Has anyone here read it? (Here's a link) I admit, I thought that I would read this thing in two to three days. It took me 3 days to make it through 20 pages because I'd have to re-read a sentence or paragraph at least 4 times before I "got it" only then later having to start the whole chapter over again. I got a question... Russell gives the following definitions: Why "with respect to the relation 'immediate predecessor'?" It's bugged me for days and I think I'm missing something. It took me long enough to get through the background information needed to define number... now this.
Defining a successor function is a way to derive (the) integers. The successor function is recursive since succ(n) = succ(succ(n-1)). If predecessor is the inverse function, you're away... Ed: for example, a well-known ring of integer/natural counting numbers is definable as U = succ(1) UU = succ(U) UUU = succ (UU) = pred(1) pred(U) = 1 ... for U a unitary 'slice function' of a certain mathematical object.
Sorry noodler, but I don't think you helped me at all. Maybe you did, but I don't see it. First of all, Russell hasn't defined what "successor" actually means until later in the chapter.. he's trying to create a broad definition that we all feel is close to the true definition we need.. and then working to define everything in there. Unfortunately, I don't know why he is mentioning "immediate predecessor" when he's still "going forward." For example... it's understood, though not yet "know" at this time, that the posterity of 10 is the set {10, 11, 12, ...}. I don't see this list "going backward." Please Register or Log in to view the hidden image!
Hmm. Posterity and heredity suggest a "before and after'. The posterity of 10 I think just means it's posterior to the set. To my way of seeing it, 10 is the head of a list - given that special place because the tail is its "inheritance". As to a "true definition" I can't say what that is (not sure if anyone can). It's obviously true that 1 and 0 are different numbers, I guess,,,
Wait... what you just said gave me an idea. I read over the whole thing and I think I finally picked apart the grammar of his statement. How dumb am I? The posterity of 0 is the set {0, 1, 2, 3, ...} while the posterity of 0 with respect to the relation "immediate predecessor" is {1, 2, 3, ...}... which is also the posterity of 1. Does this seem correct?
Yep, then posterity and predecessor are 'congruent' as are heredity and successor. That is: they correspond to the same "numerical relation" (whatever that might mean)... Ultimately you have to "see" a number; a description only gets so far as a working model. Like numbers in digital computers - they are actually units of charge in capacitive devices; capacitors "understand" electrons, the numbers are levels of charge. Write the symbols 1 and 0 which map to "fully charged" and "fully discharged". These states never actually occur in capacitors because they're considered empty at < 33% and full at > 67% charge; every number is an approximation.
The problem with Peano's arithmetic is that there existed an infinite collection of "things" that we can call numbers and that satisfy his 5 axioms. So, Russell and Ferge gave a definition of a number: The number of a class is the class of all those classes that are similar to it. -and- A number is anything which is the number of some class. He points out that numbers themselves are metaphysical in nature so they cannot be defined in the same sense that "car" or "tree" can be defined. So, instead, it's better to collect collections of objects into groups that are similar to each other. "Similar" has been defined, too...
Like the set {{Tree},{carbon atom}, {bottle of water}, {dad's car}, ... } is a class of classes. We'll call this class "1" and any class, like B = {...} that is similar to the members in "1" is said to have the number "1."
Or the set of pitch classes in music theory along with a set of rules for pitch changes. The successor of the key of C is C#, etc. The successor of a minor triad is a major triad in some list and so on. The relations let you derive other sets of relations; I guess numbers are about how elements in sets are related, set-wise (which is a recursive definition). I think that means numbers are recursive; this is "trivially true" for the integers for instance.
Absane: I can recommend a book, edited by Steven Hawking, called "God Created the Integers". It purports to examine the most important mathematical ideas that have been developed; it starts with Euclid, covers Descartes, Newton, Fourier, Gauss, Weierstrass, Riemman and other important figures. Even reading just the intro to each chapter, penned by the editor is worthwhile - Hawking is very good at mathematics and explains the essential ideas concisely. The biggest problem I had with it was the archaic language and terminology - the publishers note that no attempt was made to translate the idiom. The most interesting thing about the book, to my way of thinking, is the last chapter on Alan Turing and it presents his original thesis and his paper on the Entscheidunsprobleme = decidability problem. (I may have mispelled that, my German is terrible). Kronecker was the mathematician who coined the phrase Hawking uses for the title. Anyways, the last chapter kind of sums it up: all writable numbers are recursions.
This appears to be about the cardinality of a set; when do two sets have the same size/# of elements? And what does that mean:- Are the sets {1 2 3} and {a b c} in the same class, because they have the same number of elements? The answer depends on what you mean by "number"; the first set has 3 elements, does the second have 3 or c elements? Is 3 the same as c? If they are similar in class then they have similar cardinality - the number of the class is the class of numbers of elements. Similarity or cardinal equality depend on how the elements are written as recursions (which terminate). So in other words, if you can write symbols (with terminators) for elements in classes, the # of symbols is the "classifying number of the class".
