Kevin Karn wrote: > Patricia Shanahan wrote: >> Stephen Montgomery-Smith wrote: >>> guenther vonKnakspot wrote: >>>> Stephen Montgomery-Smith wrote: >>> >>>>> Now, one of the respondents said that Norm had rejected the axiom of >>>>> infinity, and so how is he supposed to do number theory. But nowhere in >>>>> his article has he rejected the axiom of infinity. He has rejected the >>>>> whole notion that axioms are the way to go. I would paraphrase what he >>>>> said slightly differently - axioms (might) describe the natural numbers, >>>>> but they don't define them. There is clearly a problem that there are >>>>> numbers between 1 and googolplex that we can never write down or >>>>> meaningfully describe. What makes us think that they are really there? >>>>> The evidence is at best empirical. >>>> Think about your above paragraph for a minute. Do you really think that >>>> the Number 2 is more real than some natural number which has never been >>>> mentioned or even thought of? Do you really think that "2" is the >>>> Number 2 ? "2" is the representation in one particular language of the >>>> name in one particular language of the Number 2. The Number 2 is >>>> neither its name in any language nor the representation of any such >>>> name in any particular language. The Number 2 is something which does >>>> not exist physically and can not be experienced by the senses. It is an >>>> abstract concept as are all other numbers. >>> I think your opening "think about it" sums it up best. The answer is >>> not obvious to me. I do personally hold some Platonic belief that each >>> of the natural numbers, even all of those between 1 and googolplex, have >>> an abstract existence. But I regard my viewpoint as more a statement of >>> faith (perhaps something that is quote self evident unquote) than >>> anything else. I think that questioning my viewpoint is a reasonable >>> thing to do and not at all crankish. >> Isn't there also a simplicity argument here? The set of all natural >> numbers, with no gaps and no end, is in many ways an easier thing to >> reason about than the set of natural numbers that someone, somewhere, >> has used or will use. > > You may be right, but your argument is incompletely formed, and I don't > trust it. It's not clear what you mean by "easier to reason". How do > you measure "ease of reasoning"? And it's not clear why ease of > reasoning (whatever that is) should be a priority. > > My main question is this: If we accept your simplicity argument, how do > we draw the line between orgies of metaphysical speculation (on the one > hand) and actual science (on the other)? It seems that the > metaphysicians would argue (according to your approach): Our aim is > ease of reasoning, i.e. *simplicity*. We're building vast castles of > useless logic and nested infinities which have no relevance whatsoever > to the real world, because it's *simpler* to proceed that way. That > seems wrong to me. >
I'm not proposing simplicity, at the cost of extra entities, as a way of selecting among scientific theories, but as a way of choosing mathematical abstractions.
Ultimately, I don't think the subject of this thread even asks the right question. It should be "Set theory: Should you use?". I don't even know what it means to believe set theory.
Viewing mathematics as an art form, a simple, clean theory without a lot of special cases, yet leading to rich structures and results, wins on elegance grounds.
Viewing mathematics as a tool for science and engineering, there is a long history of nice, simplifying abstractions such as imaginary numbers turning out to be incredibly useful. As an end user of mathematics, I am a strong believer in "If it ain't broke, don't fix it". I don't want mathematicians in any way inhibited from exploring apparently esoteric theories, because I don't know any way to predict which ones will turn out to be useful in the future.
I am very familiar with the difficulties of arithmetic with a bounded range. You lose all sorts of useful properties, such as associativity and commutativity of addition. Adding up numbers in one order may lead to an overflow, adding them in a different order doesn't.
The problem only gets worse, and harder to reason about, if there are gaps.
That is the sort of thing that makes me consider the gapless, endless natural numbers simpler and cleaner than anything that could result from trying to limit them to some practical subset.