Does conceivability imply possibility?

[AltF4]

This question comes up in particular with the "Modal Argument" for the supposed proof of existence of non-physical minds. One of the explicit claims made is that:

It is conceivable that one's mind might exist without one's body.
therefore
It is possible one's mind might exist without one's body.

Or, in short, that conceivability entails possibility. I want to argue against this.


First, a quick matter on definitions:

"Possibility" can mean a few different things in different contexts. It can sometimes mean "extraordinarily improbable". Such as if I ask you "Is it possible for someone to jump to the planet Mars?" You would be well within your rights to answer "No". Because "impossible" here is really just shorthand for "highly improbable".

This is not how we intend possibility, here.

There is also the matter of what is possible based off of the laws of physics. So I may ask you "Is it possible for a two classical objects to collide such that their combined final momentums are greater than their combined initial momentums?" Your answer could rightly again be no.

But again, this isn't how we really want to think of possibility for now. Perhaps our understanding of the laws are incorrect, Or perhaps they differ in other regions of space. It could still be possible.

So we set a high bar for impossibility, here. Something is only impossible when it cannot ever happen in any universe, ever. And we can be really, really sure of that fact.


Counterexample:

The halting problem. Specifically, the problem of finding a computer program which can search other computer programs for infinite loops in their source code.

(A little background) Computer programs can have many bugs in them. One kind of bug is an infinite loop. A careless programmer might accidentally write a program which loops back on itself infinitely, thus apparently "freezing". It happens all the time.

One might want to create a program, then, which can search the source code of programs to try and find if any infinite loop bugs exist therein.

Turns out that this is impossible. It's called the halting problem. There is no way to know, not even in principle, whether the bug-finding program will ever "halt" or finish. For better detail, read the Wikipedia page on the halting problem. (Or ask me if you prefer)

But of course a solution to the halting problem is perfectly conceivable. One can easily imagine having a program which searches other programs for infinite loop bugs. Indeed many such programs DO exist for other kinds of bugs. There is nothing inconceivable about it.

But it is impossible. Not "impossible" due to being impracticably. Not contingently impossible due to physics. It's 100% impossible due to mathematics itself. (Information theory) You cannot ever make such a program, in any universe that can ever exist.

Therefore conceivability does not imply possibility.

 

[Dre89]

The Halting problem is not conceivable though.

Someone unknowledgeable postulating that it's possible, and it being conceivable are two different things.

Conceiving the 'idea' of something and conceiving something are two different things. For example, I can conceive of the idea of an infinitely long wall. However, I can't actually conceive the infinitely long wall- in that in my mind I can't picture the wall in its infinite length, I can only conceive of part of it. Another example would be a square circle. You can entertain the idea, but you can't actually create a visual image of a squared circle.

If something can be conceived of properly, not just the idea of it, then it's possible.

 

[AltF4]

So I actually agree here on one point: The human mind is not a perfect thing. It screws up all the time, and is frequently wrong and incomplete. When you conceive of a thing, your mind naturally skips all the impossible parts of the thing. You can easily come to an incorrect or incomplete conception of something. Or you can think that you have a proper conception of something when really you don't.

But we have a serious epistemological problem then, don't we? How can you possibly differentiate between something which is "properly" conceived of, and something which you only think you are properly conceiving?

In what sense can the statement "X is conceivable" ever be made with certainty suitable of a proper proof? The possibility that X is in fact inconceivable, and you were wrong about being able to conceive of it is always present.

It seems to me that you either:

a) Cannot ever make statements about conceivability with certainty worthy of a proof.
or
b) Lower the bar for our definition of conceivability to include that which is impossible. (Which is how most people use the word.)

 

[rvkevin]

Dre, are you saying that in order for something to be conceivable, you must have a detailed physical account of what it would look like? For example, for the halting problem, it is not sufficient to simply be able to conceive of a program that would give the correct output, but you must actually conceive of the actual code and for that code to work in order for it to count?

 

[Dre89]

Alt- We can know whether something is genuinely conceivable if it can be genuinely conceived of.

I think what you're trying to say is that the criterion for this conceivability is iffy. It isn't actually. It's actually pretty simple. You have to be able to conceive of the whole proposition.

Consider the proposition 'X can cause itself into existence'. For that to be conceivable, you somehow have to imagine X existing, then causing itself. Of course, it already exisit before its causation, so it's impossible.

I know that seems simple, but there is no sophisticated point to make here.

Rv- Not necessarily. The thing is most of the time we simply conceive the idea, this is because most things we conceive are unquestioned, so we don't need to conceive of everything properly everytime we think about it. We only need to do such conceiving when we are trying to prove whether or not it is genuinely conceivable.

As for the program, you wouldn't need to conceive of all the programming, because that isn't the problem. The problem is you can't conceive of a program actually doing the function.

I can conceive of a humanoid robot, even though I don't know the programming, because I know the programming is theoretically possible. I can't, however, conceive of a robot which builds itself, because this isn't a question of programming, as it would have to exist prior to its programming to create itself.

I know that wasn't very clear but I hope it makes sense.

 

[AltF4]

Alt- We can know whether something is genuinely conceivable if it can be genuinely conceived of.

I think what you're trying to say is that the criterion for this conceivability is iffy. It isn't actually. It's actually pretty simple. You have to be able to conceive of the whole proposition.

But how can you ever know this?

It happens all the time where people think that they understand something perfectly and thoroughly. Even to the point where they put their lives at risk on its basis. Only to discover that they did not, in fact, have a complete or correct conception of the idea.

How can you ever say with certainty that you completely an unerringly conceive of anything?

That bar is so high that nothing of non-trivial complexity can possibly be said to be "conceivable". Surely not the case of non-physical minds. (As the thread is in reference to) You simply cannot tell me that you are able to thoroughly and systematically account for every smallest working detail of the nature of minds and their interactions with bodies. And until you do, it falls short of "proper conceivability". Since any one of those details may turn out to be impossible.

 

[rvkevin]

Rv- Not necessarily. The thing is most of the time we simply conceive the idea, this is because most things we conceive are unquestioned, so we don't need to conceive of everything properly everytime we think about it. We only need to do such conceiving when we are trying to prove whether or not it is genuinely conceivable.

As for the program, you wouldn't need to conceive of all the programming, because that isn't the problem. The problem is you can't conceive of a program actually doing the function.

I can conceive of a humanoid robot, even though I don't know the programming, because I know the programming is theoretically possible. I can't, however, conceive of a robot which builds itself, because this isn't a question of programming, as it would have to exist prior to its programming to create itself.

I know that wasn't very clear but I hope it makes sense.

I don't find the anologies helpful at all. The reason that they are theoretically impossible is because they include apparently contradictory propositions. In other words, something can't exist and exist at the same time. However, for the computer program, the issue is not whether you can have a program that can write itself; its whether this program can perform a specific function. Well, I can conceive of a program that would, when ran, search the system and return a binary result of Yes/No as to whether there is a bug in it, but as Alt notes, this is physically (and mathematically?) impossible. Isn't this us conceiving the program performing said function? There just happens to be a black box where the coding is, which is where the problem results; but as you said, the specifics aren't significant here. Since the problem lies in the black box, and not in the general concept. The general concept is conceivable, but not possible which is Alt's point.

 

[Dre89]

But you're not actually conceiving the entire process.

Most of the timewe just conceive of ideas, because it's more efficent than conceiving the entire process. The only time we need to conceive of the process is if it's something we're trying to prove as true.

So if you tried to conceive of the entire process of the bug program you'd fail.

 

[blazedaces]

I know this is not helping the discussion, but I just wanted to clarify that the halting problem, as is stated in the wiki is that "Given a description of a computer program, decide whether the program finishes running or continues to run forever. This is equivalent to the problem of deciding, given a program and an input, whether the program will eventually halt when run with that input, or will run forever".

Furthermore, it was proven only that you cannot solve the halting program for any possible program.

But, Alt, under your definition, "the problem of finding a computer program which can search other computer programs for infinite loops in their source code" is solvable. You did not say find all infinite loops.

You could just search the entire program for text containing "while(true)" for example. You might want to add an if-statement to look for the word "break" or "goto" inside the loop... This plus human eyes and you could write the program. Like I said, it wouldn't find all loops, and it may even be wrong, but it's very easy to conceive of very poorly written code (and I've seen my fair share of sloppy, horrible code) that this program might find an infinite loop inside.

Plus, if you have knowledge of the description of a program, you could give a decent estimate for how long the program should take to run. Again, a program could be written to "trick" your loop-finder by taking an exceedingly long time, but besides those cases, your program would work very well for finding common infinite loops. I mean, half the time that's exactly how I conclude there's an infinite loop in my code.

I run a short portion that should take microseconds and see it keep on going... then I go on to debug the program and see what might be the cause of the loop.

Anyway, just wanted to throw that in there. I love trying to solve challenging problems, especially in code so I guess I couldn't help myself...

-blazed

 

[AltF4]

Quite right, blazed. The Halting problem is specifically restricted to general purpose algorithms (Ones which work for any input) and always succeed. It's trivial, as you describe, to make a program which does not always work, or which does not work for all inputs.

 

[Theftz22]

Like Dre says, it's questionable whether such a program is really conceivable. That is to say, it is questionable whether you can form a mental image of a general algorithm to solve the halting problem for all possible program-input pairs.

When there is disagreement about whether something is conceivable, the dispute can be solved by depicting that thing. For, as I say, conceivability is a necessary condition to depictability. For instance, the mind existing without the brain can, and already has been depicted numerous times. Thus, the mind existing without the brain is conceivable.

If you can depict such a program, then you can show it to be conceivable.

 

[AltF4]

You're just punting the problem to the word "depicting". You then have to show that "depictability" necessarily implies possibility. Which you haven't done. And I would argue that it doesn't for the same reasons as the rest of the argument.

And I haven't the foggiest clue what you mean by that word. Do you mean "to be able to form a mental picture of it"? That would be silly. Why would that be any different than conceivability?

 

[AltF4]

Another example related to the halting problem:

While a Turing Machine is not capable of solving the Halting Problem, other more powerful machines can. Such as a Zeno Machine. A Zeno machine (inspired by Zeno's Paradox) does not perform computations in static time. Rather, each computation takes half as much time as the last. In other words, the speed of the machine grows exponentially as it runs.

Such a machine is capable of executing an infinite number of commands in a finite amount of time. And thus is able to solve the Halting Problem. There is an entire mathematical description of every inch of Zeno Machines. How they operate, exactly what problems they are and are not capable of solving, etc... You can even write algorithms (computer programs) for them.

And yet they are in-constructible. You cannot ever build a Zeno machine. Not even in principle. They are impossible.

I fail to see how this does not fit both the definitions of "conceivable" and "impossible".

 

[Dre89]

Exactly, you can't genuinely conceive of building the zeno machine.

It's just a different type of problem that prevents genuine conceivability.

 

[Theftz22]

You're just punting the problem to the word "depicting". You then have to show that "depictability" necessarily implies possibility. Which you haven't done. And I would argue that it doesn't for the same reasons as the rest of the argument.

You're misunderstanding my point. I wasn't trying to show that conceivability entails possibility, I was merely challenging your argument against it. Specifically, I was saying you haven't shown such a program to be conceivable. I argued that since depictability entails conceivability, you could prove the premise by depicting it, but otherwise the conceivability of such a program remains unproven.

So whether or not depictability entails possibility will depend on whether or not conceivability entails possibility. Since I have shown that depictability entails conceivability, transitively, if conceivability entails possibility, then so does depictability. Now I haven't tried to argue for that antecedent yet, I've just challenged your objection to it.

And I haven't the foggiest clue what you mean by that word. Do you mean "to be able to form a mental picture of it"? That would be silly. Why would that be any different than conceivability?

No, to form a mental picture of something is conceivability, depictability would be to form a physical picture of that thing such as on a piece of paper, sculpture, animation, etc.

Edit: As for the Zeno machine, I'm not even sure it's logically impossible. Physically impossible, sure, but I at least don't see any logical contradiction, so there's no prima facie reason for supposing it is logically impossible.

 

[AltF4]

I repeat my objection that if you raise the bar on conceivability to mean "Must account for every single detail of the workings and nature of the object" then you:

a) Also raise the bar past any non-trivial example. Notably well past saying that minds separate from bodies is conceivable. Sure you can picture it in your head. But so can I about a Zeno machine and a solution to the Halting Problem. In order to be "properly conceivable", you have to account ever every intricate detail of the working and nature of the thing. Which you simply cannot do with minds separate from bodies.

b) This is clearly not the common understanding of the word "conceivable". And others use this misunderstanding to their advantage. One might otherwise easily let pass the assumption that minds separate from bodies is conceivable.


And about the Zeno Machine. The machine can be fully understood in every detail. I can even draw you a functional block diagram of each of its components. There are no missing "black boxes" that are left as magic. This passes every test of conceivability I can think of. Simply declaring that it is inconceivable on the basis that it is impossible is to beg the question.

EDIT: Oh, and Underdogs, it is logically impossible. The Zeno machine (and indeed any machine capable of solving the Halting problem) violate the Church-Turing thesis. Which states that any decidable problem (a problem which is capable of being solved) can be solved on a Turing Machine. Or, put another way, there can be no machine that exists with greater computational ability than a Turing Machine. (In terms of the number of problems it is capable of solving)

 

[Bob Jane T-Mart]

I think this conceivability implying possibility argument relies heavily upon the assumption that humans have an intuitive grasp upon reality. Unfortunately, as theory of relativity and quantum mechanics has shown us, this assumption is heavily flawed. On the scales of the very small, the very large and the very high energy, the human mind cannot intuitively understand what happens.

I would also like to point out that our definition for conceivability, ie. you need to a have a full picture of how it will function, before it can be considered conceivable, has a significant problem. What if you are conceiving something based upon inaccurate information? You have the idea, entirely in your head, down to every nut and bolt, but the design of it is flawed because the information you have based your design on is flawed. Is this still considered conceiving something?

And if it isn't, then in order to conceive anything properly, you're going to need to know everything about the subject, which as of yet nobody does. This makes conceivability implying possibility a useless proposition for any discussion ever known to man.

I think AltF4 has made a similar point before, about making mistakes. If you make one mistake in your conception of whatever it is, you have not properly conceived of it. Again, this makes this notion pretty much useless.

As for examples to the contrary of conceivability implying possibility, what about perpetual motion machines? There have been numerous designs for these things, but we now know that they are impossible due to thermodynamics. Faster than light travel? I can conceive of rocket, with a huge amount of fuel, that just keeps accelerating and accelerating. I could even give you a drawing if you want. But I know that it's not possible because of special relativity.

 

[Dre89]

Bob I made the distinction between conceiving of an idea of something, and conceiving of something.

Conceiving an idea would be like thinking of an infinite regress- you can think of an infinite regress, but you can't actually picture an entire infinite regress, only part of it.

It's possible to conceive of false ideas.

Conceiving of something genuinely is actually mapping out the entire thing. You can't genuinely conceive of an infinite regress or a scquared circle, just the idea of them.

I think you and Alt are assuming conceivability is supposed to be self-evident. Conceivability can be contentious, and proving conceivablity may require very complex argumentation.

Also, conceivability implies logical possibility, but inconceivability doesn't imply impossibility. For example it's impossible to genuinely conceive of God, and probably certain things in quantum mechanics, but that doesn't mean they're impossible.

 

[AltF4]

So, Dre, will I no longer see you use the modal argument to "prove" the existence (or even possibility of) non-physical minds? As it seems pretty obvious that non-physical minds interacting with physical bodies does not come close to the high standard of conceivability that we've come to.


Back to the Zeno machine: I'm not sure what kind of argumentation you would like, Dre, to prove that the Zeno machine is conceivable. It is fully mathematically described. It is a Turing machine whose operations half in execution time each cycle. So it consist of:

- A memory tape of arbitrary length. With finitely many symbols to write to the tape.
- A head which can read/write to and from the memory tape
- A "transition function". IE: A computer program. Just a list of instructions for the read/write head.
- A store of the current state. (Finitely many states)

That describes a Turing machine. The computer in front of you right now is essentially nothing more than a glorified one of these. To make a Zeno machine, you just add the condition that rather than each operation being equal in execution time, each one takes half as much time as the last.

Then, using lots of complicated techniques beyond the scope of this post, (read into combinatorics and theoretical computer science for more info) you can even make programs for these machines. And run these programs on theoretical Zeno machines. (Not actual Zeno machines, because you can't build one.) You can analyze exactly what set of problems are solvable on them, and what kind of efficiencies they can reach on any problem.

NOTE: Be careful when Googling around for Zeno machines and other super-Turing computation devices. Sometimes there are cranks around on the Internet claiming to have built one. They're liars. Or uninformed. Just like how there are plenty of people on the Internet claiming to have built perpetual motion machines, or cold fusion. Just ignore them.

 

[Theftz22]

I repeat my objection that if you raise the bar on conceivability to mean "Must account for every single detail of the workings and nature of the object" then you:

To conceive of something you must be able to picture every feature of that thing. To know exactly how those features work isn't necessary, on my meaning of the word conceivable. Still, I can't conceive of a program to solve the halting problem. I'm not sure whether or not I can conceive of a zeno machine, partly because I'm struggling to understand what exactly it is.

a) Also raise the bar past any non-trivial example. Notably well past saying that minds separate from bodies is conceivable. Sure you can picture it in your head. But so can I about a Zeno machine and a solution to the Halting Problem. In order to be "properly conceivable", you have to account ever every intricate detail of the working and nature of the thing. Which you simply cannot do with minds separate from bodies.

I'm able to picture every feature of the mind existing without the brain, though I can't necessarily think of how exactly that would work, but that's just not part of my definition of conceivability.

And about the Zeno Machine. The machine can be fully understood in every detail. I can even draw you a functional block diagram of each of its components. There are no missing "black boxes" that are left as magic. This passes every test of conceivability I can think of. Simply declaring that it is inconceivable on the basis that it is impossible is to beg the question.

So from what I can understand a Zeno machine is simply a machine whose problem solving speed increases exponentially. Here I think you have a misunderstanding when you say:

Such a machine is capable of executing an infinite number of commands in a finite amount of time. And thus is able to solve the Halting Problem.

For at no point will the zeno machine ever execute an infinite number of commands in a finite amount of time. As the computer goes along successively performing more tasks and adding them to the running total, the number of tasks performed will always be a finite number. There is no finite number which, when added to a finite number will give you infinite. In fact, there is no finite number which, when multiplied by another finite number will give you an infinite. And so at no time will the computer perform an infinite number of tasks in a finite time. I'm also not sure what you mean when you say that being able to perform an infinite amount of commands in a finite time would solve the halting problem, but that's probably just my lack of knowledge on the subject.

EDIT: Oh, and Underdogs, it is logically impossible. The Zeno machine (and indeed any machine capable of solving the Halting problem) violate the Church-Turing thesis. Which states that any decidable problem (a problem which is capable of being solved) can be solved on a Turing Machine. Or, put another way, there can be no machine that exists with greater computational ability than a Turing Machine. (In terms of the number of problems it is capable of solving)

So I don't think it's been shown that it would solve the Halting problem, now I have no idea what a Turing machine is or why it is able to compute every computable problem, but I assume that you think because a turing machine can't solve the halting problem, and because it can solve every solvable problem, therefore the halting problem is not solvable. So since the zeno machine solves the halting problem, it must be impossible. But like I said, I don't think the zeno machine does solve the halting problem. It does not perform an infinite number of calculations in a finite time.

 

[AltF4]

I know that all this involves a lot of heavy and mathematical subject material in a rather obscure area of research. Sorry for that, try to bear with me.

Wikipedia does a good job of explaining how a Zeno machine can solve the Halting problem. It even gives a program which you can run on a Zeno machine to do it.

And yes, the Zeno machine does an infinite number of computations in a finite amount of time. It's similar to Zeno's Paradox. (where it gets the name) It works like this...

Consider the following sum:
https://upload.wikimedia.org/wikipedia/en/math/d/1/4/d141720bee6abcb33d5b75963745fe1d.png
(Sorry, the image has transparent background and doesn't paste well onto SWF)

Try doing it on your calculator yourself. Each time you add a new term in the sum, you get closer and closer to 1. Approaching it asymptotically. Keep this concept in your mind for the next part...

Consider the Zeno machine, who's operations half in execution time each iteration. Imagine that the first operation takes 1/2 a second to complete. The second takes 1/4 of a second to complete. The third takes 1/8 of a second to complete. And so on...

Then you let the machine run for one full second, and ask yourself: "How many operations has the machine just executed?" And the answer will be an infinite number. This is how a Zeno machine can execute an infinite amount of commands in finite time.


Why Zeno machines are logically impossible, not just physically so: Again, Wikipedia hits on this topic fairly well. (The first entry) Which in turn links to a number of associated research.

Consider asking a Zeno machine to compute the sum of a divergent series. Such as:

1 - 1 + 1 - 1 + 1 ...

This is a divergent series. Meaning that it has no sum. There is no asymptote which the sum approaches.

A Zeno machine could compute all infinite terms of the sum in a finite amount of time. But what would it return as a result? The Zeno machine purports to solve problems to which there are no solutions.

 

[Theftz22]

Alt, please have patience as I am going to do some research on this topic so I can understand the argument fully before assessing it. So that I can be sure I have the structure of your argument correct, would you say this is an accurate formulation?


1. A Zeno Machine is conceivable.
2. A Zeno Machine is logically impossible.
3. Therefore, conceivability does not entail possibility.

The justification for premise 1 is as follows:

It is fully mathematically described. It is a Turing machine whose operations half in execution time each cycle. So it consists of:

-A memory tape of arbitrary length. With finitely many symbols to write to the tape.
-A head which can read/write to and from the memory tape
-A "transition function". IE: A computer program. Just a list of instructions for the read/write head.
-A store of the current state. (Finitely many states)

That describes a Turing machine. The computer in front of you right now is essentially nothing more than a glorified one of these. To make a Zeno machine, you just add the condition that rather than each operation being equal in execution time, each one takes half as much time as the last.

Then, using lots of complicated techniques beyond the scope of this post, (read into combinatorics and theoretical computer science for more info) you can even make programs for these machines. And run these programs on theoretical Zeno machines. (Not actual Zeno machines, because you can't build one.) You can analyze exactly what set of problems are solvable on them, and what kind of efficiencies they can reach on any problem.

The sub-argument for premise 2 would be formulated something like this:

1. The Zeno Machine can complete an actually infinite number of calculations in a finite time.
2. A machine that can complete an actually infinite number of calculations in a finite time can solve the halting problem.
3. The halting problem is logically impossible to solve.
4. Therefore the Zeno Machine is logically impossible.

Justifications for each of the premises:

1.

Consider the following sum:
https://upload.wikimedia.org/wikipedia/en/math/d/1/4/d141720bee6abcb33d5b75963745fe1d.png

Try doing it on your calculator yourself. Each time you add a new term in the sum, you get closer and closer to 1. Approaching it asymptotically. Keep this concept in your mind for the next part...

Consider the Zeno machine, who's operations half in execution time each iteration. Imagine that the first operation takes 1/2 a second to complete. The second takes 1/4 of a second to complete. The third takes 1/8 of a second to complete. And so on...

Then you let the machine run for one full second, and ask yourself: "How many operations has the machine just executed?" And the answer will be an infinite number. This is how a Zeno machine can execute an infinite amount of commands in finite time.

2. Taken from wikipedia:

Zeno machines allow some functions to be computed that are not Turing-computable. For example, the halting problem for Turing machines can easily be solved by a Zeno machine (using the following pseudocode algorithm):

begin program
write 0 on the first position of the output tape;
begin loop
simulate 1 successive step of the given Turing machine on the given input;
if the Turing machine has halted, then write 1 on the first position of the output tape and break out of loop;
end loop
end program

3: The Church-Turing thesis states that all solvable problems are solvable by a Turing Machine. A Turing Machine cannot solve the halting problem. Therefore the halting problem is not solvable. I take this to be logically impossible, or necessarily true, because mathematical truths are necessary truths.

 

[AltF4]

The statement:

2. A machine that can complete an actually infinite number of calculations in a finite time can solve the halting problem.

Is actually not quite true. Merely being able to compute an infinite number of commands doesn't by itself give you the ability to solve otherwise undecidable problems. Consider the problem:

What have I got in my pocket?

It is a question, and it has a correct answer. But no amount of computing power will help you in finding the answer, not even a Zeno machine.

The proof that a Zeno machine can solve the halting problem is by writing a computer program that can do the task. Which is actually really simple. The Wikipedia page has an example of such a program in pseudocode that is only a few lines long.

Also:

3: The Church-Turing thesis states that all solvable problems are solvable by a Turing Machine. A Turing Machine cannot solve the halting problem. Therefore the halting problem is not solvable. I take this to be logically impossible, or necessarily true, because mathematical truths are necessary truths.

I want to retract a statement I made earlier that essentially said this. The Church-Turing thesis is called a thesis (and not a theorem or law or something similar) is because it is unprovable. Not unproven. But in fact provably unprovable. While it's widely and universally accepted, maybe we don't want to rest the argument here on this.

I think a much more convincing argument as to why the Zeno machine is logically impossible is that it claims to be able to solve unsolvable mathematical problems. Such as the summation of a divergent series. A divergent series is one which has no sum. Such as Grandi's Series

A Zeno machine could perform the summation of all infinite terms of the Grandi's series in a finite amount of time, and return an answer. What would the machine return as a result? To a question that has no answer. This is clearly a contradiction. And all without having to invoke merely physical objections as to the impracticality of building a Zeno machine.

 

[Theftz22]

The statement:



Is actually not quite true. Merely being able to compute an infinite number of commands doesn't by itself give you the ability to solve otherwise undecidable problems. Consider the problem:



It is a question, and it has a correct answer. But no amount of computing power will help you in finding the answer, not even a Zeno machine.

Well that's not an objection to the premise that "a machine that can complete an actually infinite number of calculations in a finite time can solve the halting problem". What that is an objection to is the premise that " A machine that can complete an actually infinite number of calculations in a finite time can solve the any problem". But that's just not the premise in question.

The proof that a Zeno machine can solve the halting problem is by writing a computer program that can do the task. Which is actually really simple. The Wikipedia page has an example of such a program in pseudocode that is only a few lines long.

Well that's why I put that code in the justification for premise 2.


So the new sub-argument for premise 2 is:

1. The Zeno Machine can complete an actually infinite number of calculations in a finite time.
2. A machine that can complete an actually infinite number of calculations in a finite time can sum all of the terms in Grandi's series.
3. It is logically impossible to sum all of the terms in Grandi's series.
4. Therefore the Zeno Machine is logically impossible.

What are the justification's for premises 2 and 3?

 

[AltF4]

I would actually drop the halting problem portion entirely. The alternative is a lot easier to understand, I think. And maybe is more sound, too.

So the whole thing looks like:

1) The Zeno machine is conceivable.
2) The Zeno machine is logically impossible.
3) Therefore, conceivability does not imply possibility.

With:

1a) The Zeno machine is merely a Turing machine with one (and only one) extra twist. Turing machines are definitely conceivable. The computer in front of you now is one.
1b) The "twist" is that rather than each individual computation taking constant time, we suppose that each computation takes half as much time as the previous one.

2a) Consider a mathematical problem that has no solution. Not one which is hard, or one for which a solution is not known. A math problem that provably has no answer. One such problem is the sum of the Grandi series. The series is divergent, meaning that it has no sum.
2b) A Zeno machine would be able solve the sum of the Grandi series. (A contradiction) You just tell it to start performing the calculations beginning at the first. Then in a finite amount of time, it will have computer all infinite terms of the series.

 

[Theftz22]

I'm still struggling to see why you think the Zeno machine could sum a divergent series if it is true that a divergent series has no sum. If it truly had no sum, the machine could not return anything. If it did return something, then the series must have a sum.

Anyway, wikipedia seems to say that a divergent series can be summed:

http://en.wikipedia.org/wiki/Divergent_series#Theorems_on_methods_for_summing_divergent_series

 

[Budget Player Cadet_]

I'm still struggling to see why you think the Zeno machine could sum a divergent series if it is true that a divergent series has no sum. If it truly had no sum, the machine could not return anything. If it did return something, then the series must have a sum.

...That's kind of the point. It's able to provide a "correct" answer which is logically impossible, therefore it itself is logically impossible. It's like a hypothetical machine that can divide by zero and come out with a "correct" answer: it's completely ludicrous and mathematically impossible, therefore the machine itself is logically impossible.

 

[AltF4]

Anyway, wikipedia seems to say that a divergent series can be summed:

http://en.wikipedia.org/wiki/Divergent_series#Theorems_on_methods_for_summing_divergent_series

What it says is:

In specialized mathematical contexts, values can be usefully assigned to certain series whose sequence of partial sums diverges.

So, for example, you can use an averaging function and assign the Grandi series a "sum" of 1/2. Which may be useful in some circumstances. But this is not the same thing as saying it actually has a sum. Only convergent series do.

To further show you why you can't give a divergent series a sum, consider Gandi's series again:

1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 ...

What does the series look like if we merely rearrange the terms with parentheses?

1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 ...
(1 - 1) + (1 - 1) + (1 - 1) + ...
0 + 0 + 0 + 0 + 0 + ...
So the sum appears to be equal to zero.

But what if we rearrange the parentheses differently?

1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 ...
1 + (-1 + 1) + (-1 + 1) + (-1 + 1) ...
1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 ...
So the sum appears to be equal to one

The truth is there there is no sum. So trying to find one will always result in contradiction.

 

[Theftz22]

I guess there are two objections I have to the argument.

1. I'm not sure that a zeno machine could perform an actually infinite number of calculations in a finite time. The zeno machine is adding finite numbers of calculations onto a growing amount of total calculations while being multiplied by a finite exponential increase in speed. The machine is converging on infinity as a limit, but it will never perform an actually infinite number of calculations in a finite time. Any finite quantity added to, or multiplied by, another finite quantity will only yield a finite quantity.

2. Even if it could perform an infinite amount of calculations in a finite time, the zeno machine couldn't solve for the sum of Grandi's series. The zeno machine could add up all the infinite number of terms in the series and get a sum, but it wouldn't have "solved" the problem. As you noted, simply by rearranging parenthesis, the sum can differ. But then the answer that the machine would return would just be one of many possible sums, and we don't even need a machine to do that, you just did that yourself in the previous post! Does that mean you are logically impossible? Obviously not. The only reason I could see this being in some way significant is if the zeno machine's answer is somehow to be preferred over the other answers. But why? It would only be preferred if it is correct. But if it is correct, then the series really has an objective sum after all!

 

[Bob Jane T-Mart]

I think you and Alt are assuming conceivability is supposed to be self-evident. Conceivability can be contentious, and proving conceivablity may require very complex argumentation.

Look. I'm pretty sure that if you can draw something in fair detail it's conceivable. Here is an image of the "overbalanced wheel", a perpetual motion machine, that surprise, surprise, doesn't work.

They've also made models of it.

I don't see how the overbalanced wheel can be considered inconceivable, if you can make models of it.

Explain to me what is wrong with this conception of this device. It works in theory:

On one side of the wheel the hammers extend straight out, making a longer lever arm (the distance from the center of the wheel to the hammer's head). On the opposite side of the wheel, the hammers lay close and make a short lever arm. Since the lever ratio is greater where the hammers stick out, it's unbalanced and makes the system want to tip, and thus the wheel to rotate. As it rotates around, the next hammer falls into place continuing the motion "perpetually". Of course, we now know that such a device will never actually work, but the subtle reasons why it won't work can be elusive.

Even the idea of it working is conceivable, if you want to argue that it's not the device that is inconceivable, and instead is concept of perpetual motion. I'd just like to say that Perpetual motion is conceivable. That's why people built it in the first place.

Also, conceivability implies logical possibility, but inconceivability doesn't imply impossibility. For example it's impossible to genuinely conceive of God, and probably certain things in quantum mechanics, but that doesn't mean they're impossible.

I was attacking the assumption that this conceivability = possibility idea is based upon. It is based upon the idea that humans have an intuitive grasp of reality. And the fact is that we don't.

 

[AltF4]

I guess there are two objections I have to the argument.

1. I'm not sure that a zeno machine could perform an actually infinite number of calculations in a finite time. The zeno machine is adding finite numbers of calculations onto a growing amount of total calculations while being multiplied by a finite exponential increase in speed. The machine is converging on infinity as a limit, but it will never perform an actually infinite number of calculations in a finite time. Any finite quantity added to, or multiplied by, another finite quantity will only yield a finite quantity.

Then tell me this: how many operations will the Zeno machine have executed after 1 second of running?

(Assuming the first takes 1/2 a second, the second 1/4, the third 1/8, etc...)

2. Even if it could perform an infinite amount of calculations in a finite time, the zeno machine couldn't solve for the sum of Grandi's series. The zeno machine could add up all the infinite number of terms in the series and get a sum, but it wouldn't have "solved" the problem. As you noted, simply by rearranging parenthesis, the sum can differ. But then the answer that the machine would return would just be one of many possible sums, and we don't even need a machine to do that, you just did that yourself in the previous post! Does that mean you are logically impossible? Obviously not. The only reason I could see this being in some way significant is if the zeno machine's answer is somehow to be preferred over the other answers. But why? It would only be preferred if it is correct. But if it is correct, then the series really has an objective sum after all!

No, don't confuse the language. No sum "can differ". It is not the case that there are multiple sums and any of them are valid. It is the case that there is no sum.

Try thinking of the problem another way, like this:

Consider the "running tally" approach to the Grandi's series.

1 (sum of 1)
1 - 1 (sum of 0)
1 - 1 + 1 (sum of 1)
1 - 1 + 1 - 1 (sum of 0)
1 - 1 + 1 - 1 + 1 (sum of 1)
1 - 1 + 1 - 1 + 1 - 1 (sum of 0)

So the sum depends entirely on the last term in the series that you added. But in the Grandi's series, there is no last term. (It's an infinite sum) So there is no sum.

Now consider what it would mean to say that you added up all the terms of the Grandi's series. Which is not in principle impossible by itself. Convergent series can be summed just fine. Such as:

.9 + .09 + .009 + .0009 + .00009 + ... = 1
1/2 + 1/4 + 1/8 + 1/16 +1/32 ...= 1
etc...

But not a divergent series. Because the sum depends on the last term, of which there is none.

 

[Theftz22]

Then tell me this: how many operations will the Zeno machine have executed after 1 second of running?

(Assuming the first takes 1/2 a second, the second 1/4, the third 1/8, etc...)

I don't know. At a high enough term in the sequence, my calculator simply rounds the time it takes for each calculation down to zero. But it literally doesn't make sense to say that a machine performs a calculation in zero time. Time is essentially the measure of the secession of events. Given that a machine performing a calculation is an event, it's just incoherent to say that a machine performs a calculation in zero time.

No, don't confuse the language. No sum "can differ". It is not the case that there are multiple sums and any of them are valid. It is the case that there is no sum.

Try thinking of the problem another way, like this:

Consider the "running tally" approach to the Grandi's series.

1 (sum of 1)
1 - 1 (sum of 0)
1 - 1 + 1 (sum of 1)
1 - 1 + 1 - 1 (sum of 0)
1 - 1 + 1 - 1 + 1 (sum of 1)
1 - 1 + 1 - 1 + 1 - 1 (sum of 0)

So the sum depends entirely on the last term in the series that you added. But in the Grandi's series, there is no last term. (It's an infinite sum) So there is no sum.

Now consider what it would mean to say that you added up all the terms of the Grandi's series. Which is not in principle impossible by itself. Convergent series can be summed just fine. Such as:

.9 + .09 + .009 + .0009 + .00009 + ... = 1
1/2 + 1/4 + 1/8 + 1/16 +1/32 ...= 1
etc...

But not a divergent series. Because the sum depends on the last term, of which there is none.

Fair enough.

 

[AltF4]

No calculation is performed in zero time. Quite to the contrary, every calculation is performed in non-zero finite time.

For any calculation, it will take exactly f(x) = 1 / ( 2^x ) time. Where x is the position of the calculation in the sequence. Your calculator may round down to zero, but that's besides the point.

 

[Budget Player Cadet_]

I don't know. At a high enough term in the sequence, my calculator simply rounds the time it takes for each calculation down to zero. But it literally doesn't make sense to say that a machine performs a calculation in zero time. Time is essentially the measure of the secession of events. Given that a machine performing a calculation is an event, it's just incoherent to say that a machine performs a calculation in zero time.

That's, as far as I have gathered, the point. This machine is logically impossible. It cannot exist.

EDIT: Oh. Never mind then.

 

[Bob Jane T-Mart]

Can I just point out that nobody has addressed my point about perpetual motion machines?

 

[Theftz22]

Look. I'm pretty sure that if you can draw something in fair detail it's conceivable. Here is an image of the "overbalanced wheel", a perpetual motion machine, that surprise, surprise, doesn't work.

They've also made models of it.

I don't see how the overbalanced wheel can be considered inconceivable, if you can make models of it.

Explain to me what is wrong with this conception of this device. It works in theory:



Even the idea of it working is conceivable, if you want to argue that it's not the device that is inconceivable, and instead is concept of perpetual motion. I'd just like to say that Perpetual motion is conceivable. That's why people built it in the first place.

The perpetual motion machine is physically impossible given the contingent laws of nature acting in our universe, but of course there is some logically possible world in which gravity, etc. acts differently and thus perpetual motion is possible. So the example of a perpetual motion machine does not threaten the move from conceivability to logical possibility.

I was attacking the assumption that this conceivability = possibility idea is based upon. It is based upon the idea that humans have an intuitive grasp of reality. And the fact is that we don't.

Conceivability entailing possibility is very different from saying our intuitions are reliable. For one, conceivability is merely the ability to form a mental image of something, whereas saying something is intuitive just means it really feels or seems like its true. Secondly I'm not arguing that conceivability entails actual possibility, just logical possibility. So even if intuitiveness were the same thing as conceivability, the conceivability to possibility thesis is not saying that we have an intuitive grasp on actual reality.

No calculation is performed in zero time. Quite to the contrary, every calculation is performed in non-zero finite time.

For any calculation, it will take exactly f(x) = 1 / ( 2^x ) time. Where x is the position of the calculation in the sequence. Your calculator may round down to zero, but that's besides the point.

I don't see the point then. I don't know how many calculations are done in a minute. Some very large finite number. Maybe you could show the calculation that gives an infinite with a finite time input?

 

[AltF4]

I ask this seriously (not sarcastically, it's hard to tell sometimes online): Have you taken university level calculus?

It seems like you're having a very similar hangup about infinite series that a lot of people do, before you take calculus. You wouldn't be the first person! It used to be believed that the sum of any series (of positive numbers) must be infinity. This was only proven wrong with the invention of calculus.

It's in a similar vein to the following fact:

.9999999... (repeating) = 1

Initially your reaction is that they cannot possibly be the same thing, but they are. You can view the left hand side of the equation as an infinite series, which looks like:

.9 + .09 + .009 + .0009 + .00009 + ... = .9999999...

This is again an infinite series who's limit (as the number of terms approaches infinity) is 1. That means that as the number of terms gets closer and closer to infinite, the sum gets closer and closer to 1.

And now here's the calculus part:

When the number of terms is infinite, then the sum reaches 1.

Alternatively, consider the function:
f(x) = 1/x

Imagine moving along the curve backward from 1 to 0. As you do so, the function gets closer and closer to infinity. Then, at x=0, the function reaches infinity.


So when you have a machine (the Zeno machine) performing calculations with halving duration, it can perform an infinite number of calculations in a finite amount of time. Each calculation takes f(x) = 1 / (2^x) time. (Where x is the position in the sequence).

And you can fit a countably infinite number of these calculations in a finite amount of time.

 

[Bob Jane T-Mart]

The perpetual motion machine is physically impossible given the contingent laws of nature acting in our universe, but of course there is some logically possible world in which gravity, etc. acts differently and thus perpetual motion is possible. So the example of a perpetual motion machine does not threaten the move from conceivability to logical possibility.

Well, yes it is logically possible, but what use is that notion? None, if it's not physically possible, it's not possible. The idea that something could be logically possible without being physically possible is irrelevant, because it's not possible anyway.

Conceivability entailing possibility is very different from saying our intuitions are reliable. For one, conceivability is merely the ability to form a mental image of something, whereas saying something is intuitive just means it really feels or seems like its true. Secondly I'm not arguing that conceivability entails actual possibility, just logical possibility. So even if intuitiveness were the same thing as conceivability, the conceivability to possibility thesis is not saying that we have an intuitive grasp on actual reality.

Conceivability entailing possibility rests upon the idea that we somehow understand logic/reality such that when we can only build a mental image of an object that is possible.

Again logical possibility is an irrelevant term. It's basically saying, "LET'S PRETEND THAT WE LIVE IN A DIFFERENT UNIVERSE WHERE IT IS POSSIBLE, THEN IT'LL BE POSSIBLE, POSSIBLY." So what? It has no utility. If this is what conceivability entailing possibility proves then it becomes irrelevant and not worth any consideration for any practical purpose.

And what if the laws of logic/math were to change in different possible universes? As far as I can tell, it's perfectly possible, (although admittedly, I can't tell much). If this is possible, conceivability entailing possibility is by definition true, because everything is possible anyway. At this point the notion again becomes pointless as everything is already possible.

 

[Theftz22]

I ask this seriously (not sarcastically, it's hard to tell sometimes online): Have you taken university level calculus?

I don't plan on taking calculus when I go to college. I am currently almost failing in precalc as a high school junior anyway.

It seems like you're having a very similar hangup about infinite series that a lot of people do, before you take calculus. You wouldn't be the first person! It used to be believed that the sum of any series (of positive numbers) must be infinity. This was only proven wrong with the invention of calculus.

It's in a similar vein to the following fact:

.9999999... (repeating) = 1

Initially your reaction is that they cannot possibly be the same thing, but they are. You can view the left hand side of the equation as an infinite series, which looks like:

.9 + .09 + .009 + .0009 + .00009 + ... = .9999999...

This is again an infinite series who's limit (as the number of terms approaches infinity) is 1. That means that as the number of terms gets closer and closer to infinite, the sum gets closer and closer to 1.

And now here's the calculus part:

When the number of terms is infinite, then the sum reaches 1.

Alternatively, consider the function:
f(x) = 1/x

Imagine moving along the curve backward from 1 to 0. As you do so, the function gets closer and closer to infinity. Then, at x=0, the function reaches infinity.


So when you have a machine (the Zeno machine) performing calculations with halving duration, it can perform an infinite number of calculations in a finite amount of time. Each calculation takes f(x) = 1 / (2^x) time. (Where x is the position in the sequence).

And you can fit a countably infinite number of these calculations in a finite amount of time.

I'm afraid I'm going to have to duck out of this debate. I'm over my head with the mathematics and honestly debating math is about as fun as watching paint dry to me. Besides, I don't really need the different modal properties of the mind and the brain to show that they are different, there are many more properties that the mind and brain don't share. Maybe I'll make a thread on that at some point.

Well, yes it is logically possible, but what use is that notion? None, if it's not physically possible, it's not possible. The idea that something could be logically possible without being physically possible is irrelevant, because it's not possible anyway.

Actual possibility and logical possibility are quite distinct concepts actually. Actual possibility refers to whether or not something could happen in the actual world, where logical possibility refers to whether something happens in some possible world. So don't equivocate on those two notions.

Conceivability entailing possibility rests upon the idea that we somehow understand logic/reality such that when we can only build a mental image of an object that is possible.

Yes, and that's quite different from defending the reliability of intuition.

Again logical possibility is an irrelevant term. It's basically saying, "LET'S PRETEND THAT WE LIVE IN A DIFFERENT UNIVERSE WHERE IT IS POSSIBLE, THEN IT'LL BE POSSIBLE, POSSIBLY." So what? It has no utility. If this is what conceivability entailing possibility proves then it becomes irrelevant and not worth any consideration for any practical purpose.

I'm afraid you underestimate the applicability of modal logic. Modal logical referring to possible worlds, necessity, and identity actually has huge use in prominent areas of philosophy. All the back from Aristotle it was accepted that knowledge is justified true belief. Then Gettier came around and used modal logic to show that possibly knowledge is not identical to justified true belief, and since identity is a necessary property, most philosophers believe Gettier radically changed our understanding of knowledge in the actual world. In fact this method of appealing to possibility is something we use all the time, when someone gives you a definition or a categorical statement we immediately try to formulate possible hypothetical counterexamples in a subconscious use of modal logic. In fact your recent post in the determinism thread where you consider possible universes, or possible worlds, did exactly that. Or consider a case in which we are considering the existence of a necessary being, say god. Then modal logic again comes into play because all you have to do is show that that entity is logically possible to show that it actually exists, or that its non-existence is logically possible to show that it actually does not exist. Finally consider the mind-brain identity thesis. That's what we've been discussing all along in the context of modal logic to show that they are not identical in the actual world because they are possibly not identical. All of this just goes to show that modal logic and possible world semantics has a bigger impact than you think and are relevant to the actual world.

And what if the laws of logic/math were to change in different possible universes? As far as I can tell, it's perfectly possible, (although admittedly, I can't tell much). If this is possible, conceivability entailing possibility is by definition true, because everything is possible anyway. At this point the notion again becomes pointless as everything is already possible.

Almost all philosophers have always maintained that the laws of logic and math are necessary. At least I can't even understand what it would be for them to be different. What would it mean for it to be true for 2+2 to equal 5? Given the definitions of 2, +, and 5 I just don't see what it means. It's just flatly incoherent.

 

[Bob Jane T-Mart]

I'm afraid you underestimate the applicability of modal logic. Modal logical referring to possible worlds, necessity, and identity actually has huge use in prominent areas of philosophy. All the back from Aristotle it was accepted that knowledge is justified true belief. Then Gettier came around and used modal logic to show that possibly knowledge is not identical to justified true belief, and since identity is a necessary property, most philosophers believe Gettier radically changed our understanding of knowledge in the actual world. In fact this method of appealing to possibility is something we use all the time, when someone gives you a definition or a categorical statement we immediately try to formulate possible hypothetical counterexamples in a subconscious use of modal logic. In fact your recent post in the determinism thread where you consider possible universes, or possible worlds, did exactly that. Or consider a case in which we are considering the existence of a necessary being, say god. Then modal logic again comes into play because all you have to do is show that that entity is logically possible to show that it actually exists, or that its non-existence is logically possible to show that it actually does not exist. Finally consider the mind-brain identity thesis. That's what we've been discussing all along in the context of modal logic to show that they are not identical in the actual world because they are possibly not identical. All of this just goes to show that modal logic and possible world semantics has a bigger impact than you think and are relevant to the actual world.

While modal logic may be highly useful, I don't think the notion of conceivability implying possibility is useful at all. It's like solipsism, it doesn't really go anywhere.

Almost all philosophers have always maintained that the laws of logic and math are necessary. At least I can't even understand what it would be for them to be different. What would it mean for it to be true for 2+2 to equal 5? Given the definitions of 2, +, and 5 I just don't see what it means. It's just flatly incoherent.

Well... That's the point. There may be possible universes that are incoherent to us with all the laws of logic and math changed. And if this is the case, then anything is possible, if you pick the right universe. I'm just pointing out that if this is the case, the debate is moot as everything is possible.