Can "Infinity" ever be more than a mathematical abstraction?

[someguy1]

And, pray tell, what qualifies you to dismiss his scientific hypothesis out of hand?

It's not a scientific hypothesis. The quote you gave me was not a scientific hypothesis. it's just part of Tegmark's MUH idea.

I don't know why you feel that assuming Tegmark is right, it would stop all further discussion.
Why should that necessarily be the case?

It's not the case. Nor did I say any such thing. I said that invoking Tegmark in discussions of this type sucks all the air out of the discussion, as it did a couple of weeks ago in this thread. It's been a much better thread now that people have left Tegmark out of it. You should go back and reread the first dozen pages of the thread if you want Tegmark chat. You'll get no more from me.

 

[Write4U]

It's not a scientific hypothesis. The quote you gave me was not a scientific hypothesis. it's just part of Tegmark's MUH idea.

It's not the case. Nor did I say any such thing. I said that invoking Tegmark in discussions of this type sucks all the air out of the discussion, as it did a couple of weeks ago in this thread. It's been a much better thread now that people have left Tegmark out of it. You should go back and reread the first dozen pages of the thread if you want Tegmark chat. You'll get no more from me.

By all means let's stay comfortable in our discussions of grand concepts.

And you still haven't explained why it should make a difference at all?

 

[iceaura]

Yet the mathematical real numbers may be partitioned into a union of singletons, and a singleton set containing a single real number isn't anything like a continuum. It's a point.

As far as I know, you cannot define a partition the real number line into single points. That is one of the implications of the existence of noncomputable numbers, for example.

If you're denying that the measure of the computable reals is zero and the measure of the noncomputable reals (in the unit interval, to normalize the probability measure) is 1, then again you are denying known math.

And if, instead, I am simply pointing out that you cannot identify or specify or "target", as with a dart, a noncomputable number, then I'm ok and in complete agreement with known math.
You can't throw a dart and hit a noncomputable number. Not in theory, not in practice.

There are Cauchy sequences that don't converge.

You cannot construct a Cauchy sequence (i.e. model an existing physical reality) that does not converge to a computable number.

If you take the standard real line and delete all the noncomputable numbers, the Intermediate value theorem becomes false. The computable real line is incomplete. There are Cauchy sequences that don't converge. That's the definition of what it means to have holes in the real line.

Then you need a new definition - because that one will lead you to errors of intuition, such as being able to throw a dart and have it hit one of those "holes".

Look, I don't mean to interfere with the rest of the discussion here. I'm in general agreement with your posting. It's just that if we are talking about an infinity being "more" than a mathematical abstraction, we are in the realm of philosophy or metaphysics or meaning.

 

[Write4U]

Then you need a new definition - because that one will lead you to errors of intuition, such as being able to throw a dart and have it hit one of those holes

Schrodinger's cat?

 

[someguy1]

As far as I know, you cannot define a partition the real number line into single points.

$$\displaystyle \mathbb R = \bigcup_{x \in \mathbb R} \{x\}$$.

You can do that for any set whatever. Every set is the disjoint union of its singleton subsets.

That is one of the implications of the existence of noncomputable numbers, for example.

You're wrong about that but I think I can guess what you think you mean. We do not need to identify or name or compute each element of a set. For example consider the set of noncomputable real numbers. That's a perfectly well defined set with no elements that can be named or uniquely characterized.

And if, instead, I am simply pointing out that you cannot identify or specify or "target", as with a dart, a noncomputable number, then I'm ok and in complete agreement with known math.

We say that the noncomputable real numbers in the unit interval have Lebesgue measure 1. Interpreted as a probability (which is one of the standard applications of Lebesgue measure), it means that a randomly chosen real (in the unit interval) is noncomputable with probability 1.

Of course the metaphor of "throwing darts at the line" is just a casual visualization. One doesn't actually throw darts, nor is the mathematical real line physically realizable. This is strictly a mathematical fact. But it's a fact.

You can't throw a dart and hit a noncomputable number. Not in theory, not in practice.

Well not in practice, of course. But in theory, why not? Imagine the real line like the one from high school analytic geometry class. One point is like another. When you look at the mathematical real line, almost all the points have locations or decimal expressions that are not the output of any computer program. There are uncountably many real numbers and only countably many Turing machines.

Another example from high school. When you study the function $$y = x^2$$ you have a perfectly good idea in your mind of how that function works. But if it's defined for all real numbers $$x$$, then almost all of its inputs and outputs are noncomputable. What of it? Apparently my arguments against the physical reality of noncomputable numbers have influenced you to think that they are not mathematically real. If so perhaps I was too effective. Noncomputable real numbers are full-fledged real numbers and there are lots of them.

You cannot construct a Cauchy sequence (i.e. model an existing physical reality) that does not converge to a computable number.

Of course you can, for example finite approximations to Chaitin's Omega would work. But in fact every real number is the limit of a Cauchy sequence of rationals. This is essentially the way the real numbers are constructed. This is math. Talk of "physical reality" is confusing math with physics. I wonder if you're talking about physics but saying you're talking about math. That would explain some of the things you're saying.

Then you need a new definition - because that one will lead you to errors of intuition, such as being able to throw a dart and have it hit one of those "holes".

I'm just reporting how the math works. Why would I need a new definition? I'm using the standard mathematical definitions agreed on by every mathematician in the world going back to Dedekind's construction of the real numbers in the 1870's.

Look, I don't mean to interfere with the rest of the discussion here. I'm in general agreement with your posting. It's just that if we are talking about an infinity being "more" than a mathematical abstraction, we are in the realm of philosophy or metaphysics or meaning.

I have never discussed infinity as being anything more than a mathematical abstraction. I'm arguing against actual infinities in the real world.

But for the record, and going beyond this discussion, I see no reason why some future Newton or Einstein won't find a use for the transfinite numbers, just as we found a use for the "nonsensical" idea of non-Euclidean geometry. Ideas in math that at first seem purely abstract and totally useless often become indispensable in the physical sciences. History is on the side of weird math.

ps -- The phrase "throwing darts at the real line" is from a set-theoretic argument put forth by a guy named Freiling. http://jdh.hamkins.org/freilings-axiom-of-symmetry-graduate-student-colloquium-april-2016/ It's a heuristic argument against the continuum hypothesis. Throwing darts at the real line is just a metaphor for choosing a random real number.

 

[someguy1]

By all means let's stay comfortable in our discussions of grand concepts.

And you still haven't explained why it should make a difference at all?

Sorry I didn't follow the referent of "it" in your second sentence. I just don't want to get bogged down in Tegmark again. A couple of weeks ago this thread motivated me to read half of his MUH paper, along with a couple of informed but critical reviews. Nothing I read changed my mind or gave me any new information that would alter my opinion that Tegmark's ideas are provocative and interesting but definitely not science, if by science you mean that which can be observed. Physics, not metaphysics. But really, I just don't want to argue about Tegmark any more. I already said my piece about Tegmark and if someone thinks I'm wrong, that's ok.

 

[Write4U]

Sorry I didn't follow the referent of "it" in your second sentence.

By it I meant, if it makes any difference when assuming that the universe is a collection of values and functions which follow some form of mathematical order which makes it possible for humans to symbolize these values and functions and codify them in a comprehensive representation (model) of the universe. Forget Tegmark.
Question is: Would it change anything in science?

 

[someguy1]

By it I meant, if it makes any difference when assuming that the universe is a collection of values and functions which follow some form of mathematical order which makes it possible for humans to symbolize these values and functions and codify them in a comprehensive representation (model) of the universe. Forget Tegmark.
Question is: Would it change anything in science?

Values and functions. How do we know the universe has values and functions? If I'm doing the quadratic equation I know what is my function and what are my values. Values and functions are concepts that belong to formal systems.

When you say the world has values and functions, that's where you lose me. That's philosophy.

Perhaps it's not that the universe has order, but that humans impose order on an essentially random universe. We see a lot of local nonrandomness, but maybe it only seems that way. I don't know. I just don't think anyone else knows.

What is the nature of the ultimate order of the universe? It's a good question. I don't think it's a question of science. Some would consider it a question of religion. "Let there be light." Or "We all live in a computer." That's a trendy idea floating around. It's essentially a theological claim, not a scientific one.

Would it change anything in science? I don't know. If the universe is values and functions, where are they? They're in the mathematical model, but where are they in the world? When we measure a number, where is that number? In the thing we're measuring? Or in our apparatus? Or in our minds?

Why are so many people convinced the numbers themselves are in the things we measure?

 

[Write4U]

We see a lot of local nonrandomness, but maybe it only seems that way. I don't know. I just don't think anyone else knows.

Actually we see a lot of universal nonrandomness, everywhere we look.
The Fibonacci sequence is just one recurring pattern, which is observably present from local micro to universal macro states.

Why are so many people convinced the numbers themselves are in the things we measure?

We see patterns and they are so convincing and numerous that Theism actually has assigned intelligent design to the observed order and verifiable orderly processes.

Science is not quite as speculative, but from continued observation and testing is forced to recognize an underlying hierarchy of orders, which become expressed through specific behaviors based on specific (sometimes variable) values and functions, which we can represent with a symbolic language, maths.

E = Mc^2 is a mathematical equation expressing equal values. (Necessity and Sufficiency)

p.s. note: I am not saying specific numbers. That's another issue.....

 

[iceaura]

You can do that for any set whatever. Every set is the disjoint union of its singleton subsets.

My bad. For "define" read "describe" - usefully, here.

That's a perfectly well defined set with no elements that can be named or uniquely characterized.

It is not "well-defined" for our purposes here. We are doing things like choosing and talking about holes.

Of course the metaphor of "throwing darts at the line" is just a casual visualization.

So is the metaphor of "holes". Such casual visualizations should be avoided if they mislead.

Throwing darts at the real line is just a metaphor for choosing a random real number.

You can't choose a non-computable number in any way analogous to an accomplished physical process, or useful as an abstraction of a physical process.

Of course you can, for example finite approximations to Chaitin's Omega would work.

Let's see one.

"You can't throw a dart and hit a noncomputable number. Not in theory, not in practice."
- - -But in theory, why not?

Because you have no means. All your means yield a computation.

I'm just reporting how the math works. Why would I need a new definition?

Because you are talking about holes in the mathematical models of physical reality, and claiming they prevent infinities from existing in the real world.

I have never discussed infinity as being anything more than a mathematical abstraction. I'm arguing against actual infinities in the real world.

Ok. And there are some issues with that argument, such as its requiring actual continuity to not exist, and non-computable numbers be assigned to positions on
real world physical ranges or scales, and so forth.

But if it's defined for all real numbers xxx, then almost all of its inputs and outputs are noncomputable. What of it?

Not a single one of them is available to you as an input or output.

But in fact every real number is the limit of a Cauchy sequence of rationals. This is essentially the way the real numbers are constructed.

It is the way they are defined. The non-computable ones are not constructed at all.

This is math. Talk of "physical reality" is confusing math with physics.

Whether or not that is a confusion is the subject of the thread.

 

[someguy1]

My bad. For "define" read "describe" - usefully, here.
.

I would like to go through your post point-by-point but as I read through it you are arguing against well-known standard math, such as the fact that every real is approximated by a sequence of rationals. I can't make out where you're coming from by denying basic math. I'd sort of rather not engage till I can get a better understanding on why you're denying all the basic well-known facts about the real numbers.

I do wonder if perhaps this is the first time you've thought about how strange the mathematical real numbers are. Which is just my point. Physics uses the real numbers to model various continuous phenomena. But the mathematical real numbers have many features that make them quite murky when you try to claim they are "true" about the world.

But I hope that we can at least come to agree on the basic facts about the real numbers, such as that there are uncountably many of them (so that most of them must be noncomputable) and that every real number, computable or not, is the limit of a sequence of rational numbers. Many such sequences in fact. Also the business with the holes. If you delete a real from the real line, there will be a Cauchy sequence that "should" converge to that point but doesn't, because the point's not there. That's a hole. You can drive the graph of a continuous function through that hold and falsify the Intermediate value theorem. There goes your notion of continuity.

If you remove the noncomputable reals from the real line you have a line that is almost all holes.

These things are not opinions I can argue with you about. They're basic facts about the real numbers. I hope that we can come to agreement on these facts, which I am doing my best to explain to you. After that we can talk about how they may or may not relate to the real world. But you can't argue with how the real numbers are, that won't be productive. These facts are well-known, they're in books and they're on Wikipedia. Better if you try to understand the real numbers.

The real numbers are complete. https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers

If you take out even one point, they fail to be complete and are no longer a model of continuity.

 

[Speakpigeon]

The value of a shirt lies in the fact it keeps you warm and protects your skin from sunburn. These are properties that lie completely outside the amount you have to pay for it.

This is idiotic. You're suddenly talking about a different sense of the word "value". Apart from you, we all know the word has different senses. You asked about the difference between the words "value" and "number". Given we were talking here about the mathematical concept of infinity I assumed you were asking about value and number in the context of mathematics. I provided dictionary definitions to check whether that's what you meant. You just abstained from replying properly as polite people and honest debaters do and just went into more abstruse considerations. And then all of a sudden, you go on a tangent! You're a laugh. It's impossible to have a serious conversation with you. You never answer questions and your replies either don't make sense or are just irrelevant.
And I fail to see how the sense of the word "value" you've apparently just discovered here could possibly relate to the topic of this thread. I could ask you to explain yourself but that would be a waste of time because you don't ever explain yourself.
More worryingly, this suggests you believe that the kind of values we find in the context of mathematics and physics are very much like the value of a shirt in keeping you warm and protected from sunburns. In effect, you've moved from supporting the idiotic and nonsensical thesis that the physical world is mathematically abstract to the complete opposite but as idiotic and nonsensical thesis that mathematical abstractions are as concrete and subjective as the idea that a shirt keeps you warm.

The term "value" has a much greater range of application than the term "number".

You have not a clue what you're talking about.
Still, you sure are a crafty number:

Number
15. Slang A person or thing singled out for a particular characteristic: a crafty number.

You are making the same mistake the SCOTUS made with "Citizens United".
Money is NOT free speech, it is a means of exchange (quid pro quo).
The value of "free speech" is not numerical, it is a "right" by law, which cannot be bought.

This is as idiotic an argument as anyone has ever seen. This thread is about the mathematical concept of infinity, not about free speech.

There's not one thing to retrieve from what you've said on the difference between value and number. Just pathetic. I tried to take your question seriously and all you do is just go from one irrelevance to another tangente. You're just a waste of time.
EB

 

[Write4U]

This is idiotic

No, it is short sighted on your part. This is precisely what I have been trying to explain to you, but you refuse to see.
You are the one who is questioning my posts and slapping me with derogatory titles. Are you now denying me the chance to clarify as well as trying to discredit my mental abilities. How very Trumpian of you.

And I fail to see how the sense of the word "value" you've apparently just discovered here could possibly relate to the topic of this thread.

I have been using both terms "values" and "functions" as undefined but constant (sometimes variable) universal potentials for the past ten years, precisely because of the confusion between the subjective and objective meaning of the term "number".

And as far as context is concerned, IMO, the difference between numbers and values is very much pertinent to the discussion.
Numbers are not transcendent, they are human symbols. But values are, they exist in the abstract without human permission.

 

[iceaura]

I would like to go through your post point-by-point but as I read through it you are arguing against well-known standard math, such as the fact that every real is approximated by a sequence of rationals. I can't make out where you're coming from by denying basic math

Where have I denied any math?

But I hope that we can at least come to agree on the basic facts about the real numbers, such as that there are uncountably many of them (so that most of them must be noncomputable) and that every real number, computable or not, is the limit of a sequence of rational numbers. Many such sequences in fact.

We've never left that basic agreement.

Also the business with the holes. If you delete a real from the real line, there will be a Cauchy sequence that "should" converge to that point but doesn't, because the point's not there. That's a hole.

There is no way to delete a specific non-computable number from the real line, or present the Cauchy sequences that no longer converge, while maintaining analogy or abstraction of any physical process. Your "hole" is not analogous to a physical gap or hole.

I do wonder if perhaps this is the first time you've thought about how strange the mathematical real numbers are.

I've been pointing out that your papering over their strangeness with metaphors like "holes" and "throw a dart" seems misleading in this thread.

If you remove the noncomputable reals from the real line you have a line that is almost all holes.

Or has no holes at all, depending on your use of that metaphor in whatever argument is at hand.

These things are not opinions I can argue with you about. They're basic facts about the real numbers.

The thread is about the existence of infinities and continuities in the real world, not the real numbers.

 

[someguy1]

The thread is about the existence of infinities and continuities in the real world, not the real numbers.

I was never confused about that. Q-reeus claimed arbitrary real numbers were physically instantiated and I did have to take some time to clarify that subject. To which end I brought up noncomputable real numbers, which are now troubling you in ways I can't put my finger on.

You still refuse to distinguish between the real numbers and their use in physics, and your most recent post is so full of mathematical misunderstandings -- which you don't realize are such -- that I hesitate to respond point-by-point to your misstatements.

I'll sleep on all this. I only wanted to let you know how your most recent two posts are striking me. Not as arguments to be refuted, but as misunderstandings and misstatements of standard facts so extreme as to render me unsure of how to proceed at all. I don't mean to accuse you of anything, just to explain how your posts are striking me.

Your posts strike me as anti-mathematical or a-mathematical because you understand my point at some level -- that the real numbers are problematic as the model for physical continuity -- but that you're allowing your thinking to become confused while you're processing it. That's how I'm seeing all this. Maybe that will help you to help me understand you better.

 

[Write4U]

I'll sleep on all this. I only wanted to let you know how your most recent two posts are striking me. Not as arguments to be refuted, but as misunderstandings and misstatements of standard facts so extreme as to render me unsure of how to proceed at all. I don't mean to accuse you of anything, just to explain how your posts are striking me.

I look forward to your considered critique. I hope you will find that what may seem as unusual perspectives are not quite as off the mark as may appear at first glance.

Like, "I've been pointing out that your papering over their strangeness with metaphors like "holes" and "throw a dart" seems misleading in this thread." That's exactly how mathematicians talk about it. Holes and throwing darts. I gave you a specific reference to one of the world's leading set theorists discussing another set theorists use of the phrase "throwing darts at the real line," and that had no effect on your thinking.

What you quoted was offered by Iceaura. My only contribution to "holes" and "throwing darts" is a single post asking : Schrodinger's cat?
I need not explain that thought experiment and I hope you can see the more fundamental contextual reason why I offered it for consideration.

 

[Speakpigeon]

We do not need to identify or name or compute each element of a set. For example consider the set of noncomputable real numbers. That's a perfectly well defined set with no elements that can be named or uniquely characterized.

As I understand it, the limitation of our mathematics is that there is no exact representation of a non-computable number in terms of computable numbers. Yet, it's a fact we can represent at least some, and in principle at least a countable infinity of non-computable numbers in terms of each other. So, what's this obsession with non-computability?

Computable and non-computable numbers form a continuum. This continuum should be understood entirely as a mathematical abstraction. It just happens that we sort of have a privileged epistemology with the countable set. What of it? And more importantly, this doesn't say anything as to actual physical quantities. Even the fact that we have a privileged epistemology with the countable numbers doesn't say anything as to actual continuums in the physical world if they exist.

In particular, in a physical continuum, the distinction between computable and non-computable elements just doesn't make sense. Elements in a continuum are all of exactly the same nature because our representation of the physical world isn't the physical world.

So what is missing here, is an explanation of why the distinction between computable and non-computable, which is entirely a mathematical abstraction, should be seen as a limitation on the actual physical world.
EB

 

[someguy1]

I look forward to your considered critique. I hope you will find that what may seem as unusual perspectives are not quite as off the mark as may appear at first glance.
What you quoted was offered by Iceaura. My only contribution to "holes" and "throwing darts" is a single post asking : Schrodinger's cat?
I need not explain that thought experiment and I hope you can see the contextual reason why I offered it.

I thought I was replying to iceaura but if I'm misquoting that's all the more reason for me to go to bed.

By the way you asked me about values and functions. That topic seems a little vague to me, I'm not jumping in. I don't know that there are values and functions in the world. I only know for sure they're in our mathematical models.

 

[Speakpigeon]

Numbers are not transcendent, they are human symbols. But values are, they exist in the abstract without human permission.

I already explained twice to you why you're wrong here. The number of children in the courtyard is a physical reality whether or not we humans have a symbol to represent it.
You just don't understand any of what is explained to you.
A waste of time.
EB

 

[someguy1]

As I understand it, the limitation of our mathematics is that there is no exact representation of a non-computable number in terms of computable numbers. Yet, it's a fact we can represent at least some, and in principle at least a countable infinity of non-computable numbers in terms of each other. So, what's this obsession with non-computability?

I left the topic a while back. I only used it to make a point to Q-reeus that not all real numbers could be physically instantiated in a manner consistent with known physics. Having made that point, plus told a few more people about the noncomputable numbers, which I find interesting for other reasons, I was done. iceaura is troubled that they don't seem to be proper members of sets or something, and he's wrong about that, but I'm trying to work through that with him. At least that's what I think is going on. It's a strange conversation IMO.

Computable and non-computable numbers form a continuum. This continuum should be understood entirely as a mathematical abstraction.

Yes of course.

It just happens that we sort of have a privileged epistemology with the countable set. What of it?

The only time I mentioned countability was to prove that most real numbers are noncomputable. The proof is that there are only countably many Turing machines. That's the only time I mentioned it.

And more importantly, this doesn't say anything as to actual physical quantities.

Right, but I'm not even talking about that. There seems to be considerable confusion about the mathematical real numbers themselves.

Even the fact that we have a privileged epistemology with the countable numbers doesn't say anything as to actual continuums in the physical world if they exist.

Agreed. Nor are the standard mathematical real numbers, the ones everyone learns in high school, necessarily how the physical continuum, if there even is one, actually is. There is a large philosophical and scientific literature on what the right mathematical model of the physical continuum should be. People should not be so dogmatic, go look up Weyl, and Brouwer, and Bishop, and a lot of other people.

In particular, in a physical continuum, the distinction between computable and non-computable elements just doesn't make sense.

In the physical realm, the distinction is vital. A noncomputable number carries an infinite amount of information. It's ruled out by contemporary physics. So now there is a problem mapping the mathematical real numbers onto the real world. It's not a major problem, it's just something to understand.

Elements in a continuum are all of exactly the same nature because our representation of the physical world isn't the physical world.

I don't know what that means. Aren't some numbers integers, others rational, others irrational, some computable and others not? Isn't some number 3 and some other number 47? Aren't some whole numbers odd and some even? Some reals are computable and some aren't. What of it? Why on earth is this troubling people?

So what is missing here, is an explanation of why the distinction between computable and non-computable, which is entirely a mathematical abstraction, should be seen as a limitation on the actual physical world.

Because it takes a finite amount of information to specify a computable number; and an infinite amount of information to specify a noncomputable number. That gives them very different natures if you try to project them onto the real world.

And IF you accept my premise that noncomputable reals can't be physically instantiated; THEN you have to conclude that physical continuity is very different than the continuity of the real numbers.