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

[Write4U]

someguy1 said:
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.

I'm not sure about the implications, but Max Tegmark, proposes that all of reality can be explained by the use of just 33 numbers and a handful of fundamental equations . This is one of the most attractive part. It is mathematically elegant that so much can be explained by so little.

In fact David Tong showed that the current knowledge of the universe can be expressed in a single equation, which is pretty amazing.

 

[Speakpigeon]

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

Then we would have no words to express this difference.
You certainly fail to express it.

A number is an arbitrary symbolic representation of a value.

No.
The number of children in the courtyard isn't an "arbitrary symbolic representation of a value"

In that respect they are the same.

No. We don't use "value" and "number" in the same contexts. We talk of the value of the Pound, not of the number of the Pound. The number of trees in the courtyard doesn't mean the same as the value of trees in the courtyard.
I would say that if you want to clarify your idea, you'd need to find and look carefully at an appropriate example.

A number can only be represented as a numerical glyph.

No. Any pair of things can be used to represent the number 2.

But a value is an abstract algebraic potential, which we have codified with our numbers.

The number of children in the courtyard is also an abstract concept.
The value on an angle is the number of degrees, say, 153 for example. Nothing potential in that value. This number is a value. It's the value of the angle in terms of degrees. It's a value inasmuch as it represents the relation between the particular angle and one degree, which is itself an angle. But exactly the same number can be considered in relation to something else. For example, 153 pounds can be the value of a shirt. So, 153 is the value of this shirt in Pounds. It's a number and a value. Only because that's how we use it to express the relation between this shirt and the Pound.
There's a value, in the usual sense, when we compare something to something else, usually a unit, most often a monetary unit. So, there's something material in that relation. My weight in kilos is a value. The value of my weight in kilos. It's also a number of kilos, just like there's a number of children in the courtyard. Values and numbers are abstractions of physical properties. They don't depend on the way we express numbers. We use numerical figures, symbols, to represent numbers and values. You seem to confuse the figures we use to represent numbers with the number themselves.

A value can be expressed as an expressed physical potential or attribute, other than just a number.

Please give an example.

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

If there was a clear difference, you'd already have explained what it is. And you haven't.
EB

 

[Write4U]

The number of children in the courtyard isn't an "arbitrary symbolic representation of a value"

The point is that a number is codified as a cypher.

A value can represent a range of physical properties, without necessarily having to be expressed as a number but as a quality. Numbers are used mainly in relation to a quantity or of specific values.

One can make a general point that:
Values deal primarily with qualities. They are variable.
Numbers deal primarily with quantities. They are static.

 

[Speakpigeon]

Is anyone even worrying about measuring something exactly?

How could we possibly know we're measuring anything exactly. What do you think that would even mean?
We do a measure and all we can do is compare it to another measure we've done, and this in itself is yet another measure, just a measure. We're stuck in being ourselves and not what we measure, which is probably just as well.
Still, we usually believe there's something like a material world out there and we usually think of it as having mathematical properties, i.e. mathematically expressible relations between its parts. The number of fingers on my right hand is the same as the number of fingers on my left hand. These are my own fingers and yet I can't even be sure there are anything like five fingers anywhere in my body, let alone whether I have a body. I'm left to assume I have one and so far, it seems to work wonders. So, I've just measured the number of fingers on my left hand by comparison to those on my right hand. Seems exact to me. But whether it is exact depends on what you mean by exact. Could I possibly find a different value? Well, I certainly hope not but who know. So, I'll take that as exact. Measuring other things may be different. We'll find a different measure each time we'll make one.
Still, that doesn't mean the physical world has no mathematical properties. If it has, then it may be the case that the relation between two different things in the world could be expressed, if only we could measure it, by a non-computable number. Maybe, maybe not. I don't know of any reason to assume it's not possible. This would mean the physical world somehow necessarily contains infinities. Big deal. And that, maybe, all our measures can only be approximate has nothing to do with that.
EB

 

[Write4U]

No. Any pair of things can be used to represent the number 2.

Yes, but it cannot give you the value that 2 represents other than there are 2 trees.
2 big trees have a greater biomass value than 2 small trees.
In mathematics the number "two" is always represented symbolically by the number "2" in various communcation methods. But it never does represents a value of three, unless the value is qualified as a property of the object being counted.

 

[Write4U]

We do a measure and all we can do is compare it to another measure we've done, and this in itself is yet another measure, just a measure. We're stuck in being ourselves and not what we measure, which is probably just as well.

I thought differential equations gave us a fairly precise knowledge of behavior.

Differential (mathematics)
From Wikipedia, the free encyclopedia

In mathematics, differential refers to infinitesimal differences or to the derivatives of functions.[1] The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology.

 

[Speakpigeon]

The point is that a number is codified as a cypher.

So are values. I provided examples. You didn't.

A value can represent a range of physical properties, without necessarily having to be expressed as a number but as a quality. Numbers are used mainly in relation to a quantity or of specific values.

That's also true of numbers. If there's five children in the courtyard, this fact is independent of whether I express this number using numerical figure.

One can make a general point that:
Values deal primarily with qualities. They are variable.

No. Values can be quantities. The value of a shirt is 153 Pounds. It's a quantity of Pounds. It's also therefore a number of Pounds.

Numbers deal primarily with quantities. They are static.

No. The number of children in the courtyard isn't static.
And numbers are primarily qualities. That there are five children in the courtyard is the expression of the relation with all sets of five things.
Try again.
You really suck at replying.
EB

 

[Speakpigeon]

Yes, but it cannot give you the value that 2 represents other than there are 2 trees.
2 big trees have a greater biomass value than 2 small trees.
In mathematics the number "two" is always represented symbolically by the number "2" in various communcation methods. But it never does represents a value of three, unless the value is qualified as a property of the object being counted.

Sorry, but you're not making sense and you're not even trying to hard.
A value is a number. We talk of value rather than number in particular cases. Mainly, we talk of value for the number which specifies the relation between a quantity and some relevant unit. But the value is still just a number. It's the number of those units. The value of a shirt may be 153 Pounds. It just means that the shirt has the same value as 153 Pounds. And this works both way. The value of the Pound is 1/153 that of the shirt. So a value is a number expressing the quantitative relation between different things that have some quality or property in common. There is the same relation between a shirt and the Pound as there may be between a distance of 153 metres and the metre as a unit of distance. Yet, we don't say that the value of a distance is 153 metres. So, the distinction in usage is just that, usage (and syntax). There's no fundamental difference.
And numbers in general always express a relation between things. Number 2 can be used to express the relation there is between a pair of shoes and somebody's feet. Or the valence of atoms. All pairs of things share this quality of there being two of them. It's universal.
EB

 

[Speakpigeon]

I thought differential equations gave us a fairly precise knowledge of behavior.

I feel very comfortable with the idea that a mathematical model might be an absolutely exact representation (description, specification, etc.) of certain relations between physical things.
Still, measurement is something else.
We do a measure and all we can do is compare it to another measure we've done, and this in itself is yet another measure, just a measure. We're stuck in being ourselves and not what we measure.
EB

 

[Speakpigeon]

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.

Sorry, I don't understand this idea that what we "observe" are rational numbers. We observe the physical world and it is what it is irrespective of what we may think. At least that's the usual way we think of it. So, you may think non-computable numbers don't exist but the physical world doesn't care what you think.
I don't see any reason that non-computable numbers could not possibly express relations between physical quantities. I don't see any reason that a spatially finite region of the universe could not contain an infinite amount of information.
Our measures are usually expressed in numerical figures. We always stop at a finite precision. Our measures are always approximate so there would be no point expressing a measure by anything but a finite-precision number. That doesn't mean at all that what we somehow "observe" are rational numbers. We observe physical quantities and we use finite-precision numbers to express our measures.

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.

Measures are irrelevant here. The question is not whether there's anything non-computable or somehow infinite that would correspond to our measures but whether there is any good reason to exclude the possibility that actual quantities in the physical world be non-computable or somehow infinite.

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.
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.
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 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.
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.
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.

You still haven't shown it would be somehow impossible to have a non-computable quantity in the physical world. And my point was that if there was, we might still be able to measure quantities using rational numbers and have exact measures. That's a point of principle since we're not going to be able to have exact measures. So, you need to explain what it would mean exactly for a non-computable number to express a physical quantity. If you can't, then I'm not sure you even understand what you're talking about.
EB

 

[someguy1]

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.

Ok, still not following point. I agree that below the Planck scale it's meaningless to talk about physics. The world might be quantized at the Planck scale, or there might be interesting things going on below the Planck scale that our present physics doesn't let us investigate.

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.

Yes of course the standard real numbers give insight, but that doesn't imply that the universe is accurately modeled by the real numbers. You should check out some of the alternatives like the intuitionistic continuum, the computable or constructive real line, the hyperreals, the surreals, and various other models of the continuum. Charles Sanders Peirce (with that spelling) noted that it's *impossible* for the mathematical real numbers to model the continuum, because a continuum is indistinguishable from any of its parts. 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. This is a pretty good argument against the standard reals being the proper model of the continuum.

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

Now this I can't understand at all. There are only countably many computable numbers. 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. The computable real line is *almost all holes*. If you don't understand this please ask, because when you say there are no holes in the computable real line you're stating a mathematical falsehood.

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

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. Of course that's your right, but you won't get far in conversations with people who know the math.

 

[someguy1]

Sorry, I don't understand this idea that what we "observe" are rational numbers. We observe the physical world and it is what it is irrespective of what we may think.

That's a metaphysical assumption. You don't know that we observe the physical world. We observe our lab apparatus and build models. What's "out there" is a matter of metaphysics. If you still don't understand this point there's nothing else I can say.

At least that's the usual way we think of it.

No that's not true. It's a casual way of speaking, but that doesn't prove anything. Who is this "we?" It wouldn't include most philosophers of science, nor would it include thoughtful scientists.

So, you may think non-computable numbers don't exist but the physical world doesn't care what you think.

This is a strange argument.

I don't see any reason that non-computable numbers could not possibly express relations between physical quantities. I don't see any reason that a spatially finite region of the universe could not contain an infinite amount of information.

https://en.wikipedia.org/wiki/Bekenstein_bound

How many times do I have to post the same link? A noncomputable real instantiated in the physical world would violate the laws of physics.

Our measures are usually expressed in numerical figures. We always stop at a finite precision. Our measures are always approximate so there would be no point expressing a measure by anything but a finite-precision number. That doesn't mean at all that what we somehow "observe" are rational numbers. We observe physical quantities and we use finite-precision numbers to express our measures.

Ok. But then we will never observe a noncomputable real number. And for that matter we will never even observe an INTEGER, if we are measuring a continuous quantity. We'll never observe ANYTHING exactly. So how can ou be so sure what's "out there" in the unobservable world? Things that are unobservable are by definition not in the purview of science.

Measures are irrelevant here. The question is not whether there's anything non-computable or somehow infinite that would correspond to our measures but whether there is any good reason to exclude the possibility that actual quantities in the physical world be non-computable or somehow infinite.

Besides the known laws of physics. Nothing. If you are convinced actual infinities exist, then I can't argue with you. I can only note that your thesis violates known physics. That's not to say physics won't change in the future.

You still haven't shown it would be somehow impossible to have a non-computable quantity in the physical world.

Correct. I haven't shown that, because I don't know. I HAVE shown that a physically instantiated infinity violates the KNOWN LAWS of physics. What the future laws may be, nobody can say. The next Newton or Einstein hasn't shown up yet.

And my point was that if there was, we might still be able to measure quantities using rational numbers and have exact measures.

All measurement is approximate. If they forget to tell you that in high school science class, bad on your teacher. All measurement is approximate. You're just making stuff up now.

That's a point of principle since we're not going to be able to have exact measures. So, you need to explain what it would mean exactly for a non-computable number to express a physical quantity.

No I don't. You do, since you're the one claiming it can happen. A noncomputable number carries an infinite amount of incompressible information.

If you can't, then I'm not sure you even understand what you're talking about.

If so, then by all means you should stop talking to me. Or make a better case for your own argument, which violates known physics.

 

[someguy1]

I'm not sure about the implications, but Max Tegmark, proposes that all of reality can be explained by the use of just 33 numbers and a handful of fundamental equations .

We know what Tegmark "proposes." That's quite a bit less than what anyone can prove using actual science. Of course there's a science to selling books, perhaps that's the science Max is good at.

 

[Speakpigeon]

That's a metaphysical assumption. You don't know that we observe the physical world. We observe our lab apparatus and build models. What's "out there" is a matter of metaphysics. If you still don't understand this point there's nothing else I can say.
No that's not true. It's a casual way of speaking, but that doesn't prove anything. Who is this "we?" It wouldn't include most philosophers of science, nor would it include thoughtful scientists.
This is a strange argument.
https://en.wikipedia.org/wiki/Bekenstein_bound
How many times do I have to post the same link? A noncomputable real instantiated in the physical world would violate the laws of physics.
Ok. But then we will never observe a noncomputable real number. And for that matter we will never even observe an INTEGER, if we are measuring a continuous quantity. We'll never observe ANYTHING exactly. So how can ou be so sure what's "out there" in the unobservable world? Things that are unobservable are by definition not in the purview of science.
Besides the known laws of physics. Nothing. If you are convinced actual infinities exist, then I can't argue with you. I can only note that your thesis violates known physics. That's not to say physics won't change in the future.
Correct. I haven't shown that, because I don't know. I HAVE shown that a physically instantiated infinity violates the KNOWN LAWS of physics. What the future laws may be, nobody can say. The next Newton or Einstein hasn't shown up yet.
All measurement is approximate. If they forget to tell you that in high school science class, bad on your teacher. All measurement is approximate. You're just making stuff up now.
No I don't. You do, since you're the one claiming it can happen. A noncomputable number carries an infinite amount of incompressible information.
If so, then by all means you should stop talking to me. Or make a better case for your own argument, which violates known physics.

No, I don't think it does.
OK, you've made your points and I made mine. It seems there's is nothing to add except to repeat ourselves.
EB

 

[Write4U]

No. Values can be quantities. The value of a shirt is 153 Pounds. It's a quantity of Pounds. It's also therefore a number of Pounds.

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.

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

 

[Write4U]

We know what Tegmark "proposes." That's quite a bit less than what anyone can prove using actual science. Of course there's a science to selling books, perhaps that's the science Max is good at.

So was Carl Sagan, anything wrong with selling and popularizing science? That's not a valid argument against his scientific proposition.

It is a matter of perspective. Suppose he had said that all values and functions in the universe are theoretically reducible to numbers and equations. Would that bring the same opposition to bear?
That statement cannot be denied, else all of science will ultimately fail.

 

[someguy1]

So was Carl Sagan, anything wrong with selling and popularizing science? That's not a valid argument against his scientific proposition.

We were talking about Tegmark many pages ago, and my sense is that mention of his name sucks all the air out of the discussion. Tegmark doesn't have a scientific thesis, just an interesting speculation. I don't think that's a controversial statement. I believe that Tegmark confuses the map with the territory and I've said so many times already in this thread. Carl Sagan did science. So does Tegmark, when he's doing science. When he's making claims that can't be proven, he's not doing science, but he is selling books. I didn't say that selling books is bad. I said that selling books that aren't about sciene isn't doing science. Unless it's doing the science of selling books. By that logic Jacqueline Susann (author of the fabulous trash novel Valley of the Dolls) deserved the Nobel prize.

It is a matter of perspective. Suppose he had said that all values and functions in the universe are theoretically reducible to numbers and equations. Would that bring the same opposition to bear?

That is what he says. He's wrong about the nature of science. Or to be more fair to Tegmark, he doesn't say that. He says that IF this THEN that. He's not asserting his premises, only working out their logical conclusions. His argument is more sophisticated than the understanding of some of his readers.

That statement cannot be denied, else all of science will ultimately fail.

Perhaps science has already failed or is in fact in the process of failing. Perhaps people like Tegmark are the reason why.

 

[arfa brane]

Avogadro's number is approximately 6.022×10^23, a large sum almost immeasurable, greater than the number of grains of sand on earth or even the number of stars in the universe. A "mol" of water molecules, for example, is only a few teaspoons of liquid. Because Avogadro's number is linked to a series of other physical constants, its value can be expressed to other units, such as the kilogram.

A value for Avogadro's number has been obtained by counting the number of atoms in a sphere of one kilogram of high purity Si-28. When silicon crystallizes, it forms cubic cells of eight atoms each. It is possible to calculate the number of atoms in such a sphere by examining the relationship between the total crystal volume and the volume occupied by each silicon atom, which can in turn be calculated by measuring the cubic cell. A new value has been obtained for Avogadro's number with an uncertainty of less than 20 atoms per billion, less than the 30 atom uncertainty in its 2011 value.

--https://www.quora.com/Is-Avogadros-number-exact

Note that the error in 6.022×10^23 = 6,022 ×10^20 is a number with 20 digits, 20-30 atoms per billion then (within this 20-digit number) is a large number.
Just sayin'.

 

[arfa brane]

I placed the speed of light in the not-known-exactly category, when it does have an exact value if you base the metre on the distance light propagates (in some rational number of seconds).

 

[Write4U]

I said that selling books that aren't about sciene isn't doing science.

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

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?