Boltzmannbrain

1 hour ago, joigus said:

I agree with main arguments developed by @Genady, @studiot, and @Lorentz Jr. I particularly liked Studiot's summary. I would call his argument about closed and open sets --as well as those that are neither open nor closed-- a "topological approach."

A crash course in topology would include concepts such as,

Topology: Existence of an inclusion relation in a set, ⊆ --contains--, ⊊ --does not contain. => neighbourhoods of a point.

Limit point --o accumulation point--: A point in a set that has neighbouring points also in the set that are arbitrarily close to it.

Interior of a set: All its point are limits points of the set --if I remember correctly--.

Boundary of a set: The set of all the limit points of its exterior

Closure of a set: The union of the set and ist boundary

...

etc.

With these rigorous topological definitions, when applied to the real numbers, we can prove they constitute a topological space, and, eg, the set [0,2]={xinRsuch that0≤x≤2} contains its boundary --and it is, therefore, closed; while the set, eg, (0,2)={xinRsuch that0<x<2}

Ok, this is helpful.  I will look at some of these concepts more carefully.  Thanks.

2 hours ago, studiot said:

The word point is much overused.

I debated with myself how to avoid it as far a possible to avoid confusion.

Totally agreed.

But I do not like the use of the word end in relation to lines as it can be imprecise.

An open interval has no end.

 

 

I am sorry you missed my main point (there is that word again, but with a different meaning this time)

So it means that my explanation was not good enough so I will try to do better.

But it is difficult to know what you know as you obviously have met the idea of open and closed intervals before since you used an alternative notation in your responses.

I carefully avoided [0,2) etc because it is easy to overlook which bracket is which and you can never be sure whether the writer meant it or not. The reversed square bracket stand out, don't you think ? Also curved brackets are used to denote sets.

Anyway you clearly understand that part.

 

My main point was that lines and numbers are not the same, although they have some properties in common, which allows one to exemplify the other when only the common properties are of concern.

But sets of numbers have lots of other properties where they cannot be represented as lines.

So mathematicians seek more general approaches.

If there is a number that is greater than any other number is the set then the set is bounded.

In fact we say it is bounded above and can say (similarly it is bounded below if there is a number less than any other number in the set.)

So the set (1, 2, 3, 4, 5, 6)  is bounded above by the numbers 6, 7, 8, 9, 10...

We call any of these an upper bound.

Also the upper bound may be an element of the set or it may not.

When the upper bound is an element we call it the maximum of the set.

 

Now there is a theorem, which I will not prove, called the least upper bound theorem.

"If any set of numbers is bounded above it has a least upper bound"

In our example 6 is the least upper bound of our set and is also the maximum.

But our set is also finite so it is easy to see this.

Finite sets means that the count of elements is finite: Infinite sets have an infinite count of elements.

The boundedness theorems apply to finite and to infinite sets, (But not the max and min)

Infinite sets can also be bounded.

The set of all the elements of the negative exponential e^-x, from x=0 to x = ∞  is bounded above by 1 and bounded below by 0, although the set is infinite becasue the count of x values is infinite.

Note the x = ∞ 'end' is never reached  or as I prefer there is no right hand end to this line.

So this line has no minimum.

The left hand end depends whether we include or exclude x = 0 in the set  (closed or open )

If we include x = 0 then the upper bound of 1 is also the maximum,

But if we exclude it then again the upper bound is never reached and the set has no maximum either.

 

 

Thanks a lot for this.  I did take an advanced calculus course that covered maximums, minimums, boundaries, etc, but I have long forgotten most of it.

Using "bounded" instead of "ends" does answer some of my concerns, but I still fell like there is more for me to learn about this part of mathematics.  In other words, I still have ideas that are not answer by what I know up to this point about this topic.  I will look at my old notes.

2 hours ago, Lorentz Jr said:

It's hard to comprehend because you're trying to apply the logic of finite numbers to infinities. What is an "end point"? You have a row of points, and it's the one at the end. But there may not be any "end" to the points when there's an infinite number of them.

Yes, but what keeps me curious is when infinite becomes finite.  For example, an infinite number of numbers/points has a finite distance. 

5 hours ago, Genady said:

The phrase "the points end at 1.5" is wrong. The correct phrase is, "the interval end at 1.5". This does not contradict not containing the point 1.5.

I agree.  I wasn't saying that 1.5 has to be in the interval.  I was saying that since the interval of points (if I can say this) ends at 1.5, then how is there no end point.

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Yes, this is correct. The length is 0.5 in both cases.

This doesn't seem right.  Imagine a line segment [0, 10].  Assume that the segment can revolve about the point 0.  At the halfway point, at 5 units, we we break the segment, keeping the number 5 on the part that revolves.  We revolve the broken segment to some degree then reattach another 5 at the end of the part that doesn't revolve.  The segment [0, 5] cannot revolve past the segment [5, 10] anymore.  The one 5 is taking up the space (even though it is 0 space) that the other 5 needs to revolve through.

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Can you point to any contradiction.

 

Yes, with what you say here, "... it ends at 1.5, but it does not contain the end point. It contains everything before 1.5 ...". 

So the points end at 1.5, but there is no end point?  If that is not a contradiction, then it is a least a very hard thing to comprehend.  

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I don't see any geometrical issue at all. The length of the interval stayed the same as before.

Are you saying that some interval like [1, 1.5] is the same length as the interval [1, 1.5)?

On 2/28/2023 at 10:49 AM, studiot said:

Genady and Lorenz have made some good comments but perhaps I should answer your question more formally.

The line segment does not change its geometry  - you misunderstand.

Lines, line segments and numbers are all three different things.

Lines and line segments are geometrical objects.

Numbers are not.

 

Sorry, I meant points.

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The real number line is a representation of all real numbers, placed into one-to-one correspondence with the real line as an assembly of points.

This is what I am having a hard time understanding.

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The correspondence goes all the way to 'point set theory' where the elements of a set (also called points) can be one of three types.

Interior (also called accumulation points) , boundary or isolated.

Genady has already described two of these, though he called a boundary point an end point.

A set which includes its boundary points is called a closed set.

A set which does not include its boundary points is called an open set.       

An isolated point is both closed and open.

 

As regards line segments, a line segment has two boundary points and is called an interval or line interval.

An interval can be open or closed, but since it has two boundary points it may include one and not the other and we either say the interval is half open or half closed.

 

The whole real number line is open.

The line segment or interval from say zero to plus infinity or minus infinity is half open.

The interval from zero to say 2 is closed if it includes both 0 and 2 and open if it includes neither.

 

These concepts are very important when you study limits and calculus, so they have their own special notation.

Here is a diagram, showing the geometric and algebraic representation of the types interval  (0,2) the rounded brackets are the general form when the type doesn't matter.

Now my main issue is that these points that make up a line segment do not behave like points/objects in a row.  Is that maybe a what I am getting wrong?  Or can these points be seen as being in a row?

On 2/28/2023 at 9:19 AM, Genady said:

The difference is being an end point vs being an internal point. An internal point has neighbors on both sides, but an end point has a neighbor on one side only.

OTOH, you could take out any internal point, say, point 1.5. Then you get two open ends: on the left and on the right of 1.5.

I finally have time to respond.

But let's just think about this geometrically and logically for a moment.  In the case where a line segment ends at 1.5, we are able to take 1.5 from the end of a line segment.  And then we say that it no longer ends.  How does that make any sense?

Thanks for the replies Genady and Lorentz Jr, but I am interested in one aspect of this problem.  I am interested in why the line segment should change its geometry after taking off a number from one end of it.  It seems like the number 2 in the example has a special quality geometrically speaking.  If all there is are is numbers then why is 2 an end point but not anything else?

The real numbers cannot have a next number, but I don't understand how that can be logical.

For example, consider the segment inclusively from 1 to 2, so there are the numbers 1 and 2 at each end of the segment.  We can take off a number like 1.3 or 2 from it.  If we take the number 2 away, we are left with something like the segment 1 to the limit 2 - 1/x as x goes to infinity (or whatever it is). 

So my ultimate question is, why can we take off the end of the segment if it is something that we call 2, but we can't take off another number?  The segment only has real numbers; what makes 2 so special that it can end a segment and be removable?  

19 minutes ago, Genady said:

I understand what you mean. But I think that this is unjustified. I see consciousness as physical. It might turn out equivalent to activation of a specific circuit in the brain, when consciousness is a subjective perception of this activation. Just like your subjective perception of a specific color is hidden from everyone else, but nevertheless is not unphysical and not irreducible to neuron processes.

I put equivalent in bold because now we are getting to the heart of the issue.  By "equivalent" are you saying that they are the same thing (interchangeable)?  If so, then this statement becomes an argument that the consciousness (mental aspect of it, not the physiological aspect) does not actually exist.  If yes, then two things are happening simultaneously: one, as you said, "an activation of a specific circuit"; two, subjective perception arises.  The former is a physical description of the system, and the latter is not a physical description of the system.

Please tell me your answer to my question above.

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Computational irreducibility is something completely different. Computational irreducibility is reduced by performing, or simulating, the computation.

So computational irreducibility is reducible?

6 hours ago, dimreepr said:

Reductionism fails because of tomorrow and becomes less and less relevant with every succeeding tomorrow.

You can't reduce consciousness, mostly because we can't understand the concept beyond ourselves; but also it's binary, because at the middle the spectrum consciousness is a switch and therefore it's a binary question; how do you reduce a binary question?

Yes, that is a good point.

7 hours ago, Genady said:

I don't see what makes it so different. Do you?

Yes, I definitely see what makes it different, its very nature.  It is not physical.  Physical properties affect the physical in one way or another.  The consciousness does not.  It shouldn't be there; it is unpredictable.  Also, a normal physical property is observed.  The consciousness observes; it does not get observed.  Those are two completely different things.  

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I rather think that it is another yet unexplained property of a system and that it is a consequence of the system's structure.

And I agree that when these physical properties are put in the correct process, a consciousness emerges.  But given our understanding of fundamental physics, there is nothing known to this day that would predict such a property.  It is irreducible now.  I put "now" because I suppose maybe we just have to add consciousness to our models once we understand exactly what brings about consciousness.

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Another analogy might be a programming language. It consists of a finite number of simple instructions. When put together in a certain way they calculate number pi.

I don't know enough about how reducibility applies to computer science.  From Wolfram,

"Computations that cannot be sped up by means of any shortcut are called computationally irreducible. The principle of computational irreducibility says that the only way to determine the answer to a computationally irreducible question is to perform, or simulate, the computation". 

Computational Irreducibility -- from Wolfram MathWorld

10 minutes ago, Genady said:

Not necessarily. It looks quite straightforward to me: reductionism allows for a whole to have a property which the parts lack. For example, a composition of non-white colors makes white color. Compositions of sinusoidal functions make functions which are not sinusoidal. Molecules have properties which atoms lack.  Similarly, some composition of some unconscious parts makes something conscious, why not?

An implication of reductionism is that the whole should be predicted by its parts.  We can predict systems like the ones you mention.  And if the system is too complex to predict, we usually do not observe anything as different as a conscious.  The properties that arise are normal physical types of properties that have not yet been modeled.  But the consciousness is not anything like the unexplained physical properties that are in complex systems.  

29 minutes ago, Genady said:

How do we know this?

If we are assuming that things like atoms are not conscious (by consciousness I mean awareness) then we pretty much know.  Although there is a theory IIT trying to reduce consciousness that appears to be just panpsychism, but it is still pretty radical.

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This does not contradict reductionism. Why wouldn't it be possible to predict consciousness based on a knowledge how the unconscious parts are arranged?

Maybe we are talking about 2 different definitions of consciousness. 

An implication of reductionism is that you can always make correct predictions about a system by knowing its parts.

An example of this is for science.  Most of the time science can make predictions by knowing the parts well enough.

But where hard reductionism fails is for the consciousness.  The parts that come together are thought not to be conscious.  So we have the opposite of reductionism, emergentism, or, another way to put it, irreducible.      

3 minutes ago, zapatos said:

Life experience and just common sense. 

So your life experience and common sense has led you to believe that you need a haircut in order to breath?  What is going on here?  What am I missing, seriously?

1 minute ago, zapatos said:

Yes, I was there when you did.

Then why am I wrong?

6 minutes ago, Genady said:

I don't see what makes you think so.

I interpreted the OP to be asking, at least in part, about what human nature is and what it wants.  That is why I gave the answer that I did.  I probably misunderstood what you were getting at.

16 minutes ago, zapatos said:

Wrong again. If you can expand the definition of things to suggest they lead to pain or pleasure, then I can expand the definition of things to suggest they lead to breathing. 

 

Yes, and I gave examples that I do not believe are done in order to breath.

1 minute ago, zapatos said:

I didn't say you did.

Right, I misread your post.  Then those would be examples of something you might do that is not so you can breath. 

9 minutes ago, zapatos said:

If we are going to use terms that broadly, then I guess everything I do is for breathing.

I sit in class so that I can get a job so I can drive to work so that I can shop for food so that I can cook so that I can eat so that I can continue to breathe.

etc.

 

I disagree.  I don't watch t.v. or get a haircut so that I can breath. 

1 minute ago, Genady said:

Eating things that taste good is a basic bodily function.

This sounds like physicalism.  Keeping the topic of dualism out of it, I think we are talking about the same thing.

1 minute ago, zapatos said:

Driving to work.

Paying bills.

Sleeping.

Cooking.

Shopping.

Vacuuming.

Sitting in class.

Reading the news.

Going to church.

Changing diapers.

Cutting the grass.

How many would you like me to list?

Hmm, most of these things I do to avoid pain.  For example, I cook to avoid the "pain" of hunger.  And some others are to avoid emotional pain. 

2 minutes ago, Genady said:

Everything except basic bodily functions.

Everything?!  What if I want to eat ice cream for the pleasure it gives me?  Is that not doing something for pleasure? 

17 minutes ago, zapatos said:

What are you basing this belief on?

Life experience and just common sense.  I mean what typical things can you think of that is not a function or pleasure or pain?

19 minutes ago, iNow said:

That part’s not altogether wrong. Dopamine 

Thank you for your valuable time.  I can only imagine what you had to put off to impart your wisdom onto me. 

2 minutes ago, Genady said:

Not exactly. I'm saying that we do things for various reasons, and pleasure or pain are responses for our success or failure to do what we wanted.

Then I disagree.  I believe that almost everything we do is for pleasure or to avoid pain.  But I do believe that altruism exists too, but very rarely in comparison. 

4 minutes ago, Genady said:

Isn't it the other way around? I.e., whatever we're determined to increase gives us pleasure when it increases, and whatever we're determined to decrease gives us pain if it doesn't decrease.

So you are saying our determination is a more fundamental reason why we do things?

3 minutes ago, iNow said:

That was not, however, the claim you made. The one I challenged. And which you refuse to support. 

Doing things to pump our own dopamine is not equivalent to:

"and not just for themselves, but for everyone."

I assume you are taking umbrage with the "everyone" part.  You can see it everywhere: charities, socialism, "media police", cancel culture, etc.

1 hour ago, iNow said:

Citation needed. 

Also needed: Clarity on how you’re measuring pleasure and pain and how you define “try.”

Citation: me

Try: to make an attempt (from Webster's Dictionary)

But seriously, it seems that the most fundamental motivation for almost everything that we do is determined by pleasure and pain.