Is Some of the Platonic Mathematical World Inaccessible to Us?

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Discussion Overview

The discussion revolves around the accessibility of the Platonic mathematical world as proposed by Roger Penrose, particularly in light of results from Gödel's incompleteness theorem and Cantor's work on infinity. Participants explore the implications of these mathematical concepts on the notion of discovering versus inventing mathematics.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant reflects on Hofstadter's quote regarding formal systems lacking a typographical decision procedure, suggesting this implies some aspects of the Platonic mathematical world may be inaccessible to us.
  • Another participant expresses enthusiasm for Hofstadter's book, indicating a shared appreciation for its insights.
  • A participant mentions their preference for the mathematical content over the philosophical discussions in Penrose's "The Road to Reality," indicating a divergence in focus among readers.
  • There is a suggestion that those familiar with Cantor's diagonalization argument may find connections to Gödel's incompleteness principle, hinting at deeper mathematical relationships.

Areas of Agreement / Disagreement

Participants express varied perspectives on the philosophical implications of mathematical concepts, with some focusing on the mathematical rigor while others engage with the philosophical aspects. No consensus is reached regarding the accessibility of the Platonic mathematical world or the implications of decision procedures in formal systems.

Contextual Notes

Participants reference complex mathematical concepts such as Gödel's incompleteness theorem and Cantor's work on infinity, which may involve assumptions and definitions that are not fully explored in the discussion.

Who May Find This Useful

Readers interested in the philosophical implications of mathematics, the nature of mathematical discovery versus invention, and the connections between different mathematical theories may find this discussion engaging.

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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Page 72 of GEB ,~^ :approve:

Keep reading. your trip has only just begun, that is a great book.
 
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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.
 
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.
 

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