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

[Speakpigeon]

Then when we make a measurement and the wave function collapses, the result must be computable. Else we'd violate the CUH, the Bekenstein bound, and the Church-Turing-Deutsch thesis. This is not a point of math or philosophy. It's a point of known physics.

It may be a point of "known physics" but it's not something we would know of the physical world itself.
You can't on the one hand insists that the mathematical models used in physics aren't the world itself and then, on the other hand, claim that we know the CUH, the Bekenstein bound and the Church-Turing-Deutsch thesis to be true of the world.
So, again, we're back to my point that we just don't know either way.
You should refrain from claims suggesting we would know that infinity doesn't or can't exist in the physical world. Expressing you belief it doesn't exist should be good enough.

So, how does the wave function know to land on a computable number? Does anyone understand how this works?

You cannot claim that measures in QM produce computable numbers only.
You yourself claim measures can only be approximate, how would we know they produce only computable numbers? That's absurd.
EB

 

[arfa brane]

I've given some thought to the integers being real numbers. It's a basic thing to lay out the integers along the real line, just separate them by the same "real" unit distance.

But you can lay the integers along the "unreal" line, just separate them by any distance, as long as it's countable you're quids in. That is, topologically you can think of the integers as being like knots in a long string with arbitrary gaps between--then it's about distinguishing them, however you identify them.

Moreover, "any" distance can't be uncomputable.
p.s. therefore cannot be infinite.

 

[Write4U]

Can anyone explain the difference between "values" and "numbers", because IMO, that's where the apparent problem arises.
Numbers are manmade symbols for values , which are properties of extant countable quanta, no?

What difference do actual numbers make, as long as they accurately predict the behavior of universal constants?

 

[Speakpigeon]

Can anyone explain the difference between "values" and "numbers", because IMO, that's where the apparent problem arises.
Numbers are manmade symbols for values , which are properties of extant countable quanta, no?

What difference do actual numbers make, as long as they accurately predict the behavior of universal constants?

In ordinary usage, there's no significant difference.
You seem to misconstrue the difference between the mathematical and the ordinary senses as a difference between the words themselves.
Just look at the dictionary definitions:
Mathematics
value
6. a quantity or number expressed by an algebraic term.
7. a. magnitude; quantity: the value of an angle.
1. an assigned or calculated numerical quantity.
quantity
3. something that serves as the object of an operation.
number
b. a member of any of the following sets of mathematical objects: integers, rational numbers, real numbers, and complex numbers. These sets can be derived from the positive integers through various algebraic and analytic constructions.

Usual sense
value
2. an amount, esp a material or monetary one, considered to be a fair exchange in return for a thing; assigned valuation: "the value of the picture is £10 000".
a numerical quantity measured or assigned or computed; "the value assigned was 16 milliseconds"
quantity
1. c. an exact amount or number: "the quantity of material recycled in a month".
amount
5. quantity: "a great amount of intelligence".
numbers
7a. a large quantity; a multitude: "a large number of people visited the fair".

No significant difference between the usual senses of these words.
EB

 

[iceaura]

I guess I don't understand how anyone measures '2' of anything from a continuous distribution. We measure 1.9999 +/- an error tolerance. The question of whether there really "is" some actual value there is mysterious, even in classical physics.

And that is the motivation of this question:

The number 2 is the limit of the infinite sum 1 + 1/2 + 1/4 + - - -

Does the number 2 exist, as more than a mathematical abstraction?
Is it possible to have 2 things, or does the impossibility of adding one thing and a half thing and a quarter thing and so forth in an infinite series mean we can never actually have two things?

If this is equivalently mysterious in the physical world, we have an answer of a kind: infinities exist in the real world in the same mysterious sense in which they exist in the mathematical world, and vice versa. We would have an equivalence, and could choose its label.

No such thing as I understand it. Only approximations. Nobody has ever measured an exact frequency.

It would be produced, not measured.

 

[phyti]

Infinite is an idea related to measurement.
Infinity is not a number since it has no value. It is a relation about numbers stating there is no limit to a magnitude. A common example is the set of integers. The set has a beginning but is 'open ended/without limit/unbounded', NOT finite.
Another instance of the meaningless term is found in the definition of a limit, such as:
the limit of (some sequence of n terms), as n approaches infinity, = u. By definition the sequence can never equal the limit u, because if it did, the statement would be false!
Then there's the question of, how do you ‘approach infinity'? At night, on your tip toes, or maybe while it's sleeping?
It's a contradiction in terms. You can't ‘approach infinity' anymore than you can approach the horizon, or the carrot on the stick.
So much for rigorous or precise definitions.

Since mathematics is a language, it’s subject to the same limitations as any language. The terms are by definition, and those definitions are in terms of other definitions, etc. The ultimate reality is, there are no fundamental independent definitions. They are circular or accepted without proof, as postulates or axioms.
Couple this with the fact that human experience has no provable examples of anything without end. Despite the mind being one of the most complex organisms known, it has a tendency to naively conceptualize and oversimplify to obtain a first approximation or preliminary grasp of an idea.
How many times have we heard, “it’s more complicated than we originally thought”, when experience doesn't agree with prediction.

The challenge is for anyone to measure a stick with one end, i.e. it's 'infinitely' long.

 

[Speakpigeon]

Infinite is an idea related to measurement.
Infinity is not a number since it has no value. It is a relation about numbers stating there is no limit to a magnitude. A common example is the set of integers. The set has a beginning but is 'open ended/without limit/unbounded', NOT finite.
Another instance of the meaningless term is found in the definition of a limit, such as:
the limit of (some sequence of n terms), as n approaches infinity, = u. By definition the sequence can never equal the limit u, because if it did, the statement would be false!
Then there's the question of, how do you ‘approach infinity'? At night, on your tip toes, or maybe while it's sleeping?
It's a contradiction in terms. You can't ‘approach infinity' anymore than you can approach the horizon, or the carrot on the stick.
So much for rigorous or precise definitions.
Since mathematics is a language, it’s subject to the same limitations as any language. The terms are by definition, and those definitions are in terms of other definitions, etc. The ultimate reality is, there are no fundamental independent definitions. They are circular or accepted without proof, as postulates or axioms.
Couple this with the fact that human experience has no provable examples of anything without end. Despite the mind being one of the most complex organisms known, it has a tendency to naively conceptualize and oversimplify to obtain a first approximation or preliminary grasp of an idea.
How many times have we heard, “it’s more complicated than we originally thought”, when experience doesn't agree with prediction.

The challenge is for anyone to measure a stick with one end, i.e. it's 'infinitely' long.

All very thoughtful but somewhat beside the point. The OP's question is "Can "Infinity" ever be more than a mathematical abstraction?". You're not really replying.
Sure, it seems you're saying "no, it can't". Well, is that it?
Your explanation is a mixed bag so I'm not going to go into the detail of all the points you raise. But one point is that "mathematics is a language". Sure, I can agree with that, it's a kind of language. So? How is that relevant? Does that mean that you think infinity can't be more than a mathematical abstraction just because mathematics is a language?! Would that in turn imply that there are no two oranges in the universe because 2 is a mathematical abstraction and mathematics is a language? Or is it that there are indeed two oranges somewhere in the world but there is no number 2. Something I would agree with except perhaps to say that if we're part of the physical world and we can have the number 2 in mind then ipso facto the number 2 has to exist in the physical world. Still, irrespective of that derail, do we not have then the situation where you would have to accept that although you think infinity doesn't exist in the physical world, there's no good reason to believe that there isn't somewhere in the physical world some infinite number of things, like perhaps particles, frequencies, points in time or space, whatever?
That's a question.
EB

 

[Write4U]

No significant difference between the usual senses of these words.

No, not between the words, but in the expression of the words in reality.

A number is an arbitrary symbolic representation of a value. In that respect they are the same. A number can only be represented as a numerical glyph.
But a value is an abstract algebraic potential, which we have codified with our numbers. A value can be expressed as an expressed physical potential or attribute, other than just a number.

This can still be presented as an equation, but from different perspectives. I believe there is a clear difference in those perspectives.

 

[someguy1]

OK, I'm going to try to have a debate with you on a specific point. I hope you take the time to understand the issue.

Ok. I've been thinking about all this so let me state my current viewpoint up front.

1) All measurement is approximate so all we will ever observe in the laboratory or in nature is a rational number with an error interval. Nobody will ever measure anything exactly, integer or rational or noncomputable. So the question is really meaningless.

2) The Schrödinger equation is a theoretical model that takes all real number values in an interval. It's theoretical, not actually real. When the wave function collapses, we measure some rational with an error interval as in (1). So again the question of the existence of any kind of exact value of anything is meaningless in science. It's a metaphysical assumption to say that our approximate measurements correspond to anything real "out there." People have been arguing about the "meaning" of QM for a century now and there's no agreement or consensus. One day it's, "Shut up and calculate," and the next day it's multiverse theory. Nobody knows and probably nobody will ever know.

I'm not a specialist but I would assume that the product of a non-computable number by any non-zero rational number is itself a non-computable number. Sounds fairly obvious to me. Do you agree?

Yes. The computable numbers are a subfield of the reals. The sum and product of computable real is computable. So if a rational times a noncomputable were computable, we could divide both sides by the rational and get a contradiction. That is if q is rational, n is noncomputable, and c is computable, then qn = c gives n = c/n, noncomputable on the left side and computable on the right.

Let's assume now that all existing EM frequencies as we mesure them today are in fact non-computable numbers.

As noted this is not possible. We can only measure a rational with an error interval. Nobody will ever measure a noncomputable number in the lab not only because nobody will ever measure ANYTHING exactly in the lab, but also because no physical apparatus could measure infinitely many decimal places. So the assumption's meaningless.

Suppose further that all frequencies are in fact the product of the same unique non-computable number by some rational number. In other words, this one non-computable number is in fact a common factor to all frequencies.

Again, a meaningless or metaphysical assumption to which nobody can sensibly reply. However the idea that there might be exactly one, or perhaps exactly two, or perhaps countably many noncomputables is an interesting tangential point which I won't get into, except that Turing noted that oracles -- black boxes that solve noncomputable problems -- can be modeled as noncomputable numbers, and you can line them up one after another like the ordinals. Off-topic but interesting.

I would assume that such a situation would be compatible with the actual measures done of EM frequencies given that there's always a margin of error in measures.

I just don't know what that means. If we measure 1/2 +/- 1/10000, there are noncomputable reals in that interval but we have not "measured" them.

Suppose now we change our unit system so that the non-computable number which is assumed as common factor to all frequencies is now taken as the unit for EM frequencies. I seems to me that expressed in this new unit system and given my assumptions, all frequencies would be now rational numbers.

Yes, under your assumptions, which I believe don't actually refer to anything meaningful, I believe you are correct. I do see where you're going but I just don't know enough physics to comment further.

So, given those assumptions, would you agree that all frequencies would be rational numbers?

I guess so. But again it's a metaphysical assumption that what we measure in the lab represents anything at all "out there." Are there really photons jiggling around with exact numeric values to the jiggles? Personally I have no idea.

 

[someguy1]

I've given some thought to the integers being real numbers. It's a basic thing to lay out the integers along the real line, just separate them by the same "real" unit distance.

But you can lay the integers along the "unreal" line, just separate them by any distance, as long as it's countable you're quids in. That is, topologically you can think of the integers as being like knots in a long string with arbitrary gaps between--then it's about distinguishing them, however you identify them.

Moreover, "any" distance can't be uncomputable.
p.s. therefore cannot be infinite.

Quids in? Sorry I'm a Yank. I haven't learned to speak English.

The relation between the integer 6 and the real number 6 is a bit murky actually. As sets they're very different. We "identify" a particular subset of the reals as being an exact copy of the integers, so we casually say that "the integers are a subset of the reals." Strictly speaking this is not true. Now one dives down a philosophical rabbit hole with buzzwords like structuralism and category theory.

But you are right that integers are arbitrary. If you have a line you mark one point as zero, it's entirely arbitrary. You mark a second point as 1, and this also is entirely arbitrary. After that you can build out the integers, rationals, and reals. The "actual" distances are completely arbitrary, defined only in terms of the choice of the first two points.

 

[someguy1]

If this is equivalently mysterious in the physical world, we have an answer of a kind: infinities exist in the real world in the same mysterious sense in which they exist in the mathematical world, and vice versa. We would have an equivalence, and could choose its label.

Confusing math with physics. 1/2 + 1/4 + 1/8 + ... = 1 is true in the mathematical real numbers. What that equation refers to in the real world I have no idea. The Planck scale would render the experiment meaningless within a couple of hundred terms or less.

It would be produced, not measured.

If a frequency is "produced," who produces it? God? Our meta-programmers in the sky? Max Tegmark? You're confusing metaphysics with physics IMO, an ongoing theme in this thread. We have to distinguish what we can measure and observe, which is physics; from what we think it "means" about the world "out there," which is metaphysics.

 

[someguy1]

I thought myself there was no illusion on this point:

Isn't that clear enough?

To me and to you yes. To those who believe that the values taken by the Schrödinger equation refer to something in the real world "out there," it's unclear. I'm trying to straighten them out. You'd think people would understand that nobody really knows what's "out there" or if the question is even meaningful. But based on this thread, many people are unclear on this point.

 

[someguy1]

As you see it, obviously.

Obviously!

No, but they can be good enough models to cover e.g. your objection re continuum of frequencies. Once you get the distinction between 'infinitely many existing all at once', and 'an infinity of possible instantiations'. I have consistently only claimed the latter. While a single photon as wavepacket must be expressed as a Fourier integral, the continuum of frequencies implied does not mean an infinite amount of accessible information is encapsulated. It means that discrete entity is somewhat fuzzy. It means it will trigger a hit by any real-world detector necessarily itself having a finite detection bandwidth. It becomes meaningless to then talk of infinite vs finite in that context. George Ellis again - you can never prove physical infinities exist.

I appreciate that you are engaging directly with what I'm saying, and now it seems that we haven't got much of a disagreement. You agree that the models are indeed models and not "ultimate true reality," a philosophically murky concept. As long as we're talking about models I have no objections at all. If you want to model time as a continuous real variable t, I'm fine with it. I only have a problem if someone says that t actually takes on some noncomputable value. That's against the laws of physics because that measurement would encapsulate an infinite quantity of information. We could use that measurement as an oracle to solve noncomputable problems. I find myself in complete agreement with this paragraph of yours and have no objections at all. The models are continuous, the reality is fuzzy. This I believe.

I stand by it. And it seems to me you continue to confuse a need for infinitely many over some span, with complete arbitrariness of any instantiation(s) over that span.

The model has a need for infinity, not reality itself. Are we disagreeing again? If there's even one noncomputable physically instantiated, that violates all known physics and computer science. I don't even need two, let alone uncountably many, to make that point.

Barking up the wrong tree. You keep misunderstanding my position.

I understand you when you speak clearly as you did in the above paragraph. But now you are still arguing a point I'm not understanding, since you seem to have agreed that the models are continuous but the actual instantiations of measurements aren't.

You can't avoid philosophy and maths in making that claim. Try substituting the word any for all in above. See a difference?

Actually no, since any and all mean exactly the same thing most of the time. "Any even number is divisible by 2" and "All even numbers are divisible by 2" mean the same thing. So no, I'm afraid I do not see a difference. Perhaps I'm missing something in some particular context but in general "any" and "all" are synonymous. Any bloke on the street would agree. All blokes on the street would agree. Same meaning.

Which bound assumes BH's exist. See my thread here: http://www.sciforums.com/threads/new-vector-theory-of-gravity-challenges-gr.160900/

I'll take a look but I'm pretty much physics'd out for the time being. I watched a couple of serious MIT QM lectures and followed quite a bit of it based on some knowledge of functional analysis even thought I know no physics. I learned that there's a continuous time parameter in the Schrödinger equation, leading to much confusion about the "reality" of the space of possible measurements. I'm satisfied with your initial paragraph in which you seem to agree that the reality of the continuous measurements is theoretical and not physically real. Unless I'm still misunderstanding you.

Regardless of one's stand on that matter, I have no issue with there being only finite accessible information within a finite region.

Well then now I really am happy! The Schrödinger equation gives a continuous range of real number possibilities, but noncomputable outcomes are never observed. I can certainly live with that. The smartest people in the world disagree on what QM means, I'll leave it to them.

 

[arfa brane]

Quids in? Sorry I'm a Yank. I haven't learned to speak English.

Quids in: in a position that should lead to a profit (rough meaning). A quid is an old-fashioned name for a British pound.

And about laying out the integers, suppose you have n apples, then it doesn't really matter how they get "laid out" or arranged, there are still n apples you can count. But if you want n - 1 distances, all the same between n apples, then you're embedding them "in" the reals, probably approximately. (in fact the probability is pretty close to 1 that the distances will all be approximate)

 

[iceaura]

Confusing math with physics. 1/2 + 1/4 + 1/8 + ... = 1 is true in the mathematical real numbers. What that equation refers to in the real world I have no idea. The Planck scale would render the experiment meaningless within a couple of hundred terms or less.

You appear to be confusing measurement with meaning. . The existence of the Planck scale rests on exactly the same metaphysical basis as my post - it assigns numbers to distances.

If a frequency is "produced," who produces it? God?

Machinery, physical entities set in motion by anything.

You're confusing metaphysics with physics IMO, an ongoing theme in this thread.

I'm connecting metaphysics with physics, which is kind of a normal thing to do.

The Schrödinger equation gives a continuous range of real number possibilities, but noncomputable outcomes are never observed.

There is no way to observe a non-computable outcome.

 

[someguy1]

Quids in: in a position that should lead to a profit (rough meaning). A quid is an old-fashioned name for a British pound.

Thanks, I never actually heard "quids up" before. When the Queen had tea with Donald Trump today I wonder if they needed an interpreter. Wish I'd been a fly on the wall.

And about laying out the integers, suppose you have n apples, then it doesn't really matter how they get "laid out" or arranged, there are still n apples you can count. But if you want n - 1 distances, all the same between n apples, then you're embedding them "in" the reals, probably approximately. (in fact the probability is pretty close to 1 that the distances will all be approximate)

Right, integers as discrete quantities and integers as locations on a continuous line. Philosophically these are very different things. We can exactly count 5 bowling balls but we can never measure exactly 5 meters. I'm not sure how sensible it even is to claim that an integer is a real number. It's only true if you don't think about it too much.

 

[Seattle]

Is anyone even worrying about measuring something exactly?

 

[someguy1]

You appear to be confusing measurement with meaning.

I'm arguing strenuously against confusing measurement with meaning, but I see I'm not making myself clear. No matter, I can't think of anything new to say.

The existence of the Planck scale rests on exactly the same metaphysical basis as my post - it assigns numbers to distances.

Not understanding this sentence. The Planck scale sets the limits of our ability, even in principle, to measure time and space. I believe that is physics and not metaphysics. I'm not following your meaning here.

Machinery, physical entities set in motion by anything.

When you wrote that, didn't you think I might ask you what sets the machinery in motion? And didn't you realize you'd get stuck an infinite regress and either have to say, "Damn, I guess God did it after all," or, "Damn, I have no freaking idea why anything happens or what causality is."

Please consider yourself so asked.

I'm connecting metaphysics with physics, which is kind of a normal thing to do.

Well earlier Q-reeus characterized me as "intensely philosophical and mathematical." Which I would frankly take a great compliment; but that I gather was intended to be mildly pejorative. If you are connecting metaphysics with physics, by definition you are doing philosophy and not science. Which is ok by me, but to the extent anyone is doing metaphysics, they are not doing science.

What's normal, I think, is for some scientists to be unaware they are doing metaphysics when they say, "The world is the way I have modeled it." Rather than, "I've just developed a kickass model that gets 6 decimal places of accuracy when I slam a proton into a wall of taxpayer money."

I read that in Feynman's QED, I think I mentioned this earlier, they did some experiment, this might have been about the crazy renormalization I've heard about -- "getting the infinities out of the equations" -- exactly what we've been talking about. So after Feynman and Schwinger and Tomonaga got the Nobel prize and Freeman Dyson didn't, they did some experiment and got 12 decimal places of accuracy between their theory and the experiment. 12 decimal places. Now that's great if you're an engineer or a physicist. But to the extent you think the real numbers are actually real, you need to realize that a real number has infinitely many decimal places, and that your theory in no way describes what might be happening out there. Some genius not yet born is going to shock everyone with a new paradigm and it's off to the races again. Science is never final.

I want to communicate this: The mathematical real numbers are exceedingly strange and mathematicians don't even have a complete handle on them. For example we don't know how many real numbers there are. That's the Continuum hypothesis, which has driving the development of set theory from its very beginnings to the state of the art research of today.

So when you as physicists build models of the world that have time as a continuous real-valued variable, you are making a choice of modeling tool, and you need to be very careful of the ontological assumptions you're unconsciously making. Because if you studied the real numbers the way mathematicians do, you might not believe in them at all as a model of time.

And there are alternatives. Brouwer's intuitionistic real line and its modern incarnation as the constructive real line. The hyperreals with their cloud of infnitesimals, dubbed a "monad" after Leibniz,floating around each real.

So the standard real numbers aren't the last word in real numbers. There's no reason at all to think the standard real numbers we use in math and physics have anything to do with the true nature of the continuum. There are good reasons to think it doesn't. So just stop putting such religious faith in the real numbers. They are very strange and if you really believed they represented anything physically real, you'd have to be prepared to answer a lot of hard question about which set theoretic principles you regard as having ontological truth.

There is no way to observe a non-computable outcome.

I think we are all in agreement about this.

I am fascinated by the noncomputable numbers. They're not very well known. They include all the rationals and a lot of the familiar irrationals. In fact any real number you have ever heard of and ever WILL hear of are computable. That's because the noncomputable numbers can never have names. How weird is that!

[I lied slightly for brevity. Some noncomputable numbers can nevertheless be defined, which is a subtly different concept. Chaitin's Omega is one such. It's the probability that a random Turing machine will halt. If it were computable it would solve the halting problem, which can't be solved. So Omega can't be computable. But we can easily describe it and talk about it. It's as naturally occurring a noncomputable number as you'll find. https://en.wikipedia.org/wiki/Chaitin's_constant]

Now if you take the real line, the continuous real line of our imagination, it has no holes in it. If you take out all of the noncomputable numbers, all that's left is a paltry countable set of computable numbers, a set of measure zero. If you threw a dart at the real line you'd hit a noncomputable real with probability 1.

I think this should be taken into account by proponents of the Computable universe hypothesis. In a computable universe, your real line must be extremely sparse, a virtual desert of computable locations in a sea of nonexistence. And what does that say about your physics? Well nothing really since physics can never measure a noncomputable real anyway. So you can always build a MODEL with only computable numbers and functions. But you can't conclude anything about the true nature of the world.

 

[iceaura]

Not understanding this sentence. The Planck scale sets the limits of our ability, even in principle, to measure time and space.

And that limit is a distance, to which is assigned a number. Nothing smaller than that distance, or "Planck length", can be measured.
That does not mean that the concept of smaller distances is meaningless, however, nor does it mean that smaller distances do not exist or are not "real" in some sense.
It does mean that a specific distance has been assigned a specific real number. That provides us with a number for all other distances, no matter how small.

There's no reason at all to think the standard real numbers we use in math and physics have anything to do with the true nature of the continuum.

There is good reason to think they have much to show us about the observable nature of the continuum - that as entities in a mathematical system we use to perceive or apprehend the physical world, they provide valuable information.

If you take out all of the noncomputable numbers, all that's left is a paltry countable set of computable numbers, a set of measure zero.

That doesn't mean there are holes in the line.

If you threw a dart at the real line you'd hit a noncomputable real with probability 1.

You would not. Nothing that you do physically or in theoretical analogy with a physical phenomenon can identify a specific noncomputable number.

 

[arfa brane]

Is anyone even worrying about measuring something exactly?

Newton's gravitational constant, Avogadro's number, the speed of light, and a few other constants are not known exactly.

But they are known to be constant, how come? Further, why is it ok that, for instance, the error in Avogadro's number is large, a large integer more ova, about which we know only how many powers of 10. But suppose we assume that the average value of each digit, in this unknown number is 5, then we have 5 to a large power of 10 'things', approximately.

You can get a feel for what exactness means in quantum measurements, at IBM's qx site, where you set up some gates and run a quantum program. The program has to run a reasonable number of times so the result is statistically significant. More than 100 times, say.