Ask HN: Can EVERYTHING be represented by the mathematics of today?
5 comments
It depends on what you mean by "everything", "represented", and "mathematics"...
But no.
For example, there are theories which can't be proven (https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...).
Moreover, there are a lot of things we don't know we can prove, which we may or may not be able to prove later by inventing new mathematical notations and discovering new theories. See: Collatz Conjecture (https://en.wikipedia.org/wiki/Collatz_conjecture) which is unproven; Fermat's Last Theorem (https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem) which was proven but took over 300 years and new mathematical developments; solved open problems and developments in 2022 (https://www.quantamagazine.org/the-biggest-math-breakthrough...)
There's also the philosophical question on how to fully "represent" anything in math notation. Like, I can't exactly describe my computer and include every single detail, in part because you can't observe some things without changing them (https://en.wikipedia.org/wiki/Uncertainty_principle)
But no.
For example, there are theories which can't be proven (https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...).
Moreover, there are a lot of things we don't know we can prove, which we may or may not be able to prove later by inventing new mathematical notations and discovering new theories. See: Collatz Conjecture (https://en.wikipedia.org/wiki/Collatz_conjecture) which is unproven; Fermat's Last Theorem (https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem) which was proven but took over 300 years and new mathematical developments; solved open problems and developments in 2022 (https://www.quantamagazine.org/the-biggest-math-breakthrough...)
There's also the philosophical question on how to fully "represent" anything in math notation. Like, I can't exactly describe my computer and include every single detail, in part because you can't observe some things without changing them (https://en.wikipedia.org/wiki/Uncertainty_principle)
Most modern mathematicians don't view mathematics as "representing" any aspect of reality. Instead, the prevailing philosophy today is that mathematics deals with objects that are created/defined by mathematics itself, and that any apparent correspondence to reality is a useful heuristic, not a fundamental truth.
As a basic example, consider a circle, which is a mathematical object defined as the set of points with a fixed distance from a given point. There are no circles in the physical world. There cannot be, because there are no "points" (objects without extent) either. Instead, there exist arrangements of matter that approximate circles in some more-or-less loosely defined sense.
As a basic example, consider a circle, which is a mathematical object defined as the set of points with a fixed distance from a given point. There are no circles in the physical world. There cannot be, because there are no "points" (objects without extent) either. Instead, there exist arrangements of matter that approximate circles in some more-or-less loosely defined sense.
I don't think so because I have no idea how to think about a psychicdelic trip in mathematics.
Reality of today badly implements modern mathematics, but it’s improving.
There's the conventional point that things other than abstract structure and abstract relationships can't, in some sense, be represented by mathematics at all. You can make a mathematical model that is isomorphic to some part of reality (including cultural and social details) but one can argue that what we mean by all kinds of human concepts relates to concrete, not abstract, things and experiences, which are probably not representable in mathematics. That doesn't just include "love" or "beauty" but also things like "Earth", "Barack Obama", "French", "professor", "Internet", or "Mother's Day".
Those isomorphisms can sometimes be very precise and very useful, which is why we can have, for example, physics simulations (which are based on isomorphisms between a mathematical model and a physical situation, which can be strong and precise enough to make detailed and useful predictions about how such a physical situation will evolve).
Some people believe that reality itself is mathematical at its root, in which case all of these things actually are either ill-defined (like it's not possible to guarantee that objects are or are not included in the category) or are mathematical objects, but we would still not expect to have access to those mathematical theories themselves because they would be below the level of our reality.
Apart from that, mathematical theories have limitations such as incompleteness (where consistent theories are typically not able to answer all questions that can be posed in them).
Just thinking about mathematical research, I presume that there are mathematical structures which we could represent with existing mathematics, but which we haven't realized are interesting or important yet, so we don't have a habit of doing so and don't have an established vocabulary for those. Like groups prior to the development of group theory!