Is infinity a constant or a variable ?

[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

 

[slider142]

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.

What do you mean by "the system cannot...comprehend" mathematically? Please elaborate on your post.

 

[Georgepowell]

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

Thank god religious threads are banned

 

[de_brook]

What do you mean by "the system cannot...comprehend" mathematically? Please elaborate on your post.

I mean't that there are systems in which we are restricted to work with cetain variables. They could be considered as too large for our infinity or too small for a zero. For instance if you are working with a system in which most of what you encounter are infinitesimal values such as nano values and you encounter kilo-value once, you notice that there is a jump. This, we can say kilo-value is an infinity when compared to the sysytem we are working with

 

[slider142]

I mean't that there are systems in which we are restricted to work with cetain variables. They could be considered as too large for our infinity or too small for a zero. For instance if you are working with a system in which most of what you encounter are infinitesimal values such as nano values and you encounter kilo-value once, you notice that there is a jump. This, we can say kilo-value is an infinity when compared to the sysytem we are working with

Ah, that is definitely a physical/engineering infinity, not a mathematical one. Just checking. :)

 

[de_brook]

Ah, that is definitely a physical/engineering infinity, not a mathematical one. Just checking. :)

Alright, what do you think it is? cos i know you quite agree that infinity is a symbol and it does not have a fixed value it just tells us about something very large

 

[slider142]

There are many examples of infinities that are considered constants listed in this thread that have nothing to do with variables or allowing some variable to vary. They have precise definitions and algebraic properties and can be considered "fixed" in their respective systems.

 

[HallsofIvy]

Alright, what do you think it is? cos i know you quite agree that infinity is a symbol and it does not have a fixed value it just tells us about something very large

Again, that is not mathematics- it may well be some application of mathematics, but mathematics does NOT use "infinity" to mean "very large".

 

[CRGreathouse]

I'll chime in with the other two here: each of the various infinite numbers have precise, fixed meanings in mathematics. It's not just a synonym for "big".

 

[AUMathTutor]

Right. de brooks's definition is the Physics definition... I've heard it several times before.

 

[deiki]

My explanation is quite informal, but I hope it helps.

Well it's not a constant, but roughly speaking, it's just a positive number when you approach given conditions ( usually the Weierstrass limit conditions ) that will grow bigger and bigger ... and bigger ... and bigger ... and bigger ... and bigger ...

Now if you consider the expression ∞+2, when ∞ will grow bigger and bigger as you approach given conditions, ∞+2 under the same conditions will grow bigger and bigger ... and bigger ... and bigger ... and bigger ... and bigger ...

So ∞+2 = ∞

 

[CRGreathouse]

deiki: That's a fine explanation for physicists, but it won't cut it in math. Infinity isn't growing, it just it. Further, just to muddy the waters, \omega+2>\omega.

 

[Moo Of Doom]

Further, just to muddy the waters, \omega+2>\omega.

To make it clear, here \omega is the first infinite ordinal.
To muddy the waters further, we also have \omega + 2 > 2 + \omega = \omega.

 

[aaryan0077]

I mean't that there are systems in which we are restricted to work with cetain variables. They could be considered as too large for our infinity or too small for a zero.

What if they are not comparable?

 

[aaryan0077]

For instance if you are working with a system in which most of what you encounter are infinitesimal values such as nano values and you encounter kilo-value once, you notice that there is a jump. This, we can say kilo-value is an infinity when compared to the sysytem we are working with

NO!
We never say that kilo-value here is "infinity" when compared to those Infinitesimal value.
We still say that it's too large, but we never say it is infinite compared to those Infinitesimal values.
Did you ever heard a physicist working with Schrödinger eq. and dealing in Planck numbers saying that universe started infinite years ago, NO! He will still say universe started (nearly) 13.7 billion yrs ago.

 

[aaryan0077]

Alright, what do you think it is? cos i know you quite agree that infinity is a symbol and it does not have a fixed value it just tells us about something very large

I think slider and HallsofIvy gave quite appropriate answer to your question.

 

[aaryan0077]

that will grow bigger and bigger ... and bigger ... and bigger ... and bigger ... and bigger ...

So, Are you treating it as variable?
If it is so, then why aint it is waning rather than just waxing?

 

[aaryan0077]

it's just a positive number

JUST A POSITIVE NUMBER
I think you should read the whole thread before; then post something, because if you have read all the posts, you won't have said just a positive no.

 

[aaryan0077]

So ∞+2 = ∞

That's what I said in OP, what do you mean by repeating this?

 

[aaryan0077]

Okay everyone, I discussed it somewhere else, and there's one point I found quite interesting,
I'd like to share that one.

"It is really nothing and so something immeasurable .So it is none other than Infinity itself !It is everything and nothing too.
It is immeasurable and , boundless.It is nothing ; but something ; a being that is a non-being ! That alone is infinity.
For instance , space is emptiness. It can be filled or can be vacuum.and so something can be put into it .Something can extend into it .If there is NO SPACE , NO VACUUM for something to be put into, a thing can not exist .All things need space to exist in.But the space that things need to exist is actuallyemptinesss , vacuum, NOTHING .But if that "NOTHINGNESS" is not there , where will anything exist in this universe ?So all thatexitst , needs this space - this nothingness to exist in.But this nothingness does not existbecausee of anything else .It needs nothing to exist in! It depends on nothing for its existence ; but everything else in thisUniversee or anywhere , say in hell or heaven, needs this nothingness to exist in.Even the Gods and all Avatars needed this space , the emptiness to exist .but for this, even the Gods and their Avatars do not exist!"