Is there even such a thing as a strictly random number. — Wittgenstein
https://en.wikipedia.org/wiki/Kolmogorov_complexityKolmogorov randomness defines a string (usually of bits) as being random if and only if it is shorter than any computer program that can produce that string. To make this precise, a universal computer (or universal Turing machine) must be specified, so that "program" means a program for this universal machine. A random string in this sense is "incompressible" in that it is impossible to "compress" the string into a program whose length is shorter than the length of the string itself. — Wiki
Quite.Which of the two questions do you answer? both? — Mephist
Yeah, I got that, but don't see the problem with it - if you look at infinite lines as stretched out finite lines.I thought this is obviously absurd. OK, if I have to say why: because infinite lines always intersect if they are not parallel, and finite segments can be not parallel and not intersect — Mephist
if you don't allow the existence of infinite sets, you have to treat segments as a different kind of thing than a set of points.
But in the galois field, they treat the segment made of points but don't use infinite sets, the one mentioned in the article.Can you send me any article,book recommendations that make your position clear to me cause l may be confusing your point here, I hope not. — Wittgenstein
owever, the statement, " an infinite number of elements " is a contradiction in terms as infinity is clearly not a number — Wittgenstein
Secondly, the symbol ∞ is not irrelevant or immaterial, because it is definitely mentioned in usable reduction rules that can successfully extend classical arithmetic. For example: a + ∞ = ∞ and also a * ∞ = ∞, with for example, a ∈ ℝ. The symbol clearly has some kind of "absorbing" effect on other elements in its domain — alcontali
Let's consider the formalist view of math. I think that mathematics is primarily based on substitution where we replace a set of symbols with another set of symbols which are equal or equivalent in some cases. How do we decide that ? By "meaning" l meant the criterion for substituting one expression to another. Formalism has axioms and there are rules of inference etc. It cannot work without them.In the viewpoint that math is about symbol manipulation formalisms, we may not even be interested in the concept "meaning" as some kind of correspondence factor with the real, physical world.
The problem with using the infinity symbol is that there are infinities bigger than others. It is a single character but can we substitute it with numbers ? Consider the real line, all the real number lie on it but infinity doesn't. We can by some fancy definitions extend it to hyper-real and have the rules of adding numbers to infinity like a+infinity=infinity etc. Can you generate this symbol by any finite amount of operations ? I dont think we can and in my opinion formalism is basically about operations on symbols ? Therefore by allowing infinity, we sort of compromise the formal system. This is the basic idea behind the constructivist approach, if l am wrong, you are more than welcome to correct me.For a starters, the symbol ∞ is just one character and not an ever-growing sequence of characters. Hence, it is perfectly suitable for participation in symbol-manipulation formalisms. An ever-growing sequence of characters, however, would be a problem, because in that case our symbol-manipulation algorithm may not even terminate.
I think that when we introduce the infinity symbol, we will have to drop associative law and commutative law too. There is a theorem by rieman which says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, or diverges. This doesn't apply to finite series for a reason and that is different laws regulate the symbols which have finite connotation and those which have an infinite connotation to them.With the various reduction rules available, the symbol ∞ could actually be useful when you seek to produce a closed form output result from a particular input expression. The domain does not even need to be numerical and the algebraic structure not necessarily a field. The rule templates will undoubtedly still be consistent.
The real system has been extended to the hyper real and with it's own extended rules for operation but can we construct the equal or even equivalent of this symbol by same set of operation. By introducing the symbol into the rules and not being able to generate it from the real numbers is cheating. Is this extension valid ?In other words, a field or other algebraic structure can successfully be extended with the symbol ∞ while maintaining consistency and while satisfying the pattern in existing rule templates for the symbol. So, infinity may indeed not be a number but it is certainly a legitimate extension element in numerical algebraic structures.
However, the statement, " an infinite number of elements " is a contradiction in terms as infinity is clearly not a number but sets have definite number of elements and on other hand infinity is not definite.How can we justify the existence of infinite sets. — Wittgenstein
How can we justify the existence of infinite sets. — Wittgenstein
In case of sets, we use natural numbers to determine the cardinality but putting that aside, l would say there is 1-1 correspondence between all real numbers and a point on real number line, infinity doesn't lie there. How do you define numbers ? Most people do define a set in mathematics as a collection of well defined and distinct elements. Infinity is not an element even in the infinite natural set( or any other infinite sets). If you regard infinity as number, that implies that it is finite, since all numbers are finite.Hence a contradiction in terms.How do you define numbers ( the real numbers ) ? How can you justify infinity as a number ?Depends on how you define the word "number". If you define it narrowly, to mean "a natural number", then you're right, infinity is not a number. But that's not how most people define it. Likewise, if you define the word "set" narrowly, to mean "a finite collection of elements", then you're right, there are no infinite sets. But again, that's not how most people define it.
I never compared the physical world with the mathematical one. Even if you consider the "conceptual existence " we cannot construct infinity even with all the symbols and operations in a system. This is not a physical limitation but a conceptual one. The concept of infinity does not allow it to be constructed out of numbers. Consider the case of halting problem precisely that of determining, from a description of an arbitrary computer program and an input, whether the program will finish running (i.e., halt) or continue to run forever. It was proven to be impossible and this is a conceptual restraint not an empirical one. Similar case applies to infinity.Make sure you don't confuse conceptual existence with empirical existence. The existence of the concept of infinite set is one thing and the existence of infinite collections of physical objects out there in the world is another. The former kind of existence is clearly real, the latter can be disputed.
I think that mathematics is primarily based on substitution where we replace a set of symbols with another set of symbols which are equal or equivalent in some cases. How do we decide that ? By "meaning" l meant the criterion for substituting one expression to another. — Wittgenstein
Formalism has axioms and there are rules of inference etc. It cannot work without them. — Wittgenstein
It is a single character but can we substitute it with numbers ? — Wittgenstein
Consider the real line, all the real number lie on it but infinity doesn't. — Wittgenstein
Therefore by allowing infinity, we sort of compromise the formal system. This is the basic idea behind the constructivist approach, if l am wrong, you are more than welcome to correct me. — Wittgenstein
I think that when we introduce the infinity symbol, we will have to drop associative law and commutative law too. — Wittgenstein
By introducing the symbol into the rules and not being able to generate it from the real numbers is cheating. Is this extension valid ? — Wittgenstein
If we look closely, you are in fact using k an arbitrary sub expression repeatedly. Let's say abcRefg where R is a relation or to simplify pRq, where the R can be an equal symbol or an inequality. Let's say we want to generate numbers from this system by a function F(x), where the input is a string of characters and the output is a number.Say that there exists the following rewrite rule in the system: kk* --> k+, with k any arbitrary subexpression, then we can rewite the expression xyabc(abc)*rs --> xy(abc)+rs. This has no "meaning". The resulting expression is just the result of the mechanical application of the rewrite rule on the original expression.
By alleging infinity to be a platonic abstraction doesn't help us understand its nature at all. I don't think there is a platonic world where all mathematical ideas can be found and the existence of non euclidean geometry proves that we have to create new maths a lot times by simply dropping some axioms ( parallel line axiom) and hence modify our system.Infinity itself is a Platonic abstraction that is compatible with numbers, which are themselves also Platonic abstractions. Numbers are themselves no real-world objects either. Infinity is compatible when you can extend the rules for arithmetic to support the inclusion of infinity. while not damaging the algebraic structure.
I just read about elliptic curves, the point at infinity isn't something lying at infinity. They use that term when they draw an elliptic graph on a 2d plane however in 3 dimensional geometry a line does intersect the elliptic curve at 3 points, case closed. Even, the real line can have a point at infinity by simply curling it around and making it meet at and end. This is in no way related to infinity as a concept. I think there is a misunderstanding here . I don't know much about elliptic curves but consider the aymtotoes of a hyperbola(x^2/a^2-y^2/b^2)=0 and then you get two aymtotoes and we say that they intersect the hyperbole at infinity, that does not mean they do. They keep getting closer and closer. You can never give the point of intersection there.Elliptic curve arithmetic has obviously nothing to do with the real, physical world. It was not abstracted away from the real, physical world. Elliptic curve arithmetic is a Platonic abstraction that has characteristics and properties that turn out to be interesting, while adding a point at infinity is not only a requirement for consistency, but it also happens to work absolutely fine.
Can you generate infinity from that function ? I dont think so, unless you think we can construct an infinite number of characters in a string. — Wittgenstein
Let's consider your example KK*-->K+, how about this case ab(ab)*-->cd+ . Would that be rule if and only if ab=cd — Wittgenstein
An absolute formal view will lead to many problems and there is also another problem with this language as it allows a function to take itself as an argument, that may lead to paradoxical self referential statements. Like a set of all sets for example. — Wittgenstein
By alleging infinity to be a platonic abstraction doesn't help us understand its nature at all. — Wittgenstein
When the need arises they let p+0=0 so l dont see how the point at infinity is related to infinity that we are discussing here. — Wittgenstein
Mathematics and logic is one thing ... but reality is temporal and spacial. — luckswallowsall
If you regard infinity as number, that implies that it is finite, since all numbers are finite. — Wittgenstein
Even if you consider the "conceptual existence " we cannot construct infinity even with all the symbols and operations in a system. — Wittgenstein
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