There cannot be two theories for an error-free set of data

[bahman]

Consider an error-free set of data. By error-free, I mean that the data is precise to the last digit. Let's call the set of data "y". Now assume that there are at least two theories that can explain the set of data. Let's call the theories A and B. This means that there are two unidentical functions related to theories. This means that y=A(a,b,c,...) where the set {a,b,c,...} is the set of variables. In the same manner y=B(a,b,c,...). Now let's subtract A from B. This means that A(a,b,c,...)-B(a,b,c,...)=0 since they predict the same the same set of data. Let's call C(a,b,c,...)=A(a,b,c,...)-B(a,b,c,...). C(a,b,c,...) however has to be zero for all changes in the variables. Therefore, C(a,b,c,...) is a NULL function. Therefore two theories are identical.

[Impenitent]

theory one is nothing
theory two is nothing

nothing is nothing

-Imp

[wtf]

Consider an error-free set of data. By error-free, I mean that the data is precise to the last digit. Let's call the set of data "y". Now assume that there are at least two theories that can explain the set of data. Let's call the theories A and B. This means that there are two unidentical functions related to theories. This means that y=A(a,b,c,...) where the set {a,b,c,...} is the set of variables. In the same manner y=B(a,b,c,...). Now let's subtract A from B. This means that A(a,b,c,...)-B(a,b,c,...)=0 since they predict the same the same set of data. Let's call C(a,b,c,...)=A(a,b,c,...)-B(a,b,c,...). C(a,b,c,...) however has to be zero for all changes in the variables. Therefore, C(a,b,c,...) is a NULL function. Therefore two theories are identical.

There are infinitely many functions that pass through the same set of points, even if there are infinitely many points.

There are infinitely many continuous functions passing through the same set of points. Infinitely many differentiable functions.

Your idea is wrong.

As an example, the sine function has infinitely many zeros, but it's not the zero function.

[bahman]

Consider an error-free set of data. By error-free, I mean that the data is precise to the last digit. Let's call the set of data "y". Now assume that there are at least two theories that can explain the set of data. Let's call the theories A and B. This means that there are two unidentical functions related to theories. This means that y=A(a,b,c,...) where the set {a,b,c,...} is the set of variables. In the same manner y=B(a,b,c,...). Now let's subtract A from B. This means that A(a,b,c,...)-B(a,b,c,...)=0 since they predict the same the same set of data. Let's call C(a,b,c,...)=A(a,b,c,...)-B(a,b,c,...). C(a,b,c,...) however has to be zero for all changes in the variables. Therefore, C(a,b,c,...) is a NULL function. Therefore two theories are identical.

There are infinitely many functions that pass through the same set of points, even if there are infinitely many points.

How could you justify this?

[wtf]

How could you justify this?

First let's do it for two zeros.

(*) Consider a function that inputs a real number and outputs a real number, such that f(1) = f(2) = 0.

How many function graphs could you draw that satisfy that?

Any squiggly line that crosses the x-axis at x = 1 and x = 2 would work. It could go way up then come way down. Or it could squiggle a lot. It doesn't even need to be continuous. To be the graph of a function, all it has to do is have SOME value for every input value of x; and pass the vertical line test: exactly one point on the graph directly above each x-value.

With me so far? I didn't bother to draw pictures, but hopefully this high school math is familiar to you.

Clearly there are infinitely many functions that satisfy (*).

Now just replicate that along the entire x-axis, with zeros at every integer value.

I hope you can see that there are infinitely many functions that all have the same zeros, even if there are infinitely many zeros. And by the subtraction idea that you used, there are infinitely many pairs of distinct functions that have the same zeros.

Does any of this ring a bell? You need the definition of a function, and the vertical line test, both of which are part of the standard high school math curriculum.

If you require continuity, or differentiability, or n-times differentiability, the argument still works.

[Skepdick]

Consider an error-free set of data. By error-free, I mean that the data is precise to the last digit. Let's call the set of data "y". Now assume that there are at least two theories that can explain the set of data. Let's call the theories A and B. This means that there are two unidentical functions related to theories. This means that y=A(a,b,c,...) where the set {a,b,c,...} is the set of variables. In the same manner y=B(a,b,c,...). Now let's subtract A from B. This means that A(a,b,c,...)-B(a,b,c,...)=0 since they predict the same the same set of data. Let's call C(a,b,c,...)=A(a,b,c,...)-B(a,b,c,...). C(a,b,c,...) however has to be zero for all changes in the variables. Therefore, C(a,b,c,...) is a NULL function. Therefore two theories are identical.

This is Plato's problem of one and the many...it's undecidable and subject to your meta-theoretical beliefs.

Ignore all the details because that's just a red herring. Go abstract. You there? Not abstract enough!

Take a mathematical object A. A number? A vector? A set? The continuum? A set of continuums? A space? A category? A theory? A system of theories? WHO CARES!

You have object A.
You have two explanations for A: E1(A) and E2(A).

Lets suppose that you have some notion of an "identity proof". And that you've proven E1 and E2 are "identical" with respect to that notion.
Lets also suppose that you have an infinitely many explanations. So E3, E4, E5... And you have an infinite set of proofs that all of those explanations are "identical" with respect to your notion of "identity".

Question: Are all of the identity-proofs themselves identical?

Axiom K says: All the identity proofs are essentially trivial - all the explanations in your infinite set of explanations are all the same
Univalence axiom says All the identity proofs are themselves unique/canonical.

TL;DR the explanations may well be different in their internal language/encoding/definition/structure, but they may be extensionally equivalent.

The truth on the matter? It's whatever you want it to be... because Mathematics is invented.

[Age]

For any 'data' there can be any number of theories.

'Theories' are more or less just assumptions or guesses about some thing. So, obviously, there can be any number of assumptions or guesses about what 'data', itself, 'means'.

Also, one person, itself, can have many theories about what a so-called 'error-free set of data' 'means or implies', exactly.

[bahman]

How could you justify this?

First let's do it for two zeros.

(*) Consider a function that inputs a real number and outputs a real number, such that f(1) = f(2) = 0.

How many function graphs could you draw that satisfy that?

Any squiggly line that crosses the x-axis at x = 1 and x = 2 would work. It could go way up then come way down. Or it could squiggle a lot. It doesn't even need to be continuous. To be the graph of a function, all it has to do is have SOME value for every input value of x; and pass the vertical line test: exactly one point on the graph directly above each x-value.

With me so far? I didn't bother to draw pictures, but hopefully this high school math is familiar to you.

Clearly there are infinitely many functions that satisfy (*).

Now just replicate that along the entire x-axis, with zeros at every integer value.

I hope you can see that there are infinitely many functions that all have the same zeros, even if there are infinitely many zeros. And by the subtraction idea that you used, there are infinitely many pairs of distinct functions that have the same zeros.

Does any of this ring a bell? You need the definition of a function, and the vertical line test, both of which are part of the standard high school math curriculum.

If you require continuity, or differentiability, or n-times differentiability, the argument still works.

I am talking about a continuous set of data. Of course, you can have infinite number of functions that passes through discrete set of data.

[bahman]

Consider an error-free set of data. By error-free, I mean that the data is precise to the last digit. Let's call the set of data "y". Now assume that there are at least two theories that can explain the set of data. Let's call the theories A and B. This means that there are two unidentical functions related to theories. This means that y=A(a,b,c,...) where the set {a,b,c,...} is the set of variables. In the same manner y=B(a,b,c,...). Now let's subtract A from B. This means that A(a,b,c,...)-B(a,b,c,...)=0 since they predict the same the same set of data. Let's call C(a,b,c,...)=A(a,b,c,...)-B(a,b,c,...). C(a,b,c,...) however has to be zero for all changes in the variables. Therefore, C(a,b,c,...) is a NULL function. Therefore two theories are identical.

This is Plato's problem of one and the many...it's undecidable and subject to your meta-theoretical beliefs.

Ignore all the details because that's just a red herring. Go abstract. You there? Not abstract enough!

Take a mathematical object A. A number? A vector? A set? The continuum? A set of continuums? A space? A category? A theory? A system of theories? WHO CARES!

You have object A.
You have two explanations for A: E1(A) and E2(A).

Lets suppose that you have some notion of an "identity proof". And that you've proven E1 and E2 are "identical" with respect to that notion.
Lets also suppose that you have an infinitely many explanations. So E3, E4, E5... And you have an infinite set of proofs that all of those explanations are "identical" with respect to your notion of "identity".

Question: Are all of the identity-proofs themselves identical?

Axiom K says: All the identity proofs are essentially trivial - all the explanations in your infinite set of explanations are all the same
Univalence axiom says All the identity proofs are themselves unique/canonical.

TL;DR the explanations may well be different in their internal language/encoding/definition/structure, but they may be extensionally equivalent.

The truth on the matter? It's whatever you want it to be... because Mathematics is invented.

What do you mean by "A: E1(A) and E2(A)"?

[Skepdick]

What do you mean by "A: E1(A) and E2(A)"?

You have two explanations for A: E1(A) and E2(A).

You have two explanations for A... E1 and E2.

You have two explanations for A.

E1 and E2.

[wtf]

sfy (*).

I am talking about a continuous set of data.

1) Nice of you to let us know about your additional hypotheses after the fact.

2) Data usually implies data collection. How do you collect a continuum-many data points?

3) If all you mean is that two functions that have the same value at every input across the continuum are the same function, you have merely stated a blatant triviality.

What was your point here?

[Skepdick]

2) Data usually implies data collection.

And "collection" implies set. Like the dataset of of Real numbers.

How do you collect a continuum-many data points?

The same way you examine continuum-many input-output pairs between two real-valued functions to identify/assert the existence of a bijection, I guess.

3) If all you mean is that two functions that have the same value at every input across the continuum are the same function, you have merely stated a blatant triviality.

Q.E.D

What was your point here?

Whatever his point... you seem to believe that f(x) = g(x) is a "blatant triviality" IFF f(x) and g(x) have the same value for every real-valued input.

So you must also believe that proving (f(x) - g(x) < 0) ∨ f(x) - g(x) > 0) is uninhabited for all real values is another "blatant triviality"
You must believe that decidable inequality for arbitrarily-small epsilons; and infinite-precision real values is "blatantly trivial" too.

That's a peculiar belief. And an even more peculiar proof-system.

[wtf]

Whatever his point... you seem to believe that f(x) = g(x) is a "blatant triviality" IFF f(x) and g(x) have the same value for every real-valued input.

That's the very definition of equality of functions.

[Skepdick]

That's the very definition of equality of functions.

That's the definition of one kind of equality. Extensional equality. There's other kinds. Like intensional equality.

Intensional equality implies extensional equality. The inverse is not true.

Surely you understand why the difference matters when you start differentiating; or integrating these "equal" functions?

[wtf]

That's the definition of one kind of equality. Extensional equality. There's other kinds. Like intensional equality.

I doubt the OP had your personal hobby horse in mind.

I won't be further engaging with your monomania.