Godel show the that all inductive/axiomatic logic is incomplete, as you've indicated. Math is not knowledge. Math is method. May be best looked at as a language. We can have knowledge of math. Logic is not a measure of truth. It is a measure of coherency, and coherency is imperative to avoid self-contradiction as well as building and maintaining shared meaning.

The reason why statements made in common parlance aren't 'provable' in a logical sense of proof is because logic presupposes truth. Logical proofs prove coherency. Coherency is not enough for truth. The Scopes Monkey Trials show this nicely, as do any other number of absurdity arrived at from following a valid form of argument but starting with an absurd premiss.

Gettier said nothing about these concerns. They're irrelevant.

Godel show the that all inductive/axiomatic logic is incomplete, as you've indicated. — creativesoul

That is not exactly what Gödel proved. There are (simple) axiomatic systems that are complete, such as the Presburger arithmetic.

Gödel did the following. He constructed a virtual/abstract machine, with associated language, that is capable expressing the rules of the (more complex) Dedekind-Peano arithmetic (this is our ordinary arithmetic, actually).

He created a numerical bytecode for this virtual/abstract machine to map these language expressions into numbers. Hence, properties about these language expressions became properties of numbers. At that point, Gödel used that language to express a statement that is (logically) true but not provable, i.e. a so-called Gödel statement or Gödel number:

S = "S is not provable" ---> the bytecode is powerful enough to express this

S is indeed not provable, because there is no way in which you can derive that from the Dedekind-Peano axioms. Therefore, what S actually says is (logically) true.

Hence, there cannot possibly exist a procedure to enumerate all possible valid and logically true statements (i.e. numbers that correspond to a logically true valid language expression) and decide if the theorem that the statement represents, is provable in that system, yes or no.

It also shows that (logically) true is not the same as provable, and even that the fact that a statement is (logically) true does not imply that it will be provable in any way.

Therefore, any axiomatic system that embodies the axioms of number theory or anything else that requires a language of similar complexity, is not possibly complete, aka: It is incomplete.

Math is not knowledge. Math is method. — creativesoul

Well, in Immanuel Kant's lingo, as in Critique of Puree Reason, mathematical theorems are synthetic (=knowledge-increasing) statements a priori (=do not make use of sensory experience) that are derived axiomatically from a construction logic of analytic statements a priori.

Hence, a math theorem is an arrow:

P => Q

with P the axiomatic construction logic
with Q the theorem that necessarily follows from these axioms

Therefore, Q is a belief justified by P.

In other words, a mathematical theorem satisfies the restricted Platonic definition for knowledge, i.e. a justified belief.

Logic is not a measure of truth. — creativesoul

Agreed. Logic has nothing to do with correspondence-theory truth. Logic is just a bit of lattice algebra. It is a symbol-manipulation formalism that has nothing to do with the real, physical world. It lives in its own abstract, Platonic world, just like all mathematics.

Logical proofs prove coherency. Coherency is not enough for truth. — creativesoul

Agreed. I also complete subscribe to Bertrand Russell's criticism on the coherence theory of truth.

Gettier said nothing about these concerns. They're irrelevant. — creativesoul

I agree that Gettier said nothing about these concerns. He attacked the "true", the "T" in JTB in his particular way. I have attacked it in another way. I have just pointed out that the "T" in JTB is not sustainable in mathematics either. Not one mathematical theorem satisfies the correspondence-theory requirements for the term true. Mathematical theorems are simply not isomorphic with the real, physical world, if only, because they live in their own abstract, Platonic world.

Sounds like you're a mathematician... physicist or something?

:smile:

I reject Kant's notions of a posterior and a priori. There is no knowledge without thought/belief. Thought/belief begin simply and accrue in complexity. All thought/belief are existentially dependent upon sensory experience. There is no line to be drawn between knowledge with and knowledge without.

I knew I didn't have Godel quite right, but the gist(I thought) was that he was critiquing inductive logic/reasoning. Deductive is another matter altogether.

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I also complete subscribe to Bertrand Russell's criticism on the coherence theory of truth. — alcontali

Is that not a problem then? Math is all about coherence.

Sounds like you're a mathematician — creativesoul

Not in a professional sense. Just as a hobby. I would never join the academia. They require a growing amount of ideological orthodoxy from their staff, which is unattainable for me.

physicist or something? — creativesoul

In many ways a physicist (=scientist) is the exact opposite of a mathematician.

A physicist seeks to create models that as much as possible satisfy the correspondence theory of truth. A mathematician never does that, because doing that is a constructive heresy in mathematics.

Mathematics never says anything about the real, physical world. It only says things about the abstract, Platonic world specifically constructed for that purpose; which is obviously never the real, physical world.

Therefore, mathematics and science are opposite and strictly exclude each other.

I have made my money in software engineering, but now I am semi-retired. I've recently started spending seven hours per day on physical exercise. I might come out of retirement some day, but possibly also not.

I reject Kant's notions of a posterior and a priori. — creativesoul

These are just definitions: a priori is without making use of sensory information, while a posteriori is just the opposite, i.e. empirical.

All thought/belief are existentially dependent upon sensory experience. — creativesoul

Symbol-manipulation formalisms are devoid of sensory experience. They are just blind operations on language symbols. It is what Immanuel Kant considers to be pure reason (=no sensory input). In fact, Kant originally deemed mathematics not to be pure reason, because (Greek) geometry (Euclid's Elements) revolves around solving visual puzzles. In the meanwhile, mathematics has completely abandoned the use of visual puzzles. Everything revolves around symbol manipulation, i.e. language only procedures. So, nowadays, unlike in Greek antiquity, it is pure reason.

I knew I didn't have Godel quite right, but the gist(I thought) was that he was critiquing inductive logic/reasoning. — creativesoul

Gödel was critiquing Bertrand Russell's impossible optimism in his Principia Mathematica, which gave the absurd impression that mathematics would eventually be able to provably answer all possible questions. It worked like a red cloth on Gödel, who sought to demonstrate the fundamental limitations of Russell's formalisms.

Don't use the quotation marks. — creativesoul

Yeah, I hadn't noticed the quote button when you select a fragment.

Is that not a problem then? Math is all about coherence. — creativesoul

Yes, but never in the real, physical world. Math constructs an abstract world of which the implementation guarantees coherence. It is not hard to get coherence, when you simply force the matter. It would not be possible to do that in the real world.

I reject Kant's notions of a posterior and a priori. — creativesoul

Well, Kant just defines these things: a priori meaning without the use of sensory information and a posteriori, the opposite of that (i.e. empirical).

All thought/belief are existentially dependent upon sensory experience. — creativesoul

Mathematics is pure symbol manipulation, i.e. language expressions. It does not take any sensory input. Therefore, it is pure reason. Kant criticized the practice in classical geometry (Euclid's Elements) to solve visual puzzles. So, he considered it not to be pure reason. Nowadays math is pretty much algebra only. So, Kant's issue with math has been addressed.

I knew I didn't have Godel quite right, but the gist(I thought) was that he was critiquing inductive logic/reasoning. — creativesoul

Gödel was criticizing Bertrand Russell's Principia Mathematica and its impossible optimism. It gave the wrong impression that mathematics would some day be able to solve all problems. Quod non.

Is that not a problem then? Math is all about coherence. — creativesoul

Yes, but not coherence in the real, physical world. It is about constructing abstract, Platonic worlds that are coherent by design.

Gödel was critiquing inductive logic/reasoning. — creativesoul

There used to be this presupposition that if a proposition is (logically) true, there must necessarily exist a proof for it, somewhere to be discovered. It is, in fact, the essence of David Hilbert's Entscheidungsproblem.

Back then, at the end of the 19th century and until the 1930ies, people were obnoxiously positivist and scienticist. Some people still are today, actually. It is also the era of the arrogant God of gaps ideology and the ugly modernist Corbusier buildings, because hey, soon, we will know it all.

So, in the 1930ies, Gödel, Turing, and Church set out to prove that none of that would ever happen. It led to the notions of Gödel incompleteness and Turing-complete knowledge, which is the maximum knowledge that can ever be attained.

Well, Kant just defines these things: a priori meaning without the use of sensory information and a posteriori, the opposite of that (i.e. empirical). — alcontali

I know what he means. I also know that there is no such knowledge to be had.

Yes, but not coherence in the real, physical world. It is about constructing abstract, Platonic worlds that are coherent by design. — alcontali

All of it happens in this world. Talk of different worlds is unnecessary and confusing at best, aside from entertaining possibility, which has it's good use, although until Kripke convinced me of it, I had cast modality aside. I still reject the necessary/contingent distinction.

Popular line of thought, but false on several levels.

We have some access to what's happened and/or is happening. That's all we need(assuming a meaningful claim) to check if the claim is true(or not). — creativesoul

And all the stuff under that we have access to category you refer to is justification. It is not some other, different stuff, it is stuff that justifies our position. You are responding as if I am takign some kind of skeptical position or other. I am not. I am just saying that whatever we base our conclusions on is justification.

Rhetorical drivel based upon a gross misunderstanding of truth and the irrevocable role that it plays in all thought/belief and statements thereof, including your own — creativesoul

It was polemic without much argument, yes. Fine. But you simply asserted

We have some access to what's happened and/or is happening. That's all we need(assuming a meaningful claim) to check if the claim is true(or not). — creativesoul

without explaining what this is, nor did you respond to the argument I made before that. Do you have anything but an assertion that I am wrong? Can you come down out of the clouds of abstraction, and explain what access to truth you mean, that would not be included in the category 'justification'? How do you divide up justification and access to truth? some specifics.

Mathematics is pure symbol manipulation, i.e. language expressions. It does not take any sensory input. Therefore, it is pure reason. Kant criticized the practice in classical geometry (Euclid's Elements) to solve visual puzzles. So, he considered it not to be pure reason. Nowadays math is pretty much algebra only. So, Kant's issue with math has been addressed. — alcontali

Math consists of symbols made meaningful solely by virtue of our attribution. Math says nothing about common parlance. All symbolic notation is existentially dependent upon common parlance. Truth is presupposed within all thought/belief. Math is utterly irrelevant to the role that truth plays in all human thought/belief. Math cannot take account of language, and thus is a useless tool for taking account of human thought/belief.

Truth is presupposed long before one learns the rules of counting. Math is irrelevant to this discussion.



Kant's notion of pure reason leads to a denial of animal thought/belief. Kant does not draw and maintain the distinction between linguistic thought/belief and non-linguistic thought/belief. That distinction can only be taken into proper account by virtue of thinking in terms of existential dependency and elemental constituency. Kant does no such thing. Rather...

Kant is following Hume's definition of Pure Reason, and as a result starts off on the wrong foot, so to speak...

Pure Reason, as set out by Hume, is somehow implied to be remarkably different from emotion, and it must be or else Hume's philosophy falls apart at the seams. Hume is wrong. Reason is not existentially independent of emotion. Knowledge of all human thought/belief is the seam ripper here.

It is indubitable that Hume thought/believed that pure reason was somehow separate and distinct from emotion. That's the common understanding of what Hume meant, and it is a correct understanding of Hume. Hume meant that. His mantra "Reason is slave to the passions" shows this clearly. Hume works from the unspoken premiss that reason is somehow existentially independent of emotions. The problem is that that is false.

There is a gross misconception of human thought/belief that has been very hard at work for a very long time. It's time to force it's retirement. I suspect two underlying issues are to blame for that all too common misconception. One is the long standing notion that we are superior to other animals, and that that superiority is shown - and clearly so - by the way humans think. I would agree with that on it's face, particularly regarding the breadth of complexity that language affords human thought.

Another is that all human thought/belief is inextricably bound up in language. It's not.

However, thought/belief(and thus reason) is different in terms of element constitution and existential dependency. Pure Reason is thinking about thought/belief, and as such it can be rightfully and meaningfully said to be existentially dependent upon pre-existing thought/belief. From this we can also know that whatever thinking about thought/belief is existentially dependent upon, so too is pure reason.

Not all thought/belief is existentially dependent upon language. All reason is. All reason is also existentially dependent upon rudimentary thought/belief. All rudimentary thought/belief consists of correlations drawn between different things. At the earliest stages of prelinguistic human development, all of us are drawing correlations between external and internal things. Thought/belief begins simply and grows in it's complexity.

All reason is existentially dependent upon pre-linguistic thought/belief. All prelinguistic thought/belief consists - in part - of internal emotion(fear, contentment, discontentment). All reason is existentially dependent upon emotion. That which is existentially dependent upon something else, cannot be independent from it.

Thus... there is no such thing as 'Pure Reason' except and aside from being the name of a product(figment) of the Humean imagination.

Can you come down out of the clouds of abstraction, and explain what access to truth you mean, that would not be included in the category 'justification'? How do you divide up justification and access to truth? — Coben

Who said anything about access to truth?

:brow:

Access to what's happened and/or is happening is what I wrote. I added to that. It went neglected. Odd given the charge. Did you offer an argument/explanation that I left neglected?

How much explanation is needed here? Tell a three year old that is waiting upon return that their mom is home and they'll go look to see for themselves(if it is true).

Why is it so hard for anyone else to understand?

You're attempting to dismiss, discard, and/or discount truth. That will not go unchallenged.

Therefore, the JTB definition for knowledge is wrong. It must be JB instead.

This is the same conclusion as Gettier's, but obtained in a different way. — alcontali

That's not Gettier's conclusion. Gettier's conclusion is that "[the JTB definition] does not state a sufficient condition for someone's knowing a given proposition."<sup>1

1</sup> https://www-bcf.usc.edu/~kleinsch/Gettier.pdf

Math consists of symbols made meaningful solely by virtue of our attribution. — creativesoul

The meaning of symbols comes from their relationship with other symbols. It is not possible to attribute anything to them because they often do not correspond to anything in the real, physical world. For example, what meaning could we possibly attribute to the S and K combinators in the SKI calculus? They are mentioned in the reduction rules. That is all there is to them.

All symbolic notation is existentially dependent upon common parlance. — creativesoul

You can write the symbols in full too, as complete words. They often mean absolutely nothing in common parlance. What would be the common parlance for a "tower of radical field extensions" in the Galois correspondence? There is absolutely nothing at all that corresponds to this in the real, physical world.

Math is utterly irrelevant to the role that truth plays in all human thought/belief. — creativesoul

Agreed. It has no direct incidence on correspondence-theory truth.

The reason why math plays an outsized and even dominant role in science and even in daily life, is because math is consistent by design, while the real, physical world is deemed to be consistent by assumption. The requirement of consistency forbids particular things from happening in the real world. All mathematical models in applied math or science exploit this consistency correspondence.

Math is irrelevant to this discussion — creativesoul

Math is knowledge, but math is not correspondence-theory true. Hence, math is a justified belief but not a justified true belief. Therefore, it raises exactly the same problem as the one Gettier raised.

Hume works from the unspoken premiss that reason is somehow existentially independent of emotions. — creativesoul

I'd rather agree with Hume. Reason is just a tool suitable for propositional inference. Even a machine can verify whether a new proposition is indeed provable from a set of other propositions.

Discovering new knowledge propositions is something else. We certainly do not exclusively use reason or knowledge for that. If we had knowledge on how to discover new knowledge, we would actually already have the new knowledge, and then would not need to discover it.

All reason is existentially dependent upon emotion. — creativesoul

It may originate from there somehow, but once you program the propositional inference engine in software, it no longer has anything to do with emotion. The Isabelle reasoning engine is not emotional.

In particular, Isabelle's classical reasoner can perform long chains of reasoning steps to prove formulas.

Thus... there is no such thing as 'Pure Reason' except and aside from being the name of a product(figment) of the Humean imagination. — creativesoul

Pure Reason even exists in software.

That's not Gettier's conclusion. — Michael

I did not make that conclusion directly based on Gettier's work or examples, but on the argument in the debate.org discussion about JTB:

He wrote:

Simply put" Justified belief" is enough. As stated so clearly by Gettier, it is possible for a proposition to be simultaneously true and false in a similar way as Shrodingers cat can be both alive and dead before obtaining incontrovertible evidence to prove it is one or the other. Knowledge does not equal truth so to add that into the definition of knowledge is tenable.

As I wrote, I came via another route to the idea that "Justified belief" is enough. A mathematical theorem is a justified belief that is not correspondence-theory true. So, with mathematics clearly being considered knowledge, this requirement of correspondence-theory truth is unsustainable.

Furthermore, Immanuel Kant's synthetic statements a priori are also knowledge without any reference to sensory input or the real, physical world. Hence, synthetic statements a priori are impossibly correspondence-theory true.

I did not make that conclusion directly based on Gettier's work or examples, but on the argument in the debate.org discussion about JTB:

He wrote:

Simply put" Justified belief" is enough. As stated so clearly by Gettier, it is possible for a proposition to be simultaneously true and false in a similar way as Shrodingers cat can be both alive and dead before obtaining incontrovertible evidence to prove it is one or the other. Knowledge does not equal truth so to add that into the definition of knowledge is tenable. — alcontali

Who wrote this? Clearly not Gettier. Gettier himself didn't conclude that knowledge is justified belief.

Who wrote this? Clearly not Gettier. Gettier himself didn't conclude that knowledge is justified belief. — Michael

This person is clearly someone with a background in physics.

Werner Heisenberg already pointed out in 1927 a massive flaw in the correspondence-theory truth about the position (p) and velocity (v) tuple of an electron: (p,v).

An electron may indeed have a (p,v) tuple, but if you know one element in the tuple, you can impossibly know the other element. That is his notorious uncertainty principle. Heisenberg received the Nobel prize in 1932 for arguing this. It forced everybody in nuclear physics to switch over to quantum mechanics, to the utter dismay of Albert Einstein, who hated this.

Schrödinger's cat is another gigantic problem with the correspondence theory of truth. The state of a particle is indeterminate until the external world observes it:

According to Schrödinger, the Copenhagen interpretation implies that the cat remains both alive and dead until the state has been observed.

The correspondence theory of truth simply starts falling apart at the level of nuclear physics. In fact, both Heisenberg and Schrödinger rediscovered what Kant had already insisted on: Das Ding an sich ist ein Unbekänntes.

In other words, what Gettier argued in his own work sounds trivially obvious to nuclear physicists.

I did not use the physics argument, because I do not think like a physicist. I came to the same conclusion by looking at the nature of mathematics. That is a completely different route, but one that suits me much better than physics, which I certainly recognize as a sound discipline, but one that I do not really like, because I prefer the abstract, Platonic worlds of mathematicians to the real, physical world of scientists.

This person is clearly someone with a background in physics. — alcontali

OK, but he clearly misunderstood Gettier because Gettier didn't conclude that knowledge is justified belief. Gettier only concluded, in his own words, that the JTB definition "does not state a sufficient condition for someone's knowing a given proposition."

OK, but he clearly misunderstood Gettier because Gettier didn't conclude that knowledge is justified belief. Gettier only concluded, in his own words, that the JTB definition "does not state a sufficient condition for someone's knowing a given proposition." — Michael

His argument is surprisingly closer to the examples Gettier gave, than it looks like. Gettier manually concocts what physicists call an entanglement, and then concludes, "Oh My God, it goes wrong now!".

Well, these Gettier entanglements naturally occur all the time in quantum physics. No need to fabricate them!

That is why he is not really interested in Gettier's actual examples, which even look relatively silly compared to the ones that spontaneously occur in nature.

Remember, in the first case, Smith was - by Gettier's own admission - justified in believing that he would get the job. Smith thought to himself "I am going to get the job". Smith believed that he had secured the job. That is to think of and/or about oneself, not another.

That sort of thought/belief is self-reflective. It cannot be about anyone else. It turned out to be quite false. Yet Gettier's sleight of hand was invoking the rules of entailment as his own justification for changing an undeniably self-reflective thought/belief into belief about someone else. That move is unjustifiable here.

"The man with ten coins in his pocket will get the job".<----------that is not a valid move for either Smith or Gettier. — creativesoul

a) I am a postman.
b) Mary is married to me.
c) Therefore, Mary is married to a postman.

Isn't this valid?

Who said anything about access to truth? — creativesoul

I did, in the post you responded to, first. Since you were not specific about what parts you disagreed with and that was one of the parts, I asked you about it.

You're attempting to dismiss, discard, and/or discount truth. That will not go unchallenged. — creativesoul

Nope, I am not. I am looking at the specific model or definition of knowledge, JTB, and given the way it is used being critical of using the two adjectives justified and true. It is in that specfiic context, the way justification is used in contexts with JTB, that I think using true is problematic. There are other contexts where I have no problem with true and truth.

And you haven't challenged me. You have just told me I am wrong.

Remember, in the first case, Smith was - by Gettier's own admission - justified in believing that he would get the job. Smith thought to himself "I am going to get the job". Smith believed that he had secured the job. That is to think of and/or about oneself, not another.

That sort of thought/belief is self-reflective. It cannot be about anyone else. It turned out to be quite false. Yet Gettier's sleight of hand was invoking the rules of entailment as his own justification for changing an undeniably self-reflective thought/belief into belief about someone else. That move is unjustifiable here.

"The man with ten coins in his pocket will get the job".<----------that is not a valid move for either Smith or Gettier.
— creativesoul

a) I am a postman.
b) Mary is married to me.
c) Therefore, Mary is married to a postman.

Isn't this valid? — Michael

The postman that Mary is married to is me. The man with ten coins in his pocket who will get the job is me. That's what Smith believes in either situation.

That doesn’t answer my question. Is the argument valid?

Not Smith's belief. That is what the subject matter is about... belief that is justified and true. My argument here is that the belief is being misrepresented.

You're attempting to dismiss, discard, and/or discount truth. That will not go unchallenged.
— creativesoul
Nope, I am not. — Coben

This coming from one who said "Truth is for the Pope"...

Yes, you are.