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.Mathematics and its proofs are presented as a priori truths. Physical observation and experimentation is empirical. — god must be atheist
Both mathematics and science use their procedure to justify their knowledge. So, in both cases, it is about following the correct procedure. In that sense, in both cases, knowledge is justified by formalisms. — alcontali
I wasn't looking at the difference between mathematics and science in this case.
I was rather interested in what they have in common — alcontali
I wonder if computability and epistemology are ultimately not one and the same thing? — alcontali
So, I believe that the core of knowledge-justification always consists of "paperwork", regardless of what knowledge it is about. — alcontali
(My answer to the above was clearly "not", as you know now.) — god must be atheist
a problem is solvable if there exists an effective procedure for deriving the correct answer. — alcontali
It is obvious that a claim can only have the status of knowledge if there somehow exists an effective procedure to verify its justification. — alcontali
Even the truths of the two systems are different. In the empirical world, there are no truths. Only approximations. In the a priori world, the truths are perfect. — god must be atheist
Justification/Rules comes with experience. — Harry Hindu
Small point, maybe not relevant: getting a correct answer and verifying that it is correct are different procedures. — tim wood
And yet we know of unprovable truths. — Banno
Epistemology is broader than computability. — Banno
In mathematics, a legitimate knowledge claim implies the existence of an effective proof procedure for the claim.
— alcontali
And yet we know of unprovable truths. — Banno
So what? — 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. — Banno
The theory needs to be consistent to be usable, but you are not allowed to prove that it is, because in that case this theory is provably inconsistent. — alcontali
Where is that from? — Banno
Gödel's second incompleteness theorem states no consistent axiomatic system which includes Peano arithmetic can prove its own consistency. Stated more colloquially, any formal system that is interesting enough to formulate its own consistency can prove its own consistency iff it is inconsistent. — Wolfram
OK - that's new to me. — Banno
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.I certainly agree that this is the case in the empirical domain. Science certainly works like that, even though mere experience is clearly not enough as a justification. In addition, such justification will still have to satisfy the entire framework of regulations of the scientific method, i.e. paperwork.
On the other hand, justification in the axiomatic domain does not require experience. It is based solely on provability, which is different kind of paperwork. — alcontali
"Axiomatic" knowledge without any reference to the real world is useless. When untethered from the what we observe of the world, our knowledge is meaningless. What you call "axiomatic" knowledge is really just the rule we learned by observing the world. Some people have an issue with distinguishing between following/breaking a rule with the rule itself. Rules are meaningless without a world in which they are followed or broken. — Harry Hindu
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.
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I wonder if computability and epistemology are ultimately not one and the same thing? — alcontali
... a modification of Turing's assumptions does bring practical computation within Turing's limits; as David Deutsch puts it:
"I can now state the physical version of the Church–Turing principle: 'Every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means.' This formulation is both better defined and more physical than Turing's own way of expressing it." — The Church–Turing–Deutsch thesis
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.If you come from the world of science, which is staunchly empirical, you will naturally tend to think that mathematics should be a bit like science and primarily deal with the physical universe. I can imagine that mechanical, construction, -or chemical engineers will also naturally be attracted to an empirical-constructivist take on mathematics.
If you come from the world of programming and its theoretical approach, i.e. computer science, you will not think like that. In that case, you are already used to high levels of meaningless and useless Platonicity. You should be quite used already to high-level structures that are fundamentally divorced from the senses.
Look for example at this example: AbstractObjectFactory. It is a structure-defining absurdity. To what could that structural abstraction possibly correspond in the physical universe? In fact, this source code does not even "do" anything, which is unusual for software, because the idea is that it would otherwise execute some code, but it doesn't even do that.
Absurdity is what naturally emerges out of lengthy abstraction processes. You obtain structures that mean nothing and that are essentially useless. So, yes, high-level abstract structures are naturally useless and meaningless. I am used to that. It is my professional life to deal with that kind of things. That is probably why I can appreciate the beauty of general abstract nonsense, the flagship of mathematics.
Total nonsense can be breathtakingly beautiful as long as it is consistent. It is mostly a question of developing enough intuition for that. Seriously, structural nonsense can even be pleasant to look at. — alcontali
I 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.Look for example at this example: AbstractObjectFactory. It is a structure-defining absurdity. To what could that structural abstraction possibly correspond in the physical universe? In fact, this source code does not even "do" anything, which is unusual for software, because the idea is that it would otherwise execute some code, but it doesn't even do that. — alcontali
I wonder if computability and epistemology are ultimately not one and the same thing?
If it is consistent, then it is incomplete. If it is complete, then it is inconsistent.
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