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, π, i, ϕ, 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).
*π* is the* identity of periodicity*. There’s much more going on with 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**