You might check out "infinitesimal calculus" via Google. Lots of interesting sites. Robinson introduced
his views in 1960's. Kiesler has a text about it. If I recall correctly infinitesimal was introduced as an
infinite sequence that converges to zero.
Someone back then felt it was a useful concept...and I think it has its place and should not have been dismissed.
I don't think you can do away with it. Because we can think of it, maybe it does exist.
It's not zero. It's not a number either (it's not a static value). It's a concept. The idea of a potential '1' to be reached at the end of an endless bunch of decimal zeroes (0.000... )
What's wrong with it? It gives a 'skin' if you will, to the beginning of quantity and distinguishes it from zero, which has perfectly, no potential for quantity.
I don't mind shaking things up a bit here.
I don't think that the basic concept of infinity require a masters degee to understand.
And I am certain that there is honest debate and not half the established consensus on some of the things I put forth as you make out to be.
Some things are not 'surface intuitive' and require an imagination. Some things I think are flatly intuitive. Grasping infinity can be incredibly difficult, and is only made easy when you accept it's not a number, but then it can become very easy. But I do know that the idea of infinity is to express unlimited and all-encompassing, so when I see it in relation to numbers, to me it means one simple thing (an impossible end to numbers and the concept of labeling all potential quantity with a name). It can be thought of as a package outside of which can exist nothing, and inside of which exists everything that can ever exist.
The fractions I put forth are nonsensical. They don't pretend to be numbers, and I'm not meaning them to be or claiming them to be numbers. I'm sorry I gave that impression. A number is a finite, measurable quantity (adding further...located between the infinitesimal and the infinite both of which are not numbers either).
Any fraction that has infinity involved with it, by definition cannot be a number. I have no disagreement and I believe I understand why that is for the same reason you do.
I'm trying to adapt a concept to a shorthand way of expressing it because some things seem unnecessarily muddled, possibly by overthinking them.
I do believe I have a very good understanding of infinity, which is not a number.
I disagree that 1/infinity is not infinitesimal...there are probably as many people defending me on that who are much further along in math than I am, than people who are defending the idea it is not. And why doesn't that make sense to you? Even the number 1 when used with infinity is symbolic and not so much a number at that point, because any finite number will do when placed next to infinity (they all appear to converge on zero). Perhaps we can focus on that piece first, since it's a very interesting subject.
I don't believe zero is a number...but I have a slightly different idea of a number in that case.
I think of a number as a existent quantity, so as to distinguish it from a 'quantity of zero'.
OK, I better start off simple and see if we can agree on five basic ideas.
Zero is absolutely no quantity, no space, no area, no length (on a number line), etc. It does not 'exist' because it takes up none of those things. That's how I think of zero. Mathematically, zero over infinity.
Infinitesimal is an entity that exists between nothing and something. Something more than zero, but not enough to be finite or measurable. Zero plus something immeasurably small. The least quantity needed for quantity to exist. Scaled with any finite number (no matter how small) and zero, it would virtually appear to be in the same spot as zero (but of course, is not). Mathematically, one over infinity.
Finite numbers take up quantity, space, area, length (on a number line), what have you and are limited. It's what we can deal with, see, touch, measure, conceive of, etc. Mathematically, any finite number over any finite number.
Infinite is an entity that exists between something and everything. Something less than infinity, but too much to be finite or measurable. Infinity minus something. A growing, uncountable quantity that doesn't encompass all quantity. If scaled with zero and infinity on a number line, it would appear virtually in the same spot as infinity. Mathematically, infinity over one (expresses the idea of approaching because the one in the denomnator is countable - you can start counting).
Infinity absolutely encompasses all quantity (that can ever exist), the unlimited universe of quantity that cannot begin to be measured or even approached. Scaled with any finite number (no matter how large) and zero, that number virtually looks like zero (but of course, is not). Mathematically, infinity over zero (expresses the idea of unapproachable, because the zero in the denominator is not countable - you can't even start counting).
I know, breaking some established ideas on the last two. But seems to me there has to be a practical use for "infinity over one" and "infinity over zero", and a way to appreciate a distinction between the two ideas.
In these fractions, one becomes merely a symbol for a finite quantity, because it's being placed next to infinity, which is a symbol...the importance of what finite number is used is not that important, but one is used because it is a compact, perfect representation of finite.
Just like the rest of you, I don't think I have it all figured out, but I have been thinking about this subject for many years and I tweak things a bit as I find that my definitions aren't good enough to express what I'm trying to say.
OK, you are not counting to a finite quantity. You are counting to no quantity. Show me zero dollars and zero cents and then explain why I should accept you have SOME (a finite quantity) of money.
Well, you can't because if you did, it would be at least a penny in this case. Zero dollars is equal to zero apples is equal to zero planet Jupiters. If those things are absolute equals (they are) then zero is not a true number.
Zero has a unique characteristic that sets it apart from numbers we normally think of as numbers. One or two apples, ten apples, half an apple, we're still talking apples...zero of something...what are we talking about? Battleships? Could be. Does it really matter at that point? No. It matters only if there's a number of something involved, meaning a non-zero quantity.
In order for any of those three things I mentioned NOT to be the same, they have to be a finite quantity, something other than zero. Zero does strange things when you insist it's a number. Why not have infinity be called a number? You can put it on a number line also. I don't think that's the sole justification for zero to be a number.
Like I said, it does have a place on the number line...I'll never disagree why it's there...I just disagee on what it is. It's a very underappreciated entity in mathematics.
Its basic identity is 'diminished' when people insist the infinitesimal is the same.
If zero is a number then infinity can be a number too.
Ever count to zero?
That's why I say it's not.
If zero or any other number are on the line with infinity, you are done. You cannot put any other numbers on the line in their 'proper' location.
I never said anything about putting infinity and the infinitesimal on the number line together...you basically cannot do it.
Definition of infinite: quantity that is dynamic and is constantly approaching infinity (what most people mistakenly think of when they say 'infinity' is really what I define as infinite...infinity is unapproachable, infinite is the futile effort to do that)
I never said that infinitesimal or infinite are numbers...if I did, my pardon, but they are not zero or infinity either...that's exactly why they have their own identity.
When you complicate things unnecessarily, you put off understanding them...so don't ask yourself why things are such a mystery when all you have to do is accept the obvious.
My opinion is that the whole subject has been unnecessarily complicated and possibly some distinctions (such as the idea of infinitesimal and infinite having their own identities
separate from zero and infinity) haven't been explored enough.
This idea that zero is a number is misguided. No more or less defined as a number than infinity is but for the opposite reason. Yes, it is routinely placed on the number line because it is the origin (does not mean it's a number as you can easily put infinity on an admittedly less-useful number line as well). You can also add infinity to the number line with the zero as long as no other numbers are added.
Conversely, you can have a number line that has any finite number on it, plus infinity on it...at that point you cannot place zero on it because it doesn't have a proper location. It will appear to be in the same place as the finite number you put on it if infinity is present on it.
Zero and infinity will sqeeze the other off a proper location on the number line if either of them is included on the number line along with any finite number. Infinity because of the impossible scale, and zero because
it starts to appear in the same position as any finite number, which of course it's not.
You should be able to explain accurately why anything I said is wrong if it's not making sense to you. If you can't, maybe you should just accept the simplicity of it.
True, infinity is not a number...neither is zero. That would be my argument. In order to be a number, it must exist and also be in the finite realm.
Zero has a problem...it doesn't exist...Infinity has a problem...it's not finite.
These ideas work together well in pairs and are opposites of each other:
Zero and infinity -- no quantity whatsoever and all quantity
Infinitesimal and infinite - approaching zero and approaching infinity
I roll my eyes everytime I see someone saying ".000 ... 1" equals zero. It makes as little sense as someone saying "999 ..." equals infinity.
The idea of marking the difference between being the extreme and approaching the extreme by allowing those identities are nothing to be ignored,
and I think you cannot understand zero and infinity correctly without accepting them.
Why not just make up your own symbol for it. There is no way to represent infinitesimal...1/∞ (contrary to my older belief), doesn't actually make any sense, and this idea has been dropped anyway (can look at things like wikipedia).
Besides for that, to the best of my knowledge, we don't have a character that can truly represent infinitesimals by themselves, we can only really understand the idea of it as the smallest possible number. Making up a character for it could solve that problem, as then we have something to represent it. Just an idea.
Epsilon is the proposed symbol for the infinitesmal...along with that, there should be a separate symbol for the infinite, because the infiite is to infinity what the infinitesimal is to zero. Same idea in reverse.
Does the mathematics community look at the number line the way I do, thus (I'm only using the positive half, but it also applies to the negative half):
5 basic representations (for lack of a better word) of numbers in a traditional number line (aside from positive/negative):
1. zero (itself)
2. infinitesimal (forever approaching, but never reaching zero)
3. all finite numbers between the infinitesimal and the infinite (including all fractional, irrational and trancendental numbers)
4. infinite (forever approaching, but never reaching infinity)
5. infinity (itself)
I have seen some discussion where people insist that zero and infinitesimal, and infinity and infinite are virtually
the same and allow approximations to zero and infinity, but I think that leads to errors in understanding them.
Literal zero and literal infinity should be allowed to stand alone without approximations. They both represent a pure idea.
I understand there is a proposed symbol for the infinitesimal (epsilon - ε). I like it.
Is there also a proposed symbol for the infinite (which, as far as I know is best represented by ∞ - ε, not ∞ - 1)?
Last, this part is interesting about the number line (I wrote this last night):
(Let me know if you think this is off base, or I'm not quite
describing the idea I'm trying to convey in the best way.)
Say you have a number line as such (note that there are no arrows on the ends of the line, the ends of the line are in fact - and + infinity):
-∞ ---------------------------------------------------------------------------------------------------- +∞
Q: where would you put the zero?
A: anywhere you want to in between; infinity has no scale
so you don't have to worry about exactly centering it;
every possible number in between would crowd adjacent
to the zero anyway, making it impossible to plot any
other number, no matter how large it was; the other
numbers would neither be on top of the zero (equal to
it) or away from the zero (visually indicating what
portion of infinity they are)
Q: after putting in the zero, can you put in any other numbers?
A: no; it is impossible to put any other number on the
line, if you want to avoid imposing a scale; if scale
is not important, you can do anything you want;
the reason why you cannot put any other number on the
line, is that you cannot get close enough to the zero
to avoid the problem of dedicating a portion of the
line to that number that visually indicates its
proportion of infinity - that would not represent the
nature of infinity accurately; and of course, you cannot
put any number in the same spot as the zero, because
you are saying that zero is equal to that number,
which would not represent the nature of zero accurately
Q: instead of zero, can I put in a different finite number?
A: yes, but that is the only other number you can put
on the line, (again, without imposing a scale)