Recent content by philiprdutton

sortof ... Yes: I am interested in comparing a formalization of the theory of "counting" (not the theory of computation... or are they the same?? haha don't answer that! ) with the peano arithmetic. I think it is a valid comparison. More precisely to rephrase my thoughts and your...

building How do you make parts of the two systems synonymous? Let's say you were going to build both systems from the ground up: A and B. Build it "modularly" as in: "build piece by piece". Start with the common pieces. How far can you get before you loose commonality?

half way there Maybe the supplied algorithm uses Peano arithmetic and computes a natural number. However, I am not convinced that the axiomatic Peano system indeed does any "marking." "Inside" the axiomatic systems there is no marking. How can there be? "Marking" requires a time component...

counting... what is it? Clearly stated: when humans or machines count, is there anything in the process that can be mapped over to the Peano axiomatic system? Or: Is any "slice" (or part) of the Peano axiomatic system mappable to the human/machine counting process...

number systems Given the above, I now have a revised quote to offer the world: "Counting is all too easy. Figuring out how to talk about where you started or where you stopped is the hard part." - Philip Ronald Dutton I think the Peano axiomatic system fails in regards to tricking the...

exploring While being lost in exploration, I've almost forgot what I was avoiding. :) If you use the time based metronome then what is the reference point? There is no clear way to encode it like there is with the Peano axioms.

I am just simply saying that when a human counts out loud or repeatedly says "da, da, da, da,...,da" then, the human is essentially in the process of building the number system. If I stop counting then I have not finished building the system but that's beside the point right now. Because I am...

temporal aspects I have taken some time off from this topic of prime numbers. Having recently re-read some of the earlier posts, I found the above quote particularly clean and clear in terms of communicating ideas. In earlier posts I was talking about using a bare Peano axiomatic system as a...

hmmm Perhaps on a technical note I should like to mention that you have just implied that numbers can not be defined separately from the operations on them such as addition, multiplication, etc. A long time ago, many posts back, I tried to explore this notion of defining the numbers without...

oh nice... Very interesting thought! This is kind of where I am trying to go in terms of discussion (and maybe latter someone can throw in some "rigor").

orderings are there Actually, I disagree with you when you say that "oderings are not part of the Peano axioms." The problem is that recursion uses an ordering whether you like it or not. It does not "look" like our standard notion of "ordering" but I believe it is there.

kind of my point Sure Peano does not mention the number line. The problem that I have is that the "structure"... or shall I say "Scaffolding" of the traditional number line and the peano system are so so so so so so so close. I wanted to explore the link- try to build the common baseline...

meaning I think they have an "equivalence" of sorts. The binary number definitely has meaning: "10001010". It has meaning if you known the numbering scheme. That is to say, it has positional data, and it has an imposed "order" due to the positional data. Actually, the positional data is...

not seeing it Well, I just do not see how a numbering system has anything to do with an axiomatic system. Binary numbering for example is totally independent of the Peano axioms. So, I don't see how it can be considered a model of Peano. If anyone knows the rules of the numbering system...

talking Okay I think I am starting to get it : ) Basically (correct me if I am wrong) the Peano system defines the numbers in terms of recursion, it let's you "do stuff" with the numbers, BUT it does not equate a symbol (or combination of symbols) to each unique number. I'm guessing the...