A Piece of Pi For Dessert During Covid Shutdown

[Aris Marghetis]

mathematics, a system of absolute logic.

O I love mathematics.
Unfortunately, the logic seems to break down when dealing with the concept of infinity...

I'm genuinely curious here (psychology is so fascinating): because the logic seems, to you, to break down, are you leaning towards the logic being suspect, or your perception of said logic?

[Dilip Panjwani]

https://www.newscientist.com/article...he-real-world/

[Erik Malmsten]

When I encounter things like that, Godel's Incompleteness Theorem comes to mind: https://en.wikipedia.org/wiki/G%C3%B...eness_theorems

In mathematics you can't avoid paradoxes, and many of them involve infinity.

I was waiting for the Godel theory to show up.

In chess the only truths are checkmate, stalemate or insufficient mating material. There are not infinite possibilities. Yet all theorems of play are incomplete, filled with exceptions to the "rules." We are emotionally attracted to natural looking moves, yet there may be unnatural ones that may lead to the truth. Bishops may employ a diagonal argument, rooks a straight forward one. Castling may be safe or may mean oh-oh. Patzers are attracted to the big pieces, while grandmasters can sense the pressure of the spaces around pawns. We can get stuck in an undecidable position with unanswerable questions.

One may believe any formal theory they want, and seek out to prove as many winning games as possible, yet they won't win them all. No one theory can consistently lead to the absolute truth in all games. One must let go of one's ego attachment to a theory to open up to the possibility of seeing the truth. We can take comfort in that even Magnus Carlsen doesn't have infinite knowledge and that we have to be a better player in some other parallel universe.

[Garland Best]

Unfortunately, the logic seems to break down when dealing with the concept of infinity...

Oh, you don't need infinity to make logic break down. For instance is the sentence "This sentence is false." true or false? And therein lies the rub.

Over 100 years ago Bertrand Russell attempted to write a treatise that placed all of mathematics - from set theory to arithmetic, to geometry to calculus - on a set of definitions and logical deductions. His goal was to have a system of logic that could be used to test any mathematical conjecture and rigorously prove whether the statement was true or false, given enough time.

Shortly afterwards Kurt Godel used Bertrand's own system to PROVE that either you CANNOT prove every mathematical statement true or false, or your math allows paradoxes like "This sentence is false". What's more the guy PROVED that you cannot come up with ANY mathematical system that avoids this dilemma. It's a mind-blowing example of creative thinking.

30 years ago I read the book "Godel Escher Bach" by Douglas Hofstadter. This massive book spent hundreds of pages showing how Godel was able to prove this, while somehow being entertaining to read (well at least to me :-)).

By the way, this also has applications in computer science, with what is called a Turing Engine. In theory, any computer system, including Alphazero (See how I brought it back to chess there?) can be described in a Turing engine. The question with Turing engines are "Can you predict if a program will eventually halt, or run on forever, simply by reading the code?" Godel's theorem proves that programs exist that you can never prove will halt or not.

Well I blathered on this topic long enough. Suffice to say, mathematics is fascinating and full of paradoxes like this, and are all well worth exploring.

[Fred Harvey]

Also, if you wish to eliminate infinities, please tell me what the largest integer is...

It is whatever you wish it to be! In the continuum in which we all exist, integers, like infinity, do not exist. I rather like six and seven-eighths......

[Fred Harvey]

This thread is actually becoming interesting! As a physicist, I tend to lean to reality. With a major in math, i believe math is merely an attempt to codify the continuum we live in, and can never totally represent that continuum, hence the fact that the concept of infinity is unproveable. IMHO to continue to debate this becomes meaningless mumbling full of sound and.....well you know the rest. I look forward to Ed Snidehouse proving me a total fraud, but have a sneaking feeling he is lost somewhere in infinity......

[Garland Best]

I believe math is merely an attempt to codify the continuum we live in, and can never totally represent that continuum, hence the fact that the concept of infinity is unproveable.

I think you are stating that you cannot prove or discover that there exists an infinite quantity in nature. Otherwise your statement is nonsensical. Infinity is a concept. It's defined. There is nothing to prove. These are constructs. There may not be an infinite quantity of anything in the universe, that does not make the concept itself any less real. The set of integers is infinite. There is no disputing that. If you are going to try, don't bother. You may as well dispute the existence of the square root of two, or the square root of negative one.

[Ed Seedhouse]

I think you are stating that you cannot prove or discover that there exists an infinite quantity in nature. Otherwise your statement is nonsensical. Infinity is a concept. It's defined. There is nothing to prove. These are constructs. There may not be an infinite quantity of anything in the universe, that does not make the concept itself any less real. The set of integers is infinite. There is no disputing that. If you are going to try, don't bother. You may as well dispute the existence of the square root of two, or the square root of negative one.

Actually I believe some astrophysicists believe that our universe is spatially infinite and has no boundaries in four dimensions.

If mathematics is all wrong then physics is too, since it thinks we live in a four dimensional universe. But no one can see this fourth dimension so it cannot exist and therefore physics must be wrong. At least according to Freud Smarmy.

[Aris Marghetis]

You may as well dispute the existence of ... the square root of negative one.

Well played!!

[Dilip Panjwani]

The notion of things (like the points on a line) being infinitesimally small (with their inverse being infinite), is no longer popular, as the quantum nature of reality postulates that at a certain level of smallness, you cannot further divide stuff!
On the other side (the largeness side), while the set of integers is 'potentially' infinite, each number within it is finite, and the 'potential' infiniteness is something we can dream about, but never realize in reality...especially as our space-time will end at some point with a universal black hole...

[Ed Seedhouse]

The notion of things (like the points on a line) being infinitesimally small (with their inverse being infinite), is no longer popular, as the quantum nature of reality postulates that at a certain level of smallness, you cannot further divide stuff!
On the other side (the largeness side), while the set of integers is 'potentially' infinite, each number within it is finite, and the 'potential' infiniteness is something we can dream about, but never realize in reality...especially as our space-time will end at some point with a universal black hole...

There will be no universal black hole but even if there would it won't be the end as black holes all eventually evaporate due to Hawking radiation. In the end all that will be left are photons and other massless particles. Penrose has postulated that this will result in a new "big bang" or actually an infinite series of big bangs since there is nothing to stop the process repeating endlessly.

[Dilip Panjwani]

There will be no universal black hole but even if there would it won't be the end as black holes all eventually evaporate due to Hawking radiation. In the end all that will be left are photons and other massless particles. Penrose has postulated that this will result in a new "big bang" or actually an infinite series of big bangs since there is nothing to stop the process repeating endlessly.

Close enough to what I have been postulating: INFINITE cycles of universal big bangs and black holes...each cycle with its own space-time...

[Ed Seedhouse]

The notion of things (like the points on a line) being infinitesimally small (with their inverse being infinite), is no longer popular, as the quantum nature of reality postulates that at a certain level of smallness, you cannot further divide stuff!

It doesn't postulate anything of the sort. For one thing gravitation is not quantized and a fundamental limit to size in spacetime at the low end would require that it be quantized. As yet however there is no evidence of this. A lot of physicists have tried but all have so far fallen short. Einstein still rules gravitation and his theory does not postulate that the gravitational field (which is spacetime) is quantized.

[Ed Seedhouse]

Close enough to what I have been postulating: INFINITE cycles of universal big bangs and black holes...each cycle with its own space-time...

Penrose is a lot more qualified to speculate than you or I. And even his speculation is highly controversial. What matters in the end is the evidence.

[Neil Frarey]

Happy B-Day M.C. Escher :))))

https://en.wikipedia.org/wiki/M._C._Escher