2 + 2 =....

StrugglingSceptic · Nov 5, 2005

Tinker Grey said:
PI doesn't exist?
e doesn't exist?
sqrt(2) doesn't exist?

Not according to Kronecker. In response to Lindemann's proof that pi is transcendantal, he said "What use is your beautiful investigation regarding pi ... when irrational numbers do not exist."

Kronecker felt that all mathematical objects should be constructible via finite processes, whereas the constructions of the reals by Cauchy sequences and Dedekind cuts require that actually infinite sets and classes of actually infinite sequences be legitimate mathematical objects. Kronecker seems to have rejected this.

FreezBee · Nov 5, 2005

StrugglingSceptic said:
Kronecker felt that all mathematical objects should be constructible via finite processes, whereas the constructions of the reals by Cauchy sequences and Dedekind cuts require that actually infinite sets and classes of actually infinite sequences be legitimate mathematical objects. Kronecker seems to have rejected this.

So Kronecker and Berkeley would have agreed (except they didn't live at the same time). Though they must have disagreed on the virgin-birth question, since if God created the whole numbers, and everything else is the work of humans, then...

cheers

- FreezBee

Tinker Grey · Nov 5, 2005

I would guess that some mathematicians would disagree with Kronecker on this point.

I seem to recall that someone proved that there are infinitely more irrational numbers than rational ones.

If there are no irrational numbers then there are no circles (at least perfect ones) nor are there even squares (since their diagonals are often of irrational lengths).

Odd.

StrugglingSceptic · Nov 5, 2005

FreezBee said:
So Kronecker and Berkeley would have agreed (except they didn't live at the same time). Though they must have disagreed on the virgin-birth question, since if God created the whole numbers, and everything else is the work of humans, then...

Berkeley's criticisms of the Calculus were well justified and never adequately dealt with. I do not think they have anything to do with Kronecker.

StrugglingSceptic · Nov 5, 2005

Tinker Grey said:
I would guess that some mathematicians would disagree with Kronecker on this point.

Try nearly all of them .

I seem to recall that someone proved that there are infinitely more irrational numbers than rational ones.

Cantor proved this result, and indeed produced the first comprehensive theory of infinite sets, as well as seminal work in transfinite arithmetic. This earned him plenty of derision from Kronecker (who was a professor of Cantor's). Kronecker referred to Cantor as a `corrupter of youth' because of his theories.

If there are no irrational numbers then there are no circles (at least perfect ones) nor are there even squares (since their diagonals are often of irrational lengths).

This does not follow. The Greeks got by just fine without irrational numbers. Why should the existence of a circle presuppose numbers which can measure both its circumference and radius?

Odd.

JesusZone · Nov 5, 2005

2+2 cannot add up to 5...But in God term, 1+1=1, when it comes to marriage.

Tinker Grey · Nov 5, 2005

StrugglingSceptic said:
This does not follow. The Greeks got by just fine without irrational numbers. Why should the existence of a circle presuppose numbers which can measure both its circumference and radius?

I confess I haven't given much time to these thoughts.

But, the direction of my thinking is this: The diameter is measurable; the circumference is measurable; the relationship between those measurements exist OR the measurements don't exist.

In the physical world, those distances are finite and the accuracy and precision of our measurement are subject to our instruments.

In the theorethical world, we can describe the relationship of adjacent points all equidistant from a non-adjacent point.

WRT a square, surely the distance exists between any two points -- in this case, between the vetices opposite one another. That it takes infinite precision to accurately describe it doesn't lead me to the conclusion that the number doesn't exist.

Anyway, I'll settle for the genius of Cantor over my own (such as it is).

philadiddle · Nov 6, 2005

caitsey said:
Hey, I'm in the midst of studying for my Philosophy exam tomorrow... and I came across a good question (that all you Phil buffs probably have questioned before, but bah, I haven't! lol)
Anyways...

Does God have the power to make 2 + 2 = 5?

Because of my beliefs I'm tempted to say 'yes' right away, but with further thought...I have trouble backing that answer up. Hmm....

God took a few loaves of bread and some fish to feed thousands. It added up to more then it should have, and in the end there were baskets of extra food. (Matthew 14)

you could also look at it the other way and argue semantics. It's actually a meaningless question because the question itself is a contradiction and therefore collapses on itself. It doesn't pertain to reality as we know nor any type of potential reality.

Lucretius · Nov 6, 2005

philadiddle said:
God took a few loaves of bread and some fish to feed thousands. It added up to more then it should have, and in the end there were baskets of extra food. (Matthew 14)

or maybe God was just bad at math.

StrugglingSceptic · Nov 6, 2005

Tinker Grey said:
I confess I haven't given much time to these thoughts.

But, the direction of my thinking is this: The diameter is measurable; the circumference is measurable; the relationship between those measurements exist OR the measurements don't exist.

In the physical world, those distances are finite and the accuracy and precision of our measurement are subject to our instruments.

In the theorethical world, we can describe the relationship of adjacent points all equidistant from a non-adjacent point.

But how do we know we can measure the diameter and circumference in the theoretical world?

WRT a square, surely the distance exists between any two points -- in this case, between the vetices opposite one another.

Why should it? Certainly the line between any two points exists, but how do we know we can measure that line?

That it takes infinite precision to accurately describe it doesn't lead me to the conclusion that the number doesn't exist.

If actual infinities (as in, actually infinite decimals) are illegitimate, then how would you conclude that irrationals do exist?

Anyway, I'll settle for the genius of Cantor over my own (such as it is).

This controversy over the infinite is pretty much history. The generation of mathematicians following Cantor quickly forgot Kronecker, and now infinite set theory permeates modern mathematics.

philadiddle · Nov 6, 2005

Lucretius said:
or maybe God was just bad at math.

i'm surprised at you. in the C&E forum you give good replies but all i get from you in this forum is a response reflecting your ignorance on the concept of miracles.

FreezBee · Nov 6, 2005

FreezBee said:
So Kronecker and Berkeley would have agreed (except they didn't live at the same time). Though they must have disagreed on the virgin-birth question, since if God created the whole numbers, and everything else is the work of humans, then...

... except that Jesus was the ONE (excuses to all non-Christians). Now everything figures

StrugglingSceptic said:
Berkeley's criticisms of the Calculus were well justified and never adequately dealt with. I do not think they have anything to do with Kronecker.

Neither do I - but I suppose that people can agree on a subject, even for different reasons

StrugglingSceptic said:
The Greeks got by just fine without irrational numbers. Why should the existence of a circle presuppose numbers which can measure both its circumference and radius?

Hmm, it's some 20 years ago I studied mathematics, so my memory may serve me badly, but I believe to remember that the discovery of the impossibility of such numbers became a problem for the Pythagoreans (or maybe it was some other group).

Now, for a circle we have that

Circumference C = 2*PI*R, where R = Radius

and for a square we have that

Diameter D = sqrt(2)*S, where S = Side length

What kind of sense do these equations make, if both sides do not exist at the same time?

As I recall, the Greek mathematicians never measured in order to avoid the problems of imprecise measurements - however they did acknowledge the existence of both sides of an equation at the same time. If a circle has a radius, it does have a circumference at the same time, and if a square has a side length, it does have a diameter at the same time.

What did Kronecker have to say about this?

cheers

- FreezBee

StrugglingSceptic · Nov 6, 2005

FreezBee said:
Hmm, it's some 20 years ago I studied mathematics, so my memory may serve me badly, but I believe to remember that the discovery of the impossibility of such numbers became a problem for the Pythagoreans (or maybe it was some other group).

Yes, the discovery of incommensurable magnitudes, in particular the side of a square and its diagonal, is attributed to the Pythagoreans.

Now, for a circle we have that

Circumference C = 2*PI*R, where R = Radius

and for a square we have that

Diameter D = sqrt(2)*S, where S = Side length

What kind of sense do these equations make, if both sides do not exist at the same time?

Well, the Greeks would never have written those equations. All of their mathematics was presented in terms of geometric constructions.

As I recall, the Greek mathematicians never measured in order to avoid the problems of imprecise measurements - however they did acknowledge the existence of both sides of an equation at the same time. If a circle has a radius, it does have a circumference at the same time, and if a square has a side length, it does have a diameter at the same time.

Greek geometry talks of lines being measured by other, shorter lines, which divide them exactly. They discovered that the side of a square and its diagonal cannot be measured in this way by any other shorter line -- the two are incommensurable.

Incommensurability is usually described in terms of irrational numbers, but this is an anachronism, since the Greeks didn't have such a concept. And this brings me back to my original point: the Greeks got by just fine constructing circles and squares, without a concept of irrationals.

What did Kronecker have to say about this?

Nothing would have prevented Kronecker from constructing geometrical figures. Geometric constructions can be governed entirely by a collection of axioms (as is the case with Euclid's Elements) which make no reference to numbers.

cheers

- FreezBee

Tinker Grey · Nov 6, 2005

StrugglingSceptic,

I freely admit that I'm out of my depth.

Would you give your answers to the questions you posed to me? I am interested in what you have to say.

PKJ · Nov 6, 2005

I would just answer that the question is silly, self-contradicting, loaded, and useless.

Kinda like asking "Does the current king of France have a dog?"

StrugglingSceptic · Nov 6, 2005

Tinker Grey said:
StrugglingSceptic,

I freely admit that I'm out of my depth.

Would you give your answers to the questions you posed to me? I am interested in what you have to say.

My questions were only there to show that your conclusions don't necessarily follow, and that Kronecker could reject the existence of irrationals while still believing that circles exist. He just wouldn't be able to associate a circle's radius and circumference with numerical values.

That said, if there are any specific questions you have, I'd be happy to try to give an answer, though I should admit that I haven't actually read any of Kronecker. What I have said about him in this thread is just common trivia about him.

Tinker Grey · Nov 6, 2005

StrugglingSceptic said:
My questions were only there to show that your conclusions don't necessarily follow, and that Kronecker could reject the existence of irrationals while still believing that circles exist. He just wouldn't be able to associate a circle's radius and circumference with numerical values.

That said, if there are any specific questions you have, I'd be happy to try to give an answer, though I should admit that I haven't actually read any of Kronecker. What I have said about him in this thread is just common trivia about him.

Thank you for your kind response.

Is there a link to something I could read that argues that irrationals do exist? Or, is it that the concept of irrationals work so well to explain our world that most mathematicians find the argument silly?

StrugglingSceptic · Nov 7, 2005

Tinker Grey said:
Thank you for your kind response.

Is there a link to something I could read that argues that irrationals do exist? Or, is it that the concept of irrationals work so well to explain our world that most mathematicians find the argument silly?

Not at all. I should think most mathematicians regard the arguments as extremely important historically, and most undergraduates are presented with them on analysis and set theory courses.

The arguments are generally explicit constructions of the set of real numbers in terms of rational numbers. That is, you build new objects from the rational numbers which you declare to be the real numbers, after showing how to define arithmetic for them and showing that they satisfy all the expected properties. Then, so long as someone believes that rationals exist, they are forced to conclude that reals also exist, because the reals can be constructed from them.

The first such construction was given by Richard Dedekind, and if you google for "dedekind cuts" you should be able to get an idea of what it involves (it can be quite technical, but a quick browse hasn't turned up anything more palatable).

The construction requires taking a real number to be a pair of infinite sets. Another construction (due to Cantor) requires taking a real number to be a class of infinite sequences. So both of these constructions require taking an infinite set to be a legitimate mathematical object, and for some philosophers and a small minority of mathematicians this is not acceptable (it does not appear to have been acceptable to Kronecker either).

nb_christseeker · Nov 7, 2005

Abbadon said:
How do we know God isn't constantly changing values such as "2", or "Pi", or "triangle" constantly, we just aren't capable of realizing it, or our memories of it change, or something?

Because deep down, I think we all know that's stupid. Or maybe my definition of stupid is constantly changing the more I hear certain people talk.

nb_christseeker · Nov 7, 2005

I mean, what is stupid? Braille on drive-up atm machines. now that's stupid. but someone with money thought it was smart. so having money doesnt mean you cant be stupid.

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