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11

All mathematical formalizations of (intuitive) computability are known to be equivalent, in particular they are all equivalent to computability on the universal Turing machine. So technological implementation is irrelevant. The Church–Turing thesis states that this coincides in scope with what is "computable by a human being" unconstrained by limitations of ...


9

It think there's something confused in this question. Mathematical constants like pi are exact. Pi is the ratio between the circumference and diameter of a perfect circle on a flat plane, and the fact that we cannot calculate an exact value for it does not mean that the constant itself is not exact. Planck's constant might suggest that we cannot have a ...


8

Here is some historical context. In Grundlagen der Arithmetik (1884) Frege introduced his ill-fated Axiom V, now known as the axiom of unrestricted comprehension: every predicate defines a class of objects that satisfy it, called its extension (Frege's own formulation is more technical). This led to the set of all sets and then to the Russell's paradox in ...


6

An Intuitive Walkthrough of PA as a formal system *Peano Arithmetic are a set of axioms in first order logic that describe how arithmetic of the natural numbers works. A first order formal language is a collection of variables, constants, logical symbols (such as negation, conjunction, etc.), parentheses, function letters, and predicate letters. Function ...


5

The issue is complex and any "significant" answer is hardly reducible to the Yes/No pattern. In modern mathematics, 2+2=4 is a theorem of arithmetic provable from Peano axioms. In a nutshell, assuming the definition of 1 as "the successor of 0" and of 2 as "the successor of 1" and ... and of 4 as "the successor of 3" (and thus "the successor of the ...


5

The notion is important in mathematical logic and model theory, but not in classical mathematics, including real analysis as traditionally understood. Definable predicates are generally important in the theory of formal systems because they show how expressive they are, for example Tarski's theorem on the undefinability of truth states that in a consistent ...


5

In a nutshell, the issue arises from the definition of number of as a second-order concept (i.e. a numerical quantifier) in Die Grundlagen der Arithmetik (1884). Consider e.g. 0xϕ(x)=df Card[xy] (y ≠ y) [ϕx], that reads: To assert 0xϕ(x) is to say that the objects that are ϕ are in one-to-one correlation with the objects that are not self-identical, i.e....


5

You can address this issue through significant figures, which indicate the level of precision with which a value has been measured. If I say that I walked 2 miles, that's a rough estimate - I'd be justified in saying that if I had walked anywhere from 1.5 to 2.4 miles. If I say that I walked 2.000 miles, that is a much more precise number - I'd only be ...


4

Your question is paralleled by the reaction of the Roman world to Indian ('Arabic') numerals. Accountancy was done in Roman numerals until the 1800s, exactly because of suspicions about the 'realness' of zero. While mathematicians just got on with using the far more powerful and compact Indian numbers. They can be proven to be equivalent, so it just comes ...


4

For category theory, see Natural number object. For set theory, see Set-theoretic definition of natural numbers. And see Category of sets for a link between the two theories. See also the post Categorical foundations without set theory and Philosophical Significance of Category Theory. From a mathematical point of view, the proposed construction of the ...


4

"Intuition of pure number" is the intuition Poincare inherited from Kant's a priori form of perception in time. Kant recognized two forms of perception that produce synthetic a priori, and therefore "rigorous", knowledge, space an time. The former gives rise to the geometric intuition, and the latter to the arithmetical one. However, after the discovery of ...


4

It is fair to say that the concepts are "equivalent" in the modern mathematical practice. However, they have different histories and different overtones of meaning. The notion of natural or counting numbers goes back (officially) to Pythagoreans and unofficially to prehistoric times, Ishango bone, a counting artifact, is 20,000 years old. We are dealing with ...


4

The point raised in the quote is not the same as the question that you are asking. In the quote: It is a difference whether we define what one is, and then we define what two is, and so on, or whether we define the abstract concept of (natural) number (as pointed out by Mauro Allegranza in the comments). Of course we can say that natural number refers to ...


3

The computable numbers are not technology dependent. A universal computer can simulate any finite physical system to any desired degree of accuracy. And it can simulate not just the input and the output, but also the stages intermediate between the input and output with any desired degree of accuracy: http://www.daviddeutsch.org.uk/wp-content/ItFromQubit....


3

Kant only had three epistemic categories, analytic a posteriori are highly problematic (even Kripke talks only about necessary a posteriori). As for π it was originally defined as a ratio of the circumference to diameter, and only two thousand years later related to numbers and decimal expansions. Still, one could define it as a number using one of many ...


2

Most simply, computable numbers -- including computable problems -- are capable of being solved by a computing device such as calculators, computers, and, well, humans if you are so inclined philosophically. Broadly, philosophy may approach questions of metacomputability: What is a computable machine or device? Are humans computing devices/machines? What is ...


2

Kant does touch upon it obliquely in the first critique; Cavailles expands upon it - he uses circles whereas Kant used triangles. First pi as comments have already pointed out is not defined by some decimal expression; it's defined geometrically as a ratio. Hence a priori, but also synthetic as it must take into account the subjects synthetic construction ...


2

To paraphrase Joel Hamkins answer (pointed out by user4894) on the notion of undefinable reals. The naive account of undefinability points out there is only a countable number of ways we can describe a number, but there is an uncountable number of reals, hence there must be reals that we can't describe; however the notion of definability is problematical: ...


2

Running off the idea that there are undefinable numbers because there are countably infinite definable real numbers and uncountably infinite real numbers, one result that could be considered "important" is that no continuous chaotic system can be perfectly modeled by a Turing machine. Thus, if a truly chaotic dynamic system exists, it could not be part of a ...


2

For mathematical statements the standard interpretation of "necessity" is "logical necessity", and "possible world" is a model of a theory to which the statement belongs. The theory in question here is presumably the Peano arithmetic, so one can derive that 2+2=4 is necessary from the fact that it is a theorem of Peano arithmetic, and the Gödel's ...


2

The statement "some are" means "there exists at least one". The statement "there exists at least one" is consistent with the statement "all are". This consistency between Some and All has to be accounted for. Thus the I statement, "Some S are P" becomes: "The ratio of subject S, in relation to predicate P, (1) is larger than zero percent, but (2) is less ...


2

Firstly, I apologize for a somewhat long answer to a seemingly innocent short question. Since this is primarily a philosophical question, I have removed my science hat, so here goes. Starting with the fundamental subject of enumeration, the convention of all numbers, including integers and reals, is a result of the explicit human realization of the ...


2

I'm not sure about your approach but I'd agree that the issue of precision is an important philosophical topic that deserves more attention. I don't know of anyone who explores this as a philosophical issue. It is not just to do with Pi. No location on the number line can be identified with precision if all locations are infinitely divisible. EDIT: Doh! I ...


1

There are really two questions here. One is a pure math question "do real numbers exist without the real number line?" The answer is most certainly yes. The properties of real numbers can be defined in a way completely independent of geometry. The number line is merely a simple geometric construct which can be used to explain some of those properties. ...


1

This is not a simple affirmative. First, we have to understand exactly what you mean by "set". There are many Set Theories, with different objectives. Second, categories are not necessarily sets. This is a very complicated topic; someone could argue that it makes no sense to talk about categories that are not sets, but the language of categories is in ...


1

Absolutely all numbers (not only real or positive), define subjective boundaries. Therefore, out there, positive, negative, fractions, imaginary, etc., are just subjective ideas aimed to discretize nature. In consequence, factually, there's no "less than one", because there is not even a "one". Out there, it is all interaction. This requires explanation. ...


1

Let's consider electrical charge. We have positive and negative charges, and both are actual things. Positive charge is not the absence of negative charge, and vice versa. Yet positive and negative charges cancel each other. Given an atom of helium, we have two protons and two electrons, so two positive charges (OK, six, if you want to give quarks ...


1

You're asking a philosophical question, based on examples from physics and mathematics. There's an important distinction between the latter two: Mathematics is not a science - it is the language of science, but math is based on axioms (i.e. a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments). It'...


1

Vihart from youtube has a video explaining this. The answer to your question is answered at 4:00 to 6:57. Here it is below: https://www.youtube.com/watch?v=X1E7I7_r3Cw


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