Is Some of the Platonic Mathematical World Inaccessible to Us?

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In summary, the conversation revolves around the book GEB by Hofstadter and its mind-blowing content. The speaker mentions a quote from the book about formal systems and how it relates to Penrose's Platonic mathematical world. They also bring up their previous study of Canter's work and its connection to the concept of infinity. The conversation ends with a discussion about the cover of the book and the speaker's preference for the mathematical content over the philosophical aspects.
  • #1
wildman
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I am reading Hofstadter's book GEB and am in a state of being blown away. My universe has just under gone a huge expansion... I now know why people spend their lives studying this stuff. It is staggering.

On page 72, Hofstadter quotes a result: "There exist formal systems for which there is no typographical decision procedure." This follows from the result that "There exist recursively enumerable sets which are not recursive."

This made me think of Penrose's Platonic mathematical world (The Road to Reality page 20). Let's for a moment assume that this mathematical world exists and that we discover things out of it instead of just inventing math. It would seem that the results quoted in GEB means that some of this Platonic world is inaccessible to us. Correct?

Now I studied Canter last year and read a beautiful proof that demonstrated that the are an order of infinity more irrational numbers than rational numbers. The second question is: Are the formal systems for which there exist no typographical decision procedure orders of infinity "larger" (inexact word, but I'm not sure what to use) than the systems where a decision process exist? This seems to be intuitively what should be true but as Hofstadter says, you can't always trust your intuition in such things.
 
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  • #2
Page 72 of GEB ,~^ :approve:

Keep reading. your trip has only just begun, that is a great book.
 
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  • #3
My copy of GEB is from the original printing. A few of the pages are falling out. The cover is much uglier than the modern reprint...it's mostly this ghastly tan color. But the book was a huge eye-opener, yes.

I skipped over the philosophical mumbo-jumbo at the beginning of Road to Reality and went straight to the actual math, which is much more interesting to me. I've already done enough philosophy in my life to know what I think of ontology...I don't really need to know what Penrose thinks on it.
 
  • #4
If you have studied Cantor's diagonalisation argument, then I believe you will enjoy the middle section where he uses it to derive Godel's incompleteness principle.
 

FAQ: Is Some of the Platonic Mathematical World Inaccessible to Us?

What is "Exploring the Inaccessible: A Journey Into Hofstadter's GEB"?

"Exploring the Inaccessible: A Journey Into Hofstadter's GEB" is a book written by Douglas Hofstadter, a cognitive scientist and professor at Indiana University. The book is a companion to his earlier work, "Gödel, Escher, Bach: An Eternal Golden Braid", and delves deeper into the themes and concepts presented in the original book.

What is the main focus of the book?

The main focus of the book is to explore the concept of "incompleteness" and how it relates to various fields such as mathematics, music, and language. Hofstadter also delves into the concept of self-reference and its implications in artificial intelligence and consciousness.

Is this book suitable for someone with no prior knowledge of "Gödel, Escher, Bach"?

While it helps to have some background knowledge of "Gödel, Escher, Bach", "Exploring the Inaccessible" is written in a way that is accessible to readers with no prior knowledge of the first book. Hofstadter provides a brief overview of the main concepts from "Gödel, Escher, Bach" to help readers understand the context of the new material.

What makes "Exploring the Inaccessible" different from "Gödel, Escher, Bach"?

"Exploring the Inaccessible" goes into more depth and detail on the topics presented in "Gödel, Escher, Bach". It also includes new material and updates to some of the ideas presented in the original book. Additionally, it focuses more on the philosophical and psychological implications of the concepts, rather than just the mathematical and logical aspects.

Is this book only for scientists or academics?

No, "Exploring the Inaccessible" is written for a general audience and does not require any specialized knowledge or background in science or mathematics. However, readers with an interest in these fields may find the book particularly engaging and thought-provoking.

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