Philosophy Now Forum

Posted: **Thu May 05, 2022 11:37 pm**

1. All logical systems require axioms.

2. These axioms are unproven except through the proofs in which they are used.

3. The proof justifies the axiom and the axiom justifies the proof.

4. Proof and axiom, as circular, thus become interchangeable: a proof is an axiom and an axiom is a proof.

5. This interchangeability, as equivocation, results in obscurity as it is self-referencing therefore the axiom is no longer the axiom and the proof is no longer the proof as both become indefinite.

6. Axioms/proofs are only that which are accepted to deal with this indefiniteness, thus logic requires intuition.

7. Intuition thus expands the circle of axiom=proof to axiom=proof=intuition.

8. However intuition is not universal as the axiom=proof would not have to be taught if it where such.

9. In the equivocation of "axiom=proof=intuition" the intuition not being universal necessitates the axiom/proof as not universal.

10. This absence of universality necessitates a multiplicity of logics, all of which may not agree.

11. This absence of agreement in logics results in the axiom of "Logic" being obscure yet logic was used to deduce this.

12. Logic=no logic

Posted: **Sun May 08, 2022 2:43 am**

Eodnhoj7 wrote: ↑Thu May 05, 2022 11:37 pm 1. All logical systems require axioms.

Yes. Agreed.

Logic is fundamentally foundationalist.

Eodnhoj7 wrote: ↑Thu May 05, 2022 11:37 pm 2. These axioms are unproven except through the proofs in which they are used.

Axioms are indeed unproven. There is really no "but" to that.

Eodnhoj7 wrote: ↑Thu May 05, 2022 11:37 pm 3. The proof justifies the axiom and the axiom justifies the proof.

No. Disagreed.

The emergent properties of a formal system, such as consistency (or completeness), do not prove the axioms. The lack of consistency merely makes their particular combination unusable as the foundation for a formal system, since no theorem can legitimately follow from an inconsistent set of premises.

Eodnhoj7 wrote: ↑Thu May 05, 2022 11:37 pm 4. Proof and axiom, as circular, thus become interchangeable: a proof is an axiom and an axiom is a proof.

No. Disagreed.

Replacing an axiom by a non-trivial theorem that follows from it, will not necessarily work, because the axiom will not necessarily follow from such theorem.

A logical inference arrow is directional. The arrow does not necessarily point in the reverse direction simultaneously, even though it sometimes does.

P => Q does not necessarily imply Q => P.

The following is not generally true:

(P => Q) <=> (Q => P)

Eodnhoj7 wrote: ↑Thu May 05, 2022 11:37 pm 6. Axioms/proofs are only that which are accepted to deal with this indefiniteness, thus logic requires intuition.

Disagreed.

Mathematical formalism insists that axioms do not need to be intuitive. It is not a requirement.

Wikipedia on "mathematical formalism" wrote: In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess."[1] According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all.

Formalism is the standard counterargument to the idea that mathematics would be "about X", such as for example "intuition":

Mathematics is not "about X" because mathematics is not about anything at all.

Posted: **Thu May 12, 2022 12:44 am**

godelian wrote: ↑Sun May 08, 2022 2:43 am
Eodnhoj7 wrote: ↑Thu May 05, 2022 11:37 pm 1. All logical systems require axioms.
Yes. Agreed.

Logic is fundamentally foundationalist.

Eodnhoj7 wrote: ↑Thu May 05, 2022 11:37 pm 2. These axioms are unproven except through the proofs in which they are used.
Axioms are indeed unproven. There is really no "but" to that.

Eodnhoj7 wrote: ↑Thu May 05, 2022 11:37 pm 3. The proof justifies the axiom and the axiom justifies the proof.
No. Disagreed.

The emergent properties of a formal system, such as consistency (or completeness), do not prove the axioms. The lack of consistency merely makes their particular combination unusable as the foundation for a formal system, since no theorem can legitimately follow from an inconsistent set of premises.

Eodnhoj7 wrote: ↑Thu May 05, 2022 11:37 pm 4. Proof and axiom, as circular, thus become interchangeable: a proof is an axiom and an axiom is a proof.
No. Disagreed.

Replacing an axiom by a non-trivial theorem that follows from it, will not necessarily work, because the axiom will not necessarily follow from such theorem.

A logical inference arrow is directional. The arrow does not necessarily point in the reverse direction simultaneously, even though it sometimes does.

P => Q does not necessarily imply Q => P.

The following is not generally true:

(P => Q) <=> (Q => P)

Eodnhoj7 wrote: ↑Thu May 05, 2022 11:37 pm 6. Axioms/proofs are only that which are accepted to deal with this indefiniteness, thus logic requires intuition.
Disagreed.

Mathematical formalism insists that axioms do not need to be intuitive. It is not a requirement.

Wikipedia on "mathematical formalism" wrote: In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess."[1] According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all.
Formalism is the standard counterargument to the idea that mathematics would be "about X", such as for example "intuition":

Mathematics is not "about X" because mathematics is not about anything at all.

1. Axioms are justified through their proofs. Without the proof the axiom does not exist. 1 does not exist without 1+1=2 or 2,3,4,5...

2. 1 is justified as an axiom because of the proof(s) 1+1=2, 1+1=1=3, etc.

3. An axiom results in a proof. The proof, dues to its foundations in axiom(s), in turn becomes self-evident thus becomes an axiom. The axiom(s) becomes a proof and the proof in turn becomes an axiom as the proof becomes a foundation for further proofs.

4. The truths of math/logic not being about "anything at all" contradicts them being the "manipulation of strings" as the "manipulation of strings" is a thing.

Posted: **Thu May 12, 2022 3:12 am**

Eodnhoj7 wrote: ↑Thu May 12, 2022 12:44 am 1. Axioms are justified through their proofs. Without the proof the axiom does not exist. 1 does not exist without 1+1=2 or 2,3,4,5...

A theorem follows from the axioms. A theorem is itself not an axiom. Furthermore, in Peano Arithmetic Theory, the number "1" is not an axiom.

- zero is an axiom
- the successor function is an axiom.

"1" is represented as S(0), i.e. the successor to zero.

In which arithmetic theory exactly is "1" an axiom?

Peano Axioms

Peano's axioms wrote: The nine Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic".[3] The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second-order statement of the principle of mathematical induction over the natural numbers, which makes this formulation close to second-order arithmetic.

Eodnhoj7 wrote: ↑Thu May 12, 2022 12:44 am 3. An axiom results in a proof. The proof, dues to its foundations in axiom(s), in turn becomes self-evident thus becomes an axiom.

You seem to confuse the terms "axiom" and "theorem":

Wikipedia on the term "axiom" wrote: An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the philosophy of mathematics.

Wikipedia on the term "theorem" wrote: In mathematics, a theorem is a statement that has been proved, or can be proved.

The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.

A theorem has a proof while an axiom does not.

Eodnhoj7 wrote: ↑Thu May 12, 2022 12:44 am 4. The truths of math/logic not being about "anything at all" contradicts them being the "manipulation of strings" as the "manipulation of strings" is a thing.

A string does not have to mean anything. For example:

concat("abc","d") = "abcd"

This logic sentence is true. Still, the strings that it refers to, i.e. "abc", "d", and "abcd", do not mean anything. This logic sentence is merely a formalist truth.

In that sense, this true logic sentence is "not about anything at all".

Posted: **Thu May 12, 2022 7:59 am**

godelian wrote: ↑Thu May 12, 2022 3:12 am A theorem follows from the axioms. A theorem is itself not an axiom.

The difference is in connotation,not denotation.

If you are doing regular mathematics axioms are something assumed true.
If you are doing reverse mathematics theorems are something assumed true.

In the language of logic: both axioms and theorems can act as premises; or conclusions.
Inputs to functions; or outputs of functions.

godelian wrote: ↑Thu May 12, 2022 3:12 am You seem to confuse the terms "axiom" and "theorem"

You don't seem to differentiate between Mathematics and Reverse Mathematics.

Anything that's an axiom could be a theorem. Anything that's a theorem could be an axiom.

Subject to choice.

godelian wrote: ↑Thu May 12, 2022 3:12 am A theorem has a proof while an axiom does not.

Proofs are programs.

You can write a program/proof that shows a theorem holds within some axioms; or you can write a program/proof that some axiom are sufficient, while others are necessary for a theorem to hold.

godelian wrote: ↑Thu May 12, 2022 3:12 am A string does not have to mean anything. For example:

concat("abc","d") = "abcd"

Then how come you have given the string concat("abc","d") = "abcd" a computational meaning ?

godelian wrote: ↑Thu May 12, 2022 3:12 am This logic sentence is true.

It's neither true nor false. The sentence has no truth-value until you assign it one - it has no truth-value until you evaluate it.

concat("abc","d") evaluates to "abcd" for some implementations of concat(). It may evaluate to something else for other implementations of concat.
If you are insisting that the sentence is true, then you are necessarily constraining the domain of discourse ONLY to those implementations of concat() for which the sentence evaluates to true e.g you have promoted ' concat("abc","d") = "abcd" ' to an axiom.

There are, of course, implementations of concat() for which the expression is false.

Code: Select all

In [1]: string='concat("abc","d") == "abcd"'

In [2]: def concat(a, b):
   ...:     return b+a
   ...:

In [3]: eval(string)
Out[3]: False

godelian wrote: ↑Thu May 12, 2022 3:12 am In that sense, this true logic sentence is "not about anything at all".

It's not about anything in particular. It's about the way everything relates to everything; and the way all entities interact.
Very many interesting things begin to happen the moment I ask you what concat("abcd","a") evaluates to. You could answer "abcd"; or you could answer "abcda".

e.g it could be that concat() uses strings to represent sets; or lists.

Posted: **Thu May 12, 2022 9:51 am**

Eodnhoj7 wrote: ↑Thu May 05, 2022 11:37 pm 12. Logic=no logic

Uh. No.

Logic ≡ no (no Logic)
no (Logic ≡ no Logic)

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

Posted: **Thu May 12, 2022 1:35 pm**

Skepdick wrote: ↑Thu May 12, 2022 7:59 am If you are doing regular mathematics axioms are something assumed true.
If you are doing reverse mathematics theorems are something assumed true.

In the language of logic: both axioms and theorems can act as premises; or conclusions.
Inputs to functions; or outputs of functions.

Ok, agreed. Reverse mathematics reverses theorem with axiom.

Skepdick wrote: ↑Thu May 12, 2022 7:59 am You don't seem to differentiate between Mathematics and Reverse Mathematics.

The default interpretation for these terms is still Mathematics and not Reverse Mathematics. Without further qualification, you cannot just reverse the meaning of axiom and theorem. You need to specifically mention that it is about reverse mathematics when doing that.

Skepdick wrote: ↑Thu May 12, 2022 7:59 am Proofs are programs.

You can write a program/proof that shows a theorem holds within some axioms; or you can write a program/proof that some axiom are sufficient, while others are necessary for a theorem to hold.

Ok. Agreed.

There is even formal proof for "proofs as programs and programs as proofs" (Curry-Howard correspondence) but to me the proof is not trivial because it apparently makes use of advanced combinatory logic. I have had a look at it but I could not wrap my head around it.

Skepdick wrote: ↑Thu May 12, 2022 7:59 am Then how come you have given the string concat("abc","d") = "abcd" a computational meaning ?

It's neither true nor false. The sentence has no truth-value until you assign it one - it has no truth-value until you evaluate it.

concat("abc","d") evaluates to "abcd" for some implementations of concat(). It may evaluate to something else for other implementations of concat.
If you are insisting that the sentence is true, then you are necessarily constraining the domain of discourse ONLY to those implementations of concat() for which the sentence evaluates to true e.g you have promoted ' concat("abc","d") = "abcd" ' to an axiom.

There are, of course, implementations of concat() for which the expression is false.
Code: Select all
In [1]: string='concat("abc","d") == "abcd"'

In [2]: def concat(a, b):
   ...:     return b+a
   ...:

In [3]: eval(string)
Out[3]: False

Okay, truth is only defined in the context of the interpretation of a formal system. I was assuming that the expression was true in a particular further unspecified context. I wasn't claiming that it was always true in every context.

Skepdick wrote: ↑Thu May 12, 2022 7:59 am It's not about anything in particular. It's about the way everything relates to everything; and the way all entities interact.
Very many interesting things begin to happen the moment I ask you what concat("abcd","a") evaluates to. You could answer "abcd"; or you could answer "abcda".

e.g it could be that concat() uses strings to represent sets; or lists.

Yes, the formalist idea is merely that such logic sentences are "not a body of propositions representing an abstract sector of reality."
In other words, it does not need to correspond with anything in the physical universe. This is not a requirement.
Unlike science, mathematics is indeed an exercise in coherentism and not in correspondentism.

Posted: **Thu May 12, 2022 5:51 pm**

godelian wrote: ↑Thu May 12, 2022 1:35 pm The default interpretation for these terms is still Mathematics and not Reverse Mathematics.

The default interpretation is that there are no default interpretations.

godelian wrote: ↑Thu May 12, 2022 1:35 pm Without further qualification, you cannot just reverse the meaning of axiom and theorem. You need to specifically mention that it is about reverse mathematics when doing that.

I am not reversing it. I am just interpreting it from a logical viewpoint. This being a philosophy forum and all. That may be a more default interpretation than the one you've chosen.

In mathematics axioms are premises - theorems are conclusions.
In reverse mathematics theorems are premises - axioms are conclusions.

They are just duals.

godelian wrote: ↑Thu May 12, 2022 1:35 pm There is even formal proof for "proofs as programs and programs as proofs" (Curry-Howard correspondence) but to me the proof is not trivial because it apparently makes use of advanced combinatory logic. I have had a look at it but I could not wrap my head around it.

That formalism depends on a very constrained notion of a "program".

Combinatory logic is stateless - it has no memory, just permutations.
Sequent logic has state (memory).

The sort of problems you can solve computationally with no memory vs the sort of problems you can solve with 1 bit of memory are very different.

godelian wrote: ↑Thu May 12, 2022 1:35 pm Okay, truth is only defined in the context of the interpretation of a formal system. I was assuming that the expression was true in a particular further unspecified context. I wasn't claiming that it was always true in every context.

Sure, but the point stands. concat("abc","d") == "abc" is a predicate. The implementations of concat() is a free variable.

There are many valid solutions to that equation.

godelian wrote: ↑Thu May 12, 2022 1:35 pm Unlike science, mathematics is indeed an exercise in coherentism and not in correspondentism.

I am not so sure about that. We have operators in formal languages. Operators do work.

DOING stuff seems to correspond to machines in the real world.

Posted: **Thu May 12, 2022 6:32 pm**

Skepdick wrote: ↑Thu May 12, 2022 9:51 am
Eodnhoj7 wrote: ↑Thu May 05, 2022 11:37 pm 12. Logic=no logic
Uh. No.

Logic ≡ no (no Logic)
no (Logic ≡ no Logic)

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

If logic is obscure, due to its multiplicity of meanings, then logic as contradictory is logic as illogical.

However:

1. In reference to your point the following argument can be made:

a. There is a totality of things.
b. This totality is one.
c. This one totality is without comparison, as it is everything, thus is indefinite.
d. This indefiniteness is the same as nothing, as there is no contrast, therefore 0.
e. 1=0 and A=-A.

And then:

2. Logic is dependent upon further axioms/proofs beyond the aforementioned axioms/proofs therefore is connected to that which is illogical as the axioms/proofs required unjustified axioms/proofs beyond it. As connected they equivocate because their identities are interwoven.

For example the statement 1+1=2 requires objects, which can be anything and infinite in number, beyond it but these objects are not yet identified. As such these unidentified objects are illogical as they have no identity yet. However let us say the objects are identified and (1+1+2)=A. A is identified but has further unidentified mathematical statements beyond it as A can be broken down to further parts or is a fraction of something much larger. What is defined is connected to that which is undefined and their identities are interwoven.

A simpler statement can be made in that "all actualities contain potentialities and all potentialities contain actualities".

Posted: **Thu May 12, 2022 6:36 pm**

godelian wrote: ↑Thu May 12, 2022 3:12 am
Eodnhoj7 wrote: ↑Thu May 12, 2022 12:44 am 1. Axioms are justified through their proofs. Without the proof the axiom does not exist. 1 does not exist without 1+1=2 or 2,3,4,5...
A theorem follows from the axioms. A theorem is itself not an axiom. Furthermore, in Peano Arithmetic Theory, the number "1" is not an axiom.

- zero is an axiom
- the successor function is an axiom.

"1" is represented as S(0), i.e. the successor to zero.

In which arithmetic theory exactly is "1" an axiom?

Peano Axioms

Peano's axioms wrote: The nine Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic".[3] The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second-order statement of the principle of mathematical induction over the natural numbers, which makes this formulation close to second-order arithmetic.

Eodnhoj7 wrote: ↑Thu May 12, 2022 12:44 am 3. An axiom results in a proof. The proof, dues to its foundations in axiom(s), in turn becomes self-evident thus becomes an axiom.
You seem to confuse the terms "axiom" and "theorem":

Wikipedia on the term "axiom" wrote: An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the philosophy of mathematics.

Wikipedia on the term "theorem" wrote: In mathematics, a theorem is a statement that has been proved, or can be proved.

The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.
A theorem has a proof while an axiom does not.

Eodnhoj7 wrote: ↑Thu May 12, 2022 12:44 am 4. The truths of math/logic not being about "anything at all" contradicts them being the "manipulation of strings" as the "manipulation of strings" is a thing.
A string does not have to mean anything. For example:

concat("abc","d") = "abcd"

This logic sentence is true. Still, the strings that it refers to, i.e. "abc", "d", and "abcd", do not mean anything. This logic sentence is merely a formalist truth.

In that sense, this true logic sentence is "not about anything at all".

1. I am not talking about theorems I am talking about proofs. 1+1=2 is a proof as it points out the relations of 1 and 1. 1+1=2 justifies 1 as an axiom. 1 justifies 1+1=2 as a proof (which in turn is an axiom as 1+1=2 is an axiom as well as a proof). In normal grade school arithmetic the fact the numbers are used as counters makes 1 and its successors as axiomatic.

2. A string means the relations of the symbols being used; its meaning is in relations. Math means symbols and symbols are a thing.

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