For me, the word "doubt" applies to a conscious state, not a subconscious state. — jgill

Then why bring subconsciousness into the discussion? Ok. Is the conscious state the result of the brain process?

I don't believe we are aware of all the information that enters our mind. — jgill

We, our conscious minds to be more precise, are not aware of all the information that we perceive through our sensory systems.

If that is the case what the subconscious processes may indeed inform us - in what seems to be an act of free will. — jgill

Isn't the subconscious process deterministic? Doubts are not allowed in a deterministic system. That is true since a deterministic system moves from one state to another unique state later. So there is only one state available for a deterministic system at any given time. There are two states available to choose from when we have doubts though.

Even if we ignore quantum indeterminacy... As you say, we don't know what is going on in our own brains in any detailed way, so how can you rule out subconscious bias as being what amounts to a coin flip in your head? — wonderer1

Do you mind explaining what you mean by the subconsciousness? Does it have a mind or is it a deterministic entity?

Also, I don't recall you acknowledging the the sort of indeterminacy that can result from system complexity. IIRC you have a physics background, so perhaps it would be worthwhile for you to consider the relevance of the three body problem to the complex molecular environment of a brain. — wonderer1

I know problems related to a system with three or more particles. There is no analytical solution for such a system and the system could be chaotic depending on the initial condition.

Which brings us back to the role of ignorance in attributing things to free will.

We don't know anything remotely approaching the exact initial conditions of our brains and all the environmental factors which play a determing role in what happens in our brains. Furthermore, there is lots of good evidence for the powerful role of subconscious processes emerging in our conscious thought. — wonderer1

Correct. We don't know about the exact condition of neural activity of our brain but we know that it is deterministic. There is however a problem in the deterministic worldview so-called doubt. Options are real in the case we have doubts and a deterministic entity cannot deal with a situation when there are doubts.

How is a "mind" a better explanation than subconscious processes? — wonderer1

The subconscious process cannot resolve the conflict when we have doubt in a situation. That is true since the options are real when we have doubts and we don't have any reason to choose one option over another option.

Can coins be tossed in a deterministic world? — wonderer1

Yes, we can toss a coin. I however have to add that the outcome of tossing a coin is not known to us due to our ignorance about the initial condition of the coin and the situation of the environment. If one knows the exact initial condition of the coin when it is tossed and the situation of the environment, such as wind, then one can know the outcome of tossing the coin.

If so, then why can't a neural mechanism do something analogous to tossing a coin? — wonderer1

The neural mechanisms are well-defined and deterministic. The is no agent in each neural point with a coin available to it.

You are traveling through a maze and reach a fork. Here you experience a maximum degree of doubt (uncertainty), and the consequences of making a wrong decision are large. Now you ponder and then make a decision. Is this free will? — jgill

I used the maze example to ensure we can agree that the options are real. That is a huge step in the discussion of free will since many people simply argue that one of the options is an illusion and you cannot choose it. Why do people argue such a thing? Because they believe in determinism and within determinism options are not allowed. I also used the maze example to ensure that we have no reason to choose one path over another yet we can decide and choose one of the paths. That is to me the very definition of free will: "A decision is either based on a reason or not, in the first case we are dealing with an unfree decision, and in the second case we are dealing with a free decision". By this definition, I simply set up a dichotomy so given that one of the definitions is related to the unfree decision we are left with another definition for free decision.

Or does some internal neural mechanism in your subconscious "toss a coin"? — jgill

If we accept that neural mechanisms are deterministic then subconsciousness cannot toss a coin. That is true since the outcome of tossing a coin is not known whereas the outcome neural mechanisms are well defined and known.

Hi and welcome to the forum!

I would say that you need to pick a topic you are interested the most and focus on it. Go to the related subforum and read through the names of threads to see if you like a subject. Then read through the thread you are interested in. If you don't understand something then simply google it or ask within the thread (the mention command Ctrl+m is your friend). People here are knowledgeable and kind. Hopefully, you will be able to contribute to a discussion you are interested in soon.

Perhaps "a reason" would be clearer than "reason." — T Clark

Thanks for the correction.

Even then I'd be tempted to argue your point, but then we'd just get sucked into another of those unresolvable arguments that results whenever we dive into the bottomless pit of free will. — T Clark

It is alright if you don't want to engage in the discussion of free will. I don't think it is bottomless considering the argument provided in OP.

Most of our decisions are not based on reason, for example when I decide to order a cheeseburger rather than pate de foie gras or when I turn left on Washington Street without thinking about it. — T Clark

Are you talking about unconscious decisions?

Preferences and values are not generally rational. — T Clark

Correct. By reason, I generally mean a cause for an action, whether it is feeling, preference, value, rationality, etc.

You were the one who postulated a mind distinct from the brain, and that brains cannot produce random number generators is part of the point. The supposedly freely choosing mind only requires that the doubtful maze choice be made without reason; that is to say with doubt. Since the mind is distinct from the brain as per your definition, I'm saying that it is not necessary for this mind to make meaningful choices, as that would require reasoned intent. Thus, the mind could be choosing in the doubtful maze scenario according to things that have nothing to do with intentioned choices, but rather something like a coin flip or random number generator (even if those aren't totally random). When generalized, this conflicts with any sort of conception of free will there might be; that the mind must operate the way a brain does is not required to fit your definition of free choice. — ToothyMaw

Correct. What the mind does in a doubtful situation is random and it is similar to tossing a coin.

And a question I should've asked earlier: are you saying that the freely choosing mind has freedom of choice in situations in which there is no doubt? — ToothyMaw

That is a very good question! I believe so but I don't have a solid argument for it. Most of the time we make decisions even without being aware of them. For example, think of a situation that you are deriving on a familiar road, and your conscious attention is on the music playing in your car. You make tons of decisions while you are deriving, like turning left or right at a junction without being aware of them. But suppose that a cat jumps into the road that you are deriving. Suddenly, your mind is alerted and takes control of the situation. You press the brake and stop the car to avoid hitting the cat and probably killing it. So I think that the mind plays a role in such a situation.

Exactly right. :100: :up: @wonderer1

I could model the choice the mind makes in the "doubtful" maze situation on a random number generator, or a coin toss, and it could still have the kind of freedom you describe; there is not necessarily the kind of reasoned intention that is required for a mind to be making a meaningful choice. — ToothyMaw

To the best of my knowledge, there is no pure random generator but a pseudo-random generator. You can read more about pseudo-random generator here. Regardless, the brain cannot produce a random generator to decide about a situation when the outcomes of options are not known. How about a coin toss? You can use a coin to choose a path in the maze. You however don't need it since you have the ability to freely decide.

Would you say that having freedom is dependent on being ignorant about some things? — wonderer1

No, I would say that our freedom allows us to decide when we are ignorant about the outcomes of the options.

There are multiple senses of the word "indeterminism" and indeterminism in the sense discussed in the following article is relevant here:
https://www.researchgate.net/publication/300312282_Indeterminism_in_System_Science — wonderer1

It seems to me that the author of the manuscript is a computer scientist and not a particle physicist so I won't buy his words. I read the manuscript once and I could see the author's ignorance in the field of physics. The standard model is our best theory that describes reality well and it is experimentally tested. The particles, fermions, in this model, interact with each other through forces, bosons. The form of forces are well known and they are deterministic. By deterministic I mean that you can derive a set of equations of motion for the field operator of fermions and the field operator of bosons.

Brains are enormously complex entities that aren't deterministic in the sense that, given complete information about a brain and some rather enormous amount of the environment in which the brain exists, we could make perfect predictions about what will happen in that brain in that environment. — wonderer1

The brain is a complex entity but it is made of particles in which particles interact with the well-known forces. Therefore, I don't think that the brain is an indeterministic entity.

I see room to disagree with this as an absolute in many if not most cases of general uncertainty. Should I pass the semi-truck in front of me? (the closest oncoming car appears to be miles away, though it would be safer not to) Should I really blow another $10 on another lotto ticket this week? (the odds of winning anything are astronomically low, though it's always possible) Etc. Though, your specific example of a (I would say generally uncommon) scenario where there is truly zero background information on the likelihood (or degree) of benefit or detriment of one option over the other, like a gamble, makes for an interesting thought experiment. — Outlander

I use the example of the maze to ensure we can agree that the outcome of options is not determined in this case. Once this is established I can discuss the rest of my argument. There are other cases which we are uncertain about the outcomes of options (the two examples you provided and many others).

I'd agree with that. But what of the most simple organism capable of traversal, say, a snail crawling through a log (or something that presents an identical physicality to your maze scenario)? Assuming it just doesn't turn around (or crawl up the wall as snails so often do), it will likely either end up going left or right absent of any relatable "mind-thought" process, wouldn't it?

People however will just "wing it", per se, and pick one to avoid losing time and ensure the destination, whatever it may be, is reached. By which I mean, I'd assume there would be very little deep thought on the matter other than "keep going" and "just pick one" if there is truly no information available or apparent distinction between one choice and another. — Outlander

Yes, that is correct. The point is however that we can choose one option over another when there is no information available on the outcomes of options.

I dabble in psychology but am certainly not a physiologist (how the body and "mind" work together and conversely affect one another). The brain allows "us" or "you" access to retrieve/recall our experiences and knowledge thus forming an identity or "consciousness" which can be referred to as a separate "entity" or a "mind". — Outlander

By mind, I don't mean the brain or anything like identity which is formed in the brain as a result of neurobiological processes.

But is it really? Like, what if, somehow, a person was raised in a sealed, pitch black room with zero interaction with any living being from infancy to adulthood (naturally with food and water), would they have a "mind"? — Outlander

They certainly have a mind, by mind I mean an entity that can experience what is produced by the brain. His/her ability to experience however is very limited (please see the next comment).

Would they be "conscious" in the way we consider human consciousness having no real sensory experiences or knowledge? — Outlander

Our ability to experience reality well develops over time since the time of infancy. There is ongoing research on this topic. For example, this article discusses how our visual ability developes over time.

What was the second case? Or if the maze was the second, what was the first? — Outlander

The first case is when a decision is based on a reason. We say that the decision is unfree in this case. And the second case is when a decision is not based on a reason. We say that the decision is free in this case.

I like your argument but I would say that the conclusion that follows is that D is an inadequate definition of a continuum. A continuum cannot be completely described with points. — keystone

Yes, my definition of the continuum is not adequate. Another poster gave a definition continuum close to mine but it is correct. I can search the thread and find the definition for you if you are interested.

Yes and thanks.

Yes exactly. — fishfry

Thanks for the clarification. Can we calculate the contribution of the $\omega$th term to the series?

It is alright guys. I learned lots of things from both of you and I am thankful for it. I don't think that personal attacks are useful though when people can discuss things openly, so one party learns and another party teaches. I have to say that the process of learning is time and energy consuming so it is alright if one person needs the explanation of things a few times.

You're very confused and resistant to the explanations given you to cure your chronic confusion. — TonesInDeepFreeze

@fishery already explained how to resolve the problem of indexing here.

Wrong. I explained the difference between them. Knowing the definition of 'the continuum' does not provide knowing the definition of 'continuous'. — TonesInDeepFreeze

Do you mind elaborating?

Thanks. So you simply extend the natural number to the extended natural number and resolve the problem of indexing.

1, 2, 3, 4, 5, 6, ...

Is that not an infinite sequence? — fishfry

That is an infinite sequence. I am however interested in the sequence first mentioned by Zeno in Dichotomy Paradox in which the infinite member exists. Each member of the above sequence is finite, so you cannot use the above sequence to give indexes to all members of the sequence in Dichotomy Paradox since the infinite member exists.

You mean sequence. A series is a sum. — fishfry

Thanks for the correction.

The arrow paradox is that the arrow does not move but that it moves.

/

Average speed is distance/time. In Zenos's paradox, both are finite. — TonesInDeepFreeze

Please accept my apology. My, argument here was for Dichotomy paradox. You need to replace the arrow in that post with the runner, Atalanta.

(x+y)/2 is the arithmetical mean of {x y}, not the geometrical mean. — TonesInDeepFreeze

Thanks for the correction.

What does that mean? — TonesInDeepFreeze

I mean you cannot give indexes to all members of an infinite series.

I don't know. First you would need to define "speed in all infinite steps". — TonesInDeepFreeze

I can define the speed in $i$th step as follows: $v_i=\frac{(\Delta x)_i}{(\Delta t)_i}$ where $(\Delta x)_i$ is the length of $i$th interval and $(\Delta t)_i$ is the time duration it takes the runner (I am referring to Dichotomy paradox) to move $i$th interval. The series however has infinite steps so I cannot define the speed in all infinite steps since $i$ is a natural number.

My point was that he didn't ask for a definition of 'the continuum'. The takeaway for you is to not conflate 'the continuum' with 'continuous'. — TonesInDeepFreeze

Continuum is a continuous series. He understands what continuous is if he understands what continuum is.

I didn't say that he needs to read a book. I said the definition is in chapter 1 of such books. — TonesInDeepFreeze

Thanks. @fishery gave a definition for a continuum from wiki: "Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two distinct elements there is another (and hence infinitely many others), and complete, i.e., which "lacks gaps" in the sense that every nonempty subset with an upper bound has a least upper bound."

The arrow paradox says each is zero, as in time "points". Yet there is still the forward motion of the action, driven by energy — Gregory

I am not sure whether he was familiar with the concept of speed or not. But, the average speed in the interval $i$ can be calculated as $v_i=\frac{(\Delta x)_i}{(\Delta t)_i}$ where the $(\Delta x)_i$ is the length of $i$th interval and $(\Delta t)_i$ is the time duration it takes the arrow to move $i$th interval. So everything is clear for now. The problem is however with the index $i$ which cannot be infinite since it is a natural number yet we know that infinite steps exist.

The speed of Achilles is 10meters/1second. The speed of Tortoise is 1meter/1000seconds. — TonesInDeepFreeze

Could you calculate the speed in all infinite steps?

Correct meaning you understand that the rationals are dense but not continuous? — fishfry

If by dense you mean that there exists a point between two arbitrary points then I understand that applies to the set of rational numbers. The link you provide is technical for me and I have to put more effort into understanding it.

Haven't we been doing that all along? Not sure what you mean. — fishfry

By that, I mean that there exists a point between two arbitrary points in which the between is defined as the geometrical mean.

The set of standard real numbers, as you yourself have defined it since the first post in this thread, when claiming it doesn't exist. I believe you've now come around to accepting that it does exist. So that's the mathematical continuum. The real numbers.

ps -- Technically, what I've described is a linear continuum.

Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two distinct elements there is another (and hence infinitely many others), and complete, i.e., which "lacks gaps" in the sense that every nonempty subset with an upper bound has a least upper bound.
— Wikipedia — fishfry

Ok, that definition seems good and simple for @tim wood. Thanks for providing the definition.

And the Zeno paradox does not threaten mathematics. — jgill

How could you index an infinite set of steps?

I haven't seen a conceptual analysis that concludes it is discrete, but my impression is that it's typically assumed to be continuous. — Relativist

Well, if space is continuous then it means an infinite number of steps exists yet we cannot complete them. The same applies to time in the example of the infinite staircase.

Is it your opinion, as a physicist, that chaotic systems are not (in principle) reducible to deterministic laws of physics? My impression is that the math related to chaotic systems is pertains to identifying functional patterns to make predictions. That, at least, seems to be the nature of weather forecasts - it's not that the movement of air molecules is fundamentally indeterminstic, rather it's that it's that the quantity of data that would be needed to identify the locations and trajectory of each molecule is orders of magnitude too large to be practical to compute. — Relativist

The laws of physics are deterministic but that does not mean that the chaotic behavior does not exist. It means that any error in the calculation of physical variables leads to a significant deviation from what we observe and what the calculation provides. The source of the error in the case of weather forecast is twofold: (1) The error in the estimate of physical variables in the initial point and (2) Using a discrete approach to solve a set of continuous equations.

We've been considering it at least fifty times already in this thread. What about it do you want to say? — TonesInDeepFreeze

I want to say that you could sweep all points of the continuum using that definition.

He didn't ask for a definition of 'the continuum'. 'the continuum' is a noun. He asked for the distinction between 'continuous' and 'discrete'. 'continuous' and 'discrete' are adjectives.

'the continuum' has been defined at least three times already in this thread.

'continuous function' is the defined as usual in chapter 1 of any Calculus 1 textbook.

Other senses of 'continuous' depend on context. And definitions of 'discrete' depend on context. — TonesInDeepFreeze

He asked for a definition of continuous and discrete in plain English. Could you please provide the definition in plain language without referring him to read a Calculus book?

Step (the verb) = the act of setting ones foot onto the next step (the noun; a thing).

The set of actions maps to the set of things.
The stairway consists of the set of steps, which we're stipulating as being infinite. Unlike the staircase, the acts of stepping don't exist (they are actions). — Relativist

Ok, I see what you mean and I agree.

Ok. The surface of a table-top. Discrete or continuous? A sandy beach? Or the surface of a liquid? — tim wood

You are talking about physical objects that have extensions in space so their location is not definable unless you talk about their center of mass. Do you know what the center of mass is? If not think of an ice cube. The center of an ice cube is its center of mass. The center of mass of the ice cube is definable though hence you can define the location of the center of mass of the ice cube. Now, you can move the ice cube along a line. This means that its center of mass moves from one point to another point along the line. So, by now you have a definition of a point, the center of mass of the ice cube, and a line, its motion along the line.

Certainly by your definition the number line continuous, but made up of discrete points - how can that be? — tim wood

Mathematicians work on abstract objects like points and lines all the time. They define a line as a set of dimensionless points and show that things are consistent. Whether these objects are real or not is subject to discussion.

It would seem that "discrete" and "continuous" are abstract convenient fictions their utility depending on usage in context. Thus when misused you might bet on the tortoise, but I'll bet on Achilles every time. — tim wood

Well, the Zeno paradox certainly threatens mathematics, especially the continuum concept. I also bet on Achilles since my common sense tells me he will win.

Many important metaphysical questions have implications for the physical world. Metaphysics tries to figure things out with conceptual analysis (which can include math and logic) and intuition. In this case, it appears the process can't reach a definitive conclusion. — Relativist

What about the conclusion that spacetime is discrete?

But I wonder: is it really hopeless for physics? You said that treating spacetime as discrete would lead to errors if it's actually continuous. Couldn't this be tested? — Relativist

Yes, weather forecast for example. Any chaotic system in general. Even nonchaotic systems show the error in the long term.

I wasn't claiming it disproved the existence of infinitely many stairs, but it proves that an infinite number of steps cannot be completely traversed in a sequence of of steps of finite temporal duration. — Relativist

Correct. So we are on the same page.

This is in spite of the fact that the set of steps (the activity) maps 1:1 to the set of physical steps that comprise the stairway. — Relativist

Isn't the set of steps the set of physical steps? If yes why do you use a one-to-one map?

The more important conclusion is that there's a logical disconnect between this logical mapping and the analogous temporal process; IOW, the mapping doesn't fully describe the temporal process; something is missing - and it would be worthwhile to develop a mathematics that accounted for this. — Relativist

I cannot figure out what you are trying to say here. Do you mind elaborating?

Correct. How about considering the point between two arbitrary points, namely a and b, to be mean, namely (a+b)/2? If not, could you please define the continuum for @tim wood in plain English?

Although it's a metaphysical question, it pertains to the physical world. — Relativist

Yes, it has an implication. I think it means that spacetime is discrete.