On 6/8/2015 7:30 PM, LizR wrote:
On 9 June 2015 at 14:00, meekerdb <meeke...@verizon.net <mailto:meeke...@verizon.net>> wrote:On 6/8/2015 4:16 PM, LizR wrote:On 9 June 2015 at 05:31, meekerdb <meeke...@verizon.net <mailto:meeke...@verizon.net>> wrote: On 6/8/2015 1:03 AM, Bruno Marchal wrote:or that maths exists independently of mathematicians.That even just arithmetical truth is independent of mathematician. This is important because everyone agree with any axiomatic of the numbers, but that is not the case for analysis, real numbers, etc.Everyone agrees on ZFC in the same sense. So does that make set theory and its consequences real? Reality isn't defined by what everyone agrees on.Tell it to Bruno, I was just following him. If it was then the religious majority throughout history would have been right.What makes ZFC (or whatever) real, or not, is whether it kicks back.Mathematics doesn't kick back - except metaphorically. Are you claiming an alien in another galaxy wouldn't find that arithmetic works?
No. Is that what you mean by "kicks back"?
I'm not making any metaphysical claims about the status of maths, merely saying that most mathematicians would, I think, agree that two people working independently can make the same mathematical discovery by different routes, and that some maths has real-world applications, and that when it does, it works.
Arithmetic is a hard example to discuss because it is so simple and probably even hardwired into our thinking by evolution (crows can supposedly add and subtract up to six), but it's not really so inevitable as it seems. In order to count you have to discern distinct objects and group them in imagination into a whole: So you count the players on a college football team (U.S.) and you get 105. Then you count the number on the basketball team of the same school, 35, and you add them to the football team you get 140 - but that may well be wrong. Of course you will say that's just a misapplication; but that's the point, that arithmetic is an abstraction that is invented to apply to certain cases and it is no more "out there" than other aspects of language. I agree that it's hard to imagine an intelligent species that doesn't perceive discrete countable objects and didn't invent arithmetic to describe them; maybe some plasma being on the surface of the the Sun that thinks only in continua.
(But I'm not sure how much kicking back you need from something, maybe being independently discoverable and working isn't enough?)Is it something that was invented, and could equally well have been invented differently, or was it discovered as a result of following a chain of logical reasoning from certain axioms?I'd say ZFC and arithmetic were both invented and then an axiomatization was invented for each of them. I'm not sure what "invented differently" means?...getting to the same axiomatization by a different historical path? Or inventing something similar, but not identical, as ZF is different from ZFC.It means that two people starting from the same axioms and using the same system of logic came up with two different results (and neither made a mistake).
That would mean either the axiom system was inconsistent or there was a mistake in logic. Note that Graham Priest has written several books on para-consistent logics, ones in which there can be contradictions but don't support /ex falso quodlibet/.
If within a given system A always leads to B, then it's reasonable to say B is discovered - like, for example, a certain endgame in chess leading to a particular set of possible conclusions.
?? At first reading I thought you meant A logically implies B, which means B is implicit in A. And so I thought the example was a chess endgame in which every move is forced (except resignation), A would be the board position and B the sequence of endgame moves. But then you say B is a set of possible conclusions. Since chess is a finite game the starting position already leads to a */set/* of possible conclusions.
But if within a system A can lead to B, C, D etc then it's reasonable to say it's invented,
So does the fact that Peano arithmetic lead to many different theorems mean it's invented? Does the fact that it's incomplete and can have infinitely many new axioms added to it mean it's invented?
I don't think your criterion for distinguishing invented from discovered reflects common usage.
like a competition to finish (within the grammatical system of English) a poem that begins "And now the end is near..."
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