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.

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.

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.

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 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.

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.

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.

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.

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.

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.

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.

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!

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.

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.

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.

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.

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

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?

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?

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.

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.

So as I argued, continuity is measurable as the absence of discreteness. The fact you can choose to truncate your decimal expansion in search of some specific numberline value only shows you didn’t exhaust its capacity for discreteness and thus also failed to demonstrate it is as securely continuous as you might want to assume. — apokrisis

Let's imagine a line where cuts have been made to mark all rational points (I don't believe this is possible, but let's go with it for now). I believe you cannot mark any more points on this line. If you throw a dart in between the rational points then you will hit an indivisible line segment. That is as discrete as it gets, and even then the line is securely continuous.

What is the dimension of purely empty abstract space? One might say that it is infinite dimensional, another might say that it is 0-dimensional.
— keystone

I think the maths of manifolds and topology would want to give a more sophisticated answer than that. — apokrisis

Agreed. I don't think my statement there was critical to my argument though.

But a string has a width. And so you can eventually chop it so much that the width exceeds the length. At which point, your analogy is in trouble. — apokrisis

True, the element proportions change with further divisions but still you can't cut a string out of existence. I think your argument doesn't attack the essence of my argument.

So rather than finding a point on a line, we create two lines with a cut that leaves them with a point sealing their bleeding ends, and some kind of gap inbetween … that is not a point, just the absence of even points now? An anti-point perhaps? Or what? — apokrisis

If we cut y=0 with the 'knife' x=0, then there is a void between the two newly produced line. We then have line x<0, void x=0, and line x>0. For all practical purposes this void is a point, the only difference is philosophical: that the void is not an object. It is the absence of an object (continua).

The numberline instead always exhibits its twin reciprocal properties of being both limitlessly integrated and limitlessly differentiable. — apokrisis

Both of which are performed on continua, not points.

The numberline could be other. It could be just a swamp of vagueness. It could be a fractal Cantor dust for instance, where you could never know whether you land on a cut or a line. — apokrisis

The cuts are 0-dimensional so they are illusions of convenience. If you throw a dart at the line you will always hit the line, never the cut. The cuts have measure 0 after all. Instead of a number line, let's consider the curves x=0, y=0, and y=x^2 in 2D Euclidean space. These curves cut each other at (0,0) so we have one 'point' in this system. Is there vagueness? Yes. Without constraining the curves at all points the system is more topological than geometrical. But what's wrong with that? When anyone or any computer draws this system it's always imperfectly drawn, but by labelling the curves with their equations, we know precisely what will happen if we make additional cuts.

I’m just arguing that the numberline debate is another example revealing that any holist ontology has to be triadic. — apokrisis

In part you're preaching to the choir since my view is triadic (computers/cut/continua). What is the triad in your view? Is it continuous/discrete/vagueness?

What is the usual problem of an object-oriented ontology that I'm facing?
— keystone

It binds you to a monistic and reductionist conception of nature. — apokrisis

I suppose the triad that I'm proposing isn't solely object-oriented. It has a subject (computer), a verb (cut), and an object (continua).

So zoom in on the Planck scale and you find the same metaphysics I have described. — apokrisis

I don't want to dive into what happens at the Planck scale, in part because I'm not informed enough on the topic and in part because my understanding is that we can't observe the universe at this scale.

@apokrisis

I'm just rereading through your older messages and I have a few additional comments:

• I see neither the point as infinitesimal nor the line as infinite. In terms of length, the point is exactly 0 and the line is some positive number. If you're talking about something of infinitesimal length, you are talking about some tiny line segment. If that's the case then you are only talking about continua.
• infinite = 1/infinitesimal is very problematic for hopefully obvious reasons. Why not write something that makes sense, like 1,000,000 = 1/0.000001. If you do this then it's clear that you're working solely with continua.
• The inverse relation between points and continua is that the point is nothing and the continuum is something.
• Points and continua are not the measure of each other. Nothing cannot be used to measure something (0+0+0+0... always equals 0). Whereas something can be use to measure nothing (e.g. 5-5=0). There is an imbalance here in this relationship suggesting that continua are more fundamental.
• The structure of a continuum is not defined by points, it is defined by an equation(s). Aside from being impossible, you would never try to provide an infinite list of points to completely describe a line (Cantor). You would just provide an equation - a finite string of characters to perfectly describe how points would emerge if cuts are made.
• I don't understand how continua + equations are vague. If I say I'm thinking of a plot containing the curves x=0, y=0, and y=x^2 you know exactly what I'm thinking of.
• We cut and glue continua. We can't do either with points.

See? You started an entertaining discussion that drew in some pretty good thinkers. Probably better than paying a PhD student. :cool: — jgill

This is true, and I sincerely appreciate everyone's responses.

I still don't get it. I don't see any advantage in your way of seeing things. For me, pi is clearly a number. — T Clark

What I'm saying is that pi isn't the string of digits that begin with 3.14. Pi is the algorithm(s) such as the infinite series beginning with 4/1 - 4/3 + 4/5 - 4/7 + .... Since we can't actually complete the computation of an infinite series, we never produce a number. So let's just say that pi is the algorithm. The beauty of the algorithm is that it's definition is entirely finite (I just wrote it in finite characters) and it's execution is potentially infinite (i.e. it would compute to no end). There's no need to say that pi has any association with actual infinity. If we say that it's a decimal number then we must say that it has actually infinite digits.

Seems like you're asking for an abstract, human invention to match up with your understanding of reality. It doesn't work that way. As they say, the map is not the terrain. — T Clark

I don't need mathematics to align with reality. I just think reality has a clever way of avoiding actual infinity and making sense. Reality is a good sign post. If a computer can't do the math (in principle), maybe there's something wrong with the math.

A number is not an object. It doesn't have a physical existence. Also, it's not beyond my comprehension. That way of seeing things has always made sense to me. — T Clark

A number is an object of computation. Computers do stuff with numbers. I don't think you can fill your head with all the digits of the decimal expansion of pi. The best you could do is fill your head with an algorithm for calculating pi. That's what I'm saying exists - the algorithm.

Abstract entities, i.e. all human concepts, are always simplified reflections of the world. I can't think of any that aren't. That's why math is so wonderful. — T Clark

Interesting view. I doubt that many hold this view. I think the traditional view is the inverse, that abstract objects are the ideals and reality is just an approximation of the ideal. Nevertheless, I don't hold either view. The concept of a unicorn is not a simplified reflection of any real world object.

Particles and waves are different kinds of physical entities. One is extended, spread out, in space and the other is found in a specific location. That's contradictory, and, just like numbers, both are simplified, abstract ideas. The fact that they seem contradictory, at least to most people, is a failure of human imagination. — T Clark

Imagine me flipping a coin. While it's in the air is it heads or tails? I'd say it's neither. Instead it has the potential to be heads or tails. Only when it lands does it hold an actual value. In between quantum measurements objects are waves of potential. When they are measured they hold an actual state. I see no reason why the potential should behave the same as the actual so I see no contradiction. In fact, I think if they behaved the same then change would be impossible.

I think, like many mathematicians, you are expecting math to have a precise correspondence with reality. That never works. — T Clark

I'm not expecting that, I just believe that truth rhymes.

That's kind of a circular argument:

You - Mathematics shouldn't include elements with infinite properties because that doesn't match reality. Nothing infinite actually exists.
Me - There are qualified people who believe that infinite phenomena exist.
You - They've been fooled by their reliance on mathematics which include infinite elements. — T Clark

The universe has a wonderful way of avoiding actual infinities. Maybe we could to the same in math. If we do, maybe people would be less open to the unsupported idea that the universe is infinite.

You might be interested in Norman Wildberger on YouTube. He seems to hold positions on infinity similar to yours. — emancipate

I really like Norman Wildberger. I think his issues with the foundations of mathematics are valid. However, I do not like his resolution to the issues. He still believes that points are fundamental. At the bottom he explains natural numbers using tic marks, which is no different than spaced points on a number line.

I'm basically warning against logicism — sime

I see. I'm not really heading down the logicism route at the moment.

the algorithm for approximating sqrt(2) to any desired degree of approximation can itself be used to denote sqrt(2) without being executed. — sime

Agreed. That is why I what to give existence to the algorithm and not the completed output of the algorithm (which would be for example the complete decimal expansion of pi).

An object is at rest. It is not moving.

Now the object is moving at a velocity V.

How many different velocities did the object move at, to get from zero velocity to V velocity?

If your answer is not "infinite" then you don't deserve the name "mathematician". — god must be atheist

In a quantum reality we can only talk about it's velocity when measurements were made. Since we can only ever make a finite number of measurements in any given time interval, I would answer 'a finite number'. I believe that the answer in pure mathematics should be the same.

Never could a continuum be decomposed into points
— keystone

For physics, isn't that the driving force behind quanta, to put a stopper into space leaking out ? — magritte

Somewhat related, my understanding is that the planck length applies to measurement, not space itself. My understanding is that the continuity of space is still to be understood. Until then, it is possible that there is no limit to how small one can divide space.

With my view many paradoxes (Zeno, Dartboard, Liar's, etc) are easily resolved
— keystone

I take it you must mean dis-solved? — magritte

Yes.

∞∞ isn't and object like for example an elephant or the number 10100 or the word "elephant", it's simply a shorthand for the procedure 1. n = 0; 2. print n; 3. n = n + 1; 4. go to 2]. :chin: — Agent Smith

I agree. However, I would go one step further and say that infinity-laden objects such as real numbers are also not objects like the number 10100. Instead, they are simply a shorthand for a procedure.

I have a feeling that some ideas like ∞∞ and nothing cause brain damage - Cantor lost his mind (theia mania) and spent his later years in a lunatic asylum for instance. These concepts & paradoxes of which there are many seem to have a deletorious effect on the brain/mind - constantly mulling over them may lead to a nervous breakdown. Such ideas are more than our brains can handle at present. And yet ... there have been no reports of an epidemic of mental problems among mathematicians. Why I wonder. — Agent Smith

Or maybe a certain type of unstable mind searches to understand infinity as it grapples for an absolute to hold on to.

Yep. So construction gets replaced by constraint. And then my point is you go the next step of seeing construction and constraint as the two halves of the one system. — apokrisis

The need to complicate things by forming your triadic metaphysics is unclear to me. You say that the idea of an absolute continuity as the alternative is offensive to the ontic intuition. Please explain why.

We start with the highest dimensional continuum of interest.
— keystone

Which would be the "infinite dimensional" continuum — apokrisis

What is the dimension of purely empty abstract space? One might say that it is infinite dimensional, another might say that it is 0-dimensional. What matters is what you do in that space. If you're in elementary school and learning to draw X-Y graphs, then the highest dimensional continuum of interest is 2D. I see no need for one to assert the existence of continua of which they're not interested in.

So I am saying I wouldn't deal with the metaphysics of the number line in isolation. It is illustrative of the far bigger conversation we need to have about how holism in mathematical conception plays out. The same principles have to cover mathematical structure in general — apokrisis

One step at a time :)

This is a rather basic level of discussion. Again, how could it even be a continua unless it could be cut? How could it even be a 0D point except as the positive absence of any dimensioned extension? — apokrisis

The continua that I'm describing can be cut. Consider cutting a string (continua). When doing so, end-points emerge. I see no need to say that infinite points reside within the string.

So if you want to apply the strength metaphysics to questions about mathematical structure, you have to count to three in terms of "fundamental things". — apokrisis

In reality I think that the three fundamental things are space, strings, and observers.
In math I think that the three analogous fundamental things are continua, cuts, and computers/minds.

Or in other words, no matter how many times I cut up a piece of paper, never will it vanish to nothingness.
— keystone

But each piece also gets more pointlike. — apokrisis

No, no matter how many times you cut a continua it never becomes more point-like. In a similar way, no matter how many times you cut a string it never becomes 'nothing-like'...since it always remains 'something-like'.

The cut has to be sandwiched between the two ends of two lines. Each end of the line is a point. At what point does the point marking the cut – that is, the absence of a point at that point – get marked off from the other two points marking the starts of a pair of now separated continua? — apokrisis

I would draw it like this: ----o o---- (note that the o is like an open interval)
Of this diagram, the cut is this: o o (note there's nothing actually there, the point is not an actual object)

So it is easy to picture just forever cutting a line. Or instead, just forever gluing points. — apokrisis

I can picture cutting a line and gluing lines back together. I can't picture gluing points together...that's just gluing nothing. I don't see the need for points being actual objects.

The thing is that we can't go the limit.
— keystone

But the fact that we can approach the limit – both limits – with arbitrary closeness is how we know they are there. The limit is precisely that which isn't reachable in the end. But it certainly defines the direction we need to keep going from the start. — apokrisis

0+0+0+0+0+0+.... approaches 0. Nothing comes from nothing, no matter how much of it you have. Continua are not the limiting case of points.

Now consider the following summations:
5
2.5 + 2.5
1.25 + 1.25 + 1.25 + 1.25
...

I can keep going down this route whereby each term gets smaller and smaller, but the overall sum of each line remains 5. Something evolves to something, no matter how many times you cut it up. Points are not the limiting case of continua.

There is not a duality between points and continua where they define each other. Continua are fundamental whereas points are not.

With this parts-from-whole construction, objects are finite and processes are potentially infinite...and there are no paradoxes.
— keystone

Again, this suffers all the usual problems of an object-oriented ontology. Reality is better understood in terms of relations – processes and structures. — apokrisis

I don't understand your objection. What is the usual problem of an object-oriented ontology that I'm facing?

I don't understand why you want to challenge this. I use approximations to pi all the time. When I want a quick and dirty approximation of the area of a circle inscribed in a square with sides x, I use 3/4 * x^2. I can round pi off anywhere I like depending on the precision I need. To say that irrational numbers are not really numbers doesn't make any sense to me. Of course they are. — T Clark

You say there exists a number called pi with infinite digits and you use a truncated approximation of it when you calculate the approximate area of a circle.

I say that what exists is a (finitely defined) algorithm called pi that doesn't halt but you can prematurely terminate it to produce a rational number to calculate the approximate area of a circle.

The difference is that you are asserting the existence of an infinite object, something beyond our comprehension. My approach seems more in line with what us engineers actually do, so why bother asserting the existence of something impossible to imagine if you don't even need to?

I really don't get this. I have no problem imagining continuity arising from discreteness. I learned, saw it, got it, in 6th grade algebra. — T Clark

Do you believe that 0+0+0+0+... can equal anything other than 0? If not, then how can you claim that 0-length points could be combined to form a line having length?

Holding two apparently contradictory ideas in your mind at the same time is a required skill, e.g. waves and particles. It's no big deal. I learned that, saw that, got that in 12th grade physics. — T Clark

Sounds like double-think from 1984. There are no contradictions in wave-particle duality.

What advantage is there in seeing things your way. Expecting abstract concepts such as mathematical entities to have some sort of ontological reality doesn't make sense. Mathematicians love math for math's sake. Engineers such as me just want something that works - no ontological interpretation necessary. I assume the same is true for most scientists. How does your way work better than the way it is handled normally? — T Clark

With my view many paradoxes (Zeno, Dartboard, Liar's, etc) are easily resolved. While this should be enough, it also aligns far better with what us finite beings actually do. Also, while I haven't gotten into it here, it makes quantum physics less weird. Ultimately, I'm talking about the philosophy of mathematics, not the application of it. The day to day mathematics of engineers don't change with this new foundation.

It makes ontological sense to me. I do agree that is a useful, abstract simplification. Really, all math is. All reality is. — T Clark

Yes, all reality is void of actual infinities. So why do we need to believe that reality is just an approximation of some ideal infinity-laden object that we can't comprehend or observe? Why can't we stop at reality?

This may be true, but I don't think everybody qualified to have an opinion agrees with you. There are physicists who believe the universe is infinite. That doesn't really make sense to me, but a lot of things that don't make sense to me turn out to be true, so I'll remain agnostic. — T Clark

They think it's possible only because modern math welcomes actual infinity. If mathematicians rejected actual infinity then I'm sure physicists would be less inclined to accept it.

from 0D points to 1D lines – doesn't fix the deeper issues. You just set yourself up for the same puzzle at the next geometric level — apokrisis

I'm not saying that 1D lines are the fundamental object and all other objects are constructed from them. As you say, we run into the same issue when constructing 2D surfaces from 1D curves. I am proposing that instead of constructing the whole from the parts, that we construct the parts from the whole. We start with the highest dimensional continuum of interest. Think about how we draw a graph on paper. We don't make use of pointillism. We draw a square on our paper and then cut up that square by drawing some lines, for example the line x=0 and y=0. Only when these lines are drawn does the point (0,0) emerge. The more lines we draw, the more points emerge. Might the same apply to objects in the abstract world? Might continua be fundamental instead of points?

I mean it doesn't even make sense to talk about 0D points except in the context, or in contrast, with the presence of the 1D line, right? — apokrisis

But conversely, we can talk about a 1D line in the absence of points. In the example above, when I draw the first line and assign to it the function x=0 I haven't drawn any points yet. All I've done is draw a line and described how that line will interact with other lines IF they are added to my drawing. And I can keep adding more lines to my drawing to no end, adding more and more points, but never will I have infinite points. Or in other words, no matter how many times I cut up a piece of paper, never will it vanish to nothingness. Never could a continuum be decomposed into points. For this reason, I have to disagree with you when you say that continua as foundational objects are offensive to the ontic intuition.

This would see the discrete and the continuous as being each others limiting case. — apokrisis

The thing is that we can't go the limit. We can't complete the computation of the infinite series 0+0+0+0+... but as far as I can tell it looks like it should add to 0. I find it hard to believe that in some cases it adds to 1 and in other cases it adds to 2. But this is what we're doing when we assemble 0-length points to create lines of length 1 and 2.

Instead, to me it makes more sense to start with a continuum. Start with a string of length 10. Cut it in half, then you get strings of length 5. Cut one string in half, then you get two strings of length 2.5. Keep going and as you go strings of potentially infinite different lengths emerge. There's substance there from the start. You don't have to go the limit to have useful objects when starting with a continuum. With this parts-from-whole construction, objects are finite and processes are potentially infinite...and there are no paradoxes. This is in direct contrast to set theory where the whole is constructed from the parts and the objects of study (sets) are actually infinite.

A limitation of that conceptualisation, is that it asserts what might be considered an unnecessarily rigid ontological distinction between functions (intension) and data (extension), which is surely a matter of perspective, i.e the language one uses. — sime

Can you explain this to me from a computer programming perspective? In your comparison, is the data the output of the function? A function can return a function, but it can also return another object type, like a string. In the latter case, there is a type distinction between between the function and its output, but I don't see how this is unnecessarily rigid. I suspect I'm missing your point.

Also, recall incommensurability; the length of diagonal lines in relation to square grid have a length proportional to sqrt(2). The decimal points of sqrt(2) are only "infinite" relative to the grid coordinates. — sime

Can you conceive of a computer that can display a line of exact length sqrt(2)? In reality something will give, whether it's the gridlines or the diagonal, some imperfection (or uncertainty) will be inserted into the system to allow everything drawn to be 'rationally'. Might the abstract world face the same constraints? If we are certain that the grid lines are perfect, why can't we just claim that we are uncertain of the actual length of the diagonal and instead label it with a potentially infinite algorithm for calculating the length (corresponding to sqrt(2))? In the end, the math is the same, but the philosophy is different since I'm not assuming that sqrt(2) is a number and I'm not assuming that unending processes can be completed.

That said, it could be argued that the concept of exact and correct computation, whereby a computer program or function specification is translated by man or machine to a precise and correct result of execution, is an ideal platonistic notion that is incompatible with the austere epistemic and metaphysical conservatism of finitism. — sime

Yes, I agree to some extent. As mentioned above, if we move away from actual infinity I think we need to allow some uncertainty to creep into our systems. But what's so bad about that? Why do we need our computers to complete an endless computation of sqrt(2)? When we work with sqrt(2) we never actually work with the decimal expansion.

I was speaking of currently accepted set theory, not challenges of it. — jgill

Oh sorry, I see what you're saying. I think I have a grasp of how real numbers play into accepted set theory but it is challenging for me to envision the existence of a set of all natural numbers. Without assuming its existence, accepted set theory doesn't get far off the ground.

That makes the real numbers a challenging and intriguing subject. — jgill

Maybe not as challenging as you think. I think the application of real numbers would remain largely unchanged, it just requires a reinterpretation. If convention says that in between the rational points lies irrational points, I would say that in between the rational points lies continua (e.g. little lines). With this reinterpretation,

• Proofs the irrational numbers/points exist could be reinterpreted as proofs that continua exist.
• Proof that length comes from irrational numbers/points could be reinterpreted as proof that length comes from lines.

There's a simplicity and intuitive appeal to this reinterpretation. What challenge do you see?

All mathematics is about "potential" entities. So what we gonna do? Round pi off to 3.14? 3.14159? How many decimal places do I need to get to the real pi? — T Clark

Why can't we just say that pi is not a number? Instead, it is an algorithm (e.g. pick your favorite infinite series for pi) used to generate a number. This algorithm is potentially infinite in that we can never complete it, but we can certainly interrupt it to generate a rational number. If you interrupt it, maybe you'll get 3.14. Actual infinity only comes into play if you claim that the algorithm can be completed, in which case it would generate a real number - a number with actually infinite digits. This is what I would like to challenge.

History shows that is a bad standard by which to judge a concept. — T Clark

Perhaps I should have written that I believe it is impossible to imagine assembling points to form a continuum. 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.

I accept ideas like the set of real numbers and associated cardinalities — jgill

The set of real numbers is at the center of my discontent. It puts points as the foundational building block of mathematics and I find it hard to imagine that something (an n-dimensional continuum) can be constructed from nothing (0-dimensional points).

I have never used infinity as anything more than unboundedness. To all intents and purposes my mathematics has been infinity free. — jgill

This is why my concerns are more philosophical than mathematical. I think changing our philosophy (removing points from the foundation) will have little impact on the actual mathematics that we do.

The forum has had a number of discussions about this topic, but that's no reason for you to avoid bringing it up in a new thread or resurrecting an old thread. — jgill

If I can't find a mentor, I might do just that.

The tone of the OP does not suggest Cantor's theological nonsense. — jgill

I do feel that there is a little bit of Cantor's nonsense implied in any view that supports actual infinities. 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.

Calculus is all about infinity. — T Clark

I would argue that calculus done right (with limits) is all about potential infinities.

Thanks @Andrew M! You’ve cleared up much of my confusion. I understand that a lot of the mystery of the DCQE experiment can be addressed by fully appreciating what happens at entanglement. Delayed choice is actually a bit of a misnomer. I’ll need to let this continue to sink in further but I’d say you’ve addressed my initial question so thanks. Case closed!

At the act of entanglement the photons 'decide' how they're going to act — keystone

No wait…the first entangled photon’s behaviour is random but once measured, the other photons behaviour is determined? So in the DCQE are you saying that once the phase of the signals interference pattern is selected the fate of the idler photon is determined?