lie strewn along

real number line
Georg CantorInfinities of infinities lie strewn along the length of the real number line. As Georg Cantor discovered, some infinities are bigger than others. (Does that mean some absolutes are more absolute than others?).

Very sad story: Insanity came to visit Georg toward the end of his life. Occupational hazard of contemplating the infinite too closely.

Some infinities are completely contained within others, some partially. Some uncountables are countable, ie some sets of uncountables have a countable number of members. The set of infinities that are not infinite has no members, has 0 members. Therefore the set of infinities that are not infinite is countable.

I'm not sure but I think I read somewhere that the infinity of real numbers between any two consecutive integers, eg 1 and 2, is greater than the infinity of all the integers up and down the entire length of the number line. You can put that in your pipe and smoke it.

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puny human said...

Impossible! How can one infinity be larger or smaller than another? No wonder Mr. Cantor went insane. I can barely contemplate the evening news, which is finite after all, without courting mental illness.
Best to you!
Puny

masterymistery said...

puny human, yes it seems strange but apparently there are different 'orders' of infinities. I don't get it either.

MM

Faycin A Croud said...

Numbers make me want to put something in my pipe and smoke it! Preferably something that will make me forget about the numbers.

mgeorge said...

How did I miss this? Perhaps a subconscious recoiling from what I had to study - or try to - many moons ago.

The concept of infinite sets of numbers is integral to maths. E.g. in calculus, we consider an infinite number of thin slices under a curve bound by 2 points (e.g. 2 integers) along an axis. Beyond the galaxy of real numbers lies imaginary numbers having an additional "imaginary" component (in the form n1 + n2 x i, where i is "square root of -1"). To reiterate.

Modern science rests on a foundation of such maths, helping us to explore the universe at the very large and very small extremes in an almost routine fashion. Modern maths also powers the technology behind many amenities we take for granted.

However, one is "at sea" with no conforting analogies soon after leaving the familiar shores of basic maths such as geometry and algebra that the ancients bequeathed to us. We see this in these quotes from establised scientists and mathematicians:

- As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. - Albert Einstein, Geometry & Experience, 1921
- The fact that our universe lends itself to mathematical treatment is not a fact of any great philosophical significance. A mathematical web of some kind can be woven about any universe containing several objects. - Prof. Bertrand Russell, adapted
- In mathematics you don't understand things. You just get used to them. - Prof. John von Neumann
- Mathematics is not only real, but the only reality. All you can say about a sub-atomic particle is to cite its mathematical properties. - Martin Gardner, 1994, adapted

masterymistery said...

Hey Faycin, sorry i didn't respond to your comment earlier. I put too much in my pipe and it ended up smoking me.

masterymistery said...

hi mgeorge, I hated learning maths in high school --- it's taught so very badly... as a set of rules to be memorised ...a grab bag of dusty abstractions of no interest to a teenager whose hormones are pushing zir away from abstraction.

And of course, it's one of those things that to engage with effectively requires a bit of work and effort --- something we're not too good at these days.

Those creepy "personal trainers" have made at least one smallish contribution to the betterment of humandkind --- the meme "no pain no gain".

Much food for thought: There does seem to be much disagreement among scientists/thinkers about the role of maths in reality, I mean underpinning reality, is another way of saying it.

The quotes are great, but there's one missing, and I can't remember very much about it or who said it but the gist has to do with the incredibly unlikely fact that Reality seems to have been designed by a mathematician, and that it's totally unreasonable the extent to which maths lies "under the hood" of the universe so to speak. The very opposite of the Russell quote.;

So I'm much more in the GArdner camp, not to mention Douglas Hofstader and Daniel Dennett particularly in relation to that wonderful book they co-wrote: "Godel, Escher, BAch".

Actually, responding to your comment makes me think of another topic that I could/should have included in the blog-post: "renormalisation" of infinities (I think it was Feyman, or Schwinger, or both) who found how to remove those troublesome endlessnesses out of the way of the project to include quarks in the standard model.

This discussion seems to be heading in the direction of the Anthropic Principle, and I don't feel like going there, at least not now.

Thanks for stopping by and for your illuminating commments.

MM

mgeorge said...

Hi mastermystery,

Yes, one scientist said the universe was looking less like a great clockwork and more like a great thought.

Russell was not giving a mere opinion. Together with Alfred North Whitehead, he had attempted to set out all known mathematics from scratch. It turned out that it had to first principles, not scratch. Meaning that there were those axions or assumptions - quite self-evident to you and me but philosophical anomalies all the same.

As he was not quite respectable enough for some people due to his biting views on various social issues, this great finding sank into obscurity until recycled by Kurt Godel in the field of logic, as what is now know as the Incompleteness Theorem.

We instinctively seek pattern and design, and yearn for meaning, certainty and purpose - preferrably one that provides a starring role for ourselves.

"I have approximate answers, and possible beliefs, and different degrees of certainty about different things, but I am not absolutely sure of anything... I do not feel frightened by not knowing things, by being lost in a mysterious universe without having any purpose... Most of your actions are based on incomplete knowledge..." - Prod, Richard Feynman, Pleasure of Finding Things Out, pub. 1999

Regards.

masterymistery said...

mgeorge, thanks for your comments. I've been fascinated by Godel and the theorems for a number of years. I've often wondered whether Incompleteness could be applied to language, just as much of a formal system as arithmetic, eg.

Such that the "truths" or "axioms" of language can only ever be identified let alone proven by stepping outside of the system into a meta system, a higher level of abstraction.

But what are the truths of language? Meaning. Simply, what the words mean. One can define the meaning of words inside the system, eg by saying "a clock is a device for measuring the passage of time" but then the inevitable follow up questions begin "what's time"? "what's a "device"" "what is 'passage'" ? etc

And then you're into an infinite regression, whereby if one uses words to establish firmly the meaning of words, one never arrives at the "bedrock" meaning, there's always another "are we there yet?" to answer.

And the only way out of the infinite regress is to step outside the system itself, language, in order to clarify meaning and significance.

I've written a lot about the problems of language eg ooga booga: what does it really mean?

Proably got a whole heap of it wrong --- I'm just an enthusiastic amateur!

B said...

The really odd thing about the cantor set is that it is infinitely complex, but finite in dimension - like the numbers between 0 and 1. Just to throw a conversational hand grenade in here, but check out Aleph (http://en.wikipedia.org/wiki/Aleph_number), the size of sets - possibly a number greater than inifity

Since I can probably get away with them, here's a Douglas Adams quote:
"The car shot forward straight into the circle of light, and suddenly Arthur had a fairly clear idea of what infinity looked like. It wasn’t infinity in fact. Infinity itself looks flat and uninteresting. Looking up into the night sky is looking into infinity—distance is incomprehensible and therefore meaningless. The chamber into which the aircar emerged was anything but infinite, it was just very very very big, so big that it gave the impression of infinity far better than infinity itself."

masterymistery said...

B, Cosmic Rapture was set up as a place for people to lob conversational hand grenades in. So, thanks for the grenade!

I had a look at the URL, and now my head hurts! No wonder Cantor went insane.

Douglas Adams is always welcome in these pages, and the quote is particulaRLy relevant to the discussion.

Thanks for stopping by --- I hope you will come and visit again soon. is what