But the whole point Wittgenstein's argument on the autonomy of mathematics systems is that a mathematical proposition is internally tied to its proof/proof system — Richard B
https://en.wikipedia.org/wiki/Model_theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold).
If dealing with autonomous calculi then no matter how similar the rules of the two systems might be, as long as they differ - as long as we are dealing with distinct mathematical systems - It make no sense to speak of the same proposition occurring in each. The most that can be concluded is that parallel propositions occur in the two systems which can easily be mapped onto each other. — Richard B
Hence Godel was barred by virtue of the logical grammar of mathematical proposition from claiming that he had constructed identical versions of the same mathematical proposition in two different systems. — Richard B
https://web.mat.bham.ac.uk/R.W.Kaye/publ/papers/finitesettheory/finitesettheory.pdf
The work described in this article starts with a piece of mathematical ‘folklore’ that is
‘well known’ but for which we know no satisfactory reference.
Folklore Result. The first-order theories Peano arithmetic and ZF set theory with the
axiom of infinity negated are equivalent, in the sense that each is interpretable in the
other and the interpretations are inverse to each other.
Perhaps the first and most obvious conclusion is that statements concerning the equiv-
alence of ‘Peano Arithmetic’ and ‘ZF with the axiom of infinity negated’ require some
care to formulate and prove. It is certainly true that PA and ‘ZF with the axiom of infin-
ity negated’ are equiconsistent for just about any sensible axiomatisation of the latter,
in the sense that interpretations exist in both directions.6 Probably this is the ‘folklore
result’ that most people remember. But for the finer result with interpretations inverse
to each other, careful axiomatisation of the set theory is required. A category theoretic
framework for interpretations is useful to direct attention to these refinements.
Why have you got it in for the math professors? — Tarskian
But why hate on the math professors? — fishfry
These are all mathematical truths, but they're not very interesting mathematical truths. — fishfry
leaving only the beautiful sculpture that is modern mathematics — fishfry
https://en.wikipedia.org/wiki/Hilbert%27s_program
Statement of Hilbert's program
The main goal of Hilbert's program was to provide secure foundations for all mathematics.
Completeness: a proof that all true mathematical statements can be proved in the formalism.
Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.
...
Kurt Gödel showed that most of the goals of Hilbert's program were impossible to achieve.
Out of the uncountably infinite and random universe of mathematical truth
Right, and despite their work being concluded for quite some time now, people several times smarter than both you and I combined still hold math and science as tools of precision and meaningful discovery. — Philosophim
I find this point more interesting. Why? — Philosophim
https://www.marxists.org/subject/marxmyths/john-holloway/article.htm
In speaking of Marxism as ‘scientific’, Engels means that it is based on an understanding of social development that is just as exact as the scientific understanding of natural development. For Engels, the claim that Marxism is scientific is a claim that it has understood the laws of motion of society. This understanding is based on two key elements: ‘These two great discoveries, the materialistic conception of history and the revelation of the secret of capitalistic production through surplus-value, we owe to Marx. With these two discoveries Socialism becomes a science. The claim that Marxism is scientific is taken to mean that subjective struggle (the struggle of socialists today) finds support in the objective movement of history. The notion of Marxism as scientific socialism has two aspects. In Engels’ account there is a double objectivity. Marxism is objective, certain, ‘scientific’ knowledge of an objective, inevitable process. Marxism is understood as scientific in the sense that it has understood correctly the laws of motion of a historical process taking place independently of men’s will. All that is left for Marxists to do is to fill in the details, to apply the scientific understanding of history. The attraction of the conception of Marxism as a scientifically objective theory of revolution for those who were dedicating their lives to struggle against capitalism is obvious. At the same time, however, both aspects of the concept of scientific socialism (objective knowledge, objective process) pose enormous problems for the development of Marxism as a theory of struggle.
Again, hyperbole. I can assure you if we were able to predict how everything in the universe worked — Philosophim
No, nothing you have discovered here has shaken the foundations of math or science. — Philosophim
What method did you use to find out that its true? — Philosophim
The case for a higher authority, an absolute authority, has to be argued philosophically. Not religiously, that is, not according anything so instantly assailable. — Constance
It sounds as though you yourself hold some rather specific and rigid beliefs that likewise are not entirely objective in their genesis. — Pantagruel
Just like I wouldn't grab a wrench if I were studying the atomic level of the universe, one shouldn't use certain language and terms when dealing with the foundations of knowledge and mathematics. — Philosophim
The hyperbole just isn't true. — Philosophim
Gödel’s famous incompleteness theorem showed us that there is a statement in basic arithmetic that is true but can never be proven with basic arithmetic. That is just the beginning of the story.
it took other people to fix the inconsistency in his proof just to then generate further issues in these updated proofs. — Lionino
In modal logic, modal collapse is the condition in which every true statement is necessarily true, and vice versa; that is to say, there are no contingent truths, or to put it another way, that "everything exists necessarily".
And everytime when someone makes an universal statement that ought to apply to everything, watch out! — ssu
I think you’re mis-using the word there. If everything were chaotic, nothing would exist, and if everything were perfectly ordered, nothing would change. Existence requires both. Beyond that, I can’t see the point, if there is one. — Wayfarer
Hippasus, who found irrational numbers was ostracized and when he drowned at sea, it was the "punishment of the Gods". — ssu
Out of interest, what type of believer are you? Muslim or Christian, or something less specific? — Tom Storm
Charles V's "Edict of Blood" of 1550 in the Burgundian Netherlands
No one shall print, write, copy, keep, conceal, sell, buy or give in churches, streets, or other places, any book or writing made by Martin Luther, John Oecolampadius, Huldrych Zwingli, Martin Bucer, John Calvin, or other heretics reprobated by the Holy Church.
...
That such perturbators of the general quiet are to be executed, to wit: the men with the sword and the women to be buried alive, if they do not persist in their errors; if they do persist in them, then they are to be executed with fire; all their property in both cases being confiscated to the crown.
Perhaps some can see this as chaotic, but math itself is quite logical and hence quite orderly. Unprovability or uncomputability doesn't mean chaotic. Math is orderly, we just have limitations on what to compute or prove. — ssu
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.
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]
This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution[8] and is fully determined by their initial conditions, with no random elements involved.[9] In other words, the deterministic nature of these systems does not make them predictable.[10][11] This behavior is known as deterministic chaos, or simply chaos.
Even if it obviously c is a natural number and has a precise point on the number line, not some range, we cannot prove c exactly. — ssu
The problem rises because we just assume that everything in math has to be provable. — ssu
https://en.wikipedia.org/wiki/Hilbert%27s_program
Statement of Hilbert's program
- A formulation of all mathematics; in other words all mathematical statements should be written in a precise formal language, and manipulated according to well defined rules.
- Completeness: a proof that all true mathematical statements can be proved in the formalism.
- Consistency: a proof that no contradiction can be obtained in the formalism of mathematics. This consistency proof should preferably use only "finitistic" reasoning about finite mathematical objects.
- Conservation: a proof that any result about "real objects" obtained using reasoning about "ideal objects" (such as uncountable sets) can be proved without using ideal objects.
- Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.
The existential claim carries the onus probandi (generally, existential claims are verifiable and not falsifiable, universal claims are falsifiable and not verifiable), it's not for someone else to disprove. — jorndoe
This is a far cry from the point that math can be difficult to put into words. The proof is in the very fact you're able to post online consistently for us to read your posts. That was all capable through math. — Philosophim
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.