Yeah, cheers. What bothers me is that it is clear from my notes that it was something I had forgotten.

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

That strikes me as over reach. How is "the cat is on the mat" computable, that we might believe, or even know, that it is true?

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. — Zelebg

Being inconsistent allows a system to prove anything: (p & ~p) > q; that's not very helpful. Hence, there is a natural preference for being consistent and incomplete.

I can't spell the danged thing, but I know what it is. — god must be atheist

Yeah, cool. I know what solipsism is, and can spell it, too. It should be rejected on the grounds that the level of doubt required exceeds what is reasonable; an argument found in various places, but perhaps best articulated in On Certainty.

You might put it in your reading list.

You can't establish any degree of certainty on solipsism vs. accepting that what you experience is actually the physical world.

For your information, I don't read. I think. You should try that too, sometimes.

That strikes me as over reach. How is "the cat is on the mat" computable, that we might believe, or even know, that it is true? — Banno

Is "the cat is on the mat" formally justifiable (=epistemology)?

If it is, there is a formal justification procedure to produce that justification. In that case, carrying out such formal justification procedure is a matter of computability.

If it is not, then neither the justification exists nor any need to carry out any procedure to produce it.

Justification is the paperwork while computability is the procedure to fill out the paperwork.

You can't establish any degree of certainty on solipsism vs. accepting that what you experience is actually the physical world. — god must be atheist

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

One can believe, and even be certain, of whatever one wants.

Hence to say that no one can be certain of such-and-such is to misunderstand what certainty is.



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

Is "the cat is on the mat" formally justifiable (=epistemology)? — alcontali

So the word "formally" bugs me. What precisely is the difference between a formal justification and any other justification? Moreover, does an insistence on formal justification simple rule out empirical justification?

So the word "formally" bugs me. What precisely is the difference between a formal justification and any other justification? — Banno

In this context, it just means "objectively verifiable", which automatically implies that a procedure to carry out such verification can be documented.

Moreover, does an insistence on formal justification simple rule out empirical justification? — Banno

Certainly not in science. The idea is rather that repeating the experiment should be straightforward, which implies that it must be possible to document a mechanical procedure for that.

On the other hand, if there is no mechanical verification procedure possible, then any such justification cannot be objective either.

it just means "objectively verifiable" — alcontali

I had understood that what is to count as "objectively verifiable" is itself one of the main issues in epistemology. When ought one believe such-and-such?

I had understood that what is to count as "objectively verifiable" is itself one of the main issues in epistemology. — Banno

It depends on the knowledge-justification method. Mathematical justification ("provability") is eminently and even mechanically verifiable. Scientific justification ("falsifiability") is also verifiable, even though it requires repeating the experimental tests, often manually or at least partially so.

So, mathematics has epistemic paperwork ("proof") and so does science ("experimental test report"). Verifying that paperwork is a procedure. Hence, it is fundamentally a question of computability.

Other disciplines may not produce mechanically verifiable paperwork with their knowledge-justification method. Those disciplines can therefore be considered epistemically unsound.

When ought one believe such-and-such? — Banno

In the context of a sound knowledge-justification method, there is no need to believe any particular person. Only the result of the mechanical justification-verification procedure matters. In other words, if it matter who says it, then what he says, does not matter.

Scientific justification ("falsifiability") is also verifiable, — alcontali

But... verifiable is exactly what a falsifiable hypothesis is not.

But... verifiable is exactly what a falsifiable hypothesis is not. — Banno

This is not about verificationism. We are not trying to verify the claim itself. We are trying to verify its paperwork.

A claim is justified if the required paperwork, i.e. the justification, is attached to the claim. From there on, we merely verify that the paperwork satisfies the epistemic regulations for the claim.

For a scientific claim, it means that the paperwork in annex contains a reproducible experimental test report, meaning that the claim is indeed falsifiable. We must indeed verify the claim's falsifiability. This is obviously not the same as verifying the claim itself.

Ok. I do not agree that the scientific process is algorithmic in the way you describe; nor, even, that it ought be.

The first point is about the history of science; and I would point to, say, Feyerbend as showing how science is a human, indeed a political process.

The second point is logical. That a proposition is falsifiable is not the same as it's being true; and hence, there will be verifiably falsifiable propositions that are false, yet unfalsified.

I do not agree that the scientific process is algorithmic in the way you describe; nor, even, that it ought be. — Banno

If scientific evidence -- represented by its paperwork -- is objective then there exists a mechanical procedure to verify such paperwork. One step in this procedure must indeed consist in repeating its experimental test. Even though that step is necessarily a physical activity, there must also exist a procedure for carrying it out. Hence, the verification of the paperwork is entirely objective, deterministic, and procedural. Otherwise, it is not even legitimate scientific evidence.

Verifying the legitimacy of scientific evidence is therefore a computability problem.

The first point is about the history of science; and I would point to, say, Feyerbend as showing how science is a human, indeed a political process. — Banno

In his books Against Method and Science in a Free Society Feyerabend defended the idea that there are no methodological rules which are always used by scientists. He objected to any single prescriptive scientific method on the grounds that any such method would limit the activities of scientists, and hence restrict scientific progress. In his view, science would benefit most from a "dose" of theoretical anarchism.

Feyerabend was also critical of falsificationism. He argued that no interesting theory is ever consistent with all the relevant facts. This would rule out using a naïve falsificationist rule which says that scientific theories should be rejected if they do not agree with known facts. — Wikipedia on Feyerabend

In my opinion, Feyerabend's epistemological anarchism is a dangerous point of view. It would prevent us from determining whether a proposition is scientific or not, because there would no longer exist a benchmark for that. Hence, it is his approach that would restrict scientific progress, simply, by removing the restrictions on the progress of snake oil. Feyerabend's view on science is a dangerous throwback in time because it reopens the door for accepting mere alchemy as science.

In my opinion, Feyerabend's epistemological anarchism is a dangerous point of view. It would prevent us from determining whether a proposition is scientific or not, because there would no longer exist a benchmark for that. Hence, it is his approach that would restrict scientific progress, simply, by removing the restrictions on the progress of snake oil. Feyerabend's view on science is a dangerous throwback in time because it reopens the door for accepting mere alchemy as science. — alcontali

Sure. I agree.

But that's just to say that the consequences are challenging; none of this discounts what Feyerabend says.

You havn't shown that he is wrong.

If scientific evidence -- represented by its paperwork -- is objective then there exists a mechanical procedure to verify such paperwork. One step in this procedure must indeed consist in repeating its experimental test. Even though that step is necessarily a physical activity, there must also exist a procedure for carrying it out. Hence, the verification of the paperwork is entirely objective, deterministic, and procedural. Otherwise, it is not even legitimate scientific evidence. — alcontali

One step.

Anotehr step involves one accepting the observations of the verification. That is, forming a belief. But various folk - from Feyerabend to Quine and Duhem, have shown that there is a component of choice involved in accepting any hypothesis.

That is, the process is not algorithmic.

And that to me undermines your enterprise in this thread.

We no longer follow visual procedures in mathematics. — alcontali

Not true. For example, I just posted a research note in which I gave what is primarily a geometric (visual) argument that the iteration of a linear fractional transformation form converges to a limit for a portion of the complex plane.

The second point is logical. That a proposition is falsifiable is not the same as it's being true; and hence, there will be verifiably falsifiable propositions that are false, yet unfalsified. — Banno

You might want to elaborate with an example. In appearance, it looks like word salad. I would think that if a proposition is falsifiable it is not the same as it being false. In other words there is a procedure for determining falseness, but it hasn't been applied yet. I haven't been following the thread, however, and must be missing a technical definition. Confusing. :worry:

Falsification was first developed by Karl Popper in the 1930s. Popper noticed that two types of statements are of particular value to scientists. The first are statements of observations, such as 'this is a white swan'. Logicians call these statements singular existential statements, since they assert the existence of some particular thing. They can be parsed in the form: there is an x which is a swan and is white.

The second type of statement of interest to scientists categorizes all instances of something, for example 'all swans are white'. Logicians call these statements universal. They are usually parsed in the form for all x, if x is a swan then x is white.

Scientific laws are commonly supposed to be of this form. Perhaps the most difficult question in the methodology of science is: how does one move from observations to laws? How can one validly infer a universal statement from any number of existential statements?

Inductivist methodology supposed that one can somehow move from a series of singular existential statements to a universal statement. That is, that one can move from ‘this is a white swan', “that is a white swan”, and so on, to a universal statement such as 'all swans are white'. This method is clearly logically invalid, since it is always possible that there may be a non-white swan that has somehow avoided observation. Yet some philosophers of science claim that science is based on such an inductive method.

Popper held that science could not be grounded on such an invalid inference. He proposed falsification as a solution to the problem of induction. Popper noticed that although a singular existential statement such as 'there is a white swan' cannot be used to affirm a universal statement, it can be used to show that one is false: the singular existential statement 'there is a black swan' serves to show that the universal statement 'all swans are white' is false, by modus tollens. 'There is a black swan' implies 'there is a non-white swan' which in turn implies 'there is something which is a swan and which is not white'.

Although the logic of naïve falsification is valid, it is rather limited. Popper drew attention to these limitations in The Logic of Scientific Discovery, in response to anticipated criticism from Duhem and Carnap. W. V. Quine is also well-known for his observation in his influential essay, "Two Dogmas of Empiricism" (which is reprinted in From a Logical Point of View), that nearly any statement can be made to fit with the data, so long as one makes the requisite "compensatory adjustments." In order to falsify a universal, one must find a true falsifying singular statement. But Popper pointed out that it is always possible to change the universal statement or the existential statement so that falsification does not occur. On hearing that a black swan has been observed in Australia, one might introduce ad hoc hypothesis, 'all swans are white except those found in Australia'; or one might adopt a skeptical attitude towards the observer, 'Australian ornithologists are incompetent'. As Popper put it, a decision is required on the part of the scientist to accept or reject the statements that go to make up a theory or that might falsify it. At some point, the weight of the ad hoc hypotheses and disregarded falsifying observations will become so great that it becomes unreasonable to support the theory any longer, and a decision will be made to reject it.

In place of naïve falsification, Popper envisioned science as evolving by the successive rejection of falsified theories,rather than falsified statements. Falsified theories are replaced by theories of greater explanatory power. Aristotelian mechanics explained observations of objects in everyday situations, but was falsified by Galileo’s experiments, and replaced by Newtonian mechanics. Newtonian mechanics extended the reach of the theory to the movement of the planets and the mechanics of gasses, but in its turn was falsified by the Michelson-Morley experiment and replaced by special relativity. At each stage, a new theory was accepted that had greater explanatory power, and as a result provided greater opportunity for its own falsification.

Naïve falsificationism is an unsuccessful attempt to proscribe a rationally unavoidable method for science. Falsificationism proper on the other hand is a prescription of a way in which scientists ought to behave as a matter of choice. Both can be seen as attempts to show that science has a special status because of the method that it employs.

Falsified theories are replaced by theories of greater explanatory power. — Banno

The replacement of a theory due to counterexample (original theory wrong), vs the replacement of a theory due to the development of a better, more encompassing theory (original theory correct, but superseded). Business as usual in science.

You make the distinction between theory and statement.

You make the distinction between theory and statement. — jgill

Yep.

there is a component of choice involved in accepting any hypothesis ... That is, the process is not algorithmic. — Banno

If accepting/rejecting a hypothesis is not algorithmic, then anybody may accept or reject a hypothesis on merely subjective grounds. If that is possible, then the hypothesis cannot be sound knowledge. Furthermore, this situation does not occur in mathematics and it generally does not occur in science either. It may occur in other academic disciplines, but then the question becomes: Are these fields even legitimate knowledge?

For example, for over 70 years, there were two versions of economics, one of which was the Soviet one. If that situation is possible, then the question becomes: Is economics actually legitimate knowledge? At the same time there was clearly no separate Soviet version of mathematics nor of science.

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.

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

How do you decide what goes into an "axiom pack"? @alcontali

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. — Harry Hindu

Pardon me for interjecting HH, but I thought children could perform [a priori] mathematical abstracts/computation with little empirical observation? Meaning, we have the innate ability to comprehend abstract's without experiencing anything they relate to in the world.

I thought then,' learning the rule' a priori is mutually exclusive.

How do you decide what goes into an "axiom pack"? — fdrake

You can create absolutely arbitrary axiom packs and use those instead. There is nothing wrong with that.

If the language in which it is expressed is Turing-Complete, then you can use it to describe any computable procedure. For example, you can perfectly-well load the language+axiom pack of the SKI combinator calculus instead of PA, if what you want to do, are algorithms.

PA is just a very standard pack of axioms. Same for ZFC. There are obviously alternative number and set theories, i.e. alternative axiom packs. There is not necessarily anything wrong with those. I guess that the standard packs may allow for deriving "more interesting" theorems than other axiom packs.

Furthermore, the reason why we often (but not always) use number and/or set theory cannot be explained from within mathematics.

Aah, so I can stipulate {alcontail is wrong about the significance of axioms to justifications in natural language} and derive that and have it be true because axioms are arbitrarily stipulated and nothing more can be said. Right?

It's even computably verified, just restate the axiom.

Aah, so I can stipulate {alacontail is wrong about the significance of axioms to justifications in natural language} and derive that and have it be true because axioms arbitrarily stipulated and nothing more can be said. Right? — fdrake

It will obviously be true within the model that satisfies your axiomatization. This is never the physical universe, since your axiomatic theory is not the theory of everything.

This is generally like that.

For example, none of the models for PA is the physical universe. Therefore, not one statement that PA proves, necessarily says anything about the physical universe. PA only proves statements that are true in its models.

It will obviously be true within the model that satisfies your axiomatization. — alcontali

Unless I interpret the statement as false and study the consequences. :chin:

But yes, there is a component of choice in setting up any formal system; like a Turing machine. Therefore, if you're right that:

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

all derived theories from the formal specification of computability are not legitimate knowledge.

EG: We can stipulate that we could equip a Turing machine with an oracle (a black box that allows you to output the correct result from whatever procedure you specify, even one which is undecidable) and derive another concept. This produces a useful theory in studying decision problems.

But why accept Turing machines without oracles vs Turing machines with oracles for your computability definitions? You can arbitrarily stipulate either.

The crux of the issue is that the axiom choice isn't arbitrary in all senses; it's arbitrary in the sense of setting up a formal system, it's not arbitrary in the sense of setting up a formal system to express an intuition, investigate a system (like a series of chemical equations, an electronic input-output machine like a computer) or describe behaviour in the real world.

The choice between different axiomatic systems for different purposes is not an algorithmic one, it satisfies different constraints (useability, prediction, interest, describes stuff well) and is not therefore "merely subjective".

all derived theories from the formal specification of computability are not legitimate knowledge — fdrake

If you take an arbitrary axiom A and a theorem S for which you can prove in proof P that it necessarily follows from A, then the sentence X="A $\Rightarrow$ S" is legitimate formal knowledge.

Sentence X can be utterly useless, and probably also meaningless, but it is nevertheless a justified (true) belief, with the term "true" referring to the fact that it is logically true in the model(s) for the theory embodying axiom A.

Therefore, the knowledge in the theory of computability T is not T itself, nor any arbitrary theorem S, but sentences of the type: T $\vdash$S, i.e. "T proves S", along with proof P that justifies this sentence.

You're talking about what follows if you accept the axioms as true. Not about what justifies stipulating them in the first place. There are good axiomatisations and bad axiomatisations given purposes. If you want to stipulate a system which contains usual arithmetic, 1+1=2 better be a theorem...

"Every triangle has angles which sum to two right angles" accept or reject and on what basis? The latter basis is a sense of justification deeper than your portrayal of acceptance/rejection of axioms as arbitrary.

Sentence X can be utterly useless, and probably also meaningless, but it is nevertheless a justified (true) belief, with the term "true" referring to the fact that it is logically true in the model(s) for the theory embodying axiom A. — alcontali

A person living on a sphere arbitrarily stipulates that all triangles on the surface of their sphere sum to two right angles. Since he is clever, he concludes the parallel postulate using the rest of Euclidean geometry. On this basis, you say"Of course his stipulation is justified, the choice between axiom systems is entirely arbitrary, and look, he derived the parallel postulate from it". But he lives on a sphere, so the axiom turns out to be false.

We live in a more complicated world, so we do not have easy ways to tell how relevant our stipulations are for our purposes except by investigating their consequences; be they as theorems of formal systems, as predictions, as enabling insightful description... The axioms of formal systems are not immune to these considerations, and are thus not arbitrarily chosen or chosen algorithmically.

Why would a computer choose the Turing machine formalism over the arbitrary decision procedure formalism to talk about computation? It couldn't, without having some criterion.

Is that criterion arbitrary? No, it depends on what we're studying. Are we like Euclid, living on a sphere?

The axioms of formal systems are not immune to these consideration, and are not arbitrarily chosen — fdrake

An axiomatic theory does not need to be useful. Since its model is not the physical universe, it is automatically also not meaningful. Therefore, I reject these considerations.

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

and moreover are not choosable algorithmically — fdrake

Axioms are not chosen algorithmically. On the contrary: there is no justification for choosing any particular set of axioms -- not even an algorithm -- and there shouldn't be one.

Why would a computer choose the Turing machine formalism over the arbitrary decision procedure formalism to talk about computation? It couldn't, without having some criterion. — fdrake

We almost never choose the Turing machine formalism. Approximately all computers in use are based on the Von Neumann architecture.

Is that criterion arbitrary? — fdrake

The Von Neumann architecture has taken off like wildfire. There may be reasons for that, but not one that can be explained by using a formal system. Hence, this real-world phenonemon falls outside the realm of what mathematics is supposed to study.