The Tree gives rational approximations to the GR. There is no infinite digit RL. How would you use infinite decimal digits to define the GR?

You can call RL anything you like, but that does not mean it is so under foundation theory.

This is just a scheme associated with ratios of Fibonacci numbers and the Euclidean algorithm and continued fractions. You can call the GR the "last" RL if you like. Perhaps there are those who would agree with you. Just not in the mathematical community.

How would you use infinite decimal digits to define the GR? — jgill

Wikipedia says that the decimal representation of the Golden Ratio is 1.618033988749894...
https://en.wikipedia.org/wiki/Golden_ratio

Do you disagree with this?

I don't think this works. In your link there is a proof of the fact that the Golden Ratio is irrational. As such, how are we going to describe it with the positive rationals? There is no "last" set of turns that can describe the GR, since there will always be more to describe. This akin to how Zeno's Achilles can never catch the tortoise.

Moreover, there are some infinite sets that have more elements than others and some infinite sets can be denser than others. A common example of this is that there are infinitely more irrational numbers than rational numbers. However, if we could fully describe one irrational with the rationals than it stands to reason that we should be able to do this with others through a different series of turns. However, that can't be the case given the aforementioned.

On a more philosophical note, I think there might be another problem. The Golden Ratio is a relationship that maintains between parts of a whole (e.g., segments of a pentagram, the ancestry of male bees, etc.) This is true even in the abstract sense. It is not an abstract object, but rather a property of abstract objects, ratios being necessarily relational. That is, it exists only as it is instantiated in other abstract entities, making it a sort of "second-order abstract" entity.

In the formalist interpretation of mathematics, where "an entity is what it does," the ratio is just one of the relationships that define an entity. In the Platonist/realist view, I'm not sure exactly what it is (same goes for computation).

The complete series of decimals or turns wouldn't be the ratio though, even if such a series was finitely possible. If symbols, human names for abstract objects, were those things, then writing down the unique descriptor "the first number that violates the Goldbach Conjecture," would be the same thing as "discovering" it, and should entitle us to some prize money and a place in the history of mathematics.

Now, there are definitely both philosophers and physicists who challenge the reality of infinities, in which case, perhaps such a "last sequence" can exist. That's a broader question though.

Wikipedia says that the decimal representation of the Golden Ratio is 1.618033988749894...
https://en.wikipedia.org/wiki/Golden_ratio

Do you disagree with this? — keystone

This is a numerical approximation to a geometric concept, like 3.14159... approximates another ratio, pi.

Since these expansions are non-ending they do not completely describe the mathematical entities they represent. On the other hand when these entities are used in a numerical sense the tail ends of these expansions are chopped off according to the degree one wishes to approximate an answer.



:up:

the Golden Ratio is irrational. As such, how are we going to describe it with the positive rationals? There is no "last" set of turns that can describe the GR, since there will always be more to describe. — Count Timothy von Icarus

Do you think a Cauchy sequence of positive rationals can be used to describe the Golden Ratio? If so, whats the fundamental difference here?

However, if we could fully describe one irrational with the rationals than it stands to reason that we should be able to do this with others through a different series of turns. However, that can't be the case given the aforementioned. — Count Timothy von Icarus

On the Stern-Brocot tree, might irrationals be all the infinite strings which do not end in R_repeated or L_repeated?

It is not an abstract object, but rather a property of abstract objects, ratios being necessarily relational. — Count Timothy von Icarus

Do you think some irrational numbers have conditional existence while others do not? Or are you making a claim about numbers in general requiring abstract objects to exist?

The complete series of decimals or turns wouldn't be the ratio though, even if such a series was finitely possible. — Count Timothy von Icarus

Am I understanding correctly that you believe there is no decimal representation of the Golden Ratio? If that's the case, do you believe there are 2 solutions to the equation x^2-x-1=0? Are you saying something subtle here, such as there are 2 solutions but we can't represent them in decimal form?

Since these expansions are non-ending they do not completely describe the mathematical entities they represent. — jgill

How do you finitely and completely describe these mathematical entities (irrational numbers)?

How do you finitely and completely describe these mathematical entities (irrational numbers)? — keystone

Pi is the ratio of circumference to diameter of a circle. The Golden Ratio can be defined as the ratio of a particular line segment to another - you can look it up on Wikipedia. Other irrationals, have at it.

I had never heard of the Stern-Brocot tree before you brought it up. But my knowledge of number theory is poor.

Pi is the ratio of circumference to diameter of a circle. The Golden Ratio can be defined as the ratio of a particular line segment to another - you can look it up on Wikipedia. Other irrationals, have at it. — jgill

Pi and the golden ratio are special in that sense. You can't do that with almost all other irrational numbers (unless you say that one of the line segments is the unit line, in which case irrational number, x, is defined as the ratio x/1....not particularly satisfying).

Since these expansions are non-ending they do not completely describe the mathematical entities they represent. — jgill

Almost all other irrational numbers require non-ending descriptions, such as Cauchy sequences. Do you take the view that Cauchy sequences cannot completely define an irrational number? Or do you take the view that the vast majority of irrational numbers cannot be completely described?

Do you think a Cauchy sequence of positive rationals can be used to describe the Golden Ratio? If so, whats the fundamental difference here?

I think both can be used to describe the Golden Ratio, just as we can use the English language to describe it. They can describe the ratio to arbitrary precision.

That said, because the irrationals require an infinite amount of information to encode, they would require an infinite series of rationals. However, infinity is not a rational number because it is undefined as to its status as an integer, thus I don't think an infinite series can encode an irrational number while remaining itself rational (see also the points on Cantor above re: the irrationals having more elements and density on a number line).

On the Stern-Brocot tree, might irrationals be all the infinite strings which do not end in R_repeated or L_repeated?

It seems the same problem mentioned above might hold here since we have another infinite entity. In this case, we are using an infinite part of an infinite tree, but I don't know if that changes things.

Do you think some irrational numbers have conditional existence while others do not?

That's a tough question. I was referring specifically to ratios being conditional on the existence of something else. A ratio is necessarily a relationship and any relationship between two things is contingent on those things existing. A ratio cannot exist "of itself," which is why I call it "second-order abstract."

E.g., the Golden Ratio is only instantiated in other abstract objects in the same sort of way that numbers are only instantiated in quantities of items in the world (physical examples of the GR are imperfect approximations). This of course entails that the numerical representation of GR is related to, but not identical with the GR, since the GR is a ratio which can only obtain in virtue of other objects.

But against this you could say that, if the objects involved in the relationship exist necessarily, then surely the relationships that follow from their existence are also necessary.

This is a fair point. I suppose in a formalist interpretation the existence of objects depends on axioms and axioms are only selected contingently.

I don't have a preferred philosophy of mathematics. Leaning towards a sort of Hegelian dialectical logicalism, I would say that some objects do appear to be conditional on a dialectical progression; i.e., "x must come before y." This is a view were universals/abstract entities do not exist without particulars and particulars are reliant on universals for their existence (a circular relationship). If the world moves from the most basic differentiations on, then abstract objects are contingent on this progression.

I don't find intuitionism inviting because everything experienced is mental. So, saying "abstract objects exist, but are constructed mentally" seems to reduce to "abstract entities exist." The only difference is positing some sort of unknowable, and thus entirely irrelevant noumenal world.

Or are you making a claim about numbers in general requiring abstract objects to exist?

Wouldn't numbers be an abstract object? In any event, I think this comes down to my contention that the irrational number corresponding to the GR is not identical with the GR unless it is instantiated as a ratio.

Am I understanding correctly that you believe there is no decimal representation of the Golden Ratio? If that's the case, do you believe there are 2 solutions to the equation x^2-x-1=0? Are you saying something subtle here, such as there are 2 solutions but we can't represent them in decimal form?

I'm saying a decimal number is just an encoding of a pattern/abstraction, it isn't identical with it. I believe this point is essential to understanding the P = NP dilemma. Computation has an abstract existence. Perhaps, we could even say that some mathematical relationships are only instantiated through computation or are somehow dependent on it? I don't know if I would go that far yet.

If any given encoding of a number was truly identical with it, then anyone with a working knowledge of arithmetic and Arabic numerals should find the equation: 〖(35^2*1/57)〗^3+1,912* ∛592.704 readily identifiable with its decimal form or any other of the infinite ways that same number could be encoded in an equation. According to quantitative information theory, just glancing at the equation should allow you to instantly reduce your uncertainty about the identity of the number in decimal form to 0.

Obviously, this is not how things actually work, we don't read Euclid's axioms once and instantly know geometry. Throw a sufficiently complex algorithm that is supposedly equal to a relatively simple number at a supercomputer and it might not finish the computations until the heat death of the universe. Stepwise computation seems necessary to relationships between mathematical objects; complex relations have to be broken down into logically simpler ones to be manipulated. Dialectically, we could say we have the specific encoding juxtaposed with the universal abstraction, which is a bundle of relations. Computation is the sublation of the abstraction into quasi-concrete symbols through which the encoding is actually understood... or something like that.

Do you think a Cauchy sequence of positive rationals can be used to describe the Golden Ratio? — keystone

I take it that by 'describe' you mean in the sense of 'represent' as ordinarily understand to be a denumerable decimal (or binary, whatever) expansion.

Meanwhile, GR [Phi], like any real number, is the limit of a Cauchy sequence of rationals.

might irrationals be all the infinite strings — keystone

Real numbers are not sequences. Real numbers are equivalence classes of Cauchy sequences of rationals. And a real number is the limit of a Cauchy sequence of rationals.

[EDIT: When I wrote that, I was missing the point of an alternative where reals could be sequences from the S-B tree.]

On the Stern-Brocot tree, might irrationals be all the infinite strings which do not — keystone

There are no irrationals that are nodes of that tree. Moreover, every node n of that tree is finitely many nodes away from any other node with which n shares a path in the tree.

So, you can't just magically add Phi as a node to this tree. If you want Phi to be node of a tree, then you need to adduce some other tree. And, of course, adding a node that is not connected to any other node results in something that is not a tree. So you can't just say, "Take the tree and append on to it such and such a node, even though the node is not connected to any other node of the tree". Period.

[EDIT: When I wrote that, I was off-track by missing that the notion was to consider reals as paths, not, as I mistakenly thought, as nodes.]

I don't know why you are particularly interested in this tree and Phi. We could simplify by taking the complete infinite binary tree. It starts by splitting into nodes 0 and 1 and then from every node splits again by adding a sequence with an added 0 or added 1. There's a formal definition too, but you get the picture. Now, every node of that tree is a finite sequence of 0s an 1s, and every node of that tree represents a rational number between 0 and 1. Then you might say, "But I'd like the tree also to represent an irrational number". But there is no irrational number represented by a node of that tree. Period.

/

Golden Ratio is irrational. As such, how are we going to describe it with the positive rationals? — Count Timothy von Icarus

I take it that by 'describe' you mean in the sense of 'represent' as ordinarily understand to be a denumerable decimal (or binary, whatever) expansion.

Meanwhile, there is no finite sequence of rationals such that Phi is the limit of that sequence. But there is a denumerable sequence of rationals such that Phi is the limit of that sequence.

[Phi] is not an abstract object, but rather a property of abstract objects — Count Timothy von Icarus

Phi is a real number. If one takes mathematical objects to be abstract objects, then Phi is an abstract object.

There are two different, but compatible notions:

(1) Phi is a certain real number. (viz. (1+sqrt(5))/2)

(2) Two real numbers, x and y, are such that x/y = Phi. (i.e., (x+y)/x = x/y.)

I don't think an infinite series can encode an irrational number while remaining itself rational — Count Timothy von Icarus

'sequence' ("string") and 'series' mean different things.

A series is a sum of a denumerable sequence of terms such that there is a limit to the sequence of the successive finite sums. [Note: That definition requires that the sequence has a limit, so a series and its limit are the same object, thus excluding the "undefined" sense of a series that does not converge. This is different from definitions that allow that there are nonconvergent series, which, formally, does not make sense (where we are not using the Fregean "scapegoat method) since the sequence is not the series and if the sequence does not have a limit then there is nothing that can be the value of such a "series" in the sense of a series being a certain kind of sum.]

Of course, if a series is irrational then it is not rational.

However, infinity is not a rational number because it is undefined as to its status as an integer — Count Timothy von Icarus

You were using 'infinite'. 'is infinite' is an adjective (a predicate); a set is infinite if and only if it is not finite. But 'infinity' is a noun. There are points of infinity - positive and negative - on the extended real line, but that's not what's in play here.

A ratio is necessarily a relationship — Count Timothy von Icarus

This is again conflating the two different but compatible notions. (1) A ratio of two real numbers is a real number. It is not a relationship between real numbers. (2) Two real numbers x and y such that x/y = z are in a relationship - a set of pairs of real numbers - in particular, {<r s> | r/s = z}.

Every real number is a ratio. For every real number x, there exist real numbers y and z such that x = y/z. Given certain real numbers, such as Phi, it is convenient to state equalities of those numbers as ratios, but the existence of a real number is contingent on the existence of an equivalence class of Cauchy sequences (or, with a different definition of 'real number', on the existence of a Dedekind cut), since that is what a real number is. Especially note that '/' is a defined symbol. So any definition of 'Phi' that uses '/' can be reformulated so that '/' is not used.

I call it "second-order abstract." — Count Timothy von Icarus

'second order' has certain meanings in mathematics. I guess your usage is in some philosophical but not mathematically defined sense.

In the formalist interpretation of mathematics, where "an entity is what it does," — Count Timothy von Icarus

Where can I read that that is a formalist interpretation?

:up:

I worked in the analytic theory of continued fractions years ago, and one of my forebearers was Omar Khayyam, the Persian poet and mathematician ca 1100 ad . He may have devised the continued fraction expansion of the equation

${{x}^{2}}-x-1=0$ => $x=1+\frac{1}{x}=1+\frac{1}{1+\frac{1}{x}}=\cdots =1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots }}}$

Which is a lot more palatable than Stern-Brocot. Well, in my opinion. :cool:

I don't think an infinite series can encode an irrational number while remaining itself rational — Count Timothy von Icarus

Are you suggesting that there's no way to completely describe an irrational number? Why does the infinite series have to remain rational?

I'm saying a decimal number is just an encoding of a pattern/abstraction, it isn't identical with it. — Count Timothy von Icarus

Oh...so would you argue that '2' and the numerical property held by a pair of apples is not identical?

What is your definition of 'completely described'?

Anyway:

Phi is explicitly defined:

Phi = (1+sqrt(5))/2

And:

If x not= y, then 2 = card({x y})

Why are these things even in question?

Meanwhile, going back to the original post, no irrational is a node of that tree, and if you want a tree that has irrationals as nodes, then you need to adduce a different tree, and it would not be by saying something like, "add another node to that tree as a limit node of other nodes". That's not what trees are.

[EDIT: When I wrote that, I was off-track by missing that the notion was to consider reals as paths, not, as I mistakenly thought, as nodes.]

On the other hand, if "RLRLRL..." converges to Phi, then, of course, it's fine (but redundant) to say that Phi is the limit of that sequence.

To answer that question, we would need a mathematical definition of 'describe'.

Meanwhile, GR [Phi], like any real number, is the limit of a Cauchy sequence of rationals. — TonesInDeepFreeze

I suppose I should use is instead of describe.

Real numbers are not sequences. Real numbers are equivalence classes of Cauchy sequences of rationals. And a real number is the limit of a Cauchy sequence of rationals. — TonesInDeepFreeze

"Other common definitions of real numbers include equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts, and infinite decimal representations. All these definitions satisfy the axiomatic definition and are thus equivalent." https://en.wikipedia.org/wiki/Real_number Do you agree with this? If so, let's please focus on the infinite decimal representation as I want to compare it with the infinite Stern-Brocot sequence.

So, you can't just magically add Phi as a node to this tree. — TonesInDeepFreeze

I agree that you can't just add a node to the tree. My impression is that finite SB strings describe 'destinations' (numbers) and infinite SB strings describe 'journeys' (unending processes with no destination). My issue is that I don't see how decimals are any different. Why can't we say that (non-repeating) infinite decimals are journeys that are described by unending processes (e.g. limits) and not 'destinations' (numbers)?

We could simplify by taking the complete infinite binary tree....But there is no irrational number represented by a node in that tree. Period. — TonesInDeepFreeze

Your tree works too, but I find the right-left string useful for my argument instead of just using the numbers at the nodes. While irrationals do not correspond to any node in your tree, they do describe a paths on that tree (from the top all the way down), no?

What is your definition of 'completely described'? — TonesInDeepFreeze

Loosely speaking, I mean that the definition is not an approximation but instead a perfect description. Or perhaps, I should simply say that the complete description of the number is the number.

common definitions of real numbers — keystone

equivalence classes of Cauchy sequences of rationals. Yes.

or

Dedekind cuts. Yes.

or

decimal representations. (Actually, I think two sequences. A finite sequence for the integer part and a denumerable sequence for the part after the decimal point). But, while, of course, every real has a decimal representation, it is not common to define 'is a real number' that way. First we have to have a rule for when a real has more than one representation. Also, I don't know how easy are the definitions of addition and multiplication compared with the definitions of those operations with equivalence classes of Cauchy sequences or Dedekind cuts. So I prefer to use equivalence classes of Cauchy sequences or Dedekind cuts, and usually I prefer Dedekind cuts since they are used in Enderton's 'Elements Of Set Theory', which is my go to reference.

They are equivalent in the sense that they each provide a complete ordered field, and all complete ordered fields are isometric with one another.

/

'destinations' and 'journeys'. I am not familiar with those as mathematical terminology.

Anyway, I don't say that a sequence is a real number. A real number is an equivalence class of Cauchy sequences, and every real number is a limit of various sequences of real numbers (and a limit of various sequences of rational numbers).

/

No, the paths are not real numbers. First, a path is a sequence of edges, not a sequence of nodes. Second, a sequence of nodes is not a real number. Rather the limit of the sequence is a real number.

[EDIT: When I wrote that, I was missing the point of an alternative where reals could be sequences from the S-B tree.]

/

A description is a linguistic object. A description is not a real number. However.

(1+sqrt(5))/2 is a real number.

'(1+sqrt(5))/2' is a description of that real number.

Also, there is the notion of a definite description. A definite description is an expression of this form:

'The x such that P(x)'. Where 'P(x)' is a formula with no free variables other than 'x' and such that it is a theorem that there is exactly one x such that P(x).

A definiens for a definition of a constant to stand for a real number can be reduced to a definite description. For example, defining the constant 'Phi':

Phi = the x such that x = (1+sqrt(5))/2
or reduced:
Phi = (1+sqrt(5))/2

My impression is that finite SB strings describe 'destinations' (numbers) and infinite SB strings describe 'journeys' (unending processes with no destination). My issue is that I don't see how decimals are any different. Why can't we say that (non-repeating) infinite decimals are journeys that are described by unending processes (e.g. limits) and not 'destinations' (numbers)? — keystone

I don't see what the problem is here. You can say what you want, when you want. But asking mathematicians to go along with your ideas of synatax and grammar is another matter. You tell a math person such and such is a "journey" or a "destination" and they might nod their heads and say, Well, you might say that informally. The more precise language of set theory is what they would normally speak in this intellectual environment.

I was a math professor for many years and I am not offput or disturbed by your speculations. I have encountered such notions before, many times, and have never felt discomfited by these discussions. But if I were teaching a class in foundations (thankfully, I never did) I might discuss these ideas in more detail.

Who are you trying to convince here? Philosophers who consider definitions optional?

This sort of argument is at most peripheral to most serious mathematical discussions. If you are curious about deeper, somewhat more sophisticated concerns about the foundations of the subject, look up some of @Metaphysician Undercover's posts about the ultimate nature of mathematical objects, points vs contours, infinitesimals, etc.

look up some of Metaphysician Undercover's posts — jgill

Sorry, but that's an invitation to a crazy train.

Yes.

If 4 shares an identity with 2+2, 3+1, 5+ -1, 8/2, etc. then the P≠NP problem doesn't make sense and we also are left with the "scandal of deduction," the conclusion that deductive reasoning produces no new information, where information is defined loosely in the way used for Shannon Entropy, i.e., a reduction in uncertainty. That is, seeing Euclid's postulates once, I should be completely certain as to the correct answer for any properly encoded Euclidean geometry problem. This falls flat on its face in the real world.

If a unique description of an abstract objects, e.g., a number, is that number, and actually grants the full information about that object the way information theory suggests, then the phrase "the 981x10^131st prime number after zero," should be equivalent to that object and be able to be used in computation. It can't be. No computer ever made could identity such a number in decimal form despite the phrase perfectly defining an actual natural number because the calculation requires more resources than the visible universe.

That said, we can do computational with that number. We can say it is less than its successor and greater than its predecessors because we can work on the information in compressed form. We do this all the time in computation. But my argument is that the logical operations of computation have a being as real as abstract objects and can't simply be ignored as a "finite, somehow less real thing we do to real abstract objects."

Consider this, there are infinitely more irrational numbers than rationals (despite their being infinite rationals) because each irrational is infinite and a new one can be made by flipping a digit. Thus the rationals cannot be set in a 1:1 correspondence with the irrationals.

Now consider that there are infinite ways to encode all rational and irrational numbers in an equation (let's just stick to the reals for now). That is, for 1 we can do 1, 2-1, 3-2, 4-3, etc. with infinite ways of representing the number using addition, infinite ways to represent it with just subtraction, new infinites when involving fractions, multiplication, exponents, etc.

The set of all equations that are "equal" to any number X is infinitely larger than the set of reals as they cannot be set in 1:1 correspondence. There is an infinite number of encodings for each member of the number set. The decimal form of any number is just a proper unit set of the encoding set, and thus all decimal numbers a proper subset of the encoding set.

Thus, no computer, even an infinite one, can ever recognize every equation as identity, it MUST engage in stepwise logical manipulation, not because of physical limits, but because encodings cannot be set in a 1:1 correspondence with numbers.

I didn't hit on the grounds for a possible proof of this until I was thinking about it on a run last night, but feel free to tear it down. It might not be very solid.

Sorry, but that's an invitation to a crazy train — TonesInDeepFreeze

I know. But his quest for the smallest particle of space is no more absurd than Tegmark's mathematical universe IMHO. :cool:

Choo choo! All aboard the crazy train!

If 4 shares an identity with 2+2, 3+1, 5+ -1, 8/2, etc. then the P≠NP problem — Count Timothy von Icarus

That's pure extemporization.

In ordinary mathematics, '=' is taken to have a fixed semantics such that:

4 = 2+2

means that

4 and 2+2 are the same object.

means that

'4' and '2+2' denote the same object.

That's the way it is in virtually all mathematics.

You don't get to declare otherwise.

You can have your own philosophy, but yours doesn't correspond to mathematics as understood by mathematicians.

And this has nothing to do with P vs. NP, which is a problem in mathematics that understands that 4 is the same object as 2+2.

If a unique description of an abstract objects, e.g., a number, is that number [...] then [unacceptable consequent] — Count Timothy von Icarus

Right, a description of a number and the number are not the same objects.

The set of all equations that are "equal" to any number X is infinitely larger than the set of reals as they cannot be set in 1:1 correspondence. — Count Timothy von Icarus

Wrong. An equation is a certain kind of formula. In an ordinary mathematical theory (such as set theory, which is the ordinary theory for the subject of equinumerosity) there are only countably many formulas. But there are uncountably many real numbers. It's true that the set of equations is not 1-1 with the set of reals, but it's the set of reals that is the greater.

In the formalist interpretation of mathematics, where "an entity is what it does,"
— Count Timothy von Icarus

Where can I read that that is a formalist interpretation? — TonesInDeepFreeze

I'm still interested. Where can I read a "formalist interpretation" that "an entity is what it does".

every real has a decimal representation, it is not common to define 'is a real number' that way. — TonesInDeepFreeze

If you're talking about a number like sqrt(2) then I agree. However, for indescribable real numbers, I imagine that it would be less work to exhibit a single infinite string of digits than to exhibit an equivalence class of Cauchy sequences. Of course, 'less' can be confusing when talking about infinite work.

Also, I don't know how easy are the definitions of addition and multiplication compared with the definitions of those operations with equivalence classes of Cauchy sequences or Dedekind cuts. — TonesInDeepFreeze

As for arithmetic on the Stern-Brocot tree, the algorithm is relatively simple (but computationally expensive - I rewrote it in Python if you want to play with it) (https://www.sciencedirect.com/science/article/pii/S1570866706000311#:~:text=The%20Stern%E2%80%93Brocot%20tree%20is,usual%20ordering%20of%20the%20rationals.). You can take any pair of strings (composed of L's and R's) and perform arithmetic on them, whereby it gradually consumes the strings working from the left to the right. As for infinite strings, you can interrupt it at any point to get a partial answer, whereby the longer you wait, the more characters you'll have in the output string.

No, the paths are not real numbers. First, a path is a sequence of edges, not a sequence of nodes. Second, a sequence of nodes is not a real number. Rather the limit of the sequence is a real number. — TonesInDeepFreeze

Feeding the aforementioned algorithm the string RL, it will treat it exactly as the golden ratio. If RL looks like the golden ratio and it behaves like the golden ratio, why do you not say that it is the golden ratio? I think we both agree that RL corresponds to a specific path on the Stern-Brocot tree, not a node. If the algorithm treats RL as the golden ratio, then it seems reasonable to say that the golden ratio (and all real numbers) are paths on the Stern-Brocot tree. I like this view because it clearly distinguishes between rational and real numbers. Rationals are the nodes with finite characters (e.g. R=2). Reals are the paths with infinite characters (e.g. RLR=1.9). And since nodes and paths are distinct objects, we don't feel inclined to say that 2=1.9).

A description is a linguistic object. A description is not a real number. — TonesInDeepFreeze

I am good with how you distinguished between the two.

Who are you trying to convince here? Philosophers who consider definitions optional? — jgill

This is a chat forum, not a journal. We should be allowed to spitball here.

@Count Timothy von Icarus

I appreciate your responses but Tones captured my concerns in a more eloquent way than I could have put it so I'll refrain from repeating his points.

I'm not talking about defining a particular real number. I'm talking about defining the PROPERTY 'is a real number'. Such a definition is of the form:

x is a real number <-> F(x)
where F(x) is a formula with no free variables other than 'x'.

So, three competing definitions:

x is a real number <-> x is a convergence equivalence class of Cauchy sequences of rational numbers

x is a real number <-> x is a Dedekind cut

x is a real number <-> Eid(x = <i d> & i is an integer & d is a denumerable decimal sequence)
[or something like that]

/

As for arithmetic on the Stern-Brocot tree — keystone

I'm not talking about that tree in that context. I was talking about the three competing definitions of 'is a real number' and how easy or difficult it is to define the operations for real numbers based on those definitions.

No, the paths are not real numbers. First, a path is a sequence of edges, not a sequence of nodes. Second, a sequence of nodes is not a real number. Rather the limit of the sequence is a real number.
— TonesInDeepFreeze

Feeding the aforementioned algorithm the string RL, it will treat it exactly as the golden ratio. If RL looks like the golden ratio and it behaves like the golden ratio, why do you not say that it is the golden ratio? — keystone

I take it that by 'RL', you mean the particular denumerable path. That is not a real number by any of the three competing definitions of 'is a real number'. (1) It is not an equivalence class of Cauchy sequences. (2) It's not a Dedekind cut. (3) It's not an integer and a denumerable decimal sequence.

Regarding (1), a real number is the limit of infinitely many Cauchy sequences of rationals, so if 'is a real number' would be defined as just one particular Cauchy sequence of rationals, then which of the infinitely many should it be? We don't have an answer to that question. So, instead, we take real numbers to be a whole equivalence class of the Cauchy sequences of rationals, where 'equivalence' is in the sense of 'mutually converging' (or whatever the actual technical term should be).

[EDIT: What I said in that paragraph is correct, but I was missing the point of an alternative where reals could be sequences from the S-B tree, which would eliminate the need to define as equivalence classes.]

all real numbers [...] are paths on the Stern-Brocot tree — keystone

'is a path on the left side of the SB tree' as a fourth competing definiens? It would be of 'is a real number between 0 and 1 inclusive'.

But, for example, what real number would be the edge {1/2 1/3}? And are you sure that every irrational number is one of the denumerable paths? And that the sequence of nodes of every denumerable path converges to an irrational number? Aren't there denumerable paths that stay constant on a single rational number?

we don't feel inclined to say that 2=1.9 — keystone

But 2=1.9. If your method entails that that is not the case, then I doubt that your method actually provides a complete ordered field.

/

PS. CORRECTION:

I initially misconstrued you. I was not reading carefully enough. I made the point that no irrational is a node on the tree. That is true, but not relevant, since your point (which I failed to read correctly) is that irrationals may be certain paths (not nodes).

Who are you trying to convince here? Philosophers who consider definitions optional?
— jgill

This is a chat forum, not a journal. We should be allowed to spitball here. — keystone

That's a strawman. He didn't say that all discourse has to be at the level of a mathematics journal.

Here's what you need to provide for your SB proposal:

rigorous definition of 'is an SB_real' (then let SB_R = {x | x is an SB_real})
(and you'll have to provide for the negative real numbers too, though I guess that wouldn't be too hard)

rigorous definition of 'SB_<'

rigorous definition of 'SB_+'

rigorous definition of 'SB_*'

rigorous proof that <SB_R, SB_<, SB_+, SB_*> is a complete ordered field


We also don't yet have a rigorous (not just ostensive) definitions of the SB tree, 'R' and 'L'. But I don't doubt that there are ones, though complicated they probably are, so we could at least provisionally work with the ostensive definitions we know.

Also, you might want to consider taking reals not as paths but as sequences of nodes on paths. Perhaps it's easier to talk about sequences of nodes rather than sequences of edges, or at least it's more familiar.

Choo choo! All aboard the crazy train!

lol, exactly. But I only suggest it because we already are on a crazy train. When we try to apply formalizations of information quantified as a reduction in uncertainty we get the patently false assertion that receiving a clear message in a code we understand, such as the equation (√1913 • π ) ÷ 1.934 , makes us just as certain of what the result of that equation is as having received the decimal number. That is, we should instantly know which numbers are greater than the result and which are less than it and the identity of all equations that result in the same number. This is clearly not the case. Digital computers are much quicker, but if you throw a complex enough algorithm at them it would take them trillions of years to complete it, even when the function of program(input) = a number that is quite simple to represent.

It's a broader part of P ≠ NP. There is really a separate problem with the P ≠ NP that is not related to specifically reducing complexity classes. It's related to the problem of systematic search when two equivalent terms nonetheless cannot be recognized as such without step-wise transformations, particularly when these can take more computational resources than appear to exist in the visible universe.

The bottom half of the Stanford Encyclopedia of Philosophy article on Philosophy of Information covers this, or you can also look up the closely related "scandal of deduction."

And this has nothing to do with P vs. NP, which is a problem in mathematics that understands that 4 is the same object as 2+2.

I don't think most mathematicians particularly care that much about the philosophy of mathematics. At least that is what I've heard in enough lectures and books on the topic to believe it.

There are several major schools in philosophy of math and some deny that numbers as distinct objects even exist, so I don't get how they can be too offended here. Especially since I will allow that 2+2 = 4 in a sense that the are numerically equivalent, they are just not equivalent vis-á-vis computation.

Right, a description of a number and the number are not the same objects.

Is "the successor integer of 2" a description but 2 + 1 is not? Is "three" not a number but 3 is? A mark on a paper, symbols on a Turing Machine tape and pixels are not numbers. In the mathematics of computation/theoretical computer science this is necessarily true. We don't have to reduce encodings to some one true description, although we do have the shortest description in any one appropriate language, Kolmogorov Complexity.

Wrong. An equation is a certain kind of formula. In an ordinary mathematical theory (such as set theory, which is the ordinary theory for the subject of equinumerosity) there are only countably many formulas. But there are uncountably many real numbers. It's true that the set of equations is not 1-1 with the set of reals, but it's the set of reals that is the greater.

I think you are confusing the set of all computable functions with the set of all equations. The concept of a solution set will be helpful here: https://www.varsitytutors.com/hotmath/hotmath_help/topics/solution-sets

A solution set is the set of all solutions for an equation. For example, "x + 1 = 1 + x" has the reals as its solution set. We are talking about the set of all equations, which is the size of the set of all solution sets for all equations. Thus, the SS of "x + 2 = 2 + x" is also the same size as the reals. We can use complex numbers as well, since "x + π = π + x" also = R. We can do this with addition for all the reals, but also do it with multiple addition operations, division, exponents, etc. And of course we have some equations that have only a few or one member in the solution set. The set of all solution sets is not the set of all computable functions, which I'll agree is countable.

So as you can see, there are infinitely more equations than reals, which means that no computer, even an infinite one that allows for infinite operations (which is generally disallowed anyhow), can place encodings and numbers into a 1:1 correspondence. There will always be multiple equations specifying a single number (whereas with computable functions there are not enough functions for all the reals).

Thus, not even an infinite computer can take any one number and treat it like an identity name for all the corresponding equations equal to that number via a preset repertoire (since they cannot be in 1:1 correspondence). This is basically the same thing as saying their are more possible solutions to combinations of numbers than numbers, which is prima facie true, and I imagine it can be shown combinatoricaly as well. Not only this, but in some well known cases the equation -> number relationship is many to many, not many to one.

Now many mathematicians might not care about the scandal of deduction, which is fine, but it's a serious problem created by current definitions nonetheless. Notably, for centuries extremely skilled mathematicians argued that any positive X ÷ 0 = ∞ and had good arguments for that conclusion. It's not like major shifts don't happen because current established practice ends up resulting in absurdity.

"And this has nothing to do with P vs. NP, which is a problem in mathematics that understands that 4 is the same object as 2+2."

I don't think most mathematicians particularly care that much about the philosophy of mathematics. — Count Timothy von Icarus

They don't need to care about the philosophy of mathematics to know that 2+2 is 4.

"Wrong. An equation is a certain kind of formula. In an ordinary mathematical theory (such as set theory, which is the ordinary theory for the subject of equinumerosity) there are only countably many formulas. But there are uncountably many real numbers. It's true that the set of equations is not 1-1 with the set of reals, but it's the set of reals that is the greater."

I think you are confusing the set of all computable functions with the set of all equations. — Count Timothy von Icarus

No, you are confusing what I said and meant with something you want to say or mean.

What I said is exactly correct: (1) There are only countably many formulas. (2) There are uncountably many reals. (3) Therefore, there are more reals than there are formulas.

We are talking about the set of all equations, which is the size of the set of all solution sets for all equations. — Count Timothy von Icarus

That is clearly incorrect. You know worse than nothing about this subject.

An equation is a formula. There are only countably many formulas. A solution set for an open formula with only one free variable is a subset of the set of reals. A formula may have uncountably many members in its solution set, but still there are only countably many formulas. One more time:

(4) The set of equations is a subset of the set of formulas. There are only countably many formulas, so there are only countably many equations.

(5) A solution set for a formula with only one free variable is a subset of the set of reals. A solution set for a formula may have uncountably many members.

So as you can see, there are infinitely more equations than reals — Count Timothy von Icarus

No, only as you are deluded. Your delusion is in conflating the cardinality of the set of equations with the cardinalities of solution sets for equations. Again: There are only countably many equations, but some equations have uncountably many solutions.

It might be easy to look at the problem a simpler way.

If we had developed a Platonism of processes instead of objects, we might find nothing weird with saying each unique computation is it's own unique entity. These computations would share a relationship in having a "common output," in the same way numbers can have a "common denominator."

Human biology is adapted to generating a world of discrete objects for itself. Physics increasingly casts doubt on the reality of this perception. Platonism casts a long shadow on mathematics, and there we tend to think of objects as fundemental, not relationships or processes. Hence statements like "φ is a rational number, thus the Golden Ratio is a rational number." If we focused on relations instead of objects we would say, "that is ridiculous, you cannot have a ratio of nothing to nothing else as a relationship, φ and all numbers are just useful symbols for representing relationships, not irreducible things in themselves." But instead we think of the ratio was reducible to the numerical object.

Which has obviously worked out pretty well, except for the ludicrous results of assuming any algorithm = x IS x as concerns finite computation instantiated in the world or our knowledge of uncomputable numbers.

We say that an algorithm for finding Ω ≠ Ω due to the decision problem, and yet Ω + 1 = Ω - 1 is true. That is, we can work with the uncomputables in compressed form, and we work with hard to compute numbers effectively in compressed form without identifying their numerical value all the time.

This suggests to me that computation is not simply a statement of identity (else we should jettison the uncomputable numbers).



Missed this. Mathematics: A Very Short Introduction has a good description in the intro. It's on LibGen, but if you want something that is open access you can check on intuitionism versus Platonism versus formalism, etc. https://plato.stanford.edu/entries/philosophy-mathematics/

Here is a description of some of the problems I mentioned with P ≠ NP vis-á-vis identity and Kolmogorov Complexity
https://plato.stanford.edu/entries/information/#AnomParaProb