Zero divided by zero revisited

[alan1000]

Whoops, a blooper: I posted "not customary to supply supporting arguments" incorrectly. Pls disregard that post, my mistake.

[Age]

As a final comment, I thank Skepdick for making the only reply in the entire thread which demonstrated any actual knowledge of mathematics, or evidence of critical thinking skills, or understanding of the philsophical method.

Do you still believe that zero can be divided by zero?

If yes, then what is the answer, exactly?

Age, have you understood a single sentence I posted????

"alan1000" have you understood the single thing that I have pointed out and shown here?

Also, and again, do you still really believe that zero can be divided by zero?

if yes, then what is the answer, exactly?

[godelian]

All four positions can be supported by respectable mathematical arguments. What's your preference?

An objective answer to this question -- and not as a matter of subjective preference -- was gradually developed during the 19th century:

https://en.wikipedia.org/wiki/Continuous_function

A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy defined continuity of y = f ( x ) as follows: an infinitely small increment α \of the independent variable x always produces an infinitely small change f ( x + α ) − f ( x ) of the dependent variable y (see e.g. Cours d'Analyse, p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see microcontinuity). The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s, but the work wasn't published until the 1930s.

Like Bolzano,[1] Karl Weierstrass[2] denied continuity of a function at a point c unless it was defined at and on both sides of c, but Édouard Goursat[3] allowed the function to be defined only at and on one side of c, and Camille Jordan[4] allowed it even if the function was defined only at c. All three of those nonequivalent definitions of pointwise continuity are still in use. [5] Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854.[6]

https://en.wikipedia.org/wiki/Limit_of_a_function

Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x. We say that the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, the output value can be made arbitrarily close to L if the input to f is taken sufficiently close to p.

(ε, δ)-definition of limit

One would say that the limit of f, as x approaches p, is L, if the following property holds:

for every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < |x − p| < δ implies |f(x) − L| < ε.

Symbolically:

( ∀ ε > 0 ) ( ∃ δ > 0 ) ( ∀ x ∈ R ) ( 0 < | x − p | < δ ⟹ | f ( x ) − L | < ε ) .

So, either the notions of continuity and limit will provide an objective answer to the question, or else, the answer is simply undefined.

There is absolutely no scope for any personal preference in this matter.

On the contrary, any attempt at proposing personal preferences on this subject ignores or even denies two centuries of mathematical work by numerous grandees in the field and must be rejected categorically for reasons of being incompetent.

[Skepdick]

An objective answer to this question -- and not as a matter of subjective preference -- was gradually developed during the 19th century:

The final answer to a non-trivial problem in mathematics is always a philosophical choice. That is simply the nature of the beast.

I think you have some rethinking to do

[Skepdick]

So, either the notions of continuity and limit will provide an objective answer to the question, or else, the answer is simply undefined.

There is absolutely no scope for any personal preference in this matter.

Not all functions are continuous....


https://math.andrej.com/2006/03/27/some ... ontinuous/

[godelian]

An objective answer to this question -- and not as a matter of subjective preference -- was gradually developed during the 19th century:

The final answer to a non-trivial problem in mathematics is always a philosophical choice. That is simply the nature of the beast.

I think you have some rethinking to do

The choice of theory, i.e. its axioms, is a philosophical choice. However, once you write something like "0/0", you have implicitly made that choice already by choosing the axiomatization of the real (or the complex) numbers. Therefore, you have effectively painted yourself into a corner. The philosophical degrees of freedom are mostly gone at that point. If you want to come up with a different answer from the standard one, you will need to explicitly modify the context of problem in such a way that it leaves enough space for making subjective choices. Otherwise, you will find yourself locked in pretty much completely.

Example of creating some freedom for personal choices:

5 x 3 = 2

(in a finite field of 13 elements)

So, 5 x 3 doesn't need to be 15.

[Skepdick]

The choice of theory, i.e. its axioms, is a philosophical choice. However, once you write something like "0/0", you have implicitly made that choice already by choosing the axiomatization of the real (or the complex) numbers. Therefore, you have effectively painted yourself into a corner.

Why do you assume such a mode of reasoning? Why can't I start with theorems first? I want the expression to be true.

At worst I can't axiomatize it. So what? I got the identity axiom. And thus I declare that "1/0 = 0" is identical with True.

https://xenaproject.wordpress.com/2020/ ... ory-a-faq/

[godelian]

Why do you assume such a mode of reasoning? Why can't I start with theorems first? I want the expression to be true.

Ok, reverse mathematics is indeed a legitimate activity.

However, it is an obscure niche.

You'd better clarify upfront that this is what you want to do. The default assumption is that you want to prove theorems from axioms and not that you intend to discover axioms that will make your theorem true.

At worst I can't axiomatize it. So what? I got the identity axiom. And thus I declare that "1/0 = 0" is identical with True.

https://xenaproject.wordpress.com/2020/ ... ory-a-faq/

If you accept that "1/0=0" you'd better find a way to prevent this claim from backfiring:

1/0=0. => 1=0*0 => 1=0

In the paper that you referenced about doing this in the Lean theorem prover, they carefully prevent the above chain of reasoning from ever executing. Fine. If it will never explode in their faces, then why not?

[Skepdick]

Why do you assume such a mode of reasoning? Why can't I start with theorems first? I want the expression to be true.

Ok, reverse mathematics is indeed a legitimate activity.

However, it is an obscure niche.

Niche? It's the default way to use mathematics if you are a physicist or a software engineer.

The theorem defines the invariants you want to preserve e.g symmetries. Then you go and find a model; or construct a model.

You'd better clarify upfront that this is what you want to do. The default assumption is that you want to prove theorems from axioms and not that you intend to discover axioms that will make your theorem true.

Why do you assume that either one of those things is the "default"? I simply view them as two possible choices. Two different ways to use mathematics.

f: A -> B
f: A <- B

Potato/potatoh. It's just arrow-reversal.

1/0=0. => 1=0*0 => 1=0

Why do you pre-suppose that "/" has a multiplicative inverse?

In the paper that you referenced about doing this in the Lean theorem prover, they carefully prevent the above chain of reasoning from ever executing. Fine. If it will never explode in their faces, then why not?

Because nobody ever draws any inferences from the expression. And nobody ever calls the "/" operator with 0 as the second parameter.

If they did - it would trigger an exception and the program would terminate. Exceptions/exception-handling is a foreign concept to Mathematicians.

They don't know how to surface information nested arbitrarily deep in their abstractions.

[godelian]

Why do you pre-suppose that "/" has a multiplicative inverse?

Concerning algebraic structures:

(multiplication,division)

(defined,defined) -> field
(defined,undefined) -> ring
(undefined,defined) -> ???

Concerning an algebraic structure in which the division is defined but not the multiplication, I have never thought of that possibility. It is of course conceivable but certainly not common. Does such algebraic structure even have a name? Any examples? Notable properties?

[Skepdick]

Why do you pre-suppose that "/" has a multiplicative inverse?

Concerning algebraic structures:

Algebraic? Who said anything about algebraic.

We are dealing with the geometry of syntax. Formal semantics.

(undefined,defined) -> ???

??? is right! Whatever name you choose to give it - it exists.

Concerning an algebraic structure in which the division is defined but not the multiplication, I have never thought of that possibility. It is of course conceivable but certainly not common. Does such algebraic structure even have a name? Any examples? Notable properties?

It's not only conceivable - it exists. The Lean proof assistant. Its one notable property is that it's NOT abstract

It has been reified.

https://en.wikipedia.org/wiki/Reificati ... r_science)

[godelian]

It's not only conceivable - it exists. The Lean proof assistant.
It has been reified.

So, we need a set for which the division is closed under division but not under multiplication.

The reverse is true for the integers.

What would be an example of a set the is closed under division but not under multiplication?

Is this set also closed under addition and subtraction, with these operations being each other's reverse?

[Skepdick]

So, we need a set for which the division is closed under division but not under multiplication.

Do we need a set? You seem to be pre-supposing set theory as fundamental.

Also what do you mean by "not closed"? Do you mean open, clopen or neither?

Remember - negation is non-trivial under constructivism. Unless LEM holds.

[godelian]

So, we need a set for which the division is closed under division but not under multiplication.

Do we need a set? You seem to be pre-supposing set theory as fundamental.

Also what do you mean by "not closed"? Do you mean open, clopen or neither?

Remember - negation is non-trivial under constructivism. Unless LEM holds.

In my impression, it is quite inconvenient to do abstract algebra without assuming set theory, if only, because pretty much the entire literature on the subject assumes it.

If you remove this connection, you will probably have to proceed without any documentation.

Unless there exists material that does abstract algebra without assuming that an algebraic structure contains a carrying set, the whole exercise will end up being unsupported.

[Skepdick]

In my impression, it is quite inconvenient to do abstract algebra without assuming set theory, if only, because pretty much the entire literature on the subject assumes it.

In my experience it is not at all useful to do any abstract reasoning using a theory which lacks a built-in deductive system.

And I find it particularly constraining on thought to be bound by the after-thought to set theory that is first order logic.

If you remove this connection, you will probably have to proceed without any documentation.

Unless there exists material that does abstract algebra without assuming that an algebraic structure contains a carrying set, the whole exercise will end up being unsupported.

Yeah sure: https://homotopytypetheory.org/bookhttp ... .org/book/

What object you choose to use to fulfil the role of a carrying set; and how deeply you choose to nest your carriers is entirely up to you.

One option is to use the universe-type: https://ncatlab.org/nlab/show/type+universe

Personally, I find it rather limiting that set theory has only two data types: sets and elements. How do you even begin to construct a multiset out of that?