Epistemology versus computability
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.
You will immediately notice the surprising overlap with epistemology, i.e. the theory of knowledge, which studies how knowledge is being or should be justified.
It is obvious that a claim can only have the status of knowledge if there somehow exists an effective procedure to verify its justification.
In Karl Popper's seminal publication, Science as falsification, he famously pointed out that scientific status for a claim exclusively depends on the possibility to carry out an effective procedure to falsify the claim, i.e. experimental observation and testing.
In mathematics, a legitimate knowledge claim implies the existence of an effective proof procedure for the claim.
I wonder if computability and epistemology are ultimately not one and the same thing?
You will immediately notice the surprising overlap with epistemology, i.e. the theory of knowledge, which studies how knowledge is being or should be justified.
It is obvious that a claim can only have the status of knowledge if there somehow exists an effective procedure to verify its justification.
In Karl Popper's seminal publication, Science as falsification, he famously pointed out that scientific status for a claim exclusively depends on the possibility to carry out an effective procedure to falsify the claim, i.e. experimental observation and testing.
In mathematics, a legitimate knowledge claim implies the existence of an effective proof procedure for the claim.
I wonder if computability and epistemology are ultimately not one and the same thing?
Comments (141)
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.
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.
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.
You could have fooled me with this EARLIER statement by you:
Quoting alcontali
(My answer to the above was clearly "not", as you know now.)
Quoting 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.
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".
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.
And yet we know of unprovable truths.
Epistemology is broader than computability.
Perhaps you confuse being true with being justified. There are obvious empirical truths - such as that you are reading this post.
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.
Quoting tim wood
Agreed. Most mathematicians assume that P is not NP (without proof, though).
Quoting 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.
Quoting 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.
Did you mean "it is necessarily incomplete"?
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.
So what?
Quoting 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.
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.
Quoting 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.
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 provably consistent, then it is inconsistent.
Yes. True.
That is known as Gödel's second incompleteness theorem.
I mentioned the proof strategy for this (which is actually trivial) in a previous answer.
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 ...
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.
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.
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.
Quoting The Church–Turing–Deutsch thesis
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.
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.
Quoting 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.
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’.
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?
Well, in Critique of Pure Reason, Immanuel Kant pointed out the existence of a type of knowledge that is not empirical. It is synthetic a priori. At the same time, he rejected classical Greek geometry as NOT being synthetic a priori, because it is highly visual, as it is an exercise in fiddling with visual puzzles.
In the meanwhile, mathematics has changed. It has migrated from visual fiddling to pure symbol manipulation. Nowadays, its essence is language only. We no longer follow visual procedures in mathematics.
Therefore, I disagree with relying on empiricism in mathematics. The progress in mathematics in the last few centuries has only been possible by removing its dependence on visual input. Mathematics has now finally become pure reason only.
Quoting Harry Hindu
For a starters, we simply end up cutting off the real-world origins of mathematical theories, if there were any to begin with:
Quoting Wikipedia on abstraction in mathematics
Secondly, quite a bit of mathematics does not have a real-world origin. For example, where in nature can you find something like look-ahead left-right parsers? Where in nature can you find Turing machines? Von Neumann machines?
These things are abstraction only. They started studying them in mathematics because these at first imaginary devices were potentially useful for computing. If they had limited themselves to what is readily visible in the surrounding universe, we would simply never have had computers. Nature does not have them to begin with.
Quoting Harry Hindu
Well, for example, even C/C++ header files contain mostly definitions that are not even meant to ever execute. For example, what is chromium/base/barrier_closure.h supposed to do? Even the source code of something like a web browser such as Google Chrome contains seemingly absurd abstractions that are concept heavy while being low on actual code to execute. In other words, it is not even meant to do anything. It just structures things in one way or another ...
Also, is that realization a thought or a feeling, cognition or intuition? Is it maybe bound to language? And if it is, perhaps by acquiring an understanding of some language, then the logic of such sentence automatically becomes self-evident?
So, how to make an algorithm “understand”, and then to understand words like “I”, “know”, or “exist”? But, what “to understand” actually means? And even before that, what the word “means” really means?
On closer inspection the relation between computation and epistemology seems to be more like the relation between a certain fact and complete mystery.
The practical impossibility of distinguishing an unknown pseudo-random process from a really random process provides us with ammunition for rejecting an absolute ontological distinction between randomness and lawfulness. In which case, the question as to whether nature is computable or not is meaningless.
Instead we need only invoke a game-theoretic distinction between a controlled process referring to a process we create and control ourselves using an algorithm, versus an uncontrolled process that is part of nature that we only query and make speculative computer models of.
Try solipsisism or syllopsysosm. I can't spell the danged thing, but I know what it is.
Try that. That refutes the entire truth value of observation.
If that won't convince you that I was right, then try the thought that everything you know, and stored as memory, both conscious and unconscious, and both subconscious and super-conscious, in fact your entire life experience and life just started a minute ago. Or this instant. There is no proof for this, but it is conceivable.
Under these two possible considerations empirical truths are shmafu. Only a priori truths can exist for a 100 percent degree certainty in any possible arrangement of the physical world.
Ay-vey, Immanuel. Just because you can see it, it does not mean it can't be a priori existant. What a narrow-minded little block-head that Immanuel was. Or square head. Or take your choice of synthetic a priori geometrical shape, and apply it to Immanuel Kant's head shape. You can't lose.
the only unprovable truths I know about math are its axioms and its axiomatic behaviour. (Such as 2+7=4.) Do they, the axioms of math, qualify as being outside of computability? I hardly think so, but I actually can't decide that.
Even though I very much appreciate Immanuel Kant's Critique of Pure Reason, I am still not one of his cheerleaders. On the one side, Kant says that pure mathematics has the potential for being pure reason:
Quoting Kant in Critique of Pure Reason on mathematics
But Kant denies that the visual puzzles in classical Greek geometry are pure reason:
Quoting Kant on geometry and its visual puzzles
At some point, Kant engages in infinite regress by demanding a justification for the axioms in mathematics from "transcendental philosophy". Of course, that will never work:
Quoting Kant demanding a justification for axioms
Kant believes in metaphysics, i.e. in infinite regress, while I absolutely don't:
Quoting Kant insisting on dogma-less views, i.e. insisting on infinite regress
Since its very beginning, 2500 years ago, metaphysics has never produced anything of value or anything actually worth knowing. The reason for that, is, that the method of infinite regress is faulty. It literally leads to nowhere.
We are visual creatures. We think in mostly visual forms. Our thoughts have form and those forms are the same as all of the sensory impressions we are capable experiencing. Mathematics is just different puzzles using different visual scribbles.
Quoting alcontali
It hasn't changed. Languages are visual scribbles and sounds. If the procedures you follow aren't visual, then how do you know you're following a procedure? What form does your mathematical procedure take? How would you describe the experience of performing a mathematical procedure? In describing it you will be using visual scribbles on a screen to reference the visuals in your head. If you say the experience is more like talking it out - talking to yourself in your mind, then it is taking an auditory form rather than a visual form. It seems to me that you are saying that mathematics is done unconsciously.
Quoting alcontali
I don't know what pure reason is unless it takes some form for me to know that I am engaged in pure reasoning. How do you know that you are engaged in pure reason as opposed to relying on empiricism if they both don't appear differently to you in your mind - visually.
Quoting Wikipedia on abstraction in mathematics
What form do these underlying structures, patterns, properties, phenomena take? Structure, pattern, phenomena and properties are all visual terms.
Quoting alcontali
Well, I don't see humans, or their inventions, as being separate from nature. So abstractions are natural products of our minds and our minds are products of natural selection (evolutionary psychology and computational theory of mind). Our minds are the software and our bodies are the hardware. The computer is the best analogy for the mind that we've had in our history of thinking about the mind and its relationship with the world.
Quoting alcontali
Definitions are not executed. Functions are executed and reference those definitions. No programmer would put code that isn't used somewhere in the program as code takes up memory. It would be a waste of memory space and programmers try their best to streamline their code so that it runs efficiently and isn't a memory hog. Any other "code" that isn't executed would be remarks for us humans to be able to understand what the code is used for.
By visual procedures, I mean a procedure in which the use of circles, lines, triangles, polygons, graphs, and similar visual representations are essential. Nowadays, only the algebraic symbol manipulations are essential. Mathematics is now essentially language only. For example, you do not need to create any drawing to solve the roots of a quadratic equation. In fact, that was the first non-visual, language-only procedure that appeared in the Middle Ages, in the Liber Algebrae by Algorithmi. Nowadays, mathematics has completely algebraized, including geometry.
Quoting Harry Hindu
You can represent language visually with written letters but you can also represent it verbally with sounds. You cannot do that with a line, triangle, circle, or polygon. In algebra, the visual aspect is not essential.
Quoting Harry Hindu
Their canonical description is in language only; while language is not necessarily visual. Language also has an isomorphic auditory representation. Language is not considered an empirical input.
:confused:
And algebraic symbols have curves and circles and lines. 6 + 0 = 6
You draw the symbols on paper, or repeat them with auditory symbols in your head. The spoken word, "six" is the same as the visual 6. How is it that we can represent the same idea with two different empirical forms?
Quoting alcontali
How did we learn that "the first non-visual, language-only procedure that appeared in the Middle Ages, in the Liber Algebrae by Algorithmi" if we didn't see it written out on paper? When I ask you for evidence for your claim, how would you show it?
Quoting alcontali
Quoting alcontali
I already pointed this out. Auditory forms are still empirical forms. Language and math take the form of visual or auditory symbols. How did you learn your native language without eyes and ears, or tactile sensations if you are blind? How do you use language without making sounds or visual scribbles?
Quoting Harry Hindu
Do you turn into a p-zombie when you perform mathematical calculations in your head?
Language-only communication also uses visual representations but of text and symbols only. It is not considered empirical input.
Yes, but what is computer without display screen? All the electrons moving around electronic components is like electrochemical signaling in our brains. Information without inherent meaning, something that needs to be decoded or integrated in some way, at some place where it all comes together to form subjective experience or qualia - that parallel to a computer screen which displays the mental content and at the same time perceives it, somehow.
The problem with computers is that it is all mechanical actually, in a sense that in principle you could make a PC powered on water instead of electric current and replace electronic components with wooden contraptions to produce the same kind of computation. Imagining this computer makes it more obvious why many say it is impossible computation could ever explain mind phenomena such as subjective experience and mental content.
A server. They run the Internet. This website runs on a server in some datacenter. There's no display connected to it. When the IT folks need to access it they log in remotely over the network.
Haven't followed the discussion but just happened to notice this. Computers without display screens are extremely common.
You're not reading my entire post. I asked how you learned and use language without using your eyes and ears. When you read instructions on how to assemble your new bicycle, you use your eyes and the instructions are the input and your actions in assembling the bicycle is the output.
Try reading the rest of my post (the input) with your eyes closed, and then reply (the output) with your eyes closed.
Betelgeuse will go supernova in 2 years and obliterate the Earth.
You can't escape being both empirical and rational in using language or mathematics.
A server.
A display screen is only one type of output that a computer can use. They can also produce sound, printing on paper, 3-D printing, tactile sensations in VR gloves, etc. As long as you have an electronic device processing information (input to output), you have a type of computer.
Quoting Zelebg
Yes, but what is looking at the computer screen in your brain? This is the infinite regress of the homoculus in your head - the cartesian theater. There is no screen being looked at. There is only the working of your short-term memory. That is what consciousness is - this work getting done of processing information coming in through the senses and producing output with your intent and actions.
Quoting Zelebg
This is more along the lines of direct realism vs. indirect realism. Is it brains "out there", or minds? When I look at you, I see a physical body, not a subjective experience. When I look at myself, I don't just see a body. I experience a body. There is this "subjective experience" - my mind. Is the world like my mind, or like bodies (mental or physical)? Are brains just how minds simulate other minds? I don't like to say what idealists say, and say that everything is mind. Everything is mind-like, or of the same "substance" as mind. But to say that everything is mind, or conscious, would be anthropomorphic. I think a better term would be "information". Not physical or mental. Physical and mental is a false dichotomy that leads to dualism. Everything is information.
“Computer without monitor” was a metaphor, and the point is that when we look at the brain we see input and processing, but not where or what the result and "output" is.
So again, to understand qualia and mental content, analogy between mind and computer is not complete until we discover a thing that is analogous to computer output, such as display screen, for example.
Words refer to things, and that is exactly what the word “physical” and “mental” differentiate - actual things from their abstract representations.
Information carries no inherent meaning, it needs a context or decoding against or within which it can be understood or perceived.
So? That does not mean that language can only be used to describe the physical universe. It can also be used to describe imaginary universes. You can use language to write science fiction. You can use language to describe an idea for something that does not exist yet. Your eyes never saw it. Your ears never heard it.
Quoting Harry Hindu
I don't see how it demonstrates that language would be an empirical input. I reject that point of view.
Are you being purposely obtuse?
I'm not talking about what the words are about. I'm talking about the words themselves. You would never know about those imaginary universes if you didn't have eyes to see the scribbles in the paperback sci-fi novel, or ears to hear a reader read the scribbles.
Of course we see the output. They are the nerve signals that get sent to the limbs to take action, or to the mouth to speak, etc. I did say that the output was our intent and actions.
Quoting Zelebg
Words can refer to imaginary things or illusions. Does "god" refer to something? I think you'll find a lot of disagreement about whether it does or not.
What do you mean by "physical" and "mental"?
Information is meaning. It is the relationship between causes and their effects. Effects carry information/meaning about their causes.
Obviously the output I was talking about is qualia.
Yes, and in that case it’s the other way around - mental or abstract existence of ideas is actual, while their representations can become physical.
What I said. https://www.merriam-webster.com/
Certainly not. https://www.merriam-webster.com/
That is not sure at all. People who are blind and/or deaf, still think. Sensory input is not a requirement for thought.
Qualia is not output. It's the input.
Quoting Zelebg
I have no idea what this means.
Your links are imply that I should be looking these trends up in a dictionary. I'm contesting those definitions. Why don't you provide some good reasons for continuing to use these philosophically antiquated terms.
Then what form do their, and your, thoughts take? How do you know you're thinking?
Computers do not require empirical input either.
You could learn how to accept a text stream, character by character, through a device that makes a short stroke on your palm to represent a zero, and a long stroke for a one. Every collection of six strokes represent one 6-bit character in base64. That would be your input. Next, you can output what you thought about the input by moving your index finger on a touchpad to produce short and long strokes. The process would be slow, but it would work absolutely fine. No vision nor sound needed.
Quoting Harry Hindu
I already pointed out that blind people use braille to feel the words. The tactile sensations are empirical and are the input.
Quoting Wikipedia on empirical evidence
If the fact that knowledge is transmitted through sound, vision, or tactile sensations makes it empirical, then non-empirical knowledge cannot exist. I do not subscribe to that kind of view. I prefer to use Kant's characterization of knowledge.
Of course you do, because you keep avoiding this question:
Quoting Harry Hindu
What form does "All bachelors are unmarried" take in your mind? How do you know that you're thinking it? Is it just hearing the words in your mind, seeing the words in your mind, or seeing images of bachelor's and married men? You seem to be saying that you were born knowing "All bachelors are unmarried".
The common understanding is as following:
Quoting Wikipedia on the distinction between empirical and ab initio
You are questioning the validity of very, very basic epistemic principles. I really do not see what you want to achieve by doing that. Frankly, I do not think that it will lead anywhere.
What form does the thought, All bachelors are unmarried, take in your mind? How do you know when you are thinking it and when you aren't?
It wasn't my example by the way. I was just quoting from a canonical text. You are questioning and rejecting very basic principles in epistemology. That is ok, but I am not the right person to make useful comments in that regard. For once, I am actually happy with the mainstream beliefs in this matter.
Quoting 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?
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.
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.
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.
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.
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?
In this context, it just means "objectively verifiable", which automatically implies that a procedure to carry out such verification can be documented.
Quoting 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.
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?
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.
Quoting 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.
But... verifiable is exactly what a falsifiable hypothesis is not.
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.
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.
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.
Quoting Banno
Quoting 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.
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.
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.
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.
Quoting 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:
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.
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.
Yep.
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.
How do you decide what goes into an "axiom pack"? @alcontali
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.
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.
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.
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:
Quoting 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".
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 [math]\Rightarrow[/math] 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 [math]\vdash[/math]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.
Quoting 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?
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".
Quoting 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.
Quoting fdrake
We almost never choose the Turing machine formalism. Approximately all computers in use are based on the Von Neumann architecture.
Quoting 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.
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.
Quoting 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.
Quoting alcontali
Read: most of what epistemology studies falls outside the realm of what mathematics (and specifically computability theory) is supposed to study..
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.
Quoting fdrake
Quoting alcontali
I think you found your own answer, then.
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...
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.
Quoting alcontali
So, we shouldn't trust you to know when a formal system is relevant for epistemology or not...
Who is "we"?
The MU puzzle ultimately goes to the core of the epistemology of mathematics.
Quoting 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.
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.
Quoting 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.
I was just replying to something you wrote.
Quoting 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.
Quoting 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.
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).
Yep.
The question becomes an ought, not an is.
Quoting alcontali
If you mean that science is not certain, then, yes, obviously.
Quoting alcontali
I've shown that it does.
Quoting 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.
I agree. What is the flight from conceptual content to a dead machine?
Quoting Banno
Here's the answer, a flight from the uncertainty of everything stained by social human being.Quoting 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.
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.
Quoting 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.
Quoting 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.
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 same holds true for readers.
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?
Quoting 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.
Short story: it doesn't.
Quoting 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.
Quoting 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".
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.
Quoting 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.
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.
Quoting mask
That may be overly ambitious.
Quoting mask
I think that it can be used to mechanically verify the paperwork that epistemology says must be present.
There is nothing "behind"; no separation between word and thing. The Map is not the territory, but we can still talk about the territory.
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?
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.
Actually, that is not what you have always claimed. Here's the proof:
Quoting 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.
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.
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.
Those last few posts of yours are terrible. They make no sense.
Maybe a logical reframing of this might be that it's possible to be certain of things that we do not believe that we believe?
IE: Possibly [(X is certain that P) and not (X believes that X believes that P))]
If belief as a modality collapses, this is equivalent to:
Possibly[ (X is certain that P) and not (X believes that P)]
A certainty might be an unknown known, or the truth-maker to the truth-bearing proposition. We believe only in truth bearers; statements; not their truth makers. What a statement is about and what makes it true is not the statement itself, it is merely equivalent to the statement content insofar as it is expressible.
Edit: the explanatory paragraph might make certainty range over more than statements; over perceptual events and environmental behaviour, and would require an account of the connection between believing that P and the certainty acting on the truth-maker of P.
Something we know but do not know that we know?
Then, by JTB, it is believed, but we do not believe that we believe it. The scope of each belief statement differs.
I agree! "X is certain about P => X believes that P". This doesn't address whether we can be certain of things that are not numerically identical to things we can state. Like perceptual events, or what makes "I am certain that I can ride a bike" true.
Quoting Banno
Yes! I imagine that we can be certain of procedural and perceptual knowledge, in some cases we can state that we have such knowledge: "I know how to ride a bike" and "I can recognise a starling when I hear its song", but it seems to me the collection of procedural and perceptual knowledge I am certain of is much larger than the procedural and perceptual knowledge that I could declare that I am certain of (and thereby believe, from the above).
Under the account that "X believes that P" entails "X can declare that "I believe that P"", these items of certain procedural and perceptual knowledge would not be believed by X (modus tollens) since they could not be declared by X.
I'm not going to talk about things we can't talk about. I suggest you don't, either. It's a very common problem for philosophers, easily remedied.
So, that leaves us with stuff we can talk about.
Oh I have no compunctions about talking about things which are not numerically identical to language items. All that matters is that we can treat what we're talking about consistently; a qualitative identity. I saw it raining today, that doesn't mean my perception was rain, or that "I saw it raining today" the statement was rain, all that matters is an equivalence between them.
The link between a list of rules for baking a cake and the know how of baking a cake, say, and the discrepancies between the list and the know how. Do you find it inconceivable that X knowing how to bake a cake includes X having procedural knowledge that they could not state?
EG, here's a cooking guide for sunny side up eggs:
This makes sense. I know how to gently crack an egg into a pan. But if I were to try and list what goes into that knowing how, I'd write lots of vague things like:
Make sure you don't crack the surface with too much force.
Make sure you don't elevate the eggs too high above the pan.
Gently tap the egg against a hard surface to weaken the shell.
I also know how to recognise an egg sizzling in a pan. But how could I describe the components of hearing sizzling? A undulating high pitched noise that sounds like a series of slaps? If I were to train someone to recognise sizzling noises, I would need examples; ie, knowing how to use the word "sizzling" piggybacks off a competence of recognizing sizzling noises; the components of which I cannot state.
You can't even tell me what it is you can't say, so don't try.
It seems to me your standards for someone demonstrating that there are components of know how which cannot be stated are to state them; that they must be constructed as an example. I think that a more appropriate standard would be that know how components which cannot be stated should be labelled as a class, and only some elements of the class cannot be stated.
Consider:
(1) Recognising sizzling noises.
(2) Knowing how to crack an egg gently into a bowl.
I can't split up (1) further, especially not exhaustively; maybe I make a sizzling noise or use an onomatopoeia, do I need to be able to do either to recognise sizzling noises? Nah. I can write a rough description of (2), but exhaustively detailing what it means to crack an egg gently in a way where every subtask has an associated item of declarable knowledge... No.
A person's ability to describe how to do something is much different from both their ability to do something and how they do it, and why should we expect that every component of know how has an item of declarative knowledge associated with it such that "X knows how to do procedural knowledge component Y of task Z implies X can state f( Y )" in these circumstances? To my mind, know how descriptions are much coarser grained; for an arbitrary task and agent, only some of the components of some overall tasks are such that their agent can make statements about them.
I concur. The first of the last few posts was sensible, though, methinks. The one before the one I juxtaposed two of your declared beliefs.
For the record I don't consume street drugs (coffe, granted, but not even nicotine or alcohol). I don't know how to check out the time stamp on these posts. I must have been dead tired when I wrote them, and the next morning I realized they were gibberish.
But it was too late to change them by then.
I thought of stating "Please disregard my last few posts", but I thought I would leave that joyful job to you. Thanks for coming through.
Quoting Banno
This is where you confused me. Your statement is a non sequitur to my point. I tried to address the relationship between your apparent "wisdom" and my point. I failed, because there is no relationship there.
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Personal opinion starts here, unrelated to the topic:
You've given me the advice to read stuff. I suggest you start doing precisely that, starting with the forum posts of others, and attempting to understand the points of your debating opponents.
I realize that you are more into other stuff. You're more into making sweeping statements that are unrelated to the topic, spewed out randomly or else spewed out at points where it is critical in the debate to make a stand. You do it, in my opinion, because you are incapable of fathoming the meaning of the posts of others.
If I read you right, this is not actually what I claimed.
Let me give it a try.
1. I sense things in the physical world.
2. I assume that what I sense is the physical world.
3. I realize at the same time that my senses may provide me with illusions, not feedback on the physical world.
4. I am not able to establish to any degree of certainty which is true: my assumption in 2 or my imagination regarding a possible illusion in 3.
This is beyond @Banno. He believes that what he believes is true. I shall leave Banno to that, and he can merrily go about the landscape unfettered by any criticism by me of his believing that what he believes is true.
Do you think that there is some point on which we disagree? Which point?
Quoting fdrake
It seems to me that this is exactly wrong; and that this is the point I was making in suggesting that we not try to list the things we cannot talk about... like egg sizzle, if you like.
Because in listing egg sizzle as something about which one cannot talk, we talk about it...
Indeed, we are talking about it.
So I find your point here opaque. If you are saying that we can't describe the sound of an egg sizzling, the onomatopoeic word "sizzle" undermines even that. Further, that's not the same as saying we can't talk about the sound of an egg sizzling.
Perhaps you mean that we cannot make the sound of an egg sizzling? But I can - by sizzling an egg...
So, where does that get us?
My overall argument is for the claim that it is possible to be certain of things that we do not believe. "Do not believe" as in "lack belief in" rather than "believe the negation of".
For this, I introduced a distinction between certainty and belief. Belief applies only to statements; and can thus only be a component of declarative knowledge. Certainty applies to statements and competences; and thus a certainty can be a component of declarative or procedural knowledge.
I assumed that declarative knowledge consists solely of statements, irrelevant of how those statements are produced. Procedural knowledge consists of competences; abilities to do activities reliably in appropriate contexts.
For a given item of procedural knowledge, call it a competence to do a task. I assume that every task consists of subtasks, and that all subtasks of a task must be able to be done competently (and reliably in appropriate circumstances) by an agent in order for the agent to know how to do the task. In a formula, task competence entails subtask competence for every subtask of the task.
For a given task, we can label it; knowing how to ride a bike, knowing how to recognise a starling by its song. But we need not be able to label or state every subtask that goes into the task in order to be competent at the task; to know how to do it. Knowing how to recognise the birdsong of a starling is a network of interplaying competences which may be grouped into the general descriptor "knowing how to recognise birdsong", and in that regard we may make items of declarative knowledge about the know how of recognising birdsong; whenever we may aggregate subtasks of the task into task components. Like, say, splitting up baking a cake into an ingredient mixing phase and a phase involving an oven.
We can talk about recognising a starling from birdsong. We may be able to state some subtasks of recognising a starling from birdsong, but this is not required to have the competence. In this regard we can talk about knowing how to recognise birdsong without being able to state every subtask it entails.
Items of declarative knowledge need to be statable, as statements need to be able to be stated to count as statements. Items of procedural knowledge need not be statable, as the statability of the subtasks of a task is not required to know how to do that task.***
If something is believed, it is a statement. If something is not stateable, it cannot be believed.
As I've stipulated, certainty can be applied to all items of procedural knowledge; I am certain I know how to bake a carrot cake, I am certain I know how to do all of that entails. I also stipulate in all cases that certainty of a task distributes over its subtasks; to be certain that one knows how to bake a carrot cake entails that one is certain that one knows how to do all activities (subtasks) that entails.
This was what I had in my head so far and was trying to reason towards.
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I would like to add the following.
Given all this, it may still be the case that we may only be certain of items of procedural knowledge that we can associate with an item of declarative knowledge; that is, we can only be certain of procedural knowledge items that we can state that we know how to do. For this, I think the paragraph marked with *** would need to be strengthened to:
Items of declarative knowledge need to be enumerable, as statements need to be able to be constructible using the rules of a language to count as statements of that language. The subtasks of a task which are procedural knowledge are not in general enumerable, as the enumerability of the subtasks of a task is not required to know how to do that task.
If it is possible to associate an item of declarative knowledge with the procedural knowledge of the subtasks of every task, then the items of procedural knowledge are necessarily enumerable, which goes against the previous paragraph. (Though I do I believe it is possible to associate an item of declarative knowledge with the larger tasks themselves; as task labels.) An analogy here is taking the integer part of a real number; you can't pair off the integer parts with the reals uniquely. The subtasks play the part of real numbers, the integers play the part of tasks. There is still a sense of equivalence in play; two real numbers (subtasks) are equivalent when they have the same integer part (are part of the same task or subtask aggregate). This is where I was going with the numerical identity weakened to qualitative equivalence stuff; there's too many variations of activity within a competence to state (numerically un-identical activities within the task), but we can label the collection of variations as the competence (qualitatively equivalent activities insofar as they all form part of the same task).
(Read, I claim that there are task components that we cannot state explicitly but can still aggregate, quantify over or incompletely summarise in a manner that does not completely determine each subtask by expressing its propositional content in a statement. "I know how to bake a cake" is true iff I know how to bake a cake, and I know all that baking a cake entails, but each subtask would require an "I know how..." statement, and there are too many.)
In that regard, we can be certain of things (items of procedural knowledge) that we do not believe.
As an intuition pump; characterise belief as a propositional attitude, which is a disposition (towards a statement). We lack dispositions towards most subtask components as they are done without impinging upon access consciousness. So we lack propositional attitudes towards some subtasks, so in particular we do not believe them. Intuitively, we're so immersed in them we don't even need to believe them.
Quoting fdrake
I am certain I can flip an omelet.
So, if I understand you aright, you claim that I can be certain I can flip an omelet, and yet neither believe nor disbelieve the statement: "I can flip an omelet".
You sure about that? Do I have you wrong?
Let me flip the question around on you. Can you state all that you do when flipping an omelet?
I suppose you want more detail. But then the question becomes: how much detail will suffice? What is to count as sufficient? And the answer is: enough to be satisfied; if you are never satisfied, then that's not my problem.
And I am going to maintain that if one is certain that one can flip an omelet, then one believes one can flip an omelet; and this despite one possibly never vocalising that belief. And I will maintain this because it is a performative contradiction to be certain of one's capacity and yet not believe.
This is a statement that you flip the omelette, not a description of how you flipped the omelette.
Quoting Banno
A description, maybe. But the description is never the event it describes (it merely counts as it for some purpose).
It's pretty clear that the content of a description of how something is done must under-determine how it is done; for any given description of how something is done, one must know how to do each subtask entailed in the description that the description is split into. Precisely this inability to go on forever or with completely exhaustive detail is what ensures the description underdetermines the competence. This is the distinction between a description of how to tie a tie, a video of tying a tie, and the knowledge of how to tie a tie.
As an exercise, in trying to describe how to tie a tie, you have to split it up into subtasks and then associate each subtask with a description. The splitting occurs partly because the description is attempted, but it requires that there are subtask components of tying a tie to describe accurately in the first place. The words stop at some point, but do not terminate in the acquisition of the competence.
Count as sufficient for what purposes? For teaching someone how to tie a tie? Words alone won't do. You're imagining a competence from the perspective of already having it, like "I flip the omelette", rather than learning how to use those words; how to make "I can flip an omelette" true - learning how to flip an omelette.
This speaks about being certain of that which can be stated, not which can be done. I agree entirely here, but still think it is appropriate to associate certainty with know how in general, not just declarative knowledge.
Quoting Banno
Done.Quoting fdrake
Yep. So we agree.
I'm certain I can flip an omelet. Yep.
Not a problem.