Same topic as https://thephilosophyforum.com/discussion/15304/mathematical-truth-is-not-orderly-but-highly-chaotic/

Is it ok if we just copy-and-paste the replies? Or should we link to them?

The overwhelmingly vast majority of truth cannot be expressed by language — Tarskian

This is not right. Perhaps "the vast, vast majority of subsets of natural numbers cannot be expressed by language," but judgments of true or false only apply to propositions. Propositions are linguistic entities - they can all be expressed in language. If it can't be expressed in language, it isn't a proposition and if it isn't a proposition, it can't be true or false.

judgments of true or false only apply to propositions — T Clark

The following is a legitimate proposition:

The set {6,8,11} is a subset of the natural numbers.

It is true or false.

If it can't be expressed in language, it isn't a proposition — T Clark

The following proposition is tautologically true:

Every subset of the natural numbers is a subset of the natural numbers.

The problem is that most individual subsets of the natural numbers cannot be expressed by language. Some can but most cannot.

The ineffable propositions are still true propositions because all of them are true given the tautology mentioned above.

Propositions are linguistic entities — T Clark

Propositions that can be expressed by language are indeed linguistic entities. The ones that cannot be expressed by language, however, are not. For example, the general case of "Subset X of the natural numbers is a subset of the natural numbers" is true, irrespective of whether X can be expressed by language or not.

Propositions that can be expressed by language are indeed linguistic entities. The ones that cannot be expressed by language, however, are not. — Tarskian

There are no propositions that can't be expressed in language.

Subset X of the natural numbers is a subset of the natural numbers — Tarskian

This is just a restatement of the tautological proposition "All subsets of the natural numbers are subsets of the natural numbers."

I can see you and I are not going to agree on this. I'll give you the final word.

I can see you and I are not going to agree on this. — T Clark

The distinction between countable and uncountable infinity, originally introduced by Georg Cantor, has always been controversial.

https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory

Controversy over Cantor's theory

Initially, Cantor's theory was controversial among mathematicians and (later) philosophers.

As Leopold Kronecker claimed: "I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there."

Many mathematicians agreed with Kronecker that the completed infinite may be part of philosophy or theology, but that it has no proper place in mathematics.

Logician Wilfrid Hodges (1998) has commented on the energy devoted to refuting this "harmless little argument" (i.e. Cantor's diagonal argument) asking, "what had it done to anyone to make them angry with it?"

When first confronted with the matter, I do not think that anybody right in his mind agrees on this. It is just too controversial. The first reaction is usually, disgust. It takes quite a while before someone can actually accept this kind of thinking.

When first confronted with the matter, I do not think that anybody right in his mind agrees on this. It is just too controversial. The first reaction is usually, disgust. It takes quite a while before someone can actually accept this kind of thinking. — Tarskian

Any thoughts on why?

Is it a blow to people's egos to face the limitations of human thought?

Any thoughts on why? Is it a blow to people's egos to face the limitations of human thought? — wonderer1

In my opinion , it decisively divorces mathematical reality from physical reality, which is otherwise its origin.

Humans, but also animals, have quite a bit of basic arithmetic and logic built into their biological firmware, if only, for reasons of survival. To the extent that mathematics stays sufficiently close to these innate notions, people readily accept its results.

There is no notion of infinity in physical reality. In that sense, Cantor's work is rather unintuitive. You have to learn to think like that. It does not come naturally.

Human language is countably infinite because:

its alphabet is finite
every string in human language is of finite length — Tarskian

You seem to be forgetting that languages can evolve and it's use can be arbitrary. We can always add more letters to the alphabet and we only communicate what is relevant. Why would we need a word for every natural number if we never end up finding a use for those numbers? If the universe is finite then there is no problem here. If it isn't then the universe at least appears to be consistent in that the physical laws are the same no matter where you go in the universe. Novelty would be the only aspects of the universe needing new terms to describe them.

What is one example of a subset of the natural numbers that cannot be expressed by language?

Also note that mathematical notation is a kind of extension to the natural languages.

The overwhelmingly vast majority of truth cannot be expressed by language — Tarskian

Assuming this statement is true, what do you think is its philosophical significance?

Assuming this statement is true, what do you think is its philosophical significance? — 180 Proof

:up:

What is one example of a subset of the natural numbers that cannot be expressed by language? — hypericin

There is a one-to-one mapping between the subsets of the natural numbers and the real numbers. So, we can represent a subset of the natural numbers by its corresponding real number.

We construct the real number as the Ricardian number r:

https://en.m.wikipedia.org/wiki/Richard%27s_paradox

Richard's paradox

The paradox begins with the observation that certain expressions of natural language define real numbers unambiguously, while other expressions of natural language do not. For example, "The real number the integer part of which is 17 and the nth decimal place of which is 0 if n is even and 1 if n is odd" defines the real number 17.1010101... = 1693/99, whereas the phrase "the capital of England" does not define a real number, nor the phrase "the smallest positive integer not definable in under sixty letters" (see Berry's paradox).

There is an infinite list of English phrases (such that each phrase is of finite length, but the list itself is of infinite length) that define real numbers unambiguously. We first arrange this list of phrases by increasing length, then order all phrases of equal length lexicographically, so that the ordering is canonical. This yields an infinite list of the corresponding real numbers: r1, r2, ... . Now define a new real number r as follows. The integer part of r is 0, the nth decimal place of r is 1 if the nth decimal place of rn is not 1, and the nth decimal place of r is 2 if the nth decimal place of r[n] is 1.

The preceding paragraph is an expression in English that unambiguously defines a real number r. Thus r must be one of the numbers r[n]. However, r was constructed so that it cannot equal any of the r[n] (thus, r is an undefinable number). This is the paradoxical contradiction.

The Ricardian real number r is defined as undefinable and therefore the corresponding subset of the natural numbers cannot be expressed in language either.

Assuming this statement is true, what do you think is its philosophical significance? — 180 Proof

If you look at the epistemic JTB account for knowledge as a justified true belief, it means that the overwhelmingly vast majority of true beliefs are ineffable and cannot possibly be justified.

Hence, most truth is not knowledge.

The fact that some truth can be justified is the rare exception and not the rule.

If you look at the epistemic JTB account for knowledge as a justified true belief ... — Tarskian

Perhaps your OP topic only indicates that "the epistemic JTB account" is inadequate in some way.

If you look at the epistemic JTB account for knowledge as a justified true belief, it means that the overwhelmingly vast majority of true beliefs are ineffable and cannot possibly be justified. — Tarskian

They are ineffable, so they have no opportunity to be beliefs at all, and therefore no occasion to be justified.

Ineffable truths are never believed. And I guess, numerically, most truths are ineffable. But all of these ineffable truths seem quite irrelevant too.

They are ineffable, so they have no opportunity to be beliefs at all, and therefore no occasion to be justified. — hypericin

Then there is still the next level: the beliefs about these ineffable beliefs which are not necessarily ineffable. There is a large literature about Richardian numbers even though these numbers are undefinable.

But all of these ineffable truths seem quite irrelevant too. — hypericin

Well, it's a bit like the axiom of infinity, i.e. insisting on the existence of an ineffable cardinality. At first glance, it also looks irrelevant.

https://institucional.us.es/blogimus/en/2022/01/is-infinity-really-necessary

One of the axioms of mathematics is that there exists an infinite set. Without this axiom our mathematics would be much weaker. Many of our theorems would fall like a house of cards. Newton or Gauss would probably have hesitated to accept our axiom (although without being aware that they were using it). We have accepted it for our comfort. Faith, that some people say …

Originally, most mathematicians utterly rejected the axiom of infinity and Cantor's work in general:

As Leopold Kronecker claimed: "I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there."

The ineffable sequence of infinite cardinalities is an essential axiomatic belief in contemporary mathematics, no matter how much it sounds like philosophy or theology.

:up: :up:

Perhaps your OP topic only indicates that "the epistemic JTB account" is inadequate in some way. — 180 Proof

I don't think that JTB is inadequate.

Most truth cannot be known in terms of JTB. That is not a flaw in JTB. The nature of reality is simply like that.

If we happen to know some truth, then it is the rare exception and not the rule.

The nature of reality is simply like that. — Tarskian

You have not shown that this is the case (i.e. a belief that is neither justified nor true).

The overwhelmingly vast majority of truth cannot be expressed by language.

Oh, my goodness me. How shocking.

Now, can you give an example of one those the truths?

Just one will do. Then we will have an idea of what we are dealing with. Of the import of this startling, enigmatic observation.

Hmm.

Now, can you give an example of one those the truths? — Banno

Consider the following proposition:

The set X is a subset of the natural numbers.

This is trivially true for an example subset such as {5, 67, 257}.

There are an uncountably infinite number of such subsets. However, there are only a countably infinite number of sentences in language. Therefore, for most subsets X of the natural numbers, this true sentence cannot be expressed in language.

Yes, yes, all that. So what? Give an example of one of these unstatable true sentences...

Now, can you give an example of one those the truths? — Banno

Not on this message board, obviously. But there is a rumour that the mystical can be made manifest. That is what Zen is about, is it not? And the Dao, and the holy.

Talk is cheap and very limited, so one is obliged to wave a hand in the general direction of the uniqueness that is everywhere, all the time.

Give an example of one of these unstatable true sentences... — Banno

Construct a Richardian number and map it one-to-one to a subset of the natural numbers. This subset is ineffable:

https://en.m.wikipedia.org/wiki/Richard%27s_paradox

The preceding paragraph is an expression in English that unambiguously defines a real number r. Thus r must be one of the numbers r[n]. However, r was constructed so that it cannot equal any of the r[n] (thus, r is an undefinable number). This is the paradoxical contradiction.

The paragraph expresses a number, not an unstateable truth.

The overwhelmingly vast majority of truth cannot be expressed by language — Tarskian

Assuming this statement is true, what do you think is its philosophical significance? — 180 Proof

Here’s my unsolicited attempt to answer your question: writing tentatively, with the need for corrective refutation:

Tarskian’s premise suggests to our understanding that: the mapping from verbal language to experience is categorically incomplete.

In turn, this tells us that sine qua non rules about the volume and thoroughness of verbal databases of information evolve without completion, and that therefore the epistemological project is likewise an evolving project without completion.

One of the important consequences of an epistemological project that never completes is knowledge of truths that cannot be proven. This leads to speculation about the science, math and language databases all being open. If so, it may be the case there is no complete systemization.

If it can be surmised that no systemization of science, math and language can be complete, then it might follow that no correspondence between them can be complete.

In turn, this might suggest the need for a radical overhaul of our definition(s) of truth: if correspondence is always incomplete, then the cognitive vector (thinking about science, math and language) like the physical vector, with its position and momentum coordinates, might be uncertain per Heisenberg.

The paragraph expresses a number, not an unstateable truth. — Banno

Every property of this unstateable number is itself an unstateable truth.

Example: Number r is a real number.

If number r is unstateable then this sentence is also unstateable, no matter how true this sentence may be.

https://iai.tv/articles/most-truths-cannot-be-expressed-in-language-auid-2335

Most truths cannot be expressed in language

14th December 2022
Noson S. Yanofsky | Professor of computer science at Brooklyn College

There are more true but unprovable, or even able to be expressed, statements than we can possibly imagine, argues Noson S. Yanofsky.

I actually took the example of the subsets of the natural numbers literally from Yanofsky:

http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf

Yanofsky argues that the fact that the sentence is ineffable automatically makes it unprovable. This is indeed the case for an individual sentence. The truth of entire set of sentences, however, is provable.

There are truths about sets of sentences that apply to each individual sentence while we do not have the ability to express by language most of such individual sentences.

I’m thinking Tarskian is trying to tell you that the inexpressible truth is the paradox of a Ricardian number being simultaneously: a member of the natural numbers/not being a member of the natural numbers, as based upon the unresolvable paradox.

If you claim the paradox itself is the statement of the “inexpressible” truth, then you’re trashing the principle of non-contradiction, and your logical systems crash.

Human language is countably infinite because:

its alphabet is finite
every string in human language is of finite length — Tarskian

But it isn't.

But it isn't true.

But it isn't true, manifestly.

But it isn't true, manifestly you can go on forever.

But it isn't true manifestly you can go on forever and ever.

And I told him "But it isn't true, manifestly you can go on forever and ever."

We had a discussion, and I told him ""But it isn't true, manifestly you can go on forever and ever."

etc.

How do you know that what you believe in is true if you can't express it?

...a member of the natural numbers/not being a member of the natural numbers, as based upon the unresolvable paradox. — ucarr

There's noting novel in the natural numbers not being enumerable. What this shows is that the list from which r is derived cannot be constructed.

What I would like is something that shows these unstatable truths to have some sort of significance. Trouble is, if they have significance (note the word), that significance is statable...

That there are unstatable trivialities is not significant.

"The overwhelmingly vast majority of truth cannot be expressed by language" is ambiguous. Is it to be understood, as I think @Tarskian does, as saying that there are true statements that cannot be stated, (a contradiction), or is it to be understood as that while any particular truth can be stated, not every truth can ever be stated, which is a simple consequence of there being transfinite numbers.

Hence my question - give an example of a truth that cannot be stated. "r is a real" is a truth that can be stated.

I suspect this underpins what was said by and . And sets a puzzle to 's restriction on thought - the paradox of being unable to tell us of something that cannot be said.