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

[Write4U]

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

W4U said,
A value can be expressed as an expressed physical potential or attribute, other than just a number.

Please give an example.

LOL, I did and you complained because I showed you that I was right.

I understand very well what you are stating, but you are so far from understanding what I am stating that it is almost discouraging. You have a narrow vision, perhaps it is time to expand it.

If you want to talk about infinity then you better treat the term and its implications with a little more respect and humility than you have so far.

Infinity is not a number, it is a value.....and the human symbol for that value is:

Don't see that number in the decimal system do we?......

 

[Speakpigeon]

Infinity is not a number, it is a value.....and the human symbol for that value is:

Don't see that number in the decimal system do we?......

Sorry, you're not making sense, here.
EB

 

[Speakpigeon]

In the physical realm, the distinction is vital. A noncomputable number carries an infinite amount of information.

That's just not true. What carries "an infinite amount of information" isn't the physical number itself, which is just the physical quantity, but our particular representation of it in terms of symbolic numbers and in a way which isn't computable.
It just means our representation isn't very good, which should not really come as a surprise since it's not the physical world itself.

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?

Elements in a continuum just come one after the other without a care about our mathematical distinctions. So, what could be the import of our distinction between integer, rational, irrational, computable, non-computable numbers on the elements of a physical continuum. Why would there be a difference in nature between those elements?

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.

No. That's not true.
We do need an infinite amount of information to express a non-computable number in terms of a computable number. That's true but the reverse is also true. We need an infinite amount of information to express a computable number in terms of a non-computable number.
Second, we in fact only need a finite amount of information to express a non-computable number, as long as we do it in terms of another non-computable number and provided it is properly selected (and there's at least a countable infinity of appropriate non-computable numbers to choose from).
So, why exactly should non-computable numbers be of a different nature?
All there is is a mathematical bizarrerie that has no effect on the physical world because mathematics isn't the physical world.
And the bizarrerie is entirely due to our epistemological perspective. It's been just convenient for us to use numbers the way we have done so far. It's a practical problem. We started out counting on our fingers. That's all there is to it. We've extended our arithmetic beyond its provincial origin and now we find there is a bizarrerie? We can't properly describe the universe just by counting on our fingers? Well, too bad.

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.

No, I don't accept that premise because you still haven't explained what it would mean for a non-computable number to be instantiated and what would be the physical difference between an instantiated number that would be computable and one the wouldn't be computable.
EB

 

[Write4U]

W4U said,
Infinity is not a number, it is a value.....and the human symbol for that value is:

= Infinity
Don't see that number in the decimal system do we?......

Sorry, you're not making sense, here

It's not a number, it is a symbolic representation of a value which has no beginning and no end. It is a value of the "Wholeness and the Implicate Order" as theorized by David Bohm, who called it "insight intelligence", but was speaking of a hierarchy of orders or pseudo-intelligent mathematical constants from which "enfolded" probabilities become "unfolded" in reality. The symbol we choose was for the value (1 possible workable model) of an endless continuity in the existence of the expressed physical and metaphysical reality and which yields an equation to the value of the Wholeness.

 

[Speakpigeon]

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".

Could you give actual examples of the ""confusion" you're talking about?
I want convincing quotes from well-known scientists with the appropriate link to check if necessary.
EB

 

[Speakpigeon]

It's not a number, it is a symbolic representation of a value which has no beginning and no end. It is a value of the "Wholeness and the Implicate Order" as theorized by David Bohm, who called it "insight intelligence", but was speaking of a hierarchy of orders or pseudo-intelligent mathematical constants from which "enfolded" probabilities become "unfolded" in reality.

Sorry, you're not making sense, here.
EB

 

[Write4U]

Could you give actual examples of the ""confusion" you're talking about?
I want convincing quotes from well-known scientists with the appropriate link to check if necessary.
EB

Definition of number
1a (1) : a sum of units : total

• the number of people in the hall

(2) :complement 1b(1)

• … the whole number of Senators …
• —U.S. Constitution

(3) : an indefinite usually large total

• a number of members were absent

• the number of elderly is rising

(4) numbers plural : a numerous group :many

• numbers died on the way
• —Marjory S. Douglas

(5) : a numerical preponderance (see preponderance 2)

• There's safety innumbers.

b (1) : the characteristic of an individual by which it is treated as a unit or of a collection by which it is treated in terms of units

• there is a limited number of such laboratories
• —P. D. Close

(2) : an ascertainable total

• bugs beyond number

c (1) : a unit belonging to an abstract mathematical system and subject to specified laws of succession, addition, and multiplication

• a numberdivisible by 2

; especially : natural number
(2) : an element (such as π) of any of many mathematical systems obtained by extension of or analogy with the natural number system
(3) numbers plural : arithmetic

• Teach children their numbers.

2: a distinction of word form to denote reference to one or more than one

• A subject and its verb should agree in number.

; also : a form or group of forms so distinguished
3numbers plural
a (1) : metrical structure : meter

• … most by numbers judge a poet's song.
• —Alexander Pope

(2) : metrical lines : verses

• These numbers will I tear, and write in prose.
• —Shakespeare

b archaic : musical sounds : notes
4a : a word, symbol, letter, or combination of symbols representing a number

• Spell out the numbers one through ten.

b : a numeral or combination of numerals or other symbols used to identify or designate

• dialed the wrong number

c (1) : a member of a sequence or collection designated by especially consecutive numbers (such as an issue of a periodical)

• just received issue number 8 of the magazine

(2) : a position in a numbered sequence

• You're number 7 on the waiting list.

d : a group of one kind

• not of their number

: one singled out from a group : individual: such as
a : girl, woman

• met an attractive number at the dance

b (1) : a musical, theatrical, or literary selection or production

• The actors broke into a song and dance number.

(2) : routine, act

• The comedian's number had the audience laughing out loud.

c : stunt, trick
d : an act of transforming or impairing

• tripped and did a number on her knee

e : an item of merchandise and especially clothing

• put that black velvet number with the sequins on the blonde dummy
• —Bennett Cerf

6: insight into a person's ability or character

• had my number

7numbers plural in form but singular or plural in construction
a : a form of lottery in which an individual wagers on the appearance of a certain combination of digits (as in regularly published numbers) —called also numbers game
: 2policy 2a
8numbers plural
a : figures representing amounts of money usually in dollars spent, earned, or involved

• We won't be able to stay in business with numberslike these.

b (1) : statistics 2; especially : individual statistics (as of an athlete)

• Her numbers make her the most valuable member of the team.

(2) : rating 3c
9: a person represented by a number or considered without regard to individuality

• at the university I was just a number

10: lifetime 1a —used with up

• the old feeling that comes to men in combat … that your number was up
• —Geoffrey Norman

— by the numbers
1: in unison to a specific count or cadence
2: in a systematic, routine, or mechanical manner

• a program run not by the numbersbut with concern for the participants

Table of Numbers

https://www.merriam-webster.com/dictionary/number

OTOH:

Definition of value
1: the monetary worth of something : market price
2: a fair return or equivalent in goods, services, or money for something exchanged
3: relative worth, utility, or importance

• a good value at the price

the value of base stealing in baseball, had nothing of value to say

4: something (such as a principle or quality) intrinsically valuable or desirable

• sought material values instead of human values
• —W. H. Jones

5: a numerical quantity that is assigned or is determined by calculation or measurement

• let x take on positive values

a value for the age of the earth

6: the relative duration of a musical note

7a : relative lightness or darkness of a color : luminosity

7b: the relation of one part in a picture to another with respect to lightness and darkness

https://www.merriam-webster.com/dictionary/value

I think this describes pretty well the areas which are common to both terms and definition, but also the areas which give each unique and independent aspects in and off themselves.

 

[Speakpigeon]

https://www.merriam-webster.com/dictionary/number

OTOH: https://www.merriam-webster.com/dictionary/value

I think this describes pretty well the areas which are common to both terms and definition, but also the areas which give each unique and independent aspects in and off themselves.

Could you give actual examples of the ""confusion" you're talking about?
I want convincing quotes from well-known scientists with the appropriate link to check if necessary.
EB

 

[someguy1]

That's just not true.

You make some excellent points. I don't claim to know all the answers or even understand the questions. I'll just toss out some thoughts.

What carries "an infinite amount of information" isn't the physical number itself, which is just the physical quantity, but our particular representation of it in terms of symbolic numbers and in a way which isn't computable.
It just means our representation isn't very good, which should not really come as a surprise since it's not the physical world itself.

Hmmm, yes, I have to agree. If there are pointlike thingies "out there" and we agree that out measurements are not inherent in the things themselves, then this is a very good point. After all, you can take any noncomputable number and translate it (by a noncomputable distance) and bring it to a computable location on the number line. So the noncomputability doesn't actualy inhere in the point itself, but rather in our description of it.

Now I happen to believe this, or at least believe that it's possible. That these "values and functions" that @Write4U talks about are not properties of things, but rather properties of our measurements. When we measure the frequency of red light, we're not measuring a quality of the photons, but just imposing our own minds and lab apparatus on reality, and telling ourselves we're measuring something "out there."

This goes against a lot of what scientists believe, that we are measuring what's "out there." I don't know the answer to this one. But if we accept that there are undifferentiated mathematical points "out there", then it doesn't matter if we label some of them noncomputable. Perhaps you're right.

I think the argument against the physical existence of noncomputable quantities is an argument against the CUH and the Church-Turing-Deutsch thesis. You noted earlier that these aren't necessarily true, and I agree. But a lot of people believe them; and to the extent that people believe them, it's fair to note that the idea of computability and the idea of continuity are somewhat at odds.

Elements in a continuum just come one after the other without a care about our mathematical distinctions.

Semantic quibble, between any two points there's another one. They don't come one after another. Think maple syrup, not bowling balls.

So, what could be the import of our distinction between integer, rational, irrational, computable, non-computable numbers on the elements of a physical continuum. Why would there be a difference in nature between those elements?

Good point, if we agree that science doesn't describe reality, it's only something we're making up about things we can never name or understand. Your viewpoint is extreme here. Not wrong of course, but it will put you into some trouble with those who think science is about studying what's "out there." If we can't apply everyday notions of mathematics to the world, that undermines all of science. You should think about this and tell me if you take this as a reasonable criticism.

We do need an infinite amount of information to express a non-computable number in terms of a computable number. That's true but the reverse is also true. We need an infinite amount of information to express a computable number in terms of a non-computable number.

Yes but I'm lost because you just denied that concepts like rationality or being an integer even apply to the real world. I hope you see that while you might be right, that position undermines science. You have to make sure you understand the implications of your own idea.

Second, we in fact only need a finite amount of information to express a non-computable number, as long as we do it in terms of another non-computable number and provided it is properly selected (and there's at least a countable infinity of appropriate non-computable numbers to choose from).

Yes that's true, but plenty of pairs of noncomputables are NOT computable multiples of each other.

So, why exactly should non-computable numbers be of a different nature?

If the noncomputability lies only in our minds or labeling or notation, no difference at all. So if math is only in our minds and not in the objects themselves, you are going to have an army of scientists to argue with. I'll just sit back and watch.

All there is is a mathematical bizarrerie that has no effect on the physical world because mathematics isn't the physical world.

Now you're joining me in arguing against Tegmark. I suspect your viewpoint and mine are not that far apart.

And the bizarrerie is entirely due to our epistemological perspective. It's been just convenient for us to use numbers the way we have done so far. It's a practical problem. We started out counting on our fingers. That's all there is to it. We've extended our arithmetic beyond its provincial origin and now we find there is a bizarrerie? We can't properly describe the universe just by counting on our fingers? Well, too bad.

I tend to agree. But Tegmark and many others think the math is "real" in some way. I'm arguing against that point of view and now you're arguing with me, so I'm pretty confused. But your points in this post are correct, or at least valid.

No, I don't accept that premise because you still haven't explained what it would mean for a non-computable number to be instantiated

I'll leave that to those who claim they can be.

and what would be the physical difference between an instantiated number that would be computable and one the wouldn't be computable.

If there's a physically intantiated noncomputable quantity, we could never measure it and we could never even write a program to approximate it. I'd say that would be a problem for science and a fatal blow to the Tegmarkian "the world is mathematics" point of view.

But I'm in way over my head now. I agree with most of what you said here.

 

[arfa brane]

Infinity is not a number, it is a value

Actually infinity is neither a number nor is it a value. You might be able to argue that infinity is uncomputable, but not that it's an uncomputable number (since it would then have a value, even an uncomputable one).

I think it makes more sense to talk about the complexity of numbers, as Chaitin does, in terms of algorithmic complexity. That is, roughly, how complex an algorithm needs to be to specify a given number. Algorithms then describe those numbers we can obviously compute, so those numbers which are germane to science, to 'measurements' etc.

But just as obviously, not all numbers. AC lets us throw the darts, type of thing.

 

[Write4U]

Could you give actual examples of the ""confusion" you're talking about?
I want convincing quotes from well-known scientists with the appropriate link to check if necessary.
EB

Well, you see, "if I can't speak directly to the scientist, I don't bother"; your words.
I gave you Webster's definitions on both terms. Ibelieve that should suffice.

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.

What you are espousing here is Tegmark. Now do you understand the meaning of being confused about the difference between values and numbers?
We don't observe the numbers, we observe the expressed values. What is the number of water, air, fire, earth, a square, a circle, a rectangle? Those are all properties of the elements and mathematical patterns. The value is the expressed reality as an object.

 

[Write4U]

Write4U said:
Infinity is not a number, it is a value

Actually infinity is neither a number nor is it a value. You might be able to argue that infinity is uncomputable, but not that it's an uncomputable number (since it would then have a value, even an uncomputable one).

Is that not what I said? Infinity's value is not a mathematical value, it's value lies in infinite potential or permittivity, it is a timeless uncountable physical state, which apparently resulted in the emergence of the universe.

 

[someguy1]

Is that not what I said? Infinity's value is not a mathematical value, it's value lies in infinite potential or permittivity, it is a timeless uncountable physical state, which apparently resulted in the emergence of the universe.

Well as it happens that's another thing I have a mathematically-influenced opinion about.

Now you claim that infinity apparently resulted in the emergence of the universe. That's quite a statement.

But when you say that infinity is "a timeless uncountable physical state," first, you're ignoring the mathematical definition of infinity; and second, you are making yet another metaphysical claim: That you know what resulted in the universe. I'm sure you do. But that's not a rational position that can be argued against. It's essentially a theological claim. As such, it may be properly ignored in scientific discussions.

But when you define infinity, you are saying either that "I know all about the work mathematicians have done on infinity in the past 140 years and I choose to completely ignore it." Or, you are simply ignorant of said work.

Mathematicians can define what it means for a set to be infinite. They study a huge hierarchy of levels of infinity, transfinite cardinals and ordinals, infinite numbers so big they can't even be proved to exist.

Now you might say, "Well, that's not the infinity I mean." Or you might say, "Oh really? Tell me more."

So please tell me where you're coming from. Are you making an essentially theological point? Or perhaps you have not seen the mathematical definition of infinity.

Here it is. A set is infinite just in case it can be put into one-to-one correspondence, called a bijection, with a proper subset of itself. So the natural numbers 0. 1, 2, 3, 4, 5, are infinite because as Galileo noted in 1638, they can be placed into bijection with the perfect squares 0, 1, 4, 9, 16, 25, ... And the set {a, b, c, d, e} is finite because there is no such bijection, as you should verify.

So just let me know why you're waxing poetic about a concept that's been mathematically tamed and formalized for 140 years.

https://en.wikipedia.org/wiki/Galileo's_paradox

 

[Write4U]

Well as it happens that's another thing I have a mathematically-influenced opinion about.

Now you claim that infinity apparently resulted in the emergence of the universe. That's quite a statement.

But when you say that infinity is "a timeless uncountable physical state," first, you're ignoring the mathematical definition of infinity; and second, you are making yet another metaphysical claim: That you know what resulted in the universe. I'm sure you do. But that's not a rational position that can be argued against. It's essentially a theological claim. As such, it may be properly ignored in scientific discussions.

But when you define infinity, you are saying either that "I know all about the work mathematicians have done on infinity in the past 140 years and I choose to completely ignore it." Or, you are simply ignorant of said work.

Mathematicians can define what it means for a set to be infinite. They study a huge hierarchy of levels of infinity, transfinite cardinals and ordinals, infinite numbers so big they can't even be proved to exist.

Now you might say, "Well, that's not the infinity I mean." Or you might say, "Oh really? Tell me more."

So please tell me where you're coming from. Are you making an essentially theological point? Or perhaps you have not seen the mathematical definition of infinity.

Here it is. A set is infinite just in case it can be put into one-to-one correspondence, called a bijection, with a proper subset of itself. So the natural numbers 0. 1, 2, 3, 4, 5, are infinite because as Galileo noted in 1638, they can be placed into bijection with the perfect squares 0, 1, 4, 9, 16, 25, ... And the set {a, b, c, d, e} is finite because there is no such bijection, as you should verify.

So just let me know why you're waxing poetic about a concept that's been mathematically tamed and formalized for 140 years.

https://en.wikipedia.org/wiki/Galileo's_paradox

It must have been bed-time......

But I must admit that I have never prayed in my life.. I am an atheist and stating that the universe apparently emerged from the implicated mega quantum event the instant of the BB, is not necessarily a theological claim.

IMO, a theological claim must include a sentient motivation, else it is just an unknown permittive condition.

 

[Speakpigeon]

Well, you see, "if I can't speak directly to the scientist, I don't bother"; your words.
I gave you Webster's definitions on both terms. Ibelieve that should suffice.

Could you give actual examples of the ""confusion" you're talking about?
I want convincing quotes from well-known scientists with the appropriate link to check if necessary.
EB

 

[Speakpigeon]

What you are espousing here is Tegmark. Now do you understand the meaning of being confused about the difference between values and numbers?
We don't observe the numbers, we observe the expressed values. What is the number of water, air, fire, earth, a square, a circle, a rectangle? Those are all properties of the elements and mathematical patterns. The value is the expressed reality as an object.

Sorry, you're not making sense, here.
You appear to be terminally unable to explain yourself.
EB

 

[Speakpigeon]

Actually infinity is neither a number nor is it a value.

Infinity is initially an attribute. The attribute of a number. We say for example that there's an infinite number of points or that time is infinite because we assume there's no limit to the number of seconds in the future. It's the number of points or of second which is infinite, so infinity is first and foremost an attribute.
As such, it may also be a proper quality, for example if there is an actually infinite number of points in time or in space. In this case, to the attribute of infinity that we ascribe to time or space would correspond an actual infinite number of points in time or space. In this case infinity would be something like a quality or property of the physical world.
If so, there's also no difficulty in considering infinity as a number, provided we can conceive a consistent way of dealing with infinity as a number. We already have special numbers, such as 0 and 1, primes, Integers, Reals etc., so why not infinity. That's something for mathematicians to consider. No need to be dogmatic about that.
But I agree infinity is not a value, except if by value we just mean number. For example, the value of the function 1/x when x tends towards zero could be said to be infinity.
EB

 

[arfa brane]

Infinity is initially an attribute. The attribute of a number.

I agree with the first, but not the second statement.
For instance, where is the center of the "external" face of a planar graph?
Where is the point at infinity on the Riemann sphere?

Infinity looks a lot like a place, rather than a value, or any kind of number.

We say for example that there's an infinite number of points

Because of continuity.

. . . to the attribute of infinity that we ascribe to time or space would correspond an actual infinite number of points in time or space. In this case infinity would be something like a quality or property of the physical world.

Yah. Or there might not be an actual infinity of points, space and time might be discrete, not continuous (after all, we will only ever know about real intervals of either, which means measureable intervals).

If so, there's also no difficulty in considering infinity as a number, provided we can conceive a consistent way of dealing with infinity as a number.

A difficult task, since infinity is also some attribute of location (i.e. where is infinity? not how big is it).

But I agree infinity is not a value, except if by value we just mean number.

There ya go.

For example, the value of the function 1/x when x tends towards zero could be said to be infinity.

Now you're talkin' (about limits of functions, and again, continuity).

 

[iceaura]

As such, it may also be a proper quality, for example if there is an actually infinite number of points in time or in space. In this case, to the attribute of infinity that we ascribe to time or space would correspond an actual infinite number of points in time or space. In this case infinity would be something like a quality or property of the physical world.

In an attempt to keep it as simple as possible with an example, this post:

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?

It's an honest question. One of the possible answers is "no" - that since the correspondence of the number 2 to some physical situation is contingent on assumptions or conventions known to be ad hoc or heuristic rather than theoretically unalterable and uncontradictable reality, the number "2" is a mathematical abstraction only.

The relevance would be that "infinity" and "2" share the same reality.
My own guess is that math works as material from which we can construct virtual sensory organs we lack, and that perception via math is essentially equivalent (in its relationship with reality, whatever that may be) to perception via sensory organ. So that we can "see" infinities - they are as real as densities, masses, temperatures, wavelengths, pitches, colors, etc.

 

[phyti]

Speakpigeon #255;

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.

---

Life is supported by the local portion of the universe and continues independently of the astronomically distant portions. There is no need of a larger universe.

The mathematician will tell you a finite interval of the 'real' number line contains an 'infinite' number of points. The truth of the statement depends on the physical existence of a continuum

corresponding to the mental concept of a continuum. Chemistry demonstrates matter is discrete, and measurements depend on physical objects as references. Energy is also quantized and does not have a continuous spectrum. This supports the conclusion that measurements will be finite. Plank solved the 'black body' problem with h. Where is the need for anything continuous?

example:

1. Divide a pile of 1000 seeds into 3 equal piles, the 'real' number fans will give you, 333.33 seeds per pile. The 'integer' fans will give you 2 piles of 333, and 1 pile of 334.

Which answer is 'real'? The problem says divide the pile, not the seeds.

2. Divide a metal rod 4" long into 2 equal length pcs. Are we sure there are the same number of molecules in each half? When a rod is cut, does the cut fall between molecules? (can't cut molecules)

3. An engineer can specify a dimension to 3 decimals, but can it be formed to exact specifications? No, they only work with 'significant' digits.

4. Approximations such as non-rational numbers, pi, e, ... etc, are always incomplete, and become meaningless after the decimal positions become smaller than the diameter of a fundamental particle.

When we count, the integers correspond precisely to the quantity of things counted.

So what is 'real' in the world of measurement?

Hopefully this clarifies my position regarding infinity, it can't be quantified, nor conceptualized by the finite mind. The one ended stick can never be measured if the 2nd end can't be found!

The continuum, you can't make something from nothing (unless you're a supreme being).