It sounds as though you yourself hold some rather specific and rigid beliefs that likewise are not entirely objective in their genesis. — Pantagruel
Well, yeah, I rigidly believe that we should not give powers to people that only Allah should have, and if Allah does even not exist, then so much the better. — Tarskian
If you refer to "an universal statement that ought to apply to everything", I would agree (assuming I understood your point).That may very well be in violation of Carnap's diagonal lemma:
"For each property of logic sentences, there exists a true sentence that does not have it, or a false sentence that does."
But then again, it still needs to be a property of logic sentences. For example, a property of natural numbers can apply to all natural numbers. — Tarskian
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.
If you allow infinite strings, of course, you can have uncountably many strings. That's the difference between positive integers, which have finitely many digits; and real numbers, which have infinitely many. That's why the positive integers are countable and the real numbers uncountable. It's the infinitely long strings that make the difference. But natural language doesn't allow infinitely long strings. Every word or sentence is finite, so there can only be countably many of them.
Well, instead of being able to predict just 0.1% of the facts in the physical universe, this would improve to something like 0.3%; not much more. — Tarskian
Scientism is widespread as an ideology in the modern world. Any true understanding of the nature of mathematical truth deals a devastating blow to people who subscribe to it. This is exactly why I like this subject so much. — Tarskian
In 1931, Gödel's incompleteness theorems dealt a major blow to positivism and scientism, but it was just the beginning. It is only going to keep getting worse. As Yanovsky writes in his paper:
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. — Tarskian
In my opinion, scientism needs to get attacked and destroyed because its narrative is not just arrogant but fundamentally evil — Tarskian
It is a false pagan belief that misleads its followers to accept untested experimental vaccine shots from the lying and scamming representatives of the pharmaceutical mafia; and that is just one of the many examples of why it is not hyperbole. — Tarskian
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
No, nothing you have discovered here has shaken the foundations of math or science.
— Philosophim
I did not discover anything. Gödel certainly did. Chaitin also did. Yanofsky moderately did. I just mentioned their work. — Tarskian
My problem is with positivism and scientism. I find these ideological beliefs to be very dangerous. — Tarskian
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.
It is very convincing, because it sounds scientific, and because it insists that it is scientific, and especially because you will get burned at the Pfizer antivaxxer stake if you refuse to memorize this sacred fragment from the scripture of scientific truth for your scientific gender studies exam
If you feel threatened by its chaotic nature, it means that it disturbs your ideological beliefs — Tarskian
I’m detecting a distinct political slant here. Is it Libertarianism? Trumpism? Anarchism? Would I be right to surmise that you are not a backer of climate change science?It is very convincing, because it sounds scientific, and because it insists that it is scientific, and especially because you will get burned at the Pfizer antivaxxer stake if you refuse to memorize this sacred fragment from the scripture of scientific truth for your scientific gender studies exam.
As you can see, everybody who craves credibility insists on sailing under the flag of scientism and redirect the worship and adulation of the masses for the omnipotent powers of science to themselves and their narrative. — Tarskian
I would agree with @Tarskian, especially a mix of both can be harmful, because one can come to be so dogmatic that one starts to think that model or theory of reality is far more real than just the reality itself. And this dogmatism leads people forgetting that scientific theories are only models of reality. You don't care how real life is different from the scientific model, the model itself is right.My problem is with positivism and scientism. I find these ideological beliefs to be very dangerous.
— Tarskian
I find this point more interesting. Why? — Philosophim
The true nature of the universe of mathematical facts makes lots of people uncomfortable.
Imagine that we had a copy of the theory of everything?
It would allow us to mathematically prove things about the physical universe. It would be the best possible knowledge that we could have about the physical universe. We would finally have found the holy grail of science.
What would the impact be?
Well, instead of being able to predict just 0.1% of the facts in the physical universe, this would improve to something like 0.3%; and not much more. — Tarskian
If you feel threatened by its chaotic nature, it means that it disturbs your ideological beliefs. Someone who uses them as tools of precision and meaningful discovery would never feel threatened by that. — Tarskian
It is probably best to use an example from Soviet Union but in fact modern western society does exactly the same: — Tarskian
and especially because you will get burned at the Pfizer antivaxxer stake if you refuse to memorize this sacred fragment from the scripture of scientific truth for your scientific gender studies exam. — Tarskian
I would agree with Tarskian, especially a mix of both can be harmful, because one can come to be so dogmatic that one starts to think that model or theory of reality is far more real than just the reality itself. — ssu
Yes, absolutely! All the various philosophical schools of thought have contributed each their way. Even if I criticize reductionism and favour the idea of more-is-different, there's a place for reductionism. Yet positivist are the one's that can quite easily fall into that dogmatism.Isn't your problem with dogmatism, or a misuse and/or misunderstanding of science/positivism, instead of with science/positivism itself? — Philosophim
But a sentence is not the same as a string. — Treatid
The interpretation of a sentence depends on the context/axioms. The same string in two axiomatic systems is two distinct sentences. — Treatid
However, the assertion that natural languages are countably infinite no longer holds given there are an uncountable infinite number of contexts for any given sentence. — Treatid
Most mathematical truth is unprovable and therefore unpredictable, if only because most of its truth is ineffable ("inexpressible"). — Tarskian
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
Sure. What this argument purports to show is that a natural language has no fixed cardinality. And this is what we might expect, if natural language includes the whole of mathematics and hence transfinite arithmetic.For natural language to be uncountable, you must find a sentence that cannot be added to the list. To that effect, you would need some kind of second-order diagonal argument. — Tarskian
Not I, but Langendoen and Postal. If you wish you can take up the argument, I'm not wed to it, I'll not defend it here. I've only cited it to show that the case is not so closed as might be supposed from the Yanofsky piece. Just by way of fairness, Pullum and Scholz argue against assuming that natural languages are even infinite.I didn't completely follow what you're doing, but in taking the powerset of a countably infinite set, you are creating an uncountable one. There aren't uncountably many words or phrases or strings possible in a natural language, if you agree that a natural language consists of a collection of finite-length strings made from at an most countably infinite alphabet. I think this might be a flaw in your argument, where you're introducing an uncountable set. — fishfry
But the point is that "...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" appears misguided, and at the least needs a better argument.
Your posts sometimes take maths just a little further than it can defensibly go. — Banno
http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf
Another important uncountably infinite set is the collection of subsets of the natural numbers. The collection of all such subsets is uncountably infinite. Now that we have these different notions of infinity in our toolbox, let us apply them to our concept of true but unprovable statements. All language is countably infinite. The set of statements in basic arithmetic, the subset of true statements, and the subset of provable statements are all countably infinite.
This brings to light an amazing limitation of the power of language. The collection of all subsets of natural numbers is uncountably infinite while the set of expressions describing subsets of natural numbers is countably infinite. This means that the vast, vast majority of subsets of natural numbers cannot be expressed by language. The above examples of subsets of natural numbers are expressed by language, but they are part of the few rather than the many. The majority of the subsets are inexpressible. They defy language.
https://math.stackexchange.com/questions/1206460/proving-that-the-set-of-all-english-words-is-countble
This is the question : Prove that the set of all the words in the English language is countable (the set's cardinality is אo) A word is defined as a finite sequence of letters in the English language.
Answer 1: There are 26 letters in the English language. Consider each letter as one of the digits on base 27. This mapping yields that the cardinality of your set is ≤|N|, hence this set is countable.
Answer 2: The set Sn of the English words with length n is finite (this is almost obvious). So it's also countable. Why is it finite? The set An of all sequences with length n made up of latin characters is finite as it contains 26n elements. Only some of these sequences are meaningful/actual English words. So Sn⊂An. So Sn is also finite. The set T for which you have to prove that it is countable is: T=S1∪S2∪S3∪... Now you have this theorem: "A countable union of countable sets is also countable". Applying it you get that T is also countable. Thus your statement has been proved.
Not I, but Langendoen and Postal. If you wish you can take up the argument, I'm not wed to it, I'll not defend it here. I've only cited it to show that the case is not so closed as might be supposed from the Yanofsky piece. — Banno
https://aclanthology.org/J89-1006.pdf
This book is an extended argument in support of the theses that natural languages are transfinitely unbounded collections, that sentences are not limited in length (number of words) by any cardinal number, finite or transfinite, and that no constructive grammar can be an adequate grammar for any natural language.
https://fa.ewi.tudelft.nl/~hart/37/publications/the_papers/on_vastness.pdf
However, as I mentioned before, the authors do not so much argue for “not assuming a size law” but for “assuming the negation of a size law”. For example, the rules (if any) of English do not stipulate a maximum finite length of sentences; one can easily break such a stipulation by prefixing a maximum length sentence with “I know that”. The rules of English also do not explicitly state that sentences should be finite; one can add “All English sentence should be finite in length” to the rules or not. The authors argue, quite vociferously at times, against adding that condition mostly on the grounds that it is not a purely linguistic one. However, and this is where I disagree, they then conclude that, somehow, necessarily there should be sentences of infinite length.
https://en.wikipedia.org/wiki/First-order_logic
Infinitary logic allows infinitely long sentences. For example, one may allow a conjunction or disjunction of infinitely many formulas, or quantification over infinitely many variables. Infinitely long sentences arise in areas of mathematics including topology and model theory.
Infinitary logic generalizes first-order logic to allow formulas of infinite length. The most common way in which formulas can become infinite is through infinite conjunctions and disjunctions. However, it is also possible to admit generalized signatures in which function and relation symbols are allowed to have infinite arities, or in which quantifiers can bind infinitely many variables. Because an infinite formula cannot be represented by a finite string, it is necessary to choose some other representation of formulas; the usual representation in this context is a tree. Thus, formulas are, essentially, identified with their parse trees, rather than with the strings being parsed.
Not I, but Langendoen and Postal. If you wish you can take up the argument, I'm not wed to it, I'll not defend it here. I've only cited it to show that the case is not so closed as might be supposed from the Yanofsky piece. Just by way of fairness, Pullum and Scholz argue against assuming that natural languages are even infinite. — Banno
Langendoen and Postal do not agree that "a natural language consists of a collection of finite-length strings". — Banno
Does mathematics also "consists of a collection of finite-length strings made from an at most countably infinite alphabet"? — Banno
Also, doesn't English (or any other natural language) encompass mathematics? It's not that clear how, and perhaps even that, maths is distinct from natural language. — Banno
All of which might show that the issues here are complex, requiring care and clarity. There's enough here for dozens of threads. — Banno
This was my first thought. Natural languages would seem to need to be computable, which would entail countably finite. — Count Timothy von Icarus
Any proof will contain at most a finite number of characters. At least for us finite entities.Any readable proof of Cantor's Theorem will contain at most a finite number of characters. — Banno
That's actually not Cantor's theorem (the power set of any set has a strictly greater cardinality than the set itself).Yet it shows that there are numbers with a cardinality greater than ℵ0. — Banno
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