The reason why it is so hard to prove is actually very easy to answer.

**These ***constants, identities, and variations***
being referred to in this post, and others like it, all lay embedded in
a far deeper substrate than current mathematics has yet explored**.

Mathematics has been, and always shall be my ‘first love’, and it has provided for me all of these years.

**I am not criticising mathematics in any way.**
It is my firm belief that mathematics will overcome this current
situation and eventually be quite able to examine these kinds of
questions in a

**much more ***expansive ***and ***deeper ***way.**
** **
We need to extend our examination of mathematical knowledge,

*both in depth and in scope,* out farther and in deeper than numbers (sets and categories as well - even more below) have yet done.

**I’ll introduce you to a pattern you may have already noticed in the current stage of our mathematical endeavour.**
** **
We all know there are numbers which lay outside of $Q$ which we call Irrational numbers. There are also numbers which lay outside of $R$ which we call Imaginary numbers. They have both been found,

**because the domain of questioning exceeded the range of answers being sought within the properties each of those numbers.** This pattern continues in other ways, as well.

**We also know there are abstractions and/or extensions of Complex numbers where the ‘air starts to get thin’ and mathematical properties start to 'fade away': **Quaternions, Octonians, Sedenions,…

This pattern continues in other ways:

**Holors**, for example, which extend and include mathematical entities such as
Complex numbers, scalars, vectors, matrices, tensors, Quaternions, and other hypercomplex numbers, yet are still capable of providing a different algebra which is consistent with real algebra.

**The framing of our answers to mathematical questions is also evolving.** Logic was, for example, limited to quite sophisticated methods that all were restricted to a boolean context. Then we found other questions which led to boundary, multi-valued, fuzzy, and fractal logics, among a few others I haven’t mentioned yet.

**Even our validity claims are evolving. **We are beginning to ask questions which require answers which transcend
relationship properties such as causality, equivalence, and inference in all of their forms. Even the idea of a binary relationship is being transcended into finitary versions (which I use in my work). There are many more of these various patterns which I may write about in the future.

**They all have at least one thing in common:** *each time we extend our reach in terms of scope or depth, we find new ways of seeing things which we saw before and/or see new things which were before not seen.*
* *
There are many ‘voices’ in this ‘mathematical fugue’ which ‘weaves’ everything together:

**they are the ***constants, variations, identities,*** and the ***relationships they share with each other.*
* *
The constants $e, \pi, i, \phi, c, g, h$ all denote or involve ‘special’ relationships of some kind.

**Special in the sense that they are completely ***unique***.**
For example:

- $e$ is the
*identity of change* (some would say proportion, but that’s not entirely correct).
- $\pi$ is the
* identity of periodicity*. There’s much more going on with $\pi$ than simply being a component of *arc* or, in a completely different context, a component of *area*...

**These relationships actually transcend mathematics.** Mathematics ‘consumes’ their utility (making use of those
relationships), but they cannot be ‘corralled in’ as if they were ‘horses on the farm’ of mathematics.

**Their uniqueness cannot be completely understood via equivalence classes alone.**
**They are ***ubiquitous ***and therefore not ***algebraic***.**
**They are ***pre-nascent*** to number***, ***equivalence classes, and validity claims and are therefore not ***rational***.**

These are not the only reasons.

**It’s also about WHERE they are embedded in the knowledge substrate compared to the concept of number, set, category…. ***They lay more deeply embedded in that substrate.*
The reason why your question is so hard for mathematics to answer is, because our current mathematics is, as yet, unable to decide.

**We need to ‘see’ these problems with a more complete set of ‘optics’ that will yield them to mathematical scrutiny. **
**Question on Quora**