Care to define this term? Unless you do it is impossible to understand the dichotomy.
Here, you’ve given an overview of what a ε-δ proof should accomplish, and a semi-intuitive informal ε-δ proof. That much, I still pretty much remember. What I’m looking for is the strictly formal ε-δ proof, with the sequence of algebraic inequalities leading to the QED – that’s what I only vaguely remember how to do.
Dayum, now I’m almost motivated to dig out my old calculus book (yes, I still have it!) and review that stuff again!
Although, as I recall, the sample ε-δ proofs we did were trivial examples – like prove using ε-δ :
lim[sub]x->3[/sub] (x+7) = 10
or: lim[sub]x->3[/sub] (x[sup]2[/sup] - 2) = 7
Not real obvious (to me, after all these years) how to apply that to proving:
lim { .9, .99, .999, … } = 1
Well, what if we had a number system of something-like-reals but having a minimum “quantum” value like Valmont314 seems to suggest (as best I can make out what he’s trying to say)? Is that what he (or anyone) means by “infinitesimal”?
Would all other quasi-real-numbers in the system be whole multiples of that one “quantum” value?
If we had a number system like that, how would it differ from another better-known number system commonly known as “the integers” ?
It would not obey the associative law, i.e., (ab)c would not always equal a(bc). For example:
(.0000000001 * .0000000001) * 10000000000 = 0 * 10000000000 = 0
but:
.0000000001 * (.0000000001 * 10000000000) = .0000000001 * 1 = .0000000001
A lot of people try to do math intuitively. Unfortunately, their intuitions are wrong.
A well-designed version of such a system might turn into what computer people call fixed-point arithmetic; it isn’t worthless by any stretch, but neither is it especially interesting. It’s certainly nothing new.
Who knows what Valmont314 means about anything. In terms of the hyperreals, which are well-defined, ‘infinitesimals’ are infinite sequences of real numbers that approach some value arbitrarily closely or go off to infinity, as opposed to real-valued hyperreals, which are infinite sequences of real numbers that repeat the same real value ad infinitem.
OK, that probably made no sense. Here we go with an actual explanation: A hyperreal is defined as an infinite sequence of real numbers; you compare hyperreals by checking only the sequence’s end behavior, by which I mean if there’s a finite sequence of fives and then an infinite sequence of eights, for example, the hyperreal is equal to eight.
A real number ends up with infinite repetition. An infinitesimal is one where the end behavior isn’t infinite repetition, but a constantly increasing or decreasing sequence of values. For example, an infinite infinitesimal where the end behavior is [0.1, 0.01, 0.001 …] is strictly less than any real number because, no matter what real number it is, there are an infinite number of terms in that sequence less than that real number’s value, and we only consider end behavior, ignoring any finite number of terms.
A finite infinitesimal approaches some real number, like 0 or 15; an infinite infinitesimal has end behavior like [10, 20, 30, 40 …] or [-1, -10, -100 …], so it goes off to infinity as the terms grow without bound. Note that the inverse (1/x) of a finite infinitesimal is an infinite infinitesimal, and vice-versa. Note also that we can compare infinitesimals: An infinitesimal like [0.1, 0.01, 0.001 … ] is larger than [0.1, 0.0001, 0.0000001 … ] because the second shrinks faster; similarly, [1000, 1000000, 1000000000 … ] is larger than [1, 10, 100, 1000 … ] because it grows faster.
Yes. Otherwise, you’d have numbers with absolute values less than your quantum, which, per hypothesis, aren’t allowed.
@Derleth: Thanx for the overview of what hyperreals are. Until this thread, I’ve never heard of them before – and I haven’t yet gone looking for information about them. Time for me to hit up Wikipedia, for starters, I suppose.
We can also do arithmetic on infinitesimals, simply by pairing them off such that, if we’re adding [1,2,3,4 …] to [2,2,2,2 …], for example, we add [1+2,2+2,3+2,4+2 …] to get [3,4,5,6 …], which is perfectly well-defined and larger than both of the numbers we started out with, as we expect. All of the other mathematical operations proceed in the same fashion.
The reason Robinson invented the hyperreals was to put the intuition of Leibniz on a sound footing, and dispel the ghosts of departed quantities. dx can now be understood as a finite infinitesimal d multiplying some real number x, so dx is a finite infinitesimal, as no real value x is large enough to change the end behavior of a finite infinitesimal.
Now, as per the definition of a derivative:
f'(x) = f(x + dx) - f(x)
-------------------
dx
We can do some calculus in the hyperreals. Set f(x) = x^2 and work it out to get f’(x) = ((x + dx)^2 - x^2)/dx) = (x^2 + 2xdx + dx^2 - x^2)/dx = (2xdx + dx^2)/dx = (dx(2x + dx))/dx = 2x - dx.
So f’(x) = 2x + dx. But we just said that dx is a finite infinitesimal, and therefore smaller than any real number, so it is 0 when we transition out of the hyperreals and back to the reals. So, at the end of it, f’(x) = 2x.
So you can do all of the calculus on that footing, and, as Robinson desired, get a bit further into Leibniz’s head.
Here is a PDF that goes through a standard exposition of what the hyperreals are and how to use them. It’s also a pretty good example of what mathematicians consider lucid prose, in that it can’t really be typeset cleanly in plain ASCII. 
Here is another PDF on the history and philosophy of the infinitesimal and the hyperreals.
I’d be remiss if I didn’t mention (eventually) that Calculus Made Easy is an introductory calculus textbook which develops the whole subject in terms of infinitesimals. It actually predates Robinson, so it doesn’t use the hyperreals, but it’s precisely the same general idea.
Calculus Made Easy is public domain, and it’s available for free download from Project Gutenberg.
So is multiplication done the same way? That is, is [1,2,3,4 …] x [2,2,2,2 …] = [2,4,6,8,…] ?
Or is it something more complicated? (I’m thinking of adding matrices, where you just do element-by-element, versus matrix multiplication which is more involved than just multiplying element-by-element.)
I just quickly skimmed the Wiki on hyperreals. (When you mentioned Robinson above, my first knee-jerk thought was: Raphael Robinson ? But no, not him.)
If Robinson invented hyperreals “to put the intuition of Leibniz on a sound footing,” what was the point? That is, what did this accomplish that the ε-δ definition of limits (and ε-δ proofs) did not accomplish? I could see that, if hyperreals had been developed earlier than ε-δ.
Okay, I should read all the docs you linked.
(ETA: And why would the Calculus Made Easy book have used infinitesimals rather than ε-δ, given that it was published in early 1900’s, when ε-δ goes back to mid-19th century?)
Yes. To begin with, we want multiplication and division to work like they do for the reals in terms of the commutative and associative properties.
Because mathematics has a philosophy, and an art, and Robinson’s opinion in that realm was that infinitesimals were cleaner than ε-δ stuff, that they gave a better insight into what was really going on when you talked about a derivative, for example. Infinitesimals were inarguably what both Newton and Leibniz were imagining when they did calculus, and they invented the thing!
Because infinitesimals came first, and not everyone likes ε-δ stuff. Some people think it obscures more than it illuminates.
It seems to me that what you’re trying to do, is to represent all quantities as integers, with the base unit being this “least element” think you keep talking about. So instead of 1.345 mm, you would have 1,3450,000,000,000,000,000,000,000 (an integer) of these least-length values.
You may practically be able to make something like that work for you, but your system has no way to represent irrational numbers, or even many rationals, exactly.
I was thinking the same thing and I’m not even a mathematician. The thing is, if the world is completely quantized, in terms of actual results we might not need the reals at all, at least once we have our final answer to any particular real life problem.
Whereas the universe might be quantised, at maybe the plank length, I don’t think this is a hard size - it isn’t as if the universe is gridded into little cubes of that size. Everything is probabilistic.
Anyway, a number system base upon such a notion is useless for a number of reasons. Whilst length might be quantised by one value, mass, energy, time momentum etc all have different quantisation values - so your numbers are typed - they have to carry with them their units and dimensions. Even if you normalise (assuming you can) the units you still need to carry the dimensions. If you wish to represent any sort of abstract notion, it isn’t clear what you do at all.
But, and this is key. These numbers only represent final physical values. You can’t usefully calculate with them. As has been pointed out earlier - what is suggested is essentially no different to fixed point arithmetic. Fixed point has a lot of serious limitations, with the accumulation of error being one of the most critical. Ill-conditioned numeric calculations can simply go nuts and calculations become un-viable if there isn’t significantly better precision in the intermediate stages of calculation than the inputs and final answers. Computational scientists understand this well (or had better if they want results that are better than gibberish.) It is bad enough with full floating point arithmetic, but fixed point is likely to regularly produce little more than random answers. Also, as we have seen the system isn’t closed, and most of the usual properties we associate with useful numbers fail.
Limits of sequences are related, but not identical, to limits of functions of the sort you’re dealing with.
Let s[sub]n[/sub] be the nth term in your sequence, so s[sub]n[/sub] is .999…999 with n total 9s in the expansion.
Let ε > 0 be real. We must show that there is an integer N such that for all integers n > N, the nth term of the sequence is closer to the limit L than ε, or
| 1 - s[sub]n[/sub] | < ε for all sufficiently large n.
Since 1 - s[sub]n[/sub] = (1/10)[sup]n[/sup], we may take N to be the greater of the ceiling of -log(ε) and 0, where log represents the base 10 logarithm. This is because, if n > N > 0, we have (1/10)[sup]n[/sup] < (1/10)[sup]N[/sup] <= (1/10)[sup]-log(ε)[/sup] = 10[sup]log(ε)[/sup] = ε. (The inequalities follow because f(x) = (1/10)[sup]x[/sup] is a decreasing function.) If N < 0, the implication is that ε > 1, so s[sub]n[/sup] is trivially smaller for all n.
This is the other sticking point anyway. It asserts the existence of the a couple of properties that don’t exist. There isn’t a limit of precision of the reals. This seems to be the core problem. The logic seems to stem from the curious property of the lack of a least element. This also ties back to Eric’s issue. The lack of a least element is taken to imply that there is a non-zero value that limits the reals - and that this value in some sense quantises the reals. So we get back to the problem where the notion of infinity is not understood, and arguments about 1/infinity, and infinitesimals that go is a circle. Ties back to the continuum hypothesis - which I sort of wonder if Valmont314 isn’t somehow trying to disprove - especially with the notion of a dichotomy of “symbolic representations” for numbers.
This is just the stuff Valmont314 wrote, and the subject of the existing conversation. Sadly they don’t add anything new.
At 1100 AM: Post #1000!
Yeah, I’m peeved I didn’t get to manage #999. Oner suspects these posts arrived with the explicit purpose of nailing those numbers.
AND coming this early in the month from a new member of the Class of October 2012! More than one suspects. Who could it be?
Also… According to ancient Hindu prophecy, the universe was supposed to end with a giant thunderclap when this thread reached 999 posts. What went wrong?
@President Johnny Gentle: Thanks for the more formal ε-δ proof. That’s what I was looking for. It’s been so long since I’ve done one of those, I sort of wanted a reminder of how they work.