What is the largest number that can fit in x units of space?

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Discussion Overview

The discussion revolves around the question of what is the largest number that can be represented within a limited space, where each digit and mathematical symbol occupies one unit of area. Participants explore various mathematical representations and concepts, including infinity, factorials, and notation systems, while debating the nature of numbers and mathematical definitions.

Discussion Character

  • Debate/contested
  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that infinity could be considered the largest number, while others argue that infinity is not a number.
  • 9^9^9 and 9! are proposed as potential candidates for the largest number that can be represented.
  • A participant mentions that taking a very small number and considering its reciprocal could yield a large value, indicating a search for the most efficient representation.
  • There is a discussion about the nature of infinity, with some asserting that it is an abstract concept and not a specific number, while others challenge this view by discussing mathematical points and the Planck length.
  • Participants debate the relationship between mathematical definitions and physical reality, questioning whether mathematics should conform to real-world concepts.
  • Some assert that mathematical ideas are defined by axioms and are not necessarily tied to reality, while others argue for a connection between mathematics and physical existence.

Areas of Agreement / Disagreement

Participants express differing views on the nature of infinity, the largest number representation, and the relationship between mathematics and reality. There is no consensus on these topics, and multiple competing perspectives remain throughout the discussion.

Contextual Notes

Participants reference various mathematical concepts and notations, including factorials and Cantorian set theory, without resolving the definitions or implications of these terms. The discussion also touches on the Planck length and its relevance to mathematical definitions, but no agreement is reached on its significance.

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The question: You have to write the biggest number you can with a limited space, in other words each digit (including mathematical symbols) has an area of 1 unit. What is the greatest number that can be written on an area of x units? (note the area can be rearanged, so 2^5 would be two units, and your "paper" can go in any direction)

note i haven't done the problem...
 
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infinity is not a number.
 
9^9^9 maybe.
 
possibly, and actually 9^9 is what i thought too, but there are other ways. for example taking a eally smail number, and then 1 over that number would be quite big. But what i was actually lookng for was a proof that some way is the most efficient way... thanks for your posts
 
Well, there's also:
9!...
which is pretty huge, and arrow notation:
http://mathworld.wolfram.com/ArrowNotation.html

You really need to specify things a bit better to come up with a solid answer.
 
Cyclovenom said:
infinity is not a number.
I'm not so sure about that.
Mathematically any given line segment is composed of an infinite number of points.
Given the existence of a plank length, it would seem to resolve to a specific number.
 
NoTime said:
I'm not so sure about that.
Mathematically any given line segment is composed of an infinite number of points.
Given the existence of a plank length, it would seem to resolve to a specific number.

Infinity is an abstract concept. Please explain how you can "resolve" a specific number for infinity.
 
NoTime said:
...
Mathematically any given line segment is composed of an infinite number of points.
Given the existence of a plank length, it would seem to resolve to a specific number.


A line segment is not a real object! Plank doesn't apply!

It sounds the same as "there is a limited amount of real numbers in the [0,1] interval"...
 
  • #10
NoTime said:
I'm not so sure about that.
Mathematically any given line segment is composed of an infinite number of points.
Given the existence of a plank length, it would seem to resolve to a specific number.

This makes no sense at all.

"Infinity" is simply not defined in the reals.
 
  • #11
Rogerio said:
A line segment is not a real object! Plank doesn't apply!
Is a line segment that I draw any less real than say a square or a triangle?
In other words -> Why does Plank not apply?

Rogerio said:
It sounds the same as "there is a limited amount of real numbers in the [0,1] interval"...
Infinity is a concept that was developed long before knowledge of QM.
Ma Nature seems to be saying that this is indeed the case.
Personally, I'm inclined to take Ma's word for it rather than the imagination of man.

I will note that Zeno's arrow unerringly hits the target in real life.
Does this mean that there comes a place where you can no longer half the distance?
 
  • #12
Mathematical ideas and concepts are defined by axioms and definitions -- formal statements in the language of mathematics. In particular they are not defined by "reality".


There is no "Planck length" for a mathematical line segment because it is not a logical consequence of the axioms.



As an aside, there is no evidence (experimental or theoretical) that there is a smallest unit of length in reality either; "Planck length" doesn't mean what you think it means.
 
  • #13
Hurkyl said:
Mathematical ideas and concepts are defined by axioms and definitions -- formal statements in the language of mathematics. In particular they are not defined by "reality".
I do know what an axiom is.
I also do not deny that they say exactly what you say they do.

Why should mathematics be exempt from reality?

Hurkyl said:
As an aside, there is no evidence (experimental or theoretical) that there is a smallest unit of length in reality either; "Planck length" doesn't mean what you think it means.
To my knowledge the "Planck length" was determined from experimental data.
It is also the root of QM.
Is it a discontinuity or simply a region within which a determination cannot be made seems (at least to me) more problematical.
I connect with the idea that you do not like what I have said.
OTOH I have seen a number of arguments against the existence of infinities.

Why should I take one side or the other?
 
  • #14
NoTime said:
OTOH I have seen a number of arguments against the existence of infinities.

Infinity is an abstract concept, so it does not have any concrete existence. Furthermore, infinity is not a particular number by definition. It's fine if you don't agree. Create a new word to label your concept, instead of arguing against an established definition.
 
  • #15
NoTime said:
I do know what an axiom is.
I also do not deny that they say exactly what you say they do.

Why should mathematics be exempt from reality?


To my knowledge the "Planck length" was determined from experimental data.
It is also the root of QM.
Is it a discontinuity or simply a region within which a determination cannot be made seems (at least to me) more problematical.
I connect with the idea that you do not like what I have said.
OTOH I have seen a number of arguments against the existence of infinities.

Why should I take one side or the other?

The reals are a well defined field and do not contain an element known as "infinity". That is how they are defined, and that is that.

However, "infinity" appears in Cantorian Set Theory, (surreal numbers) as the cardinality of certain sets.

Mathematics is simply a set of results obtained through an established logical method, based on a certain framework of axioms. There is no need for a mathematical structure to have any physical meaning.

And for your information, the Planck Length is not determined from experimental data. It is simply a length scale obtained by a suitable manipulation of fundamental physical constants (= \sqrt {hG/c^3} ~ or~about~10^{-35} m ). Presently, that is too small to "measure".
 
  • #16
Why should mathematics be exempt from reality?

Because mathematics does not deal with reality; it deals with the formal consequences of axioms.

An axiom is simply a logical statement; it gets the name "axiom" because of the way we use the statement.

Here are a few of the axioms of the real numbers:

a + (b + c) = (a + b) + c
a * b = b * a
a * (b + c) = a * b + a * c

There are about 10 in all, of varying complexity.


Now, it is entirely possible to define "infinity" to mean "7". If you did so, then infinity really would be a real number. However, no standard definition of infinity yields a real number. In fact, no definition of infinity yields an object that is any sort of familiar number! In particular, Infinite numbers (such as those in the Surreals, or in the Cardinals) are not called "infinity".



Any sort of connection between mathematics and "reality" falls under the purview of science. It is science that says real numbers have some sort of connection to the real world. If science determined that there was a fundamental length, then that would mean that science would no longer attempt to say that real numbers are lengths; it does not mean that the mathematical meaning of a real number should change.
 
  • #17
Hurkyl said:
Any sort of connection between mathematics and "reality" falls under the purview of science. It is science that says real numbers have some sort of connection to the real world. If science determined that there was a fundamental length, then that would mean that science would no longer attempt to say that real numbers are lengths; it does not mean that the mathematical meaning of a real number should change.
Nice statement :smile:
It addresses the heart of the matter for me.
To the degree that attempts are being made to define "reality" using mathematics, then what is the requirement for mathematics to conform to "reality"?

Gokul43201 said:
And for your information, the Planck Length is not determined from experimental data. It is simply a length scale obtained by a suitable manipulation of fundamental physical constants (= \sqrt {hG/c^3} ~ or~about~10^{-35} m ). Presently, that is too small to "measure".
The fundamental physical constants are called that because they can ONLY be determined by experiment.
Show me an equation that produces c. NOT a value for c, but c itself.

Take a simple well known case.
A bound electron has energy levels A and B.
What is the set of points in the interval AB?
I say that set is empty.
You can imagine that an electron can have any energy level between A and B.
It just is not true.
 
  • #18
wave said:
Infinity is an abstract concept, so it does not have any concrete existence. Furthermore, infinity is not a particular number by definition. It's fine if you don't agree. Create a new word to label your concept, instead of arguing against an established definition.
Good Point.

Ok. I will change my initial answer to

\pi

Nothing was said in the original question about "biggest" being a value.
The number of digits in Pi is claimed to be... nevermind
 
  • #19
To the degree that attempts are being made to define "reality" using mathematics, then what is the requirement for mathematics to conform to "reality"?

There is no requirement that mathematics conforms to reality. The general procedure is that you take experimental data and theoretical/philosophical reasoning to extract axioms. Those axioms then define a mathematical theory, and then the hypothesis is that this theory is what will describe reality.

For example, experiment led to Maxwell's equations, and from these, Einstein reasoned that the speed of light must be constant in all inertial reference frames. Einstein then took this as an axiom, which defined a new theory, then he went on to flesh out this theory and arrived at Special Relativity.
 
  • #20
NoTime said:
Show me an equation that produces c. NOT a value for c, but c itself.

Please explain what you mean. The speed of light in a vacuum, denoted by c, is a value. What else can c be, if not a value?
 
  • #21
Hurkyl said:
There is no requirement that mathematics conforms to reality.
Mathematics per se, then I agree. There is no need to conform there.

I seem to recall that one of the properties of an axiom is that it can, at least in theory, be proven wrong.
Do you disagree with this statement?
So I am not so sure about mathematics as applied to reality.

For example: Einstein's axiom could, in theory, be proven wrong.
Lots of people have tried! None have succeeded! None are likely to succeed!
But, if they did then SR would be wrong (or at least incomplete).

Could one of the problems with the variations of string theory simply be that some of the mathematical axioms employed are non physical?
 
  • #22
NoTime said:
The fundamental physical constants are called that because they can ONLY be determined by experiment.
Show me an equation that produces c. NOT a value for c, but c itself.

I'm sorry, I misunderstood what you meant by the "Planck Length is determined from experimental data". I thought you meant that the Planck Length itself was determined experimentally.

And yes, fundamental constants are determined experimentally or from other constants, which are themselves determined experimentally.
 
  • #23
wave said:
Please explain what you mean. The speed of light in a vacuum, denoted by c, is a value. What else can c be, if not a value?
If you had such a thing as an equation that completely defined the universe.
Then one solution to that equation would show that
a) There is light or EM if you prefer.
b) That light's speed is a constant.

As far as value is concerned then c=1 is quite useful.
 
  • #24
NoTime said:
Mathematics per se, then I agree. There is no need to conform there.

I seem to recall that one of the properties of an axiom is that it can, at least in theory, be proven wrong.
Do you disagree with this statement?

An axiom, by definition, can not be proved or disproved.

People have toiled and struggled with Euclid's parallel line axiom over centuries with no luck. And then a bunch of folks decided to replace it with an alternate axiom and ended up inventing alternative geometries, as a result.

The relationship with physical reality is irrelevant to the correctness of an axiom. An axiomatic system is incorrect if it leads to contradictions within its own framework.

http://mathworld.wolfram.com/Axiom.html
 
  • #25
NoTime said:
For example: Einstein's axiom could, in theory, be proven wrong.
Lots of people have tried! None have succeeded! None are likely to succeed!
But, if they did then SR would be wrong (or at least incomplete).

Could one of the problems with the variations of string theory simply be that some of the mathematical axioms employed are non physical?

You are confusing a mathematical axiom with an assumption involved in a physical theory.

If you are talking about the latter, then yes, such an assumption may prove to be incorrect. Newtonian mechanics was based upon the assumption that the Galilean Transform was correct. We now know that to be untrue.
 
  • #26
Gokul43201 said:
The relationship with physical reality is irrelevant to the correctness of an axiom. An axiomatic system is incorrect if it leads to contradictions within its own framework.
This is a much better statement than the one I made. :smile:

Gokul43201 said:
An axiom, by definition, can not be proved or disproved.
Errrrr. What part of "leads to contradictions within its own framework" doesn't mean proved wrong?
 
  • #27
Gokul43201 said:
You are confusing a mathematical axiom with an assumption involved in a physical theory.

If you are talking about the latter, then yes, such an assumption may prove to be incorrect. Newtonian mechanics was based upon the assumption that the Galilean Transform was correct. We now know that to be untrue.
Yes.

I can restate my question then.

If mathematics is used to do physics.
Is an axiom no longer an axiom, but a hypothesis?

I have never liked the word assumption :biggrin:
 
  • #28
1 = infinity

0.000(insert infinite amount of 0's here)000.1 to 1.0

Same applys to every other number or value so why even discuss it?
 
  • #29
heeeeey
not to sound like a party pooper, but can anyone actually prove how to show one number is the *largest* over a given "area"?
 
  • #30
1 = infinity

0.000(insert infinite amount of 0's here)000.1 to 1.0

Same applys to every other number or value so why even discuss it?

This makes no sense.
 

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