@GrandMinnow: Imagine having a discussion with a child. If they ask a question, one way of addressing it is to add layers of complexity to the issue such that it is beyond their grasp, pile on a dozen textbooks and say 'ask me when you know what you're talking about'. Another way is to simplify the issue to the bare essentials and provide an informal (perhaps imperfect) answer to inspire them to continue their quest for knowledge. And you may even learn something in the exercise of simplifying the issue to the bare essentials. I certainly find the repeated 'why...why...' questions from kids quite revealing of my own lack of understanding.

I'm not here to pick fights or spread misinformation. I'm here to learn. I've been quite open about my educational background and (I think) I've asked you many more questions than claimed that I have the answers. We don't need gatekeepers to 'scenic trails', we need people to help the litterers learn how not to litter. In joining this forum you did not sign up to teach others so feel free to ignore my messages, but if you're inclined to help then I welcome it. I certainly could benefit from someone with your knowledge.

FWIW, at the end of all 3 videos that I linked you to I include a message, like this one from my derivative paradox video:

"Now it’s worth repeating that the complete point-based construction is how we do math. The incomplete construction simply offers a different perspective on the paradox. We should not discard established ideas just because a different view might offer a more appealing resolution to a single paradox...Can we even do math with incomplete constructions? Or are there insurmountable problems with that approach? Let’s talk about it."

There are two issues being discussed here: (1) potential problems with the current philosophical foundations for math (2) potential problems with my proposed half-baked alternative to the philosophical foundation for math. I don't think you will enjoy us talking informally about (1) so I recommend that we set that aside.

Here are some replies to a subset of your comments which I think are most relevant:

Zeno's paradox (at least as it is usually presented) is not a formal mathematical problem — GrandMinnow

I think the beauty of paradoxes, such as Zeno's, is that they capture the essentials of a profound problem in a way that anybody can discuss. If you don't want to talk informally, or if you want to disregard Zeno's paradox due to its informal presentation that is fine, we can go our separate ways.

I watched the one about Zeno's paradox up to the point you said, "mathematicians begin with an assumption [that] [an] infinite process can be completed". What exact particular quote by a mathematician are you referring to? Please cite a quote and its context so that one may evaluate your representation of it in context, let alone your generalization about what "mathematicians [in general] begin with as an assumption" — GrandMinnow

You're right that I shouldn't have made that generalization. My apologies. Here's one quote by James Grime on Numberphiles video on Zeno's Paradox:

"I want to give you the mathematician's point of view for this, because, well, some say that the mathematicians have sorted this out........So something like this-- an infinite sum-- behaves well when, if you take the sum and then you keep adding one term at a time, so you've got lots of different sums getting closer and closer to your answer. If that's the case, if your partial sums--that's what they're called-- are getting closer and closer to a value, then we say that's a well-behaved sum, and at infinity, it is equal to it exactly. And it's not just getting closer and closer but not quite reaching. It is actually the whole thing properly."

To be fair, he follows that by saying that that's the paradox. Do you believe that infinite processes cannot be completed? If so, how can I move from A to B to C. I'll never get to C because I'll never complete the infinite steps required to get to B.

Anyway, I'm not sure where you want our conversation to go, but I'd be glad to hear your feedback on the rest of the videos that I linked if you care to give me a chance.

Again, set theory rises to the challenge of providing a formal system by which there is an algorithm for determining whether a sequence of formulas is indeed a proof in the system. So whatever you think its flaws are, that would have to be in context of comparison with the flaws of another system that itself rises to that challenge. — GrandMinnow

Agreed.

It's worth noting that the challenges in the first post of this thread have been met. But hell if I know whether the poster understands that by now. — GrandMinnow

Should we move this discussion to a new thread?

@norm: Thanks for the book recommendations, I plan to read both "Analysis By Its History" and "Groundless Grounds".

The mainstream real line is a vast darkness speckled by bright computable numbers, numbers we can actually talk about, numbers with names, while most of them are lost in the darkness and inferred to exist only indirectly. — norm

Very poetic, I like it!

But when I do math, I don't think of R in terms of that glorious set-theory mess at all...In my POV, foundations is its own fascinating kind of math. It doesn't really hold up the edifice of applied calculus, IMO. It's a decorative foundation. Humans trust tools that work most of the time. Full stop. — norm

To me it is concerning that the foundations are so disconnected from the applications. Could this be an indication that further foundational work is required? I'm not sure if you're following this thread closely but I pointed GrandMinnow to a few links on my YouTube channel. Here's one that's somewhat related to our discussion on integrals (Dartboard Paradox). You might be interested in this perspective as it offers a different perspective (granted, probably wrong and certainly half-baked). Nevertheless, I'd love to hear your thoughts on this view, especially if you can find flaws in it...but no pressure at all!

To me it is concerning that the foundations are so disconnected from the applications. Could this be an indication that further foundational work is required? — Ryan O'Connor

I only have a moment just now, but I'll respond to the point above. It seems to me (consider Hume's problem of induction) that humans just are 'irrationally' inductive animals. Foundations that come later than the edifice are not really foundations at all, despite the metaphor.

I'm no expert, but my impression was the math moved toward being totally mechanized, totally formal, totally computer-checkable. The self-image of the mathematician changed, probably because math become its own art/science, not just part of physics, etc. But writing proofs 'feels' more like convincing the intuition of another mathematician and reassuring one's own.
<running late, got to go!>

To me it is concerning that the foundations are so disconnected from the applications — Ryan O'Connor

It's unfortunate the word "foundations" is used in mathematics. Foundational set theory (that intersects with philosophy significantly) and all that revolves around it is a marvelous intellectual area, but a great deal of mathematics flows unimpeded by its pronouncements. Do most seasoned mathematicians worry about the existence of irrational or non-computable numbers? Or non-measurable sets? Not likely.

I've mentioned my own experience before: as beginning grad students sixty years ago we were required to take an introductory course in set theory (foundations). Afterwards the prof - a young energetic fellow - made the statement, "Unless you are really fascinated with the material in this course, I recommend you never take another foundations of mathematics class." I wasn't and I didn't. But I ultimately went into classical complex analysis. Had I gone into a more abstract realm of math I might have needed it.

Could this be an indication that further foundational work is required? — Ryan O'Connor

It depends on which mathematician you ask. Let's hope not. It makes little sense to ask anyone outside the profession.

I'm no expert, but my impression was the math moved toward being totally mechanized, totally formal, totally computer-checkable — norm

And I'm sure Norm would agree, that movement would drive most mathematicians out of the profession. It can't be emphasized enough how much mathematics depends on intuition, imagination, inventiveness, and a spirit of exploration. Devising and proving theorems is an art form.

It's unfortunate the word "foundations" is used in mathematics. Foundational set theory (that intersects with philosophy significantly) and all that revolves around it is a marvelous intellectual area, but a great deal of mathematics flows unimpeded by its pronouncements. Do most seasoned mathematicians worry about the existence of irrational or non-computable numbers? Or non-measurable sets? Not likely. — jgill

I liken it to physics. Many engineers get by with classical physics. They don't worry that it predicts singularities because it works great for them in their applications. But the singularities are the loose thread which suggest that classical physics is not fundamental. It doesn't lie at the foundation. We need to go quantum. If set theory properly lies at the foundation of mathematics then (I believe) it should have no loose threads (e.g. paradoxes). With that said, obviously set theory works to a significant extent so even if further refinements needed to be made to the foundation, I'm sure that the essence of set theory will play a significant role.

Afterwards the prof - a young energetic fellow - made the statement, "Unless you are really fascinated with the material in this course, I recommend you never take another foundations of mathematics class." — jgill

LoL. I'm sure the philosophy students in the room were aghast.

Do you believe that infinite processes cannot be completed? — Ryan O'Connor

I don't think in a framework of "infinite processes being completed or not completed". The notion of "an infinite process being completed or not completed" is not a notion I find meaningful; I am not burdened with it.

Moreover I already addressed this with regard to set theory:

Set theory itself (at least at the level of this discussion), as formal mathematics, does not say "an infinite process can be completed". Set theory doesn't even have vocabulary that mentions "completion of infinite processes". And the assumptions of set theory are the axioms. There is no axiom of set theory "an infinite process can be completed". — GrandMinnow

But you blow right past that. (You're too busy with things like explaining that I am free to ignore you, and extending my little metaphor of a 'trail' into an overblown conceit that is becoming ludicrous).

infinite sum — Ryan O'Connor

An infinite sum is the limit of a function. It is the unique number such that the terms of the sequence converge to that number.

The actual mathematics does not say "process" or "process completion". It doesn't need to be saddled with it. If you find problems with the intuitive view of an infinite sum as the completion of a process, then it's your intuitive framework that is problematic, not mine, since I don't have to resort to that framework and, as far as I know, mathematics may be understood without it.

So much more to unpack:

Imagine having a discussion with a child — Ryan O'Connor

Imagine a discussion among intelligent and educated adults.

If they ask a question, one way of addressing it is to add layers of complexity to the issue such that it is beyond their grasp — Ryan O'Connor

Among these educated and intelligent adults, if they post their opinions on certain matters that bear upon technical considerations, then one may point out where those technical points have been misconstrued and offer relatively concise corrections and explanations. If this is beyond the grasp of any of these educated and intelligent adults then they can consult any of a number of books that explain at a level any intelligent person can understand by simply beginning with chapter one and reading forward.

pile on a dozen textbooks — Ryan O'Connor

You have been occasionally exaggerating my points in order to knock them down. And you use other variations of the strawman argument, most saliently to me in the very first remarks in your video on Zeno's paradox, as I pointed out. (Though you have since retracted in this thread, I don't know whether you intend to correct the video itself).

To the point, I did not "pile on a dozen textbooks". You asked me about education. I gave you a list of three (plus some optional supplements) introductory undergraduate textbooks for a three step sequence: Symbolic Logic, Set Theory, Mathematical Logic. And I don't say that one has to have such a background merely to ask questions, speculate, ruminate, or convey ideas on the subject. Rather, my point is that when your philosophizing moves into matters that bear on technical points in mathematics, and you mangle those points or talk past right past them, then it's appropriate to point that out. And I recommended a few books in response to your question about education.

and say 'ask me when you know what you're talking about' — Ryan O'Connor

I never said that or anything equivalent to it.

I'm here to learn — Ryan O'Connor

Perhaps you are, but that's not all. You've also here for other people to be impressed with what you say.

We don't need gatekeepers — Ryan O'Connor

I'm not a gatekeeper in the sense of saying that people may not post whatever they want to post. You can post as you like; I don't try to stop you. Meanwhile, I hope you are not a gatekeeper saying what I may post, including criticisms of your posts. And my purposes in posting are not determined by what you think a forum needs or doesn't need.

we need people to help the litterers learn how not to litter — Ryan O'Connor

I have suggested ways you can abate your littering.

feel free to ignore my messages — Ryan O'Connor

I already responded to your sophomoric protest that I may ignore you. Of course I feel free to ignore you, and I also feel free not to ignore you. Your personal preference in the matter is not relevant to me. When you post, others may reply or not reply arbitrarily at their own prerogative.

if you're inclined to help then I welcome it — Ryan O'Connor

I've offered you help already. I've given you explanations at a pretty simple and straightforward level. But you blow right past most of the key points in those explanations. And I've given you a list of three books that constitute a truly splendid introduction to the subject on which you are posting. I am not even a mathematician, but at least I have made myself familiar with a number of books on the subject. If you are sincere about learning and benefiting from what I know, then the very best you could do is to accept my expertise in book collecting and get hold of the books I mentioned. Instead you take umbrage at the offer and whine as if you've been unduly sandbagged.

I don't think you will enjoy us talking informally about potential [problems with the current philosophical foundations for math] — Ryan O'Connor

If you don't want to talk informally, or if you want to disregard Zeno's paradox due to its informal presentation that is fine — Ryan O'Connor

I am, as time permits, interested in philosophy of mathematics and informal discussion about mathematics and the philosophy of mathematics. And Zeno's paradox is of course important in the philosophy and history of mathematics. But this is the point you keep missing: When the informal discussion bears upon, or especially critiques, the actual mathematics and the actual foundational formalizations, then it is critical not to speak incorrectly, especially from ignorance, about the actual mathematics, formal theories, and the developments in set theory and mathematical logic. I surmise that you, like cranks, find poring through the actual technical development to be onerous but you prefer to opine about it in ignorance anyway. This is witnessed by the fact that no matter how many times one suggests to a crank that he consult the actual writings on the subject, he will never even look at chapter one.

If set theory properly lies at the foundation of mathematics then (I believe) it should have no loose threads (e.g. paradoxes). — Ryan O'Connor

I addressed that already. You blew right past it.

And I'm sure Norm would agree, that movement would drive most mathematicians out of the profession. It can't be emphasized enough how much mathematics depends on intuition, imagination, inventiveness, and a spirit of exploration. Devising and proving theorems is an art form. — jgill

I very much agree (creative intuition is why it's beautiful and fun). I also agree with what you said about foundations. We never even covered constructions of R in the classroom. I actually have found 'foundations' (and mathematical logic) fascinating, but I never had the time or sufficient passion to really catch up with the present. I still love computability theory, but the most fun I've had mathematically is inventing things (like my own construction of R or various crypto systems or oddball never-before-seen (and not actually useful) neural networks. My blessing/curse is that I can't help approaching it like a sculpture. I don't care much about applications. I like beautiful machines made of pure thought.

You might be interested in this perspective as it offers a different perspective (granted, probably wrong and certainly half-baked). Nevertheless, I'd love to hear your thoughts on this view, especially if you can find flaws in it...but no pressure at all! — Ryan O'Connor

I'll try to find time for it & definitely give some friendly feedback.

the foundations — Ryan O'Connor

set theory — Ryan O'Connor

You don't know what set theory is.

You don't know about the symbolic logic in which set theory is formulated. You don't know what the language of set theory is. You don't know the axioms of set theory. You don't know the theorems of set theory and how they are derived from the axioms. You don't know the definitions in set theory. You don't know how set theory develops numbers and mathematics. You don't know how set theory axiomatizes calculus and other mathematics of the sciences. You don't know the purpose, motivation, and role of the axiomatic method. You don't know about constructive, intuitionist, predicativist, or finitist altermatives to classical mathematics.

You don't know anything about it.

Yet you have persistent critiques of it.

How do you do it?

LoL. I'm sure the philosophy students in the room were aghast. — Ryan O'Connor

That would be the empty set, Ryan. We were all math majors and the course was taught in the math department. :smile:

You don't know anything about it.

Yet you have persistent critiques of it.

How do you do it? — GrandMinnow

With a certain aplomb. I admire his spirit while avoiding his critiques. :cool:

You might be interested in this perspective as it offers a different perspective (granted, probably wrong and certainly half-baked). Nevertheless, I'd love to hear your thoughts on this view, especially if you can find flaws in it...but no pressure at all! — Ryan O'Connor

That's a well made video. FWIW, I do like the continua-based approach. Have you looked into smooth infinitesimal analysis ? It seems similar. One issue worth noting is your description of a quasi-Riemann integral as an endless process. In an actual Riemann integral, for f which is continuous on [a,b], there exists a definite sum. In other words, we know that it's a particular real number, even if we only ever approximate it (like the areas under the standard normal curve.)
In SIA, certain issues are circumvented, because every function is smooth (infinitely differentiable). Some strange logic is involved.

With a certain aplomb. I admire his spirit while avoiding his critiques. :cool: — jgill

Yes, and I can relate to theimpatience of someone who wants to talk about something now. Chances are that this thread will inspire some serious reading, and talking through things could help pick out just the right book. I mentioned a philosophy book and a math book, because for many people (whether they know it or not) it does turn out to be a philosophical issue, a perspective from which to interpret what math means. Or the issue here could be one of intuition and pedagogy and not really about nitty-gritty foundational work.


I don't mean to talk as you aren't 'here' with us. I'm curious about what kind of pure math that you have studied, if you feel like sharing. Have you wrestled with real analysis? I am nostalgic for basic real analysis on R, working through proofs of theorems about the beautiful Riemann integral. We didn't bother with constructions. We just used the axioms. I had the itch, so I learned the two classic constructions, and I was very passionate about grabbing those slippery real numbers in my intuition. It bothered me and it delighted me. (I was OK at writing proofs, but pretty good among my peers at reading them. Indeed, research can be a little dreary compared to enjoyed the condensed, finished product of generations --with only brief flashes of invention.)

At what point did we prove that it was a number? — Ryan O'Connor

Interesting! So, you think the square root of 2 could be something other than a number. Well, the square root operation is closed over real numbers i.e. a square root of a real number has to be a real number. Does that answer your question?

That would be the empty set, Ryan. We were all math majors and the course was taught in the math department. — jgill

Mathies keep the fun stuff to themselves. :P

That's a well made video. FWIW, I do like the continua-based approach. Have you looked into smooth infinitesimal analysis ? It seems similar. One issue worth noting is your description of a quasi-Riemann integral as an endless process. In an actual Riemann integral, for f which is continuous on [a,b], there exists a definite sum. In other words, we know that it's a particular real number, even if we only ever approximate it (like the areas under the standard normal curve.)
In SIA, certain issues are circumvented, because every function is smooth (infinitely differentiable). Some strange logic is involved. — norm

Thanks for checking it out! I'm glad you like the approach. I have not looked into SIA. However, just a few points on the wiki page seem concerning to me, like I have no problems with discontinuous functions but I do have a problem with infinitesimals. Nevertheless, I will check it out. For Riemann integrals, how do we know that it corresponds to a real number if we are only ever able to approximate it?

Interesting! So, you think the square root of 2 could be something other than a number. Well, the square root operation is closed over real numbers i.e. a square root of a real number has to be a real number. Does that answer your question? — TheMadFool

Yes, I think it's an algorithm for calculating a number (but the algorithm cannot be executed to completion). I'll need to look up a proof of your statement and hopefully I can understand it! Perhaps it will convince me.

I have not looked into SIA — Ryan O'Connor

Weird stuff, IMHO. Low priority in the world of mathematics.

However, just a few points on the wiki page seem concerning to me, like I have no problems with discontinuous functions but I do have a problem with infinitesimals. — Ryan O'Connor

I can understand your hesitation. As long as one demands a 1-to-1 map between individual mathematical inventions and metaphysical correlates, it's hard indeed to be satisfied. There's a thought (I think Hilbert's) that we should just mathematical systems as a whole for correspondence with reality. You mentioned something like points-at-infinity in your discussion with MU. I mentioned the convention that 0! = 1. I'm personally doubt l that there's a rigorous way to include the continuum without offending someone's intuition. Great mathematicians have wrestled with this. As I mentioned, you might like Errett Bishop more than anyone I'm aware of.

For Riemann integrals, how do we know that it corresponds to a real number if we are only ever able to approximate it? — Ryan O'Connor

The 'magic' of least upper bound axiom does the heavy lifting in R. Every nonempty subset of R which is bounded above has a least upper bound, a 'supremum.' I can look at 'lower Riemann sums' (approximations with step functions <= to f(x)) and define the supremum of that set to be what the integral means, which is to say the number that is being approximated.

A related example is the greatest lower bound or infimum of a set. Consider the set {1/n : n >= 1}. All of its members are positive. It's infimum, however, is 0. It's the only lower boundary that makes sense. Any number great than 0 misses an element, and any number less than 0 is not as close and effective a bound as 0. This kind of thinking is IMO the essence of basic real analysis. (Note also that pi is defined geometrically but in terms of the zeros of a trigonometric function, which requires a detour through the convergence of power series first. I like Rosenlicht's little Dover book, Introduction to Analysis. It's cheap and is just good.

I'll just reiterate that if you aren't that concerned with proofs (deriving theorems from axioms), then mathematicians will hardly recognize that you are even interested in (pure) math. Lots of foundations can and even must 'spit out' the same engineering math, the stuff that keeps the planes in the skies. Being a mathematician, to anyone with mainstream training, means reading and writing proofs. I don't mean to be a snob about this. I'm personally just as interested in pedagogy for applied math as pure math (I'm an anti-foundationalist with empirical leanings. I like Wolfram, etc.)

Weird stuff, IMHO. Low priority in the world of mathematics. — jgill

That's been the case in my experience too. For applications, though, the dual numbers are actually important today. Some of the autodifferentiation powering machine learning uses the 'forward method', employing dual numbers to great effect (and even hyperdual numbers.) This allows one to compute high-dimensional f(x) and grad(f(x)) at the same time at low cost.

I do have a problem with infinitesimals — Ryan O'Connor

Infinitesimals are made rigorous with non-standard analysis derived with techniques of model theory or with internal set theory.

What argument have I lost? — Metaphysician Undercover

@Ryan claims that we should poll everyone in the world; I claim we should poll the professional mathematicians; and you claim we should poll the philosophers specializing in metaphysics. Doesn't seem like there's any qualitative difference between your position and the others.

"Existence" is a word which is being used here as a predicate. — Metaphysician Undercover

Don't start that! I'm sure you know the trouble one gets into using existence as a predicate. Existence is not a predicate.

So we need criteria to decide which referents have existence in order justify any proposed predication. Naturally we ought to turn to the field of study which considers the nature of existence, to derive this criteria, and this is metaphysics. — Metaphysician Undercover

This is a mistake on your part. Doctors know more than the average person about doctoring; and baseball players know more about baseball. But metaphysicians don't know any more about existence than the rest of us. All they know is what other thinkers have said about the problem. Being a philosopher confers no special knowledge at all about what's true. That's one of the fundamental problems with philosophy. If I study math in school, I'll learn about math. If I study philosophy, I'll learn about what the great thinkers have said about philosophical problems; but I won't learn the truth about anything. So you have no credential whatsoever.

Mathematics does not study the nature of existence, so mathematicians have no authority in this decision as to whether something exists or not, — Metaphysician Undercover

Agreed. But mathematicians have total authority in terms of what has mathematical existence.

regardless of whether it is a common opinion in the society of mathematicians. — Metaphysician Undercover

That's the main criterion. And yes it's historically contingent, and yes it's somewhat unsatisfactory if one wants to believe in some kind of ultimate existence, but that's the position I'm taking.

If you are arguing otherwise, then show me where mathematics provides criteria for "existence" rather than starting with an axiom which stipulates existence. — Metaphysician Undercover

The entire history of mathematics is filled with examples, starting from the discovery of irrational numbers right through to the present day. Now you may well respond that if I'm admitting this was a discovery and not an invention, then irrational numbers were already "out there" waiting to be discovered. I have no good answer for this objection but neither does anyone else.

Sabine Hossenfelder has a current blog post on Do Complex Numbers Exist? Might be relevant, I'm not qualified to judge.

Sabine Hossenfelder has a current blog post on Do Complex Numbers Exist? Might be relevant, I'm not qualified to judge. — Wayfarer

I love her videos and articles but felt that she entirely missed the meaning of complex numbers in that video. When physicists talk about math it's always a disaster. But she's always worth watching.

ah OK - a hit, and a miss!

ah OK - a hit, and a miss! — Wayfarer

Just my opinion, I didn't want to give you a hard time for posting it, she's always worth watching. That particular video annoyed me but as you know I'm easily annoyed! It's good that you posted it and people should watch it. She does talk about the use of complex numbers in physics and that's definitely worth watching.

But metaphysicians don't know any more about existence than the rest of us. — fishfry

It seems like you don't know anything about metaphysics, which is the study of existence. Why would you think that someone who has not studied existence would know as much about existence as someone who has studied existence?

Metaphysics studies questions related to what it is for something to exist and what types of existence there are. Metaphysics seeks to answer, in an abstract and fully general manner, the questions:[3]

What is there?
What is it like?
Topics of metaphysical investigation include existence, objects and their properties, space and time, cause and effect, and possibility. Metaphysics is considered one of the four main branches of philosophy, along with epistemology, logic, and ethics.[4] — Wikipedia: Metaphysics

The entire history of mathematics is filled with examples, starting from the discovery of irrational numbers right through to the present day. — fishfry

So, where are your examples? Where is the criteria for existence found in mathematics?

hat's been the case in my experience too. For applications, though, the dual numbers are actually important today. Some of the autodifferentiation powering machine learning use the forward method, employing dual numbers to great effect (and even hyperdual numbers.) This allows one to compute f(x) and grad(f(x)) at the same time at low cost. — norm

See, I learned something from your post! Thanks, norm.

but felt that she entirely missed the meaning of complex numbers — fishfry

What would you say the meaning is? Just curious. :cool:

Hello Ryan O'Connor,

I do agree, that we have not proved the existence of irrational numbers.

My argument is not only the problem of indirect proofs, but a deeper one:
We can not be sure that our logic is totally right.

It is astonishly easy to construct an alternative logic that gives in finite cases mostly the same results as classic logic, but has totally other results with infinite cases or indirect proofs or antinomies.

I myself constructed such a logic, the layer logic.
There we have an additional parameter, the layer,
and statements are not right or wrong but have a truth value in every layer.
And this values can be different in different layers without making a contradiction.

Therefore in indirect proofs (as for the irrationalioty of root 2) we need to get the contradiction in the same layer. But analysis shows, that different layers are used,
and therefore with layer logic we do have no contradiction and no proof anymore.

The same way we can show, that the diagonalization of Cantor does not work anymore.
So we need only one kind of infinity (that of the natural numbers) in layer logic or layer set theory.

But not all is better with layer logic:
The uniqueness of the prime decomposition (over all layers) can not be proved.

So we have a nice layer set theory where the set of all sets is a set,
but arithmetics may partly be time dependent:

My newest guess is, that there is a layer for all objects (quants) that can interact
(except interacting with gravity),
and if some interact than the layer for all objects is increased.
This way we get in the layers a kind of time arrow since the big bang,
and properties (even in math like prime decomposition)
can depend from it and change with time.

Most mathematicans will not like such a world,
but it seems possible to me.

The square root of 2 could be a rational quotient with different numinator and denominator
in different layers (or times).

Some details to layer logic you can find in:
https://thephilosophyforum.com/discussion/1446/layer-logic-an-interesting-alternative

Even more (in German) here:
https://www.ask1.org/threads/stufenlogik-trestone-reloaded-vortrag-apc.17951/

Yours
Trestone

My newest guess is, that there is a layer for all objects (quants) that can interact
(except interacting with gravity),
and if some interact than the layer for all objects is increased.
This way we get in the layers a kind of time arrow since the big bang,
and properties (even in math like prime decomposition)
can depend from it and change with time. — Trestone

This is similar to what I was telling Ryan in the other thread on Gabriel's horn. The classical way that mathematicians apply numbers to spatial representations (Euclidian geometry) assumes an eternally continuous, and static, space. But modern observations have produced a new concept called spatial expansion. Therefore we need to allow that space itself changes with time, and this means that the assumption of a static space is incorrect. So if we propose a number of points in space, and these points, if connected with lines, make a shape such as a triangle or square, and then we propose some passage of time, then these same points in space will no longer make the same shape.

"Existence" is a word which is being used here as a predicate. — Metaphysician Undercover

In casual discussion, mathematicians may say things like "the square root of 2 exists". But in a more careful mathematical context, we don't say that. Instead we say, "There exists a unique x such that x^2 = 2." Then we may apply the square root operator to refer to sqrt(2).

So, indeed, in careful mathematics 'existence' is not a predicate. In careful mathematics It is not even grammatical to use 'existence' as a predicate. Instead, there is an existential quantifier that is applied to a variable and a formula (usually with that variable free in the formula); the formula specifies a "property".

Symbolically, existence is not a predicate in which we would write "the square root of x has the property of existing:

E(sqrt(2))

That is not even grammatical.

Instead, we write:

Ex x^2 = 2

Which reads, "There exists an x such that x^2 = 2".

'Ex' is the quantifier, and 'x^2 = 2' is the formula specifying the property.

Then we also derive a uniqueness quantifier and write:

E!x x^2 = 2

Which reads, "There exists a unique x such that x^2 = 2".

And that justifies using

sqrt(2)

as a term.

"1" does not refer to anything.