That differs from how I find 'classical' is used. I find that 'classical' mathematics means all and only those results that can be formalized as theorems of ZFC with classical logic. And classical logic means the first order predicate calculus including the law of excluded middle. — TonesInDeepFreeze

Well, that's interesting. I learned something. Thanks. Classical analysis of course means more or less what I said, going back to Weierstrass and Cauchy - and I forgot, the study of special functions - but in foundations classical has another meaning.

Wiki:

In the foundations of mathematics, classical mathematics refers generally to the mainstream approach to mathematics, which is based on classical logic and ZFC set theory.

:cool:

I just proved an interesting result demonstrating a sequence of sine contours on the interval [0,1] that not only converge uniformly to the line from 0 to 1 while growing towards infinite length, but converge in this way along the interval [0,1] all the way down to the point (0,0) while becoming infinite in length. I mention this only because there is interest in the relationship between 0 and infinity and somehow combining continuity and discreteness.

I'll give details if requested. :cool:

I don't see any advantage to the fact that your way of conceptualizing pi is "entirely finite." — T Clark

I think it boils down to the real line being composed of points and each point having a unique number. Instead of focusing on pi, let's return to the real line having length but its constituient points all having no length. Does that bother you? Does it bother you that a probability of 0 does not mean impossible? Or that there are the 'same number' of even integers as there are integers? These are age old paradoxes that don't bother many educated people so perhaps you see things more clearly than I do...

It is my understanding that computers do not generally store the algorithm for generating pi, they store the actual number rounded to a specified number of decimal laces. If computers think pi is a number, why shouldn't I?" — T Clark

Yes, they store and work with rational numbers...not real numbers.

When I measure light one way, it's always a wave. When I measure it another way, it's always a particle. It's not a wave that becomes a particle. It's always both at the same time." — T Clark

Are you talking about the double-slit experiment or some other experiment?

The universe has a wonderful way of avoiding actual infinities.
— keystone

Again, sez you — T Clark

I think I was unclear in what I meant. When physics equations result in singularities/infinities we take that as a sign that there's something wrong with our equations. Time and again we have made progress at understanding the universe by eliminating those singularities/infinities.

I think you and I have taken this as far as we're going to get. I don't see the need for or value of the way of seeing things you propose. You obviously disagree. Neither of us is going to convince the other." — T Clark

I don't think you're seeing my point, but fair enough.

In a quantum reality we can only talk about it's velocity when measurements were made
— keystone

we were talking in terms of Calculus, and that is a very integral and important circumstance to my question. Perhaps I should have pointed that out. — god must be atheist

I agree that in terms of the orthodox point-based interpretation of calculus that there would be infinite points along a line. However, a continuum-based interpretation which involves the exact same computations (and which I would argue is more consistent with the limit definitions) follows the same measurement restrictions as our quantum reality.

You mean the continuum is everything. That is the opposite of nothing. Then what you call continua are the line segments that are fall inbetween these two complementary extremes. — apokrisis

No, I don't believe the continuum is everything. I think that the computer/mind lies outside the continuum. For example, when you imagine a sphere your mind exists outside that sphere.

Then what you call continua are the line segments that are fall inbetween these two complementary extremes. — apokrisis

In our mind, we are neither thinking of everything nor nothing. We can only think of something. I don't believe in the existence of either extreme.

if the line is cut, then you are also talking about a lack of line with some infinitesimal length, not a 0D point. — apokrisis

When a line is cut, none of the line is lost. It is just divided. "Nothing" exists between the cuts, and nothing has no size. We can call this 'nothing' a point.

This just helps show that the idea of a 0D point is ontically problematic and in need of much better motivation than you are providing. You assume too much without providing the workings-out. — apokrisis

I'm only conveying my ideas piecemeal and even then my ideas are not formalized. I don't come to this thread with any notion that I have it all figured out. I'm greatly appreciative of your feedback here. You're the first to ever entertain my idea on cutting a continuum. (or perhaps you have the same idea)

Nothing and everything are really the same. A void and a plenum are either too empty to admit change, or too full to admit change. White noise is both every song ever written, or that even could be written, played all a once, and no song being played at all. — apokrisis

To some extent I agree with this. The part I have trouble with is your use of 'everything'. I think your 'everything' is every 'potential' thing. My 'everything' is every 'actual' thing (which doesn't include objects/events that don't exist/happen).

continua must exist as a constraint on a state of everything. — apokrisis

What exactly do you mean by this? I don't think 'a state of everything' needs to exist for 'something' to exist.

Sure. Behind it all is symmetry and symmetry breaking. Numbers are based on the maximum symmetry that is their identity operation - 0 for addition, 1 for multiplication. This first step suffices to produce the integers. Then more complex algebra gives you further levels of symmetry to populate the number line more densely with other symmetry breakings.

There are generators of the patterns. You start with the differences that don’t make a difference. Then this yields a definition of the differences that do.

Again the logic of the dialectic and the basis of semiotics. Stasis and flux are a dichotomy. Mutually dependent and jointly exhaustive. Each is the measure or the other. — apokrisis

I understand how we start with natural numbers > integers > rational numbers > real numbers, etc. I'm not sure what to make of this comment though. You return to the point that 'each is the measure of the other' so I think that's key to your argument, I'm just not comprehending it yet...

To use the usual example, when you say x=0, are you talking about 0.00…. to some countable number of decimal places. Have you excluded x=0.0000….a gazillion places later …0001? — apokrisis

When I say 0, I don't mean 0.1, 0.01, or 0.001. I mean exactly 0 the rational number. Is that still vague?

I don't think you're seeing my point — keystone

That's true and I don't think you've seen mine and I don't think either of us is going to change.

No, I don't believe the continuum is everything. I think that the computer/mind lies outside the continuum. For example, when you imagine a sphere your mind exists outside that sphere. — keystone

Huh? Weren’t we about talking about how we “picture” the continuum just as much any other mathematical object like a sphere? Non sequitur here.

In our mind, we are neither thinking of everything nor nothing. We can only think of something. I don't believe in the existence of either extreme. — keystone

Can you picture a hypersphere as easily as a sphere? Does that make you doubt that it is a constructable object? Is your whole argument going to be based on what you personally find concretely visible in your minds eye? That’s a weak epistemology that won’t get you far.

When a line is cut, none of the line is lost. It is just divided. — keystone

And when the line is joined, is nothing gained? If there is no gap due to the cut then is there no connection if there is a join?

You just seem to be saying stuff. I can’t picture a cut which doesn’t result in a gap. Are you now claiming you can see that just as vividly as a sphere? Is this an argument where we just accept your word on everything? The rules for constructing mathematical objects is becoming unclear.

You're the first to ever entertain my idea on cutting a continuum. (or perhaps you have the same idea) — keystone

It’s a standard kind of idea. For instance - https://en.wikipedia.org/wiki/Dedekind_cut

The part I have trouble with is your use of 'everything'. I think your 'everything' is every 'potential' thing. My 'everything' is every 'actual' thing (which doesn't include objects/events that don't exist/happen). — keystone

Yep. I mean an everythingness that is a universal potential and not some set of actual things.

Actualisation indeed eliminates possibilities. Which is how one would argue for infinity as an unbounded process rather than an actualised value.

What exactly do you mean by this? I don't think 'a state of everything' needs to exist for 'something' to exist. — keystone

Can you picture getting something from nothing? Can you picture being left with something having carved away most of everything?

One of these two is more picturable, no?

It is also the central principle of physics in being the principle of least action. The sum over possibilities or path integral.

You return to the point that 'each is the measure of the other' so I think that's key to your argument, I'm just not comprehending it yet... — keystone

It’s the logic of a reciprocal or inverse operation. How do we recognise the discrete except to the degree in lacks continuity. How do we recognise continuity except to the degree it lacks the discrete. One is present to us to the degree the other is absent.

Grey is the vague. It can then become white to the degree it sheds its blackness, and black to the degree it leaves behind its white. We have two ultimate limits where black = 1/white and white = 1/black.

How do we recognise the discrete except to the degree it lacks continuity. — apokrisis

If it's not a rhetorical question (and apologies to the OP if this is off topic)...

The final requirement for a notational system is semantic finite differentiation; that is, for every two characters K and K' such that their compliance-classes are not identical, and every object h that does not comply with both, determination either that h does not comply with K or that h does not comply with K' must be theoretically possible. — Goodman, Languages of Art

So not necessarily a matter of degree. Arguably a matter of discrimination. Which can be all or nothing. Witness digital reproduction. Where black and white are kept safely apart by grey, and there is no need for any collapse (or refinement) into 50 or more shades.

(Easy with those abstract nouns please, Apo...)

So not necessarily a matter of degree. Arguably a matter of discrimination. — bongo fury

I am talking about how the spectrum that allows your 50 shades of grey arises. This is confusing for sure. But after the separation of the potential, you get the new thing of the possibility of a mixing.

So we start with a logical vagueness - an everythingness that is a nothingness. We have a “greyness” in that sense. Something that is neither the one nor the other. Not bright, not dark. Not anymore blackish than it is whitish. You define what It “is” by the failure of the PNC to apply. You are in a state of radical uncertainty about what to call it, other than a vague and uncertain potential to be a contextless “anything”. It is not even a mid-tone grey as there are no other greys to allow that discriminating claim.

But then you discover a crack in this symmetry. You notice that maybe it fluctuates in some minimal way. It is at times a little brighter or darker, a little whiter or blacker. Now you can start to separate. You can extrapolate this slight initial difference towards two contrasting extremes. You can drag the two sides apart towards their two limiting poles that would be the purest white - as the least degree of contaminating black - and vice versa.

Once reality is dichotomised in this fashion, then you can go back in an mix. You can create actual shades of grey by Goodman’s approach.

I once had a holiday job mixing industrial paint in monster vats. It was amusing how my recipe for the morning for some company’s shade of white, used to paint their fridges, requires drums and drums of bright white, and then a few teaspoons of black, and indeed a touch of red, to make it “theirs”.

So once you separate, then you can create. With a vat of white and a fat of black, you can mix every shade of grey inbetween. But if you only have a vat of light grey and a vat of dark grey, your range is way more limited. And if you have two middle of the range greys, then it might be hard to know whether the two have already been mixed as your efforts at mixing look to be making no further difference.

This is the ontology. Potential gets divided, the divisions produce the further thing of a space of free possibilities - a spectrum of mixed states.

Everythingness Is turned into a set into of local particulars within some global bounding constraints, At which point, our everyday notions of t reality as a collection of medium sized dry goods takes over. We take the atomic particulars for granted and get on with constructing the mixtures that have become concretely possible. We don’t seem to need a theory of how this generalised somethingness itself came to be.

I am talking about how the spectrum — apokrisis

I know, but as usual you don't see where I'm coming from.

that allows your 50 shades of grey — apokrisis

Yours not mine.

This is confusing for sure. — apokrisis

With that attitude...

But after the separation of the potential, you get the new thing of the possibility of a mixing. — apokrisis

...and with those abstract nouns.

So we start with a logical vagueness - an everythingness that is a nothingness. — apokrisis

Do you mean, an indiscriminate application of colour words to the domain of things (or patches)?

We have a “greyness” in that sense. Something that is neither the one nor the other. Not bright, not dark. Not anymore blackish than it is whitish. You define what It “is” by the failure of the PNC to apply. You are in a state of radical uncertainty about what to call it, other than a vague and uncertain potential to be a contextless “anything”. It is not even a mid-tone grey as there are no other greys to allow that discriminating claim.

But then you discover a crack in this symmetry. You notice that maybe it fluctuates in some minimal way. It is at times a little brighter or darker, a little whiter or blacker. Now you can start to separate. — apokrisis

Do you mean, you are able to apply the words in a manner that begins to distinguish two different though still overlapping colours?

You can extrapolate this slight initial difference towards two contrasting extremes. You can drag the two sides apart towards their two limiting poles that would be the purest white - as the least degree of contaminating black - and vice versa. — apokrisis

But you're anticipating the later refinement (the bipolar continuum) and assuming it's intrinsic to the earlier distinction. I was pointing out that it isn't.

Once reality is dichotomised in this fashion, then you can go back in and mix. You can create actual shades of grey by Goodman’s approach. — apokrisis

Goodman's approach is concrete and clear. Yours is abstract and poetic.

A discrete classification in no way has to imply a continuous one.

First, there are two different notions of 'the continuum'. One is that the continuum is the set of real numbers R. The other is more specifically that the continuum is R along with the standard ordering on R, or formally the ordered pair <R L> where L is the standard 'less than' ordering on R. — TonesInDeepFreeze

Is it possible for a continuum to exist and be defined mathematically without relying on numbers?

where can one read of a notion of the real continuum as an "n-dimensional continuum"? What does it mean? — TonesInDeepFreeze

I'm referring to a curve (1D continuum), surface (2D continuum), etc.

where can one read of a notion of the real continuum as an "n-dimensional continuum"? What does it mean? — TonesInDeepFreeze

I'm not referring to the construction of the set of real numbers but construction of a line. Can you comment on whether points can be assembled to construct a line without making use of real numbers?

Suggestion: Since you are interested in formulating an alternative to infinitistic mathematics, then you would do yourself a favor by first reading how infinitistic mathematics is actually formulated, as opposed to how you only think it's formulated, and also you could read about non-infinitistic alternative formulations that have already been given by mathematicians. — TonesInDeepFreeze

I acknowledge that I could really benefit in reading more textbooks. But just practically speaking, I could waste a lot of time sinking my head in textbooks without connecting with the community/a mentor to make sure I'm headed in the right direction. There's certainly value to me in having these discussions on this thread at this intermediate point along my journey.

If by "Cantor's nonsense" you mean his religious beliefs, then it is plain, flat out false that axiomatic infinitistic mathematics implies Cantor's religious beliefs. — TonesInDeepFreeze

What are your thought's on Hilbert's Hotel Paradox? In this paradox, he describes a hotel having infinite rooms. In this story we can't describe the hotel using inductive sets. The hotel simply has actually infinite rooms. Do you think it's a gross misrepresentation of infinite sets?

I've heard people say that the paradoxes entwined with actual infinities are beautifully mysterious...I just think they demonstrate the flaws of the concept of actual infinity.
— keystone

What specific paradoxes do you refer to?

Keep in mind that no contradiction has been found in ZFC. — TonesInDeepFreeze

Most notably Hilbert's paradox of the Grand Hotel, but also the following:

• Gabriel's horn
• Galileo's paradox
• Ross–Littlewood paradox
• Thomson's lamp
• Zeno's paradoxes
• Cantor's paradox
• Dartboard paradox

I know, but as usual you don't see where I'm coming from. — bongo fury

I hope this is a prelude to you making an attempt to explain then. :meh:

...and with those abstract nouns. — bongo fury

So abstractions are banned from a metaphysical discussion. :up:

Do you mean, an indiscriminate application of colour words to the domain of things (or patches)? — bongo fury

Nope. As usual you don't see where I'm coming from.

Do you mean, you are able to apply the words in a manner that begins to distinguish two different though still overlapping colours? — bongo fury

Back to the abstract nouns I guess. Semiotics as a maximally general theory of meaning tells us signs point out the differences that make a difference, not just merely the differences. But then there must be a spectrum of differences of some kind such that there could indeed be the differences that make a difference that are then different from the differences that don't. ie: we want to be able to separate signal from noise in a crisp and dichotomous fashion.

So before acts of signification can be a thing, there must be a spectrum of differences to be thus divided into its opposing classes. And where does this spectrum arise in a fashion that can leave it also the generalised indifference which is merely the noise against which the difference that makes a difference stands out?

Confused yet? :lol:

Primal difference must be resolved back to an indifferent sameness that allows meaningful difference to exist – be actualised – as a second order contrast.

Its thermodynamics. Let loose a bunch of particles in a box. They all have different velocities and directions. In time, they will still all be moving differently, but they will have collectively arrived at some constant macrostate equilibrium. You have a baseline of indifferent difference that can now support a measure of significant difference. A negentropic fluctuation.

Whatever linguistic distinction that Goodman might want to make in terms of how the world is can now be made in a measurable fashion. We can have the particularity of actual events as we have the generalised anonymity of an indifferent ground – a state that is concrete and actual too.

So maybe you don't even realise that to make a mark on reality, you need a stable surface on which such a mark can endure. There are steps to take to reach the place in which you want to set up your metaphysical camp.

Goodman's approach is concrete and clear. Yours is abstract and poetic.

A discrete classification in no way has to imply a continuous one. — bongo fury

Goodman can kiss my arse then.

Fine. But it's not easy to axiomatize real analysis that way.

One can philosophize all day about how one thinks mathematics should be. But other folks will ask "What are your axioms?" They ask because they expect that an alternative mathematics should have the objectivity of set theory, which is utter objectivity in the sense that, by purely algorithmic means, we can definitively determine whether a purported proof is actually a proof. — TonesInDeepFreeze

I agree. IF there is any merit to my view, then the hard work hasn't even begun.

Is it possible for a continuum to exist and be defined mathematically without relying on numbers? — keystone

I don't know whether a geometric theory can pick out one particular line and definite it as 'the continuum'? I'm too rusty on the subject.

I'm referring to a curve (1D continuum), surface (2D continuum), etc. — keystone

Thanks.

The hotel simply has actually infinite rooms. Do you think it's a gross misrepresentation of infinite sets? — keystone

Hilbert's Hotel is an imaginary analogy that seems fine to me.

A bit of magic is needed to make the leap from a finite collection of points forming nothing to an infinite collection of points forming a continuum.
— keystone

As I mentioned, that is not how it is done. You would do yourself a favor by reading a good textbook on the subject so that you would have a basis to critique the actual mathematics rather than what you only imagine is the actual mathematics. — TonesInDeepFreeze

Why can't we talk naively about points combining to form a line? It seems a little disturbing that to discuss what is seemingly a very simple concept requires significant training.

Keep in mind that no contradiction has been found in ZFC.
— TonesInDeepFreeze

Most notably Hilbert's paradox of the Grand Hotel, but also the following:

Gabriel's horn
Galileo's paradox
Ross–Littlewood paradox
Thomson's lamp
Zeno's paradoxes
Cantor's paradox
Dartboard paradox — keystone

Yes, those are paradoxes. But my point is that they are not contradictions in ZFC* (and I'm not claiming that you claimed that they are contradictions in ZFC).

Zeno's paradox is actually resolved thanks to ZFC (I mean thanks to ZFC for providing a rigorous axiomatization for late 19th century analysis).

Cantor's paradox was met by ZFC by not adopting unrestricted comprehension.

Galileo's paradox strikes me a "nothing burger". I am not disquieted that there is a 1-1 between the squares and the naturals.

Gabriel's horn. I don't know enough about it.

Ross-Littlewood. Another limit problem. Doesn't bother me. Indeed, set theory provides a framework for rigorously distinguishing between terms of a sequence and the limit of the sequence .

Thompson's lamp. A non-converging sequence, if I recall. Again, rather than this being a problem for set theory, it's a problem that set theory (as an axiomatization of analysis) avoids.

Dartboard paradox. I don't know enough about it.

/

Tarski-Banach. I'm not expert on it. But my impression is that it strikes as paradoxical only when we overlook that points are not physical things. Points are abstract, used in a conceptual armature to "model" physical things but we don't contend that those physical things are actually made of abstract points. At least that's my naive layman's take on it.

/

But Lowenheim-Skolem. The problem for me is that I'm not sure that my write-up to myself about it is correct in all details. But even with the technical explanation, it does raise for me some puzzlement.

/

* A contradiction in ZFC would be a theorem of the form:

P & ~P

No such theorem has been shown in ZFC.

The existence of the set of natural numbers is derived axiomatically. Granted, the key axiom is that there exists a successor inductive set, which is an infinitistic assumption. — TonesInDeepFreeze

Wikipedia: The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC), but only by showing that it follows from the existence of the natural numbers.

I feel like you could give me a little more slack here on my phrasing.

On the other hand, the notion of "potential infinity" demands alternative axioms.

Take just the non-infinitistic axioms of set theory. What axioms does the "potential infinity" proponent add to get real analysis? — TonesInDeepFreeze

I cannot answer this question.

My point is that your description is not an accurate or even reasonable simplification of how set theory proves that there is a complete ordered field and a total ordering of its carrier set. (The carrier set is the set of real numbers and the total ordering is the standard less-than relation on the set of real numbers.)

I feel like you could give me a little more slack here on my phrasing. — keystone

Your phrasing struck me as polemical and misleading by saying "magic" and "leap", which does not do justice to the fact that set theory is axiomatic, and while the set of naturals is given by axiomatic "fiat", the development of the integers, rationals and reals is done from the set of naturals in a rigorous construction.

you would never try to provide an infinite list of points to completely describe a line (Cantor)
— keystone

Cantor doesn't do that. In fact, Cantor proved that that CAN'T be done. It's his MOST famous result.

You have it completely backwards.

What articles have you read about Cantor that have led you to your terrible misunderstandings? — TonesInDeepFreeze

I should have explained explicitly what I meant when I wrote "(Cantor)" as you interpreted my intention backwards.

"Most notably Hilbert's paradox of the Grand Hotel, but also the following:

Gabriel's horn
Galileo's paradox
Ross–Littlewood paradox
Thomson's lamp
Zeno's paradoxes
Cantor's paradox
Dartboard paradox"

The Diagonal Paradox can be extended in principle to any curve in 2D. For example, a circle of radius 1 has a circumference of 2pi, but if I apply my system of sine curves to the circumference I find that as they converge uniformly to the existing circumference, their lengths tend to infinity. Hence I am staring at what appears to be the simple circle I began with, but I now have one with infinite circumference, and hence infinite area.

Thus infinity is everywhere in plane geometry where it shouldn't be.

I consider the Paradox an aberration that results from collapsing one dimension to a lower dimension in certain circumstances and insignificant although bizarre. But Wolfram claims that this crops up in Feynman diagrams. It goes to the very nature of lines and points.

IF there is any merit to my view, then the hard work hasn't even begun. — keystone

Of course, non-infinitistic systematizations for mathematics are interesting and of real mathematical and philosophical import. And there are many systems that have been developed. Personally though, I am also interested in comparisons not just on the basis of having achieved the thing, but also in how complicated the systems are to work with, the aesthetics, and whether fulfilling the philosophical motivations are worth the costs in complication and aesthetics.

I should have explained explicitly what I meant when I wrote "(Cantor)" as you interpreted my intention backwards. — keystone

I think I see now. You didn't mean that Cantor claims that we can list the points in the line, but rather Cantor showed that we can't do that?

If you let me know that the above is correct, then I should retract what I said earlier.

If you want to argue for potential infinities over actual infinities, then the real world is surely the better place to test your case.

Arguing against maths using physicalist intuition becomes Quixotic if maths simply doesn’t care about such things. Physics at least cares. — apokrisis

I agree about the importance of the real world, and perhaps investigating the sub Planck scale is important for the deepest insights. I just don't think it's required here. Maybe I'm wrong.

What I have said is that - as the history of metaphysics shows - there are two camps of thought about the physical world. Broadly it divides into the reductionism of atomism and the holism of a relational or systems approach. — apokrisis

Thanks for this detailed explanation. These concepts are brand new to me. I would think that my views fall within holism.

You can claim to have no problem with an infinity of cuts and yet have a problem with an infinity of points. — apokrisis

I have a problem with an infinity of anything, including cuts. I believe that the only thing that is infinite is potential.

I would say the 0D point and truncated interval are in the same class of question-begging objects. Both are atomised entities lacking a properly motivated existence. — apokrisis

Okay, I accept that substance (continua) and void (0D points) and are both fundamental!

Okay, I accept that substance (continua) and void (0D points) and are both fundamental! — keystone

I'm thinking of something more irreducibly complex. A dimensionality that is "completely" void can't help but have some residual degree of local fluctuation. And likewise, a dimensionality that is "completely" full, can't help but have some residual degree of fluctuation – but of the opposite kind. Particles can appear in the coldest vacuum state. But holes or local voids can appear in the hottest vacuum states.

So you have here a system – like the Universe – with its reciprocal Planck cut-off conditions. Finitude that seals both its ends. The hot Big Bang is where there is so much of everything that there is no room at all for local somethingness – except as the smallest void-like fluctuations. Some fleeting patch of coolness.

This becomes the eternal spawning multiverse of inflation, for example, where the inflation field rages, but here and there, by a quantum fluctuation, some spot cools just slightly and that results in a phase transition. Another bubble universe – such as our Big Bang – starts to form at that place.

And then that nipped off bud of dimensionality keeps growing, keeps cooling and expanding, until it eventually flips over into its de Sitter state of a vast void – a space as cold and empty as it can quantumly get. And now it is the expression of the opposite thing of a nothingness with its residual minimum entropy particles. The void is now the hole that hosts the faintest possible sizzle of its own blackbody radiation – photons so cold that their wavelength is about 38 billion lightyears.

The point is that mathematics can conjure up all kinds of models based on simple premises. It can just take concrete starting points for granted, and take the resulting paradoxes as something to either ignore or even be a little proud of.

But physics is now pushing maths rather harder. It is time to be a bit more serious about eliminating those confusions. It is time to stop being so content with a reductionist metaphysics and to get serious about the modelling of holistic reality.

Physics and cosmology are highly concerned about how the Universe could exist – how finitude could be extracted from potential infinity.

Space, time and energy all look to have had a definite start at the Planckscale that defines the dimensionality of the Big Bang. The continuum was born of one cut-off that could separate the nothingness that could be found in the everythingness. The fluctuations that were the holes.

And then this dimensionality – by expressing the reciprocal actions of spreading and cooling – is on the long path to the other end of space, time and energy. The story gets inverted. The cosmos has become "all hole" – the largest nothingness – with only the faintest possible remaining sizzle of particle fluctuation.

So maths can have its petty wrangles over how to model infinity. It's inconsequential. But to the degree that the interpretation is holistic, it is going to be on the right side of history. And so the intuitionists and finitists feel more correct for that reason.

However what really matters – if we are interested in models of reality as it actually is – is the fact that finitude can be extracted from pure unboundedness. Closure can be extracted from openness ... if that openness is also being transformed from a vagueness to generality (in Peirce-speak).

The universe can exist as it is making the heat sink that it is falling into.

Although there are still big questionmarks. We still seem to need eternal inflation at the front end as a kind of somethingness to get the Big Bang ball rolling, and dark energy at the back end as also a kind of somethingness to deliver the de Sitter state that ensures an eternalised Heat Death cut-off at the other end.

The metaphysical riddle isn't yet solved. However the physics of the residual "somethings" has become highly constrained. And overall, they point to a holistic or pansemiotic view of existence – the triadic systems story where the container and its contents co-emerge from unbounded potential.

The small grows large. The hot grows cold. Symmetries are broken in ways that are themselves symmetric. By heading to infinity in either of these directions, you encounter the infinitesimal as a consequence.

The universe has an irreducibly complex generator in that it is a triadic and recursively self-referencing knot of relations. And any proper notion of a continuum would have to pull off that trick too.

The problem here is that the real number line is the mathematical object that was in question, surely? So as a construction, it hosts both the rational and the irrational numbers as the points of its line. — apokrisis

I would argue that the 'real number line' should instead be called the 'real line' since it's composed of more than just numbers. Consider the proof that sqrt(2) is an irrational number. I would argue that the proof only demonstrates that sqrt(2) is not a rational number and that something beyond rational numbers must exist on the real line. It does not prove that sqrt(2) IS a number. I believe that irrationals are algorithms which describe this mysterious other object - continua. For example, if we conventionally said that two curves intersect at a point with irrational coordinates I would say that they intersect but that we cannot precisely determine the coordinates of that point. All we could do is use those irrationals to identify a point with rational coordinates arbitrarily close to the intersection point. To me, this is what we do in practice. We can go down the wrong path (philosophically at least) in assuming the existence of an object that is, in principle, beyond our reach. In QM we have come to accept a certain level of uncertainty. Why can't we do the same in math?

And so the claim becomes that reality has a fundamental length – the unit one interval. — apokrisis

I don't believe there is a fundamental length since any length can be divided. If there is any fundamental unit related to rational numbers it would be the unit step along a branch in the Stern-Brocot Tree. With each step down the tree we add an L or an R to the string representation of the number above it.

I appreciate that you are using a lot of physics analogies here but I feel like you've gone to far. Your explanation (involving higher-dimensional ratios, virtual particles, etc.) seems to complicate things far more than it simplifies.

Consider the proof that sqrt(2) is an irrational number. I would argue that the proof only demonstrates that sqrt(2) is not a rational number and that something beyond rational numbers must exist on the real line. It does not prove that sqrt(2) IS a number. — keystone

We do prove "sqrt(2) is a [real] number".

More exactly:

We prove that there is a unique positive real number r such r^2 = 2, and then we prove that r is not the ratio of two integers.

In QM we have come to accept a certain level of uncertainty. Why can't we do the same in math? — keystone

I wouldn't argue that we can't. I suppose people already have made logic systems with values such as 'uncertain' that can be be applied to a different mathematics. And I can imagine that certain scientific enquires might be better served by such systems.

But that doesn't erase the rewards meanwhile of classical mathematics.