For example, in what way is the SKI combinator calculus useful or meaningful? It is obviously neither. It is merely "interesting". — alcontali

You mistake the claim that all stipulated axioms and formal systems are useful or arbitrary or relevant in every sense for the much weaker claim that some stipulated axioms and formal systems are useful or arbitrary or relevant in some sense.

I am endorsing the second claim. I believe the second claim entails:

(1) There are justifications for choosing between different stipulated formal systems.
(2) These reasons are not part of any stipulated list of compared formal systems.
(3) and are thereby not algorithmically choose-able.
(4) These justifications are not arbitrary.

Hence, this real-world phenonemon falls outside the realm of what mathematics is supposed to study. — alcontali

In a very trivial sense, yes; the world is not constituted by mathematical symbols or formal systems. In a not so trivial sense, no; some mathematical symbols and formal systems may be used to describe the world or be useful for other purposes.

Hence, this real-world phenonemon falls outside the realm of what mathematics is supposed to study. — alcontali

Read: most of what epistemology studies falls outside the realm of what mathematics (and specifically computability theory) is supposed to study..

You mistake the claim that all stipulated axioms and formal systems are useful or arbitrary or relevant in every sense for the much weaker claim that some stipulated axioms and formal systems are useful or arbitrary or relevant in some sense. — fdrake

Well, no. I do not even care if a formal system is useful or meaningful.

For example, I have just viewed a video that mentions the MU puzzle. I think that the MU puzzle is fantastic. It gave me a kick to investigate it. The MU puzzle is an axiomatization that is purposely useless and meaningless. That is probably one of the reasons why I like it so much.

In my opinion, the reason why mathematics can be very attractive is not because it is useful or meaningful. On the contrary, the more it has real-world semantics, the less it is "beautiful". In my opinion, good math looks a bit absurd.

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

You mistake the claim that all stipulated axioms and formal systems are useful or arbitrary or relevant in every sense for the much weaker claim that some stipulated axioms and formal systems are useful or arbitrary or relevant in some sense. — fdrake

Well, no. I do not even care if a formal system is useful or meaningful. — alcontali

I think you found your own answer, then.

I think you found your own answer, then. — fdrake

The absurd, useless, and meaningless MU puzzle cannot be solved. There is proof for that, i.e. justification. Hence, "The MU puzzle cannot be solved" is a justified (true) belief, i.e. legitimate knowledge.

Furthermore, there exists an entirely mechanical procedure to verify the paperwork for its justification.

Hence, the associated paperwork is both epistemically justified and computably verifiable.

By the way, Wolfram has built a complete demonstration project to illustrate the problem.

Ah yes, the MU puzzle, something which entirely resembles how humans come to conclusions using evidence and argument...

Ah yes, the MU puzzle, something which entirely resembles how humans come to conclusions using evidence and argument... — fdrake

The way in which most humans generally come to conclusions amount to stirring in a pile of total bullshit.

That is why nobody trusts people who cannot understand the proof for the MU puzzle for anything serious. They will go to great lengths to keep them from building a bridge or flying an airplane.

The way in which most humans generally come to conclusions amount to stirring in a pile of total bullshit. — alcontali

Well, no. I do not even care if a formal system is useful or meaningful. — alcontali

So, we shouldn't trust you to know when a formal system is relevant for epistemology or not...

So, we shouldn't trust you to know when a formal system is relevant for epistemology or not... — fdrake

Who is "we"?

The MU puzzle ultimately goes to the core of the epistemology of mathematics.

The MU puzzle is a puzzle stated by Douglas Hofstadter and found in Gödel, Escher, Bach involving a simple formal system called "MIU". Hofstadter's motivation is to contrast reasoning within a formal system (ie., deriving theorems) against reasoning about the formal system itself. MIU is an example of a Post canonical system and can be reformulated as a string rewriting system. — Wikipedia on MU puzzle

Through its stronghold on their language and related invariants, mathematics has a profound influence on science and engineering.

The MU puzzle may be absurd but the proof for the fact that it cannot be solved, is not. That proof is pure knowledge. That is probably why wolfram.com, "Computation meets knowledge" , is also so smitten by it. They know exactly why they say "Computation meets knowledge". As I have said already, if epistemology describes the paperwork requirements for knowledge, then computability describes the procedure to verify that paperwork.

Just look at the quote. "Reasoning within the formal system is much different to reasoning about the formal system itself". You don't even need a formal meta-language to consider differences in axiomatic systems, natural language suffices. This much more general context of natural language and human behaviour is the context in which epistemology resides, not the much more restricted context of formal languages.

There are justifications for choosing some formal systems over others in some circumstances, given a choice between two formal systems the only thing which can facilitate choice between them is embedding them both in a system of comparison exterior to both, be that system not formal (as with natural language), formal, or natural language talking about formal systems both formally (in a formal meta language) and informally (using exterior considerations; intuition, relevance; to guide the formal meta language principles and object language desirable properties).

Most of the history of mathematics, science and engineering proceeded without the idea of a formal system and the arbitrarity of their axioms and inference rules... One wonders how it could possibly be so central in all respects but arrive so late in its history.

You don't even need a formal meta-language to consider differences in axiomatic systems, natural language suffices. — fdrake

Maybe, maybe not.

I like formal metalanguages.

Tarski's convention T is an interesting take on the matter. Tarski does use formal metalanguages in his theory of truth. In fact, he pretty much has to. The metalanguage must be able to express everything the object language says. So, the metalanguage is a superset of the object language. The difference is that the metalanguage can also express statements about the object language.

Of course, this does not necessarily mean that Tarski is the only way to go about the problem. It is just that Tarski's work has left a profound impression on the subject. His fingerprints are all over the place ...

It's difficult to see if you're making an argument or making a series of unconnected statements about formal languages but not about the reduction of epistemology to formal languages, never-mind the reduction of epistemology to effective procedures.

For someone who touts the central role formal systems play in justification your posts don't read like a tightly constructed syllogism.

Maybe, maybe not. — alcontali

Demonstration that meaningful discussions of mathematical concepts can occur solely in natural language:

Alice: "I don't like the Dedekind cut construction of the real numbers from the rationals because it doesn't make completeness of the reals as obvious as the Cauchy sequence construction"
Bob: "Are you sure? The Dedekind cut construction explicitly axiomatises the holes in the real line that the rationals leave."
Alice: "But it leaves the intuitive connection to sequences by the wayside for that purpose, considering that we're teaching sequences and convergence to undergraduates before teaching them about the formal construction of the reals, surely it's better to leverage knowledge we can assume the students have?"
Bob: "The construction of Dedekind cuts only requires that students have intuitions about intervals of rational numbers, not sequences, in essence the knowledge is more elementary..."
Alice: "I guess we can agree how intuitive each is depends on the strengths of the background knowledge of each student."

Broader point: formal systems don't just have syntactic rules, don't just have formal semantics, they also have conceptual content. The conceptual content of mathematical objects and systems is what unites them over the varying degrees of formality of their presentation.

It's difficult to see if you're making an argument or making a series of unconnected statements about formal languages but not about the reduction of epistemology to formal languages — fdrake

I was just replying to something you wrote.

formal systems don't just have syntactic rules, don't just have formal semantics, they also have conceptual content. — fdrake

Not necessarily. For example, the MU puzzle's formal system does not have any conceptual content. Still, it is an important example formal system.

Another problem is that the term "semantics", which is extensively used in model theory, does not really mean "meaning" in the ordinary sense. It rather means "satisfiability". Therefore, a model is just another un-semantical/meaningless formalism. That is good, because the introduction of real semantics in mathematics would be a dangerous thing.

never-mind the reduction of epistemology to effective procedures. — fdrake

That is what it is today already. Verifying the justification's paperwork is a procedure. If there is no procedure possible for that, then the justification is unusable.

Verifying the justification's paperwork is a procedure. If there is no procedure possible for that, then the justification is unusable. — alcontali

I think you're equivocating between:

(1) If someone knows something, they obtained that knowledge through a process they can (at least) partially describe unambiguously in natural language. The description here might be called a procedure.
(2) If someone completely describes an effective procedure in natural language, it can be implemented in a suitable programming language. (Church Turing Thesis)
(3) If someone writes a proof (formal justification in a formal system), it can be represented as a computer program in a model of computer programs and vice versa (Curry Howard Correspondence).

If you accept (2) and (3), it follows that if someone describes an effective procedure in natural language, it can be represented as a proof in a formal language. But they don't have any relevance to (1). IE The claim "knowledge consists only of effective procedures" is completely independent of (2) and (3).

"A process that someone obtains knowledge from that they can at least partially and unambiguously describe" is in no way "a completely described effective procedure" even if you accept (2) and (3).

If accepting/rejecting a hypothesis is not algorithmic, then anybody may accept or reject a hypothesis on merely subjective grounds. — alcontali

Yep.

The question becomes an ought, not an is.

If that is possible, then the hypothesis cannot be sound knowledge. — alcontali

If you mean that science is not certain, then, yes, obviously.

...and it generally does not occur in science either. — alcontali

I've shown that it does.

The existence of such "component of choice" points to the fact that the body of statements, i.e. the discipline, is in fact not legitimate knowledge. — alcontali

Now you are beginning to see the problem. Yes, science is not algorithmic, and hence not certain. It's a human enterprise, subject to all sorts of politics and abuse. None of this should be a surprise. All of this makes careful thinking about scientific issues so much more valuable.

Broader point: formal systems don't just have syntactic rules, don't just have formal semantics, they also have conceptual content. The conceptual content of mathematical objects and systems is what unites them over the varying degrees of formality of their presentation. — fdrake

I agree. What is the flight from conceptual content to a dead machine?

Yes, science is not algorithmic, and hence not certain. It's a human enterprise, subject to all sorts of politics and abuse. — Banno

Here's the answer, a flight from the uncertainty of everything stained by social human being.

There is proof for that, i.e. justification. Hence, "The MU puzzle cannot be solved" is a justified (true) belief, i.e. legitimate knowledge. — alcontali

In general, my problem with prioritizing strictly formal proofs is that we forget that moving from formal proof to the real world is an act of informal interpretation. I don't see how we can get 'behind' the 'throwness' of ordinary language. Yes, we can make games like chess, but the leap from chess kings to the present king of France is something unspecified by the rules of chess. In real language, we can't strictly control the meanings of our signs. They are caught up in history and context.

In general, my problem with prioritizing strictly formal proofs is that we forget that moving from formal proof to the real world is an act of informal interpretation. — mask

Formal proof is never about the real world. Furthermore, mathematics is not directly applicable. It first has to go through a framework of empirical rules and regulations, such a science or engineering. In that sense, there is no act of informal interpretation of mathematics.

Without downstream empirical discipline that regulates the issue of correspondence with the real world, mathematics is simply not applicable.

In real language, we can't strictly control the meanings of our signs. They are caught up in history and context. — mask

Natural language is primarily used for non-knowledge which is the overwhelming majority of what is being expressed. In fact, we do not use that much epistemically-sound knowledge. It is not the main purpose of language (or communication in general) anyway.

Yes, science is not algorithmic, and hence not certain. — Banno

An important part of science can actually be verified mechanically. I propose to reserve the term "scientific" to only that part of science. In other words, there is a lot of non-science deceptively masquerading as science.

Certainty is a type of belief. It is not a type of truth. — Banno

You seem to misunderstand the meaning of "certainty". It is a relationship between belief and truth, not simply a belief.

Conceivably nothing we experience or conclude from our experiences have relations to reality. Conceivably all we experience and all we conclude from our experiences are a direct contact with and opinion about reality.

We call truth a perfect match between experience and opinions about reality.

There is no certainty (see my first paragraph) that reality is what we do or don't experience.

Therefore there is no certainty that we know the truth. We may, or we may not.

The trouble with thinking instead of reading is that you are bound to repeat the errors made by others. — Banno

The same holds true for readers.

Formal proof is never about the real world. Furthermore, mathematics is not directly applicable. It first has to go through a framework of empirical rules and regulations, such a science or engineering. In that sense, there is no act of informal interpretation of mathematics. — alcontali

I think I do understand this idealization of ultra-pure math, which I like in some ways. But if it's just chess, then why should we expect it to matter in the real world?

Natural language is primarily used for non-knowledge which is the overwhelming majority of what is being expressed. In fact, we do not use that much epistemically-sound knowledge. It is not the main purpose of language (or communication in general) anyway. — alcontali

Which supports my point, I think. If 'pure knowledge' is just formalism, how could it be important for us? I've occasionally bumped into people (not you) who think that formal logic can somehow save the world. But real logic (applied logic) is entangled with ordinary language. Ultra pure math is something like language purified of all ambiguity but also therefore any reference to the world we live in.

Its easy enough to confuse the means for the ends when the means posit their own ends.

But if it's just chess, then why should we expect it to matter in the real world? — mask

Short story: it doesn't.

If 'pure knowledge' is just formalism, how could it be important for us? — mask

Long story: Some of it may (unpredictably) meander downstream through the hands of science and engineering. From there on the question becomes: Do science or engineering matter? For both mathematics and science, usefulness is ultimately harnessed by engineering.

Ultra pure math is something like language purified of all ambiguity but also therefore any reference to the world we live in. — mask

Yes, agreed.

Without purification, however, it would be substantially less interesting to use in science or engineering. We also cannot know during the discovery process of mathematics if science or engineering will ever be able to do anything with it. That could take decades, if not, centuries.

That is why I do not particularly like the adjectives "useful" or "meaningful" in mathematics. The ever continuing abstraction process tends to remove both of those. Good mathematics is rather "interesting", "surprising", "beautiful", and/or "intriguing".

Long story: Some of it may (unpredictably) meander downstream through the hands of science and engineering. From there on the question becomes: Do science or engineering matter? For both mathematics and science, usefulness is ultimately harnessed by engineering. — alcontali

I think there's truth in that these days, but we know that historically it was the reverse. Math was purified from its immersion in applications--by Greeks as I understand it.

Without purification, however, it would be substantially less interesting to use in science or engineering. We also cannot know during the discovery process of mathematics if science or engineering will ever be able to do anything with it. That could take decades, if not, centuries. — alcontali

Fair enough. My primary point is that philosophy isn't like pure math and yet is what we have for dealing with the world strategically. Computation only gets us so far.

I think there's truth in that these days, but we know that historically it was the reverse. Math was purified from its immersion in applications--by Greeks as I understand it. — mask

Yes, apparently, Greek geometry originally came over from the Egyptian harvest taxation bureaucracy. Arithmetic came through from harvest inventory accounting clerks. It must have eventually led to memos getting circulated on how to systematize these things.

For computing it was exactly the other way around. The mathematical properties of computation were known at least a decade before they finally managed to build the first computer.

My primary point is that philosophy isn't like pure math and yet is what we have for dealing with the world strategically. — mask

That may be overly ambitious.

Computation only gets us so far. — mask

I think that it can be used to mechanically verify the paperwork that epistemology says must be present.

I don't see how we can get 'behind' the 'throwness' of ordinary language. — mask

There is nothing "behind"; no separation between word and thing. The Map is not the territory, but we can still talk about the territory.

You seem to misunderstand the meaning of "certainty". It is a relationship between belief and truth, not simply a belief. — god must be atheist

How would that work?

Are you claiming that a belief is always a belief that such-and-such is true? That's what I've long claimed.

What more is there to a certainty, that it is not simply a belief?

Are you claiming that a belief is always a belief that such-and-such is true? That's what I've long claimed.

What more is there to a certainty, that it is not simply a belief? — Banno

Think of it this way: the likelyhood that your certainty is right on (ie. that your belief is false, or else that your belief is right on target) is reflected by the degree of certainty. And the degree of certainty can't be established by any means by humans when it comes to KNOWING whether what we sense as reality is itself reality or not.

So what more is there to a degree of certainty: the possibility that our belief is false, or right on, or anywhere in-between.

Are you claiming that a belief is always a belief that such-and-such is true? That's what I've long claimed. — Banno

Actually, that is not what you have always claimed. Here's the proof:

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. — Banno

You are claiming this now, because I convinced you of its truth. I don't know whether to figuratively praise you for learning from me, or else to figuratively deduct points for claiming something that belies your earlier claim.

Are you claiming that a belief is always a belief that such-and-such is true? That's what I've long claimed. — Banno

I just realized that you claimed here a tautology. "A belief is always a belief". It can be followed by "that such-and-such is true" or by "that such-and-such is false", and it will still hold true, as you claim nothing more, that a belief is a belief.

In this very sense, I agree with you too. An apple is an apple, a god is a god, and a belief is a belief. Make no mistake about it.

Are you claiming that a belief is always a belief that such-and-such is true? — Banno

Hehe! I actually never made such a claim, because it would be false. (And yes, you can call me out on that, what with my giving you an understanding I claimed two posts up.)

I can have a belief, that my belief is actually false.

For instance, my belief is that there is no god.

Then I think, maybe there is a god. It does not manifest, but its existence is possible.

Now: do I believe my belief is true, or that my belief is false?

------------

Another example:

I experience the world. My belief is that the world I experience is real.Then I think of solipsism. All of a sudden my belief is that my belief is false.

@god must be atheist

Those last few posts of yours are terrible. They make no sense.