https://en.wikipedia.org/wiki/Mathematical_beauty
Mathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics.
http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf
There are more true but unprovable statements than we can possibly imagine.
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.
The world of mathematical truth does not look like most people believe it does. It is not orderly. It is fundamentally unpredictable. It is highly chaotic — Tarskian
A convolute argument, perhaps, but it shows that one must do more than simply assert that natural languages are at most countably infinite. Yanofsky must argue his case. " ...the collection of all properties that can be expressed or described by language is only countably infinite because there is only a countably infinite collection of expressions" begs the question. Indeed, the argument above shows it to be questionable.1. Let L be the NL English.
2. The set S0 is contained in L, where
S0= {Babar is happy; I know that Babar is happy; I know that I know that Babar is happy; . . .}
3. S1 may be constructed as follows
a. Form the set of all subsets of S0, P(S0).
b. For each element B in P(S0), form the sentence that is the coordinate conjunction of all the sentences in B.
c. Let S1 be the collection of all sentences formed in (3b).
S1 = {Babar is happy; I know that' Babar is happy; I know that I know that Babar is happy; ... ; Babar is happy and I know that Babar is happy; Babar is happy and I know that I know that Babar is happy;... ;Babar is happy, I know that Babar is happy, and I know that I know that Babar is happy;...}
4. S0 is denumerable, but S1, which is equinumerous with P(S0) is not denumerable (by Cantor's Theorem).
5. S2, S3, etc., can be constructed analogously. Each successive S has a greater transfinite cardinality than the one preceding it.
6. All of the S collections are contained within L.
7. L has no fixed cardinality. — The Vastness of Natural Language
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.
Could you say a little more about what makes an unprovable mathematical proposition true? — Joshs
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.
The world of mathematical truth does not look like most people believe it does. It is not orderly. It is fundamentally unpredictable. It is highly chaotic. — Tarskian
Why should we suppose that natural languages are only countably infinite? — Banno
a. Form the set of all subsets of S0, P(S0). — The Vastness of Natural Language
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.
Unlike what most people believe, math is not more orderly than the physical universe itself. — Tarskian
Yes, Yanofsky's paper also mentions Chaitin's work: — Tarskian
Yanofsky's paper mentions an even larger class of random mathematical truth: unprovable because ineffable ("inexpressable"). There is no way to prove truths that cannot even be expressed in language. — Tarskian
The world of mathematical truth does not look like most people believe it does. It is not orderly. It is fundamentally unpredictable. It is highly chaotic. — Tarskian
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
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
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.The world of mathematical truth does not look like most people believe it does. It is not orderly. It is fundamentally unpredictable. It is highly chaotic. — Tarskian
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 basic problem is that people simply have these ideas what mathematics should be like and don't notice that their own premises, which they hold as axioms (obviously! What else they could they be?), aren't actually true. And when those "axioms" aren't true, we end up somewhere in a paradox.Hilbert believed it so strongly that he insisted that all his colleagues should work on proving the above. A lot of people still believe it. You can give them proof that it is absolutely impossible, but they simply don't care about that. They will just keep going as if nothing happened. You can't wake a person who is pretending to be asleep. — Tarskian
Gödel didn't make it easy. In my opinion Cantor's diagonalization is an easier model. Or basically just use negative self reference with avoiding a Cretan liar situation.This is, in fact, the only hard part in Gödel's proof. The proof for the lemma is very short but it is widely considered to be incomprehensible: — Tarskian
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
And everytime when someone makes an universal statement that ought to apply to everything, watch out! — ssu
So in a way, negative self reference in my opinion is a very essential building block for logic. — ssu
The set of finite-length strings over an at most countably infinite alphabet is countable. There are countably many strings of length 1, countably many of length 2, dot dot dot, therefore countably many finite strings. — fishfry
That is the only direction that we use in engineering. We never use the other direction:
If it is true, then it is pretty much never provable. It is a rare exception, if it is. — Tarskian
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.
dangerously false pagan belief that misleads its followers into accepting untested experimental vaccine shots from the lying and scamming representatives of the pharmaceutical mafia — Tarskian
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