pariah

there is no such thing as a "highest" rational number.

i can say that with certainty.

xian friend: no you cant.

halp?

xoc

There is no such "thing" as a highest rational number, because there is no such "thing(actual exisitng object)" as a number. A number is an idea; although it may be representative of a real thing or set of real "things", it could as well represent unreal thing(s). So numbers form a kind of idealised systematic way of viewing "existence/reality/etc.", whether or not they have an actual idealized existence or not or merely are assumed is impossible to prove(I believe: a big point of difference between Plato vs. Aristotle is whether the "Idea(l)s" really existed or not).

Since OUR system of numbers gives a limitless series of rational numbers, one can say with as much certainty that "there is no such thing as a highest rational number" as "there is a 4." Proving non-existence is often fundamentally impossible(for things or ideas), only existent things/ideas prove their reality by their influence.

pariah

what about the certainty that physical objects exist? like my keyboard?

also, what works of plato/aristotle are you referring to? i would like to read them ;]

SlateGreySky

That's not affirmed by all or even most contemporary analytic metaphysicians (I'm thinking Quine and later). Most of them would view both numbers and sets as actually existing, abstract objects.

I think the key to pariahSS's question is to ask the Christian friend, "why can't I?" That might clarify their reasons for denying the (true) assertion that there is no "highest" rational number.

xoc

Well Plato goes into the "Ideas" in The Republic, Book 8 or so. I haven't read Aristotle but read of their philosophical disagreement(Aristotle was Plato's pupil at one point) on the Ideas, whether they were ultimately real or merely useful conceptions, only "abstract."

I think a good criteria for determining the existence of something is it's consistency in experience; we believe our awake selves are "real" (and the world we see) because of it's consistency in our experience; every time you go to the desk where the keyboard is, you find it, can see it/touch it(gives consistent sense phenomena) etc. That's the general distinction we often use to determine the "illusory world of dreams" (where the consistency of objects/people etc. is always changing, one person changes into another etc.) from the "real waking world." 100% certainty is not possible (as we could all be plugged into one Matrix grid of illusion, etc.) but consistency gives a reasonable credence to faith in a real objective world.

tensorproduct

I think you should ask your xian freind to tell you what they think is the highest rational number, and then add 1 to it.
This is the nature of an infinite set. So yes, you can say that with certainty.

On a related note could somebody tell me whether or not the set of rational numbers is countable.

PJPSYCO

You could try a Venn diagram. Show him the realm of ideas, and then show him where the ideas that we apply to the concrete are(ie a tiny circle in the huge circle from step one).

Theli

I would say that theoraticly we can (with our current system) create an infinite amount of numbers so there is no theoratical "largest number", but in reality there are practical limitations.
Numbers only exist as long as they are stored, that's the only answer I can come up with. So, if you wan't to find the current largest number search through all the harddrives in the world.

Demosthenes

there's no such thing as a highest rational number because given a rational number R, R + 1 is also a rational number. Since R + 1 is greater than R then R cannot be the highest rational number and so on ad infinitum.

pariah

yes he understnads the logic behind it, but he was getting at the notion that nothing exists for sure.