then also obviously there is no English language in this hypothetical world. You seem unwilling to make this same distinction when discussing propositions and truth, as if somehow they're special entities very unlike the English language.