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.

There are abstractions. These are constructed by us, doing things using words. — Banno

And this is where you're not making sense.

You say that propositions are constructed by us doing things using words but then say that there are true propositions even if we're not doing things using words. Make up your mind.

I would say that platonism best reflects the way we generally think about things like the set of natural numbers N. — frank

That doesn't make it true. As I said earlier, it's us being uncritically bewitched by grammar into thinking that a sentence such are "there are numbers" is saying something it's not.

Instead of those, look at the SEP article on philosophy of math. It shows the alternatives to platonism are logicism, intuitionism, formalism, and predicativism.

Do you want to go through those? — frank

No, because it's not relevant to what I am arguing, which concerns whether or not there are mind-independent true propositions. Whether these propositions are about mathematics or physics makes no difference. To repeat what I said above:

Some linguistic activity by a suitably intelligent mind is required for there to be propositional content, and so for there to be a true proposition, and so for there to be a truth.

This is all I am arguing.

But I think we should be careful in saying that "an utterance" is required. — J

I mentioned elsewhere that terms like "utterance" are being used as a catch-all for speech, writing, signing, believing, thinking, etc.

Some linguistic activity by a suitably intelligent mind is required for there to be propositional content, and so for there to be a true proposition, and so for there to be a truth.

You're referring to this argument?

(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
(P2) Mathematical entities are indispensable to our best scientific theories.
(C) We ought to have ontological commitment to mathematical entities.

Firstly, "having ontological commitment to mathematical entities" does not entail platonism. Immanent realists and conceptualists also have ontological commitment to mathematical entities.

Secondly, P2 appears to presuppose that nominalism is false. The nominalist might agree that mathematics is indispensable to our scientific theories but won't agree that mathematical entities are indispensable to our best scientific theories, because they believe that no mathematical entities exist.

which is that if you deny platonism of any kind, you're rejecting science in general — frank

You don't need to believe in mind-independent abstract objects to believe in mind-independent physical objects, and you don't need to believe in mind-independent abstract objects to believe that these mind-independent physical objects move and interact with one another.

It's not clear what you're asking.

Are you asking me if the sentence "we will say true things in the future" is true?

FYI, @bongo fury and @frank

Quine

Quine does not accept the existence of any abstract objects apart from sets. His ontology thus excludes other alleged abstracta, such as properties, propositions (as distinct from sentences), and merely possible entities.

So with paintings there is the landscape being painted and the painting. We say that the painting is accurate if it resembles the landscape being painted and inaccurate if it doesn't.

With language there is the landscape being described and the utterance. We say that the utterance is true if its propositional content "resembles" (for want of a better word) the landscape being described and false if it doesn't.

But according to platonists, in most situations there is the landscape being described, the propositional content, but no utterance, and that this propositional content is true if it "resembles" the landscape being described and false if it doesn't.

I don't think the notion that there is false propositional content without an utterance makes any sense, and so I also don't think the notion that there is true propositional content without an utterance makes any sense.

Even if we want to distinguish an utterance from its propositional content, an utterance is required for there to be propositional content. Propositional content, whether true or false, doesn't "exist" as some mind-independent abstract entity that somehow becomes the propositional content of a particular utterance.

So when you ask if the propositional content of an utterance was true before the utterance was made, I literally don't understand you. The propositional content only "came into being" when a meaningful utterance was uttered, which is just to say that we understand an utterance (e.g. conceptualism), and which is perhaps best explained by Wittgenstein.

You're basically saying Quine was an idiot. — frank

No, I'm saying he's wrong, just as every other conceptualist and immanent realist and nominalists says.

Surely Quine suggests we refer timelessly (non-modally) to the sentence inscribed or uttered in a future region of space-time? And we describe it (rightly by your hypothesis) as true? Is that non-sensical? — bongo fury

Yes. I think that Wittgenstein provides a much more sensible approach to language. There's no mystical connection between utterances and mind-independent, non-spatial, non-temportal abstract objects; there's just actual language-use and the resulting psychological and behavioural responses.

I don't think you're bothering to look very deeply into this. — frank

I think I’m looking into it only as deeply as it needs to be. Platonism is a result of being bewitched by language, misinterpreting the grammar as entailing something it doesn’t.

But are you denying that it's already true? — bongo fury

I’ve been over this so many times.

The word “it” in the phrase “is it true?” refers to either an utterance or an utterance-dependent proposition, and so asking if an utterance or proposition is true before it is uttered is a nonsensical question, like asking if a painting is accurate before it is painted.

Just be aware of what you're giving up if you reject mathematical realism. — frank

I’d be giving up on mind-independent abstract objects, which is of no concern.

Just look at Quine's indispensability argument in the SEP article I cited. — frank

And perhaps you could look at the epistemological argument against platonism.

I think immanent realism collapses into conceptualism — Michael

I'm not sure how. — frank

Because the immanent realist believes that "properties like redness exist only in the physical world, in particular, in actual red things."

An immanent realist about propositions would have to believe that propositions exist only in particular things, and presumably the only particular things within which a proposition can exist is an utterance. But a sound is only an utterance if there is a mind to interpret the sound as an utterance. And so it's not clear how immanent realism about propositions can be distinguished from conceptualism about propositions.

Hence it seems that with respect to propositions we must be platonists (mind-independent propositions), conceptualists (mind-dependent propositions), or nominalists (no propositions).

Only platonism allows for something that can putatively count as a mind-independent truth, and I think that platonism about propositions is more problematic than the alternatives, most likely because I think that physicalism or property dualism is more parsimonious than the theory that there is the physical, the mental, and the independently abstract.

But you know they're all true. Even the unwritten ones. Since they satisfy the equation n+(n+1)=2n+1. — fdrake

Firstly, I don’t think that n+(n+1)=2n+1 proves mathematical platonism.

Secondly, what is true? The equation? What is an equation? Is it a meaningful string of symbols?

This is where I think the grammar is causing confusion. There is both a platonist and a non-platonist interpretation of "there are unwritten equations".

As an analogy, consider something like "there are unpainted red paintings". It's certainly true in the non-platonic sense that someone could paint a red painting that doesn't exist in the present, but it's not true in the platonic sense that there exists in the present some painting that is red but unpainted.

And so "there are unwritten true equations" is true in the non-platonic sense that someone could write a true equation that doesn't exist in the present, but it's not true in the platonic sense that there exists in the present some equation that is true but unwritten.

In general there are four different positions on the topic, paraphrased from here:

1. Platonism - there are mind-independent and particular-independent abstract objects
2. Immanent realism - there are mind-independent and particular-dependent abstract objects
3. Conceptualism - there are mind-dependent abstract objects
4. Nominalism - there are no abstract objects

With respect to propositions, I think immanent realism collapses into conceptualism (propositions are particular-dependent, i.e. dependent on meaningful utterances, and meaningful utterances are mind-dependent), giving us three options:

1. Propositions are mind-independent
2. Propositions are mind-dependent
3. There are no propositions

(1) and (2) will argue that truth is a property of propositions, (3) that truth is a property of utterances.
(1) allows for true propositions (truths) without minds, (2) and (3) only for true propositions (truths) with minds.

I reject platonism. I'm undecided on nominalism and conceptualism, but the things I am saying are consistent with both.

And so the issue is forced into a juxtaposition. Better to ask how propositions are dependent on mind — Banno

Given that the crux of the recent debate is over whether or not there are truths (true propositions) without minds, it's an appropriate juxtaposition.

If there are truths without minds then propositions are mind-independent (platonism).
If propositions are mind-dependent (conceptualism) then there are no truths without minds.

There are infinite additions. — Banno

Are you arguing for mathematical platonism, or are you arguing for a non-platonic interpretation of "there are an infinite number of true additions and false additions that we could write out"?

Because I don't believe in mathematical platonism.

Returning back to my diagrams:



There are an infinite number of quoted mathematical equations that we could write out inside the World A circle, and in writing them out they are either blue (true) or red (false), but none that we can write out inside the World B circle because there's nobody in that world to assert them. Which is why there are mathematical truths and falsehoods in World A but no mathematical truths or falsehoods in World B.

This is where the platonist disagrees; he would argue that there are an infinite number of blue and red mathematical equations that we could write inside the World B circle even though there's nobody in that world to assert them.

So please clarify your position on this. Is it sensible to write out red and blue mathematical equations inside the World B circle?

There are really two parts to this discussion.

The first concerns the dispute between platonism and conceptualism – are propositions mind-independent or not?

The second concerns the dispute between realism and anti-realism (as defined by Dummett) – is a proposition’s truth value verification-transcendent or not?

This leaves us with four possible positions:

Platonism + realism: there are mind-independent propositions with verification-transcendent truth conditions.

Conceptualism + realism: there are mind-dependent propositions with verification-transcendent truth-conditions.

Conceptualism + anti-realism: there are mind-dependent propositions with verification-immanent truth-conditions.

Platonism + anti-realism: there are mind-independent propositions with verification-immanent truth-conditions.

I’m not sure how sensible the last of these is, and so perhaps we can dismiss it for now.

Of the other three, only platonism + realism allows for anything that can be considered a “mind-independent truth”.

Now there is some ambiguity with the phrase “mind-independent truth”. On the one hand it might mean “a proposition that is mind-independent and true” and on the other hand it might mean “a proposition that is mind-independently true”.

The former is just platonism.

If the latter does not mean the former then it more accurately means “a proposition that is mind-dependent and mind-independently true”, which is conceptualism, and doesn’t really seem to satisfy the intention of the phrase “mind-independent truth”, and is why I have been arguing that either platonism is correct or there are no truths if there are no minds.

Note specifically that a proposition being mind-dependent does not entail that its truth value is mind-dependent, which I think is where @frank is making his mistake.