my regular mistake is that...

[juan_gandhi]

I assume that there is an infinite number of integer numbers, while for most of programmers in this world and age there are either 2^16, or 2^32, or 2^64 of them. Tastes may vary. Of course the generations of 2^16 numbers will retire soon; new generations are sure that there are 2^32 integer numbers and 2^64 "long" numbers. They'll go too; but while they are here, it is really hard to talk to them. We are in different universes.

Comments

[spamsink.livejournal.com]

Primality is not a quality of an integer, but a quality of an element of a ring - they have chosen a familiar ring for convenience; the fact that they subtract 1 is, in that particular case, merely replacing a 1 with a 0 in a bit vector.

My point is that, if a value does not represent a quantity, it is not a number, it is a bit vector.

[ivan-gandhi.livejournal.com]

How come? Can you express minor Fermat theorem in an arbitrary ring? I'm kind of confused.

I don't know what you mean by quantity. I know how to define numbers; quanitity comes from observable objects in physical theories; they, in turn, depend pretty much on the way we view things. I hope you do not mean that an electron, a cloud, a galaxy exist "objectively", not as a part of our world model?

[spamsink.livejournal.com]

Yes, the theorem works in finite fields, according to wikipedia (Why fields, when a ring is enough? We're not dividing anything.)

I am saying that the only case you really need a potentially unbound integer is when you're counting objects (or their obvervations, or the instances of their models), but the magnitude of the count is limited by the Universe. Maybe is some other Universe you'd need 1000-bit numbers, but not in this one. All other uses of "integer numbers" in speech are for the sake of word economy and convenience. Are a 250 Gb hard drive contents an integer? One can say so, but then they can represent a rational number in [0, 1) just as well. So unless a bit vector is really used as an integer - for object counting - there is no particular reason not to say it's a rational in [0, 1), and who's to say which way is "right"?

[ivan-gandhi.livejournal.com]

I see your point. My view is different. I do not think that, since the world is finite, there is only a limited finite number of all possible states.

But... suppose we accept your model. What kind of numbers would you use to ensure finiteness? Is there such a model? Say, we consider all numbers as being module 2^270. No, it won't work. NSA? There is a finite number of elements, but there is no limit. Any ideas?