Which is the largest of these rational numbers?
If you say 5.213454, then you're still thinking in integer terms. If you say 1.5075643002 is the largest within a rational number frame of reference, then you thinking as I am right now.
Is Pi an irrational number, or have we just not effectively defined it yet? With the Halting Problem in mind, can we ever determine whether Pi is rational or irrational? On that basis, is there such a thing as irrational real numbers, or are these just rational numbers beyond the reach of our precision? Unlike most, this last question will be answered within this article.
I was taught that rational numbers were those that can be expressed as one integer divided by another - but I'm reconsidering that as numbers that lie between fixed bounds, or more intuitively, "parts of a whole".
When we deal with rational numbers in everyday life, we're not really dealing with rational numbers as I conceptualize them within this article. We're just dealing with clumps of integers.
To enumerate things, there needs to be a frame of reference. If you say "six" I'll ask 'six what?', and if you say "a quarter", I'll ask 'a quarter of what?'
Usually, the answer is something quite arbitrary, such as "king's toenails". Want less arbitrary? How about "the length of a certain platinum-iridium bar in Paris stored at a particular temperature" - feel better now?
Your cutting machine won't explode in a cloud of quantum dust if you set it to a fraction of a millimeter; within the machine's "integers", are just more smaller "integers". If you think you understand anything about rational numbers from contexts like these, you're kidding yourself, in my humble opinion.
To put teeth into integers, they have to enumerate something fundamental, something atomic. By atomic, we used to say "something that cannot be divided further"; today we might say "something that cannot be divided further without applying a state change, or dropping down into a deeper level of abstraction".
Ah - now we're getting somewhere! Levels of abstraction, total information content, dimensions of an array... think of chemistry as a level of abstraction "above" nuclear physics, or the computer's digital level of abstraction as "above" that of analog volts, nanometers and nanoseconds.
If layers of abstraction are properly nested (are they?), then each may appear to be a closed single atom from "above", rational numbers from "within", and an infinite series of integers from "below". Or not - toss that around your skull for a while, and see what you conclude.
Within a closed system, there may be a large but finite number of atomic values (or integers, in the non-ordered sense), being the total information content of that system. If rational numbers are used to describe entities within the system, they are by necessity defined as values between 0 and 1, where 1 = "the system". In this sense, 7.45 is not a rational number, but might be considered as an offset 7 from outside the system, and .45 within the system.
You might consider "size" as solidity of existence, i.e. the precision (space) or certainty (time, or probability) at which an entity is defined. If you can define it right down to specifying exactly which "atom" it is, you have reached the maximum information for the closed system. So 0.45765432 is a "larger" number than 0.5, in terms of this closed-system logic.
You can consider integers vs. rational numbers as being defined by whether you are specifying (or resolving) things in a closed system (rational numbers, as described within this article) or ordering things in an open system (integers).
What closes an integer system is your confidence in the order of the integers you enumerate. What closes a rational system is whether you can "see" right down to the underlying "atoms".
Information and energy
Can one specify an entity within a closed system with a precision so high that it is absolute, within the context of that system?
We may generalize Pauli's exclusion principle to state that no two entities may be identical in all respects (or rather, that if they were, they would define the same entity).
Then there's Heisenberg's uncertainty principle, that predicts an inability to determine all information about an entity, without instantly invalidating that information. Instantly? For a zero value of time to exist, implies an "atom" of time that zero time is devoid of... otherwise that "zero" is just a probability smudge near one end of some unsigned axis (or an arbitrary "mid-"point of a signed axis).
Can you fix (say) an electron so that its state is identical for a certain period of time after it is observed? How much energy is required to do that? Intuitively, I see a relationship between specificity, i.e. the precision or certainty to which an entity is defined, and the energy required to maintain that state.
If "things fall apart", then why? Where does the automatic blurring of information come from? Why does it take more work to create a piece of metal that is 2.6578135g in mass than one manufactured to 2.65g with a tolerance of 0.005g?
One answer may be; from deeper abstraction layers nested within what the current abstraction layer sees as being integer, or "atomic". The nuclear climate may affect where an electron currently "is" and how likely it is to change energy state; what appears to be a static chemical equilibrium could "spontaneously" change, just as what appears to be reliable digital processing can be corrupted by analog voltage changes that exceed the trigger points that define the digital layer of abstraction.
In this sense, the arrangement of sub-nuclear entities may define whether something is a neutron or a proton with an electron somewhere out there; the difference is profound for the chemical layer of abstraction above.
To freeze a state within a given layer of abstraction, may require mastery over deeper levels of abstraction that may "randomize" it.
What does it mean, to exist? One can sense this as the application of specificity, or a stipulation of information that defines what then exists. Our perspective is that mass really exists, and just happens to be massive in terms of the energy (information?) contained within it.
There's a sense of energy-information conservation in reactions such as matter and antimatter annihilating their mass and producing a large amount of energy. How much energy? Does that imply the magnitude of information that defined the masses, or mass and anti-mass? Do you like your integers signed or unsigned? Is the difference merely a matter of externalizing one piece of information as the "sign bit"? What do things look like if you externalize two bits in that way?
Like most of my head-spin articles, this one leaves you hanging at this point. No tidy summary of what I "told" you, as I have no certainty on any of this; think of this article as a question (or RFC, if you like), not a statement.