In sum:

Your 'pseudo-intervals' are okay, up to, but not including, a 'vanishingly small pseudo-interval'.

But it's unnecessarily complicated when instead all we need is:

There is an algorithm that successively outputs the S-B rows. And there are algorithms to output successive S-B finite paths, each path ending in a node. We take the comp-reals to be those algorithms, calling them real-ithms.

Then, define '<', '+' and '*' on real-ithms, and you'd be on our way to something.

In other words, for arbitrary real-ithms G and R:

G < R <-> [fill in definiens]

G+R = [fill in definiens]

G*R = [fill in definiens]

/

Your move to this yet next iteration of you proposal (same pig, new dress) only serves to DISTRACT from the fact that AGAIN you want to eat your cake and have it too. You want only finite objects, but you also want real numbers, and you wanted them to be at the "mirage last row", or the last output of after an infinite loop, but now courtesy of a "vanishingly small pseudo-interval" while those DON'T EXIST with just your finite-only approach.

The solution to this problem is therefore not a number, but instead a vanishingly small pseudo-interval. — keystone

You STILL don't get it. You just keep putting new clothes on an old pig. Every time, you reformulate but you retain the essential fallacy.

There is no single "vanishingly small pseudo-interval". There are only successively smaller pseudo-intervals on successive rows.

This is still essentially the same subject, as you still (after I asked twice) SKIPPED answering my question*, though I can see for myself that the correct answer is 'no'.

* https://thephilosophyforum.com/discussion/comment/806060

I really should not continue to reply when you so obnoxiously continue to apply the same fallacy clothed differently each time though I have explained it over and over and over and even asked you whether you understand, yet you don't reply even to that question itself.

But I'll look under the hood yet again:

As I understand, there is just one line, and then each row is a different set of pseudo-intervals for that line.

I don't see a problem with that. Except that I don't see that it improves the more simple approach:

There is an algorithm that successively outputs the S-B rows. And there are algorithms to output successive S-B finite paths, each path ending in a node. We take the comp-reals to be those algorithms, calling them real-ithms.

Then, define '<', '+' and '*' on real-ithms, and you'd be on our way to something.

In other words, for arbitrary real-ithms G and R:

G < R <-> [fill in definiens]

G+R = [fill in definiens]

G*R = [fill in definiens]

But then this:

The solution to this problem is therefore not a number, but instead a vanishingly small pseudo-interval. — keystone

No, throw that out. There is no such thing. There are only successive rows with smaller and smaller pseudo-intervals.

Your "vanishingly small pseudo-interval" is just a variation on the "mirage last row" fallacy.

The claim that there are only countably many algorithms/programs does not imply that this view has gaps in the line. — keystone

"Gaps". What I actually say: There are only countably many comp-reals, so they don't provide a complete ordered field, thus there is no isomorphism with the continuum. But the standard calculus depends on the completeness of the reals. So if you claim your proposal can do mathematics for the sciences, then you have to show how it does that. And by 'show' I don't ostensive examples, handwaving or picture stories. I mean axioms and theorems - formulas. And formulas then used to solve problems of science.

The line is there in full from the start. — keystone

You are SUCH a self-contradicting liar. You said:

There are no points on this line. — keystone

[bold ORIGINAL]

You seem to have no compunctions about insulting intelligence.

All that's really changed is the philosophy — keystone

Changed from standard analysis? No, your proposal is radically different from standard analysis, from the start: Standard analysis uses infinite sets; you disclaim infinite sets. Standard analysis has a continuum; you don't. Standard analysis has uncountably many reals; with you, it's not clear how many comp-reals there are, since there are denumerably many real-ithms but you disclaim that there are denumerable sets (but maybe you could argue that there is not a set of all real-ithms but instead a program that itself generates real-ithms).

I've moved on from talking about the output of an unending program — keystone

Right, you dressed it up differently again, now as "a vanishingly small pseudo-interval". Same pig, new dress.

If you would like to make some point about what is written in a paper, then you can say what that point is, rather than just have me scurry and sidetrack to read something that you haven't said what about it you think is important to consider relative to what I've said.

The gravamens may not be mathematical, but certain mathematical points have entered in. As to the mathematical points themselves, agreed upon definitions are crucial. If some of the same terminology is used in philosophical senses different from the mathematical senses, then each time it should be made explicit which sense is active.

Pi is mathematical. And 'is infinite' has both a mathematical sense and philosophical senses. Whatever else one may wish to say philosophically about Pi and the notion of infinity it is at least a good starting place to reference the mathematical definitions. And by doing that, I cleared up a common confusions about mathematics:

(1) There is a distinction between the adjective 'infinite' and the noun 'infinity'.

For the adjective, we define:

S is infinite if and only if S is not finite.

For the noun, we define by choosing objects to serve as +inf and -inf in the extended reals.

The adjective and the noun should not be conflated. In particular, other than the extended reals, in ordinary mathematics, there is no object named "infinity". An such infinite sets such as the set of natural numbers is not named "infinity".

(2) There is a distinction between cardinality and magnitude.

Pi has finite magnitude.

Pi has an equivalence class of Cauchy sequences of rationals is infinite (has infinite cardinality).

(3) The decimal expansion that represents Pi (as Pi is the limit of the sequence) is infinite and has no finite initial subsequence that converges to Pi.

One can say whatever philosophy about the notion of infinity, but when it's mixed with mathematics, such as Pi, Circles, sequences, etc., the one should at least get the mathematics right before invoke it in one's philosophizing.

Pi is not a circle
— Banno

Of course it’s a circle — invicta

"Pi is a circle" is a mathematical claim, not a philosophical claim.

Mathematical definitions of 'Pi' and 'is a circle' are crucial for settling a dispute about the claim "Pi is a circle".

In the public discourse, it is mentioned that the quality of the public discourse is deplorable. But it's interesting to me that you don't see mentions of the need to specifically highlight the actual informal fallacies that are ubiquitous. I don't see people talking about a need to better educate people in the logic of discussion and disagreements.

I'll take your word for it that R RL[...] yield successively more accurate approximations of Phi.

Algorithm [named 'Chasing mirage-Phi']

(1) X = R. Go to (2).
(2) X = XR. Go to (3).
(3) X = XL. Go to (2).
(4) X = mirage-Phi. [whatever that is!]

In keystoneland is mirage-Phi an infinite sequence RRLRLRL... ? No, because in keystoneland, there are no infinite sequences.

In keystoneland, is mirage-Phi a "potential" output? It would be, if it were something that exists in keystoneland.

Lesson: Of course an algorithm may have an instruction that is never executed. But if the instruction mentions an object as an output, then that object has to exist, whether as an output or a "potential" output.

Please, no more or your handwaving! You don't have a coherent proposal. If you are sincere, then state axioms, rules of inference, definitions, and prove the theorems that justify a logical construction.

I added this:

Df. if x is a member of the extended reals, then x has finite magnitude if and only x is a real number

Df. if x is a member of the extended reals, then x has infinite magnitude if and only if (x = +inf or x = -inf)

Th: pi has finite magnitude

And I would like your answer whether or not you understand this:

"[...] the limit of infinite SB paths are nodes corresponding to real numbers."
— keystone

In your proposal, you don't prove that there IS such limit. [Saying it is "potential" but not "actual" is just more hand waving by you; or, in the sense we can mathematically define 'potential' and 'actual', they still don't obviate that you have not proved that there is such a limit.]

In contrast with your mental pictures, in mathematics with Cauchy sequences, there we DO prove that there is an object that is the limit.

You can't just hand wave to a say that there is a limit; it's not enough to say that the terms of the sequence get closer to each the next - you have to PROVE that there is an object such that the terms get arbitrarily close to it.

Consider sqrt(2). In the rational numbers, there is no limit to the sequence 1, 1.4, 1.41, 1.414, 1.4142 ... But we prove that in the reals there is a limit, viz. the least upper bound of the range of the sequence. And that least upper bound is the square root of 2.

That is, we prove that, in the reals, every bounded set has a least upper bound. {x | x^2 < 2} is bounded. So it has a least upper bound. Then we prove that the least upper bound of {x | x^2 < 2} squared is 2, thus the least upper bound of {x | x^2 < 2} = sqrt(2) . And the sequence of rationals that are approximations converges to the limit, which is sqrt(2).

You can't just say, "The terms of the sequence get closer, so PRESTO POOF OF MAGIC, there's this mystical, fictional "mirage" thing that I say is the limit!" You have to PROVE that there is such a limit.

I asked you a question:

Do you agree that in principle any algorithm can be stated as a Turing machine?

Please answer that.

The set theoretic formulation of Turing machines is the lingua franca of computability. We don't need to evaluate particular computer code in particular computer languages that discussants and readers of the thread might not know. Moreso, for simple algorithms, we can convey the ideas in plain English such as:

(1) Print "Hello". Go to (2).
(2) Go to (1).
(3) Print "Goodbye". Halt. — TonesInDeepFreeze

But if you do need a real formalization, then I suggest Turning machines, which is common ground in computability.

I'm not looking forward to a response in which I would read you compounding your confusions due to ignorance of the basic mathematics and dodges to fancifulness instead while SKIPPING the decisive refutations I provide you.

/

"I haven't the foggiest" - TonesInDeepFreeze

I revised the above response, since actually I did know what you were driving at. My revised response:

(1) Print "Hello". Go to (2).
(2) Go to (1).
(3) Print "Goodbye". Halt.

That's an algorithm. The execution of it successively prints "Hello", and it never prints "Goodbye", and it never halts.

Algorithms may have an instruction that is never executed [...] — TonesInDeepFreeze

/

What I want is your view on whether it makes any sense to say what this function returns. — keystone

I'm not interested in evaluating computer code. I do know what you have in mind. It's more readable (does not require that anyone know the particulars of a particular computer language) the way I've described it.

The algorithm never halts. It prints "Hello" over and over again. It never prints "Goodbye".

as described above the code is designed to return 1. — keystone

"Designed to" is not defined. Without a definition, a reasonable sense would be psychological: what is the intention of the programmer, which is mental. That's not mathematics.

So, not to yet again detour through such subjectivity, we merely observe an objective and mathematical fact: There is an instruction in the algorithm that is never executed.

I would phrase this by saying that the output of the function is potentially 1, but it is never actually 1. — keystone

Definitions:

T is a potential output per an algorithm G if and only there is an instruction in G to print T but execution of G does not print T.

T is an actual output per an algorithm G if and only execution of G prints T.

We don't need to say 'actual output' as we can just say 'output'.

But now we're YET AGAIN back to the same mistake you keep making:

For Phi to be a potential output, Phi has to first exist. Your confused fanciful musings claim Phi is a potential output, but you don't first prove existence of Phi to have Phi as a potential output.

But you revert to an analogy with '1' as a potential output.

In context of your finitist proposal, the number 1 (or "Goodbye") already exists to be a potential outupt, but Phi (or an infinite string of "Peach Desk Happiness Sink Rain Courage [...]") does not already exist. Your analogy is clearly inapt.

As in other threads, you just keep rephrasing your most basic misconception. You shift from one imaginary picture to another in which, somehow, your own mentation allows impossible things. The imaginary pictures are not mathematics, no matter that you're able to shift from one to another after another.

The basic misconception in this thread is the same as in the other threads, such as with Thompson's lamp. In Thompson's lamp, it's an impossible situation. There is no final state in the infinite sequence of states, and the sequence does not converge to a limit state. Here, there is no final row in the tree, but there is a sequence that converges to the limit Phi, but we must FIRST have the existence of Phi for it to be the limit. And with "Hello Goodbye", successive finite strings of "Hello"s are printed, but never an infinite string of them, and "Goodbye", though it exists, is never printed. "Goodbye" is NOT like a limit there.

Disagreements about terminology are unnecessary. Discussants can instead acknowledge the clear definitions in mathematics:

INFINITE:

Df. x is finite if and only if x is 1-1 with a natural number

Df. x is infinite if and only x is not finite

Df. x is countable if and only if (x is finite or x is 1-1 with the set of natural numbers)

Df. x is denumerable if and only if (x is infinite and x is countable)

Df. +inf is an element in the extended real system such that for any real number x we have x < +inf. Note that for any specification of 'the extended real system', +inf can be any object other than a real number or -inf (see below), and +inf does not have to be infinite (in the sense of 'is infinite' defined above).

Df. -inf is an element in the extended real system such that for any real number x we have x > -inf. Note that for any specification of the 'the extended real system', -inf can be any object other than a real number or +inf (see above), and -inf does not have to be an "infinitesimal" (and 'infinitesimal' is not defined anyway, except in non-standard analysis, which is different from the extended real system).

CIRCLE:

Df. L is a circle if and only there exists a plane and member x of that plane such that L is the set of points all equidistant from x

Df. if L is a circle, then x = G(L) ('the origin of L') if and only if all members of L are equidistant from x

Df. if L is a circle, then x = C(L) ('the circumference of L') if and only if there is an increasing sequence of lengths of perimeters of inscribed polygons such that x is the limit of the sequence

Df. If L is a circle, then x = D(L) ('the diameter of L') if and only if x a line through G(L) such that x ends on both ends on members of L

PI:

Df. x is a real number if and only if x is an equivalence class of Cauchy sequences of rational numbers

Th: if x is a real number, then x is infinite

Note: The above theorem is an artifact of the construction of the reals as equivalence classes. The theorem only says that the CARDINALITY of x is infinite; it does NOT say that any real number has infinite MAGNITUDE. Cardinality and magnitude are DIFFERENT. No real number has infinite magnitude.

Df. if x is a member of the extended reals, then x has finite magnitude if and only x is a real number

Df. if x is a member of the extended reals, then x has infinite magnitude if and only if (x = +inf or x = -inf)

Th. if x is a member of the extended reals, then x has finite magnitude if and only if x is a real number

Th. if x is a member of the extended reals, then x has infinite magnitude if and only if (x = +inf or x = -inf)

Df. x = pi if and only if for any circle L, x = C(L)/length(D(L))

Th: pi is infinite

Note: The above theorem is an artifact of the construction of the reals as equivalence classes. The theorem only says that the CARDINALITY of pi is infinite; it does NOT say that pi has infinite MAGNITUDE. Cardinality and magnitude are DIFFERENT. pi does not have infinite magnitude.

Th: pi has finite magnitude

Df. if x is a real number, then s = the decimal expansion of x if and only if [fill in the definiens here of the well known notion of a certain denumerable sequence]

Th: The decimal expansion of pi has no terminal repeating subsequence

It's like the material conditional. The weaker the antecedent, the stronger the conditional, and the stronger the antecedent, the weaker the conditional.

The legal example is M&B. She reduces her claims successively:

Now she has a herniated spinal disk.

to

Now she has symptoms.

to

Now the symptoms are different.

/

Anyway, with informal fallacies, I like to take them in greatest generality while still within an essential characteristic. I don't think it's about checking for adherence to particular varying definitions of them, but rather to their basic structure. Some people might says that strawman is claiming an opponent said something he didn't say, while other people might say that strawman also extends to attacking only incidental elements of an argument that are weak. Both the strict and the less strict definitions share the essential feature of knocking down an easy target while not confronting the main force of the opponent's argument.

M&B, "moving goal posts", and "no true Scotsman" may have distinctions among them, but they're all of a shared genus.

But, sincerely, I do like your new version of the tree, accounting for the negative rational numbers. Also, in context of standard math, I like the elegance of Phi generated by such a simple instruction (I'm taking your word for it that that sequence of rationals does converge to Phi, and I'm assuming we can prove that).

Let's contrast with mathematics:

We have axioms and definitional axioms.

We provide an algorithm by which to determine whether a given string is or is not an axiom. We provide an algorithm to determine whether a given sequence of strings is or is not a proof from the axioms.

We provide an algorithm to determine whether a given formula is or is not a definitional axiom.

By asserting 'there exists an x such that blah blah about x' we at least mean that we have a proof of that formalized as a theorem.

We define 'is a natural number'.

We prove that there exist natural numbers.

We prove that there exists {n | n is a natural number}.

We prove that there exists an equivalence relation on the natural numbers to form the set of integers as a set of certain equivalence classes of ordered pairs of natural numbers.

We prove that there exists an equivalence relation on the integers to form the set of rationals as a set of certain equivalence classes on ordered pairs of integers.

We define 'is a Cauchy sequence of rationals'.

We prove there exists an equivalence relation on the Cauchy sequences of rationals to form the set of reals as a set of certain equivalence classes on the Cauchy sequences of rationals.

We define 'less than', addition and multiplication on the the reals.

We define 'is a complete ordered field'.

We prove that <R 'less than' addition multiplication> is a complete ordered field.

We prove that all complete ordered fields are isomorphic with one another.

We prove the theorems and provide definitions used in real analysis (including calculus and mathematics for the sciences).

Mic drop.

the journey corresponding to RRL converges to the point corresponding to the golden ratio. — keystone

That is egregious.

First you said reals are paths. And that is plausible.

Then you said reals are not paths but they're nodes. That is absurd since there are no nodes for irrationals.

Then you said reals are nodes that are in "row infinity". That is absurd since there is no "row infinity".

Then you said reals are programs. That is plausible, and I mentioned that there might be something like that in the mathematical literature.

Then you said you'd clean up your act and reply with something more clear.

Then you come back only to circle back again to saying reals are (or "correspond to", whatever that means) nodes in "row infinity".

Like I said, bait and switch. But with you, it's bait and switch and switchbait and switch again back to one of the earlier switches. In another thread I learned a term "Motte and Bailey fallacy", for when someone can't defend their position so they switch to an easier position to defend but not acknowledging that they've switched. But you actually switch back to the earlier impossible to defend position!

Bait: reals are paths
Switch: reals are nodes in "row infinity" as those nodes are converged on
SwitchBait: reals are programs
Switch: reals are nodes in "row infinity" as those nodes are converged on

So you SKIPPED all the explanation I gave you why the notion of "row infinity" is nonsense. And you SKIPPED extensive explanation why it is question begging to say that such nodes are the limits (per convergence) of sequences: For x to be a limit of a sequence, x has to exist to even be a limit. It is nonsense to say that a sequence converges to a point Phi when you haven't proven that there IS such a point. I've told you about that over and over, but you blithely and dishonestly SKIP it.

Then the piece de resistance:

I'll use 'actualized' for what I mean. In my view, an object is actualized if it is present in the memory of a 'computer'. I am actualized because I am present in the 'computer of the universe'. I'm thinking of a purple cow so that purple cow is actualized because it is present in the 'computer of my mind'. This statement is actualized as I type because it is present in the memory of my laptop. When a memory of an object is flushed, it becomes 'potentialized'. — keystone

That is not progress when the subject is a mathematical account of reals as programs.

as if jurors don't recognize the sounder of the arguments — Hanover

Do you find that jurors can usually be counted on to reason logically? I don't know about the courtroom, but I find that in everyday life, people are very often extremely irrational when discussing matters of public controversy. I'm curious what you find in the courtroom.

it would have been poor advocacy on B's behalf. — Hanover

No doubt about that. But the flip side of that: If a lawyer sees that he or she could get an advantage by using sophistry (not falsehoods, but rather fallacious reasoning that would trick the jurors), should the lawyer do it on behalf of the client? If the lawyer doesn't and even though it is the best chance for the lawyer's client, then isn't the lawyer failing to provide the most effective advocacy? I've always been curious how lawyers and the "art of the law" view this.

The old cliche is "The facts and the law". But it should be "The facts, the logic and the law".

A little knowledge (from Wikipedia) can be a dangerous thing. — jgill

Oh, you said it so well.

Do you consider this a valid function? — keystone

I like this better:

(1) Print "Hello". Go to (2).
(2) Go to (1).
(3) Print "Goodbye". Halt.


That's an algorithm. The execution of it successively prints "Hello", and it never prints "Goodbye", and it never halts.

Algorithms may have an instruction that is never executed because executions always loop ahead of the instruction.

That's different from saying that an algorithm executes infinitely but then finally provides an output after executing infinitely.

An algorithm outputs successive finite approximations of Phi. But it never outputs the infinite expansion of Phi. That would be a supertask, while you disavow that there are supertasks! You are again trying to have your cake and eat it too. Grow up. And maybe have the courtesy not to keep looping back to an already refuted starting point in this discussion, which is to say, to not exercise your right as a crank to SKIP whatever refutes you.

Algorithm to emulate conversation between keystone and TonesInDeepFreeze:

(1) Print "keystone says: Phi is a node in the tree." Go to (2).
(2) Print "TonesInDeepFreeze says: Phi is not a node in the tree, because first, in your proposal there is not an infinite tree, and second, even in standard mathematics, there are no nodes in that tree that are irrational numbers, and third, there is no "row infinity" and so no "row infinity" that has nodes for Phi or any other irrational number, and fourth, paths don't "converge" to points the way you incoherently imagine they do." Go to (1).
(3) Print "keystone says: I see now. I should learn some mathematics." Halt.

Instruction (3) is in the program, but never executed.

You said reals are programs. Fine. As far as I can tell, those are computable reals. Fine. So go ahead and write a template for programs that are reals. Then define 'less than', addition and multiplication of programs. Then show how to do calculus with this. But, meanwhile, looping back to junctures that have already been refuted is stupid and obnoxious.

Depends on the context? — fdrake

Of course. I'm not claiming to talk about every possible context. This is at the level of some generality that we would understand not to be completely universal.

I don't think I've ever seen that.

Anyway, that starts as a reductio ad absurdum (or modus tollens, whatever) argument against a bailey X, which is common, but not the only form of argument against X.

I also don't trust that it's rightly construed as just a fallacy of inference — fdrake

I don't think t it is a fallacy of inference, in a strict sense. It's a form of dishonest argumentation.

trying to point the fallacy out will appear as castigation. — fdrake

What can one do then? Not flagging it is letting the other party get away with dishonest argumentation. I see that a lot: Person A commits M&B and person B just keeps rebutting each successive M, but it's never made explicit that the person A is trying to pull a fast one on us. I wish journalists and such were more explicit:

Jake Blitzer Anderson: Senator Sneakyspeak, your bill mandates that everyone with chronic indigestion will be forced, without due process, to move to the Aleutian Islands, but that's unconstitutional.

Senator Sneakyspeak: No, I'm just saying that those people can get better medical care there.

Jake Blitzer Anderson: But medical care in the Aleutian Islands is no better than in the rest of the country.

Senator Sneakyspeak: No, I mean they have nutritionists there to help with chronic indigestion.

Jake Blitzer Anderson: Okay, that's all the time we have. Tomorrow we'll interview Governor Doublecross about his election promise to turn litter on city sidewalks into freshly minted doubloons.

That's not the way it should go. Jake Blitzer Anderson should say, "No, Senator, don't switch what we're talking about. Your bill would move everyone with chronic indigestion to the Aleutian Islands. You have to defend THAT, not some weaker idea."

The broadest form of the argument in support of an absolutist view on the right to free speech is that it is through argument that we reach higher truth. Only through the exchange of ideas can society evolve its knowledge base. Should we wish to evolve intellectually as a society, we cannot stifle speech, but must let the ignorant speak freely so that their ignorance can be corrected. — Hanover

I'm not sure I like that notion. It premises freedom on its utility, while I think of freedom axiomatically as an end in and of itself. It would be regrettable if free speech did not contribute to our collective knowledge and enlightenment, but if that were the case, shouldn't we still value free speech as more fundamental than whatever outcomes come from it? It seems safe to say that often free speech results in the good guys losing, as, for whatever reason, lies, misinformation and speciousness prevail in the court of public opinion, but still, we are glad that everyone gets to speak, as not only do I want freedom for myself, but I want freedom for others, as I would take restricting their freedom as odious in and of itself, but also, transitively, a restriction on my freedom to hear them, even if in principle alone. Put simply: I want my freedom, and I want everyone to have their freedom too - because it's right. The benefit of greater understanding is great too, but it's secondary to the enjoyment of the mere pure right itself.

I think the "free speech because free speech is how ideas progress" argument is weaker than "free speech because it is an inalienable right no matter it also is good for the progress of ideas". With the former, when free speech does not result in better ideas prevailing, then the door is opened to saying, "Well, I guess that didn't work out, did it?"

You point out to Group C, the neutral jury, that A doesn't seek the truth (i.e. justice), but just seeks a preferred outcome regardless of the facts and is therefore not to be trusted. — Hanover

That's an interesting example. And excellent for illustrating M&B.

This is getting away from the purpose of your example, but I want to show another angle on it too:

Suppose, after reducing and reducing, she does finally support a reduced argument M that is still sufficient for her case. Then, as a juror, I might also reason this way:

Yes, she resorts to M&B, so her trustworthiness is in doubt. However, no matter her trustworthiness, she did finally support M, and that, more than her trustworthiness, is what finally counts. (Also, it's pretty much a given that the defense team also is not so much interested in the truth unless the truth happens to support their defense, so that, while the defense team presumably wouldn't lie, it probably wouldn't refrain from specious arguments if they helped and it could get away with them.)

where a Group A refuses to admit their stronger claim has failed and that their position is admittedly objectively weaker, there isn't the proper repercussion where a controlling Group C meaningfully condemns them. Instead, Group A just grows stronger, each member proud of their group's shameless advocacy of a desired outcome. — Hanover

That is well said. It is a vitally important point. In all kinds of "discourse" [scare quotes because it is more like noise and not coherent discourse], especially in politics and public affairs, people get away with massive amounts of dishonest argumentation all the time. It is heartbreaking.

less a fallacy than a strategy in getting a desired outcome — Hanover

They're separate. There's the fallacy and there is the motivation for deploying the fallacy.

First, I don't claim always to live up to the being a most high-minded discussant.

Second, I don't know what you think is low-minded about my saying:

it is may be more informative to hear a refutation of a defensible claim than a refutation of a preposterous claim. — TonesInDeepFreeze

And I don't know why you think my entire reply is preposterous.

Among all the nonsense you posted, this is a good item to especially highlight:

as we approach row infinity, the journey corresponding to RRL converges to the point corresponding to the golden ratio. — keystone

Now we are back to where we were near the beginning of this thread!

Aside from, 'journey' undefined, and there being no full tree, and convergence in this context not being defined: There IS NO point in the tree corresponding to Phi.

After I've posted so much to explain to you why your notes are nonsense, you regress back to your original fallacy!

You are a consummate crank: You cop out with undefined terminology. You assert contradictions but seem not to realize it. And most of all, the very essence of a crank: You SKIP refutations that have been given you and just go back to reasserting what has been refuted as if it was not refuted, not even RECOGNIZING the arguments that refute you, thus putting the conversation in a circle.

You still systematically resort to sophistry. But I will spend some of my finite supply of patience anyway, though I shouldn't.

First, do you agree that any program you can write can in principle be stated as a Turing machine?

At every row we have a blue line corresponding to the real number line. — keystone

At every row you have a finite set of rational numbers ordered by 'less than'. That's not the real number line.

You say "corresponding to the real number line". There you go again, invoking an infinitistic object (the real number line) that you claim doesn't exist. That's a cheat. You want to develop calculus without infinite sets. So you can't then smuggle in infinite sets. You can't say that something corresponds to something else that you yourself claim does not exist! Please, don't continue to insult my intelligence.

Row infinity does not exist. From now on, when I say 'approaching row infinity' it will be an informal placeholder corresponding to an unending journey down the tree. — keystone

(1) I am wary that you'll smuggle in the existence of a "row infinity" anyway. It would be more clear to just stay away from the threat of such sophistical pits and instead just use defined terminology.

(2) "unending journey". Back to undefined terminology again.

(3) There is no "tree" in this context (in the sense of all the tree).

(4) So you are taking "approaching row infinity" as informal for "unending journey down the tree", so an informal rubric to stand for another informal rubric. As they say in show business, the jokes write themselves.

I have an issue with infinite sets but I see value in the concept of equivalence classes. Instead of discussing what an equivalence class might mean in the absence of infinite sets, please grant me slack to use this concept loosely. While each node only contains the reduced form of a rational number, it actually correspond to an equivalence class of integer pairs. — keystone

In other words, "I don't accept that there are infinite sets, but I'm going use infinite sets anyway".

No, I don't grant you that "slack". The very point is that you propose to do calculus without infinite sets. If you're going to use infinite sets anyway, the all bets are off. You're wasting our time.

As we progress down the tree — keystone

No, in your context, there is no tree. There is only a program. That program never outputs what you are calling "the tree". Even as the program runs with no upper bound of steps, it never outputs what you are calling "the tree". Only, at any given step, the program outputs a finite number of rows. Please stop trying to have your cake and eat it too.

let's imagine that the real number line is still present in its entirety at every row. — keystone

In other words, let's imagine a contradiction. Let's imagine that the real line does not exist but that the real line exists.

Who all are the people that constitute the "we" who say the tree doesn't exist but there is a program held in their mind?
— TonesInDeepFreeze

When I imagine the Stern-Brocot tree, I visualize the first few rows of the tree and then internally mutter 'and so on as governed by the rules for constructing it'. That 'and so on' is not a formal program, but it refers to an algorithm which we can fit into our finite brains and be comprehended. — keystone

(1) You believe the tree does not exist. Yet, earlier in your post, you refer to "the tree", so one must think you are referring to something that does exist.

You need to state some axioms, then definitions and proofs. Otherwise, you're doomed to perpetually post nonsense.

(2) I asked who are the "we". Your reply is that it's you.

With the real number line now embedded in the tree, hopefully it is reasonable to say that as we approach row infinity, the journey corresponding to RRL converges to the point corresponding to the golden ratio. — keystone

What an obnoxious mess that is.

(1) There is no real line in this context. There's only a program that does not output the real line but only successive finite proper subsets of rationals.

(2) "The tree" (as in the full tree) does not exist in this context. Only, at any given stage of output, finitely many rows.

(3) The real line is not "embedded" in the tree in this context. Moreover, 'embedded' has a mathematical definition. But you haven't defined your own sense.

And even in standard mathematics, one would have to say in what way the real line is "embedded" in the three. The real line is the set of points of the form <x 0> where x is a real number, along with the less than ordering. The S-B tree (or your variant) do not have a row that is the real line. And no row has irrational nodes anyway! But, of course, we do prove the existence and define 'the real line' irrespective of the S-B tree anyway.

You propose a program, to avoid committing to infinite objects. Fine. But then you invoke infinite objects anyway. That is stupid and childish.

(4) You said you use 'row infinity' in this sense

"when I say 'approaching row infinity' it will be an informal placeholder corresponding to an unending journey down the tree."

So let's plug it in and see what we get:

as we unending[ly] journey down the tree, the journey corresponding to R RL[...] converges to the point corresponding to the golden ratio.

Aside from, 'journey' undefined, and there being no full tree, and convergence in this context not being defined:

There IS NO point in the tree corresponding to Phi. We've been through this already!

Now we are back to where we were near the beginning of this thread!

Your idea of reals (the computable reals, I think) being programs might be okay. But your latest post does not contribute a coherent extension of the idea nor much hint of making it rigorous. Your latest arguments sputter at the level of nonsense from the start.

I'll use 'actualized' for what I mean. In my view, an object is actualized if it is present in the memory of a 'computer'. — keystone

That's too vague for me to take as mathematics. Let me know if you ever get around to moving past word poetic metaphors and instead present some actual mathematics.

Let's say we color the potentialized points on the real number line red and the actualized points green. — keystone

Better yet, howzabout you give mathematical definitions for 'potentialized' and 'actualized".

I don't understand why we would need to adjust anything in our formalization of calculus. — keystone

Because the standard formalization uses infinite sets. Meanwhile, you don't have formalization.

Get a book on mathematics, why don't you?

Probably, the idea of reals as programs has been written about. I wonder whether it's covered in the subject of computable analysis.

And it seems interesting to apply the notion of programs to the S-B tree. I can see the basic idea: Each program would generate sequences of Rs and Ls. A real number would be a program. For a rational real, execution of the program would halt. For an irrational real, execution of the program would not halt.

But we have to keep in mind that this only accounts for the computable reals. So we'd have to explain how to formalize calculus with only the computable reals. I think that has been written about, though I don't know enough about it.

Refraining from sophistry is a basic requirement. Charity and Steelman Variation are "going the extra mile".

do we extend democratic rights to radical anti-democrats, e.g., fascists? Surely not [...] — Jamal

I don't know why you say that. On the contrary, usually it is recognized that democratic rights extend even to people who advocate against democratic rights.

anti-democratic elements, like written constitutions — Jamal

I'm not familiar with the notion that constitutions are anti-democratic. On the contrary, usually it is recognized that democracies need to have constitutions.

because we value the principle of charity we shouldn’t extend charity to irrational interlocutors — Jamal

By Charity in this context I only mean the form of discourse. While Charity may be charitable in the general sense of 'charity', I don't think that's its main attraction, which is that Charity contributes to better enquiry as it may go more directly to finding of truth, as it is may be more informative to hear a refutation of a defensible claim than a refutation of a preposterous claim.
.

M&B

X asserts B and/or argues for B.
Y refutes either B or X's argument for B.
As if to pretend that neither B nor X's argument for B were refuted, X instead either asserts a more defensible M and/or argues for M.
X is dishonest not to concede that B and/or X's argument for B were refuted.

Strawman

Y asserts M and and/or argues for M.
As if to to refute either M or Y's argument for M, X instead refutes a B that is less defensible than M.
X is dishonest to pretend that either M or Y's argument for M were refuted.

Charity

X asserts a not very defensible B and/or argues for B.
Y does X the favor of addressing a more defensible M instead.

Siege

X asserts a not very defensible B and/or argues for B.
Y refutes B and/or X's argument for B, and Y does not do X the favor of instead addressing a more defensible M.

Steelman Variation
Y asserts B and/or argues for B.
X disagrees with B and/or Y's argument for B, but X provides an even better argument for B and then refutes even that argument.

/

If X resorts to M&B, I think Y's best reply is "I'll address M instead, as long as you are clear that B was refuted and that we're moved to talking about M instead, which is different from B." That is to say, "I'll extend Charity but you need to not resort to M&B."

/

In high-minded academic debates one expects that the participants don't commit M&B or Strawman. And that truly high-minded participants extend Charity, even Steelman Variation and don't commit Siege. To me, that is a mark of intellect and enlightenment

On the other hand, in a forum such as this, one can count on M&B and Strawman being posted ubiquitously. Sometimes Siege too. But Charity or Steelman Variation only rarely. In my opinion, cranks rarely deserve Charity or Steelman Variation.

PS. As I mentioned, in my first few posts I mistook your initial proposal. In those posts I have now added edits in brackets, and marked as 'Edit', to note my mistake.

VARIOUS

Infinite paths do not actually have destinations but neither does 1/x have a value at x=infinity. — keystone

There are two different notations:

(1)

lim 1/x [x = 1 to inf] = 0

There 'inf' does not stand for an object named 'inf'. Rather, it is an informal placeholder as we may replace it with a formulation that does not invoke 'inf':

Let f = {<x 1/x> | x is a natural number greater than 0}.

lim f = 0

No mention of 'inf' there. Saying "to inf" is merely a figure of speech and does not imply that there is an object named "inf".

(2)

An extended system where there are the objects -inf and +inf.

Note that those may be any objects other than real numbers. They don't have to actually be infinite sets.

Ordinarily the stipulative (and I stress 'stipulative') definition of the division operation yields:

1/inf = 0

But there is value in saying that 1/x approaches 0 as x approaches infinity. — keystone

We don't just "say" it. We prove it.

there is value in saying that the path R L[...] approaches 1 as we descend the tree and approach 'row infinity'. — keystone

(1) You've cited two different versions of the S-B tree. Please choose one and stick with it.

In one version, 1/1 is the initial node and it is not obtained by any path. In the other version, 0/1 and 1/0 are the initial nodes, and 1/1 is the sole rational that is obtained twice by two different path, viz. R from 0/1 and L from 1/0.

In any case, 1/1 is not reached by by any finite or denumerable sequence R L[...].

And again, please choose one of the two versions and stick with it.

(2) Please stop saying "row infinity". There IS NO row infinity. We PROVE that there is no row infinity. It is not coherent mathematics to keep saying "row infinity" no matter that you put single scare quotes around it or call it a "mirage" or "fiction".

In both of these examples, infinity is a useful fiction. — keystone

Wrong. In the case of standard mathematics, we PROVE the existence of whatever objects we use. But in your case, you just hand wave that somehow there are "fictions" not defined even as abstract mathematical objects that explain your arbitrary claims.

however you want to phrase [the notion of "row infinity"]. — keystone

I don't phrase it in any way. Because I don't have the intellectual mathematical dishonesty of trying to get by with nebulous undefined hand waving terminology.

paths (as described by infinite sequences of rational numbers) — keystone

Just to be clear, I still don't know whether you understand that paths are not sequences of nodes.

it doesn't seem like a big jump to say that the limit of a SB path is analogous to the limit of a Cauchy sequence, or in other words, that the limit of infinite SB paths are nodes corresponding to real numbers. — keystone

For crying out loud, I gave you DETAILED explanation why that is incoherent.

One more time: With Cauchy sequences, there IS an object that is the limit. But with your bull, there is no object (except you preposterously and egregiously hand wave that these objects are "mirages" or "fictions").

I gave the example of sqrt(2), but you SKIPPED.

AGAIN, you can't just take a sequence and say that there is a limit; it's not enough to say that the terms of the sequence get closer to each the next - you have to PROVE that there is an object such that the terms get arbitrarily close to it.

When you just say that there is a "limit" of the sequence of paths (or now it's nodes, you keep switching). You have to PROVE that there is such a limit.

In the rational numbers, there is no limit to the sequence 1, 1.4, 1.41, 1.414, 1.4142 ... But we prove that in the reals there is a limit, viz. the least upper bound of the range of the sequence. And that least upper bound is the square root of 2.

That is, we prove that, in the reals, every bounded set has a least upper bound. {x | x^2 < 2} is bounded. So it has a least upper bound. Then we prove that the least upper bound of {x | x^2 < 2} squared is 2, thus the least upper bound of {x | x^2 < 2} = sqrt(2) . And the sequence of rationals that are approximations converges to the limit, which is sqrt(2).

You can't just say, "The terms of the sequence get closer, so PRESTO POOF OF MAGIC, there's this mystical, fictional "mirage" thing that I say is the limit!" You have to PROVE that there is such a limit.

The bottom line being that if your magical mystical mirages are merely fictions, not proven mathematical objects, then you don't have any math about them!

I don't know how I can make this any clearer for you.

it doesn't seem like a big jump to say that the limit of a SB path is analogous to the limit of a Cauchy sequence — keystone

It is a HUGE jump, because 'limit of a Cauchy sequence' has an exact mathematical definition, while 'limit of a path' in this context is undefined and your notions about it are INCOHERENT. See above

infinite paths do not end at any node (in a similar way that Cauchy sequences do not end at any rational number — keystone

There is a similarity only if we ignore, as you do, the decisive difference that the Cauchy sequences DO converge to objects that we DO prove to exist, as we don't rely on magic wand waving as you do.

And you need to drop the word "end". The Cauchy sequences do not end at a limit. The sequences do not have an end. Rather they CONVERGE. Every time you say things like "end" you only entrench in your mind a basic misconception and then add it to the landfill of misinformation on the Internet.

The issue then is that in the S-B tree, we only have nodes corresponding to the rational numbers. — keystone

First you said the rationals are the finite paths. In that case each node is the terminal node on a finite path. So a rational is a finite path and has its corresponding terminal node.

The nodes corresponding to the real numbers are fictional. — keystone

Your "fictional" is meaningless. Anyone can say, "Here are my objects that are solutions. They're fictional objects that I made up. I like them. So there ... my math! By the way, I have an answer to Goldbach's conjecture! There is an even number that is not the sum of two primes. It's a fictional number I made up. I call it 'the Emerald number'. Behold my math!" It's bull.

Comparatively, with the real number line, we do not distinguish between the rational and irrational points on the real number line. They are of the same essence. — keystone

Oh, yes, "essence". Yeah right, everyone knows what an essence is in mathematics.

Meanwhile, I'll inform you:

A rigorous formulation is that rational numbers are equivalence classes of integers. Then real numbers are equivalence classes of Cauchy sequences of rational numbers. So, strictly speaking, the set of rationals is disjoint from the set of reals. But there is an embedding of the rationals in the reals. So, in another sense of 'rationals', we take the rationals to be the image of that embedding.

Why is it important to correct you on this? Because garbargy undefined rubrics like 'essence', 'mirage', etc. only invite fallacious inferences based not on math but on mere suggestibility.

This disagreement between the S-B tree and the real number line is what I'm trying to highlight. — keystone

You're only highlighting your ignorance, confusion and dishonesty.

I've explained more than once that there is NO "disagreement" between the S-B tree and the continuum. They are both proven to exist from the mathematical axioms without inconsistency.

Indeed, if we countenance a proposal to take real numbers as the paths in the S-B tree, then presumably we get a number system isomorphic with the standard treatment. There's not a quandary about this.

I can write a program that generates all rows and stops — keystone

No, you can't.

However, I cannot execute it. The program is no less of a program just because the output doesn't exist. — keystone

If someone says I "wrote a program to do X but it doesn't do X", I wouldn't know what that would mean.

Ordinarily, by "program that does X" we mean that it does X.

The [fictitious] output is used to describe the program/execution (not the other way around). — keystone

There you go again with your "fictitious" stuff. It's bull.

A program is not specified by an outcome. A program is specified by instructions, and those instructions entail outcomes upon inputs. One such algorithm provides for printing natural numbers increasing in size and never stopping. But there is no algorithm that provides for printing every one of the natural numbers and then stopping. That is, there is a program such that for any natural number, the output will write that natural number; but there is no program that will finish writing all the natural numbers. That is, the set of natural numbers is computably enumerable but it is not finitely enumerable.

DID SOMEBODY SAY 'PATHS'?

I wanted to treat the real numbers as unending journeys along a path but you were inclined to treat them as the paths themselves. — keystone

You are SUCH a liar.

To be clear, my point here is not to claim that one should be discouraged from revising one's position or proposal, but rather that you're lying that it was my idea, prior to yours, to take the reals as paths.

From the very first post in this thread:

A number on the tree can be described by the right/left turns needed to get there from the top.
For example:
R = 2/1
RL = 3/2
RLR = 5/3

If we continue down the tree with this alternating pattern RLRLRLRLRLRL... we approach the Golden Ratio.

Is there anything wrong with completing this tree and saying that the infinite digit RL[...] is the Golden Ratio? — keystone

You were asking, obviously basically rhetorically, about an irrational number being a denumerable* sequence. First you mentioned a sense in which a denumerable sequence of rationals (each rational is a node coded itself as a binary finite sequence of Rs and Ls) approaches a real, but then you asked about just taking the denumerable sequence itself to be that real. There was no mention of "journey". As you wrote it, you were suggesting taking reals to be the denumerable sequences themselves.

* Denumerability is indicated by '...', which is ordinary informal notation for a denumerable sequence.

The notion of reals being infinite sequences of S-B branching came from YOU not me.

might irrationals be all the infinite strings which do not end in R_repeated or L_repeated? — keystone

Right there, you suggested reals as infinite strings (i.e. denumerable sequences). Literally, "infinite strings" (not "journeys") is what you said.

The notion of reals being infinite sequences came from YOU not me.

https://thephilosophyforum.com/discussion/comment/803226

There I mistakenly rejected your idea. I mentioned the standard approach, which is fine to do, but also I misconstrued that you meant that an irrational could be a node. Ironically though, even though at that juncture you weren't suggesting irrationals as nodes, later you advocated for irrationals as "mirage"-like limits. In the link above, I mistakenly took you to mean nodes, and I unnecessarily explained that there are no such nodes. But later I also explained that there are no "mirage"-like limits either. Or if the real is to be taken as a limit of such a sequence, then the object that is that limit must have been proven already to exist.

The notion of reals being infinite sequences came from YOU not me.

Why can't we say that (non-repeating) infinite decimals are journeys that are described by unending processes (e.g. limits) and not 'destinations' (numbers)? — keystone

"journey" and "destination" had not been given mathematical definitions by you - you had not stated a difference between a path and a "journey" or a difference between a node and a "destination". The crux of your point could only be understood to be that a real is an infinite string as that is what you literally suggested in the previous quote.

While irrationals do not correspond to any node in your tree, they do describe a paths on that tree (from the top all the way down), no? — keystone

There it is, right there: Associating reals with paths. Now, explicitly 'paths'.

The notion of reals being paths came from YOU not me. You are lying when you say it is the reverse.

No, the paths are not real numbers. First, a path is a sequence of edges, not a sequence of nodes. Second, a sequence of nodes is not a real number. Rather the limit of the sequence is a real number. — TonesInDeepFreeze

There, I was still adhering to the standard approach, in which it turns out that every real is a limit of a sequence of rationals. So clearly I was not pressing you to say that reals are paths, which you had just done yourself anyway. I hadn't yet realized the plausibility of the notion, and I was mistakenly RESISTING that notion, not pressing you to adopt it.

You have it backwards: It was not I who first suggested reals as paths; it was YOU who first suggested it. You're lying when you say it is the reverse.

If RL[...] looks like the golden ratio and it behaves like the golden ratio, why do you not say that it is the golden ratio? — keystone

There it is again, right there: You were arguing that reals may be taken to BE paths.

The notion of reals being paths came from YOU not me. You are lying when you say it is the reverse.

I think we both agree that RL[...] corresponds to a specific path on the Stern-Brocot tree, not a node. If the algorithm treats RL[...] as the golden ratio, then it seems reasonable to say that the golden ratio (and all real numbers) are paths on the Stern-Brocot tree. — keystone

There it is again, right there in plain quotes and explicitly 'paths': "(all the real numbers) are PATHS" [emphasis added]

The notion of reals being paths came from YOU not me. You are lying when you say it is the reverse.

I take it that by 'RL', you mean the particular denumerable path. — TonesInDeepFreeze

So by that time, based on all you posted, I said I take it that you mean the reals are paths.

I didn't first suggest that idea. YOU did. You are lying that it was initially my idea and not yours.

CORRECTION:

I initially misconstrued you. I was not reading carefully enough. I made the point that no irrational is a node on the tree. That is true, but not relevant, since your point (which I failed to read correctly) is that irrationals may be certain paths (not nodes). — TonesInDeepFreeze

There I even corrected myself to align with YOU that reals are paths, as I had earlier misconstrued you to be taking reals to be nodes.

I didn't first suggest that idea. YOU did. I was even mistakenly arguing against the plausibility of it. You are lying that it was initially my idea and not yours.

The SB tree might offer something here since it appears that each real number has a single path which can correspond to a sequence of rationals [as you hinted]. — keystone

Again, paths.

The notion of reals being paths came from YOU not me. You are lying when you say it is the reverse.

I think it's clear that every decimal number can be captured by a SB string (of L's and R's) but that is no proof. — keystone

Again, paths.

The notion of reals being paths came from YOU not me. You are lying when you say it is the reverse.

Every path (whether finite or infinite) leads to a different number. Finite paths lead to rational numbers. Infinite paths lead to irrational numbers. Or I think you'd be more comfortable saying that infinite paths that don't end in R or L lead to irrational numbers. I think of the limit of the tree as the real number line as depicted here: — keystone

There you slipped to undefined "leads to". And "limit of the tree", while there is no such thing defined in this discussion.

what is a point if not a node on the tree? Sqrt(2) does not converge towards any node on the tree. However, it appears to converge to a node that exists at 'row infinity'. Of course, there is no 'row infinity' which is why I relate it to a mirage. — keystone

Right there you switched from saying a real is a path to saying a real is a node at "row infinity" which is a "mirage".

First, as I've explained in spades, that's nonsense.

Second, this is where it would have helped if you had clearly said, "I am abandoning reals as paths and I am switching to reals as mirage nodes."

If we are redefining 'is a real number' as 'is a path in S-B' (I would prefer 'is a sequence of nodes on a path in S-B') — TonesInDeepFreeze

Again, I was following YOUR view that the reals are paths, as it was not clear to me that you had actually abandoned that proposal.

I think it's clear that every decimal number can be captured by a SB string (of L's and R's) but that is no proof. — keystone

'string' again. If not a paths, then what other sequences? Later, I gave reasons why sequences of nodes would be better, but, if I recall correctly, you ignored that, so I kept on with paths. So, by that time you were suggesting two proposals at the same time:

reals are paths
reals are "mirage" nodes that the paths converge to

The first notion is plausible.
The second notion is nonsense.

In any case a path is not a "mirage" node, so it is incoherent to suggest that reals are both paths and "mirage" nodes.

Every path (whether finite or infinite) leads to a different number. Finite paths lead to rational numbers. Infinite paths lead to irrational numbers. Or I think you'd be more comfortable saying that infinite paths that don't end in R or L lead to irrational numbers. I think of the limit of the tree as the real number line as depicted here: — keystone

No, I am UNcomfortable with that. I don't use the word 'leads'. It is not mathematically defined here.

And you need to mathematically define what you mean by the continuum being a "limit" of the tree, and prove that it is.

As we've informally agreed, irrationals (e.g. sqrt(2)) are unending paths on the SB tree. — keystone

Putting aside in what sense this is formal or not, there you again explicitly said that reals are paths. LITERALLY "irrationals are unending paths".

With this view, the objects of concern are the nodes (not the paths). — keystone

So it was paths, then "mirage" nodes, then paths again, then nodes again.

Can we say that the rational/irrational numbers are the limits of their corresponding paths. — keystone

"limit" is UNDEFINED by you in this context. I guess you're still talking about your nonsense "mirages".

As such, the object of study should not be the complete output of the program (which cannot be generated) but instead the program itself whose execution is potentially infinite. — keystone

And there you moved from reals being limits (after they were previously paths) to reals being programs.

So you started by taking reals as paths, and I eventually went along, then very recently you quickly and briefly switched to nodes. Then paths again, and now they're not paths or nodes but programs. Then you lie by saying it was my notion, not yours, that reals are paths.

You are such a piece of work.

FALSE TELLERS

"Just remember it's not a lie if you believe it".
Look up 'lie' in Merriam-Webster.
— TonesInDeepFreeze

Lie - to make an untrue statement with intent to deceive.
Perhaps cranks are deluded, but our intentions are pure. — keystone

Now you are even lying about lying. I mentioned a dictionary definition, and instead of reporting back the whole definition, you truncated it to just the sense you like while leaving off the senses that dispute you:

The two other senses you left off:

": an untrue or inaccurate statement that may or may not be believed true by the speaker or writer

: something that misleads or deceives"

It is not uncommon for people to believe their own lies. People convince themselves of untrue claims, and over time they entrench themselves deeper and deeper into the falsehoods. And when they are confronted with refutations of those falsehoods, they respond by entrenching themselves even deeper. They intentionally ignore all clear evidence and irrefragable arguments contrary to their false beliefs so that they can continue to propagate the falsehoods. That's not just lying; it is systematic lying. And when a person makes a ludicrous claim but has no basis for it, even if they irrationally believe it, we may still call it a lie. "There are transmitters in vaccines that let the government know where you are and who you are with at all times". Even if a person believes it, it's a lie, not a mere mistake.

And you know who epitomizes all of that? Cranks.

It is wonderful for you to have just provided such an ironic example where you deceptively left off the part of the definition of the word 'lie' itself!

And if you were to reply that you didn't mean to deceive but that you forgot to read the rest of the definition or whatever, then you're still lying about the definition, because failing even a modicum of diligence to get the facts right but nevertheless spouting egregious falsehoods is another form of lying.

But anyone can make a mistake, right? Sure, of course, people get things wrong and sometimes because they were not thorough in checking the facts or whatever, and that doesn't ordinarily deserve to be called 'lying'. But when it is a pattern, over and over and over, such as with cranks, then it deserves to be said that it is lying.

So, no, cranks do not have "pure" intentions.

I think it is a virtue to be able to adjust one's view upon new evidence. — keystone

Of course. But what is annoying with you is that you don't clearly say that you've changed your plan.

First you talked of reals as paths. Then you talked of reals as "limits" of paths (though that is not defined). Then you said actually there are only programs. But the way you casually slip around among those plans makes it difficult to follow and know really at each juncture what it is you actually claim. If you are to keep changing, then it would be helpful and just the least of consideration to say something like "Previously my approach was X. But now I see that X doesn't work. So from now on my approach is Y". Otherwise, it's an undo strain to follow your continual swerves.

This is not a bait and switch. — keystone

First it is was paths, then suddenly and briefly nodes, then paths again briefly, then programs.

It is unreasonable for you to disallow programs from the discussion, especially given that computer science is so closely tied with mathematics — keystone

As you added in an edit, I don't. I don't disallow computable analysis or any other rigorous mathematics. But I also don't disallow myself from commenting in full when you're spouting bull.

I should have acknowledged that my ideas and method of communication are evolving. However, I wouldn't call this a bait and switch since I'm not trying to trick you. — keystone

Regarding moving from paths to nodes to programs, it's not a question of tricking. Rather, it's that you are so confused and sloppy that you shift from one half-baked idea to another instead of organizing a coherent proposal or even point of view. That careless shifting adds up to misdirection even if not intended. And there is your continual disinformation about mathematics, some of it so egregiously produced as to be lies.

And there is this whopper lie:

I wanted to treat the real numbers as unending journeys along a path but you were inclined to treat them as the paths themselves. — keystone

I'll address that in another post.

"INCONSISTENT"

You need to stop using the word 'inconsistency' with your own private meanings.
— TonesInDeepFreeze

Granted, the branches of the S-B tree never actually intersect [...] — keystone

That and the rest of your paragraph have nothing to do with my point, which is (AGAIN) that 'inconsistency' has an exact mathematical definition. A theory is not inconsistent because it doesn't comport with your hand waving! A theory is inconsistent if and only if it there is an S and its negation that are both theorems of the theory.

until I can support arguments like this with something more concrete, I'll refrain from using the term inconsistency. — keystone

Not just "something more concrete". You need to prove that there is a sentence such that both it and its negation are theorems of the mathematics in which both the S-B tree and the continuum exist. You can't fudge it. It's flat out the case that NOTHING else justifies claiming inconsistency.

PHILOSOPHY

There is no actually infinite object corresponding to the S-B tree. Nobody has ever seen the actually infinite object with their minds eye. What we see with our minds eye is the program. — keystone

Oh no no no, you don't get away with that.

First of all, it's ludicrous to say "we" see a program when considering the S-B tree. Who all are the people that constitute the "we" who say the tree doesn't exist but there is a program held in their mind?

Of course, EVERY mathematical object in this context is abstract. No one has seen the number 0 nor any other number. No one has ever seen a line or plane or space. No one has ever seen Euclidean space or any other space, finite or infinite.

To say what I mean by saying "exists", I merely need to say that there is such and such an existence theorem. Of course, one may have philosophical views that certain things exist independently of formalizations. But all that is required at minimum are theorems. So when I say "exist", without opining on philosophical notions, I mean at least there is an existence theorem.

Now, it is true that, at least in principle, one can picture all the members of a finite set together but not of an infinite set. Moreover, aside from capability in principle, there are finite sets that humans can't picture - take any sufficiently large finite set, not even just abstractions, but concretes. It doesn't even have to be very large. Try closing your eyes and picture seventeen similar looking but different objects as distinct and all at once.

More importantly, one can also understand the meaning of a predicate such as 'is a natural number' and then take the set of natural numbers to be the abstract mathematical object that "stands for" that property.

Anyway, this is a diversion from the mathematical subject we've been talking about. The subject is not philosophical senses of 'exist'. That's a fine subject, but nothing in it alters that when one says "the S-B tree and the continuum exist" one may reduce that to the fact that the mathematical axioms prove that statement. So, for the record, notwithstanding any other philsophical bents, when I say "X exists" I mean that, at least, the axioms prove the theorem "X exists".

For reference, these are the approaches that have been discussed here:

STANDARD

(1) Reals are Dedekind cuts.

Advantages:

* Easy to visualize.

* 'less than', addition and multiplication have been defined.

Disadvantage:

* Unintuitive that a real is the entire set of all the rationals less than the real.

(2) Reals are equivalence classes of Cauchy sequences of rationals.

Advantages:

* Intuitive that sequences rather than entire "less than sets" are involved.

* 'less than', addition and multipication are handily defined.

Disadvantage:

* Unintuitive that a real is an entire set of equivalent Cauchy sequences of rationals.

(3) Reals are sequences of rationals.

I have heard about this, but don't know how it works as a definition or how 'less than', addition and multiplication are defined. Also, I don't know why we couldn't take reals in (0 1) to be the ordinary binary(or decimal, whatever) expansion (excluding sequences of all 1s after some index, or all 9s after some index, whatever). But since that is not one of the two standard methods - (1) or (2) - I can only guess that there's a complication in this method.

Advantage:

* Intuitive that a real is a particular sequence.

S-B TREE

(4) Reals are paths.

This seems plausible to me.

Advantage:

* Intuitive that rational reals are a finite path and irrational reals are a denumerable path.

Disadvantages:

* What about fractions not in lowest terms? I guess division would be defined in a special way. And I think we might end up with having to deploy equivalence classes of integers, which is what we do anyway in the standard definition of the rationals.

* Need to define the negative reals.

* Need to define 'less than', addition and multipication.

* Sequences of nodes are slightly more intutive (familiar) than sequences of edges.

(5) Reals are sequences of nodes.

This seems plausible to me and better than (4).

Advantage:

* Intuitive that rational reals are finite sequences and irrational reals are denumerable sequences.

* Sequences of nodes are slightly more intutive (familiar) than sequences of edges.

Disadvantages:

* What about fractions not in lowest terms? I guess that division would be defined in a special way. And I think we might end up with having to deploy equivalence classes of integers, which is what we do anyway in the standard definition of the rationals.

* Need to define the negative reals.

* Need to define 'less than', addition and multipication.

(6) Reals are limits of paths.

This is NONSENSE. "limit of path" is not defined. If paths converge to something that is a real number, then those things they converge to must already be defined and proven to exist.

No Advantage, because it is NONSENSE.

Disadvantage:

* It isNONSENSE.

(7) Reals are programs.

I think there is mathematics for this, but I don't know enough about the details - either the definition of 'is a real' or the definition of 'less than', addition and multiplication.

Advantage:

* Perhaps it's more friendly to finitism, but I don't know that infinite objects can be actually dispensed with.

Disadvantages:

* Unintuitive that numbers are programs.

* Might be very complicated to formalize.

* I guess it would only account for the computable real numbers. So it might be complicated to formalize calculus.

Four posts to follow. Untangling your confusions and lies.

Unfortunately, it's likely that you'll reply with even more confusions and lies.

It is a fascinating subject. I wish I were up to speed to absorb and appreciate that paper.

We'll go with our continuing lead story:

Now, you claim there is an inconsistency there. So PROVE that there is an inconsistency there. Otherwise, you are making the claim utterly without basis; you are fabricating, which is to say you are lying.
— TonesInDeepFreeze

With my view now outlined above, I am arguing that it is inconsistent to say that the real number line is composed entirely of actual points. — keystone

That's yet more dishonest obfuscation by you.

For about the 1000th time: Inconsistency is having both a statement and its negation as theorems.

"actual" is not a terminology of the theory.

So there is no "inconsistency" regarding it.

You need to stop using the word 'inconsistency' with your own private meanings. Inconsistency is an exact rigorous notion in mathematics. When you appropriate the term to use it with your own private and undefined meaning, you disservice honest and coherent discussion.

"Just remember it's not a lie if you believe it".

Look up 'lie' in Merriam-Webster.

he nodes are projected down and at the limit we have nodes for all real numbers forming the continuous real number line in totality. — keystone

You mangle terminology. Choose a definition and stick with it.

First you said the reals are the paths. Now you say they are nodes.

And you use "limit" in an undefined way.

Your insouciance in not making definitions and sticking to them invites confusion and is annoying.

Can we say that the rational/irrational numbers are the limits of their corresponding paths. — keystone

(1) What is a limit of a path? I can see what the limit of a sequence of nodes is, but you would need to define what you mean by a limit of a sequence of edges.

(2) You said you wanted the reals to be the paths. Now you want to switch to something else. Let us know when you've reached a stable decision.

(3) Your notion doesn't even make sense, as follows: You are trying to define 'is a real number' per the S-B tree. But you need also to prove the existence of the objects that meet that definition. You haven't proven the existence of whatever the limits are supposed to be.

Look, you have the existence of the paths. And you have the existence of the sequences of nodes. But you don't have the existence of whatever you think are going to serve as "the limits". If you want to have the things that are going to serve as the limits, then you need to prove they exist. And not "mirages"'. Or give a mathematical definition of 'mirage'.

One can write a finite (but complete) computer program to create the entire S-B tree. — keystone

To be clear, since you write ambiguously "create the entire tree". Yes, there is a program such that, for any n, the program will generate up to and including the nth row and stop. But there is no program that generates all rows and stops. I take it that you agree.

One can write a finite (but complete) computer program to create the entire S-B tree. — keystone

A program can generate any arbitrary finite part down the tree, but it can't create "the entire" tree.

Again, the tree is an infinite object. And if you consider it to be the limit of the finite stages, then it still has to exist to BE that limit!

Meanwhile, you seem now to regard the reals as limits of successive finite sequences of finite paths. I don't see a problem in that. But, again, it entails that those limits are THEMSEVLES objects - existing. Indeed, they are the denumerable paths. You can't say that the denumerable paths are limits but that they don't exist.

The notion of limits in this context is infinitistic. The limits of sucessive finite sequences of finite paths are the denumerable paths. If you want them to be our real numbers, then they exist and it's just jejune to say [paraphrase], "They don't "actually" exist but they exist as "mirages"". That's just undefined verbiage, nice for poetry but it's not mathematics. Please don't insult intelligence that way.

Look at sqrt(2) in standard mathematics. It is the limit of a sequence of rationals, but it has to exist ITSELF to BE there to be the limit. We can't coherently say, "There are just the finite approximations and the sqrt(2) is the mirage (something that does not exist but only seems to exist) at the end of those approximations." No, for the sqrt(2) to be a real number, it has to exist.

Now, you can say that there are only successive finite sequences of rationals getting closer to one another progressively in succession. You could cook up a theory in which it works that way. But then you cannot say that there is a limit that is the sqrt(2).

As such, the object of study should not be the complete output of the program (which cannot be generated) but instead the program itself whose execution is potentially infinite. — keystone

Ah, so you don't want the S-B tree after all. You want instead a program that generates successive finite number of rows.

But does the tree exist for you or not? Please don't answer with yet more wiffle waffle undefined terminology. Please just say whether it exists or not.

I understand the notion of the entire tree as a "limit" of the finite sequences of accumulations of rows. But a limit in mathematics is itself an existing object. It's not a "mirage". You can't have it both ways: You can't say BOTH (1) there is not an entire tree but only finite approximations and (2) the entire tree is the limit of those approximations. To be the limit of a sequence is to already exist to be the limit. Sequences don't converge on something that doesn't itself exist.

If the paths don't exist in their entirety, then those real numbers that are supposed to be those paths don't exist, or whatever you think are supposed to be the "limit" nodes don't exist.

But let's say the object of study is the program itself. Okay, but then pray tell how do you extract from that study of the program real analysis for the mathematics for the sciences? I think there are rigorous theories (I guess they can be made fully rigorous?) that do that kind of thing. But you don't provide a clue how you would do it.

Actually, this is a huge bait and switch by you. You said that the real numbers are to be the paths in the tree. But now you don't want to have the existence of those paths, so you switch to saying "study the program". I was game for talking about the initial proposal, but now you've switched to something undefined to the point of nebulousness.

I don't mind a computer program that prints all natural numbers. — keystone

To be clear, you mean a program that at any stage generates up to the nth natural number for some n and always goes to the next stage and never halts. That is called a 'recursive enumeration'.

'we' don't actually work with the infinite sets themselves, 'we' work with the 'algorithms' used to generate them — keystone

There is mathematics formulated along those lines.

But that doesn't entail that there is inconsistency in the standard approach!

Do your thing, whatever it is, but you should lay off spreading disinformation that there is inconsistency in infinitistic mathematics.

What I don't like, because it's a lie, is your claim that the existence of the real line contradicts the existence of the S-B tree.
— TonesInDeepFreeze
You are evading my point.
— TonesInDeepFreeze

Let's table this for a moment as I better understand your interpretation of real numbers on the SB tree. — keystone

I'm not tabling it for anything. It is flat out incorrect that

There is an inconsistency in claiming that both (1) the real number line exists and (2) 'row infinity' does not exist. — keystone

The S-B tree is clear enough by the ostensive definition we have of it. (Eventually though we should have a rigorous definition.)

And we agree that there is no row of the S-B tree that is not indexed by a natural number. (I.e., we agree that there is no "row infinity" of the S-B tree.)

And I understand the notion of the irrational reals being the denumerable paths of the S-B tree.

So we don't have to wait to see what my interpretation is.

These are consistent with one another:

The S-B tree exists.

The set of finite paths of the S-B tree exists and there is a 1-1 correspondence between the set of finite paths and the set of rational numbers. So the proposal is to take the rational real numbers as the finite paths.

The set of denumerable paths of the S-B tree exists. And there is a 1-1 correspondence between the set of denumerable paths and the set of irrational real numbers. So the proposal is to take the irrational real numbers as the denumerable paths.

There is no row that is not indexed by a natural number. (I.e. there is no "row infinity".) Therefore, no irrational real number has a path with a final node.

Now, you claim there is an inconsistency there. So PROVE that there is an inconsistency there. Otherwise, you are making the claim utterly without basis; you are fabricating, which is to say you are lying.

/

Do you think the real number RL (phi) is the path itself or the limit of the path? — keystone

I'm accepting whatever coherent proposal YOU are making. You want to make phi the path. I've said that that is fine with me.

Just to be clear on what we're talking about:

'graph' is defined different ways. I use the definition by which a graph is a certain kind of triple. It follows from the definition that:

Every graph is a triple <V E f>. V is the set of vertices (aka 'the nodes'), E is the set of edges, and f is a function on E that assigns to each edge either an unordered pair or ordered pair of vertices.

Watch out now, here comes something quite pedantic:

<V E f> =
<<V E> f> =
{{<V E>} {<V E> f}} =
{{{{V} {V E}}} {{{V} {V E}} f}}

So the only two things that are literally in the graph (i.e., are members of the graph) are {{{V} {V E}}} and {{{V} {V E}} f}.

I mention that to explain why I personally don't like to say "the paths are in the tree" but rather "the paths are of the tree".

But it would be unduly pedantic indeed to disallow that conversationally we use the word "in" more loosely (i.e., not just for membership), and that most mathematicians don't say "of" but say things like:

"The nodes are in the graph", "the edges are in the graph", the "paths are in the graph" and "the rows" in the graph".

So I'll go along with that usage.

Now the bullet points:

Every path is a sequence of edges.

A path is NOT a sequence of nodes.

Every tree is a graph.

The S-B tree has both finite and denumerable paths.

As I understand the proposal, the finite paths are the rational real numbers, and the infinite paths are the irrational real numbers. (I suggest that it might be better to make the rational reals the finite sequences of nodes and the irrational numbers the denumerable sequences of nodes. I.e. "eliminate the middleman".)

of course such a sequence does not converge to sqrt(2), since sqrt(2) is a sequence and not a node.
— TonesInDeepFreeze

(RL) converges to the node corresponding to 1. — keystone

Actually 1/1 is the one exception. 1/1 is not represented by a path but by a node, since there is no path leading to the node 1/1.

RL (phi) does not converge to any node. — keystone

At least for right now, I won't quibble with the notion of a path converging to a node. But I don't see what your reply has to to do with my point that sqrt(2) is a sequence not a node, except that you give phi as another example.

You brought the conversation to a level of complexity/formality that I wasn't comfortable — keystone

And that does not entail that I did that as something similar to a "No true Scotsman" ploy (as I surmise you were suggesting).

The reason why I prefer the S-B tree view is that it's more understandable to amateurs like myself. — keystone

There are attractions in using the S-B tree:

Every fraction is in lowest terms. (But we will still have to deal with justifying, e.g., 2/4 = 1/2. There is no node 2/4, so there is no path to 2/4, so 2/4 is not a rational number. So what is it? This is why the standard approach takes rationals to be equivalence classes.)

It's intuitive that the rational reals would be the finite paths and the irrational reals the denumerable paths. And that also means that, e.g., 1/3 doesn't need to be mentioned as represented by a denumerable sequence.

Real numbers can be sequences and not equivalence classes of sequences.

I'm trying to understand what the real line is from the perspective of the S-B tree. — keystone

The continuum is <R S> where R is the set of real numbers and S is the standard 'less than' ordering on the set of real numbers. There are some details to work out in taking the set of real numbers to be the set of paths (but tossing in 1 also, as 1/1 has no associated path), but it seems to be a plausible idea. But then you have to define the 'less than' ordering.

But you are also trying to impugn the standard theory, which you have objected to for being infinitistic. But the S-B approach is no less infinitistic. Then, irrationally, and even defeating your own purpose ("cut your nose to spite your face") of using the S-B tree, you say there is an inconsistency. And, aside from S-B, it's fine to propose a finitistic or constructive alternative or computable analysis or non-standard analysis or even a theory with a paraconsistent logic. But that doesn't permit propounding the lie that the standard theory is inconsistent or inconsistent with the S-B tree (indeed, the S-B tree itself is within the standard theory).

The mathematical axioms prove the existence of the S-B tree.

The mathematical axioms also prove the existence of a complete ordered field with the carrier set being the set of Dedekind cuts.

The mathematical axioms also prove the existence of a complete ordered field with the carrier set being the set of equivalence classes of Cauchy sequences of rationals.

Those two complete ordered fields are isomorphic.

We can also try to figure out showing that we have a complete ordered field with the carrier set as the set of paths in the S-B tree. And, since all complete ordered fields are isomorphic with one another, the one based on the S-B tree would be isomorphic with the others too.

The set of paths in the S-B tree is not part of the S-B tree. But that doesn't contradict anything else in the mathematical theory from the axioms. The S-B tree exists, and lots of things other than the S-B also exist. The set {p | p is a path in the S-B tree} is one of them.

I don't know, but maybe you are thinking that there is ONLY the S-B tree and anything other than the S-B tree doesn't exist (except as a "mirage" or whatever crank dodge concept you invoke)?

If that is the case, then dump that thought. The S-B tree exists; the Dedekind cut real numbers exist: the Cauchy sequence real numbers exist; and the set of paths in the S-B tree exists (and possibly it too as a basis for a complete ordered field).

the computation — keystone

There are two separate matters:

(1) The definitions of the operations.

For addition, this is of the form:

x=y = z <-> P
where P is a formula in which only x, y and z occur free and such that we have the theorem:

AxyE!zP

If you propose the tree as a basis for the real numbers, then you have to provide such a definition.

Please tell me whether you understand that. Actually, don't bother, because I already know you know nothing about mathematical definitions.

(2) Computations. That's not what I have been talking about.

As we've informally agreed, irrationals (e.g. sqrt(2)) are unending paths on the SB tree. — keystone

I don't take it as merely an informal proposal. If it's going to hold up, then it will be formalized.

sqrt(2) is a real number. A real number doesn't converge. A sequence converges. — TonesInDeepFreeze

I had edited that to:

If we are redefining 'is a real number' as 'is a path in S-B' (I would prefer 'is a sequence of nodes on a path in S-B'), then of course such a sequence does not converge to sqrt(2), since sqrt(2) is a sequence and not a node. — TonesInDeepFreeze

unlike with our other thread, you cannot defend your position by making your arguments more and more complex — keystone

Not only are you lying about there being an inconsistency, but you're moving on to mischaracterize what I posted in other threads. And indeed you even more obnoxious, because what I did in the other threads was to generously give you increasingly detailed explanations of things you're ignorant about. As you continued not to grasp the basic mathematics, I generously explained it to you in yet more detail. That is not me resorting to defending my arguments by making them more complex. You are a real piece of work.

I, on the other hand, am defending my position by making my augments more and more simple. — keystone

Yes, it's a simple lie that the existence of the real line is inconsistent with the existence of the S-B tree.

Again, you have evaded the point:

Since you claim there is an inconsistency, then PROVE it.

I suppose one way for you to end this would be by explaining why the SB tree (and my argument) oversimplifies the issue. — keystone

What? Why should I do that? I don't claim that the S-B tree "oversimplifies" anything. Now, you're also resorting to strawman.

If you don't like my description of the real number line as it relates to the SB tree — keystone

What I don't like, because it's a lie, is your claim that the existence of the real line contradicts the existence of the S-B tree.

can you propose a better description using the SB tree? — keystone

What? I have no problem with saying that set of real numbers is the set of paths in the S-B tree, with whatever other finer qualifications need to be given to make it all work out rigorously. And then defining an ordering to define the continuum. I said explicitly that I can imagine it all working out.

But your lie is saying that there is contradiction between the existence of the continuum and the existence of the S-B tree.

For example, the square root of 2 does not remind me of a mirage. It is not problematic that it is the limit of a sequence of rationals but is not one of the entries in that sequence.
— TonesInDeepFreeze

It does not bother me that there are (unending) paths on the SB tree having no destination node. I also think there's value in saying that such a path is approaching a 'virtual' node. — keystone

(1) You are evading my point. I'll say it again (whether regarding ordinary analyis or an S-B proposal):

It is not problematic that a real is the limit of a sequence of rationals but is not one of the entries in that sequence.

(2) "virtual node". More undefined figurative language. The last refuge of the crank.