Mathematics and its proofs are presented as a priori truths.

Physical observation and experimentation is empirical.

Therefore you're right, when you say computability and epistemology are different. One depends on a priori findings of truth, the other, empirical ways o finding the truth.

Even the truths of the two systems are different. In the empirical world, there are no truths. Only approximations. In the a priori world, the truths are perfect.

Mathematics and its proofs are presented as a priori truths. Physical observation and experimentation is empirical. — god must be atheist

Both mathematics and science use their procedure to justify their knowledge. So, in both cases, it is about following the correct procedure. In that sense, in both cases, knowledge is justified by formalisms.

I wasn't looking at the difference between mathematics and science in this case.

I was rather interested in what they have in common: they both have a large bureaucracy of procedures.

So, I believe that the core of knowledge-justification always consists of "paperwork", regardless of what knowledge it is about.

Both mathematics and science use their procedure to justify their knowledge. So, in both cases, it is about following the correct procedure. In that sense, in both cases, knowledge is justified by formalisms. — alcontali

It's like saying, "A duck lays an egg, a penquin lays an egg, therefore both duck and penguin are chicken."

Aside from that, science does not follow a formalism.

Other than that, you are spot on correct.

I wasn't looking at the difference between mathematics and science in this case.

I was rather interested in what they have in common — alcontali

You could have fooled me with this EARLIER statement by you:

I wonder if computability and epistemology are ultimately not one and the same thing? — alcontali

(My answer to the above was clearly "not", as you know now.)

So, I believe that the core of knowledge-justification always consists of "paperwork", regardless of what knowledge it is about. — alcontali

If you believe that, nobody can sway you from it. Belief defies everything. Some people believe in the Easter Bunny; some, in Jesus the Christ; some, that the empirical world is actual reality; some that everything that has any resemblance to anything else are equal to each other.

(My answer to the above was clearly "not", as you know now.) — god must be atheist

Yes, in the current state of affairs, they are not.

The common factor in the terms "scientific method", "axiomatic method", and "historical method", is clearly the term "method", which is a synonym for "procedure".

The core of their epistemology is their "method", i.e. their procedures. So, knowledge seems to be justified by paperwork procedures ...

Right you are. You seemed to have discovered that methods exist for each discipline. But that does not reduce them to "paperwork".

What about a computation that results in a number with an infinite number of decimal values? It seems to me that word-use along with number-use are approximations. Unless there is some new mathematical discovery, our binary symbols (words and numbers) can only approximate the analog world.

Knowledge is an approximation - using existing rules to interpret current sensory data. Our rules are our justifications. We have a rule that water on the window is an indication that it is raining. Having experienced water on the window along with the state-of-affairs it raining numerous times is justification that it is raining. If you only experienced rain once with water on the window being the indicator, you don't have justification, or a rule. Justification/Rules comes with experience.

a problem is solvable if there exists an effective procedure for deriving the correct answer. — alcontali

It is obvious that a claim can only have the status of knowledge if there somehow exists an effective procedure to verify its justification. — alcontali

Small point, maybe not relevant: getting a correct answer and verifying that it is correct are different procedures. This an important part of the discussion of NP-complete problems. For problems, like the traveling salesman problem, that are computationally formidable, an (in theory) approach is to guess an answer, and then check to see if it is correct, the idea being that verification is generally quick and simple. Lots of online discussions of NP-completeness.

(And it seems to me that at least with the traveling salesman, verification would be as difficult a problem as solving it in the first place.)

In mathematics, a legitimate knowledge claim implies the existence of an effective proof procedure for the claim. — alcontali

And yet we know of unprovable truths.

Epistemology is broader than computability.

Even the truths of the two systems are different. In the empirical world, there are no truths. Only approximations. In the a priori world, the truths are perfect. — god must be atheist

Perhaps you confuse being true with being justified. There are obvious empirical truths - such as that you are reading this post.

Justification/Rules comes with experience. — Harry Hindu

I certainly agree that this is the case in the empirical domain. Science certainly works like that, even though mere experience is clearly not enough as a justification. In addition, such justification will still have to satisfy the entire framework of regulations of the scientific method, i.e. paperwork.

On the other hand, justification in the axiomatic domain does not require experience. It is based solely on provability, which is different kind of paperwork.

Small point, maybe not relevant: getting a correct answer and verifying that it is correct are different procedures. — tim wood

Agreed. Most mathematicians assume that P is not NP (without proof, though).

And yet we know of unprovable truths. — Banno

Only if the theory is consistent.

If it is possible to prove that such theory is consistent, then it is necessarily inconsistent (second incompleteness theorem).

So, the first incompleteness is fundamentally ambiguous: a first-order theory resting on enough arithmetic is inconsistent and/or its model contains at least one unprovable truth.

Epistemology is broader than computability. — Banno

At first glance, yes. That is indeed my first impression too.

However, in practical terms, effective justification will always get translated into a paperwork procedure. At that point, it simply degenerates into computability. If it is not possible to create paperwork for the justification, then in all practical terms, there will be no justification. So, if epistemology does not equate computability, it will actually not work.

Only if the theory is consistent.

If it is possible to prove that such theory is consistent, then it is necessarily inconsistent (second incompleteness theorem). — alcontali

Did you mean "it is necessarily incomplete"?

Did you mean "it is necessarily incomplete"? — Banno

It is inconsistent and/or incomplete.
Therefore, it could also be complete and inconsistent.
Hence, there may not be an unprovable truth as long as the theory is also inconsistent.

Sure. If it is inconsistent, it explodes, and hence anything follows.

So what?

In mathematics, a legitimate knowledge claim implies the existence of an effective proof procedure for the claim.
— alcontali

And yet we know of unprovable truths. — Banno

If the theory is consistent, it contains unprovable truths. If, as you say, "a legitimate knowledge claim implies the existence of an effective proof", then this could not be.

So what? — Banno

The theory needs to be consistent to be usable, but you are not allowed to prove that it is, because in that case this theory is provably inconsistent.

So, yes, we really need consistency but we are also not allowed to prove it.

If the theory is consistent, it contains unprovable truths. If, as you say, "a legitimate knowledge claim implies the existence of an effective proof", then this could not be. — Banno

Yes, and that is a problem.

Still, the incompleteness theorem does not give you access to such unprovable truth.

The theory can obviously not prove of any particular truth that it is unprovable.

The incompleteness theorem only vaguely and very ambiguously says that such elusive, unprovable truth exists, without telling you what exactly it is, and only on the condition that the theory is not inconsistent, but that is in turn something which you are not allowed to prove.

In fact, if you omit any of the material or formal conditions for the incompleteness theorem, you will effectively be asserting a falsehood. It will just not be true.

For example, "I know for sure that there are unprovable truths" (P) implies "I know for sure that the theory is consistent" (Q), and is therefore a falsehood.

P --> false
Q --> false
~Q or P --> true
Q => P --> true

So, the correct way of stating the incompleteness theorem is:

"If the theory is consistent then its model contains unprovable truths" or
"The theory is inconsistent or its model contains unprovable truths"

In fact, this follows from Carnap's diagonal lemma. There exists a logic sentence s as such that:

s <--> isNotProvable(%s)

Which we can write as:

(~s and ~isNotProvable(%s)) OR (s and isNotProvable(%s))
---------------------------- (A) OR --------------------- (B)

If you look at (A), it says (~s and isProvable(%s)). So, it says that "s is false and s is provable", which means that the theory is inconsistent, because it proves a falsehood.

If you look at (B), it says (s and isNotProvable(%s)). So, it says that "s is true and s is not provable", which means that the theory contains an unprovable truth.

Note that the diagonal lemma does not tell you what s is. It only says that there exists at least one s in A or in B. Hence, you still don't know what s is. The lemma itself cannot tell you what it is.

Imagine that you can prove ~A. So, you can prove that the theory is consistent. In that case, B is provable. So, it means:

isProvable(B)

= isProvable(s and isNotProvable(%s))
= isProvable(%s) and isProvable(isNotProvable(%s))
= isProvable(%s) and isNotProvable(%s)

That is a contradiction.

So, if B is provable, then the theory contains a contradiction. So, in that case, it is inconsistent. Therefore, you are not allowed to prove ~A (="the theory is consistent"). If you can do that, then the theory is automatically inconsistent.

The theory needs to be consistent to be usable, but you are not allowed to prove that it is, because in that case this theory is provably inconsistent. — alcontali

If it is consistent, then it is inconsistent?

No. If it is consistent, then it is incomplete. If it is complete, then it is inconsistent.

If it is consistent, then it is inconsistent? — Banno

If it is provably consistent, then it is inconsistent.
Yes. True.

Where is that from?

Where is that from? — Banno

That is known as Gödel's second incompleteness theorem.

Gödel's second incompleteness theorem states no consistent axiomatic system which includes Peano arithmetic can prove its own consistency. Stated more colloquially, any formal system that is interesting enough to formulate its own consistency can prove its own consistency iff it is inconsistent. — Wolfram

I mentioned the proof strategy for this (which is actually trivial) in a previous answer.

OK - that's new to me.

OK - that's new to me. — Banno

Starting from Carnap's diagonal lemma, the proof strategy for Gödel's first incompleteness theorem is really easy. Once you have established the first incompleteness theorem, the second one follows almost trivially.

In fact, it is the same for Tarski's undefinability theorem. It also follows almost trivially from Carnap's diagonal lemma.

Therefore, the hard part is the proof for Carnap's diagonal lemma. That is where all the meat is ...

I certainly agree that this is the case in the empirical domain. Science certainly works like that, even though mere experience is clearly not enough as a justification. In addition, such justification will still have to satisfy the entire framework of regulations of the scientific method, i.e. paperwork.

On the other hand, justification in the axiomatic domain does not require experience. It is based solely on provability, which is different kind of paperwork. — alcontali

Provability requires observation. Axioms take some kind of form in the mind, or else how do you know you have them? The forms they take are the forms you have observed.

Were you just born knowing 3 + 0 = 3, or did you have to observe anything to acquire this "axiomatic" knowledge? "Axiomatic" knowledge without any reference to the real world is useless. When untethered from the what we observe of the world, our knowledge is meaningless. What you call "axiomatic" knowledge is really just the rule we learned by observing the world. Some people have an issue with distinguishing between following/breaking a rule with the rule itself. Rules are meaningless without a world in which they are followed or broken.

"Axiomatic" knowledge without any reference to the real world is useless. When untethered from the what we observe of the world, our knowledge is meaningless. What you call "axiomatic" knowledge is really just the rule we learned by observing the world. Some people have an issue with distinguishing between following/breaking a rule with the rule itself. Rules are meaningless without a world in which they are followed or broken. — Harry Hindu

If you come from the world of science, which is staunchly empirical, you will naturally tend to think that mathematics should be a bit like science and primarily deal with the physical universe. I can imagine that mechanical, construction, -or chemical engineers will also naturally be attracted to an empirical-constructivist take on mathematics.

If you come from the world of programming and its theoretical approach, i.e. computer science, you will not think like that. In that case, you are already used to high levels of meaningless and useless Platonicity. You should be quite used already to high-level structures that are fundamentally divorced from the senses.

Look for example at this example: AbstractObjectFactory. It is a structure-defining absurdity. To what could that structural abstraction possibly correspond in the physical universe? In fact, this source code does not even "do" anything, which is unusual for software, because the idea is that it would otherwise execute some code, but it doesn't even do that.

Absurdity is what naturally emerges out of lengthy abstraction processes. You obtain structures that mean nothing and that are essentially useless. So, yes, high-level abstract structures are naturally useless and meaningless. I am used to that. It is my professional life to deal with that kind of things. That is probably why I can appreciate the beauty of general abstract nonsense, the flagship of mathematics.

Total nonsense can be breathtakingly beautiful as long as it is consistent. It is mostly a question of developing enough intuition for that. Seriously, structural nonsense can even be pleasant to look at.

And yet we know of unprovable truths. — Banno

Another small point. Godel's aren't unprovable; instead they're provable, which is exactly the point and significance of them. It's the what-it-takes of the proofs that's the matter of interest.

The central concept in computability is that a problem is solvable if there exists an effective procedure for deriving the correct answer. That is pretty much what the Church-Turing thesis says.
...
I wonder if computability and epistemology are ultimately not one and the same thing? — alcontali

I think so. The Church–Turing–Deutsch thesis takes this a step further and states that the universe itself is mathematically isomorphic to a quantum Turing machine.

... a modification of Turing's assumptions does bring practical computation within Turing's limits; as David Deutsch puts it:

"I can now state the physical version of the Church–Turing principle: 'Every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means.' This formulation is both better defined and more physical than Turing's own way of expressing it." — The Church–Turing–Deutsch thesis

The stance that 'knowledge consists of instructions and the ability to follow them' is the epistemological philosophy known as constructivism. In my view this is misguided, because

i) Most of our knowledge and inferences consist of analogies and analogy-making, rather than consisting of recipes and the ability to follow them. Indeed, the Church-Turing thesis is purely the expression of an analogy between mathematical formalism and practical rule-following by humans.

ii) Constructivism cannot be self-justifying without pain of infinite regress; so-called 'constructive' reasoning actually consists of implicit analogical inferences expressed as axioms that lack further constructive justification or explication.

iii) In practice most of our so-called 'constructive' inferences are outsourced to external oracles we call 'calculating devices'. But unless one holds strongly realist beliefs in causality and identifies the logical description of a machine as an expression of a physical hypothesis, the output of a calculator cannot be said to be 'constructed' from it's inputs. Indeed, a central function of logic is to be able to describe the world in an agnostic fashion without committing to speculative physical theories.

Classical Logic together with Model-Theory and the Axiom of Choice accommodates our analogical leaps of faith known as "truth by correspondence" that stem from our non-deterministic interactions with nature better than intuitionistic logic, since the latter is purely the expression of games of solitaire played according to known rules.

If you come from the world of science, which is staunchly empirical, you will naturally tend to think that mathematics should be a bit like science and primarily deal with the physical universe. I can imagine that mechanical, construction, -or chemical engineers will also naturally be attracted to an empirical-constructivist take on mathematics.

If you come from the world of programming and its theoretical approach, i.e. computer science, you will not think like that. In that case, you are already used to high levels of meaningless and useless Platonicity. You should be quite used already to high-level structures that are fundamentally divorced from the senses.

Look for example at this example: AbstractObjectFactory. It is a structure-defining absurdity. To what could that structural abstraction possibly correspond in the physical universe? In fact, this source code does not even "do" anything, which is unusual for software, because the idea is that it would otherwise execute some code, but it doesn't even do that.

Absurdity is what naturally emerges out of lengthy abstraction processes. You obtain structures that mean nothing and that are essentially useless. So, yes, high-level abstract structures are naturally useless and meaningless. I am used to that. It is my professional life to deal with that kind of things. That is probably why I can appreciate the beauty of general abstract nonsense, the flagship of mathematics.

Total nonsense can be breathtakingly beautiful as long as it is consistent. It is mostly a question of developing enough intuition for that. Seriously, structural nonsense can even be pleasant to look at. — alcontali

I come from "both" (they are not separate) the "world" of science and of programming (computer science). Programming is useless until you put the program into a computer to be executed. Before that, it is simply a list of rules to follow independent of any rule-follower. There are even rules to writing a program in a certain computer language. Those rules are meaningless until you follow them in writing a program. In other words, rules without any causal relationship are meaningless. Rules without the reason to have those rules in the first place is meaningless.

It is illogical to severe empiricism from rationalism, or to think of them as opposing views. Making an observation entails using your eyes and brain - making sense of what it is that you are looking at. It is one process, not two separate ones that can be done without the other.

Like I said, you weren't born knowing 3+0=3 because you needed to observe this rule in order to know there is a rule and then observe how such a rule is useful in the world. The rule itself stems from our own observations of individual things and the need to quantify those individual things that share similarities. So these "axiomatic" domains themselves require at least two observations - one to learn the rule and the other to learn what the rule is for.

Look for example at this example: AbstractObjectFactory. It is a structure-defining absurdity. To what could that structural abstraction possibly correspond in the physical universe? In fact, this source code does not even "do" anything, which is unusual for software, because the idea is that it would otherwise execute some code, but it doesn't even do that. — alcontali

I other words, it doesn't qualify as software. If it doesn't execute, or do anything, then the programmer didn't follow the rules for writing a program in that particular language. It's merely observable scribbles on a screen.

I wonder if computability and epistemology are ultimately not one and the same thing?

Once the axioms have been postulated it is all about mechanically following the rules and procedures. But what about ontological axioms based on intuition, self-evident, when some things just make innate, unexplainable yet somehow still logical sense.

It seems your question is actually asking whether consciousness is a computable function able to produce results such as imagination and intuition. And if not, then they are not one and the same, but where exactly is the difference amounts to what is called ‘the hard problem of consciousness’.

In other words, how would you arrive by computation to possibly the only certain epistemological and ontological true statement: “I think, therefore I know I exist”?

If it is consistent, then it is incomplete. If it is complete, then it is inconsistent.

Is consistent and incomplete supposed to be any better than inconsistent and complete? They look kind of the same to me, like partial truth is still a lie, in a sense that it can misguide you just the same.

But is there a reason we should think incompleteness theorem actually applies to anything but a bunch of narrow and specific mathematical abstractions?