Yielding a picture from a puzzle

[Michali]

Seebs mentioned a use of the equals sign where it is used to describe a relationship where one thing "yields" another thing, and this is different than something being identical with another thing. Probably rightly so.

Now, firstly, let's analyze this concept and maybe it will shed some light on the whole ordeal.

First question:

Is it that functions yield such and such, or is it that such and such yields a function? In other words when we notate f(x)=y... how do we linguistically express this expression?

Second question:

Is this relationship anything like the "emergence" concept? For instance, are the pieces of a puzzle yielding the entire picture? What are the specifications surrounding that?

[Farren]

And the winner is:...

Some functions are reversible and some arent. Lets examine a simple example used by programmers sometimes in pseudo-random-number-generation functions, the MOD function, which returns the remainder of a number divided by another number. This essentially removes information about the former number.

Simply put, it works like this y=MOD(3,x) means y is equal to the remainder when the value of x is divided by 3. So if x=30, y=MOD(3,30) then y=0, because 30 divided by 3 yields a remainder of 0. But y=MOD(3,60) also yields y=0, because 60 divided by 3 also yields a remainder of 0.

From the above we can see that y=MOD(x,3) yields the same result for many different values of X. So knowing Y does not reveal the exact value of X, only the set of possible values of x (...-6,-3,3,6,9....).

But knowing X reveals the exact value of Y.

Now lets imagine a deterministic universe where X is effectively "the past" and Y, "the future". In a universe which follows the above, simple rule, the future is knowable from the present, but the past can only be reduced to a set of possibilities. There is a future-past asymmetry.

And no, this asymmetry is not related to "emergent" order. Emergence implies that a pattern of many units becomes evident (emerges in our observation), in the collective functional interaction of lots of units , that is not apparent in the behaviour of any one, individual unit. It has nothing to do with qualities of the individual elements of non-recursive sets (which is what the asymmetry I described creates).

[Iacchus]

Are you referring to the nature of synchronicity here? This is what I've noticed when I work with it. That it comes together like pieces of a puzzle which, ultimately come to form a picture. I refer to it a bit more in this post.

[Michali]

Originally Posted by Farren

And no, this asymmetry is not related to "emergent" order. Emergence implies that a pattern of many units becomes evident (emerges in our observation), in the collective functional interaction of lots of units , that is not apparent in the behaviour of any one, individual unit. It has nothing to do with qualities of the individual elements of non-recursive sets (which is what the asymmetry I described creates).

So, functions are not related to emergence phenomena, you say?

Is it reasonable to say that seeing all the pieces correctly placed together tends to yield seeing one big picture?

[naturalist.atheist]

Originally Posted by Michali

Seebs mentioned a use of the equals sign where it is used to describe a relationship where one thing "yields" another thing, and this is different than something being identical with another thing. Probably rightly so.

Now, firstly, let's analyze this concept and maybe it will shed some light on the whole ordeal.

First question:

Is it that functions yield such and such, or is it that such and such yields a function? In other words when we notate f(x)=y... how do we linguistically express this expression?

In the mathematical sense a "function" is a mapping with specific properties such as "one-to-one" and "onto". However in this age of computers the concept of a function and the "=" gets a bit muddy, with the "=" and a function becoming a kind of operation as opposted to a mathematical identity.

Second question:

Is this relationship anything like the "emergence" concept? For instance, are the pieces of a puzzle yielding the entire picture? What are the specifications surrounding that?

Mathematically the answer is no since "=" is not a relationship. It is an "identity". Mathematically when we say f(x) = y we identify f(x) as y but we are none the wiser unless we can define y in terms of x.

[Michali]

Originally Posted by naturalist.atheist

In the mathematical sense a "function" is a mapping with specific properties such as "one-to-one" and "onto". However in this age of computers the concept of a function and the "=" gets a bit muddy, with the "=" and a function becoming a kind of operation as opposted to a mathematical identity.

It seems like this mapping description is something like "x -> y" where this "->" is meant to be interpreted as what exactly? So let's just assume that X and Y have this mapping between them, but what is the best way to describe this relationship?

In other words, let's say I have x=(1234) and y=(abcd).

You might suggest f(x)=y could be represented with the following:

1-a
2-b
3-c
4-d

What exactly will the concept of function relate to in the above diagram? Is it the hyphen(abstract) playing the role of function, or are 4 hyphens together playing the role of function. Is the entire diagram playing the role of function? Or are only the principle features playing the role of function, and if so, what are they.

In other words, how should we use "function" in English? That's what I think I'm searching for.

Mathematically the answer is no since "=" is not a relationship. It is an "identity". Mathematically when we say f(x) = y we identify f(x) as y but we are none the wiser unless we can define y in terms of x.

Right I agree that is less of a relationship and more like identity myself. But reading your sentence here closely, and it seems as though you don't think f(x) yields y, but that f(x) is identical to y. Am I right?

[naturalist.atheist]

Originally Posted by Michali

It seems like this mapping description is something like "x -> y" where this "->" is meant to be interpreted as what exactly? So let's just assume that X and Y have this mapping between them, but what is the best way to describe this relationship?

In other words, let's say I have x=(1234) and y=(abcd).

You might suggest f(x)=y could be represented with the following:

1-a
2-b
3-c
4-d

What exactly will the concept of function relate to in the above diagram? Is it the hyphen(abstract) playing the role of function, or are 4 hyphens together playing the role of function. Is the entire diagram playing the role of function? Or are only the principle features playing the role of function, and if so, what are they.

In other words, how should we use "function" in English? That's what I think I'm searching for.

One way to understand it is to consider a sub-class of functions called discrete functions.

Given a discrete set called X which contains the set of integers I there is another set Y which also contains a set of integers where there is a mapping between a member in the set X to a member in the set Y. You could think it as something like this,

For a range of integers that enumarate the integers in X, X(1), X(2), X(3), ...... and X(1) could be 1, X(2) is 2 and so forth.

There is a pairing between the two sets created by a mapping

X(1) -> Y(1) [ 1 maps to Y(1)]
X(2) -> Y(2)
X(3) -> Y(3)

And so on.

In the continuous case the sets become all numbers. How the numbers map specifically defines the specific function. The mapping or function is onto if there is a member of X for every member in Y. A function is one-to-one if for a given X(n) there is always only one member of Y that it maps to (not two or more values). This allows us to assume that if Y(n) = Y(m) then n = m.

Right I agree that is less of a relationship and more like identity myself. But reading your sentence here closely, and it seems as though you don't think f(x) yields y, but that f(x) is identical to y. Am I right?

You are right. Mathematically when you use the "=" it means that the left side is identical to the right side. The "=" is also used widely in computer programming but it is not the same thing as the "=" in mathematics. When the "=" is used in programming it is essentially a memory operation not a mathematical identity.

[Michali]

Ok, so a memory operation in programming you say. Well do you mean that the computer could store the "left side" and recall it for later or something? Or do you mean that we remember the "left side" occuring before the "right side"?

Thus you might mean, "In programming, the equals sign is like the meaning of the word 'yield', in that, the left side occurs before the right side." Does this mean they are not identical?

[Iacchus]

1 + 1 = 2. You have to do the math first, before it "yields" the number 2.

[naturalist.atheist]

Originally Posted by Michali

Ok, so a memory operation in programming you say. Well do you mean that the computer could store the "left side" and recall it for later or something? Or do you mean that we remember the "left side" occuring before the "right side"?

Thus you might mean, "In programming, the equals sign is like the meaning of the word 'yield', in that, the left side occurs before the right side." Does this mean they are not identical?

Back in the old days of machine language or assembler there was no such thing as an "=" in the programming language. The program executed instructions that moved data around and performed arithmetic operations. In modern programming languages variables are declared that essentially map to memory locations or intermediate registers. When you write A = B in a programming language it means "take the data stored in the memory location identified in the program as B and copy it to a memory location identified in the program as A. It is not until you learn about pointers and handles that what is going on becomes obvious. (Or writing programs in assembler.) You can't really understand computers until you break down what is going on when machine language is executed by the processor. If you understand it well enough you can design and build your own processors and computer languages.

[Farren]

The dreaded dupe monster strikes again.

[Farren]

Michali,

Equivalence

In both programming and math, any statement of the form "x=f(y)" can be expressed as "f(y) yields x", meaning that once you have reduced f(y) to a value, you will have x. But I suspect this isn't really what your opening post is trying to establish.

Because its possible for something to yield something and to in a sense be something (identity) at the same time, inasmuch as the variable on the left side of the equation may represent a quality, in the real world, that materially shares existence with the quality on the right side, in the real world.

The population of the USA, for instance, being the sum of the populations of its states.

But its also possible for something to yield something without being that thing.

Your mass, for instance, being a function of your velocity in general relativity.

Another example is the weight of two objects. I weigh one thing and it's 10kg. I weigh another thing and its 10kg. I say "the weight of this is equal to the weight of that", but I don't mean that the respective masses those weights we're measuring are the same material things.

It depends on what the variables on both sides of the equation represent.

Mathematical equivalence does not imply shared material identity. We find equivalence in transformations of quantities we assign to material identities and express that using an "=" sign.

Emergence

"Emergence" is used to describe the concept that qualities may be apparent in a whole that cannot be found in its parts.

For instance, flocks of birds fly in a V-formation, but individual birds don't have any individual conception of the fact that a V-formation is ideal. They don't, like a group of rowers, deliberately co-ordinate their action to form a V-shape. Individual birds simply try to avoid colliding with each other while staying close together and that results in an optimal flying formation. So we say the V-formation "emerges" in a flock from behaviour in birds who individually have no idea what the optimal formation is.

Emergent phenomena are extremely common in large groups of organisms. Ants and termites often provide the most startling examples.

Here's a nice list of examples of emergent phenomena:

Multicellular Computing: Emergent Phenomena in Nature

[Michali]

Right, thanks Farren, I'm pretty familiar with the emergence concept. The purpose of the thread though, and what is most interesting about your post is what you mean here:

Originally Posted by Farren

It depends on what the variables on both sides of the equation represent.

Can you help me understand what you mean by your use of "represent"?

Originally Posted by Farren

Mathematical equivalence does not imply shared material identity.

I agree fully here. "Material identity" is very interesting to me as it is the very thing I'm attempting to undermine. Is this "material identity" necessarily only useful for material examples or would you agree that your distinguishment is also the way we distinguish between "3+3" and "6"?

[Farren]

Originally Posted by Michali

Right, thanks Farren, I'm pretty familiar with the emergence concept. The purpose of the thread though, and what is most interesting about your post is what you mean here:

Can you help me understand what you mean by your use of "represent"?

I mean we use numbers to represent, or quantify, qualities of things and collections of things. The speed of a bullet. The weight of an apple. The number of apples that exist.

Originally Posted by Michali

I agree fully here. "Material identity" is very interesting to me as it is the very thing I'm attempting to undermine. Is this "material identity" necessarily only useful for material examples or would you agree that your distinguishment is also the way we distinguish between "3+3" and "6"?

Well I'm using material identity to mean the physical presence of the things that have the qualities we quantify. So the material identity of 4 apples is the apples themselves. Its possibly contentious to speak of the "material identity" of, say, the speed of a falling apple, but apples do indeed fall. We're describing something material, I think, when we talk about the speed of an apple.

So I think its fair to talk about the speed of two apples having different material identities. As in "the speed of one apple may be equal in value to the speed of another apple, but it is not the same physical property in the universe, even though it is in the same type of property". Certainly in all of our computations we treat them as seperate identities.

They're different data points in the universe, different variables, even though they have the same value and describe the same quality of two different things.

But could numbers themselves be considered to have similarly separate material identities?

I don't think thats a meaningful question, because we don't treat numbers as material things. They have mathematical identities. And whether functions of numbers have the same mathematical identity as numbers they yield is an interesting question.

In a previous thread I referred to lambda calculus, which is a formal system for describing all other mathematical functions and quantities, in the minimum possible set of symbols. Lambda calculus basically provides all the operations for a Universal Turing Machine (a theoretical universal computing device), which is the theoretical model for modern computers.

In any event, in lambda calculus, everything is reduced to the interaction of functions, even the mathematical identity of numbers themselves.

By its very nature, lambda calculus represents every number as a function of other numbers, as the following Church Numerals (numbers encoded in Lambda Calculus) demonstrate:

0 ≡ λf.λx. x
1 ≡ λf.λx. f x
2 ≡ λf.λx. f (f x)
3 ≡ λf.λx. f (f (f x))
...
n ≡ λf.λx. fn x
...

In other words, in this "underlying math of math", the identity of 3 is essentially a function of a function of 1.

So if we understand Lambda Calculus to describe the underpinnings of all math, then I'd say yes, the mathematical identity of (3+3) is 6 is (2+2+2). They're just different ways of writing down the same identity.

What are you angling at with this?

[Iacchus]

Until you actually group them together, the properties of 1 + 1 are separate entities. It's the "+" sign that denotes "the act" of grouping them together.

[Michali]

Originally Posted by Farren

I mean we use numbers to represent, or quantify, qualities of things and collections of things. The speed of a bullet. The weight of an apple. The number of apples that exist.

Right, ok, but are you sure you were talking about numbers when you made that statement? After all, you said it depends on what the "variables on both sides of the equation" represent. In a previous thread, we were able to describe an operation in a simulator as being expressible with "f(x)=y". Do you mean to draw a distinction between these or were those states essentially identical to numerical expressions as well?

Once again, what did you mean by 'f(x)=y is identical dependent upon what both sides of the equation are representing.'- paraphrased.

Originally Posted by Farren

Well I'm using material identity to mean the physical presence of the things that have the qualities we quantify. So the material identity of 4 apples is the apples themselves. Its possibly contentious to speak of the "material identity" of, say, the speed of a falling apple, but apples do indeed fall. We're describing something material, I think, when we talk about the speed of an apple.

I think defining a thing as though it is separable from its qualities, you will find difficult. For instance, you consider the speed of an apple to be material, when it is actually a relative quality between you and the apple. Do you consider an apple's movement in a direction to be essential to the apple?

So I think its fair to talk about the speed of two apples having different material identities. As in "the speed of one apple may be equal in value to the speed of another apple, but it is not the same physical property in the universe, even though it is in the same type of property". Certainly in all of our computations we treat them as seperate identities.

Can you not imagine the apple could have another speed? Of course one would think two different apples having the same speed are different. But what about one apple having a different speed? Suppose I throw it. Was not the apple that went through the air identical to the apple I once had in my hand? Also, what if my brother were bicycling next to it as it flew, and he watched it holding perfectly still when I watched it traveling quickly.

They're different data points in the universe, different variables, even though they have the same value and describe the same quality of two different things.

So you say, given two apples, I have two different variables. If they are traveling at the same speed, they have a certain identical value called speed. That's fine of course, but in f(x)=y all the values on both sides of the equation are equal. Such as in the values expressed in 3+3=6.

If two apples had all the same values, aren't they the same variable? I mean, literally, every quality, including "point in space-time"?

But could numbers themselves be considered to have similarly separate material identities?

I don't think thats a meaningful question, because we don't treat numbers as material things. They have mathematical identities. And whether functions of numbers have the same mathematical identity as numbers they yield is an interesting question.

In a previous thread I referred to lambda calculus, which is a formal system for describing all other mathematical functions and quantities, in the minimum possible set of symbols. Lambda calculus basically provides all the operations for a Universal Turing Machine (a theoretical universal computing device), which is the theoretical model for modern computers.

In any event, in lambda calculus, everything is reduced to the interaction of functions, even the mathematical identity of numbers themselves.

By its very nature, lambda calculus represents every number as a function of other numbers, as the following Church Numerals (numbers encoded in Lambda Calculus) demonstrate:

0 ≡ λf.λx. x
1 ≡ λf.λx. f x
2 ≡ λf.λx. f (f x)
3 ≡ λf.λx. f (f (f x))
...
n ≡ λf.λx. fn x
...

In other words, in this "underlying math of math", the identity of 3 is essentially a function of a function of 1.

So if we understand Lambda Calculus to describe the underpinnings of all math, then I'd say yes, the mathematical identity of (3+3) is 6 is (2+2+2). They're just different ways of writing down the same identity.

What are you angling at with this?

I mean to be angling that there is no need to distinguish f(x)=y as being identical from this concept of "yielding". And then to go from there. I tried looking up lambda calculus, and I'm super interested in it, and I'm hoping you can help me understand it. It reminds me of the set theory explanation for mathematics. First, what is the lambda symbol supposed to represent?

[seebs]

There is a huge distinction between identity and other kinds of equality.

Michali, your basic problem is that everything you're doing is devoted to trying to recover your "end states" notion. You're emotionally committed to that, and any time anything proves it wrong, you start attacking it and trying to show that it doesn't exist or doesn't mean anything.

You're wrong. Give it up.

You are not going to find a magical way of eliminating the fact that there really are multiple states over time in Conway's life, and that means that you can't get away from the fact that determinism does not imply reversibility. And that means your entire argument as to why the end state causes the earlier states, rather than vice versa, is wrong.

No matter how cool it would be, no matter what a great short story it would make (I read a lovely time travel story based on it), it isn't actually true. And it won't become true no matter how many times you try to find new ways of asking the question "but doesn't that mean that everything anyone has ever told me about functions or mathematics is wrong and I'm right".

[Michali]

Originally Posted by seebs

There is a huge distinction between identity and other kinds of equality.

Michali, your basic problem is that everything you're doing is devoted to trying to recover your "end states" notion. You're emotionally committed to that, and any time anything proves it wrong, you start attacking it and trying to show that it doesn't exist or doesn't mean anything.

You're wrong. Give it up.

You are not going to find a magical way of eliminating the fact that there really are multiple states over time in Conway's life, and that means that you can't get away from the fact that determinism does not imply reversibility. And that means your entire argument as to why the end state causes the earlier states, rather than vice versa, is wrong.

No matter how cool it would be, no matter what a great short story it would make (I read a lovely time travel story based on it), it isn't actually true. And it won't become true no matter how many times you try to find new ways of asking the question "but doesn't that mean that everything anyone has ever told me about functions or mathematics is wrong and I'm right".

Then, hey, you know what, you can stay out of it. All this post is doing is attempting to undercut me on a personal level. And check out the other posters on this thread, they agree that f(x) is identical to y. You claimed it was otherwise. Now if you want to justify that, then by all means do it. Don't bring up the no-input thing, or anything else from other threads, this is all about whether f(x) is identical to y, and just what that is supposed to mean.

[naturalist.atheist]

The problem here may stem from the common practice of confusing a mathematical model of reality with reality itself. Kinda like confusing a plastic model of a 747 with an actual 747. You can learn something about 747s from a plastic model but it is just a crude approximation of the thing itself.

[Michali]

Originally Posted by naturalist.atheist

The problem here may stem from the common practice of confusing a mathematical model of reality with reality itself. Kinda like confusing a plastic model of a 747 with an actual 747. You can learn something about 747s from a plastic model but it is just a crude approximation of the thing itself.

What's the problem you're referring to?

[naturalist.atheist]

Originally Posted by Michali

What's the problem you're referring to?

I may be butting into the middle of something that I am not following very well but there seems to be some important point surrounding the "=" and a function. Both are mathematical abstractions. They only have validity by definition in the realm of mathematics. They have also found great use when trying to model reality, but it is only a model.

[Iacchus]

What about the "reality" of the model itself? You can't escape just by claiming it's an abstraction.

[naturalist.atheist]

Originally Posted by Iacchus

What about the "reality" of the model itself? You can't escape just by claiming it's an abstraction.

I am not contesting the existence of the model. There are plastic models of 747s. But you can't fly in them to Frankfurt. That appears to be one of your many thinking problems. You seem to think that if you can come up with a model it must be a perfect representation of reality simply because your model is real.

You are a very fine example of a complete idiot.

[Iacchus]

No, it is not just a model. We are in fact reality (not divorced and/or separate therefrom) looking at itself.

[naturalist.atheist]

Iacchus, thinking is something that is not for you. You are just no good at it. You are so bad at it that it is painful to read your posts. It is like listening to somebody who just can't sing. It is an ordeal. Do yourself a big favor and don't even bother trying to think. Just make declarative statements like, "I like bananas." Stick to simple ideas about you. But don't make too many of them because frankly by now no one is likely to care.