If it is looping forever, then wouldn’t it be the case of demonstrating the concept of infinity or eternity?
If it HALTed, then could it be the case of uncertainty or unpredictability? Were you able to find the way or reason for HALTing?
**I know that many mathematicians adhere to a Platonic approach, but for me mathematics has a rather phenomenological meaning. I regard mathematics simply as a particular case of philosophy. We can trace how concepts such as numbers, multiplicity, categories, infinity, and space have been examined by both philosophers and mathematicians. Like philosophy, mathematics initially proceeds largely from what is self-evident, and then relies on the tradition of its predecessors in the case of philosophy, and on their proofs in the case of mathematics. Our mathematics arises from everyday phenomenological experience. For example, if human beings existed on a microscopic scale or, conversely, on a macroscopic scale, their space—and consequently their geometrical axioms—would be different.
**
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So the issue is really digging into what we mean by “infinity.”
We have this word “infinity.” We throw it back and forth.
But I should make it clear that this thread isn’t about the easier situation where we see that a program will never finish.
It’s about the situation where we can’t figure out if the program will finish. Turing showed that there is no “good” single program that can tell us if any other program is good or not.
This suggests to me that the concept of “finishes or not” is actually quite fuzzy. If we had a good program that could decide for us, then the concept would be more definite and objective.
“Eternity” or “infinity” is the state of the looping created by the machine program execution.
We definitely know that program will either loop forever, or halt, or as you claimed we can’t figure out of the program will finish.
So, it is not the case we don’t know. We know it will be one of the 3 cases.
In real life cases, you would know it for sure if you run the program, and by observing how it operates.
But if you are talking about just making analogy, isn’t the case of the probability?
The program doesn’t know or care what will be the case. The judgement is in your mind.
Well one of those cases is that we don’t know.
I agree with you that we now know, with new intensity, that in general we can’t know.
We realized that an “objective method” that would rescue us from this ignorance is “nonsense.”
The issue in poetic terms is whether “God” knows.
For a mathematical platonist, the situation is “actually binary” up in “Heaven.” Down here in time it’s ternary or even a continuum of enacted intensities of certainty, but for the platonist “time is an illusion.” The “truth of the matter” is “secretly already here.” In other words, the future is not real as future. It is tacitly already present.
My view is that “time is real” and the future is really the future — it isn’t here yet. God did not build an infinite staircase that is “somehow out there.” Humans invented rules that allow them to keep building more and more mathematical reality in the empirical world.
I don’t think so, my friend. Zero-day exploits. The halting-problem is just a symbol for a larger problem.
But… when it comes to actual computers in the world, the math is all finite. So certain theoretical problems vanish. In theory, we can objectively determine whether a finite system halts. We can look for a repetition as it moves through a finite set of states. In practice, the number of states is astronomical, so we can’t do this in general.
Still, large programs surprise us. In a certain sense, people don’t understand how LLMs work. Yes, they can zoom in on this or that parameter or operation, but basically we are “growing” programs that are almost organic in their complexity.
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Turing machines and other models have “infinite memory,” which really changes the theoretical situation. So I’d say maybe keep that “gap” in mind. Turing machines are “pure math.” They cannot exist in the world except as an idea.
People started studying TMs because some mathematicians wanted a “God machine” that would generate all mathematical truths, which means it couldn’t contradict itself. But this God machine was itself math, so The God machine would have to be able to talk about itself. The God Machine should be able to tell us if it itself — The God Machine — never contradicted itself. That’s when things got weird.
In what sense, time is real? What do you mean by “real”? Isn’t the future in your imagination?
Not sure about God, but I agree that math is just a description system for the objects in the world.
When a machine operates, it needs INPUT from the world, and it processes the input data within the subroutines, and outputs the results. Without the output, all machine operation is just mechanical movements with no informational meanings.
Under that view, TM program is just movements of reading 0 and 1 in infinite loop. Expecting for it to HALT requires some external intervention.
That is actually a beautifully appropriate question.
What is a question ? What are you doing when you ask me or someone else a question ? Or when you look for something in the dark ?
What is the structure of that doing ? What is the structure of that “in-progress” now ?
We do of course imagine the future. But it’s more than that. Our entire existence is structured around the future.
Look at what you worry about. Try to make sense of what you are doing now. We act in the perceptually present "for the sake of " the oncoming future — in terms of the past as not just memory but our basic way of doing things. The “real” past is here now as the way we “automatically” tend to approach things.
This is “thrown projection,” more or less, from Heidegger.
Existence is thrown projection, prejudicial disclosure, visceral expectant remembering inquiry.
The nature and essence of time, and “real” have been always the hot philosophical topic, which were the independent threads in their own right. 
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Many imaginations become reality, and the future is one of them. But all future manifests into the present in the end. Before the manifestations, the future still belongs to imagination.
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Yes. So the “God” metaphor was me trying to sketch a certain way that some people think of math.
I see math as very much performed in the world by primates and their machines. The “information content” in my view is the stuff we do when we see the outputs. I build an actual bridge or missile this way rather than that way. We have to all enact the “meaning” of math symbols together.
As Saussure emphasized, signs are arbitrary. There is nothing special about the shape “3,” but we “give” it meaning by all of us treating it mostly the same way in practical life.
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So my point was that the structure of our being here is like opening a box to see what is inside. You asked me a question. You moved toward something, to bring something into view. But you used English to do it, which is your past being part of this movement into bringing something into view.
Life is something like a hand constantly peeling back a curtain. But reality is so rich that there is always more curtain to pull, always more future. There is always “darkness” around us. We imagine what might happen and try to “steer” the oncoming future in that direction.
This is not intended as a “metaphysical statement.” It really is just a pointing at the structure of human life. I am afraid of bad things happening. I work toward things I want to happen. I learn more about things. I move closer to things. I turn my back on other things. Basically I try to shape what is here for my hands and eyes and so on so that really bad stuff doesn’t end up here in the “now.”
In a way, we are something like our desired future haunting the present and borrowing its hands and eyes and ears. “I am my own future in the mode of not yet having it securely.”
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Yes, I agree. Only the part of future can be imagined. What is actually happening in the present now is actually random.
In other words, the future as a form of time can be imagined. But the actual content and detail of time seems in total randomness.
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I agree. So we enact a sense of some control of the future. No matter our theoretical beliefs, we in fact do hard work in the world with a sense that it is more likely to result in a happy future rather than a sad future.
But yeah we can always be surprised. Like huge AI programs may destroy our species one day, without most people wanting that to happen. So the tool that we wanted to help us create a nice future takes every future away from us.
But this applies to a chainsaw or a Honda. Even our tools, usually our friends, can surprise us as foes.
You might say that our beliefs are our basic tools. “Bad” beliefs lead us into disaster.
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So I get your point. Events have dual side of faces. TM program can loop or HALT in our imagination. But in reality, we don’t know.
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Just to be clear, we know well enough in some cases. I mean we have no doubt about simple programs that we can build to always halt or never halt. But as soon as programs get big, we are just trusting them like empirical objects. Then that “third state” becomes pragmatically real and even dominant.
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The fact that you pick these 3 words fascinates me.
What I think you don’t say here is that these 3 words — at least from over here — “unfold” the relationship of being and time.
I’d love to get your thoughts on this "typewritten manifesto" to see how close our views are. I happen to use the numeral “3” as an example in the paper. Now that feels like luck.
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I’ll check it out…
I use the 3 terms as shorthand for the 3 fundamental laws of logic/thought.
3 is necessary for triangulation. Its even an early biblical concept.
“No testimony is valid unless in the presence of at least two or three witnesses.”
Why?
Solve 3×j=¿
We cannot say the laws give us incomplete perspective or they provide only temporal understanding. That perspective itself relies on the laws.
You and Husserl are using the simultaneous operation of identity, distinction, and exclusion to identify, distinguish, and exclude the claim that these operations are merely temporal or incomplete.
You’re relying on their full, simultaneous structure to argue that the structure itself is partial or unfolding over time. That move is self-referential—the three laws have to be working all at once for you to even make the distinction between one moment and another.
The laws that enable thought don’t operate like a process where we piece together distinct objects step by step. They give a multi-perspectival grasp in which the parts are already held together in the whole. Thought itself can’t be treated as just another object we think about.
I get yourmanifesto. But such a degree of skepticism cuts the necessary ground that enables testing and skepticism.
The three laws let us recognize our information is incomplete. But that doesn’t mean the laws themselves are incomplete or temporal. You’re conflating what the lens reveals with the lens.
We know all possibility of the case in TM program.
If it is Loop, then it is not Halt.
If it is Halt, then it is not Loop.
If Loop, then Halt is a contradiction.
But the knowledge only exists in your intuition.
The truth is only available, when the program executes.
You seem to be denying the truth, based on the intuition.