Another poster said that the real line is the set of real numbers.

Just to be exact: The continuum is the ordered pair <R L> where R is the set of real numbers and L is the standard ordering of the set of real numbers. If we speak of a "line" then perhaps, to be most exact, we would consider the set {<x 0> } | x is a real number}. (In other words, the "horizontal axis".)

Yay!! Thanks a tonne :) — keystone

I'm gratified to know I'm being helpful. Thanks.

I agree that the program you describe halts, however, I'll focus on Turing's version, by agreeing that the program that computes 0.333... to an arbitrarily fine precision halts.
I agree that by definition that's what it means for a number to be a computable number, so by definition 0.333... is a computable number.
I agree that by that definition pi is a computable number. — keystone

Ok correct to there.

I agree that no program can compute pi to infinitely fine precision.
You might even agree with me that no program can compute 0.333... to infinitely fine precision. — keystone

"Infinitely fine precision" is imprecise. If you change it to arbitrarily fine precision, we already have well known programs for both.

Perhaps by infinitely fine precision you mean specifying all of its digits at once. Sort of like running the Turing machine for 1, for 2, for 3, ... such that all of them were run and the outputs collected into the full decimal representation of pi.

If so, we agree that no physical computer could ever do that, simply because it would require an infinite amount of computing resources: time, energy, and space. If those are finite, then so is what we can practically compute. We agree.

pi an 1/3 are both computable, so the same argument applies.

By the way, set theorist Joel David Hamkins is investigating Infinite-time Turing machines." Here's the abstract:

We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. The resulting computability theory leads to a notion of computation on the reals and concepts of decidability and semi-decidability for sets of reals as well as individual reals. — Hamkins

Is that cool, or what!

@Michael Perhaps this is of interest.

My understanding is that a program halts if it reaches a point where it completes its execution and stops running. Do you actually disagree with this definition of halt? — keystone

I agree, adding the picky detail that it is required to terminate on a Halt state. If it terminates on an error state, it doesn't count. I believe that's one of the rules. With that proviso, we agree.

You are employing a straw man argument. I'm saying that the program that computes 0.333...to infinitely fine precision does not halt, and you are saying that the program that computes 0.333... to an arbitrarily fine precision does halt. I agree with you, but your argument doesn't address my point. — keystone

You lost me. You AGREE with me but I didn't address your point.

I am not sure what you agree with and what I didn't address.

I said: No halting algorithm can print out ALL of the digits in one execution. It would take infinitely many steps. That's against the rules for Turing machines. That's obvious.

But to be computable, a number doesn't need an algorithm to print out all its digits at once and then halt. That's impossible.

Rather, all that's needed is a machine that inputs n and outputs the n-th digit.

Now that's pretty clear. And it was very sensible on Turing's part, to realize that this is the right definition, and to avoid all talk of computations that run forever. He avoided all that. All computations must halt after finitely many steps. That is the property that characterizes computation from everything that is not computation. At least in the formal sense. Of course real life programs are designed to keep running, such as web servers and operating system kernels and the like. Still, actual computers can't compute anything that's not already computable by a Turing machine, so it doesn't matter.

So tell me what you agree with and what point I didn't address, and I'll try to address it.

I believe the term 'computable number' applies to a number which can be represented by an algorithm. Am I wrong? If so, it is a very misleading name because the definition makes no mention of computers, finite resources, or anything of the sort. I would much rather call them 'algorithmic numbers' but let's stick with the current terminology. — keystone

Turing invented the abstract computer. That's what a Turing machine is.

There is no accounting for terminology, it's mostly historical accident. Open and closed sets in topology are confusing because a set can be open, closed, neither, or both. Generations of students have been confused about open sets. But the terminology is dug in deep, there is no changing it.

When we say computable, we mean relative to Turing's definition of an abstract computer. A definition that has served computer science well ever since then.

Perhaps you are making the distinction between computer science and computer engineering! Turing machines are not made out of transistors and chips and wires; real computers are.

I hope you are not going to let yourself get hung up on the notion of abstraction. The purpose of the abstract Turing machine is to allow us to reason logically about what computers can and can't do. The abstract theory informs computer engineering, it's just not all of it.

Hope this helps. You do not want to get bogged down in confusion about the realm of the abstract versus the realm of the actual.

I believe the term 'halt' applies to the execution of the algorithm. If it cannot be executed to completion then it does not halt. — keystone

Yes, correct. But if you are referring to the idea of printing out all of the digits in one shot, you are just denying or refusing to accept the standard usage of the term. It pertains to an abstract computer, not a real one.

Are you getting hung up on that?

Your failure to see the above distinction relates to one of my central complaints about the current (bottom-up) view of mathematics: mathematicians too often obfuscate the program (the algorithm) with it's execution (the generation of output by the algorithm). And it doesn't help that we call both the program and it's output the same thing: numbers. This is where I'm trying to bring clarity to the situation by redefining terms (such as what it means to be a rational vs. a real), but it turns out that such efforts just makes you think I don't know what I'm talking about. — keystone

You seem to be unhappy that abstract math doesn't climb into the wiring cabinet and start patching cables. Do I have that right?

I don't think you don't know what you're talking about. I think I don't know what you're talking about. I'm throwing out guesses. You don't like infinities, ok there's finitism. You want things to be algorithmic, ok there's constructivism. You want there to be a minimal positive real, ok that's computer arithmetic. None of it sticks. Why do you reject the doctrines you espouse?

Shouldn't the first principles be self-evident? — keystone

Of course not. That belief was demolished by non-Euclidean geometry leading to General Relativity. Kant said we have a priori knowledge of the Euclidean and Newtonian universe, and that this was true knowledge of the world. He was wrong. We are born with an intuition of Euclidean space, but that turns out to just be a local approximation to the world we live in. Who knew, right?

We experience continua and finite numbers all the time in our physical reality. The same cannot be said about points and transfinite numbers. It is the points which must be constructed from first principles. It is infinity which must be derived, not axiomatized into existence. — keystone

Ok. I get that you feel enthusiastic about this. I'm on your side. I hope you can work out your ideas. I do think they are a little half-baked at the moment. That's an honest assessment.

If you don't like the axiom of infinity you are a finitist, but perhaps not an ultrafinitist. You might be interested in learning about finitism[/ul].

Euclid's line is so simple -- breadthless length. It's hard for anyone to say that's not self-evident. And I can easily construct a point from that line - I cut it and the midpoint emerges. The bottom-up view is far less self-evident. Somehow combining sufficiently many objects of no length results in an object of length. And even though nobody has a good explanation of how this works we nevertheless proceed by saying that the continuum constructed in this fashion is paradoxically beautiful and only to be seriously discussed by the experts. Really? — keystone

If I stipulate that every mathematicians that ever lived is a bad person for doing whatever you think they did ... would it help?

Can you step back a tiny bit and see that if every smart person who ever lived is a dummy acting from bad faith ... well, maybe it's you, and not them. Maybe a lot of people already thought about these issues. Excessive grandiosity is often an indicator of crankitude.

I mean actually, if you feel that everyone else is wrong and you are right ... you should keep this to yourself! You don't want me to know you feel that way.

You expect a deeper structure to my line, such that, say, when I cut line (0,4) at point 2 that this involves identifying a pre-ordained point and isolating it by means of a cut. That's not what I'm proposing. My line has no deeper structure or additional properties beyond continuity and breadthless length. — keystone

More breathless than breadthless. (Sorry couldn't resist!)

I've come to realize that I've been heading down the wrong path by saying that my line is a bundle of 2ℵ0
2
ℵ
0
points. — keystone

When I Quote your posts, those expressions always render as one character per line. Makes it hard to read.

Anyway ok, you claimed that many times, and now you're not sure. That's fine, since your theory is a long way from being able to define $2^{\aleph_0}$. That expression is a fairly sophisticated "bottom up" construction.

I ended up here because I was defending against your arguments that my line has gaps. It turns out that my defense has just made you expect a structure to these points. It is better for me to just claim as a first principle that the line is continuous. As such, I'd like to discard the 'bundle' argument. — keystone

Abandoning the entire bundle argument? So the real line is no longer made of a countably infinite union of overlapping open intervals, each characterized by a particular computable number it contains? I thought that was a pretty good thing to achieve agreement on. You are abandoning this now?

As far as gaps go, they're important. The completeness property, aka the Least Upper Bound property, aka Cauchy-completeness, is the defining characteristic of the real numbers. Accept no substitute! If someone tries to sell you a model of the real numbers, ask them if it's complete! Mathematical shopping advice.

Instead, the structure you are looking for comes from the cutter, not the line. I can cut line (0,4) anywhere I want and label the midpoint that emerges '2'. In labelling that point '2' I am making an agreement with myself that any subsequent cut I make, I will label it to maintain the structure we have come to expect with numbers (as captured by the SB tree). For example, if I subsequently cut the line (2,4) I agree to label it with a number between 2 and 4. — keystone

Your focus on the Stern-Bricot tree is ... well some adjective anyway. Do you know the infinite complete binary tree? Start with a binary point at the top node. Underneath each node from now on is a left node called 0, and a right node called 1. The infinite tree has countably many nodes, but uncountably many paths through the tree. Each path corresponds to a real number and all the real numbers are represented by some path. Or sometimes two distinct paths, as in .0111111... and .10000... I find it helpful to visualize the real numbers that way sometimes.

But your cut idea, you just can't get to enough of the real numbers that way, with each number represented by finitely many cuts, if that's what you're doing.

In the top-down view, the cutter/mathematician plays a central and active role in maintaining structure and, moreover, actualizing objects....not unlike the the observer in QM...hmmm.... — keystone

Ah, well that's like the active intelligence of intuitionism. You should read up a bit on this too.

Here's what Wiki says:

In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality.[1] That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality. — Wiki

How did we ever survive before Wikipedia?

There's nothing actually infinite about the line. What is infinite is the potential for the cutter to make 2ℵ0
2
ℵ
0
cuts to the line. — keystone

Well you are a long way from making that many cuts when you start by denying even countable infinity!! Isn't that a little inconsistent?

But since (1) an infinitely precise computable irrational cut requires the completion of a supertask, — keystone

I don't think you should tie in supertasks. There's already enough fuzzy thinking on that topic going around.

(2) non-computable irrational cuts cannot be algorithmically defined and (3) the cutter can only ever perform finitely many cuts, this potential can never be completely actualized. When working in 1D, the mathematician will forever be stuck working with a finite set of lines and points. However, because the mathematician can continue to make arbitrarily many more cuts (i.e. any natural number of cuts), that set can grow to be arbitrarily large (i.e. have any natural number of elements). — keystone

Very difficult to get a model of the real numbers while denying infinite sets. It's been tried, really. The constructivists have a pretty interesting model. They even have their own version of completeness, even though their model is only countably infinite. Maybe you should look at constructivism.

Of course that's possible (and likely). After all, that's why these ideas are being discussed in this chat forum and not eternalized in the Annals of Mathematics. But since yelling 'I'm not crazy' only makes one sound crazier, I won't challenge this point further and hopefully the ideas will eventually speak for themselves. — keystone

You have seemingly expressed the idea that you are right and thousands of years of the smartest people in the world looking at these issues are wrong. That's always a bad sign. You should note it yourself, as an objective check on your own thought process.

I know how it sounds, that's why I'm reluctant to talk about QM and paradoxes at this time. — keystone

Ah, you have discretion. Well actually that's a point in your favor. But QM, really? That's a stretch.

When I communicate the fundamentals, you ask for the implications. When I communicate the implications, you ask for the fundamentals. — keystone

And I've understood precious little of either :lol:

My only hope is that at some point the fundaments become coherent to you, after which the implications will naturally follow. I admire you for sticking with me for this long given that you think my ideas have so far been incoherent. — keystone

I'm in shock that you're abandoning the little open intervals. I was clinging to that like a reed in the ocean.

I ask ChatGPT to give me the Latex equivalent of an expression and I insert that Latex string with the math tag. — keystone

Jeezus. That ain't workin'. Maybe some control characters in there. And don't use ChatGPT, it rots your brain. Lot of foolishness floating about in the culture lately.

Not only cannot a physical computer produce an infinite string, but a Turing machine (not an extension of the notion of a Turing machine to infinite outputs) can't do it.

/

From my non-exhaustive readings on Turing machines, I don't know what an 'error state' is for a Turing machine. Basically, a Turing machine takes an input and either halts with a state and whatever is on its tape when halted or it does not halt.

But there are different formulations of Turing machines, so I would welcome knowing of one that defines an error state.

I'm very keen to respond but there's a lot to reflect on and respond to and I'm completely overloaded with work and family responsibilities so I might not respond until next weekend. As always, I appreciate your comments and this dialogue.

I'm very keen to respond but there's a lot to reflect on and respond to and I'm completely overloaded with work and family responsibilities so I might not respond until next weekend. As always, I appreciate your comments and this dialogue. — keystone

No worries mate.

 

I've further refined my thinking on numbers and algorithms so please allow me to start from scratch here without us bringing in any of the out-of-date baggage from my past few posts. 

Fractions
I would like the objects of computation to be fractions. I have an unorthodox view of fractions. I don't see them being constructed from other number systems (such as integers) or as elements of other number systems (such as real numbers) but rather as numbers in and of themselves. The structure of the fractions are captured by trees, my preference being the Stern-Brocot tree, which places fractions at the nodes and whose basic arithmetic can be performed by navigating the Stern-Brocot tree. But when I mention the SB-tree in the following post, I acknowledge that any equivalent structure (such as the infinite complete binary tree) can be used in its place.

• When I say that 1/2 = 2/4, I mean that fractions 1/2 and 2/4 correspond to the same node on the SB-tree.
• When I say that 0.99 = 9/10 + 9/100, I mean that 0.99 and 9/10 + 9/100 are both algorithms which output the same fraction when executed.
• When I say that 0.99 ⥱ 99/100, I mean that the algorithm 0.99 outputs the fraction 99/100 when executed.
• With this view, positional number systems (such as decimal) describe algorithms on fractions, not fractions themselves.
• Only basic arithmetic is valid on fractions (specifically roots are excluded).
• Any set of fractions is incomplete (i.e. has gaps) so cannot be used in isolation to construct a line.

Reals

• When I previously said that 4 - 4/3 + 4/5 - 4/7 + 4/9 - ... does not halt, I meant that when executed, that algorithm will endlessly compute as it works towards outputting the complete sum all in one go (hence, infinite precision). However, I accept that what Turing would do is reformulate the algorithm to essentially compute a partial sum of the infinite series, output the resulting fraction of arbitrarily fine precision, and halt. I like Turing's approach so moving forward I'll treat an infinite series as this type of algorithm which halts. I acknowledge that this is a departure from classical mathematics which doesn't equate the infinite series with such an algorithm.
• It is meaningless to talk about the unique output of the algorithm 4 - 4/3 + 4/5 - 4/7 + 4/9 - ... because the output depends on the precision requirements.
• Instead of working with the equivalence class of all possible outputs from such an algorithm, I propose that we work with the algorithm itself.
• When I say that 9/10 + 9/100 + 9/1000 + ... !⥱ 1, I mean that that algorithm can never output the fraction 1.
• When I say that 0.9 ≃  9/10 + 9/100 + 9/1000 + ...  ≃ 1+ 0/10 + 0/100 + 0/1000 + ... ≃ 1.0, I mean that all 4 algorithms are asymptotically equal (i.e. they share the same limit).
• When I say that 9/10 + 9/100 + 9/1000 + ... is rational, I mean that that algorithm is asymptotically equal to a positional number algorithm whose fractional digits eventually end with repeating digits (namely, 0.9).
• When I say that 9/10 + 9/100 + 9/1000 + ... is natural, I mean that that algorithm is asymptotically equal to a positional number algorithm whose fractional digits are repeating 0's (namely, 1.0).
• When I say that an irrational has no positional number algorithm, I mean that it cannot be represented as a positional number algorithm whose fractional digits eventually end with repeating digits.
• A computable irrational, such as π, is an algorithm based on more complex arithmetic of fractions, such as 4 - 4/3 + 4/5 - 4/7 + 4/9 - ..., nevertheless being entirely represented with finite characters.
• I propose that non-computable irrationals are inaccessible and unnecessary, and that a real can only be a rational or a computable irrational.
• Following from my proposition, any set of reals is necessarily incomplete (i.e. has gaps) so cannot be used in isolation to construct a line.

Continua

• I propose that a line is a fundamental object, it is continuous breadthless length, and it is described by its endpoints. 
• I propose that we redefine the term interval from describing the points that lie between endpoints to describing the line that lies between endpoints. For example, the interval (-5,5) describes the line that resides between points -5 and 5.
• I propose that the only operation we can perform on lines (and continua in general) is to cut (or join) them AND I endeavor to argue that this operation is entirely sufficient. (Moving forward, as a matter of brevity, I won't mention joining).
• I propose that in 1D a cut entails the partitioning of a line. 
• A cut at a point entails cutting the line once, and labelling the midpoint that emerges with any fraction whose value is between the endpoints. For example, cutting (-5,5) by 2 results in (-5,2) U 2 U (2,5).
• A cut at a line entails cutting the line at two points, and labelling the midline that emerges with an open fractional interval. For example, cutting (-5,5) by (1,2) results in (-5,1) U 1 (1,2) U 2 U (2,5).
• To understand how reals fit into this picture, it is beneficial to see them as Cauchy sequences of fractional intervals, and the corresponding algorithm (when executed) outputs a fractional interval arbitrarily deep into the sequence, and halts.
• For example: 0.9 ≃ 1.0 ≃ (9/10,11/10), (99/100,101/100), (999/1000,1001/1000), (9999/10000,10001/10000), ...
• A cut of (-5,5) at 0.9 entails a cut at one of the intervals in the Cauchy sequence, which necessarily is a cut at a line.
• Instead of selecting a specific interval in the Cauchy sequence, the cut of (-5,5) at 0.9 can be generalized as a cut by (1-ε, 1+ε), where we can replace ε with a small positive number and 1-ε and 1+ε are fractions. The resulting system can be generalized as (-5, 1-ε) U 1-ε U (1-ε, 1+ε) U 1+ε U (1+ε,5).
• Any set of points is incomplete (i.e. has gaps) but the points are just one part of a continuum-based system. The system also includes lines, and as a whole the system is continuous.
• I propose that calculus is the mathematics real cuts on continua.

 

In my post directed to both you and TonesInDeepFreeze, I either directly or indirectly addressed a lot of your shared points. Below are my responses to your unshared points. Please let me know if I failed to respond to any of your points.

set theorist Joel David Hamkins is investigating Infinite-time Turing machines. — fishfry

I'm skeptically curious :)

You seem to be unhappy that abstract math doesn't climb into the wiring cabinet and start patching cables. Do I have that right? I don't think you don't know what you're talking about. I think I don't know what you're talking about. I'm throwing out guesses. You don't like infinities, ok there's finitism. You want things to be algorithmic, ok there's constructivism. You want there to be a minimal positive real, ok that's computer arithmetic. None of it sticks. Why do you reject the doctrines you espouse? — fishfry

• Computers - Clearly physical computers are limited by our finite observable universe, but I think that we are also in agreement that the abstract computers of mathematics also cannot be infinite. They can be arbitrarily large but are nevertheless finite. I'm interested in abstract computers, not physical computers. 
• Infinities - I think it's impossible to model a continuum using a finite set of indivisibles. However, I'm proposing that we model a continuum using an evolving finite set of divisibles. As we make cuts, the finite set grows, but it remains finite. As we make joins, the finite set shrinks, but never becomes empty.
• Algorithmic - I don't want to use algorithms to construct indivisibles. I want to use algorithms to deconstruct (i.e. cut) divisibles (i.e. continua).
• Minimal positive real - I'm not saying that there exists a minimal positive number. Rather, I'm saying that we can generalize the output of algorithms by using a placeholder. So when I say that we cut (0,1) arbitrarily many times to produce an arbitrarily small number, I'm not saying that we can make infinitely many cuts or that there exists an infinitely small number. Rather, I'm saying that you can pick a positive number as large as you please and divide 1 by it.

Shouldn't the first principles be self-evident?
— keystone

Of course not...non-Euclidean geometry... — fishfry

Ok fair point. Agreed.

If I stipulate that every mathematicians that ever lived is a bad person for doing whatever you think they did ... would it help? Can you step back a tiny bit and see that if every smart person who ever lived is a dummy acting from bad faith ... well, maybe it's you, and not them. — fishfry

We need to make a distinction between the core mathematical idea and language with which it's communicated. For example, the Pythagorean Theorem was known and used in various forms long before the formalization of bottom-up number-based systems. And it will continue to hold value even if we move past bottom-up number-based systems to top-down continuum-based systems. I'm not proposing that any such mathematical idea is wrong.

And even IF I'm right, it doesn't mean that bottom-up number-based systems are useless. They would remain useful in the same sense that Newtonian mechanics remains useful (it just cannot be used to describe our reality at a fundamental level). But yes, I agree that it is likely me who is the dummy. I am likely experiencing the Dunning–Kruger effect. But nevertheless ideas should be challenged on their merit, not on how unlikely it is for an important math idea to originate from an engineer on a chat forum.

Abandoning the entire bundle argument? So the real line is no longer made of a countably infinite union of overlapping open intervals, each characterized by a particular computable number it contains? I thought that was a pretty good thing to achieve agreement on. You are abandoning this now? — fishfry

I had to abandon the bundle argument, in part because it seems to imply a structure that's not there. For example, I never proposed that the real line was made of a countably infinite union of overlapping open intervals. Rather, I proposed that a computer can begin to cut the line but it will never exhaust cutting such that the line is divided into infinitely many partitions.

As far as gaps go, they're important. The completeness property, aka the Least Upper Bound property, aka Cauchy-completeness, is the defining characteristic of the real numbers. Accept no substitute! If someone tries to sell you a model of the real numbers, ask them if it's complete! Mathematical shopping advice. — fishfry

I agree that a number-based system that has gaps cannot be used for calculus. However, a continuum-based system begin with a continuous line, and if all we do is make cuts, there will never be a state of the system where gaps are present.

Do you know the infinite complete binary tree? — fishfry

I agree that the SB-tree and the infinite complete binary tree capture the same information. In fact, the binary tree might be preferable since binary is the language of computers. However, I prefer the SB-tree since it places fractions at the nodes. As described in my post to both you and TonesInDeepFreeze, I see binary numbers as algorithms operating on fractions. Since paths down the binary tree can also be seen are algorithms operating on fractions, I feel that the distinction between a node and a path is less clear with the infinite complete binary tree.

Ah, well that's like the active intelligence of intuitionism. — fishfry

Yes, I do think my view falls near the intuitionist camp.

Well you are a long way from making that many cuts when you start by denying even countable infinity!! Isn't that a little inconsistent? — fishfry

Yeah, you're right. In my reformulated view (where I posted to you and TonesInDeepFreeze) I make no mention of uncountably many cuts.

Very difficult to get a model of the real numbers while denying infinite sets. It's been tried, really. — fishfry

I think it's impossible to model a continuum using a finite set of indivisibles. However, I'm proposing that we model a continuum using an evolving finite set of divisibles.

Jeezus. That ain't workin'. Maybe some control characters in there. And don't use ChatGPT, it rots your brain. Lot of foolishness floating about in the culture lately. — fishfry

Is the problem simply that I'm using Latex? And ChatGPT is imperfect but nevertheless awesome as a co-pilot.

Ok. I get that you feel enthusiastic about this. I'm on your side. I hope you can work out your ideas. I do think they are a little half-baked at the moment. That's an honest assessment. — fishfry

I genuinely appreciate this sentiment. Given that my ideas continue to get reformulated throughout this discussion, I could only agree that they are in the baking process (and to be realistic, they are likely less than half-baked at this point). I also want to acknowledge that when the baking is complete the end product may not be anything anyone wants to eat!

In my post directed to both you and fryfish, I either directly or indirectly addressed a lot of your shared points. Below are my responses to your unshared points. Please let me know if I failed to respond to any of your points.

If one proposes a mathematics without infinite sets, then that is fine, but the ordinary mathematics for the sciences uses infinite sets, which are not derivable from the rest of the set theoretic axioms, thus requiring an axiom. — TonesInDeepFreeze

I agree that applied mathematicians often formulate their theories with infinite sets, but that is largely because those are the tools provided by pure mathematicians. In practice, applied mathematicians do not actually use infinite sets. I believe that applied mathematicians would welcome a different set of tools which allowed their theories to be reformulated free of infinite sets, while still retaining their usefulness.

For that matter, we don't physically experience breadthlessness, so breadthlessness is itself an idealization, just as infinitude is an idealization. — TonesInDeepFreeze

1D continua (i.e. lines) are a simpler version of the 3D continua we experience. Comparatively, infinitude is a more complex version of the finitude we experience. As such, the idealization of lines and the idealization of infinitude are not comparable.

The notion of length is quite coherent. In context of the reals, length is a property of segments not of points. — TonesInDeepFreeze

I agree with this. But if the segment is built entirely from points, where would the length come from if not the points?

Who said anything equivalent with "the continuum constructed in this fashion is paradoxically beautiful and only to be seriously discussed by the experts"? Specific quotes are called for. Otherwise the claim is a flagrant strawman. — TonesInDeepFreeze

I started writing a defence for that comment but given that fryfish has entertained a long conversation on the topic with me, a non-expert, I think it's better for me to just take that statement back.

It has been proposed in this thread that a sequence converges as n gets arbitrarily large. A sequence is a function. A function has a domain. If the domain is not infinite, then n cannot be arbitrarily large. — TonesInDeepFreeze

I am proposing that we not work with the infinite sequence itself (which cannot exist in a computer) but rather the algorithm designed to generate the infinite sequence (which is described with finite characters and, as per Turing, halts if executed.) As for the domain, I am proposing that we not work with an infinite set of numbers, but rather a line upon which arbitrarily many fractions can emerge by means of cuts.

One is welcome to work it out in some other way. But then the natural question is: What are your primitives, formation rules, axioms and inference rules? — TonesInDeepFreeze

I agree that a formal treatment is the ultimate destination for a math idea. I see this informal forum conversation as the journey which may be slowly taking me toward that destination or may be leading me toward the junkyard to dump my ideas. Either way, this journey is useful to me.

It was claimed that Russell's paradox is "still there". In what specific post-Fregean systems is it claimed that the contradiction of Russell's paradox occurs? — TonesInDeepFreeze

I think it's best for me to just take this claim back.

Back to the poster who claims to offer an alternative to classical mathematics: The word 'isolate' keeps coming up. What is a rigorous mathematical definition of 'isolate'? — TonesInDeepFreeze

In earlier posts I described 'potential' points which existed only in bundles that can be isolated by means of a cut. I have since scrapped that idea. Instead of proposing that cuts isolate points, I am now proposing that cuts create points.

It was claimed that the interval (0 1) is not an infinite union of disjoint intervals. It is false that the interval (0 1) is not an infinite union of disjoint intervals.Ostensively: (0 1/2) U [1/2 3/4) U [3/4 7/8) U ... — TonesInDeepFreeze

Treating your example algorithmically and using the notation described in my recent post I would say the following: (0, 1/2) U [1/2, 3/4) U [3/4, 7/8) U ... !⥱ (0, 1)
In other words, that algorithm can never output the interval (0, 1).

But I think the original discussion was about describing a line as the union of fundamental objects. Defining a line as the union of smaller lines would be a circular definition, which is why I want to take the line as fundamental.

'Dedekind cut' is not defined in terms of 'gaps', nor 'executions', nor algorithms. 'Dedekind cut' is mathematically defined and we see mathematical proofs that the set of Dedekind cuts is uncountable. — TonesInDeepFreeze

In the context of computation, this distinction is moot since we can only ever speak of arbitrarily many cuts. As such, I'm willing to withdraw my comment about uncountably many cuts made to countably many rationals.

Instead of that comment, let me reframe my position in the context of cuts as described in my recent post to you and fryfish. If we inspect the outcome of a real cut, we cannot determine what real algorithm was used to make that cut. This is because when executing real cuts we are forced to select a fractional interval at some finite depth in the Cauchy sequence of intervals, and whatever that interval may be could correspond to potentially ${\aleph_0}$ different real cuts.

If a Dedekind cut does not correspond to the cut described above, it likely corresponds to one that is infinitely precise such that it results in a cut at a point (not a line). This would be akin to the impossible and non-sensical task of selecting the final interval in the Cauchy sequence of intervals, whose interval would contain only a single number. Of course, there is no final term. From the computational perspective, this type of cut is not possible so has no relevance in the top-down view. If it holds together in the bottom-up view then fine.

Stern-Brocot is interesting. But a while back, your improvisations on it came to confused handwaving. I'm not inclined to start all over again about it with you.

Let me know when (or if) you have a system with formation rules, axioms and inference rules.

Otherwise, discourse with you is such that what obtains is just what you say obtains, without your interlocuters having access to checking your arguments by the objective reference of mathematical proof.

In practice, applied mathematicians do not actually use infinite sets. — keystone

Depends on what you mean by "applied". Ordinary mathematics, even as used for basic applied interests such as speed and acceleration use ordinary calculus, which is premised in infinite sets.

/

In context of ordinary mathematics, a line in 2-space is a certain kind of set of ordered pairs and a line in 3-space is a certain kind of set of ordered triples. If one wishes to propose an alternative, then they can set it up coherently with axioms and definitions. On the other hand, if you wish to limit yourself to ostensive, imagistic musings then I wouldn't begrudge you from entertaining yourself that way; only that count me out, as I have better avenues of intellectual engagement.

/

You ask where does length "come from" if not points. The mathematical definition of 'distance between points' is given by a well known formula in high school Algebra 1. Look it up.

Who said anything equivalent with "the continuum constructed in this fashion is paradoxically beautiful and only to be seriously discussed by the experts"? Specific quotes are called for. Otherwise the claim is a flagrant strawman.
— TonesInDeepFreeze

[...] I think it's better for me to just take that statement back. — keystone

Even better would be to figure out why you are prone to such things to begin with.

generate the infinite sequence (which is described with finite characters and, as per Turing, halts if executed.) — keystone

No, the execution does not halt. You have it completely wrong.

I see this informal forum conversation as the journey which may be slowly taking me toward that destination or may be leading me toward the junkyard to dump my ideas. Either way, this journey is useful to me. — keystone

That's nice. But, by magnitudes, less useful to me than actually reading mathematics and philosophy of mathematics. For that matter, even less useful to me than, say, watching my screen saver. But, meanwhile, I don't mind correcting falsehoods you post about mathematics itself and to comment on the irremediable handwaving in the description of your own musings.

It was claimed that Russell's paradox is "still there". In what specific post-Fregean systems is it claimed that the contradiction of Russell's paradox occurs?
— TonesInDeepFreeze

I think it's best for me to just take this claim back. — keystone

See earlier in this post.

Back to the poster who claims to offer an alternative to classical mathematics: The word 'isolate' keeps coming up. What is a rigorous mathematical definition of 'isolate'?
— TonesInDeepFreeze

In earlier posts I described 'potential' points which existed only in bundles that can be isolated by means of a cut. I have since scrapped that idea. Instead of proposing that cuts isolate points, I am now proposing that cuts create points. — keystone

You seem really not to recognize that it is a potentially infinite game with you, where you pretend to define or explain terminology or concepts in terms of yet more undefined terminology and concepts.

And who can keep up with you? So 'isolate' is now out? And 'create' is now in?

!⥱ — keystone

I guess I'm supposed to glean that that means something like "as goes to infinity" or something?

Anyway, what you said makes no sense and is not an answer to my correction of your false claim.

What I wrote:

Let f be the function whose domain is the set of natural numbers such that:

f(0) = (0 1/2)
for n>0, f(n) = [(2^n - 1 )/2^n (2^n+1 - 1)/2^n+1)

The range of f is an infinite partition of (0 1). — TonesInDeepFreeze

There is no "algorithm" or "going to infinity" there.

Purely that I proved that, contrary to your false claim, there is an infinite partition of (0 1). But you can't be bothered to actually looking at the proof to understand it (though it is very simple), as instead you're too preoccupied with handwaving and strawmaning that I've invoked an "algorithm" and a "going to infinity", which suggests of you a kind of a narcissism as you wish that other people indulge your undefined and confused musings while you ignore (worse, strawman) clear proofs and exact explanations from your interlocuter.

Defining a line as the union of smaller lines would be a circular definition — keystone

I didn't invoke any circularity. Read the proof.

'Dedekind cut' is not defined in terms of 'gaps', nor 'executions', nor algorithms. 'Dedekind cut' is mathematically defined and we see mathematical proofs that the set of Dedekind cuts is uncountable.
— TonesInDeepFreeze

In the context of computation, this distinction is moot since we can only ever speak of arbitrarily many cuts. — keystone

You wave "In the context of computation" here like it's a "Get Out Of Jail Free" card that you can use to evade that you're mischaracterizing what a Dedekind cut is.

I'm willing to withdraw my comment about uncountably many cuts made to countably many rationals. — keystone

Again, better not to be prone to making outlandishly false claims.

If we inspect the outcome of a real cut, we cannot determine what real algorithm was used to make that cut. — keystone

Again, Dedekind cuts do not invoke algorithms.

when executing real cuts we are forced to select a fractional interval at some finite depth in the Cauchy sequence of intervals, and whatever that interval may be could correspond to potentially aleph_0 different real cuts. — keystone

Learn to use mathematical terminology coherently. You suffer the misconception that throwing around jargon in inapposite imagistic clumps is mathematically meaningful.

Etc.

In my post directed to both you and TonesInDeepFreeze, I either directly or indirectly addressed a lot of your shared points. Below are my responses to your unshared points. Please let me know if I failed to respond to any of your points. — keystone

I thought you don't believe in points :-)

Computers - Clearly physical computers are limited by our finite observable universe, but I think that we are also in agreement that the abstract computers of mathematics also cannot be infinite. They can be arbitrarily large but are nevertheless finite. I'm interested in abstract computers, not physical computers.  — keystone

TMs are arbitrarily large but finite.

Infinities - I think it's impossible to model a continuum using a finite set of indivisibles. However, I'm proposing that we model a continuum using an evolving finite set of divisibles. As we make cuts, the finite set grows, but it remains finite. As we make joins, the finite set shrinks, but never becomes empty. — keystone

Murky.

Algorithmic - I don't want to use algorithms to construct indivisibles. I want to use algorithms to deconstruct (i.e. cut) divisibles (i.e. continua). — keystone

Murky. To me, anyway.

Minimal positive real - I'm not saying that there exists a minimal positive number. — keystone

You've said it repeatedly, haven't you?

Rather, I'm saying that we can generalize the output of algorithms by using a placeholder. So when I say that we cut (0,1) arbitrarily many times to produce an arbitrarily small number, I'm not saying that we can make infinitely many cuts or that there exists an infinitely small number. Rather, I'm saying that you can pick a positive number as large as you please and divide 1 by it. — keystone

Suppose I grant that you have some alternative construction of the reals. What of it? All models of the reals are isomorphic to one another.

We need to make a distinction between the core mathematical idea and language with which it's communicated. For example, the Pythagorean Theorem was known and used in various forms long before the formalization of bottom-up number-based systems. And it will continue to hold value even if we move past bottom-up number-based systems to top-down continuum-based systems. I'm not proposing that any such mathematical idea is wrong. — keystone

My point exactly. If you have an alternative view of the reals, nothing changes.

And even IF I'm right, it doesn't mean that bottom-up number-based systems are useless. They would remain useful in the same sense that Newtonian mechanics remains useful (it just cannot be used to describe our reality at a fundamental level). But yes, I agree that it is likely me who is the dummy. I am likely experiencing the Dunning–Kruger effect. But nevertheless ideas should be challenged on their merit, not on how unlikely it is for an important math idea to originate from an engineer on a chat forum. — keystone

I don't see what ideas you've challenged.

I had to abandon the bundle argument, in part because it seems to imply a structure that's not there. — keystone

It took weeks for me to understand your bundles, and just when I did, you took them away.

For example, I never proposed that the real line was made of a countably infinite union of overlapping open intervals. Rather, I proposed that a computer can begin to cut the line but it will never exhaust cutting such that the line is divided into infinitely many partitions. — keystone

Well we agree on that.

I agree that a number-based system that has gaps cannot be used for calculus. However, a continuum-based system begin with a continuous line, and if all we do is make cuts, there will never be a state of the system where gaps are present. — keystone

Ok. Not that I understand.

I agree that the SB-tree and the infinite complete binary tree capture the same information. In fact, the binary tree might be preferable since binary is the language of computers. However, I prefer the SB-tree since it places fractions at the nodes. As described in my post to both you and TonesInDeepFreeze, I see binary numbers as algorithms operating on fractions. Since paths down the binary tree can also be seen are algorithms operating on fractions, I feel that the distinction between a node and a path is less clear with the infinite complete binary tree. — keystone

"Binary numbers" aka real numbers can never be algorithms, since there are way too many of them. There are uncountably many reals and only countably many algorithms.

Yes, I do think my view falls near the intuitionist camp. — keystone

You should study intuitionism then.

I think it's impossible to model a continuum using a finite set of indivisibles. However, I'm proposing that we model a continuum using an evolving finite set of divisibles. — keystone

Sigh. I am not getting much from this latest post.

I genuinely appreciate this sentiment. Given that my ideas continue to get reformulated throughout this discussion, I could only agree that they are in the baking process (and to be realistic, they are likely less than half-baked at this point). I also want to acknowledge that when the baking is complete the end product may not be anything anyone wants to eat! — keystone

ok!

I'm not commenting on your other post, but I did note this:

I propose that non-computable irrationals are inaccessible and unnecessary, and that a real can only be a rational or a computable irrational. — keystone

Then you are a constructivist. I don't understand why you disagree.

I'm not commenting on your other post — fishfry

I understand the previous post was lengthy, and I know you don't owe me anything. However, I wonder if this marks the end of our discussion. I'm unsure how to keep it going since anything more might just be more words to skip over. If you have any advice on how to continue, I'd appreciate it. If you'd prefer to end our conversation here, I accept that and thank you for the discussion!

I understand the previous post was lengthy, and I know you don't owe me anything. However, I wonder if this marks the end of our discussion. I'm unsure how to keep it going since anything more might just be more words to skip over. If you have any advice on how to continue, I'd appreciate it. If you'd prefer to end our conversation here, I accept that and thank you for the discussion! — keystone

Didn't say that, just got a little overwhelmed by all the line items. I'll take a look at it.

But what about my point about constructivism? If you reject the noncomputable reals, you're a constructivist.

ps -- I read through your list and I don't get it. Suppose I stipulate all that. Then what? So what? I just don't get it. And I surely don't have the heart to pick apart every line item. So what should I do? I'm impressed at the energy you put into this. I really can't comment on the rest of it, because there would be no end. "I would like the objects of computation to be fractions." What am I supposed to make of that? What is a computation? What is the object of a computation? In what way is 1/2, say, the object of a computation, but pi isn't? I could go on like this for every item but what would be the point? If that list of items encapsulates your philosophy of math, I'm happy for you. I'm happy for everyone's philosophy of math. Bertrand Russell's philosophy of math turned out to be wrong, but presumably he was happy with it. I don't think I'm going to be able to make you happy regarding that particular post.

Didn't say that, just got a little overwhelmed by all the line items. I'll take a look at it. — fishfry

Oh ok, that's great to hear. Yeah, sorry for the large number of line items...

But what about my point about constructivism? If you reject the noncomputable reals, you're a constructivist. — fishfry

You're right, I'm likely a constructivist/intuitionist. I say 'likely' because there's a lot of material to go through, and I need more time to fully understand it all. However, my views align with the key principles of constructivism. My main frustration with the material I've found so far is that it doesn't seem to address what I'm talking about...

Suppose I grant that you have some alternative construction of the reals. What of it? All models of the reals are isomorphic to one another. — fishfry

With my view, reals are constructed one at a time. It is impossible to construct ${\aleph_0}$ reals, let alone $2^{\aleph_0}$ reals. Given this, it's pretty clear that I'm not constructing the familiar reals.

My point exactly. If you have an alternative view of the reals, nothing changes. — fishfry

I think it is more correct to say that I have an alternate view of continua for which reals only play a supporting role. If mathematics were reformulated to be entirely absent of actual infinities would that be significant?

I don't see what ideas you've challenged. — fishfry

I'm working towards a foundation free of actual infinities.

It took weeks for me to understand your bundles, and just when I did, you took them away. — fishfry

Okay, I was too rash to take the bundles away. I think they're a useful way for us to find common ground. One thing I need to make clear though is that when I write (0,4) I'm referring to a bundle between point 0 and point 4. I'm not referring to $2^{\aleph_0}$ points each having a number associated with them. If I cut that bundle, a midpoint will emerge and I can assign to that point any number between 0 and 4. A number is only assigned when the cut is made. How does that sound?

"Binary numbers" aka real numbers can never be algorithms, since there are way too many of them. There are uncountably many reals and only countably many algorithms. — fishfry

I comment on this in the long post which you haven't responded to.

Sigh. I am not getting much from this latest post. — fishfry

Yeah, the other post of mine was more beefy.

Let me know when (or if) you have a system with formation rules, axioms and inference rules. — TonesInDeepFreeze

I think only at that time would we enjoy talking with each other.

Depends on what you mean by "applied". Ordinary mathematics, even as used for basic applied interests such as speed and acceleration use ordinary calculus, which is premised in infinite sets. — TonesInDeepFreeze

https://www.youtube.com/watch?v=jreGFfCxXr4

Oh ok, that's great to hear. Yeah, sorry for the large number of line items... — keystone

Your line items are helpful to you, and that is the ultimate goal. Technically it doesn't matter whether I ever understand your ideas or not, as long as I am useful as a sounding board. So if you will take the glass half full approach to my not relating to your charts and graphs and lists, then you can feel free to keep posting them and my eyeballs will feel free to be glazed.

You're right, I'm likely a constructivist/intuitionist. I say 'likely' because there's a lot of material to go through, and I need more time to fully understand it all. However, my views align with the key principles of constructivism. My main frustration with the material I've found so far is that it doesn't seem to address what I'm talking about... — keystone

Check out this guy.

https://en.wikipedia.org/wiki/L._E._J._Brouwer

I must say that in my modest studies of those subjects, constructivism seems more reasonable. I'm perfectly ok with working out the consequences of restricting the real numbers to the computable reals.

But intuitionism, with its active intelligence creating sequences as they go ... that's just a little out there for my taste.

Given this, it's pretty clear that I'm not constructing the familiar reals. — keystone

You're not constructing the familiar reals? First time I'm hearing this. Maybe you're constructing the computable reals. Is that what you're doing?

I think it is more correct to say that I have an alternate view of continua for which reals only play a supporting role. If mathematics were reformulated to be entirely absent of actual infinities would that be significant? — keystone

I'm pretty sure, but have no specific info about this, that people already decided you can't do analysis, that is calculus and the theory of the reals, without the axiom of infinity. But I could be wrong. I think if you could do analysis without the axiom of infinity that would be impressive.

I'm working towards a foundation free of actual infinities. — keystone

But infinities are one of the most fun and interesting part of math! I always liked infinities. I think I just don't understand the psychology of someone who doesn't like the axiom of infinity.

Tell me, what makes you interested in trying to do math without infinite sets?

Okay, I was too rash to take the bundles away. I think they're a useful way for us to find common ground. One thing I need to make clear though is that when I write (0,4) I'm referring to a bundle between point 0 and point 4. — keystone

I would interpret that as your intuition that the open intervals with rational endpoints are a basis for the usual topology on the reals. All the open sets are unions (perhaps infinite) of open intervals with rational endpoints. But then again ... do you allow infinite unions and intersections of sets? Do you want to get rid of infinitary operations as well as infinite sets?

I comment on this in the long post which you haven't responded to. — keystone

Did I miss a post? Or do you mean the long list of definitions and principles that glazed my eyes a bit?

By the way my eyes glaze over frequently at many things. It's nothing personal.


Yeah, the other post of mine was more beefy.[/quote]

Sorry I'm still confused. Did you mean the big list?

You give me link to some unidentified video so that I would have to take my time to watch through to find out, or guess, what it is you want me to know about it.

Your line items are helpful to you, and that is the ultimate goal. Technically it doesn't matter whether I ever understand your ideas or not, as long as I am useful as a sounding board. So if you will take the glass half full approach to my not relating to your charts and graphs and lists, then you can feel free to keep posting them and my eyeballs will feel free to be glazed. — fishfry

Trying to make my ideas clearer so that your eyes might not glaze over has indeed helped me collect my thoughts. I've also benefitted in other of your recommendations, such as construtivism which I really appreciate. So thanks for the glass half full. But there will come a point where no further progress can be made if I can't produce post that you are able to digest.

Check out this guy. — fishfry

I do plan to do a deeper investigation into Constructivism and certainly Brouwer will be a part of that. Thanks.

constructivism seems more reasonable...But intuitionism...that's just a little out there for my taste. — fishfry

I, on the other hand, am particularly drawn to intuitionism because I find it to be the least 'out there'. In this perspective, what exists are not infinite, eternal abstract objects in some inaccessible realm, but rather the finite set of objects currently being 'thought' by active computers. In my view, if the number 42 is not presently within the thoughts of any computer, then 42 does not currently exist.

You're not constructing the familiar reals? First time I'm hearing this. Maybe you're constructing the computable reals. Is that what you're doing? — fishfry

In line with my intuitionistic view, I'm not constructing any infinite set, rather constructing computable reals one cut at a time. More importantly, I can stop at any point and still have a working system. There's no need to complete the impossible task of constructing all the real numbers...after all, computers do math without ever having the complete set of real numbers in memory.

I'm pretty sure, but have no specific info about this, that people already decided you can't do analysis, that is calculus and the theory of the reals, without the axiom of infinity. But I could be wrong. I think if you could do analysis without the axiom of infinity that would be impressive. — fishfry

I disagree with this decision. I believe it is possible to perform analysis without relying on the axiom of infinity. While I don't have formal rules or detailed structures yet, I possess concepts that would be found in an introductory calculus textbook, or perhaps an introductory engineering calculus textbook. Admittedly, this is a significant claim that requires substantial support...it's just that your eyes glaze over...

But infinities are one of the most fun and interesting part of math! I always liked infinities. I think I just don't understand the psychology of someone who doesn't like the axiom of infinity.

Tell me, what makes you interested in trying to do math without infinite sets? — fishfry

Cantor's proofs are quite fascinating. Many people, often labeled as "infinity cranks," argue that actual infinities are riddled with contradictions. These individuals are in the minority, as most mathematicians do not share this view. I'm intrigued by the idea of a mathematics that does not rely on actual infinity, as I believe this approach is more aligned with true mathematics. It promises to be free of contradictions and brings with it the potential for beauty and advancement.

I would interpret that as your intuition that the open intervals with rational endpoints are a basis for the usual topology on the reals. All the open sets are unions (perhaps infinite) of open intervals with rational endpoints. But then again ... do you allow infinite unions and intersections of sets? Do you want to get rid of infinitary operations as well as infinite sets? — fishfry

Why do you talk of everything, such as 'all the open sets'? I can't imagine a computer holding this infinite set in memory. I'd rather talk about what I know is possible, such as a computer which holds a few open intervals with rational endpoints. As for infinitary operations, my long post with many bullets (let's call it the bullet post) addresses my view.

Did I miss a post? Or do you mean the long list of definitions and principles that glazed my eyes a bit? — fishfry

Yes, the bullet post.

Sorry I'm still confused. Did you mean the big list? — fishfry

Yes, the bullet post.

I possess concepts that would be found in an introductory calculus textbook — keystone

Such books don't axiomatize the principles used. And those books make use of infinite sets.

I think even constructivist and intuitionist set theories have a version of the axiom of infinity. But the logic of those systems is different from classical logic, so a statement in one system might not mean what it means in another system with a different logic.

I'd like to know whether a "no complete, only potential, infinity" concept has been axiomatized in a way that would be to the satisfaction of cranks if they were ever to actually learn about such things.

Also, keep in mind the amount of complication an alternative axiomatization might be. Already with intuitionistic logic, the semantics is much more complicated than with classical logic. Of course, that price might be worth paying.

Such books don't axiomatize the principles used. And those books make use of infinite sets. — TonesInDeepFreeze

That's what I was trying to convey—I don't have axioms and formal rules, but rather concepts relevant to a typical practitioner of calculus. My approach would achieve the same methods as traditional calculus but without relying on actual infinities. I agree that current calculus texts depend on actual infinity, which I believe is both unnecessary and undesirable.

I think even constructivist and intuitionist set theories have a version of the axiom of infinity. — TonesInDeepFreeze

But isn't it more like a potential infinity?

I'd like to know whether a "no complete, only potential, infinity" concept has been axiomatized in a way that would be to the satisfaction of cranks if they were ever to actually learn about such things. — TonesInDeepFreeze

I wonder if what is missing isn't the axiomatic systems—of which constructivists offer many—but rather the relevant and accessible intuitions for a typical practitioner of calculus.

You give me link to some unidentified video so that I would have to take my time to watch through to find out, or guess, what it is you want me to know about it. — TonesInDeepFreeze

To be honest it's because I didn't feel like continuing the dialogue with you because I find some of your comments offensive.

Trying to make my ideas clearer so that your eyes might not glaze over has indeed helped me collect my thoughts. I've also benefitted in other of your recommendations, such as construtivism which I really appreciate. So thanks for the glass half full. But there will come a point where no further progress can be made if I can't produce post that you are able to digest. — keystone

I don't see why. One of the best ways you can respond to someone who brings a problem to you is to just ask them to explain it all to you in detail. By the time they're halfway done they usually solve it themselves. It's their explaining that does the trick, not my understanding. Smart bosses do that.

Also, the thing is, and I thought of mentioning this to you the other day, I don't actually care about anyone's ideas about what the real numbers are or how math should work. There's nothing at stake for me here. I enjoy trying to relate your ideas with things I know in math, but there's never going to come a point where I "digest" this. All this is a very half-baked brew, to mix metaphors.

I do plan to do a deeper investigation into Constructivism and certainly Brouwer will be a part of that. Thanks. — keystone

Welcome. If you think everything's generated by algorithms, constructivism's your math philosophy.

I, on the other hand, am particularly drawn to intuitionism because I find it to be the least 'out there'. In this perspective, what exists are not infinite, eternal abstract objects in some inaccessible realm, but rather the finite set of objects currently being 'thought' by active computers. In my view, if the number 42 is not presently within the thoughts of any computer, then 42 does not currently exist. — keystone

Computers don't have thoughts. In fact, there aren't any numbers in computers. There are electrical circuits. It's the humans that interpret certain bit patterns in certain circuits as numbers. The Chinese room does not understand Chinese.

Let me ask you a question. Suppose there is a computer in the world that contains, at this moment, a bit pattern corresponding to the character string "XLII". Then suppose there's another computer somewhere else in the world, and it contains the bit pattern 101010. And in yet a third computer, we find a bit pattern corresponding to the character string "42".

Do these three computers each instantiate the existence of the same number 42? And how would you know?

I am drawing your attention to the problem of representation. There is never a number inside the computer. There are only bit patterns. And depending on the encoding scheme, the same bit pattern may mean different things; and different bit patterns may mean the same thing.

In line with my intuitionistic view, I'm not constructing any infinite set, rather constructing computable reals one cut at a time. More importantly, I can stop at any point and still have a working system. There's no need to complete the impossible task of constructing all the real numbers...after all, computers do math without ever having the complete set of real numbers in memory. — keystone

You can do engineering like that, but you can't do math. In finitism (rejecting the axiom of infinity) we can do a fair amount of number theory, but not analysis. You can't do calculus, you can't do physics. You can do finite approximations, but the underlying theory is infinitary.

I disagree with this decision. I believe it is possible to perform analysis without relying on the axiom of infinity. — keystone

You should research that claim rather than just proclaim it. This is one of the reasons I am never going to "digest" your ideas. Many clever people have given these matters considerable thought. You should do a literature search on this idea to clarify your thinking.

Can you see that grandiose claims made without sufficient background come down to untrained feelings and intuitions? Not that there's anything wrong with that. But it supports my belief that there is nothing to digest.

While I don't have formal rules or detailed structures yet, I possess concepts that would be found in an introductory calculus textbook, or perhaps an introductory engineering calculus textbook. Admittedly, this is a significant claim that requires substantial support...it's just that your eyes glaze over... — keystone

There's a mathematician and and engineer joke in there somewhere. And eye glazing is something else. I think you have a bad idea, not in the sense that it's absolutely wrong; but in the sense that you have a very naive understanding of what's involved, so that it seems grandiose.

But that is NOT what makes my eyes glaze. Certain diagrams and definitely long lists make my eyes glaze. Eye glazing is entirely independent of reasonableness. Your claim about calculus is unreasonable. That doesn't make my eyes glaze. The diagrams and lists make my eyes glaze. Hope that's clear. Someone could show me a diagram or list that was 100% correct and brilliant, and my eyes would still glaze. Eye glazing is no measure of the quality of an idea, it's just how I process or have difficulty processing it. But grandiose claims that you personally have figured out how to do analysis in the absence of the axiom of infinity, that's a bad idea. My eyes are perfectly clear about that.

Hope I made this distinction. Some of the best ideas make my eyes glaze.

Cantor's proofs are quite fascinating. Many people, often labeled as "infinity cranks," argue that actual infinities are riddled with contradictions. These individuals are in the minority, as most mathematicians do not share this view. I'm intrigued by the idea of a mathematics that does not rely on actual infinity, as I believe this approach is more aligned with true mathematics. It promises to be free of contradictions and brings with it the potential for beauty and advancement. — keystone

Why? Infinitary set theory is perfectly clear of contradictions. Well, as far as we know. We can't prove its consistency without assuming more powerful systems.

Are you a Cantor crank by any chance? A long time ago (when I was younger and I suppose less harmless looking than I am these days) a traffic cop pulled me over. As he was writing my ticket, he asked me if I'd ever been arrested. "I just want to know who I'm dealing with," he said. That's why I ask if you're a Cantor crank. I just want to know who I'm dealing with.

Why do you talk of everything, such as 'all the open sets'? I can't imagine a computer holding this infinite set in memory. I'd rather talk about what I know is possible, such as a computer which holds a few open intervals with rational endpoints. As for infinitary operations, my long post with many bullets (let's call it the bullet post) addresses my view. — keystone

The open sets were your idea. And the standard topology on the reals is generated by the open intervals with rational endpoints. There are only countably many of those.

Nobody's saying you can't approximate things with computers. You're imagining some kind of mathematical metaphysics that isn't really there. But the constructivists have this all worked out. Did I post this Andrej Bauer article, Five Stages Of Accepting Constructive Mathematics? Give it a read, tell me if any of it makes sense to you. I read it a while back, don't remember much. I never get very far trying to understand constructivists.

https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf

Yes, the bullet post. — keystone

Ok good, thanks.

One of the best ways you can respond to someone who brings a problem to you is to just ask them to explain it all to you in detail. — fishfry

Agreed. But sometimes the person will not see the flaw in their argument unless explicitly identified by someone else.

There's nothing at stake for me here. I enjoy trying to relate your ideas with things I know in math, but there's never going to come a point where I "digest" this. — fishfry

By "digest" I didn't mean to suggest that you would accept it. But there's value in being able to entertain a thought without accepting it.

Do these three computers each instantiate the existence of the same number 42? And how would you know? — fishfry

If a human thinks of a duck and somehow in their computations the duck behaves exactly like the mathematical object 42, then (within that person's thoughts) the duck represents an instance of the number 42. As the old saying goes, "If it swims like a 42, and quacks like a 42, then it probably is a 42." I think we both agree that absent of an intentional being giving mathematical meaning to the duck (or to electrical activity within a computer), no mathematics is going on.

But if at a later time the human's thoughts of the duck do not correspond to the number 42, then the duck is no longer an instance of 42. The number 42 is contingent on thought. It's existence is temporary.

If we frame our views within this context, the difference is that you believe in an infinite consciousness whose thoughts eternally encompass all numbers. On the other hand, I believe there is no such preferred consciousness; rather, there are only finite consciousnesses whose thoughts can hold only a finite number of numbers at any given time.

In finitism (rejecting the axiom of infinity) we can do a fair amount of number theory, but not analysis. You can't do calculus, you can't do physics. You can do finite approximations, but the underlying theory is infinitary. — fishfry

Computers perform calculus, and everything they do is finite. So, you're essentially arguing that there's a disconnect between the theory and the practice. Remember, in the case of calculus, the practice came first, and mathematicians later developed an actual-infinity-based theory to justify the practice. Might it be possible that a potential-infinity-based theory could provide a better justification for the practice? This one-minute video by Joscha Bach, titled "Before Constructive Mathematics, People Were Cheating," eloquently captures my view: https://www.youtube.com/watch?v=jreGFfCxXr4

You should research that claim rather than just proclaim it. This is one of the reasons I am never going to "digest" your ideas. Many clever people have given these matters considerable thought. You should do a literature search on this idea to clarify your thinking. — fishfry

While I haven't done much research on logic, I have a reasonably strong grasp of basic classical calculus. I understand that continuity is essential for classical calculus—my view starts with continua. I also understand that limits are essential for classical calculus—my view achieves the same ends by using arbitrariness. If you don't want to entertain my ideas simply because clever people weren't able to make calculus work within a finitist framework, that's fine as well. But let's be clear—it's not that you can't digest my ideas; it's that you won't entertain them.

Can you see that grandiose claims made without sufficient background come down to untrained feelings and intuitions? Not that there's anything wrong with that. But it supports my belief that there is nothing to digest. — fishfry

I understand how my claims appear. I'd like to support my position but it's quite hard if you don't look at my figures or words. You ask for the beef but the only comments you respond to are the bun.

I think you have a bad idea, not in the sense that it's absolutely wrong; but in the sense that you have a very naïve understanding of what's involved, so that it seems grandiose. — fishfry

I believe my view is naïve in the same sense that Naïve Set Theory is naïve (minus the contradictions).

Why? Infinitary set theory is perfectly clear of contradictions. Well, as far as we know. — fishfry

Joscha Bach seems quite confident that classical mathematics is filled with contradictions.

Are you a Cantor crank by any chance? — fishfry

You’ve probably heard the story of Penzias and Wilson, who struggled with persistent background noise on their radio receiver, initially attributing it to pigeon droppings. It turned out to be the cosmic microwave background radiation from the Big Bang, earning them a Nobel Prize. I believe Cantor has interpreted his incredible discoveries as mere pigeon droppings.

The open sets were your idea. — fishfry

I don't think I mentioned open sets.

Did I post this Andrej Bauer article, Five Stages Of Accepting Constructive Mathematics? — fishfry

Funny you mention this. I skimmed through it a few days ago and then watched his YouTube lecture by that name yesterday. Now, I'm in the middle of his lecture on LEM. I'm really excited about watching his lectures.

Someone could show me a diagram or list that was 100% correct and brilliant, and my eyes would still glaze. — fishfry

Is your preferred format essay?? How did you become a mathematician and not an english major? But seriously, how am I supposed to communicate my ideas to you? This might not be the best chat forum etiquette, but would you be open to a Google Hangout? ...Please feel no need to even respond to that idea...

Agreed. But sometimes the person will not see the flaw in their argument unless explicitly identified by someone else. — keystone

I am trying to understand you.

By "digest" I didn't mean to suggest that you would accept it. But there's value in being able to entertain a thought without accepting it. — keystone

I'm trying to understand your argument or thesis or idea.

If a human thinks of a duck and somehow in their computations the duck behaves exactly like the mathematical object 42, then (within that person's thoughts) the duck represents an instance of the number 42. As the old saying goes, "If it swims like a 42, and quacks like a 42, then it probably is a 42." I think we both agree that absent of an intentional being giving mathematical meaning to the duck (or to electrical activity within a computer), no mathematics is going on. — keystone

You said that numbers get instantiated when they appear in a computation. I asked you whether one number or several numbers get instantiated when various representations exist. Who determines that they act the same? Where is that process, that brings a number into existence?

But if at a later time the human's thoughts of the duck do not correspond to the number 42, then the duck is no longer an instance of 42. The number 42 is contingent on thought. It's existence is temporary. — keystone

Is God watching all this and keeping track of everyone's version of each number? This seems like a cumbersome idea.

If we frame our views within this context, the difference is that you believe in an infinite consciousness whose thoughts eternally encompass all numbers. On the other hand, I believe there is no such preferred consciousness; rather, there are only finite consciousnesses whose thoughts can hold only a finite number of numbers at any given time. — keystone

I believe no such thing, what are you talking about? I believe in the axioms of ZF and not much else. They are purely a human artifact.

Computers perform calculus, and everything they do is finite. So, you're essentially arguing that there's a disconnect between the theory and the practice. Remember, in the case of calculus, the practice came first, and mathematicians later developed an actual-infinity-based theory to justify the practice. Might it be possible that a potential-infinity-based theory could provide a better justification for the practice? This one-minute video by Joscha Bach, titled "Before Constructive Mathematics, People Were Cheating," eloquently captures my view: https://www.youtube.com/watch?v=jreGFfCxXr4 — keystone

Yes, that's how constructivists think. Thanks for telling me it was a short vid, you got me to watch it. Math doesn't need a justification. It doesn't have to make constructivists happy.

While I haven't done much research on logic, I have a reasonably strong grasp of basic classical calculus. I understand that continuity is essential for classical calculus—my view starts with continua. I also understand that limits are essential for classical calculus—my view achieves the same ends by using arbitrariness. If you don't want to entertain my ideas simply because clever people weren't able to make calculus work within a finitist framework, that's fine as well. But let's be clear—it's not that you can't digest my ideas; it's that you won't entertain them. — keystone

You are making a grandiose claim that's likely to be false. But there are plenty of productive finitists, and the constructivists are on the march these days due to computer proof systems. But there is something to be said for infinitary math. Why shouldn't we enjoy having such a lovely theory of the infinite? What is the harm?

I understand how my claims appear. I'd like to support my position but it's quite hard if you don't look at my figures or words. You ask for the beef but the only comments you respond to are the bun. — keystone

LOL. It's hard to develop a theory of the reals without the axiom of infinity. The figures and words haven't helped much so far. You say, Start with a line. Make a cut. I don't know what these things are. You're just approximating the reals. I don't see anything to grab on to.

I believe my view is naïve in the same sense that Naïve Set Theory is naïve (minus the contradictions). — keystone

Your idea isn't naive. It's grandiose. And let's talk about something else please.

Joscha Bach seems quite confident that classical mathematics is filled with contradictions. — keystone

You say that like it's a bad thing!!

Paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating[clarification needed] way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" systems of logic, which reject the principle of explosion. — Wiki

https://en.wikipedia.org/wiki/Paraconsistent_logic

But some guy thinks standard math has contradictions. He could be right. And if it did, the contradictions would be repaired. People wouldn't stop doing infinitary math. If a contradiction were found in ZF, it wouldn't affect group theory or differential geometry .

You’ve probably heard the story of Penzias and Wilson, who struggled with persistent background noise on their radio receiver, initially attributing it to pigeon droppings. It turned out to be the cosmic microwave background radiation from the Big Bang, earning them a Nobel Prize. I believe Cantor has interpreted his incredible discoveries as mere pigeon droppings. — keystone

I don't follow the analogy you're making. Cantor has underestimated or overestimated his discoveries?

I don't think I mentioned open sets. — keystone

You have been making use of open intervals all along, haven't you?

Funny you mention this. I skimmed through it a few days ago and then watched his YouTube lecture by that name yesterday. Now, I'm in the middle of his lecture on LEM. I'm really excited about watching his lectures. — keystone

Ok. Well if I turned you on to constructivism I'm happy and I've done some good. But I can't go down that road with you too far, because I have tried to understand constructivism a few times and it just doesn't speak to me. I like infinitary math and I think that if you reject the noncomputable, you are missing a lot.

You know, that's a particular bit of philosophy I can assert. The noncomputable reals are telling us something. Infinitary math is telling us something. The history of math is expansive, never contractive. Nobody says, "Those complex numbers, they were a step too far." But they say that about infinitary math.

Is your preferred format essay?? How did you become a mathematician and not an english major? — keystone

Well I didn't become a mathematician! I got to grad school and my eyes glazed. Well there were a lot of things going on. I ended up in programming. In math when you're stuck, you're stuck. In programming you can always code something, get something running, solve a bug, do something useful. I don't have an eye glaze factor when I'm coding, but I do when I do math. Interesting, I never thought of it that way but it's true.

But seriously, how am I supposed to communicate my ideas to you? — keystone

Are you getting frustrated? I'm sorry, I thought I was helping the best I can. You're doing fine, I've understood a lot. It's a tall order to reformulate analysis without the axiom of infinity. Even constructivists have infinite sets, not not noncomputable ones. It's like if you told me you were going to do brain surgery. You might be able to learn to do it, but you are not there yet as far as I can see.

This might not be the best chat forum etiquette, but would you be open to a Google Hangout? ...Please feel no need to even respond to that idea... — keystone

No thanks, this forum's all I can handle and barely at that. Plus I hate Google. They went from Don't be evil to being evil.

I don't know why you are acting as if I'm not attending to what you say. I sense a difference of perspective that I'm not privy too. Everything seems fine at my end.

You said that numbers get instantiated when they appear in a computation. I asked you whether one number or several numbers get instantiated when various representations exist. Who determines that they act the same? Where is that process, that brings a number into existence? — fishfry

I was trying to convey that the representation itself is not important; what matters is the behavior. If in my mind x+x=2, then x behaves like 1. Similarly, if y+y=2, then y also behaves like 1. In this scenario, 1 has multiple representations (x and y) in my mind, but that isn't an issue because they both behave the same.

But I must highlight that to conclude that x=1, I don’t work through an infinite checklist, considering all possible arithmetic equations involving 1. No, I'm mindful of the consistent and finite set of rules associated with the construction and arithmetic of the SB-tree (or equivalent tree), so all I need to do is declare that x will behave like the node occupied by 1. At that moment, I bring 1 into existence and it is representation in my mind is the character x.

Is God watching all this and keeping track of everyone's version of each number? This seems like a cumbersome idea. — fishfry

We each are the god of the mathematical systems that inhabit our own minds. If we want to compare my (-inf,1) U 1 U (1,+inf) with your (-inf,42) U 42 U (42,+inf) we need to agree to the SB-tree and compare the nodes where my 1 and your 42 lie. If they correspond to the same node, then our systems are equivalent. While nobody explicitly does this, it's the unspoken agreement we make when comparing systems. I don't see why we would need a third party to arbitrate the comparison.

I believe no such thing, what are you talking about? I believe in the axioms of ZF and not much else. They are purely a human artifact. — fishfry

Since no human artifact can be infinite, is it fair to say that you believe in the axioms but not in the infinite objects they describe? If so, this directly supports my thesis—forget about the existence of infinite sets and instead focus on the (Turing) algorithms designed to generate these infinite sets.

It's hard to develop a theory of the reals without the axiom of infinity.....Even constructivists have infinite sets, not not noncomputable ones. — fishfry

I'm receptive to a constructivist approach to the axiom of infinity. If were talking about computable infinite sets in the same way that Turing talked about computable real numbers I have no problems, provided we do not assert the existence of infinite objects.

But there is something to be said for infinitary math. Why shouldn't we enjoy having such a lovely theory of the infinite? What is the harm? — fishfry

What harm is there in relying on Newtonian mechanics when it performs admirably for slow-moving objects like ourselves? Similarly, what harm is there in embracing General Relativity when the singularities it predicts are distant from our everyday experience? There's a certain beauty in their simplicity; as a mechanical engineer, I rely on Newtonian mechanics daily and will continue to do so regardless of advances in physics.

Yet, as physicists began pulling the loose threads of classical physics, a more fundamentally robust and aesthetically compelling framework emerged: Quantum Mechanics. There is something to be said for pulling the loose threads.

You say, Start with a line. Make a cut. I don't know what these things are. — fishfry

It would be much easier if you would just roll with the intuitions for a little while so we wouldn't get stuck on the first step. Let's sweep through the whole idea informally and if it has any merit then we can sweep through again and formally define things. Think of a line as a piece of string. Think of a cut as what you do with scissors to partition the string. You're making this more complicated than it needs to be.

You're just approximating the reals. — fishfry

No, I'm not. Yes, I'm referring to (Turing) algorithms that produce rational approximations with arbitrary precision, but the algorithm itself is exact. The algorithm perfectly encapsulates the essence of the real. That's why I'm emphasizing the algorithm itself, not its output.

And if it did, the contradictions would be repaired. People wouldn't stop doing infinitary math. — fishfry

I agree to both sentences! (1) That's what I'm trying to do and (2) I'm just trying to throw a 'potential' in front of the 'infinitary math'.

I don't follow the analogy you're making. Cantor has underestimated or overestimated his discoveries? — fishfry

Cantor has already received considerable acclaim, making it difficult to envision greater recognition for him. What I meant to convey is that Cantor unearthed something monumental, yet his interpretation was poop (actual infinities). I believe that in the future, it will be recognized that his true discovery lay in articulating the potential within continua and mathematics as a whole.

You have been making use of open intervals all along, haven't you? — fishfry

I'm using open interval notation to describe the bundle (line) the lies between its endpoints. This bundle cannot be described as an infinite set of individual points, because as I mentioned before, we can only talk about individual points when the line has been cut. For this reason, I'm reluctant to say that I've been proposing open sets.

The noncomputable reals are telling us something. Infinitary math is telling us something. — fishfry

I agree with this sentiment. Whether it's noncomputable reals, the halting problem, Gödel's incompleteness theorems, or the liar's paradox, they are all screaming at us that there is a potential in mathematics that cannot be fully actualized. But Classical mathematics aims to actualize everything, much like classical physics. They both suffer the same flaw...and I believe are both addressed with the same resolution: a top-down view.

The history of math is expansive, never contractive. Nobody says, "Those complex numbers, they were a step too far." But they say that about infinitary math. — fishfry

Yeah, you're going to lose some things with constructive mathematics, be it LEM or the axiom of choice. But by and large I'm proposing a much more beautiful structure. Just as classical physics was a natural stepping stone to QM, actual-infinitary-math is a natural stepping stone to potential-infinitary-math.

I don't have an eye glaze factor when I'm coding, but I do when I do math. — fishfry

I can reframe my examples in Python if that's your preference. The main drawback however is that my posts would get longer.

Are you getting frustrated? I'm sorry, I thought I was helping the best I can. — fishfry

It's frustrating to think that I'm running out of ways to communicate my ideas so I'm starting to think that the conversation might end prematurely with the least desirable conclusion (that you don't know whether my ideas are right or wrong). But perhaps it's too soon to talk about the end. I'm getting value out of every post you and I write so I'd be grateful if we keep going and just take it one day at a time.

I don't know why you are acting as if I'm not attending to what you say. I sense a difference of perspective that I'm not privy too. Everything seems fine at my end. — fishfry

It's just that at some point we'll need to talk beef and I'll need to figure out an alternative way to communicate the bullet post.

I was trying to convey that the representation itself is not important; what matters is the behavior. If in my mind x+x=2, then x behaves like 1. Similarly, if y+y=2, then y also behaves like 1. In this scenario, 1 has multiple representations (x and y) in my mind, but that isn't an issue because they both behave the same. — keystone

Yes but who determines that? You said that numbers come into existence when some computer represents them. If one computer represents "XLII" and other represents "42", does your creation engine instantiate one or two numbers? How does God or whoever decides what numbers are instantiated, immediately know whether those two expressions always behave the same? If you think about it it's a hard problem.

But I must highlight that to conclude that x=1, I don’t work through an infinite checklist, considering all possible arithmetic equations involving 1. No, I'm mindful of the consistent and finite set of rules associated with the construction and arithmetic of the SB-tree (or equivalent tree), so all I need to do is declare that x will behave like the node occupied by 1. At that moment, I bring 1 into existence and it is representation in my mind is the character x. — keystone

"... the consistent and finite set of rules ..."

Oh, you use an axiomatic BOTTOM UP system after all!!!!!

We each are the god of the mathematical systems that inhabit our own minds. If we want to compare my (-inf,1) U 1 U (1,+inf) with your (-inf,42) U 42 U (42,+inf) we need to agree to the SB-tree and compare the nodes where my 1 and your 42 lie. If they correspond to the same node, then our systems are equivalent. While nobody explicitly does this, it's the unspoken agreement we make when comparing systems. I don't see why we would need a third party to arbitrate the comparison. — keystone

If you have a set of rules (a bottom up concept) that let you know when two representations denote the same number, then why do you need the computer? Why not just accept that the rules themselves bring all possible numbers into existence already?

Since no human artifact can be infinite, is it fair to say that you believe in the axioms but not in the infinite objects they describe? — keystone

I believe in the mathematical existence of the abstract objects they describe. I make no claim that they are physically real. Just like when I read Moby Dick I accept the existence of Ahab and the whale, without making any ontological commitments outside of the context of the novel.

If so, this directly supports my thesis—forget about the existence of infinite sets and instead focus on the (Turing) algorithms designed to generate these infinite sets. — keystone

There simply aren't enough algorithms to generate all the sets. There are countably many algos and uncountably many subsets of the natural numbers.

I'm receptive to a constructivist approach to the axiom of infinity. If were talking about computable infinite sets in the same way that Turing talked about computable real numbers I have no problems, provided we do not assert the existence of infinite objects. — keystone

Ok. You should study constructivism. I can be of no assistance, I don't know much about it.

What harm is there in relying on Newtonian mechanics when it performs admirably for slow-moving objects like ourselves? — keystone

Completely wrong analogy. Infinitary math always works. It just doesn't apply to physics (as far as we know). And if it DOES someday turn out to be important for physics, it will be a good thing if the pure mathematicians have been studying it all along.

Just as Riemann developed non-Euclidean geometry 70 years before Einstein needed it for general relativity.

Or just like Diophantus studied number theory some 2200 years before public key cryptography became the foundation of online commerce.

Similarly, what harm is there in embracing General Relativity when the singularities it predicts are distant from our everyday experience? There's a certain beauty in their simplicity; as a mechanical engineer, I rely on Newtonian mechanics daily and will continue to do so regardless of advances in physics. — keystone

Then you are agreeing with my point. I read something interesting once about sailing. Sailboat technology did not get really good until sail was no longer a useful means of practical transportation. Today's high-tech sailboats were developed by purely recreational sailors, after sailboats became obsolete in commerce.

Yet, as physicists began pulling the loose threads of classical physics, a more fundamentally robust and aesthetically compelling framework emerged: Quantum Mechanics. There is something to be said for pulling the loose threads. — keystone

I disagree with the analogy. Constructivism is not a deeper form of conventional math. But I can see the argument being made. Just not by me.

It would be much easier if you would just roll with the intuitions for a little while so we wouldn't get stuck on the first step. Let's sweep through the whole idea informally and if it has any merit then we can sweep through again and formally define things. Think of a line as a piece of string. Think of a cut as what you do with scissors to partition the string. You're making this more complicated than it needs to be. — keystone

I'm fine with cutting strings. You have never explained to me how this serves as a new foundation for math.

No, I'm not. Yes, I'm referring to (Turing) algorithms that produce rational approximations with arbitrary precision, but the algorithm itself is exact. The algorithm perfectly encapsulates the essence of the real. That's why I'm emphasizing the algorithm itself, not its output. — keystone

That's fine, you're making a constructivist argument. I'm the wrong person to have that discussion with, since I have no feel for constructivism, despite making a run at the subject several times.

I agree to both sentences! (1) That's what I'm trying to do and (2) I'm just trying to throw a 'potential' in front of the 'infinitary math'. — keystone

I'm not stopping you. You have a ways to go in terms of developing a comprehensive, logical theory that I can understand. It could just be me.

Cantor has already received considerable acclaim, making it difficult to envision greater recognition for him. What I meant to convey is that Cantor unearthed something monumental, yet his interpretation was poop (actual infinities). I believe that in the future, it will be recognized that his true discovery lay in articulating the potential within continua and mathematics as a whole. — keystone

That's a stretch. Please don't be a Cantor crank. I'd be disillusioned. If you are one, it's better to keep it to yourself.

I'm using open interval notation to describe the bundle (line) the lies between its endpoints. This bundle cannot be described as an infinite set of individual points, because as I mentioned before, we can only talk about individual points when the line has been cut. For this reason, I'm reluctant to say that I've been proposing open sets. — keystone

So after all this time your interval notation does not not stand for its conventional meaning?

How nice of you to let me know.

Can you see that in terms of communication, you could have either made that clear up front, or defined a different notation, like <a,b>. Can you see why I've been utterly confused for weeks?

I agree with this sentiment. Whether it's noncomputable reals, the halting problem, Gödel's incompleteness theorems, or the liar's paradox, they are all screaming at us that there is a potential in mathematics that cannot be fully actualized. But Classical mathematics aims to actualize everything, much like classical physics. They both suffer the same flaw...and I believe are both addressed with the same resolution: a top-down view. — keystone

I would say that the noncomputable reals show us the limitations of algorithms. And, being a skeptic of simulation theory and the trendy thesis that human minds are Turing machines, this is an important plank in my platform. Algorithms are vastly insufficient to express what's important, either about math or reality. This is perhaps why the constructivists annoy me.

Yeah, you're going to lose some things with constructive mathematics, be it LEM or the axiom of choice. But by and large I'm proposing a much more beautiful structure. Just as classical physics was a natural stepping stone to QM, actual-infinitary-math is a natural stepping stone to potential-infinitary-math. — keystone

I have no doubt you have a beautiful structure in mind. Your challenge is to express it. Maybe I'm a poor sounding board.

I can reframe my examples in Python if that's your preference. The main drawback however is that my posts would get longer. — keystone

Jeez people's code fragments definitely make my eyes glaze. No code please.

It's frustrating to think that I'm running out of ways to communicate my ideas so I'm starting to think that the conversation might end prematurely with the least desirable conclusion (that you don't know whether my ideas are right or wrong). — keystone

No risk of that. I already know your ideas are wrong. Hey you set yourself up for that one :-)

But perhaps it's too soon to talk about the end. I'm getting value out of every post you and I write so I'd be grateful if we keep going and just take it one day at a time. — keystone

That's fine with me. I can't promise to meet your expectations about what you wish I'd write. I've responded sensibly and I think relevantly to each point you've made. I don't have to agree with your overall point of view, or even understand it.

But of course you could start by defining your notation.

Those parenthesis (a,b) don't stand for standard open sets. Can you see that you redefined a notation that's so universal that I had no choice but to be sent into a state of confusion about your meaning? Can you see why I've been confused for weeks?

It's just that at some point we'll need to talk beef and I'll need to figure out an alternative way to communicate the bullet post. — keystone

I make no commitment to meet your conversational expectations. What you see is what you get. You have expectations about my state of mind that are unlikely to be met. I'd ask you to accept that rather than continually expressing disappointment with my posts.

According to the Buddhists, unfulfilled expectations are the source of suffering.

Like I say, if you want to communicate, don't redefine extremely well-known standard notations without announcing it clearly.

And also, if I could make this request ... can you write shorter posts? Short and to the point.

Well I didn't become a mathematician! I got to grad school and my eyes glazed. — fishfry

In my first semester as a grad math student I was required to take a course called Introduction to Graduate Mathematics. It was basically naive set theory and at the end the professor said, "You should only continue in foundations if this course really appeals to you. How many of you find that to be the case?" I recall out of thirty students one or two hands went up. The rest of us wiped the glaze out of our eyes and went on into other areas of mathematics.

But this was 1962, back in the dark ages. And at a state university, not a top-notch school, like Harvard. Things have changed since then. When were you in a grad math program?

I agree with this sentiment. Whether it's noncomputable reals, the halting problem, Gödel's incompleteness theorems, or the liar's paradox, they are all screaming at us that there is a potential in mathematics that cannot be fully actualized. But Classical mathematics aims to actualize everything — keystone

Not true. I published papers when I was active that never assumed infinity was actualized. Fryfish and I, sometime back, argued about the use of transfinite math in analysis, particularly functional analysis. He pointed to the use of Zorn's lemma or the axiom of choice as a required tool to prove the Hahn-Banach theorem, and I replied that that was true, but by altering the hypotheses slightly, they were not required. Hahn-Banach was my only very brief encounter with transfinites in my career. But then I sought interesting theory in classical analysis - a far cry from foundations. So, your statement is not entirely correct.

But I admire your tenacity.

And also, if I could make this request ... can you write shorter posts? Short and to the point. — fishfry

I believe the main issue is that new topics are added more often than old ones are removed, leading to bloated posts. I'll not respond to a few of your comments to address this request.

I'm fine with cutting strings. You have never explained to me how this serves as a new foundation for math. — fishfry

Label the original string (-inf,+inf). Cut it somewhere. Label the left partition (-inf,42). Label the right partition (42,+inf). Label the small gap between the strings 42. Now you have a new system: (-inf,42) U 42 U (42,+inf). But you seem to get hung up on those intervals number being continuous even though I'm saying that those intervals describe continua - abstract string in this case.

If you have a set of rules (a bottom up concept) that let you know when two representations denote the same number, then why do you need the computer? Why not just accept that the rules themselves bring all possible numbers into existence already? — fishfry

Moving forward, instead of writing "computer+mind", I'm just going to write "computer".

I believe that true mathematical rules exist independently of computers. These rules are necessary truths and finite in number. If one assumes they describe actually existing objects, such objects must exist beyond our comprehension, as no computer could contain them.

However, if we assume that mathematical objects must exist within a computer, then not all mathematical objects can actually exist and it becomes a matter of a computer choosing which objects to actualize.

I believe in the mathematical existence of the abstract objects they describe. — fishfry

Please allow me to use the SB-tree as something concrete to talk around. I acknowledge that any infinite complete tree will do.

We outline the rules for constructing the SB-tree and can mentally construct it to an arbitrary depth. Everything we ever actually construct is finite. Why insist on believing in the computationally impossible — the existence of the complete SB-tree?

There simply aren't enough algorithms to generate all the sets. There are countably many algos and uncountably many subsets of the natural numbers. — fishfry

You've said that the reals correspond to unending paths down the infinite complete binary tree, so indeed, there are potentially $2^{\aleph_0}$ paths that cannot be algorithmically defined. This doesn't mean the rules for constructing the tree are incomplete; it simply means there are paths computers can never traverse. Computers cannot exhaust these rules.

Or here's how I see it. When I see the tree, I do not see paths and nodes. Instead I see a continua at each row, being cut by the numbers at each row. For example, I see the top two rows of the SB-tree as:

Row 1: 0 U (0,1) U 1 U (1,+inf)
Row 2: 0 U (0,1/2) U 1/2 U (1/2,1) U 1 U (1,2) U 2 U (2,+inf)
...

With this view, I would rephrase the conclusion as follows: computers cannot completely cut continua. Computers cannot exhaust cutting. Actually, I would go one step further and assume that computers are all that's available, so I would simply say that continua cannot be completely cut. But we know that already, you'll never cut a string to the point where it vanishes.

I'd ask you to accept that rather than continually expressing disappointment with my posts. — fishfry

Sometimes I push back as a form of defense. Nevertheless I'll try and be more mindful of this. I'm very appreciative of our conversation. Thanks!

I'm putting this comment in a separate post because I wanted my main post to be 'smallish' and I'm not expecting a response from this.

So after all this time your interval notation does not not stand for its conventional meaning? — fishfry

Here are quotes from my earlier posts. You don't have to read all bullets as they all say the same thing. I'm just trying to highlight that the confusion is not for lack of me trying.

• "Suppose I introduce a new concept called 'k-interval' to define the set of ANY objects located between an upper and lower boundary. Would you then consider allowing objects other than points into the set?"
• "Yes, the endpoints are rational, and the object between any pair of endpoints is simply a line."
• "Revisiting the analogy above, when I utilize an interval to describe a range, I am referring to the underlying and singular continuous line between the endpoints"
• "Yet, between each tick mark, there exists a bundle of 2ℵ0 points to which we can assign an interval."
• "I propose that we redefine the term interval from describing the points that lie between endpoints to describing the line that lies between endpoints."
• "One thing I need to make clear though is that when I write (0,4) I'm referring to a bundle between point 0 and point 4. I'm not referring to 2ℵ0 points each having a number associated with them. "

Not true. I published papers when I was active that never assumed infinity was actualized....so, your statement is not entirely correct. — jgill

Certain areas of mathematics, like combinatorics, are sufficiently distant from foundational issues and actual infinities. These areas transcend the label of 'classical' mathematics.

I think even constructivist and intuitionist set theories have a version of the axiom of infinity.
— TonesInDeepFreeze

But isn't it more like a potential infinity? — keystone

In classical set theory, we define 'is infinite'. I don't know whether there is a definition of 'is potentially infinite' in any theory. But, it seems reasonable to say that the notion of potential infinity is supposed to be upheld in constructivist and intuitionist set theories, which is the very reason I've brought them to your attention.

Note that Brouwer himself was not interested in formalization. But some constructivists and intuitionists have been.

You give me link to some unidentified video so that I would have to take my time to watch through to find out, or guess, what it is you want me to know about it.
— TonesInDeepFreeze

To be honest it's because I didn't feel like continuing the dialogue with you because I find some of your comments offensive. — keystone

A litte earlier:

As always, I appreciate your comments and this dialogue. — keystone

But of course, people have the right to change their mind, even for the least of reasons.

Nicely done that at the end of a post in which you did continue to comment on my remarks, you say that you don't want to continue. And however you've taken other comments by me, the one I made about the link to a video is not very much offensive, if at all. On the other hand, it is at least a bit rude to link to a video without saying what point you want to make with the video or what part of the video you have in mind, so that I wouldn't have to have to watch all of a video without even context of your point about.

Anyway, I gave a proof that you are incorrect when you claim that the interval (0 1) is not an infinite union of disjoint intervals, whether or not you want to take a minute to understand the proof.