The difference between Mathematics and Mysticism

There is no "smallest number greater than 0". This is not only because Mathematicians are self-centred egotists, or even because maths profs are Mathematicians (see above) who wish only to flunk their students.

Mathematics believes in doing well-defined things (ONLY). It's practically the science of well-defined things. Having a smallest positive number would break this. For suppose such a number e existed. Then we know e > 0, so e > e/2 > 0, which is a contradiction (e/2 is a smaller positive number!).

So no such thing exists. And you cannot meaningfully "define" something to be this nothing. There is no number 1/infinity, no number 1/aleph0, no number 1/ω, and no number 0.00...001 (where there are "infinitely many" 0's between the decimal point and the 1). None of them make sense, none of them are defined, and all of them are even more nonsensical than the idea of a smallest positive number e as above. And don't get me started on 0.99999...; it really does equal 1 (and no, 1-0.999... is not such a "smallest number greater than 0").

Grow up; deal with it; there are bigger things to worry about.

hey, what's the smallest number greater than zero?
think of a number greater than zero.
is this like a magic trick?
in a way, yes.
cool! i like magic tricks! alright, got one!
halve it
uhuh
halve it again
yup
halve it again
ok...

*** several minutes later ***

halve it again
um, how long is this going to take?

*** several hours later ***

halve it again
are you following me?!?!??!

*** several weeks later ***

halve it again
please! i can't take it anymore! can't we just call it quits and round it down to zero?!?!?!
certainly not! halve it again...
arghhhhhhh!!!!!!

The difference between infinity and the smallest number greater than zero is this:

You cannot capture infinity.

Infinite's just a simpler way of bringing into play the concept of "There is a number larger than any specific numbers which you might think of." Because of this, you can never say, and have it carry any meaning or even be provably true/false, something like infinite < 40e100, or infinite + 1 > infinite. Infinite is like oil in water - it always rises to the top, no matter how you try to capture it.

On the other hand, the smallest number greater than zero, if such a beast exists, can be caught very easily:

0<x<1
Now that we've captured it, it's very simple to find it (by using, for instance, a binary search), and, having found it, create a new smallest number greater than zero.

The reason that no such number exists is that as soon as you "find" it, you can create a new smallest number by halving it. Since math is continuous, and not granular (that is, there is always a real number between any inequal two real numbers, no matter how close together they might be), you will never reach a point where you cannot halve.

There can't be a number

  _
0.01

It doesn't make sense. Think about it: you're saying there are infinitely many zeroes, and then a one. So the one is after the last of all those infinitely many zeroes. But guess what? The whole point of infinity is that there is no "last" zero!! It's sort of like the line from The Phantom Tollbooth: "Just follow that line forever, and when you get to the end, turn left." You don't get to the end.

Does "the smallest number greater than zero" exist? Depends on your context. Obviously, in some contexts, like number theory, where "number" generally refers to natural number, there certainly is a smallest number greater than zero: 1. But that's obviously not under discussion here.

There is no real number which is the smallest number bigger than zero. CentrX asks how come it doesn't exist when infinity does. Well, there is no real number which is infinity either, so in a sense they're in the same boat. Infinity is not a real number, though it can be represented in other systems. And of course, there are uncountably many infinities of different sizes, which only confuses matters. Take the reciprocal of one of those infinities, say aleph null? Fair enough. You'll have to give that a meaning somehow, and whatever it is won't be a real number (though it may be meaningful in some other system). Is 1/aleph_0 bigger than 1/aleph_1? Um. Probably, if your system is coherent. In which case, which is really the smallest number? It would be the same as asking what's really the biggest infinity,. and just as unanswerable (though I've heard there are concepts of the "greatest" infinity, sometimes called "Omega" (with a capital Omega).

John Conway's "Surreal Numbers," as described in a book by Donald Knuth of that name, admit transfinite numbers, and even describe sensible ways they behave in arithmetic. So you can actually take the reciprocal or even the square root of infinity in his system, and it makes a certain kind of sense.

But in simplest terms, just as there is no largest number (infinity is not a number in the usual sense), there is also no smallest positive one.

We do have math capable of handling the concept of the smallest number after zero. Its a particular application of calculus called infinitesimal calculus. This method was invented by Leibniz in the mid 1670s, and published in 1684 - nine years before the earliest account of Newton's method.

Leibniz developed his calculus in order to find methods by which discrete infinitesimal quantities could be summed to calculate the area of a larger whole. This probably came from his metaphysical work on monads. Newton was working with infinitesimal changes of force and motion with respect to time.

In order to work with the smallest possible number, one must work with infinitesimals, around which the model of calculus has been developed.

To work with infinitesimals, it necessary to use the hyperreal number system. This is much closer to the math that Newton and Leibniz did. In 1960, Abraham Robinson established the framework for non-standard analysis.

First off, some definitions.

  • Internal - objects that exist within classical mathematics
  • Standard - limited objects within the internal set

There are three axioms added to ZFC theory (the basis for today's mathematics): Idealization, Transfer, Standardization. These lead to the three principles of non-standard analysis:

  • 1st principle: If E is an internal object which is defined from standard objects, then E is standard.
  • 2nd principle: All elements of an internal set are standard iff the set is finite.
  • Transfer principle: Let P(x) be an internal expression relative to x. Then P(x) is true for all x, iff P(x) is true for all standard x.
For any x (standard or not)
  • x is limited iff there is a limited integer greater than x.
  • x is unlimited iff it is greater than any limited integer.
  • x is infinitesimal iff its absolute value is less than 1/n for any limited integer n
  • x is perceptible iff x is not unlimited or infinitesimal
  • x is infinitely close to y iff x - y is infinitesimal

A new function is added called the Standard Part that operates on the hyperreal number set and maps it back into the real numbers. For 'e' designating Epsilon (an infinitesimal):
SP(1 + e) = 1
SP(e) = 0
It is not possible to take the standard part of an infinite number, but it is not difficult to take the standard part of the reciprocal.

Yes, this is a bit hazy in my memory. I happily defer to anyone who can explain non-standard analysis better than I. The point being, we do have a formal system for dealing with the smallest number greater 0, and it has been around for awhile.

A text book on infinitesimal calculus has been made available under a Creative Commons License when the book went out of print and reverted back to the professor (incidently, the one I took 2nd semester calculus under) who wrote the book. The book is available via PDF at http://www.math.wisc.edu/~keisler/calc.html








ENOUGH


The answer is 1 unless you're a freak.




No more writeups, please.



An editor who is also a mathematician writes:

I have just killed thirteen (13) write-ups in the above node, because none of them demonstrated sufficient knowledge of non-standard analysis and infinitesimals to make a useful contribution to it. Trust us, we mathematicians do know what we're doing. All your radical ideas have already occurred to us.

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