Dear Asher Klatchko

ϕ = (1+√5)/2 is the usual golden ratio.

The identity ϕ² = ϕ +1 is always true in the usual field of real numbers regardless of the geometric interpretation of the sides of Kepler's right angle triangle. Different geometries are equipped with different metrics.

In topology, the non-quantum geometry, mathematicians didn't care about quantities. But they can construct homeomorphisms or diffeomorphisms to

compare two (geometrical objects) in two different spaces.

Also, they equipped a suitable algebra with each space and study their algebraic properties as homotopy, homology, etc.

You can share your idea in this concern to show the deformation of (Kepler's triangle) to any new space which is not necessarily Euclidean; the new deformed triangle may not preserve any of its original geometric properties (length areas, angles,etc.)

Best regards

Dear Issam Kaddoura

Thanks for your response.

I think that it is fair to speculate that the field of real numbers hatched in a flat brain with extraordinary abilities to project to curved space.

Yet, the philosophical question addresses the idea of whether the numbers exists outside this brain independently, say in spacetime.

By this I indirectly refer to Sir Roger Penrose’s ideas about a Platonic universe whose mathematical truths (and objects) are engraved in spacetime.

I have heard inflation cosmology physicists going as far as to claim that inflation explains Euclid etc. Maybe it justifies it but what ifs still remain...

We hear that spacetime is an emergent quality of something more fundamental, say strings. Are strings and their dynamics possible without number?

Best fun,

Asher

the first statement is incorrect; phi is a well-defined mathematical constant and it ALWAYS satisfies the equation (and has nothing to do with the Pythagorean Theorem)

in a non-Euclidean geometry, the PT does not hold, but most common non-E geometries have models in Euclidean geometries (spherical, etc.; hyperbolic is more difficult)

there is no issue about where the numbers exist; there are spaces, spaces have these properties, numbers are the elements of those spaces

Christopher Landauer ,

if you can construct the Kepler triangle in non Euclidean space you should be short listed for the Field medal.

the kepler triangle makes essential use of the pythagorean theorem, which is not universally true in non-euclidean geometries, so the triangle cannot be found in a space that has no ``flat'' (i.e, euclidean) parts

Christopher Landauer Bravo! therefore “the first statement“ isn’t incorrect.

Although Euclidean & non-Euclidean (Linear and non linear) geometry do not contradict each others. In most practical cases both are transformable to each other. The funny thing is that, regarding these and too many other complex issues mathematicians are fighting with each other, while computer science enjoying approximate theories and employing it for practical applications. The artificial intelligence is a live and popular example.

The non Euclidean geometry also satisfy triangle inequality and parallelogram law (via infinitesimal distribution in parts for approximation ) . However, the question in this query has a good weight and need analytic study for discussion (It is not straightforward---It has some tricks to be highlighted).

Peter Breuer i don’t think that I am obliged to explain the meaning of “therefore“. I shall move on...

Muhammad Ali i do not argue against the truism of the FibonaccI identity. I just ask if the fact that one would not be able to construct it in a non Euclidian space sheds light on the question, where does it exist...

for example, let’s speculate that real numbers are some approximation of a more general entity which is better suited to describe a curved space. That being the case one would argue that real numbers and their stripped down natural shadows, or even better, prime numbers are just a mere recipe of approximating a “deeper reality“. or is it the other way around, namely g-d created the integers first... (Hawkins)

the issue seems to be the difference between the (well-defined) number phi and its equation on the one hand, and the pythagorean theorem and non-euclidean geometries on the other

there might, of course, be a different constant that is suitable for a different geometry, but that has to be proven

conflating two different issues is often the source of surprising and non-intuitive difficulties

while stephen hawking apparently did write a book that said ``god created the integers'', the quote actually goes back to leopold kronecker: ``god created the integers; the rest is the work of man'' (not exactly, of course, since the original is in german)

Christopher Landauer

I assume that the correctness of Fibonacci’s identity is timeless. That is, at any time that an anthropogenic entity would care to check its validity it should find out that it is true. Now, let’s assume that there exists a generalization of it which could also be ‘spaceless’, namely, at any geometry in any number of dimensions which could possibly exist this generalization would be found to be correct. This supposition makes the generalization spacetimeless. Okay, apart from being as wonderful as it possibly could what does it suggest as far as the actual existence of the ‘numbers’. Do they exist inside or outside spacetime? I assume of course that the existence of spacetime is indisputable...

agree with peter

the issue is the fantastic ability of mathematical notions to explain physics (see wigner, on the unreasonable effectiveness of mathematics in the ....), which confuses people into conflating the two

and what is the generalization of a number? if you are writing about phi, it is well-defined by its quadratic equation (and has value about 1.618, but easily computed to an arbitrarily large finite number of digits); a generalization might exist with a different formula and hence a different value, but what is the connection to phi that makes it a generalization (geometric, algebraic, what)?

numbers have nothing to do with spacetime (oh, that is what peter wrote 8-))

So geometry and in particular the Pythagorean theorem has nothing to do with spacetime? That's not what the Pythagoreans or Eratosthenes of Cyrene and most of all Plato thought -- https://plato.stanford.edu/entries/platonism-mathematics/.

In my view when you start to count ‘things‘ (rabbits etc.) you’re down to spacetime. That’s true for actuaries as well as for physicists. For example, Sir Roger Penrose estimates the volume of the phase space of possible universes as 10^10^123. Among which, those compatible with the second law of thermodynamics (as observed from here e.g. Researchgate), is 10^10^101. This leads to a certain probability that if a universe is uniformly chosen at random among all the possible universes, such that it appears like ours, this probability is infinitesimally small. Those are numbers that have quite a bit to do with spacetime. They may describe spacetime along some other two dozen universal constants, some of which are engraved on the headstones of the distinguished physicists Boltzmann and Hawking.

Still the question holds: in what domain do these 'numbers' exist?

mathematics is about far more than counting things; it deals in concepts that are no more about spacetime than they are about rabbits (fibonacci numbers have a cute explanation via rabbits), but they have (when done properly) an amazing consistency that often feels like it is discovered instead of invented

and there is no proof anywhere in physics that the so-called ``universal constants'' are actually constant except very locally; that is just the simplest choice (an italian friend of mine once said not entirely facetiously that it was clear that william of ockham was not a venetian, because there the simplest choice is almost always wrong)

“Greetings parents and congratulations to Kenyon’s graduating class of 2005. There are these two young fish swimming along and they happen to meet an older fish swimming the other way, who nods at them and says “Morning, boys. How’s the water?” And the two young fish swim on for a bit, and then eventually one of them looks over at the other and goes “What the hell is water?”

https://fs.blog/2012/04/david-foster-wallace-this-is-water/

Peter Breuer imagination? there appears to be lack of. If only I could “ask a computer...” A couple points though.

It has been asked what were number generalizations? since you mentioned Russell & Whitehead you may consider sets as such. among others those that are tightly connected to spacetime, would be, quaternions, and spinors. In general, as we move to non-commutatIve objects imagination may help.

i wrote a long essay in high school called ``but is it really there?'' where i went carefully through all of the constructions peter mentions (including the necessary equivalence relations), and decided (only partly facetiously) that 3/2 is not actually a number at all, it is a problem we do not know how to solve with the numbers we know (i.e., integers) - we then learn enough about the properties of and relations among these problems that we finally gain sufficient familiarity with them that we can begin to think of them as numbers, and can go on from there - it was a fun exercize - but the word ``really'' is ambiguous

on the other hand, remember that quaternions and spinors are not connected to spacetime at all; they are apparently good candidate models for some of the observations we make of properties of spacetime, but they are purely mathematical constructs

the fact that they (and so many other mathematical methods) seem to tell us remarkable and verifiable things about spacetime (eugene wigner, ``the unreasonable effectiveness of mathematics in the natural sciences'') is a happy accident that defies easy explanation, and causes people to assume that there must be some deeper connection

i allow that possibility, but do not assume it, since nobody has provided any verifiable argument for what that connection could be or how it might work (or a verifiable argument that there can be none); it remains an unverified assumption that i for one do not expect to be discovered or invented and explained any time soon, if ever, and i have never needed it in my own theoretical work (in mathematics, computation, and linguistics, but not physics)

also remember that large chunks (though not all) of modern mathematics were developed independently of and prior to the physics they were eventually applied to; the connection was made later (sometimes much later) between the formal mathematical method and the physics problem in spacetime

careful there, peter, there are some very peculiar notions about the existence of numbers that are, um, guaranteed to exist and difficult to describe - check out gregory chaitin's work on omega; it is nasty, difficult, and entirely rigorous mathematically

BTW i agree about q and spinors

i don't actually get along with the real world very well, so i don't particularly care whether reality conforms to my mathematics (though my position on this changes as it gets later at night 8-))

Are you guys competing with Chaitin on egos? No chance to win!

First let’s attempt to help the imagination problem by connecting Wallace’s address of ‘what the hell is water’ to spacetime. I think that one of the friends suggested to ask a computer... sigh!

But who needs spacetime? Isn’t it like asking, who needs chairs. After all everything is Platonically reduced to its abstract form and is elicited by dialectics...

You may consider a little wrinkle in the argument, after all, in order to gain knowledge, one would need information; which in turn is subject to a finite velocity of propagation and is therefore pretty much enmeshed in spacetime. Please note that ‘therefore’ here is in the logical sense of ‘ergo’...

If numbers are immutable abstract forms where do they exist, in the intellect? Okay, so where does the intellect exist if not in brains? But who needs brains if we have computers that ask what the hell is spacetime?

The sigh is borrowed from a reaction to a similar case where a website is attempting to make you prove that you are not a robot. Here the Platonist web designer shows you a number of objects and asks you to identify which ones are transcendental. To which you would respond ‘what the hell is transcendental?’ because he did not show you the symbol — π

Peter Breuer I imagine that too.