#10 - Foundation of Math in Crisis: why our scientific understanding is incomplete or inconsistent

In episode 10 of the Quantum Consciousness series, Justin Riddle explains that mathematics does not have a firm foundation in first order logic. While it seems intuitive that mathematics should be able to be described by simple principles, there is no simple way to explain how humans can comprehend mathematics. What happens when we try to use first order logic to explain how first order logic works? Well, we run into paradox: situations that are impossible to resolve and do not make any sense. For example, take the liar’s paradox, “this sentence is false.” There is no way to make sense of this sentence, because it leads to paradox. The content of the sentence is not supposed to refer to the truth value of the statement itself. Similarly, you cannot use first order logic to refer to the truth value of first order logic itself. Sets cannot describe sets (Bertrand Russell’s set paradox). And most dramatically, digital computers cannot even describe digital computers (Alan Turing’s halting problem). All of this leads Roger Penrose to conclude that “humans are not using a knowably sound algorithm to ascertain mathematical truth.” While we can understand things and learn mathematics, it’s really not clear how this is possible and something more than digital computers are required! Quantum mechanics offers a novel lens to this problem as the first order logic that appears in particles also has a supervening wavefunction that does not appear to follow the rules of first order logic. I don’t have the answer for you, but I sure have a lot of questions for you to consider!

~~~ Timestamps ~~~
0:00 Introduction
2:20 Types of knowledge
4:15 First Order Logic
7:57 Foundation Crisis
10:57 Gödel’s Incompleteness
13:08 Paradox
15:56 The Halting Problem
23:30 Roger Penrose’s Noncomputation
24:20 Particles ruled by waves

#quantum
#consciousness
#philosophy

Twitter: @JRiddlePodcast
Music licensed from and created by Baylor Odabashian. BandCamp: @UnscrewablePooch

[gehtdichnixan]

Thank you so much for breaking Penrose's Noncomputation down. After having heard it multiple times before and listening to this part of your video a couple more times, I finally get what he wants to say. Interesting.

[starxcrossed]

After reading lots of Iain McGilchrist I have come to the idea that the left brain is what we could call analogous to classical computer and the right brain more analogous to quantum computer. He has done lots of research into stroke patients and finds the left brain specializes in logic and speech to the degree that evolution will give you a 50/50 shot at experiencing reality based on which hemisphere you lose functioning of.

[DanBray]

Mathematics is just nothing with the absence of infinity. Numbers have no absolute existence because numbers don't consist of anything but other numbers or infinite infinitesimals, which are precisely equal to zero. Numbers go all the way up to infinity but the number infinity - 1 simply does not exist.

Also, this sentence is not a paradox because the quantum nature of our reality consists of everything being false. The truth is just also true and consists entirely of complete opposites being false. There is just nothing that is neither finite nor infinite, or zero energy without the absence of energy (Zero-point energy).

At the macro scale, the universe is entirely mathematical. Even the uncertainty is just a result of mathematics (Heisenberg Uncertainty Principle).

The complete inverse is also true. Our entire mathematical existence, is just mathematical consistencies that happen to be within the field of possibilities (uncertainty).

The existence of information, is simply not a requirement for possibilities to exist only as possibilities. At time zero, the possibilities are infinite. Within the possibilities there must be mathematical consistencies, which simply exist as entropy. The big bang happens!

The possibilities are no longer infinite because they are constrained to the mathematical consistencies, considering anything else would be mathematically inconsistent. There is still an extremely high amount of uncertainty though, which must contain mathematical consistencies. Therefore, the entropy of the universe increases further.

Without conscious awareness, the universe simply would not exist at all. The universe would simply be no information that does not exist. However, possibilities that only exists as possibilities, are enough to result in conscious experience.

When mathematical consistencies go back into the field of possibilities, the information is converting into the form that has absolute existence (no information, which is possibilities that only exist as possibilities). When the resulting uncertainty is used for inputs to manipulate our mathematical existence, it has purpose (mathematics can be used for purpose).

Infinity cannot actually be reached outside of spacetime to discover the probabilities in the Heisenberg Uncertainty Principle, therefore, the uncertainty is only the illusion of awareness. When mathematical consistencies go back into the field of possibilities, there is still the illusion of awareness but it must contain mathematical consistencies. An an illusion of awareness that is also mathematical real and can use mathematics with purpose, is simply conscious experience. Consciousness is simply a real illusion with purpose.

[TheGloryofMusic]

Gödel showed how it is possible to systematically generate all possible mathematical statements. But only a vanishingly small proportion of them are interesting. So how do mathematicians discover make their discoveries? Clearly, not by logic. Logic is used to prove theorems. It is the aesthetic or intuitive sense that is necessary. Likewise, Beethoven was able to access universal archetypes. When I first heard the late quartets, I had the feeling that I had known the music all my life. I highly doubt that AI will be able to generate artistic masterpieces.

[nrrgrdn]

I love your intros! Life is meaningful! You are real!

[gehtdichnixan]

20:40 paradox is only impossible in first order logic ;-)

[davidball]

Douglas Hoffstadter uses these ideas in his book on awareness: "I am a Strange Loop" (2007). Keep On Keeping On! D from Scotland.

[taku]

We know that computers based on 0 and 1 can simulate uncertainty (neural networks), then why can't humans be computers based on 0 and 1 that operate on uncertainty?

[gehtdichnixan]

25:28 we use digital computers and observe collapsed wave functions in experiments to then describe QM using mathematics based on first order logic. My brain is about to explode.