∞ is a free variable

What is the basis for reason? And mathematics?

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Skepdick
Posts: 14510
Joined: Fri Jun 14, 2019 11:16 am

∞ is a free variable

Post by Skepdick »

In Programming Language Theory there exists the concept/distinction between bound and unbound variables.
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. A free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not a parameter of this or any container expression. Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. The idea is related to a placeholder (a symbol that will later be replaced by some value), or a wildcard character that stands for an unspecified symbol.
In colloquial English this could trivially be understood as the distinction between defined and undefined terms.

The symbol "x" is said to be unbound by default, until another symbol; or an expression is bound to it. It represents a free variable.
Unless, and until bound to something free variables represent an unbounded entity. Something without limits, lacking value or quantification.

But that is exactly what the symbol "∞" represents!

If two symbols represent the same concept (an unbound quantity; or value) it follows by the identity axiom that the two symbols are synonymous and interchangeable.

X is identical to ∞ (x ≡ ∞)

∞ to Mathematicians, is like Truth to Philosophers; or like God to theists.

It means whatever you want it to mean.
“When I use a word,’ Humpty Dumpty said in rather a scornful tone, ‘it means just what I choose it to mean — neither more nor less.’

’The question is,’ said Alice, ‘whether you can make words mean so many different things.’

’The question is,’ said Humpty Dumpty, ‘which is to be master — that’s all.”
Impenitent
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Joined: Wed Feb 10, 2010 2:04 pm

Re: ∞ is a free variable

Post by Impenitent »

strings are more free?

if there are a finite number of memory spaces created (as huge a number as that is) within all the connected computing devices, how could a number that is infinite actually fit?

-Imp
Skepdick
Posts: 14510
Joined: Fri Jun 14, 2019 11:16 am

Re: ∞ is a free variable

Post by Skepdick »

Impenitent wrote: Tue Apr 23, 2024 12:10 pm strings are more free?

if there are a finite number of memory spaces created (as huge a number as that is) within all the connected computing devices, how could a number that is infinite actually fit?

-Imp
Suppose there is a finite number of memory spaces.
Suppose there's a single address even.
Suppose we call this address "∞"

How much can you fit in it?

As much as you want, apparently.
Impenitent
Posts: 4374
Joined: Wed Feb 10, 2010 2:04 pm

Re: ∞ is a free variable

Post by Impenitent »

even if each bit of memory represented one number in the infinite chain, there are a finite number of bits

-Imp
Skepdick
Posts: 14510
Joined: Fri Jun 14, 2019 11:16 am

Re: ∞ is a free variable

Post by Skepdick »

Impenitent wrote: Tue Apr 23, 2024 8:55 pm even if each bit of memory represented one number in the infinite chain, there are a finite number of bits

-Imp
I only need 1 bit. It can represent anything. Say - all numbers.

If you don't like that representation, it's probably because you care about something other than representing stuff...

Coincidentally "the ability to represent anything" is one of the defining features of free variables. Like ∞.
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