Is infinity a constant or a variable ?

[HallsofIvy]

Okay, so how will you define infinity?
Except that it is undefined, or say
Something without upper bound?
Though second definition looks to be defining infinity, it itself implies the non-definitive nature of infinity.


Also ,you said like 7, pie, next you will say e, though they are not "exactly" defined as
e = 2.2.7182818... approx
pi = 3.1415926.. approx
But you can say
∞ = something... approx

pi and e are exactly defined. The fact that they require an infinite number of symbols in some specific numeration system has nothing to do with their definition.

 

[CRGreathouse]

Yes one is defined, we have a set of operations and rules to perform over it, also 7 is a symbol with same thing as 1, just 7 times of it.
But for ∞ we have very limited rules, say just evaluating some limits in which something tends to ∞ we have certain rules, or in physics we have some process of renormalization (I just know the name, nothing else) which can remove certain infinities by inserting some more infinities.
So it is certainly not as "expressible", rather say "explicit" as, say "the symblol 7".

You haven't defined one for me, nor have you defended your bare assertion that ∞ is not expressible or explicit. I also have no idea what, if anything, you mean by "very limited rules".

The renormalizations of physics have little or nothing to do with math.

 

[CRGreathouse]

AWESOME! I never knew that, thanks for making me know, but I am still not sure I get it completely or not, I think I got to have a sleep before any more reasoning.

Feel free to ask any questions you like. Someone can probably address them -- maybe me, maybe someone else.

Edit: By the way, the "Aleph_Null" slider mentioned is the same as the "aleph_0" I mentioned.

 

[aaryan0077]

Okay thanks everyone for you help, but how does this ends?
I mean what's the conclusion.

 

[CRGreathouse]

Okay thanks everyone for you help, but how does this ends?
I mean what's the conclusion.

Something like
"There are lots of kinds of infinities, none of which are variables."

 

[aaryan0077]

Say,
Is infinity like what Nick89, HallsofIvy, & diazona mentioned that it is rather a concept than a number.

Or as what CRGreathouse said, that it's a symbol ∞, and has got some rules to maniupalate it.

Or is it like CRGreathouse said later that what if numbers were to loop, and we cannot figure out if infinity is less than negative ten billion or more than 7?

Or as said by Russell Berty, that it's is not a real-valued constant. [Even when the set of Reals is extended to make the “Extended Reals”, infinity is still not a real-valued constant.]
and Infinity is not a real-valued variable, or as he says in end that it's just an interval notation to represent an unbounded interval.

What's the final result?
How should this thread end?

 

[aaryan0077]

Something like
"There are lots of kinds of infinities, none of which are variables."

Can you explain it a bit, and what's the meaning of "none of which are variables." does it means they are constant?
Why are you confusing me?

 

[matt grime]

I think you're probably confused because you have asked an ill-defined question - one which cannot have a satisfying answer.

The symbol [itex]\infty[/tex] and the notion of infinity or something being infinite can have different connotations in different contexts, that is what you should have learned in this thread. The best attempt one can make at answering 'is it a constant or variable' is that they (the different meanings) are neither and nor is it a sensible question to ask in the first place. My guess as to what you mean by 'constant or variable' is that you need some physical quantity or model of it which is 'infinity'. For example, position and time will be variables to you, and the ratio of an (idealized) circle's circumference and its diameter is pi, a constant.

 

[HallsofIvy]

Can you explain it a bit, and what's the meaning of "none of which are variables." does it means they are constant?
Why are you confusing me?

What do you see as the distinction between "variable" and "constant"? What are the definitions?

 

[CRGreathouse]

The best attempt one can make at answering 'is it a constant or variable' is that they (the different meanings) are neither and nor is it a sensible question to ask in the first place.

Although I don't describe the various infinities as constants, I think that's really what they are. So I'll differ from you on this point.

But when summing up for the OP I did avoid that, sticking to the well-agreed-upon "they're not variables".

 

[CRGreathouse]

Is infinity like what Nick89, HallsofIvy, & diazona mentioned that it is rather a concept than a number.

It's not a "real number". As used in high-school calculus, it's not a number at all but a concept.

Or as what CRGreathouse said, that it's a symbol ∞, and has got some rules to maniupalate it.

In R*, "+∞" is an "extended real number" just like any other, and has rules to manipulate it.

In P¹, "∞" is a "projective real number" and has rules to manipulate it.

In C* (the Riemann sphere), "∞" is an extended complex number and has rules to manipulate it.

In ZF, "$\aleph_0$" is a cardinal and has rules to manipulate it.

...

There are lots of infinities.

Or is it like CRGreathouse said later that what if numbers were to loop, and we cannot figure out if infinity is less than negative ten billion or more than 7?

I was talking about the ∞ in C* (or the ∞ in P¹) when I said that. ∞, in that context, can be thought of as the "greatest and the least element". It has the largest size, but it's not sensible to think of it as positive or negative.

Or as said by Russell Berty, that it's is not a real-valued constant. [Even when the set of Reals is extended to make the “Extended Reals”, infinity is still not a real-valued constant.]
and Infinity is not a real-valued variable, or as he says in end that it's just an interval notation to represent an unbounded interval.

None of the "real numbers" (but one of the extended complex numbers, two of the extended real numbers, and infinitely many of the cardinals) are infinite, so whatever kind of infinity you mean it isn't real-valued.

 

[matt grime]

Although I don't describe the various infinities as constants, I think that's really what they are. So I'll differ from you on this point.

As we're both guessing what the OP means by 'constant' and 'variable', I don't think that we differ, or agree. I mean, what if I fixed a field k, and formed the polynomial algebra k[some infinite cardinals] with the rules of cardinal arithemetic? Surely they're now 'variables'? Of course if we ascribe the 'physical' meaning of infinite cardinals as equivalence classes of sets, then aleph_0 is always the cardinality of the the integers (and let's avoid any set theoretic issues, which might imply that the cardinal number of a set may 'change' and be 'variable'), so it's 'constant' now... It's a truly pointless debate.

 

[Hurkyl]

I know I was thinking more along the lines of formal logic, with 'constant' meaning a nullary function.

 

[AUMathTutor]

In mathematics, which is referentially transparent, I don't think there's much of a difference between the idea of "variable" and "constant". Perhaps the closest I could come up with is that a variable is an undetermined constant. I really have no idea how mathematics formally distinguishes between the two.

In CS, it's a lot easier. It's a constant if it's a bit pattern. It's a variable if it's a reference to a memory location (which is inherently changeable). These concepts rely on a concept of "state" which you really don't have in mathematics.

Even given the CS definition, however, the idea of infinity (or infinities) seems to correspond more closely (in a semantic sense) to constants rather than variables. If I give you a certain instance of infinity (some kind of infinity, say, aleph-nought or something) that's what it is, and it can be encoded somehow (say, using its definition). Since this definition can be encoded as a sequence of bits, and this sequence isn't a placeholder for anything (it is what it is), then it seems to me that (depending on what is meant by infinity) that infinity is a constant, not a variable.

Unless you want to get into actual vs potential infinities, in which case I'm peace out, yo.

 

[Hurkyl]

In CS, it's a lot easier. It's a constant if it's a bit pattern. It's a variable if it's a reference to a memory location (which is inherently changeable). These concepts rely on a concept of "state" which you really don't have in mathematics.

I'm not really sure at what you're thinking, but this is fairly inaccurate if applied to programming language syntax.

For example, in C, even if we decide to insist that "const" is different from what you mean by constant (despite the standard specifying that certain const variables are compile-time constants), you still have things like:
. Macros (and their arguments) have nothing to do with the abstract machine model, let alone memory locations on actual computers
. String literals are generally put into memory locations, despite being constants

Of course (IMHO) it's saner to include what C calls "const" to be considered a constant -- the terms "constant" and "varaible" are defined by the formal language.



State is relatively easy to treat mathematically; you just make everything a function defined on some abstract "state space" (which is often just a set, although it might have other properties defined for it).

 

[Tac-Tics]

In mathematics, ..., I don't think there's much of a difference between the idea of "variable" and "constant".

Variables in mathematics never vary.

Terminology is loose in both fields, but in both mathematics and computer science, there is a pretty simple definition if you're working with a wide number of variable-having systems. The rules come from lambda calculus, but easily generalize to propositional logic, algebra, and set theory.

First, variables cannot exist in a vacuum. They must always be created by a special thing called a binding form (whose name I stole from Lisp). A binding form provides three things for the variable: its name, its scope, and its purpose.

Names of variables can be pretty much anything: x, y, z, the greek letters, or full identifiers like "sin" or "cosh". They are names and nothing more.

The scope of a variable is the expression in which the variable even exists. For example, in the expression $$50n + \Sigma_k^5 k^2$$, the variable k is created by the binding form $$\Sigma$$. It only exists inside the expression $$k^2$$. To say something like $$k + \Sigma_k^5 k^2$$ is nonsense, because k simply doesn't exist outside of the sigma which creates it.

The purpose of a variable depends on the type of binding form. I list a bunch of these in another post I link to below. But they include definition, function abstraction (the lambda of lambda calculus), universal and existential quantification, summations and integrals (the "dummy" variables of both), and a few others.

Consider the expression "x^2 + 1". What is x? We don't know. We can't actually see the binding form of x in the expression we are considering, we say that it is unbounded (relative to that expression).

It might be a simple number like 2. It might be a function parameter, such as in "f(x) = x^2 + 1". It might be a dummy parameter in an integral, such as $$\int x^2 + 1 dx$$. If we can see the binding form in the expression, we say that x is bounded (relative to the expression in question).

The boundedness of variables isn't so important in mathematics, but it is absolutely critical to know for doing functional programming in computer science. The interesting thing is that there is a direct correspondence between constants in mathematics and unbounded variables in computer science. An unbounded variable IS a constant.

One implication of this is that "constantness" of a variable depends on which expression you're looking at. "Pi" by itself isn't a constant. It's not 3.14 inherently in the fabric of space and time. You have to wait until the author says "let pi be the ratio of a circumference of a circle to its diameter" and only then does pi take on a meaning. And author could just as easily say "let pi be the function which maps ordered pairs to their first coordinate" or "let pi be the function which maps integers to the count of lesser primes".

Another application is a function of two parameters, such as in multivariable calculus, which undergoes "partial" application. That is, you have a function f(x, y), and a real number c. You can create a new function g_c(x) which is equal to f(x, c). In a sense, this is a way to turn parameters into constants. Not literally, of course, but you do substitute y, which is bounded by the definition of f with another variable c, which is bounded "farther out" by "let c be a real number".


See another post explaining this in a little more detail here:
at https://www.physicsforums.com/showthread.php?t=258803


Of course, when talking about infinities, you have to keep in mind that infinity is a name we give to many things. A few of them aren't even mathematical objects. Aleph null and the cardinality of the continuum would be constant values. The infinity in "lim x->infinity" isn't really...

 

[AUMathTutor]

Well, I was really thinking more in terms of attribute vs value, so that's more in line with software modeling than programming languages, but still.

Sure, state is easy to treat mathematically. I was just pointing out the obvious fact that in mathematics the ideas of assignment and side effects don't make sense. I don't want to get into it, since it's sort of sidetracking a little, but the whole point of functional languages is to be more like mathematics in this respect.

I stand by my analogy: variable : constant :: attribute : value. Again, this doesn't make a lot of sense mathematically, and I never made any representation that this was the case. And however you slice it, the idea of constant vs variable is an easy one to make in programming languages... constants don't really need or have identity (a 7 in one part of the code isn't a different 7 than one another place, even if both are put into memory in different locations). To give the Jay Leno explanation used in intro courses, variables are like buckets and constants like what you put into them.

None of this is meant to be very precise, by the way. Thanks for pointing out where I could have been clearer, though.

 

[AUMathTutor]

Wow, that was a very informative post, Tic Tacs. I knew about the whole Lambda Calculus thing in CS, but I had no idea that mainstream mathematics incorporated these ideas so fundamentally. I guess I assumed that since the LC was such a recent development (comparatively), there was probably an older distinction mathematicians used before that. In hindsight, I guess that's why Church formalized it in the first place, and before that it wasn't really much of an issue.

Really excellent post, Tic Tacs.

 

[Tac-Tics]

Wow, that was a very informative post, Tic Tacs. I knew about the whole Lambda Calculus thing in CS, but I had no idea that mainstream mathematics incorporated these ideas so fundamentally. I guess I assumed that since the LC was such a recent development (comparatively), there was probably an older distinction mathematicians used before that. In hindsight, I guess that's why Church formalized it in the first place, and before that it wasn't really much of an issue.

As rigorous as mathematics can be, most mathematicians are actually pretty sloppy about it. Calculus would be damned to hell for its absolute abuse of notation if it weren't for the fact the entire world runs on the thing. It wasn't until the advent of computers that people really learned what rigor meant! In CS terms, "left as an exercise to the reader" simply means "some dude already coded it and it looks like it works pretty good".

Church's formalization also built tremendously on the work of other logicians and mathematicians in his time who were working furiously to axiomatizing mathematics. But it's still a very neat educational tool. I just wish I was useful for something outside of writing a Lisp.

Really excellent post, Tic Tacs.

I'm glad you liked it. Oh, but it's Tac-Tics. I'm clever like that ;-)

 

[AUMathTutor]

Oh, I get it. You see, it's funny because you took the name of a small sugar candy that freshens your breath and sort of turned it on its head, creating a sort of double entendre, if you will. Pun is the purest form of comedy. ;D

 

[Hurkyl]

The infinity in "lim x->infinity" isn't really...

Depending on the context. You're correct as it's usually taught in Calc I -- that's just a stop-gap until the student learns to use the extended real numbers (or the projective real numbers, or some other compactification depending on the application), in which case x approaches infinity just as it would any other point in a topological space, and limits such as $\lim_{x \rightarrow (\pi/2)^-} \tan x$ converges to +infinity.

 

[Georgepowell]

How to troll Mathematicians:

Talk about infinity.

 

[de_brook]

In mathematics, infinity is a symbol representing an extremely very large quantity compared to the variables you are working with such that the system cannot even comprehend. Thus, we could have different infinities for different systems. An infinity for a system A may be a finite number for a system B.

 

[AUMathTutor]

That sounds more like the physicist's idea of infinity to me.

 

[aaryan0077]

In mathematics, which is referentially transparent, I don't think there's much of a difference between the idea of "variable" and "constant". Perhaps the closest...
... you want to get into actual vs potential infinities, in which case I'm peace out, yo.

Sorry
I am not into CS

 

[aaryan0077]

Variables in mathematics never vary.

Then what else varies?

 

[aaryan0077]

Variables in mathematics never vary.

The scope of a variable is the expression in which the variable even exists. For example, in the expression $$50n + \Sigma_k^5 k^2$$, the variable k is created by the binding form $$\Sigma$$. It only exists inside the expression $$k^2$$. To say something like $$k + \Sigma_k^5 k^2$$ is nonsense, because k simply doesn't exist outside of the sigma which creates it.

Okay, but even to say $$50n + \Sigma_k^5 k^2$$ sounds stupid unless it is said that n $$\in$$ N (where N is set of natural no.), or unless one had presumed this that n $$\in$$ N

 

[aaryan0077]

The purpose of a variable depends on the type of binding form. I list a bunch of these in another post I link to below. But they include definition, function abstraction (the lambda of lambda calculus), universal and existential quantification, summations and integrals (the "dummy" variables of both), and a few others.

Consider the expression "x^2 + 1". What is x? We don't know. We can't actually see the binding form of x in the expression we are considering, we say that it is unbounded (relative to that expression).

It might be a simple number like 2. It might be a function parameter, such as in "f(x) = x^2 + 1". It might be a dummy parameter in an integral, such as $$\int x^2 + 1 dx$$. If we can see the binding form in the expression, we say that x is bounded (relative to the expression in question).

so we have "x^2 + 1". What is x? We don't know. Alright.
Say, now we have x - y = 0. ( sorry for not using much latex, I don't know much about it)
so as you said there are three things.
1. Name - x, y.
2. scope .

For scope you said "The scope of a variable is the expression in which the variable even exists." and it is about the binding form of the expression.
Here binding form may be "= 0" , or may not be (I don't know as this also include a little idea of CS), if it is not, then your point ends here.

If it is; then, where is the scope?
x $$\in$$ R, no we can also have x $$\in$$ C, and so the y.
what if x represent sin$$\phi$$, and as x = y, so do y.
So scope is not defined.

3. Purpose
this will come when scope is clear, but I don't think scope has a clear picture.

 

[aaryan0077]

See another post explaining this in a little more detail here:
at https://www.physicsforums.com/showthread.php?t=258803


Of course, when talking about infinities, you have to keep in mind that infinity is a name we give to many things. A few of them aren't even mathematical objects. Aleph null and the cardinality of the continuum would be constant values. The infinity in "lim x->infinity" isn't really...

I checked that link, you told about binding form in discussion, I think HallsofIvy gave quite logical answer both time.

For the last thing said "when talking about infinities...", this is the point I think I have to agree with you.

 

[aaryan0077]

Well, I was really thinking more in terms of attribute ...

...very precise, by the way. Thanks for pointing out where I could have been clearer, though.

CS Again?

 

[aaryan0077]

How to troll Mathematicians:

Talk about infinity.

Yeah! This is a fact. But what's your point.

 

[aaryan0077]

In mathematics, infinity is a symbol representing an extremely very large quantity compared to the variables

Compared to Variables?

 

[aaryan0077]

Thus, we could have different infinities for different systems. An infinity for a system A may be a finite number for a system B.

Also this may be possible that the infinites of two different system are "not comparable".
Or not even the finite one are comparable

 

[Georgepowell]

Yeah! This is a fact. But what's your point.

Sorry if that was offensive, I wasn't suggesting that you are a troll. Look at how many replies you have though! Only a thread on infinity could cause that.

 

[slider142]

Sorry if that was offensive, I wasn't suggesting that you are a troll. Look at how many replies you have though! Only a thread on infinity could cause that.

$0.\bar{9} = 1$?? is a crackpot magnet as well. :D