Einsteins last years with Godel were ignored for 50 years, UNTIL NOW! We still stand at the frontier

reply to post by bigfatfurrytexan


Yeah it is interesting because zero is actually "negative infinity" based on cardinal numbers. Aristotle was against "negative infinity" but the basis of the Greek Miracle is that "negative infinity" remains undefined.

This was considered the great breakthrough of Eudoxus as Princeton math professor Nelson points out Confessions of an Apostate Mathematician by Edward Nelson Department of Mathematics Princeton University pdf

The Babylonians were wizards at computation, they had at least one of the two ways of generating what we now call Pythagorean triples (as Otto Neugebauer brilliantly demonstrated from the Plimpton cuneiform tablets), they could approximate the square root of two to as many sexagesimal places as desired, but the question of the rationality or irrationality of the square root of two simply would have made no sense to them, because they had no numbers in their conceptual framework. This is not surprising, because numbers did not yet exist in those days. Numbers were invented (or revealed, as believers would maintain) by Pythagoras.

So the number of "number places" is not the issue for infinity because the Babylonians could also do that -- as it is part of "divide and average" mathematics.

But Eudoxus eliminated the dualism of number and magnitude. His idea was this: rather than say what the ratio of two magnitudes is, it suffices to define a notion of two such (possibly nonexistent) ratios being equal, and this he did by a subtle quantification over all the Pythagorean numbers. Despite the fact that it was in the Elements for all to read, this construction of the real number system was not understood – not even by Galilei. It was just last century that the notion was re-invented by Richard Dedekind. I know of no parallel to this in the history of human thought.

O.K. the "Dedekind Cut" is considered a solution by only some mathematicians. As I pointed out math professor Luigi Borzacchini recognizes a "preestablished disharmony" from this invention of Eudoxus but also Borzacchini argues that it was Archimedes who also made this Greek Miracle invention.

This article emphasizes Archimedes’ original contribution to the principle or lemma of Archimedes (in modern terms, given two homogeneous magnitudes a, b, there exists an integer n so that na > b), which tends to be attributed to Eudoxus before him. According to the author, Archimedes had to develop the lemma, because he dealt with more complex problems including the length of curved lines.

Why rob Archimedes of his Lemma? (English) Mediterr. J. Math. 3, No. 3-4, 433-448 (2006). ISSN 1660-5446

The professional math forum discussing the recent discovery of Archimedes' use of magnitudes for an "actual infinite" calculus

So the "actual infinite" is the great paradox of calculus versus the "potential infinite" used by Aristotle.

Borzacchini chimes in for clarification:

There was only a real arithmetical theoretical field: music, and there we can find the roots of the arithmetical theory of proportions (Szabo). Eudoxos' theory was tailored to face geometrical 'logoi' to be employed in astronomy or in exhaustion methods. In Aristotle we read the awareness of possible common proofs, but never in classic times it became a general theory. ...The Greeks understood with Eudoxus the problem of multiplying numbers and magnitudes, but never the problem of attaching numbers to magnitudes, except for trade.

The paradoxes of infinity remain - the Princeton professor Nelson goes into the deep debates in mathematics about infinity -- and then concludes -- "math works" just the same way an "herb" works for both a tribal culture and a modern culture - but without any need of any formal logical reasoning. This answer is deeply ironic considering again -- math (and science) works -- but for whom? Math wipes out the tribal cultures through the creation of Western geometric technology destroying local ecological-based cultures.

Nelson compares this new meaning of math to a music composer no longer making "program" music - but he is revealing again the secret origin of math from music without Nelson really understanding the secret truth of music. Nelson links to Nonstandard analysis which transfers infinitesimal "ideal" numbers into actual real numbers

O.K. so people who accept calculus have to also accept the proof for the square root of two (and other real numbers) -- but as I've pointed out this is not strict logic -- Did the Greeks discover the irrationals?

very interesting topic

very very interesting

reply to post by Moduli


There is a difference between a direct measurement of a physical observable and an inference based on logical relations which are themselves non-physical. In my experience when singularities or infinities pop up in simple scientific situations I just switch to a new conception/state and use a new framework- it is really an arbitrary practice from the point of view of a logician.

Originally posted by ZeuZZ

The problem is that today, some knowledge still feels too dangerous.
Because our times are not so different to Cantor or Boltzmann or Gödel’s time.
We too feel things we thought were solid, being challenged, feel our certainties slipping away.
And so, as then, we still desperately want to cling to belief in certainty.
It makes us feel safe.
At the end of this journey the question, I think we are left with, is actually the same as it was in Cantor and Boltzmann’s time.
Are we grown up enough to live with uncertainties?
Or will we repeat the mistakes of the twentieth century and pledge blind allegiance to yet another certainty?

Thanks for reading

edit on 7-7-2012 by ZeuZZ because: (no reason given)

Excellent post op.
Yes some knowledge and some half knowledge is dangerous.
But any knowledge will always be incomplete. Uncertainities galore.
However the question of infinities and infinite dimensions will always turn any1 topsy turvy,
unless one has a physical essence of infinities and infinite dimensions in ones physical being.