Adding to my response about the particular paradoxes. Even if we granted that they indicate flaws in the concept of infinitude, then that is a concept of infinitude extended beyond set theory into imaginary states of affairs for which set theory should not take blame. Those paradoxes don't impugn set theory itself.

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. — apokrisis

It's not about me. It's about computers in general. I can imagine a computer picturing a 4D hypershere as easily as a sphere. And if you don't accept that computers can picture things, then I can imagine a mind that lives in a 4D universe that can picture a 4D hypersphere as easily as a sphere. I can imagine a computer of arbitrarily large capacity and processing power, but I can't imagine an computer with infinite capacity and processing power.

I can’t picture a cut which doesn’t result in a gap. — apokrisis

Here's an analogy that closely relates to how I see it. Consider a magic-eye (stereogram) puzzle. In this analogy, the printing on the page is the continuum and I, the observer, am the computer. When I look at the page I never physically do anything to it. However, if I look at it just right, pieces of it appear to float above the page and form an image. The interaction between the observer and the page (the computer and the continuum) result in a beautiful outcome. In this analogy, the interaction is the act of cutting. Mathematics occurs when 'computers cut continua'. But then a moment later I get disctracted and the image vanishes. All that's left is the unobserved page and the observer.

But for the analogy to more closely align with my view of math, each time I look at the page a totally different image could pop up. The page contains the potential of infinite images, but only one image is actualized at any moment.

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? — apokrisis

Returning to my magic eye analogy, what actually exist are the page and the observer. For a brief moment a single image pops out and contingently exists as well. If all potential images popped out simultaneously then the whole page would pop out resulting in no image at all. So while a cat could potentially pop out, if it's a dog that does actually pop out, then the cat doesn't actually exist...not now. And so, there can be no set of all images.

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 — apokrisis

That is true.

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. — apokrisis

Returning to my magic eye analogy, the image pops up only because the background does not. Can the image and background be the 'measure of the other' that you're referring to? If so, then that makes sense to me.

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

I know one can easily imagine imagining the hotel (i.e. that above each floor there is another floor) but can you imagine the actual endless hotel as a whole? I'm trying to get at whether you can imagine a set of all natural numbers.

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). — TonesInDeepFreeze

I will have to trust you on your claims related to ZFC since I don't have a good understanding of it.

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

What was the 19th century analysis resolution to Zeno's paradox?

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

Hopefully you don't mind returning to Hilbert's hotel since I'd rather work on actual objects then relationship between sets. Are you not disquieted that a subset of rooms is equinumerous to the full set of rooms?

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

Wikipedia: If a dart is guaranteed to hit a dartboard and the probability of hitting a specific point is positive, adding the infinitely many positive chances yields infinity, but the chance of hitting the dartboard is one. If the probability of hitting each point is zero, the probability of hitting anywhere on the dartboard is zero.

Are you not disquieted that a probability of 0 does not mean impossible?

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. — TonesInDeepFreeze

Achilles travels half the distance from A to B in 1 second, half the remaining distance in 0.5 seconds, half the remaining distance in 0.25 seconds, etc. Does he reach B? The answer is yes, in 2 seconds. This implies he completes an infinite set of actions. However, what happens when he holds Thompson's Lamp and each step switch the state of the light on/off? What's the final state of the lamp? If he completes one infinite set of actions (Zeno) he must be able to complete the other set of actions that are paired with it (Thompson). Without resorting to axioms, does this bother you?

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

P & ~P

No such theorem has been shown in ZFC. — TonesInDeepFreeze

Right. And to be honest I don't even know if I have any beef with ZFC (since I don't fully understand it). I don't think calculus needs actual infinity to work. All I'm proposing is that we reinterpret it and keep the math unchanged. Similarly, I suspect (with no evidence to provide) that ZFC doesn't need actual infinity to work either. Perhaps it just needs a reinterpretation. For example, I have no problem forming a 1-1 relationship between n and the n^2. I just don't think there's an actual set that contains all n (and similarly all n^2). In other words, my qualms are not with the math, they are with the philosophy.

can you imagine the actual endless hotel as a whole? — keystone

Not as a physical object. On the other hand, I know so little of cosmology that I don't know how to dispel my bafflement that the universe could be finite or my bafflement that the universe could be infinite.

whether you can imagine a set of all natural numbers. — keystone

Yes, I do conceive it clearly. I conceive the abstract property of being a natural number. Then I conceive the set of all and only the natural numbers to be the set that pertains to all and only those things having said property. I know what the property is; so it takes only a shift to conceive of a set that corresponds to the property. (Of course, that can't be applied always, lest we get contradiction from unrestricted comprehension. In that case, my naive intuition just needs to adjust to accept restriction.)

That is not at all an argument that there exists such a set. It's only a description of my own intuition.

/

Or another view: If there is any mathematical reasoning that can be considered safe, then it's manipulation of finite strings of objects or symbols (whether concrete sticks on the ground, or abstract tokens). In that regard, I can see the formal derivation in Z set theory of the theorem that we read off as "there exists a set whose members are all and only the natural numbers", though the actual formal theorem doesn't have English words like that. Then, in some a worst case scenario, some crisis where my ability to conceive abstractly is terribly diminished, if I really had to, I could fall back to extreme formalism by taking the theorem to be utterly uninterpreted, but a formula nevertheless to be used in mathematical reckoning.

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.) — TonesInDeepFreeze

I'm not sure what you're referring to.

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. — TonesInDeepFreeze

In Hilbert's full Hotel, move guest 1 to room 2, guest 2 to room 3...yada yada yada...it can accept an additional guest. Nobody explained the paradox with this phrasing, but I think this captures my frustration. I feel like the 'yada yada yada' skips over the most important part. It is in this sense that I feel like some magic is happening. I understand the standard explanation that since there is no last room each guest moves to a new room, but consider an alternate interpretation: in the first step 1 there is a dislodged guest, in the second step there is a dislodged guest, in the third step there is a dislodged guest, how does the yada yada yada result in no dislodged guests.

What was the 19th century analysis resolution to Zeno's paradox? — keystone

Infinite summation: convergence of an infinite sequence to a limit.

Are you not disquieted that a subset of rooms is equinumerous to the full set of rooms? — keystone

I don't conceive of an infinite set of physical rooms.

As to sets, I already mentioned that I am not bothered that the squares (a proper subset of the naturals) is 1-1 with the naturals.

/

Dartboard paradox. I'm rusty in probability theory. I'd have to go back the books to refresh myself.

Thompson's Lamp and Zeno's paradox. I already addressed those. I don't have more to say about them.

I don't think calculus needs actual infinity to work. — keystone

It does in its common form.

But there are non-infinitistic systems too. I know a little about them, but not enough to say how well they perform.

I suspect (with no evidence to provide) that ZFC doesn't need actual infinity to work either. — keystone

It wouldn't be ZFC then.

We could delete the axiom of infinity, but then we don't get analysis.

Or we could negate the axiom of infinity, but then we don't get analysis but instead a theory inter-interpretable with first order Peano arithmetic.

I have no problem forming a 1-1 relationship between n and the n^2. I just don't think there's an actual set that contains all n (and similarly all n^2). In other words, my qualms are not with the math, they are with the philosophy. — keystone

The axiom of infinity and the results from it are mathematics. If you want a mathematics without the theorems that we read as "there exist infinite sets" and "there exists a set whose members are all and only the natural numbers", etc., then that is not just philosophical but also mathematical

consider an alternate interpretation: in the first step 1 there is a dislodged guest, in the second step there is a dislodged guest, in the third step there is a dislodged guest [etc.] — keystone

The imaginary hotelier can do that also.

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.)
— TonesInDeepFreeze

I'm not sure what you're referring to. — keystone

I'm referring to the fact that set theory proves there exists a complete ordered field and a total ordering of its carrier set. And then I highlighted that the carrier set is the set of real number and the total ordering is the standard less-than relation on the set of real numbers. I don't know how to put it more simply. Other than perhaps to add that that set together with the ordering is the continuum. I don't recall whether I mentioned it earlier in this thread, though it is utterly basic to discussion of the continuum.

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. — jgill

Hah! In the spirit of the infinite fractal coastline paradox. Nice paper.

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. — jgill

No, I think it is significant and general. To relate it to my own interests – the Cosmos – there is this same virtual vs actual wrangle over renormalisation in quantum theory. Either renormalisation is some horrible kluge to be eliminated by better maths ... or in fact finitude and its dichotomous cut-offs have to be brought into the maths of the infinite somehow.

What I would point out again is how complex numbers may have their Penrosian "magic" as they speak simultaneously to the symmetries or conservation principles of rotation and translation. The incommensurability arise at that point – the foundational distinction between spinning at a spot and moving away in a straight line.

A 0D point could be spinning, it could be moving. It has no context, so we can't yet tell. It is technically vague. No symmetries have been broken and so no symmetries have been revealed.

But as soon as we imagine anything happens, the foundational symmetry breaking is the Noether symmetries that close the world for Newtonian mechanics.

Newton starts the world already in motion. The first derivative. The zeroeth doesn't really exist even at a Gallielean level of relativity. And this Newtonian world is defined by its twin inertial freedoms – to rotate and translate. One keeps things anchored at locales. The other sends them moving and so carving out the global largeness of space.

This would be why you find that converging to a limit has this fractal coastline property. Rotation and translation want to be different. They exist by being incommensurate.

It is the old squaring the circle story. Pi is irrational as curves and straight lines are at root incommensurate. They have to be that to co-exist in the same world and so be the symmetry-breaking making that world.

Complex numbers then speak to how the two directions of free action – one closed and cyclic, the other open and expansive – can be united under a unit 1 symmetry description. They explain how the symmetry is selectively broken in a way that is special to a 3D continuum. You get the chirality and commutative order arising as something that materially matters – unique in that it can produce those local knots, or Newtonian particles, that can't unknot, in the style modelled by twistor theory or fibre bundles..

So it is not a mistake resulting from the simple collapse of a higher dimensionality into a lower one. It is about a complex collapse – a reduction of dimensionality to a 3D continuum that then sets you up in a world where the number of spin directions finally match the number of momentum directions.

The collapse produces the internal gauge symmetry where rotation and translation become the new thing – a breaking of the symmetry achieved in 3D. You can get a cosmos founded on Newtonian mechanics with the Noether dichotomy to close it as a world, make it safe for local knots of energy that can't come untied, just endlessly shuffled about like bumps in a rug.

In short, you can collapse dimensionality to SO(3) and discover it then spits out SU(2). The magic happens. Commutative order becomes a thing. Time is born. Space is anchored in a way that can be described symmetrically by spherical or Cartesian coordinates. Etc, etc. :grin:

And what of the cut off issue? I'm thinking this gives an argument in terms of the Planck scale representing a dimensional ratio.

The problem with infinities and infinitesimals is this urge to give them a concrete value. Or if not that, then they are processes without bounds. And neither answer is truly very satisfying in the light of physical reality.

The point about the Universe is that it is a story of fractal dimensionality or scalefree growth. The universe exists as a log/log story of cooling and spreading. This is both open and closed in some sense. And I've explained that in terms of it being an inversion at a deeper level.

It is going from very hot to very cold by going from very small to very large. Something is increasing as something is reducing. And if we look at a light ray, we see it is the same thing from opposite perspectives. Light waves – as your simplest sine wave or helix (the helix making the rotation~translation deal explicit) – are both stretched to their maximum possible extent and also redshifted to their maximum possible extent. They become as big as the cosmic event horizon and as cold as absolute zero.

So while we do like to measure all this using yardsticks like rulers and thermometers, it is essentially a dimensionless ratio. The Planck triad does not stand for some actual independently measurable number. It stands for a Platonic ratio - just like e scales unit 1 growth or pi scales unit 1 curvature. The cut off becomes simply the point where local spin and global translation first measurably come apart. And this is a qualitative distinction, not a quantifiable one.

The same goes for the Heat Death where everything arrives at its other end where the Planck triad are inverted - 1/planck - to give the cut off marking the other end of the effective 3D continuum. (Again, inflation and dark energy are unsolved aspects of this view.)

And this is why I argue for the numberline in these same terms - as the infinitesmal and the infinite as each other's reciprocal measure. A way to have the small and the large being both unboundedly open and yet also finitely closed.

The missing bit is that the numberline is based on naked spatial intuition. Peirce and Brouwer were trying to bring in time and energy (the two being complementary under quantum uncertainty) as the way to achieve the trick of an "open closure". Sure you can count forever. But that then takes time an energy.

You have two actions in opposition. This means that you can only expand until you run out of energy. And you can only contract until you run out of space.

Adding this to the numberline conception builds in the self-limiting finitude that the conventional spatial version lacks.

As I've said, Turing computation has run into a similar story of physical constraint. Infinity only reaches so far in a world where the gravity of being a hunk of circuits curling up into a black hole at a certain critical energy density – a cut off point.

So rather than either/or when it comes to taking sides on virtual infinity vs actual infinity, I've been pushing "both" in the dichotomistic sense for this reason. Openness and closure are what must emerge as themselves something that co-arise from the firstness of Peircean vagueness and become a combined continuity within the thirdness of Peircean generality.

A tricky business.

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. — jgill

No, I think it is significant and general. — apokrisis

One way of perceiving the sum of all paths in Feynman's path integral for paths that consistently move towards a target is to have the particle pass through a series of finite parallel plates, each with "portholes" where the particle pops through, from one plate to the next in a system where both the number of holes and the number of plates increases without bound, with plates being squeezed closer and closer together.

A path then is zig-zag, roughly like the original diagonal paradox except with random configuration. If all possible paths of this type arise, then there will be some with lengths growing towards infinity that are nevertheless contained in a closed and finite environment and go from left to right, let's say. Like the sequence of contours in my note. One meter or infinite meters? The "same" path. :chin:

I believe that irrationals are algorithms which describe this mysterious other object - continua. — keystone

I've already raised this point, asking if sqrt(2) is the exception or the rule. Higher dimensional generators could produce generators of some number that looks to be an irrational point of the line. But then these numbers - growth constants like e and phi – are ratios and so are dimensionless unit 1 values more than they are some weird real number.

The status of any regular irrational seems different. They would lack generators apart from decimal expansion. Something else is going on.

I don't believe there is a fundamental length since any length can be divided. — keystone

Can the Planck length be divided? Not without curling up into a black hole.

What you believe and what the Universe would like to tell you seem two different things. Who wins?

I appreciate that you are using a lot of physics analogies here but I feel like you've gone to far. — keystone

Reductionism in either maths or physics is showing its age. I am just exploring the holism that would give the larger view. And in being triadic, that is irreducibly complex. Tough to deal with perhaps. But it is what it is.

I can imagine a mind that lives in a 4D universe that can picture a 4D hypersphere as easily as a sphere. — keystone

Yea, nah. I'm not buying these feats of your imagination.

In this analogy, the interaction is the act of cutting. — keystone

That's taking a point of view. Performing a figure~ground gestalt discrimination. So it would indeed be a good analogy of how an observer reads information into the world. It is how the brain sees reality in terms of collections of bounded objects.

We are getting towards Peirce who completed Kant's project of creating a metaphysics that begins in the world-constructing mind. But we can only go far taking this semiotic path before wondering how to take the step to a pansemiotic view – the one where the world is "thinking itself" into definite being in ontic structural fashion.

If all potential images popped out simultaneously then the whole page would pop out resulting in no image at all. — keystone

Well the trick is that there is enough contour information to allow only the one possible stereoscopic reading once the brain has filtered out all the pixillated noise of the random colours. The hidden rabbit or seagull is merely hidden while the brain finds a way to suppress the shapelessness of the coloured pixels from the intelligibility of a depth perception-based contour.

So as a stimulus it is fixed - designed to play off two different boundary constructing visual subsystems. We are being informed of an incoherent surface by the random field of colour pixels. We have to stop looking at it as a flat incoherent surface to find the quite different depth-based reading.

The page contains the potential of infinite images, — keystone

The problem is that it doesn't. It plays on a dichotomous rivalry of brain subsystems. You have to switch off the one and employ the other. The search is for the single hidden interpretation. Only one of the two points of view can spot it.

Can the image and background be the 'measure of the other' that you're referring to? If so, then that makes sense to me. — keystone

That is the standard logic of Gestalt psychology. Figure and ground arise as a holistic calculation. The whole brain is organised by this contrast-creating logic. Neuroanatomy is a collection of useful dichotomies or symmetry breakings we can impose on the world to make in intelligible.

This is why the brain is not a computer. It works holistically. It imposes contrast so as to separate the world into signal and noise. It doesn't crunch data. It constructs meaning by suppressing randomness. To be attentively focused is to have defocused on everything else.

So some things about your magic eye analogy hold. It certainly explains why reductionism has such a grip on our psychology. The brain is so skilled at ignoring backdrops and seeing only the foreground events that we might indeed believe that suppressing meaningless noise is an effortless and costless mental affair.

But the brain has vastly more inhibitory connections than excitatory ones. It has to work on ignoring as much of the world as it can so as to then see it in an object-oriented fashion.

Putting a finger on an irrational value has this sort of extreme cost. Unless we have some generating algorithm to shortcut the whole decimal expansion, we just have to plough on using infinite resources. Every next digit is just as much a surprise at the last. Even if we might be thinking after even five decimal places, well how many more do I pragmatically need here.

So your general approach – to root the question in actual psychology – is right. And that is Brouwer/Peirce for you. There is a cost to decimal expansion where there is no shortcut algorithm. This is the Kolmogorov complexity approach I mentioned. What could be more pointless(!) than a numerical value without the constraint of a generator?

.
.
.
IF f(n) $\geq$ MAX THEN f(n) = MAX
.
.

A line of code in our (simulated) universe where f(n) is any function on n (any number) and MAX is the largest number permissible within the simulation.

Are you not disquieted that a probability of 0 does not mean impossible? — keystone

All impossible propositions have probability 0, but not all probability 0 propositions are modally impossible. For instance, the probability of number being picked by randomly selecting a random
real number between 0-5 is zero, but a number will be selected. The fact that a number will be selected is not impossible, in fact, it will actually occur in this situation.

Yes, this will seem very counterintuitive. The simplest way I can explain it in a non-technical fashion is that selecting any non-zero probability for each number will force us to add up way over 100%, because there are infinitely many other "participants" (numbers), which means the only probability we can assign to each participant is zero.

There's actually a way out of this being nonstandard analysis giving us infinitesimals and particularly nilpotent infinitesimals in several hyperreal systems (*R). These can be very elegant in several applications including this one, where we can include numbers so small that even their squares (n^2) are zero but the numbers themselves, n, may not be.

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. — TonesInDeepFreeze

If the infinitistic systemization for mathematics are more powerful, beautiful, and simple then there's not much appeal to a non-infinitistic systemization. As mentioned earlier though, I do wonder whether our infinitistic systemization can simply be reinterpreted from being based on actual infinity to being based on potential infinity. However, I'm not far enough in my learning journey to answer this.

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. — TonesInDeepFreeze

The above is correct.

https://thephilosophyforum.com/discussion/comment/737931

That message is retracted by me.

If the infinitistic systemization for mathematics are more powerful, beautiful, and simple — keystone

I don't know definitively that that is the case, but it seems to me to be so.

I do wonder whether our infinitistic systemization can simply be reinterpreted from being based on actual infinity to being based on potential infinity. — keystone

There are systems - such as intuitionistic* ones and others - that are said to embody use of potential infinity. But, personally, I don't know of one that can define within the system 'is potentially infinite', while set theory does define 'is infinite'. So, at least as far as I can tell, saying 'potentially infinite' is not yet, at least, a formalized notion but rather a manner of speaking.

* Though intuitionistic set theory does have an axiom of infinity. The problem though is that we can't directly compare statements in intuitionistic theories with those in classical theories, because the semantics are so radically different.

So, at least as far as I can tell, saying 'potentially infinite' is not yet, at least, a formalized notion but rather a manner of speaking — TonesInDeepFreeze

I had never heard that expression in a mathematics discussion before joining TPF.

Whether by that name or not, the idea goes at least as far back as Aristotle.

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. — apokrisis

In terms of using the truths of physics as a guide in the hunt for truth in mathematics, I'm totally with you. However, I feel like you're mentioning a lot of physical phenomenon but not explaining clearly how they relate to mathematics.

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. — apokrisis

I'm not puzzled by unboundedness. I can draw a line with open ends on a piece of paper and label the ends negative and positive infinity. This unbounded object is entirely finite.

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 — apokrisis

It appears that you are looking at the universe from a point-based perspective in that there's a first instant which is followed by the next instant, and so on. Just as points cannot form a line, instants cannot form a continuum of time (i.e. a timeline). We must start with the entire timeline of the universe and only then can we make cuts in the timeline (i.e. observations/measurements) at different points in time to actualize reality (i.e. collapse the wave function of the universe). And if the timeline is like the one I just drew on the paper above, it is meaningless to talk about what existed at t=0 because nothing actually existed at that time.

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

My impression is that we do not prove sqrt(2) is a number, but instead we assume it is a number by means of the completeness axiom.

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. — TonesInDeepFreeze

One would be crazy to say that classical math is wrong. I'm in no position to say this with certainty, but I believe that our finite descriptions of any given real number is valid and useful. Similarly, our finite descriptions of any given infinite set is valid and useful. I just don't think the real number or the infinite set itself actually exists, nor does it need to. We never work with the decimal expansion of pi, nor do we work with the infinite set itself (e.g. by explicitly listing all its elements). We always work with the finite descriptions. We work with algorithms. And if we want to work with an actual number, we halt the potentially infinite algorithm to generate a rational number and use that in our computation. I believe this is what classical mathematics does. At least that's what engineers do.

However, I feel like you're mentioning a lot of physical phenomenon but not explaining clearly how they relate to mathematics. — keystone

One could always be clearer. But then there is also the issue of how much you are equipped to understand. I can't be held responsible for all the work you might need to do.

However I'm not complaining. At least I'm make progress on making my ideas clearer to myself at times. :grin:

I can draw a line with open ends on a piece of paper and label the ends negative and positive infinity. This unbounded object is entirely finite. — keystone

You can draw a sign that you then interpret in a certain habitual fashion. The issue then is how does this sign relate you to the reality beyond. Does is create a secure bridge? Or is it wildly misleading?

It appears that you are looking at the universe from a point-based perspective in that there's a first instant which is followed by the next instant, and so on. — keystone

Nope. I've said I'm starting the world from its Planckscale cut off. To start the world from a point would be to start it from the singularity where all physics has been scrambled to nonsense.

And the logic is the same whether we are talking in terms of fundamental intervals or fundamental durations.

My triadic systems approach says intervals and durations are irreducibly complex entities. They are born not as some monistic given but from an emergent, self-referencing, dichotomy.

And this is Planck scale physics. It is what the maths says.

How do we measure time? In the usual approach we spatialise it. We call upon the fundamental contrast between a rotation and a translation. We imagine a length that is constrained to a self-repeating local cycle – a clock hand that goes round in a local circle. And then we use that locally fixed symmetry as the standard unit to then create some unbounded sequence of length intervals. Every time the big hand completes one sweep, the day adds another hour of temporal distance travelled.

And there is nothing in this mental picture to say why it can't continue forever. Physical change has been anchored at one of its ends by being made to spin forever in the one spot. And that then frees up the possibility of change being turned into an infinity of time steps in its other direction. Globally, time becomes as infinitely large as it likes.

So you can have a proper definition of an instant or an interval only when you have nailed the rather ambiguous or vague notion of "a change" to a dichotomous coordinate system – a symmetry breaking. One end has to be fixed. You do this by creating a cycle that just goes round and round the same 0D point. The other then is allowed to flap completely free. It can follow a straight line forever as a straight line is simply an endless repetition of steps that never repeat rather than the other thing of an endless repetition of steps that only repeat.

A moment in time – a durée - thus is an irreducibly complex object. It combines rotation and translation to create the emergent thing of a "fundamental time step". And length intervals are also the same trick. They just get stripped of the philosophical nod to energy and change (the things time must measure) and become a story of the 0D point and its 1D line.

At least with clocks you can see how repetition and difference are the partners in crime. The idea of the spatial interval becomes shrouded in mathematical mystery.

But if we add back the physics – the irreducible quantum indeterminacy found in nature – then the clock of the length interval becomes visible. Any point must vibrate. It will have a resonant motion ... because QFT says so. And we can see the reciprocal relation between location and momentum as plain as your face in Heisenberg's uncertainty principle.

Nature has a fundamental frequency. Physics says it is so. A systems metaphysics says it is only to be expected.

If maths has been left behind in this grand and still unfolding adventure, tough shit.

My impression is that we do not prove sqrt(2) is a number, but instead we assume it is a number by means of the completeness axiom. — keystone

(1) In set theory, there is no completeness axiom. Rather, we prove as a theorem that the system of reals is a complete ordered field.

(2) We assume axioms. (Or, in another view, we don't even assume them but rather merely investigate what their consequences are.) The theorems we derive from the axioms are not "assuming by means of". Okay, in a broad loose way of speaking, someone might say that the theorems are essentially just "assumptions" unpacked from the axioms. But that really muddies the matter terribly. Granted, everything we prove is, in a sense, "already in the axioms", but that obscures:

Yes, often we adopt axioms to prove the theorems we already know we want to have. But so what? That is, as they say, a feature not a bug of the axiomatic method.

And any alternative mathematics that is axiomatized is itself going to have that feature. So there's no credit in faulting set theory in particular for that.

And yes one might want for the axioms to be intuitively correct ("true") even if the theorems might be surprising. And with set theory, people's mileages vary. I find the axioms of set theory to be exemplary in sticking to only principles that are in concordance with the intuitive notion of 'sets'.

(3) So getting back to my earlier point: 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. Not the other way around as, if I recall, you suggested.

A moment in time – a durée - thus is an irreducibly complex object. It combines rotation and translation to create the emergent thing of a "fundamental time step" — apokrisis

?? Bergson's time intervals? What? Rotation & translation = the two spools? :chin:

Hah, no. Not Bergson. The more general structuralist notion of a hierarchy of scales of spatiotemporal integration. So what Stan Salthe would call cogent moments. And physics would call lightcones.

The cosmos is structured in terms of the scales at which interactions have had time to complete whatever change was going to happen. And beyond that event horizon ate all the possibilities yet to be actualised.

So time has a fundamental grain determined by c. A moment or duree is the completion of a change. And the Planck scale is the size of the smallest such moment.

It is indeed measured by a rotation and translation as it is the the time and space large enough to contain the first Planck scale energy fluctuation - a wave with the Planck frequency. So a single beat of a helical spiral. A sine wave. A rotation and translation that begin at the same size. But then a beat later, the space has doubled and so the energy halved. The fluctuation has already been redshifted. The difference between local spin and global translation is already established.

The fact we measure time with clocks is just showing how general the logic is. We have to have something that changes and yet that makes no difference to be able to measures the changes that do make the difference.

But then really accurate clocks wind up actually using atomic vibrations.

So time has a fundamental grain determined by c. A moment or duree is the completion of a change. And the Planck scale is the size of the smallest such moment. — apokrisis

Food for thought.

Wiki:

Definition of the second
In 1968, the duration of the second was defined to be 9192631770 vibrations of the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom. Prior to that it was defined by there being 31556925.9747 seconds in the tropical year 1900.[18] The 1968 definition was updated in 2019 to reflect the new definitions of the ampere, kelvin, kilogram, and mole decided upon at the 2019 redefinition of the International System of Units. Timekeeping researchers are currently working on developing an even more stable atomic reference for the second, with a plan to find a more precise definition of the second as atomic clocks improve based on optical clocks or the Rydberg constant around 2030.