Is objective truth mathematics?

[PseuTonym]

So what is wrong with my definition?

a set of mind independent rules between objects that can explain every state of reality. Isn’t this the definition of mathematics?

Let’s begin with “a set of rules.”

If you are claiming something about an entire set of rules, then we cannot test any one rule and hope for much. So, it seems that you need the entire set of rules before you can evaluate them as a whole set to determine whether or not they fulfill your requirements. Is there a complete listing of the rules in that set, or is it an infinite set? Can you give any kind of reference, such as a link to a website, or an author, title, and page number?

Suppose that we are looking at a single rule. I don’t know why we are supposed to look at a rule.

Are you thinking of a computational method?

Including the requirement “between objects” doesn’t seem helpful. Can you give an example of a rule in mathematics that isn’t a rule between objects?

Why not look at a conjecture?

If a mathematical conjecture has such a structure that it is possible for a finite computation to refute the conjecture … then somebody might discover and record data that refutes the conjecture.

It might be worthwhile to preserve that record of the counter-example, so that people won’t put in too much effort trying to prove that the conjecture is true, and so that they have an opportunity to use the data to perform the computation and persuade themselves that the data does indeed provide a counter-example to the conjecture.

Alternatively, perhaps the generally accepted premises of mathematics in the year 2016 are contradictory, and it is possible to deduce that the conjecture is true from the assumption that all of the premises are true, even though the conjecture is actually false. In that case, the effort to prove that the conjecture is true might have a revolutionary effect in mathematics. However, if nobody preserves a record of the counter-example data, then the revolution won’t occur until somebody rediscovers the counter-example.

What do you think is an appropriate subject area label for a finite amount of data that constitutes a counter-example to a mathematical conjecture? Can we file it under mathematics? Are we obligated to pretend that mathematics includes no specific facts of any significance, and that everything that merits the label “mathematics” has to be both abstract and general?

And what is the purpose of this? Is it supposed to persuade somebody that – although the history of mathematics includes errors – today the label “mathematics” is enough to guarantee that all of the statements that have been given that label are true statements?

 

[STT]

a set of mind independent rules between objects that can explain every state of reality. Isn’t this the definition of mathematics?

Good, but lets to stick to the following definition: Mathematics is a set of abstract rules between objects that can allow producing knowledge.

Let’s begin with “a set of rules.”

If you are claiming something about an entire set of rules, then we cannot test any one rule and hope for much. So, it seems that you need the entire set of rules before you can evaluate them as a whole set to determine whether or not they fulfill your requirements. Is there a complete listing of the rules in that set, or is it an infinite set?

The set of rules is not necessary infinite. The set of rules in principle should be big enough to minimally exhaust all objects. By exhaust I mean that we should be able to reach to any object by operating on another object.

Can you give any kind of reference, such as a link to a website, or an author, title, and page number?

Just think of Arithmetic. You have a set of object which are number and a set of rules like, plus, minus, etc. I think this is understandable for a person who knows mathematics.

Suppose that we are looking at a single rule. I don’t know why we are supposed to look at a rule.

I don’t understand what you are trying to say here.

Are you thinking of a computational method?

Computational method are part of mathematics too.

Including the requirement “between objects” doesn’t seem helpful. Can you give an example of a rule in mathematics that isn’t a rule between objects?

NULL operator.

Why not look at a conjecture?

I think that we need intuition to set up a set of rules and objects.

If a mathematical conjecture has such a structure that it is possible for a finite computation to refute the conjecture … then somebody might discover and record data that refutes the conjecture.

It might be worthwhile to preserve that record of the counter-example, so that people won’t put in too much effort trying to prove that the conjecture is true, and so that they have an opportunity to use the data to perform the computation and persuade themselves that the data does indeed provide a counter-example to the conjecture.

Alternatively, perhaps the generally accepted premises of mathematics in the year 2016 are contradictory, and it is possible to deduce that the conjecture is true from the assumption that all of the premises are true, even though the conjecture is actually false. In that case, the effort to prove that the conjecture is true might have a revolutionary effect in mathematics. However, if nobody preserves a record of the counter-example data, then the revolution won’t occur until somebody rediscovers the counter-example.

What do you think is an appropriate subject area label for a finite amount of data that constitutes a counter-example to a mathematical conjecture? Can we file it under mathematics? Are we obligated to pretend that mathematics includes no specific facts of any significance, and that everything that merits the label “mathematics” has to be both abstract and general?

I have no idea how this is related to our discussion.

And what is the purpose of this?

We are just trying to define mathematics.

Is it supposed to persuade somebody that – although the history of mathematics includes errors – today the label “mathematics” is enough to guarantee that all of the statements that have been given that label are true statements?

The existence of an error of course questions all of the related statements.