lao tzu wrote:
The personal insults constitute a non-starter, of course, but I'm assuming you could probably refrain in a direct conversation away from the internet.
There are no personal insults coming from me. You are mistakenly taking my objections to an entire mathematical community
personally. This would be like a Christian being
personally offended when someone points out the absurdities in the Bible. Of course, Christians often are personally offended in this way, but it's totally unjustifiable since they didn't write the Bible.
In a very similar way, for you to become
personally offended by objections to the development of the formalization of modern mathematics over last few centuries is truly absurd. You did not develop this formalism, and thus for you to become personally insulted over someone who rejects it is nothing other than absurdly silly.
For someone who calls himself "Lao Tzu" you should most certainly know better than to allow yourself to become emotionally insulted over someone's view of a formalism that you did not personally create.
lao tzu wrote:
What are these "completed infinities" you suppose mathematician's accept?
If you accept the work of Georg Cantor, you necessarily accept the concept of completed infinities because that's precisely what Georg Cantor's work is based upon. There were many famous mathematicians at the time who objected to this precisely on these grounds. Henri Poincare was one of the most vocal about this proclaiming Cantor's Set theory would be a
disease from which the mathematical community would someday recover. I totally agree with Henri Poincare. Leopold Kronecker also rejected Cantor's completed infinities with a passion, as did many others.
So I'm just siding with a the mathematicians who reject this concept. And there were many of them at the time of Cantor, and there are still many today who do not accept these notions.
In fact, you appear to be one of them who believe that you can somehow accept Cantor's formalism whilst simultaneously rejecting his necessary requirement that infinities can be completed.
In fact, that's the only way in which infinities of larger or smaller cardinal properties can even exist.
If you accept infinity as an endless process, then what sense to you make of an infinite that is "larger" than this?
One set is
more endless than another?
I hope you'll forgive any personal insult that this may seem to imply, but IMHO, that's absurd.
I totally reject the idea of completed infinities, and I totally reject the idea of infinities that are
more endless than other infinities. That too is simply absurd.
lao tzu wrote:
Hilbert's Hotel seems paradoxical to the uninitiated, but it is merely heuristic. It shows a property of infinite sets that is not shared by finite sets. You can't double the size of a finite set and still make the elements match up. But you can do this with any infinite set. We walked in knowing the finite and the infinite were different, and Hilbert's hotel shows us a specific instance of that difference. It's not the only difference; it's only an engaging difference that can serve to motivate further investigation.
I have no problem with the concept of infinity as endlessness, or as an endless process. I'm not arguing against concept of infinity in general. A set that is endless is actually uncountable, IMHO. To claim that it can be used as a standard for countability (as Cantor claims) requires that it be thought of as a completed cardinality. And it is Cantor's proposal that I object to.
I have no problem with Hilbert's Hotel.
lao tzu wrote:
Investigating further, we look at the other infinite sets we typically encounter. We know the real numbers and the rationals are different. To measure some of that difference, we've created the concept of cardinality.
We didn't
create the concept of cardinality to do this. On the contrary the concept of cardinality is older than the hills.
Moreover, there are two things wrong here:
One is that really really have no business creating a set called the "Real Numbers" and tossing irrational relationships into the set and attempting to treat irrational relationship as thought they themselves have well-defined cardinality. They don't.
So that was mistake #1.
However, given that mistake that seems historically inevitable to have happened, we still don't need the concept of cardinality to tell these two sets apart. Because the difference between them is no a difference in cardinality. They are both infinite sets and we truly only need one concept of infinity (i.e. endlessness)
Both of these sets are equally infinity. They are both endless sets. They do not have a difference in cardinal properties.
However, they are indeed quite different sets qualitatively speaking. The set of rational numbers is a set containing sets of valid cardinality.
The set of "Real Numbers" is a set that contains two different types of elements. One type of element are indeed the rationals which are themselves sets of valid cardinality, the other type of element are these irrational relationship that are not valid ideas of cardinality.
And therein lies the difference between these two infinite sets. The infinities are identically (i.e. they both contain an endless amount of elements). Neither set is more endless than the other. They are both endless.
The difference between these objects is a qualitative difference, not a difference in cardinality. And this is what becomes totally lost in Cantor's formalism.
lao tzu wrote:
Yes, it was born a century ago as Cantor's baby, but an entire host of mathematicians, including your correspondent, have been engaged in helping it grow up ever since. Dissing Cantor because you're uncomfortable with the theory of transfinite cardinals is akin to dissing Darwin because you're uncomfortable with evolution, or disputing the size of the pyramids because they were measured in a tutu. You can insult us freely, at least here in America where you're guarded by the first amendment, but it doesn't make the behavior any less rude or poorly considered as a critique of the arguments.
This has nothing at all to do with what I'm comfortable with.
I'm just making purely logical observations here.
Irrational relationship shouldn't be treated as cardinal numbers in the first place. It is neither necessary, not productive. On the contrary attempting to treat them as such leads to all sorts of utter absurdities that only serve to confuse the truth of what's going on.
You've compared this with Darwin's evolution, but that's a really bad comparison because there is much evidence for Darwin's evolution. Where is there any evidence that irrational relationship need to be thought of, or formally defined as cardinal properties of sets? That is a totally arbitrary choice what cannot be justified by nature.
In fact, this very notion brings up the question of whether or not you believe mathematics is a "science"?
So you believe that there is ONE TRUTH "out there" that mathematics reveals to us? And if so, then what criteria do you suggest we use to measure whether or not we actually have that ONE TRUTH?
And is that ONE TRUTH a
physical property of this universe (like fossil records are for Darwin's theory of evolution)?
If you are going to compare the work of Georg Cantor with the work of Charles Darwin you had better be prepared to show
physical evidence to back up Cantor's formalism.
I believe I can show just the opposite. I believe that I can show that Cantor's ideas are entirely whimsical (even he claimed that he got his ideas from God).
I can show
physical evidence from this physical universe for my understanding of irrational relationships.
Therefore my theories are more akin to Darwin's than Cantor's are.
I'm suggesting that mathematics not only should be, but actually can be made into an actual physical science.
At least with respect to numbers and ideas of quantities. Please don't ask me to drag things like Boolean Algebra into this because Boolean Algebra has nothing at all to do with numbers as an idea of a cardinal property of a set.
I'm talking about mathematics here as a formalism of ideas of
quantity which I propose is indeed an actual physical property of our physical universe.
Why do you think mathematics has worked so well when applied to the physical universe anyway?
Eugene Wigner, a physicist once wrote a paper entitled
"The Unreasonable Effectiveness of Mathematics in the Natural Sciences".
Now what sense does that even make? Why should it be
unreasonable that mathematics should be so effective in describing the natural sciences?
It would only be unreasonable if a person believed that mathematics was just a totally made-up whimsical idea of mankind (which it unfortunately has become recently).
But if mathematics were actually a science that stuck to only describing the actual quantitative physical behavior of the physical universe (as it first started out to be) then it should come as no surprise at all that mathematics describes the physical universe so well.
I claim that mathematics most certain
can be a physical science. But in order to get there we need to back track and toss out places where we had made some horrible wrong turns.
If mathematics had stuck to it's guns from the very beginning it would be science. Unfortunately irrational relationship were discovered and the mathematicians didn't know how to handle them so they treated them like cardinal properties (which physically they are NOT).
And that was a wrong turn. That wrong turn was actually made thousands of years ago, and was already well-accepted by the time Cantor came along, so I guess we really can't pin the blame on Cantor for this underlying problem.
In any case, if you are going to hold that Cantor's work is equivalent to Darwin's work, then you had better be prepared to also claim that mathematics is a science that is directly dependent upon physical evidence from the physical world, and show me how that fits in with Cantor's work.
I am certain that you won't be able to do that, because that's not what we have here.
Cantor's ideas are entirely whimsical and he himself even claimed that he got his ideas from God, not from the physical universe.
Cantor's formalism is basically his own little religious whimsical cult.
lao tzu wrote:
Keep in mind while you're sniggering at his voices from God that you're posting under the handle "Divine Insight."
Let me tell you about my screen name.
When I came to logon I couldn't think of a decent screen name. I sat there at the computer and asked the Goddess to give me some divine insight into a clever screen name. I intuitively heard her say to me, "That's a good one".
So I typed it in and here I am.
But let's not get distracted into wishy-washy religious stuff.
lao tzu wrote:
The theory of cardinals is both a sharp-edged and a blunt tool. Blunt because it can't differentiate between the rationals and the counting numbers, but sharp in that it can find a difference between the reals and either one of them, or between the reals and any of the less well known sets we speak of as "countably infinite," such as the algebraic numbers I alluded to earlier.
But it can't. Cantor's diagonalization proof is logically flawed. His list of reals cannot be a complete list. Therefore the fact that he can come up with strings of numerals that can't be on his list is no surprise at all. It doesn't make any logical statements about the cardinality of any sets at all. It's a bogus proof.
But as I have already pointed out, mixing irrational relationships in with cardinal numbers in a single sit is already a bogus idea to begin with. So the whole shebang is flawed from square one.
lao tzu wrote:
We use these theories because they work. We replace these theories with theories that work better. Are you proposing something that works better? Well, no. Do you have, for instance, a measure or criterion that allows us to differentiate between both the reals and the rationals, and between the rationals and the integers? That's what you need.
But they don't work. They only seem to work in very esoteric situations concerning abstract mathematical collections that contain objects that don't even have well-defined cardinality themselves.
Moreover, there may be something
useful in that kind of abstraction. I'm not saying that we can't studying those sorts of things. Just like I'm not saying that we shouldn't be studying Boolean Algebra. I'm simply saying that we need to recognize that these ideas are no longer ideas of proper cardinality, but instead they have become contaminated with other qualitative ideas.
Ideas that would actually be
better studied if their own qualities were actually recognized to be something
different from an idea of cardinality.
As I've already said, you can learn much more about irrational relationship when you realize their true physical nature than you can if you instead try to treat them like ideas of cardinality of sets.
lao tzu wrote:
It's not reasonable to expect anyone to trade in better for worse.
And I would never ask anyone to do that. What I have to offer is definitely better.
lao tzu wrote:
p.s. Your method, a finite truncation of an infinite list you've named a "square" via some private definition — and it is your method, not Cantor's, or any mathematician's — is insufficient to count the counting numbers, even before we expand the set to include your 3forevers, 142857forevers, etc.
You could fix the first, by the way, by extending each of your terminating "integers" with an infinite string of zeros, though it'd fit like a tire patch on a bagel. You'd be better off simply sticking with Cantor's constructions, which wasn't broke until you broke it.
To accomplish the second, map the strings to associated simplified fractions, and order them first by the sum of numerator and denominator, and last by the numerator, yielding a total order on the full set of positive rational numbers.
1/1
1/2, 2/1
1/3, 2/2*, 3/1
1/4, 2/3, 3/2, 4/1
1/5, 2/4*, 3/3*, 4/2*, 5/1
1/6, 2/5, 3/4, 4/3, 5/2, 6/1
etc.
*skip unsimplified fractions to avoid redundancy
By observation and construction, then, the set of rationals is countably infinite. QED.
I'm familiar with how the rationals can be placed into a one-to-one correspondence with the natural numbers. I'm not impressed, because there is not need to even do this anyway. It's utterly meaningless. Both the set of integers and the set of rationals are infinite.
But I can, and already did, I can show how
using Cantor's method I can create infinitely many integers that cannot be put into a one-to-one correspondence with the integers themselves.
And there is not need to use infinitely long integers.
I already did it somewhere in this thread. Let me see if I can find it.
Here it is clear back from page #1 post #3:
Here is my list. On the left I have numbered the list with integers that I'm going to line up with. To the right of each number I choose an
aribitary integer just like Cantor does in his diagonal argument.
I increase the numerical length of my integers in each row. But I am free to do this because these are
arbitrary integers just as Cantor chooses his
arbitrary reals. I could also always use longer integers if I so desire, but I do it this way simply because it makes for a neater diagram.
Here is my arbitrary list of integers
1) 1
2) 12
3) 167
4) 5785
So now I run my diagonal line down this list
1) [strike]1[/strike]
2) 1[strike]2[/strike]
3) 16[strike]7[/strike]
4) 578[strike]5[/strike]
I arbitrarily replace the following numerals.
1 = 2
2 = 3
7 = 8
5 = 6
My new integer is 2386, this integer cannot be anywhere on my list above.
In fact, I can create many more integers that cannot be on my list just by making different arbitrary substitutions above. And I can continue this forever.
What will I end up with? I will end up with infinitely many integers that
cannot be anywhere on my infinite list.
Thus showing that I have created infinitely many integers that cannot be put into a one-to-one correspondence with the integers.
This is, of course, total baloney. But this is the folly of Cantor's diagonal method.
His method is clearly flawed. It can even be used to prove that the integers can't be put into a one-to-one correspondence with themselves which is totally absurd.
So this is a proof by contradiction that Cantor's diagonalization method of trying to put sets into a one-to-one correspondence with each other fails. It's logically flawed.
We can use his method to show that we can create a list of integers that can't be anywhere on our list whilst we are attempting to map them to the integers themselves.
And I've already explained why this is impossible. It's impossible because our system of numerical notation cannot be used to create square lists that are complete. This is why this method must
always fail.
It has nothing to do with the set of real numbers being "cardinally larger" than the set of integers. They are both equally infinite.