The Hilbert vs. Brouwer Argument: The Real vs. Unreal and Its Application to Mental Health

I've been thinking a lot about intuitive mathematics and its validity. What interests me about it is how it accurately relates to the mind's capability to construct objects. This poses valid arguments that the human mind indeed has the creative power to construct things that could be described mathematically.

However, I would agree with the traditional scientific community and those who opposed Brouwer's line of thought, including Hilbert, who was angered by the assumption that this could potentially describe reality. The core of the argument between Hilbert and Brouwer was whether Brouwer could override the complete mathematical thought and claim that everything is just the creation of our mind. If nothing is real and there is no objective reality, then our traditional logic, filled with paradoxes, is simply invalid. But is that a compelling argument?

Background on Hilbert Space

To understand Hilbert's position, it's useful to know a bit about Hilbert space. Named after the German mathematician David Hilbert, a Hilbert space is a fundamental concept in functional analysis and quantum mechanics. It generalizes the notion of Euclidean space to infinite dimensions. A Hilbert space is a complete, infinite-dimensional vector space equipped with an inner product. This inner product allows for the definition of distance and angle, enabling the generalization of many geometrical and analytical concepts.

Hilbert spaces provide the mathematical framework for quantum mechanics, where states of a quantum system are represented as vectors in a Hilbert space, and physical observables are represented as operators on these spaces. This framework has been immensely successful in predicting and explaining a wide range of physical phenomena.

The Argument

At the time of their debate, the community was clearly rejecting Brouwer's ideas, supported by advancements in actual technology. Science produces algorithms and ways of looking at nature that allow us to create sophisticated machinery. Thus, it’s not proper to say traditional mathematics is invalid because it produces paradoxes. Rather, it indicates a limitation to our knowledge already described in science.

Traditional science is more rigid and grounded in its ability to evaluate our understanding of reality. It allows us to approximate using statistical and indirect observation methods. As pointed out by the uncertainty principle in quantum dynamics, direct observation of an event changes the outcome, making it impossible to observe nature directly. Yet, science has developed ways to sneak peeks at the mechanisms of nature.

I don't think we need to worry about traditional science being overridden by intuitive mathematics. The misunderstanding between Hilbert and Brouwer was that they were talking about two different things. However, it would be incorrect to say that what Brouwer was describing was "real" in mathematical terms. Just because you can construct objects in the mind does not mean they are real, though you can potentially find ways to define them mathematically.

Brouwer's mathematics, focusing purely on the mind’s constructive ability, could be a different branch. Traditional science shouldn't be intimidated by it but should continue to be grounded as a regulator in assessing reality. Just because we can create a Santa Claus in our minds doesn't mean Santa Claus is real. The same applies to various concepts generated by the mind.

There are even studies suggesting that mental constructs could be genetically conveyed across generations. The study of the mind is open for exploration and needs traditional scientists to engage with it. Traditional scientists will stay grounded and not get overly excited about the mind's possibilities, avoiding the pitfall of thinking they are gods because they can create concepts in their minds.

Scientific inquiry has been good at taming ambition within the scientific community, creating rigorous mechanisms for assessing the validity of arguments. Traditional scientists are well-equipped to explore intuitive mathematics. They can explore, play with, and study its applications while staying grounded in reality and traditional science.

People overly absorbed in their minds sometimes lose touch with reality, leading to psychological problems. There’s a certain characteristic of the mind that prefers to be grounded. Studying psychological or mind disorders could be helpful, revealing that the mind feels better within a natural environment, calming it down.

How should we understand the mind's need to be grounded in the "real"? How should we understand the impact of mind creation of the unreal and regulate it, assuming that it can potentially lead to mental health disorders? It does seem that the mind is constructed in a certain way, and instead of "creating," we rather "discover" that space. But when we interact with the unreal, we do the opposite. Are all human minds constructed the same? What drives the fascination with the unreal?