One of Godel’s biographers, Dr. Wang, a rather conservative philosopher himself, concluded that the consequences of Godel’s Incompleteness Theorem for Mathematics included at least one of the following, if not all:
1. Mathematics is inexhaustible.
2. Any consistent formal theory of mathematics must contain undecidable propositions.
3. No theorem-proving computer (or program) can prove all and only the true propositions of mathematics.
4. No formal system of mathematics can be both consistent and complete.
5. Mathematics is mechanically (or algorithmically) inexhaustible (or incompletable)
Certainly if mathematics, as the foundation of science, is without limit, that at least suggests that other aspects of reality are also without limit, inexhaustible, and contain “undecidable propositions.” There is no reason that this limitlessness should be necessarily limited to mathematics. Godel actually believed that he had demonstrated the truth of Platonism, but neglected to publish that further proof. This proof certainly does imply that “truths” are discovered from a larger field of reality rather than merely created as an arbitrary convenience.
Godel certainly believed that although brain states might be mathematically determined as measured by such things as electo-encephalograms or brain imaging techniques, nevertheless, neither of those techniques nor any other mathematically based technique could [even in theory] predict or determine the richness of consciousness. He was certainly accurate regarding the limitations of the abilities of digital systems like computers to emulate consciousness, and was a consultant to the Artificial Intelligence community until his death.
Hi Greg,
An old yogic master once described to me this mental experiment to help visualize the “infinite world”.
If you would have a microscope that can magnify for ever, what would you see with it? Of course by magnifying several trillion times, you would see the molecules, the atoms, the quantum waves/particles, etc. Beyond that (Plank scale) you would see the whole observable Universe, and if you keep magnifying eventually you would see our galaxy, and then our solar system, our planet, and in the end you would see yourself watching through the magic microscope. At this point the zoom on your microscope have to be something of the order of 10 to the power 100. And it never ends, you can magnify another 10 to the 100 and you will see yourself again. So the implication of this mental exercise is that you are everywhere. In any point of the Universe you chose to magnify, is just that you exist at a different level. The strange thing to me is that this levels are not separate worlds, they are basically the same one. And the rule applies also looking outwards. (the whole observable Universe (18 bilions light years)is just a sub-quantum granulation of a 10 to 100 bigger Universe, etc).
Why the Math wold be any different at different scales?
My speculation about mathematics being both limitless and complete is that mathematical postulates repeat and preserve themselves inwards and outwards of our observable Universe. This is hard to grasp, but if you can mentally imagine that you would shrink 10 to the power of 100, you would end up absolutely the same person in the same Universe. So you are everywhere, in every point, just at a different scale. All these different scale worlds, are not separated, they are unique as ONE that exhibits Platonic entanglement.
10 to 100 is roughly the size of the observable Universe (18 billion light years) divided by Plank scale dimension (10 to -34 m).
I found your comments really quite interesting, but I was hoping that you might expand upon them a bit. If you would, expand a bit on your comment regarding scales and Platonic entanglement.