Yes, the theorem itself, as you quoted it, does not mention truth. But from the theorem, we do go on to remark that the undecided sentence is true.

And the statement is neither true or false in the system on an even more fundamental basis than that it is undecided by the system: — TonesInDeepFreeze

There is an even more fundamental reason that the object-theory does not yield a determination of truth. That is that the object-language does not have a truth predicate. — TonesInDeepFreeze

But if we adhere to strict "truth=provability" principle, then the sentence is not true even in the metatheory, if it assigns truth to sentences subject to their provability in the object theory.

When we're talking about plain arithmetical truths, I don't know why we would have to go down the road of wondering about realism. I mean, non-realists still recognize the truth of arithmetical statements. — TonesInDeepFreeze

Anti-realists recognize arithmetical statements as true relative to particular mathematical theories, which are as fictitious as any other such theories. Realists view some mathematical systems, such as arithmetic, as representing an objective, mind-independent reality; for them the mathematical study of such systems can be likened to scientific research.

Again, I want to disclaim that this is a simplistic caricature, but here are some statements in the same spirit by mathematician G.H. Hardy:

Mathematical theorems are true or false; their truth or falsity is absolute and independent of our knowledge of them.

Pure mathematics... seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way. — G.H. Hardy

(Quoted from another work of Torkel Franzen: "Provability and Truth" (1987))

But if we adhere to strict "truth=provability" principle, then the sentence is not true even in the metatheory, if it assigns truth to sentences subject to their provability in the object theory. — SophistiCat

What "truth=provability" principle do you have in mind? What is its mathematical formulation? Meanwhile, the incompleteness theorem proves that the set of provable sentences does not equal the set of true sentences. [Often now, I'll leave tacit the usual qualifiers such as "provable in system S" and "true in the standard model".]

Anti-realists recognize arithmetical statements as true relative to particular mathematical theories — SophistiCat

True relative to models of theories. (Of course though, if P is a theorem of a consistent theory S then P is true in any model of S.)

which are as fictitious as any other such theories — SophistiCat

Not necessarily for arithmetical theories or even the arithmetical part of broader theories.

It's hard to believe that the issue really amount to categorizing mathematics, due to Godel.

Perhaps you could elaborate on computational proof. When I conjecture a theorem in complex analysis I usually turn to the many programs I've written for examples that will either suggest the conjecture is true or abruptly halt the process - if only temporarily - by demonstrating it is false in a particular case.

I don't think this is what you are discussing, however. I'm trying to see the link between actual mathematics and foundational mathematics in this regard.

It depends on the definition of 'mathematically proven'. — TonesInDeepFreeze

So much depends on definitions, it seems.

what you [Pfhorrest] said about there always being a meta-level wherein the unprovable truths within a system can be proven seems questionable. — Janus


It is the case that there is an infinite escalation of theories, each proving arithmetical truthts not provable in the lower theories.

Would this fact render all such proofs non-exhaustive and/ or trivial — Janus


The theories are not exhaustive, indeed. But I don't see why that would make the proofs trivial. — TonesInDeepFreeze

Perhaps, then, it depends on whether we have in mind mathematical or philosophical triviality.

for incompleteness, it's not just a matter of having to assume things to prove things. — TonesInDeepFreeze

Thanks. I have learned from this thread to avoid discussion of this topic in future.

Pure mathematics... seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way. — G.H. Hardy

I don’t see how this can be so. The fact that 317 is a prime number is indeed not dependent on your or my assent, but it’s regardless a fact which only a rational mind can grasp, and in that sense is what can be called an intelligible object or object of rational thought. I don’t see how this undermines idealism but rather reinforces it, in my understanding of that term.

It's a matter of quantifier order:

Godel: For any system S of a certain kind, there exist statements undecided by S.

False: There exist statements F such that for any system S of a certain kind, F is undecided — TonesInDeepFreeze

This is a great way of stating it! Thank you. :-)

I want to rephrase it a little more consistently so the simple shift in quantification order is more apparent to others:

Godel:
for any system S of a certain kind,
there is at least one statement F such that
S cannot decide F.

False:
there is at least one statement F such that
for any system S of a certain kind,
S cannot decide F.

I'm talking about sentences in the language of arithmetic. I don't know whether these matters bear upon your areas of mathematics.

I am pretty rusty on this stuff, so take this modulo a grain of salt:

The Godel sentence G "says":

"For every n, it is not the case that n is the Godel-number of a proof of the sentence with Godel-number g."

G is a sentence purely in the language of arithmetic. The "it says" about proofs and Godel-numbers is seen and proven (in the meta-theory) with regard to the construction of G per the arithmetization of syntax.

And, G has Godel-number g.

The part "it is not the case that n is the Godel-number of a proof of the sentence with Godel-number g" is a computable property. Let's call it 'C'. So G is of the form:

For all n, Cn.

Now, for concision, let's say we're looking at some particular system S.

Godel-Rosser proves "If S is consistent, then both G and ~G are not theorems of S."

And let's say that by 'true' we mean true in the standard model for the language of arithmetic. Godel did not himself have formal model theory to reference, but in context we may say that his context might as well be tantamount to it. Moreover, we could dispense the formality of models by just agreeing that 'true' means what it ordinarily means to mathematicians who don't care about mathematical logic. For example, '0+0=0' is simply true and '0=1' is simply false.

So, either G is true or ~G is true. So, on that basis alone, we know that there is a true sentence that S does not prove. But that is not constructive - it uses excluded middle and doesn't tell us specifically which one of the two is the true one.

But we can constructively (I think?) show "If S is consistent then G is true" anyway.

Now, let's look at a certain kind of arithmetical sentence. These are sentences in the language of arithmetic all of whose quantifiers are bounded. For example:

"For all n<20, if n is prime then n has a twin prime."

For such sentences, there is an algorithm to decide their truth. Moreover, it is said that from the sentence itself, we can "read off" the algorithm (please don't ask me the technical definition of "read off" - I have not yet pursued how to formalize it).

Now, where I tripped myself up earlier in this thread is that I might have conflated the fact in the above paragraph with the fact that we do easily prove "if S is consistent then G is true", as I am not clear whether that proof is one that also is "read off" from the sentence itself, in context of the construction of the sentence vis-a-vis the arithmetization of syntax.

Thanks. I have learned from this thread to avoid discussion of this topic in future. — Wayfarer

Wise decision, the dogmatic don't provide reasonable discourse.

What dogmatism do you think you have witnessed?

What dogmatism do you think you have witnessed? — TonesInDeepFreeze

Aside from whatever SophistiCat might say, it is not the case that formalism regards the incompleteness theorem in that way.

(1) Sentences are not true in a language. They are true or false in a model for a language.

(2) Sentences are not proven just in a language, but rather in a system of axioms and rules of inference.

(3) There are different versions of formalism, and it is not the case that in general formalism regards truth to be just provability.

(4) Godel's theorem in its bare form is about provability and does not need to mention the relationship between truth and proof (though a corollary does pertain to truth). The theorem is about provability, which is syntactical. Even if we had no particular notion of truth in mind, Godels' theorem goes right ahead to show that for systems of a certain kind, there are sentences in the language for the system such that neither the sentence not its negation are provable in the system. — TonesInDeepFreeze

You claim that quote is dogmatic. What is your non-dogmatic basis for that claim?

The quote is not dogmatic, I say non-dogmatically. The quote describes the way mathematical logic uses certain terminology and certain other plain facts about mathematical logic. It is apropos to mention those terminological conventions and basics of mathematical logic, since the context of the discussion is Godel's theorem, which is a subject in mathematical logic. Meanwhile, I allow that anyone is welcome to stipulate their own terminology and even to propose an entirely different framework for consideration of mathematics. However such a proposal would be subject to the same scrutiny for coherence and rigor to which mathematical logic is subject. That is it the antithesis of dogmatism.

"It is not our business to set up prohibitions, but rather to arrive at conventions." - Rudolph Carnap

Metaphysicians are musicians without musical ability. — Another Carnap quote

I hate all those Vienna Circle types. That’s one of the reasons I’m going to keep out of these discussions.

The crank claims that mathematics is wrong. Not just that he proposes different mathematical conventions and definitions, but that the more ordinary conventions and definitions are blatantly wrong onto themselves. The crank cannot understand the stipulative nature of definitions. And the crank claims to find contradiction in mathematics when the crank has only found things that are not actual contradictions but instead are things that happen to be counterintuitive to him. The crank uses sophistry, evasion, raw repetition and ignorance for his position that only his own conception is right and that mathematics is wrong. The crank fancies that he eviscerates mathematics though he does not know even the least of its basics and horribly misconstrues the few bits that he has happened to come across. Thus, the preponderance of the crank's attack is the strawman. The crank is not interested in learning about the subject on which he so tendentiously opines. Instead, he is only interested in announcing his personal truths from the soapbox. The crank never (or virtually never) admits a mistake. All of that is dogmatism.


The logician says that from certain conventions, axioms, rules, and definitions, certain things follow and certain things do not follow. And the logician allows that people may set up different conventions, axioms, rules, and definitions. And the logician might even allow that proposed frameworks may have value even though they have not yet been axiomatized. The logician admits that definitions are stipulative so that definitions themselves are not inherently true, and that we may regard enquiries that proceed with different definitions. The logician seeks scrutiny of his work and is always eager to correct any errors found in his formulations. The logician admits that certain questions are not answered and that there is much still unknown. All of that is the antithesis of dogmatism.

I hate all those Vienna Circle types. That’s one of the reasons I’m going to keep out of these discussions. — Wayfarer

Whatever the merits or demerits of Carnap's views on metaphysics, the quote I mentioned does have wisdom.

And one may have one's own reasons for eschewing a conversation, but having a dislike of certain philosophers is not much of a rational basis for rejecting a conversation about mathematical logic.

https://thephilosophyforum.com/discussion/comment/546980

Okay, I admit, that does read like a 'Goofus and Gallant'.

Ah, you've met @Metaphysician Undercover, then.

Now, let's look at a certain kind of arithmetical sentence. These are sentences in the language of arithmetic all of whose quantifiers are bounded. — TonesInDeepFreeze

Thanks. Certainly in the study of combinatorics there are conjectures in which all possible cases are finite in number and a computer program can do the job. Like the four color problem.

The declared aim of the Vienna Circle was to make philosophy either subservient to or somehow akin to the natural sciences. As Ray Monk says in his superb biography Ludwig Wittgenstein: The Duty of Genius (1990), “the anti-metaphysical stance that united them [was] the basis for a kind of manifesto which was published under the title The Scientific View of the World: The Vienna Circle.” Yet as Wittgenstein himself protested again and again in the Tractatus, the propositions of natural science “have nothing to do with philosophy” (6.53); “Philosophy is not one of the natural sciences” (4.111); “It is not problems of natural science which have to be solved” (6.4312); “even if all possible scientific questions be answered, the problems of life have still not been touched at all” (6.52); “There is indeed the inexpressible. This shows itself; it is the mystical” (6.522). None of these sayings could possibly be interpreted as the views of a man who had renounced metaphysics. The Logical Positivists of the Vienna Circle had got Wittgenstein wrong, and in so doing had discredited themselves. — Wittgenstein, Tolstoy and the Folly of the Logical Positivists

And one may have one's own reasons for eschewing a conversation — TonesInDeepFreeze

What I meant was, I will henceforth refrain from invoking Godel’s theorems to make philosophical claims.

Meanwhile, I allow that anyone is welcome to stipulate their own terminology and even to propose an entirely different framework for consideration of mathematics. — TonesInDeepFreeze

When you reject such, and insist on the other, it's dogmaticism.

When you reject such, and insist on the other, it's dogmaticism. — Metaphysician Undercover

To what do 'such' and 'other' refer?

Let me explain it clearly then, since you seem to be having trouble understanding. When someone accepts, believes in, and adheres to principles which have not been empirically proven and argues a philosophy which gives the highest esteem to such principles, that is dogmatism. Many principles employed in modern mathematics, axioms, have not been empirically proven. So to believe strongly in, and argue from such principles, even labeling those who doubt these unproven principles as cranks, is dogmatism.

Let me explain it clearly then, since you seem to be having trouble understanding. — Metaphysician Undercover

That's condescension coming from a person who can least afford it.

argues a philosophy — Metaphysician Undercover

I haven't argued a philosophy.

Many principles employed in modern mathematics, axioms, have not been empirically proven. — Metaphysician Undercover

It's dogmatic of you to preclude that interest in abstract mathematics must be dogmatism.

And I have not claimed that abstract mathematics has the kind of direct empirical correspondence that you dogmatically require. However, I do observe that it is used for, and has been a crucible for, the sciences and for the very technology you are using to be a condescending boor.

Moreover, whatever one's regard for mathematics, it is not dogmatism to point out what its actual formulations are as opposed to dogmatic attacking ignorance and misconstrual, such as yours, of the formulations.

labeling those who doubt these unproven principles as cranks — Metaphysician Undercover

I have never faulted anyone for doubts about axioms or abstract mathematics. Indeed, the literature of debate regarding doubts and criticisms of various mathematical approaches fascinates and excites me and has my admiration. What I have done though is point out when people blindly attack mathematics from ignorance, confusion, stubbornness and dogmatism. There is a Grand Canyon of difference between, on one hand, doubts and reasoned critique and, on the other hand, attacks from willful ignorance, frothing confusion, and sophomoric dogmatism.

That's condescension coming from a person who can least afford it. — TonesInDeepFreeze

Oh no, @Metaphysician Undercover has plenty of that.

He doesn't have the actual superiority to spend. He's in overdraft with just that one pathetic attempt.

Many principles employed in modern mathematics, axioms, have not been empirically proven. — Metaphysician Undercover

Mathematics is true a priori and so empirical validation isn't relevant.

There is a Grand Canyon of difference between, on one hand, doubts and reasoned critique and, on the other hand, attacks from willful ignorance, frothing confusion, and sophomoric dogmatism. — TonesInDeepFreeze

This is true. That is why in such matters, circumspection might often be called for.

I haven't argued a philosophy. — TonesInDeepFreeze

Take a look at my quote above, and the context from where it's taken. You are arguing a philosophy of truth.

empirical validation isn't relevant. — Wayfarer

That's exactly what makes arguing for mathematics as the purveyor of truth, dogmatism.

You are arguing a philosophy of truth. — Metaphysician Undercover

I made no argument for a philosophy regarding truth.

empirical validation isn't relevant.
— Wayfarer

That's exactly what makes arguing for mathematics as the purveyor of truth, dogmatism. — Metaphysician Undercover

And I didn't argue that mathematics is a purveyor truth known a priori and that empirical concerns are not relevant.

Your claim that I am dogmatic is unsupported.