√5 and Phi

[nothing]

Ok that's fine. But then do this. Say, "A standard polynomial is such and so. But for my purposes I want to define a nothing-polynomial as ..." then make your definition. In math you can make up any definition you want.

But when you change the definition of a standard term without first announcing that fact, you simply appear divorced from reality, claiming to have a polynomial with integer coefficients then presenting an expression that is nothing of the kind.

If you wish to communicate with others, then whenever you re-define standard terminology, announce that fact by clearly giving your own definition. Otherwise you can't be taken seriously. Doesn't that make sense?

No, as I don't want or expect anyone here to take me seriously - I don't even!

Please define "real integer coefficient" if it means something other than a standard integer. Else how can your reader know what you're talking about?

3 is a real integer coefficient of 3π - the usage/definition is not uncommon.

Pure word salad. Devoid of meaning.

Meaning devoid of grasping.

More of same. Don't mean to be unduly critical. But parts of your posts are lucid and parts aren't, and I hope I'm giving you useful feedback by pointing out the areas where you need to work on clarifying your exposition so that people can understand you.

Such feedback is more appropriate via pm, not public - the sentiment is not altruistic otherwise.

Nothing seems very clear to me.

There is a reason(s) for it, but not appropriate to talk about.

You might find this helpful.

https://en.wikipedia.org/wiki/Euler%27s_formula

I am aware of Euler's but do not see e^(2πi)=1

Please define "geometric" in this context. Of course pi has many geometric relations. But in terms of standard math, with the standard definitions, do you understand that phi is algebraic, pi is transcendental, and that these are mutually exclusive classes of real numbers?

I don't myself care about "standard" anything - western science, including math, is rooted in much hubris and absurd assumptions. The same article you linked to already indicates the "geometric" implications:

This formula can be interpreted as saying that the function e^iφ is a unit complex number, i.e., it traces out the unit circle in the complex plane as φ ranges through the real numbers. Here φ is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counterclockwise and in radians.

In deriving Φ by way of π the unit circle derived by way of Φ² = Φ + 1 is "geometrically" bound to the operation of π. This is broad spectrum and general, thus any desired base/power arrangement can be used for any purpose, and the relationship will be preserved because it is "geometrically" bound.

Pi doesn't move. It's a fixed real number sitting at a particular location on the number line.

...which is what makes it appear transcendental. It doesn't mean anything just "sitting" at a particular location:
it is like a pottery wheel: you can not shape pots without the wheel moving. Similarly, π is meaningless unless
in relation to some kind of motion if/when modelling anything at all in the real universe.

All in all, as we wrap the real number line around the unit circle, it keeps cycling over the same points with a period of 2pi. That's the period of the sin and cosine functions, and also of the complex exponential. The point (1,0) in the plane, or the complex number 1, gets hit by rotating around the circle 0 radians, 2pi radians, 4pi radians, and in general every 2pi radians you come back to the same spot. This is true.

Your relation to phi is much less clear and I can't understand what you're doing.

Each 2π is thus a "whole" rotational base. The relation is not different: it is the same.
All I am doing is showing that, if using (π + π√5) instead of (1 + √5) the '1' is derived naturally
as the unit circle, and intrinsically in relation to π (otherwise not the case if arbitrarily generating Φ with '1'):
every 2π concerns '1' and π√5 concerns a pentagram, with 2π being both a full circle and both symmetries of a pentagram.
I thought the implication would have been more obvious: the "marriage" of "line" and "curve" under a common base of '1'
concerning any possible Euclidean geometry.

[Skepdick]

you asked what language is describing...

Yeah... but is it describing the noumena, or your perception of them?

[wtf]

No, as I don't want or expect anyone here to take me seriously - I don't even!

Ah. Sorry I took the trouble then.

I am aware of Euler's but do not see e^(2πi)=1

e^(2pi i) = cos 2pi + i sin 2pi = 1 + 0 = 1.

If x is a real number, the function f(x) = e^(ix) wraps the real line around the unit circle with period 2pi.

ps -- Another cool way to visualize this is to imagine the real line spiraling up and spiraling down like a coil on either side of the unit circle. Each time it goes 2pi radians around, it's directly over the same post on the unit circle but a little higher or lower on the spiral.

Topologists call this a covering space. Looks something like this.



You see how there are infinitely many points above (and below) a given point on the circle, that correspond to it; just as cosine and sine of -4pi, -2pi, 0, 2pi, 4pi, 6pi, etc. are all the same.

You seem to be interested in what happens as you travel around the unit circle, which is why I'm mentioning this.

[Impenitent]

you asked what language is describing...

Yeah... but is it describing the noumena, or your perception of them?

the only thing to which we have access are the impressions...

-Imp

[Skepdick]

the only thing to which we have access are the impressions...

I agree.

How many origins are there for our impressions?

Is there "the universe" which impresses us, and then the "the abstract universe" which also impresses us?

[nothing]

No, as I don't want or expect anyone here to take me seriously - I don't even!

Ah. Sorry I took the trouble then.

I am aware of Euler's but do not see e^(2πi)=1

e^(2pi i) = cos 2pi + i sin 2pi = 1 + 0 = 1.

If x is a real number, the function f(x) = e^(ix) wraps the real line around the unit circle with period 2pi.

ps -- Another cool way to visualize this is to imagine the real line spiraling up and spiraling down like a coil on either side of the unit circle. Each time it goes 2pi radians around, it's directly over the same post on the unit circle but a little higher or lower on the spiral.

Topologists call this a covering space. Looks something like this.



You see how there are infinitely many points above (and below) a given point on the circle, that correspond to it; just as cosine and sine of -4pi, -2pi, 0, 2pi, 4pi, 6pi, etc. are all the same.

You seem to be interested in what happens as you travel around the unit circle, which is why I'm mentioning this.

It is all good: my interest in the unit circle relates to an inversive symmetry (ie. of a pentagram) which solves for the identity of the two Edenic trees, thus for any/all suffering/death in any conceivable Judaism/Christianity/Islam context.



Basically each π of 2π is one symmetrical half of a pentagram and/or circle:
the two "roots" are 'to know' and 'to believe' reflecting the two trees, and
the two "operators" are 'all' and 'not' (ie. alpha and omega) thus generates bi-orientation:
{to know all thus: not to believe} tends towards any possible all-knowing 'state', god or no-god, and
{to believe all thus: not to know} tends towards any possible belief-based ignorance(s) causing/perpetuating suffering
wherein all "believers" are rooted in some belief, thus liable to believe the opposite of what is true.



It is designed as a universal 'orientation' system that tends away from all suffering and towards all knowledge. If applied to the spiral, the pentagram has two orientation "poles" with each being some configuration between the two: knowledge and belief-based ignorance.

I intuit this relates directly to real/imaginary numbers, and endeavor to prove this true, but there are other more important things needing attention.

[Skepdick]

I've had a bit of a rough day so I am not on my best game choosing my words - if you pick up on any irritability/crankiness in my tone - it's not you....

Of course. But the representation is not the number.

Yes exactly. A representation is a pointer to the abstract concept it points to.

So it seems to me that we are on the same page then, the symbol 0 is a representation. It's a pointer.

Hence my question: Where is the referent?

I am sure you will tell me something like... "In the universe of Mathematical abstractions".... but that's not good enough.

Succ(0) is in that universe.
Pred(Succ(0)) in that universe.

In which universe is Pred(0) ?

It's not only in math that we use abstractions as perfected examples of the imperfect things of our world, right? To be human is to abstract.

I agree. But the position I am coming from is abstracting mathematics itself!

Meta-logic. N-th order functions. I am sure you have a "standard definition" for it that I don't know.

Of course TMs can represent numbers. I don't see what your problem is. In computer memory if I have an address that points to a bit pattern that represents a number, so what? I hope this isn't confusing you in any way. What is your concern with a chain of a dozen pointers that end up pointing to a memory location that holds a bit pattern that represents a number? So what?

My concern is lazy evaluation

You've accused me of being "too strict" on a number of occasions, but strictness and laziness are on a sliding scale just design choices in a programming language. In so far as I can tell you default to max(laziness) . A key flaw of lazy languages (if we are to say Mathematics is lazy) is that "Space complexity of non-strict programs is difficult to understand and predict." So you haven't so much ignored complexity theory, you have just traded time complexity for space complexity. Memory.

Suppose you have two functions:

f(x) = 0
g(x) = 0

The suppose you have some 2nd order function:

y(f(x)) = 10
y(g(x)) = 1

Do you still insist that f(x) is "equal" to g(x)? It's equal by value - sure, but there's clearly a measurement to be made (information to be obtained) such that they can be differentiated.

We have such 2nd order function in the real world. time(). Why is that function "not allowed" in the Mathematical kingdom?

In "software engineering" we have names for such things - leaky abstractions. ALL abstractions leak. If they didn't - we wouldn't be able to measure anything.

This remark doesn't make any sense at all. What do countable and uncountable infinities have to do with anything? Except to note that TMs can only represent at most countably many numbers. Most real numbers are not computable.

Any mathematical function you give me, I am going to wrap it in my time() function.

I will measure how long it takes to return a value. Is time() a computable by your definition?

By mine: sometimes.

But that doesn't invalidate set theory. It's two different things.

The question I am leading to is simply: What would invalidate set theory?

Is the construction of some N-th order system in which set theory's axioms explode sufficient?
Is the construction of some N-th order system in which the axioms of 1st order logic explode sufficient to invalidate 1st order logic?

We are talking abstraction here, right? How abstract is too abstract an explosion?

To me sets are just arrays... you know. Memory. read(), write() - side effects. Tape on a Turing Machine.
If you want to read the 1st cell on the tape you have to do 1 MHR (Move-Head-Right) operation.
If you want to read the Nth cell you have to do N MRH operations.

CPU cycles - time. All the imperatives.

But of course they are actually no such thing, even if they're called that. All physical machines are limited by time, space, and energy. So we can never arbitrarily represent all the real numbers, or even, in the age of the universe, all the positive integers. Computers are finite.
All arbitrary precision means is that you can have as many decimal places of accuracy as you like, subject to the physical limitations of the computer. You can NOT approximate every real number for the simple reason that your computer is finite. Again, surely you know that.

We need to unpack this word - "representation", because you seem to be defending the position I've always held.

Of course we can represent ALL the positive integers. Here - I have represented them.

Code: Select all

def all_integers(start=0):
  print(start)
  all_integers(start + 1)

all_integers()

We can represent anything. f(x) = y. Oracle machine!

Declaratively - this is true. It's not true imperatively/constructively.

To express the entire decimal representation of pi? You need all of them.

Ok well, I gave you a function which generates all the integers, so it seems to me the solution can be represented:

generate_pi(all_integers())

What am I missing?

But why are you going on about this? It's perfectly clear that we can crank out as many digits of pi as we want, ignoring resource constraints. That's Turing's definition of computability.

Because it's perfectly clear that f(x) = y can represent ANYTHING.

And so the question is not representation, the question is evaluation.

Revelations? Talking points? What? Tell me.

Godel! I can recurse as deep as it's needed to be recursed in order to explode any grammar/representation system. Find any equivocation.

Trivial to construct such a system in Lambda calculus,. Say I prescribe:
for all numbers x, y: x != y.

That is - I insist that ALL symbols have unique identities. You know - like numbers!
So you have no formal alphabet left for variables. Then any axiomatic mention of (x=y) explodes by default.
In such a system Peano explodes. Everything explodes if you axiomatically choose completeness.

That's what Godel said.

So if everything explodes by construction, how and why did you choose consistency?

No. Turing-computable is Turing computable, and Skepdick-computable is Skepdick-computable. You do yourself no favors by insisting that everyone else in the entire world is using the wrong definition.

No! What I am insisting on is that YOU don't know if you are using the right definition or not. But it doesn't matter either.

According to your definition, can you classify this function for me: time(all_integers())
time() is a function. It's represented so you can treat it as abstract and upload it to the "Abstract Mathematical Universe" where all of your other abstractions exist.

Is that wtf-computable or not computable?

You can call coffee tea, but if you ask for tea at the diner they're going to give you coffee. You can't start arguing with them that YOU use the word tea to mean coffee.

False analogy. I am not asking you to give me anything - I am asking you to explain how you know whether you've accidentally violated one of your axioms or not - even unintentionally. When you make an error in reality - you know because feedback loops.

Where's the feedback loop in Mathematics?

Paradox of rule-following.

Even quantum computing has the SAME power as classical computation. It runs more efficiently on some problems (as far as we know, but not proven); but in terms of computability, it's the same.

Wow!!!! That is called the Turing tarpit. https://en.wikipedia.org/wiki/Turing_tarpit

If you wrap ANY computable function in the 2nd order time() function, they are no longer "the same".
Linear speedup theorem

And if you take the expressive power of your grammar into account. then everything about the imperative paradigm matters!

So we are back at representation/grammar/richness of vocabularies.

Why are you calling coffee, tea? Computable has a standard meaning and by insisting that YOUR definition is right any the standard one is wrong, you embarrass yourself. A definition can't be wrong. If you want to talk about the limits of practical computation, call it "practically computable" or Skepdick-computable and we can have a conversation. Otherwise you're tedious.

I have told you that I don't have a definition. I have also told you that my definitions keep changing. And in the real world - that's perfectly OK!

We have English, metaphors, Google and a whole swathe of concepts in our heads. Surely we can arrive at definitions pretty rapidly - it's really not a big deal.

Namedropping some philosophical shit while acting like a crank is a waste of time.

But Mathematicians don't care about time! That's what you said.

As long as you insist on changing a common definition and then arguing with people about it, you're a crank. Your basic ideas are fine but the way you come across is very cranky.

As long as you insist that definitions matter in practice - you are an academic. You come across as very inflexible when it comes to languages/grammars.

ESPECIALLY the grammar of Mathematics.

Floats represent reals in the sense that all floats are rational numbers. That's as far as it goes. And you can't represent ALL rationals, because your computing equipment has bounded resources.

Again. Define "represent".

let f(x) be the function which produces the set of all <insert whatever abstract mathematical object you want here>

And then?

There are real numbers you can NOT approximate with floating point arithmetic running on physical hardware. It would be helpful for you to understand this point.

Let f(x) be the generator function for ALL real numbers that cannot be represented on a finite classical computer.

As long as you don't want to do physics or biology, I suppose we don't care. How much civilization are you willing to abandon to stick to your position, whatever it is? Abstractions DO exist in the world. Justice, law, traffic lights, property. We could not run our lives without abstractions. Math consists of abstractions that often do have impact in our lives.

Obviously abstractions are useful. Nobody is denying that. What I am wondering is: how far into the clouds is abstract mathematics willing to go into before they notice their feet are no longer touching the ground?"

Very odd response. Computer programming is not an example of symbolic computation? You would deny this? Either by deliberately mis-parsing it, or by stating an obvious falsehood? You say programming ISN"T symbolic computation? Wow. News to me.

Programming is Logic. A pretty high order logic at that.

Some of Mathematics isn't Logic. They parted ways somewhere around "first order logic with equality".

You "keep telling me" things that are perfectly obvious. Why do you do that?

I don't know what is and "isn't obvious" to you. And I don't know the impedance mismatch between us is.

A number of times in the interaction you've gone "Oh!", that kinda signals to me that I've told you something new.

Of course. All you're doing in all of this is talking about the different between computer science and software engineering. Why do you think this is news to anyone?

And I am telling you that there is no difference. In as much as any community has chosen their foundations/definitions/axioms - they are doing engineering.

We'll can park the discussion on how any abstract discipline gets to call itself a "science" for another time...

Then we'll all be as smart as you? You're being a crank again. You're right and everyone else is wrong.

Then I'll be the crank. My ego isn't on the line here. Nor my reputation or my future. I am rich enough.

And constructive math is fun.

Ok. What of it?

Feet off the ground...

Oh I see. So what?

Everything declarative reduces to imperative by compilation.

Relational algebra.

Besides, Skepdick-computability keeps changing.

Mathematical "=" keeps changing too. It's just language - for communicating ideas. Is why we can re-define it.

It's a function of how fast our hardware is and how clever our algorithms are. Whereas Turing-computability is eternal. It doesn't change over time.

Yeah.... See. That's bullshit. If the "abstract mathematical universe" is immutable (e.g eternal and unchangeable) no information could ever leave it under observation. No Mathematician could ever learn anything about Mathematics. You wouldn't even have a read() function.

Mutation implies memory. Variables. You got those, right? x and y? Those same things I took away from from you with stricter axioms.
So clearly you want SOME mutability even if you keep telling yourself the "eternal and unchanging" story. Maybe it's just the medium you use to do mathematics with? Pen and paper - eternal and unchanging.

Software is dynamic. Things look differently when you "animate" then in spacetime then they look abstractly.
In so far as the mind of a Mathematician can be talked about in terms of "software engineering" all the limits/distinctions that we have practically identified applies to your head.

Side effects? Got those! Variables! Why do you feign immutability?

You don't care for computer science, but you also have some kind of psychological block on believing in it or using its standard terminology.

I am using computer science. What does it mean for me "to believe" or "not believe" in it?

Meaning what? I can't connect your personal philosophy with your technical points.

Meaning: how many universes are there?

Your mind: 1
The universe of the noumena: 2

Is the "abstract mathematical universe" one of the two above or is it a 3rd universe?

Abstract mathematical objects.
https://plato.stanford.edu/entries/abstract-objects/

Full circle.

is the function Previous(0) computable?
If not - why not.
if yes - what "abstract mathematical object" does it produce?

In computer science - the "abstract mathematical object" is called an unhandled exception. New Information.

https://en.wikipedia.org/wiki/Exception_handling

This is the usual control flow stuff/engineering stuff.

But I am curious. While I consider all of those design patterns to be abstract logical objects (because they can be expressed in Logic...), do you consider them to be "abstract mathematical objects" or do you gave some exclusion criterion?

Like, for example - what is the Mathematical equivalent to call with continuation? What is the mathematical equivalent to synchronous and asynchronous functions?

[Impenitent]

the only thing to which we have access are the impressions...

I agree.

How many origins are there for our impressions?

Is there "the universe" which impresses us, and then the "the abstract universe" which also impresses us?

and the abstractions of the "abstract universe" which are beyond empirical impression...

unicorns, gods, imaginary numbers...

origins are limited only by the imagination...

some are worth more than others...

-Imp

[wtf]

I've had a bit of a rough day so I am not on my best game choosing my words - if you pick up on any irritability/crankiness in my tone - it's not you....

No worries. You wrote a long post and I'll only respond to a little of it today, but hopefully it's the heart of the matter.

So it seems to me that we are on the same page then, the symbol 0 is a representation. It's a pointer.

Yes, exactly. We not only agree on this point, but even on the metaphor. It's often occurred to me that a representation of a real number, say, is like a pointer to the actual real number.

Except that in this case, when we dereference the pointer, there's nothing there. That is, we hope that after following the pointer to the real number we will find the real number. But nobody knows what that means. And I think this is the point you're getting at. If you say, "Oh it's a Platonic real number," that position doesn't actually tell us anything at all. We might as well believe in magic fairy dust. A point you are about to make, I believe.

Hence my question: Where is the referent?

Of course. Deep question. Pondered by mathematically and philosophically oriented humans for thousands of years. We have five rocks, but what is five?

Before writing down some idle thoughts on the topics, I'd like to pose you a question. What is justice? In what world does justice live? You might point to the courts, or the cops on the beat, or the government; but that is not justice. Justice is the abstract standard that our finite, limited, human activities can only approximate, and often very badly.

Would you therefore say justice doesn't exist? But it does. Everyone knows what justice is. Even if we can't explain exactly what we mean by the world of abstract ideas, abstract ideas have some kind of existence. As does the number 5. Yes?

I am sure you will tell me something like...

Your opinion of what I believe is often diametrically opposed to what I actually believe. Such is the case here.

"... In the universe of Mathematical abstractions".... but that's not good enough.

I have perfectly well already stipulated this a couple of paragraphs up, but perhaps I never made this point with sufficient clarity so I will now.

Nobody knows what is the referent of a number. Least of all me.

Let me outline a few ways of dealing with the problem. I take no position. Personally I take math on its own terms and don't care one way or another about its ontological status. Some days I'm a formalist and some days I'm a Platonist. But when people are doing math, they're just doing math. Solve for x. Understand some abstract structure. Solve a research problem. That's mathematical work. And that is sufficient for doing math; just as knowing how to dig a ditch doesn't necessarily involve thinking about the quarks that make up the dirt.

So let's look at some possible approaches to the question of, "What is the referent of 5?"

* Platonism. The abstract 5 exists in Platonic heaven. Easily attacked. What else lives in Platonic heaven? The baby Jesus? The Flying Spaghetti monster? Purple unicorns? One is immediately in trouble suggesting that there is a supernatural realm of existence in which the true number 5 lives.

No, I'm afraid you are wrong that I have asserted this position. However, you are right that most mathematicians implicitly take this position. When Wiles proved Fermat's last theorem, he (and the rest of the world, in fact) believe that he said something true about the natural numbers.

Even though we can't say with certainty what exactly is a natural number, we know one when we see one and our formalisms are highly effective in working with them. Nobody complains that Wiles doesn't know what the number 5 is therefor FLT hasn't been proved.

So in a sense, even though Platonism is difficult to defend, in some sense it's true, at least about some mathematical objects. We believe in the natural numbers even though we can't say what they are.

Perhaps it's the collective believe that reifies them; in the same way that collective agreement reifies the meaning of green and red traffic lights.

* Pure formalism. The symbol '5' doesn't mean anything at all. Trouble is the "unreasonable effectiveness" of math. No sensible person denies that 5 exists, even if we are hard pressed to write down a definition that will please all the critics. Formalism is a good philosophy to retreat into when you're cornered, but it can't be all there is to say about math, because it's clear that math is "true" in some mysterious way we can't put our fingers on.

* Pick a canonical mathematical object to call 5. In fact in set theory, 5 = {0,1,2,3,4}. Likewise each of those numbers is the set of those natural numbers smaller than it, till you define 0 as the empty set. This works, it's the standard way of defining the natural numbers in math and all other numbers are built on these. The drawback is (1) you have to adopt all the machinery of set theory, which introduces many other philosophical problems; and (2) as Benecerraf noted in a famous paper, those sets I wrote down can't possibly "be" numbers; they're only representations. We still don't know what numbers are.

* Numbers are defined by their relations to other numbers. This is the category-theoretic approach. We can in fact do set theory without elements these days if we like. This satisfies some people's philosophical concerns.

* Maybe numbers are just however we use them. Mathematical objects are just the practices and traditions of mathematicians. I'm not sure what this position is called but it seems reasonable to me.

Just idle thoughts. But I think this is what you're getting at. Nobody knows what a number is, but this does not stop us from doing math. We don't know what justice is, and that doesn't stop us from catching criminals and sometimes punishing the innocent. You know, there's the stuff of the world and the stuff of our dreams.

ps -- I read the rest of your post and I found it incoherent as I find many of your posts. I realized that we do have some things we agree on, and others where there's no contact at all. From now on I'll just interact with the parts of your posts that make sense to me, and I'll ignore the rest. I can't for the life of me understand why you're going on about lazy evaluation, a technique in programming language theory. It doesn't seem to bear on anything. I'll just leave all this stuff alone from now on.

Not trying to be provocative, just saying what's going to help me preserve my sanity. And also to signal you as to what you're saying that I understand and can interact with, and the other stuff that suddenly veers into left field.

It's like one moment we're talking about the philosophy of numbers: What is the referent of 5?

The next moment you're talking about lazy evaluation. Might as well talk about linked lists or database index files. Just software abstractions used in real world software engineering. What of it? That's the part I don't get.

[wtf]

It is designed as a universal 'orientation' system that tends away from all suffering and towards all knowledge. If applied to the spiral, the pentagram has two orientation "poles" with each being some configuration between the two: knowledge and belief-based ignorance.

This is beyond my pay grade. If I can answer any mathematical questions I'll do my best. Frankly the wrapping of the real line around the unit circle is one of the coolest things in math. But if you can do something about reducing human suffering, I'm all for it. As far as increasing knowledge? That hasn't always worked out so well. Nuclear bombs and such. Original sin. Knowledge is a double-edged sword; and too often in our world, knowledge is aligned with power and not truth. You agree?

[Skepdick]

ps -- I read the rest of your post and I found it incoherent as I find many of your posts. I realized that we do have some things we agree on, and others where there's no contact at all. From now on I'll just interact with the parts of your posts that make sense to me, and I'll ignore the rest. I can't for the life of me understand why you're going on about lazy evaluation, a technique in programming language theory. It doesn't seem to bear on anything. I'll just leave all this stuff alone from now on.

I agree with pretty much everything you said before this so I am not going to respond on any of it. For every Mathematical denotation there is a clear philosophical interpretation. I much prefer science/engineering to philosophy so I want to elaborate on WHY I think lazy evaluation matters to Mathematics (and so I must do Philosophy).

You say that the Mathematical universe is idealised.
Mathematics only cares about pure functions. Referential transparency (e.g referencing/dereferencing are isomorphic)
Time doesn't matter - everything is static and unchanging.
I'll take you on your word and I will even aspire to that narrative in spirit.

Wittgenstein's rule-following problem (and Kripke's response to it) is about whether, either of us knows what "purity" really means,
and how, despite our best efforts at practicing 'purity' we all might be practicing 'furity' rather than 'purity'.
It's an abstract concept with no formal metric. Like justice, beauty and an infinite list of virtues that I need not enumerate.

Both of us aspire to purity. Neither of us can define it or measure it, so it's difficult to say anything meaningful about it.

So when you are doing math - are you being 'pure' or 'fure'? You don't know.
When I am doing math - am I being 'pure' or 'fure'? I don't know!
Is my 'pure' your 'fure' or vice versa?!? We don't know.

What is purity? Deep question. A more shallow one is: can we measure it?
Because that's the only kinds of questions Math can answer - quantitative ones.

So here is my metric: Purity is strictness. And lazyness-strictness is the sliding scale by which I measure it.
Model it as you wish. strict maps to 0, lazy to inf - or the inverse. It doesn't matter.

In an impure universe (the actual one), where time, memory or representation does matter functions must be as strict as possible, otherwise they leak implementation details. They leak time, which is what allows for side-channel attacks etc. So back to the pure universe....

Given the identity f(x) = g(x) there must be no other, differentiating function that you can find. No function d(x) where

d(g(x)) != d(f(x))

Now. Up to here, all of this is my own reasoning. From 1st principles + engineering experience.
Where my brain makes a leap is that - to me that seems exactly like the conclusion in the "Sometimes all functions are continuous" article.

Even though I don't understand a significant portion of the topology/homotopy jargon in it (or whether it's even important to do that in order to understand the implications), it seem to me that I am concluding roughly the same things the author is concluding.

f(x) = g(x) does not imply d(f(x)) = d(g(x))

It seems to me that what mathematicians call "discontinuities", software engineers call "leaky abstractions".
The details which we attempted to abstract away - surfacing at higher layers of abstraction. An edge case. An unhandled exception

It signals new information. It signals a false assumption somewhere underneath.

And I won't bore you with the details of why that looks a lot like the measurement problem in Physics to me, or why I think Mathematics is an attempt to model reality. The number line represents time.

When you ask me "Do you think the numbers exist" I am hearing "Do you think time exists?"

How cranky do I sound now? Or am I "stating the obvious" again?

[Skepdick]

and the abstractions of the "abstract universe" which are beyond empirical impression...

Well, somebody is abstracting the abstraction - somebody is experiencing it.

So there is clearly an abstract() procedure of some sort.
And if there is an abstract() procedure, then there is also a deabstract() procedure.

So what does deabstract(abstract universe) give us? Real universe?

[nothing]

It is designed as a universal 'orientation' system that tends away from all suffering and towards all knowledge. If applied to the spiral, the pentagram has two orientation "poles" with each being some configuration between the two: knowledge and belief-based ignorance.

This is beyond my pay grade. If I can answer any mathematical questions I'll do my best. Frankly the wrapping of the real line around the unit circle is one of the coolest things in math. But if you can do something about reducing human suffering, I'm all for it. As far as increasing knowledge? That hasn't always worked out so well. Nuclear bombs and such. Original sin. Knowledge is a double-edged sword; and too often in our world, knowledge is aligned with power and not truth. You agree?

Kind of - if replacing 'knowledge' with 'technology' then yes. However any real knowledge must concern what it is the practical negation of: belief-based ignorance that is not real knowledge. Knowledge and belief are thus as poles: all knowledge would mandate the present negation of all belief-based ignorance(s) otherwise causing/perpetuating suffering/death, including illicit use of such technologies concerned. These two poles are practically equivalent to the two Edenic trees, however "believers" are severely tempted to "believe" in some authority rather than consciously "know" who/what/where/why/when/how and/or if not to believe. The latter requires asking questions / inquiring / science (ie. 'con'science) which is precisely what is not taking place if/when in a fixed state of belief, as one will not seek anything if they "believe" they already have it.

Therefor I was/am seeking ways to mathematically express a theoretical being
which captures the 5-fold symmetry:
Head=will to operate
Arms=two conjugate {operators} 'all' and 'not' (+/-)
Legs=two reciprocal {roots} 'to know' and 'to believe'

As this 5-fold applies:

π + π√5
‾‾2π‾‾ = Φ

3π + π√5
‾‾2π‾‾ = Φ + 1

and is meaningful to me because an equation can then be made which can describe any/all possible orientations a human being can have,
given the two poles are known and the bi-orientation / bi-rotation symmetry is intrinsic.

[wtf]

π + π√5
‾‾2π‾‾ = Φ

But this is just a restatement of the defining property of Φ. You could plug in any real number whatsoever and it would still be true.

You have (pi + pi sqrt(5) / 2pi = (pi(1 + sqrt(5))) / 2pi = (1 + sqrt(5)) / 2.

Just as the pi cancelled in the numerator and denominator; you could put any real number you like in there and it would still be true. So you've said nothing here.

[wtf]

π + π√5
‾‾2π‾‾ = Φ

But this is just a restatement of the defining property of Φ. You could plug in any real number whatsoever and it would still be true.

You have (pi + pi sqrt(5) / 2pi = (pi(1 + sqrt(5))) / 2pi = (1 + sqrt(5)) / 2 = Φ.

Just as the pi cancelled in the numerator and denominator; you could put any real number you like in there and it would still be true. So you've said nothing here.

I looked up Edenic and found, "Of or suggesting Eden, the paradise of the Bible."

Is that what you mean?