What is the smallest possible positive real number?

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Discussion Overview

The discussion revolves around the concept of the smallest possible positive real number, exploring mathematical and physical implications. Participants engage in a debate about the nature of numbers, particularly focusing on the role of infinity, the relationship between physical reality and mathematical abstraction, and the constraints of symbol systems in representing physical phenomena.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Exploratory

Main Points Raised

  • One participant conjectures that the smallest positive real number could be 1/aleph-0, suggesting a finite interpretation of aleph-0 in a physical context.
  • Another participant asserts that there is no smallest real number due to the infinite nature of real numbers, challenging the initial conjecture.
  • Concerns are raised about the implications of using aleph-0 as a finite number, with a suggestion that it indicates a misunderstanding of its mathematical properties.
  • Questions are posed regarding the formal rules for constructing numbers and the relationship between physical constraints and mathematical abstraction.
  • A participant expresses uncertainty about the relevance of irrational numbers and the implications of counting in a universe with a finite number of atoms.
  • There is a discussion about the distinction between integer functions and real functions, emphasizing the different domains of these mathematical objects.
  • One participant reflects on their limited understanding of infinity and expresses a desire to explore the relationship between integers and real numbers in a physical context.

Areas of Agreement / Disagreement

Participants express differing views on the existence of a smallest positive real number, with some supporting the conjecture and others rejecting it. The discussion remains unresolved, with multiple competing perspectives on the relationship between mathematics and physical reality.

Contextual Notes

Participants highlight limitations in understanding the implications of infinity and the nature of mathematical objects, indicating that the discussion is constrained by differing interpretations of mathematical concepts and physical realities.

Jack Martinelli
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Just a conjecture...

If any realizable real is built from the integers, then the single smallest positive real is: 1/aleph-0.

See http://mathworld.wolfram.com/Aleph-0.html. In a physical context aleph-0 is finite.

Note that there are no positive integers that _exist_ (are realizable) if their value exceeds the number of atomic (in the Democretic sense) particles in the universe.
 
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Eh? You almost got to surreal numbers etc, but not quite. There is no smallest real number, what with their being an infinite number of unbounded negative ones.

Can I ask you to stop thinking about things unmathematically like this before you get a reputation for being a crank, troll, or something worse?
 
Too late. I've already got a crank reputation. Only because I'm convinced that any symbol system intended to be used in representing a physical system is constrained by the _physical_ properties -- not the symbol system. Of course -- as I think you maybe implying -- understanding the purely abstract mechanics is fundamental to any application.

see: ftp://ftp.princeton.edu/pub/harnad/Harnad/HTML/harnad90.sgproblem.html[/URL]

for a an idea of how a symbol system relates to reality.
 
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Originally posted by Jack Martinelli
Too late. I've already got a crank reputation. Only because I'm convinced that any symbol system intended to be used in representing a physical system is constrained by the _physical_ properties -- not the symbol system. Of course -- as I think you maybe implying -- understanding the purely abstract mechanics is fundamental to any application.

see: ftp://ftp.princeton.edu/pub/harnad/Harnad/HTML/harnad90.sgproblem.html[/URL]

for a an idea of how a symbol system relates to reality. [/B][/QUOTE]

Then don't post in a mathematics forum where you are wrong, by construction. Of course if you find something that is physically a number...
 
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Is it against the rules to post ideas (speculations?) that relate how the physical constrains the abstract? (in a physics forum where people discuss this kind of thing?) Note the hierarchy : physics->mathematics->number theory.

Ok. Help me out. What are the formal rules for constructing the numbers so that they are constrained?

Would you say:

1. I don't like introducing fuzzy ideas into set theory. It just makes a mess.

2. If you can't make it rigorous and formal, it's not math. Take it someplace else.

3. Numbers are not constrained by reality. They have "suficient" abstraction so that, in a physical context, they can be used to represent _any_ verifiable quantity.

Q1: is this a theorem? axiom? proof? (what's "suficient" mean?)

Q2: How do you make "verifiable" rigourous?

Q3: at what point of a mathematical construction should constraints enter? What are the rules here?

Q3: what fun is it to revisit/discuss textbook material? (Booorrrrinnngggg!) (why are you here?)

Q4: Is it possible to construct a "physical ontology" (set of axioms) where "physics proofs" are possible? Are the axiom, axioms?

Q5: Have we(mankind) forumualted the "correct" symbol system for representing physical systems? (how do you do this?)

Q6, Q6, Q7... Qn

(you obviously didn't read the paper I referred to in my prior post. And if you're not going to, then why discuss this any further. I.e., if you're not going to participate in the discussion _I started_, you probably shouldn't.)

gtg...
 
I would suggest looking at the way it works round here that you are posting in the wrong place. Try posting in Theory Development.

There are an infinite set of numbers, they satisfy the axiom of infinity in ZF(C), they form an inductive set, er, any other statements about that?

Whether or not the natural numbers and the reals are what *you* want in physics is up to *you*, but to say that aleph-0 is finite indicates a lack of understanding that I cannot comprehend, seeing as aleph-0 is not an object in a 'phyiscal context' as you put it.If you want to do maths with a 'practicall largest' number, don't call it aleph-0 would be my advice, cos that ain't what it's there for.
 
Jack, just out of curiosity--

I don't necessarily think you are onto anything that you are going to be able to do anything useful with. But still, what was your thinking process in ruling out 1/Aleph-1 as being even smaller than 1/Aleph-0?
 
As to whether I'm on to something or not, have you considered any consequences to my conjecture (math, after all, is an a-postori art/science).

My thought process... I wasn't interested in eliminating any other number - only in preventing or identifying theoretical errors. If, for example, some theory predicts some number that is greater than nelph-0 (my maximum integer) or less than 1/nelph-0 (and given some theoretical context) the theory might be in doubt. -- lots cheaper than building a super-collider.

What I know & understand about aleph-0 or aleph-1 or serious number theory couldn't fill a teaspon. My grasp of infinity-1 is no better than infinity-10^x. I do have an intuitive grasp of convergence & divergence, L'Hopital's rule etc.. I really have no intuition of an absolute infinity. You might as well be talking about sphinxes and unicorns. (what units should you use?)

Well, in my thinking, there was on other thing. I'm interested in a general limit where an integer function is distinguishable and/or indistinguishable from a real fuction. And it seems to me that, the way the reals are defined (the way I understand it) there is no way to do this. If I've got this right... there are arbitrarily close rational & irrational fractions where there are an infinite number of reals between them. In a physical context, these numbers bother me. (do they have a name?) The reason why is that in measurement you always either count events or objects or find a quotient :

(Measurement magnitude) = (target)/(reference)

And in finding this magnitude you are always dealing with counts of reference objects -- even though they are of finer & finer granularity.

If I seem ignorant it is only because I am. I am a student. For me, curiousity, thoughts, mistakes & risks preceed learning. It is my belief that teaching, nurtures, inspires, and encourages first and instructs last. Hopefully I get some of that here.
 
Ok, here's a serious problem if you can only count to as many atoms as there are in the universe: there is then no way to count how many ways there are of arranging those atoms. There is no way of saying what happens if by spontaneausly another atom manifests itself.

Irratonal numbers are by definition not fractions.

Most of your questions don't make much sense - what do you mean by integer function? If you have a function defined on the integers, it is therefore by definition different from every function that is defined on the real numbers because they have different domains (the input set).The numbers that bother you are called the real numbers.

You are also assuming that all measurements are discrete. Well, they maybe - that is quantization for you, but that is not the issue. Mathematical objects such as the real numbers are more than just the object you use to measure velocity, say. They are cauchy sequences of rational numbers, the rational numbers are the localization of the integers which is the.. got it? They are useful for physics but they "exist independently" of your physical interpretation. If you have a philosophical problem with that then you have a philosophical problem, the mathematics is still there.

Cardinals, infinite ones don't have units, ok, mathematics does not have to be physical.
 
  • #10
originally posted by Jack Martinelli Only because I'm convinced that any symbol system intended to be used in representing a physical system is constrained by the _physical_ properties -- not the symbol system.

But you titled this "The smallest real number", then referred to
"relizable" numbers (I have no idea what that means) then went back to "smallest positive real" again.

Certainly you are free to posit any number system you want- just don't claim that it is the real number system until you can prove that.

You also say "In a physical context aleph-0 is finite". On the contrary: aleph-0 is not defined in a "physical context". I might point out that even physicist prefer to work with "continuous" models when describing the thermodynamics of huge numbers of molecules- it's easier- but my real point is that "physical context" has no place in a mathematical discussion.
 
  • #11
Originally posted by HallsofIvy
But you titled this "The smallest real number", then referred to
"relizable" numbers (I have no idea what that means)


Maybe you can explain this to me:

http://www.mth.uea.ac.uk/~h720/research/files/integersequences.html

I didn't really understand it, but came away with the idea that a number i can be expressed as a digit sequence, but in the general case, an arbitrary i may not be expressed (realized) at all.

Originally posted by HallsofIvy
Certainly you are free to posit any number system you want- just don't claim that it is the real number system until you can prove that.

How do you prove a definition? -- oh wait, If I can construct a number system that compares in every way with the reals, I guess that would constitute a proof?

Originally posted by HallsofIvy
You also say "In a physical context aleph-0 is finite". On the contrary: aleph-0 is not defined in a "physical context".

I understand that now & agree.

Originally posted by HallsofIvy
I might point out that even physicist prefer to work with "continuous" models when describing the thermodynamics of huge numbers of molecules- it's easier- but my real point is that "physical context" has no place in a mathematical discussion.
Then...
Q1: Is math meaningful? (please read Harnad paper in 2nd post)
Q2: Should I just accept whatever you tell me?
Q3: Why does math work?
Q4: Can you proove that red exists? (Jack shoots from the hip.)
Q5: Is "math" what mathematicians do, or does it occur naturally?
 
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  • #12
And you're on to philosophy again. Maths is the study of mathematical objects (What is English Literature the study of, who gets to say what counts as literature, or English). As such Q4 isn't mathematical and doesn't have an answer here. Roughly one might say maths starts from reasonable hyopetheses that we can agree by common consent are 'mathematical' and deduce, induce, infer and so on using logical rules (if something is not false, it is true, though there are those who would disagree with that).

Mathematics works, internally, by design - if something didn't work it wouldn't be valid maths.

Is anything meaningful? You shouldn't just accept what we say. We all make mistakes, but if a reasonable number of reasonable people tell you the same thing then it will be "correct", especially if they offer a proof of what they say.
 
  • #13
"Q1: Is math meaningful?" (Yes, I've read Hadamard's paper)
In my opinion mathematics gives us a large number of "templates" which may or may not be useful in a certain application. Since mathematical theories work with exact values while physics (or other applications) can only be approximate, no mathematical theory ever works perfectly for a given application, and there can be several different mathematical models that work well for exactly the same application.

"Q2: Should I just accept whatever you tell me?"
Heavens no! Anyone who has seen my responses on this board can tell you that!

"Q3: Why does math work?"
See answer one.

"Q4: Can you proove that red exists? "
I can prove experimentally that light of a certain wave length that people have agreed to call "red" exists. Would you accept that.

"Q5: Is "math" what mathematicians do, or does it occur naturally?"
People who are not mathematicians can recognize "mathematical" models that work more or less well in a given situation. In that sense mathematics "occurs naturally" but, again, no mathematical model ever works perfectly for any given application.
 
  • #14
HallsofIvy said:
"Q1: Is math meaningful?" (Yes, I've read Hadamard's paper)
In my opinion mathematics gives us a large number of "templates" which may or may not be useful in a certain application. Since mathematical theories work with exact values while physics (or other applications) can only be approximate, no mathematical theory ever works perfectly for a given application, and there can be several different mathematical models that work well for exactly the same application.

Well, here is what makes sense to me. The universe is not an approximation and doesn't make mistakes. I.e., It is exactly what it is and does exactly what it does. Compare this to "The universe is only approximately correct." Or, "The universe is probably what it is."

As a reference for "what works" there is none better than all of existence. It seems to me that a symbol system is guaranteed to work if you construct a comprehensive semantic alignment between the physical and the abstract.

Do I know how to do this? Nope. But its fun to try.

My sugestion is that Math works because the universe does.

"Q4: Can you proove that red exists? "
I can prove experimentally that light of a certain wave length that people have agreed to call "red" exists. Would you accept that.

(that was a trick question...)

A blind man, who is only blind, could accept your "proof", and even find it useful, but would not find your "proof" meaningful.

(BTW, Here's another article you might find interesting...

http://www.utm.edu/research/iep/c/chineser.htm)

A sense gives us a reference for other senses of the same type. You don't have to provide a proof. It is self evident and more convincing than any proof.

"Q5: Is "math" what mathematicians do, or does it occur naturally?"
People who are not mathematicians can recognize "mathematical" models that work more or less well in a given situation. In that sense mathematics "occurs naturally" but, again, no mathematical model ever works perfectly for any given application.
 
  • #15
I have failed! I was trying to do something that is kind of number theory related that has _nothing_ to do with physics & is useless. Yes. Useless. But instead I came up with this...

see http://www.martinelli.org/misc

The whole thing is not exactly journal quality. But I'm not particularly interested in taking to that level (that would be another 80% of the effort). If, however, anyone out there is interested in going there, I'd be happy to help clean up the code.

Here is an excerpt ...

============================

Looking at long integer sequences for patterns can be very difficult. The following presents a method of constructing graphs of integer sequences in a way that makes it easy to graphically examine very large sequences. The method also has some novel applications in computer graphics.

========================

The page gives a few details of the math & includes an ActiveX control to display various examples. I didn't take the effort to add a secure signature (>$200 !) so, you'll have to trust me that my control is harmless. If you want the source, just let me know. In case you were wondering... I didn't use Java because it's too slow for me.

The controls on the curve explorer page are pretty obvious to me, but probably not to everyone else, so if you have trouble with the thing post your question here.
 
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  • #16
Nonstandard Analysis

This thread touches upon nonstandard analysis, see
http://mathworld.wolfram.com/NonstandardAnalysis.html
http://mathworld.wolfram.com/HyperrealNumber.html
As to a smallest positive real number, there isn't one,
infinitesimal numbers are not real numbers. My limited
understanding is that even if one extends the real numbers
to include infinitesimals, there is not a smallest positive
number. The subject of infinite and infinitesimal numbers
gets a bit mind boggling. For example, there are so many
levels of infinite cardinals that you can't form a set of them,
it breaks set theory. A proof I saw on the web goes like this:
suppose there was such a set of all infinite cardinals, consider
representative sets corresponding to all these infinities, now
consider the union of all these sets, and then consider the
power set of the union. The cardinality of the power set is
greater than that the union and thus is greater than all the
cardinals in the supposed set of all infinite cardinals, which
is a contradiction, so there is no set of all infinite cardinals.
So how many infinities are there? Some sort of 'inaccessible'
cardinal, and that's where my mind goes tilt.
 
  • #17
Thanks Gerald!

I was sure my fuzzy thinking was at least pointed in the right direction. I'll look into Nonstandard analysis somemore.
 

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