Nevertheless, and to all practical purposes, mathematics enables a very wide range of successful predictions, doesn’t it? The mathematical physics underlying the technology on which this conversation is being conducted provides a high degree of prediction and control, doesn’t it? Otherwise, it wouldn’t work. — Wayfarer
For the indifferent or one who finds the question incoherent it is not a matter of truth value, and that is the point. So, Joshs "none of the above": seems most apt. — Janus
I believe Chaitin made a similar point. He has a proof of Gödel's incompleteness theorems from algorithmic complexity theory. I believe he says that mathematical truth is essentially random. Things are true just because they are, not because of any deeper reason.
This sounds related to what you're saying. — fishfry
http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf
Gregory Chaitin described an innovative way of finding true but unprovable statements. He started by examining the complexity of the axioms of a logical system. He showed that there are certain statements that are much more complex than the axioms of the system. Such statements are true but cannot be proven by the axioms of the logical system. The following motto is sometimes used to explain this:
“A fifty-pound logical system cannot prove a seventy-five-pound theorem.”
In particular, basic arithmetic is a logical system that has a level of complexity and so there are certain types of statements that are true but too complex to be proven using basic arithmetic. The main point for our story is that within basic arithmetic we can always find more complicated statements of a certain type. Hence, there are infinitely many true but unprovable statements.
Cristian Calude extended Chaitin’s findings. He demonstrated that provable statements are actually very rare within the space of all true statements. In a sense, he showed that in the space of all true statements, every provable true statement is surrounded by many unprovable true statements.
In Godel, it appears consistency is assumed — tim wood
Why should we suppose that natural languages are only countably infinite? — Banno
https://en.wikipedia.org/wiki/Countable_set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers.[a] Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.
Could you say a little more about what makes an unprovable mathematical proposition true? — Joshs
He attributes to Godel this idea:
:“'Basic arithmetic cannot prove a contradiction.' — tim wood
As a bonus, Gödel described another interesting statement in the language of basic arithmetic. He was able to formulate a statement in basic arithmetic that says:
“Basic arithmetic cannot prove a contradiction.”
http://sammelpunkt.philo.at/id/eprint/2676/1/Bagaria.pdf
page 12:
Let CON(T) be the sentence ¬BewT(Í⊥Î). Thus, CON(T) says, via coding, that T is consistent.
http://www.sfu.ca/~kabanets/308/lectures/lec11.pdf
We say that a proof system P is consistent if P does not prove both A and ¬A for some sentence A. That is, a consistent proof system cannot derive a contradiction A ∧ ¬A. In the case of a proof system P for arithmetic, we get that P is consistent iff P does not prove the sentence “1 = 2” (since 1 6 = 2 can be derived in P by the usual axioms (of Peano arithmetic) for the natural numbers).
ConsP : “the sentence “1 = 2” is not provable in P ”
It turns out that this statement is also true but unprovable.
So you would have 'don't care' mapped to unknown? — Tom Storm
perhaps due simply to a complete lack of interest — Janus
Contrary to what the weekly sophist implies, choice of axioms is not arbitrary. — Lionino
As previously stated, you have not read the article you yourself linked. Congrats. — Lionino
But religion is certainly not about this. It is about ethics. What is ethics? — Constance
Well,
PA is a chaotic complex system without initial conditions. — Tarskian
looks a bit... overstated. — Banno
http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf
We have come a long way since Gödel. A true but unprovable statement is not some strange, rare phenomenon. In fact, the opposite is correct. A fact that is true and provable is a rare phenomenon. The collection of mathematical facts is very large and what is expressible and true is a small part of it. Furthermore, what is provable is only a small part of those.
That the conclusions follow from the premises can be said about every fiction book — Lionino
If you had actually read the "article" you linked, you would know that Gödel's original axioms are inconsistent — Lionino
Aren't these the "initial conditions"...? These are the Peano axioms:
Zero is a natural number.
Every natural number has a successor in the natural numbers.
Zero is not the successor of any natural number.
If the successor of two natural numbers is the same, then the two original numbers are the same.
If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers.
It's far from obvious what this has to do with chaotic systems.
I'm not following Tarskian's argument at all. — Banno
https://en.m.wikipedia.org/wiki/Non-standard_model_of_arithmetic
In mathematical logic, a non-standard model of arithmetic is a model of first-order Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934).
might have been done by any number of fanatics (Castro, Hitler, Putin, whoever) — Tom Storm
The effectiveness of math can be demonstrated through its consistency and predictability. — Tom Storm
https://www.hawking.org.uk/in-words/lectures/godel-and-the-end-of-physics
What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.
https://en.wikipedia.org/wiki/Chaos_theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities.[1] Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization.[2]
even within the single religion. It is unpredictable and inconsistent.
What do you mean by the term "existence"? — 180 Proof
https://en.wikipedia.org/wiki/Existential_quantification
Existential quantification
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)" or "(∃x)").
You are unhappy with students being taught the state of the art in their field. — fishfry
The president of the United States draws a paycheck. — fishfry
All the 18 year olds are apprenticed out to people who will pay them even though they're completely ignorant? You can't be serious. What are you talking about? — fishfry
So maybe you're against large organizations. — fishfry
We don’t prove existence ... We might be able to prove a god wouldn’t struggle, or a god wouldn’t need sleep, but we can’t prove that struggle-free, always awake god exists. — Fire Ologist
https://en.wikipedia.org/wiki/G%C3%B6del%27s_ontological_proof
Gödel's ontological proof is a formal argument by the mathematician Kurt Gödel (1906–1978) for the existence of God.
The proof[8][10] uses modal logic, which distinguishes between necessary truths and contingent truths.
Most criticism of Gödel's proof is aimed at its axioms.
the difference between ~b(G) and b(~G) — 180 Proof
not belief("God exists")
belief("God does not exist")
True, False, Not-True — 180 Proof
Math axioms can be shown to work. — Tom Storm
Religion cannot demonstrate gods. — Tom Storm
I would call that evidence of religion's disfunction. — Tom Storm
Or do you think the supposed truths held by Marxists — Tom Storm
We know that people can be galvanized by deception and undemonstrated beliefs. — Tom Storm
If the difference between faith and reason isn't obvious to people — ssu
https://en.m.wikipedia.org/wiki/Foundationalism
Identifying the alternatives as either circular reasoning or infinite regress, and thus exhibiting the regress problem, Aristotle made foundationalism his own clear choice, positing basic beliefs underpinning others.
https://en.m.wikipedia.org/wiki/Basic_belief
Beliefs therefore fall into two categories:
- Beliefs that are properly basic, in that they do not depend upon justification of other beliefs, but on something outside the realm of belief (a "non-doxastic justification").
- Beliefs that derive from one or more basic beliefs, and therefore depend on the basic beliefs for their validity.
We accept science and math because they work — Tom Storm
My point is that math demonstrates its utility — Tom Storm
We still can't demonstrate that there are any gods. We can demonstrate that math works. We seem unable to get past this point. — Tom Storm
Yes, it can be but that formulation is not popular – though it's formerly my preferred position (while quite reasonable, it's too narrow in scope): — 180 Proof
https://en.m.wikipedia.org/wiki/Atheism
Atheism, in the broadest sense, is an absence of belief in the existence of deities. Less broadly, atheism is a rejection of the belief that any deities exist. In an even narrower sense, atheism is specifically the position that there are no deities.
I'm not a foundationalist. — Tom Storm
We do not say there is no god, that would be making a positive claim. — Tom Storm
For many, atheism is about belief not knowledge. — Tom Storm
https://en.wikipedia.org/wiki/Gettier_problem
The JTB account holds that knowledge is equivalent to justified true belief; if all three conditions (justification, truth, and belief) are met of a given claim, then we have knowledge of that claim.
The quesion we are addressing is - is there good reason to belive in god the way there are good reasons to believe in math? — Tom Storm
Religion all over the world behaves like a political party - theism being incidental to its machinations — Tom Storm
https://en.wikipedia.org/wiki/Mandate_of_Heaven
The Mandate of Heaven (Chinese: 天命; pinyin: Tiānmìng; Wade–Giles: T'ien1-ming4; lit. 'Heaven's command') is a Chinese political ideology that was used in Ancient China and Imperial China to legitimize the rule of the king or emperor of China.[1] According to this doctrine, Heaven (天, Tian) bestows its mandate[a] on a virtuous ruler. This ruler, the Son of Heaven, was the supreme universal monarch, who ruled Tianxia (天下; "all under heaven", the world).[3] If a ruler was overthrown, this was interpreted as an indication that the ruler was unworthy and had lost the mandate.[4]
The difference is it misses a key factor. Demonstration of effectiveness. We have good reasons to accept math and the axioms because we can demonstrate their effectiveness. Anyone can do this at any time. — Tom Storm
https://writings.stephenwolfram.com/2014/08/computational-knowledge-and-the-future-of-pure-mathematics/
So how big is the historical corpus of mathematics? There’ve probably been about 3 million mathematical papers published altogether—or about 100 million pages, growing at a rate of about 2 million pages per year. And in all of these papers, perhaps 5 million distinct theorems have been formally stated.
We can't even agree on which gods or why gods or how gods. — Tom Storm
No one has asked for a "mathematical proof" — 180 Proof
only you have offered one that amounts to nothing more than a "higher-order modal" tautology. — 180 Proof
By "faith" I mean worship of supernatural mysteries e.g. "a god" (re: OP), not mere (un/warranted) trust in a usage or practice. Context matters. — 180 Proof
"Godlike" (e.g. Spinoza's metaphysical Deus, sive natura) is not equivalent to any supernatural god (e.g. "God of Abraham") so this "proof" is theologically irrelevant. — 180 Proof
More specifically, his argument consists of some undecidable (i.e. disputable) formal axioms — 180 Proof
https://en.wikipedia.org/wiki/Undecidable_problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer.
even if valid, it is not sound — 180 Proof
nothing nonformal, or concrete, is "proven". — 180 Proof
This might shed more light on where you think Wittgenstein went wrong. — Joshs
Could you provide your own critique of Platonic explanations of the mathematics, lie that of Goedel, or the correspondence theory of truth? This might shed more light on where you think Wittgenstein went wrong. — Joshs
https://en.wikipedia.org/wiki/Remarks_on_the_Foundations_of_Mathematics
Wittgenstein wrote
I imagine someone asking my advice; he says: "I have constructed a proposition (I will use 'P' to designate it) in Russell's symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: 'P is not provable in Russell's system'. Must I not say that this proposition on the one hand is true, and on the other hand unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable." Just as we can ask, " 'Provable' in what system?," so we must also ask, "'True' in what system?" "True in Russell's system" means, as was said, proved in Russell's system, and "false" in Russell's system means the opposite has been proved in Russell's system.—Now, what does your "suppose it is false" mean? In the Russell sense it means, "suppose the opposite is proved in Russell's system"; if that is your assumption you will now presumably give up the interpretation that it is unprovable. And by "this interpretation" I understand the translation into this English sentence.—If you assume that the proposition is provable in Russell's system, that means it is true in the Russell sense, and the interpretation "P is not provable" again has to be given up. If you assume that the proposition is true in the Russell sense, the same thing follows. Further: if the proposition is supposed to be false in some other than the Russell sense, then it does not contradict this for it to be proved in Russell's system. (What is called "losing" in chess may constitute winning in another game.)
Just as we can ask, " 'Provable' in what system?," so we must also ask, "'True' in what system?"
If you assume that the proposition is provable in Russell's system, that means it is true in the Russell sense, and the interpretation "P is not provable" again has to be given up.