From: mk_thisisit
Mathematics is a tool used to describe reality, and it is theoretically capable of describing everything with precision, including physics [02:35:00]. However, there are fundamental limits to what can be proven or known about mathematics itself [00:00:22].
Mathematical Consistency
It is impossible to prove that mathematics is consistent [00:00:22]. While one can deeply believe in its consistency, there is no formal proof for it [00:00:24]. Consistency means that it is not possible to prove a theorem and its opposite simultaneously [02:11:00]. To carry out certain arguments, such as those related to Gödel’s theorem, one needs to assume or know that mathematics is consistent [02:21:00]. The inability to prove consistency implies an element of faith in mathematics [02:26:00].
Hilbert’s Program and Gödel’s Theorem
At the turn of the 20th century, mathematician David Hilbert proposed a program to formalize mathematics [02:27:00]. The idea was to build the entire edifice of mathematics by establishing axioms and deriving everything from them, similar to Euclid’s approach [02:27:00]. However, this idea collapsed due to the work of an Austrian mathematician, Kurt Gödel [02:28:00].
Gödel’s theorem proved that if one starts with a set of axioms, there will always be statements that are true but cannot be proven from those axioms [02:28:00]. This means it is impossible to choose a finite number of “good” axioms that would be sufficient to build all of mathematics [02:54:00]. The revelation of Gödel’s theorem led some to feel it was “the end of the world” for mathematics, as it proved the impossibility of Hilbert’s grand vision [02:59:00].
Furthermore, there are axioms that do not belong to the standard set where one can add either the axiom or its opposite, resulting in interesting mathematical frameworks [01:18:00]. These abstract axioms are far removed from the kind of mathematics connected to observable or physical phenomena, so mathematicians in those fields generally do not worry about such problems [01:19:00].
Consciousness
Roger Penrose, a physicist working at Oxford, has theorized about consciousness being a result of quantum effects [01:36:00]. His argument relies on a mathematical concept connected to Gödel’s theorem [01:51:00]. In his book, The Emperor’s New Mind, Penrose argued that no algorithm, including one representing the human brain, could ever prove statements that are true but unprovable [02:24:00]. Since humans can know these statements are true, Penrose concluded that the calculation must be non-classical, possibly a quantum algorithm [02:49:00].
However, there is a known flaw in this reasoning: Penrose’s argument is only valid if one believes that mathematics is consistent [02:55:00]. As previously stated, the core point of Gödel’s theorem is that it is impossible to prove mathematics is consistent [02:11:00]. Therefore, Penrose’s conclusion based on this assumption cannot be definitively supported [02:18:00].
Other Limits and Philosophical Questions
The discussion extends to other philosophical questions about mathematics and reality:
- Description vs. Creation: Most professional mathematicians believe that mathematics discovers rather than creates, unearthing abstract concepts that already existed [00:09:31]. Often, mathematical discoveries with no apparent application later find relevance in physics, such as two-dimensional models describing particle behavior at the interface of magnetic fields [00:10:10].
- Fundamental Laws: Mathematics is sometimes used to attempt to answer fundamental questions, such as the existence of a creator or the loneliness of humanity in the galaxy [02:27:00]. Arguments are made using probabilities or fine-tuning of physical constants [02:33:00]. Penrose, for example, points to the extremely low entropy at the Big Bang as a more shocking problem than fine-tuning [02:40:00]. He proposes a cyclical universe theory in his book Cycles of Time, where the Big Bang is the final phase of a previous universe, mathematically demonstrating a change in scale from an expanded, boring era to a new beginning [02:44:00].
- Nature of Reality: Some theories, like the holographic model in physics, suggest that what we perceive is an illusion, and we live in one less dimension, with quantum phenomena creating the higher dimensions [01:48:00]. The question arises if the universe is fundamentally “complex” (in the mathematical sense of complex numbers) rather than “real,” which might unify classical and quantum physics [04:30:00]. However, human brains appear to perceive the world in real numbers, leading to the assumption that this is simply how reality is [04:32:00].
- Infinite Dimensions: Mathematicians also study spaces with infinitely many dimensions, which are important in fields like the “symmetry of all symmetries” (automorphism groups of free groups) [01:52:00].
Despite these philosophical limits, mathematics is seen as an infinite field with endless discoveries [04:08:00].