What is Truth, Logic, and Math?

[Veritas6]

What do you make of these claims about truth, logic, and math?

The objector argues against Platonic realism, the idea that: “conceptual tools like logic and mathematics must necessarily possess a kind of intrinsic existence unto themselves, independent of space and time.” How would Aristotle’s moderate realism play into these claims below?:

“it’s one thing for an individual to be ‘absolutely certain’ of a given proposition in accordance with well-specified rules, but it’s another thing entirely for that certainty to transcend all time and space across all linguistic and cultural barriers. ‘Truth’ is not a physical entity unto itself, but simply a label given to ideas in accordance with made-up goals. No proposition can ever be ‘absolute,’ in this sense, because no epistemic system is universally binding us to consistently label a given proposition as ‘true.’”

“whenever we talk about a thing like logic, we’re not talking about some intrinsic metaphysical essence of the universe. Rather, what we’re really talking about is a system of rules that operates on linguistic propositions… It is not a singular entity unto itself, nor does it occupy any particular location in the universe. It is, however, a process that people engage in. It’s something you do.”

“science doesn’t ‘presuppose’ anything. It’s a process. You either do science or you to not do science. However, part of doing science is the logical expression of your ideas. Logic is not something that is ‘presupposed’ but rather invented to express ideas in a rigorous manner.”

“[The law of non-contradiction] doesn’t [describe reality]. LNC is a rule imposed on language and propositions. Nothing more. Failure to abide by LNC is simply a failure to put words together properly… Contradictions are not allowed to mean anything in our use of language. The only reason contradictions don’t ‘exist in nature’ is because contradictions literally ‘don’t describe anything.’ They are words put together wrongly.”

“logic is a human invention that governs the manipulation of propositions. Logical rules are thus analytic in nature, and do nothing to describe objective reality. They only dictate the rules by which we TALK about reality.”

(continued below)

 

[Veritas6]

“the symmetric law of equality is nothing more than an axiomatic assertion - a self-imposed rule for the manipulation of mathematical symbols based on the definition of equality. And all this particular rule says is that the truth of an equality is independent of its order in expression. Any decent high-school level textbook on basic mathematics will openly introduce itself with the fundamental axioms of algebra - rules made up by human minds for the express purpose of consistent manipulation of mathematical expressions… This is why we say that mathematics is ‘invented’ and not ‘discovered.’ The only real ‘discovery’ that occurs in mathematics is a rigorous implementation of the rules toward their natural conclusions. Mathematical theories are really only ‘valid’ just so long as they avoid contradictions.”

“propositions can either be true or they can be false, but there is no such thing as raw ‘essence of truth’ interwoven into the fabric of space and time… a truth value is classically defined as a member of a binary set that contains the elements ’True’ and ‘False’. The purpose of this set is to serve as a kind of marker for linguistic propositions in order to help us measure their epistemic ’correctness.’ What exactly that means is open to some interpretation, but we can give it a rigorous definition through a mechanism known as a truth assignment… a truth assignment (also called a truth valuation, or an interpretation) is defined as a mapping function between the set of simple linguistic propositions and the set of binary truth values

“mathematics itself is purely axiomatic in nature - a bunch of made-up rules based on operations and relationships between elemental abstractions within arbitrarily-defined sets. Even deductive logic is the same thing: axiomatic rules designed to formalize our capacity for sorting out information and evaluating the final ‘truth’ of various propositions. So while countless hack philosophers love to brag about the ‘absolute truth’ behind a phrase like ‘2+2 = 4,’ they nearly always overlook the fact that this is simply an application of arbitrary rules toward their natural conclusions… statements that must universally be labeled as ‘true’ by all people for all time, certainly do not exist.”

(emphasis mine)

 

[Barnesy]

So while countless hack philosophers love to brag about the ‘absolute truth’ behind a phrase like ‘2+2 = 4,’ they nearly always overlook the fact that this is simply an application of arbitrary rules toward their natural conclusions**… statements that must universally be labeled as ‘true’ by all people for all time, certainly do not exist."

If you have two things and then you have another two things then there are definitely four things. Thats something that people call axiomatic. We can then make up rules that say that 2+2 equals 4. The way we write those down and where we put the numbers and how we tell people how that works might be different but 2+2 always equals 4.

 

[Veritas6]

If you have two things and then you have another two things then there are definitely four things. Thats something that people call axiomatic. We can then make up rules that say that 2+2 equals 4. The way we write those down and where we put the numbers and how we tell people how that works might be different but 2+2 always equals 4.

Yes, 2 + 2 will always equal 4. I suppose one counter could be “2”, “4”, “+”, “=“ can be defined differently (because they’re only symbols), but you would still get 4 apples from adding 2 apples to 2 more apples.

This is why we say that mathematics is ‘invented’ and not ‘discovered.’ The only real ‘discovery’ that occurs in mathematics is a rigorous implementation of the rules toward their natural conclusions. Mathematical theories are really only ‘valid’ just so long as they avoid contradictions.”

It just doesn’t make sense how the objector says it’s all “arbitrary rules” based on “natural conclusions”. What are natural conclusions? Here is one link with further information:

AnticitizenX: Part 2: Absolute Truth

 

[rossum]

Yes, 2 + 2 will always equal 4.

Not in mathematics. “2 + 2” is undefined in number bases 2 and below. 2 + 2 = 11 in base 3. 2 + 2 = 10 in base 4. 2 + 2 = 4 in bases 5 and higher.

The truth of a statement in mathematics depends on which set of axioms are in place for that statement. That can be an advantage, for example for Trinitarians: 3 = 1 is true, modulo 2.

The parable of truth goes something like this…

When truth was born, it was visited by a magical fairy and blessed with three qualities: the ability to be absolute, the ability to be objective, and the ability to refer to the external world. The condition for this blessing was that truth could only exhibit a maximum of two of those qualities simultaneously. Therefore it could be absolute and objective, but not refer to the external world (mathematical truth). It could be objective and refer to the external world, but not absolute (scientific truth). Finally, it could be absolute and refer to the external world, but not objective (aesthetic truth).

 

[Veritas6]

The condition for this blessing was that truth could only exhibit a maximum of two of those qualities simultaneously. Therefore it could be absolute and objective, but not refer to the external world (mathematical truth). It could be objective and refer to the external world, but not absolute (scientific truth). Finally, it could be absolute and refer to the external world, but not objective (aesthetic truth).

What does “2 + 2 = 11 in base 3” mean? Could 2 + 2 = 11 in base 6?

Would you agree with the statement that: “logic is a human invention that governs the manipulation of propositions. Logical rules are thus analytic in nature, and do nothing to describe objective reality. They only dictate the rules by which we TALK about reality”?

 

[neophyte]

What do you make of these claims about truth, logic, and math?

They’re meaningless sophistries. Anyone who thinks that “A can be not-A” can sometimes be a true statement has abandoned reason.

 

[rossum]

What does “2 + 2 = 11 in base 3 ” mean?

It means S(S(0)) + S(S(0)) = S(S(S(S(0)))) in terms of the Peano axioms. Alternatively, (2 x 1) + (2 x 1) = (1 x 3) + (1 x 1) expanding the “base 3” part.

Could 2 + 2 = 11 in base 6?

No. 11 in base 6 is 7 in base 10.

 

[AlNg]

What does “2 + 2 = 11 in base 3 ” mean?

The statement in quotes is misleading because the left hand side of the equation is in base 10. and the right hand side of the equation is in base 3.

 

[StephieNorthCo]

??? What would the left hand side look like in base 3?

 

[Veritas6]

No. 11 in base 6 is 7 in base 10.

Correct, so 11 cannot be 7. I’m not saying you agree with the objector (I don’t know your position), but this contradiction doesn’t describe reality according to him. All that matters is arbitrary language:

“[The law of non-contradiction] doesn’t [describe reality]. LNC is a rule imposed on language and propositions . Nothing more. Failure to abide by LNC is simply a failure to put words together properly… Contradictions are not allowed to mean anything in our use of language. The only reason contradictions don’t ‘exist in nature’ is because contradictions literally ‘don’t describe anything.’

It’s an interesting position, would you agree logic is a human invention, that these rules are: “analytic in nature, and do nothing to describe objective reality. They only dictate the rules by which we TALK about reality”?

 

[rossum]

Correct, so 11 cannot be 7. I’m not saying you agree with the objector (I don’t know your position), but this contradiction doesn’t describe reality according to him. All that matters is arbitrary language

Correct. The truth of any mathematical statement is determined by the premises surrounding that statement – the context. Change the context and the truth of the statement may change.

The meaning of any statement in a human language is determined by the dictionary for that language. The problem being that the dictionary for, say, English, is written in English itself. The ‘dictionary’ for mathematics is written in mathematics. Hence, any statement in any human language is at some level self-referential and so cannot be considered absolute.

 

[IWantGod]

The meaning of any statement in a human language is determined by the dictionary for that language. The problem being that the dictionary for, say, English, is written in English itself. The ‘dictionary’ for mathematics is written in mathematics. Hence, any statement in any human language is at some level self-referential and so cannot be considered absolute.

Mathematics isn’t arbitrary and is possible only because there is such a thing as impossibility, thus so long as you “mean” that you are adding two irreducible numbers to another two irreducible numbers it is not possible to get a quantity other than four. So it does speak to an absolute; it speaks to something that is eternally true about the essential nature of necessary reality and what it means for something to be possible.

It really doesn’t matter that you can change the meaning of numbers, because so long as you stick to what you mean you will always get an impossibility and thus a particular quantity. So while something like pure-mathematics can be changed to produce different results that do not necessarily speak of any object in reality, the activity of doing maths does speak to something fundamental to reality and it is that by which we come to understand things like the principle of non-contradiction.

 

[IWantGod]

“[The law of non-contradiction] doesn’t [describe reality]. LNC is a rule imposed on language and propositions . Nothing more. Failure to abide by LNC is simply a failure to put words together properly… Contradictions are not allowed to mean anything in our use of language.

That the principle of non-contradiction does reflect something that is true about reality is self evident in that there are absolute facts that cannot be otherwise so long as they are true. For example you know that you exist, thus so long as you know, you have no choice but to admit that the possible opposite is not true. This is self evident to us and is not just a play one words, or a semantic device with artificial rules, but rather it is something discovered in reality. We are witness to the fact that somethings are impossible and that can only be the case if there is something fundamental to reality that makes things impossible.

The only reason contradictions don’t ‘exist in nature’ is because contradictions literally ‘don’t describe anything.’ They are words put together wrongly.”

The reason why contradictions don’t exist is because reality is as such that they cannot have an act of reality. It cannot just be a failure in describing something correctly, because that would make the idea of a possibility itself a semantic device and therefore the idea of there being an error is arbitrary and therefore anythings possible beyond the artificial rules of words. Surely it was your knowledge of impossibility that made you aware of the meaningless of a statement, that there were to contraries that were absolute at the same time.

 

[IWantGod]

How would Aristotle’s moderate realism play into these claims below?:

I think Aristotle would say that possibilities are as such because of the nature of the uncaused cause. It is God’s nature that makes a thing fundamentally impossible or possible. Truth, logic, and maths is just a reflection of that natures eternal activity.

 

[EndTimes]

conceptual tools like logic and mathematics must necessarily possess a kind of intrinsic existence unto themselves, independent of space and time?

Noumenae precedes language

language attempts to convey noumenae to e.g., the human mind

Even if ∞ is presented in a manner intended to correlate parameters of God,
∞ / infinity - is not God by any stretch…

Simplified: Maths are useful Abstractive ‘Relationships’ borne in the domain of Mind…
Albeit a limited language compared w/human tongue.