Is math a Tautology

Is math a Tautology
johnson.mafoko
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Posted Feb 25, 2013 - 2:29 AM:
Subject: Is math a Tautology
Tautologies, says Wittgenstein, are senseless propositions, which means, they don't say anything about world. Axioms of logical theories(Propositional logic and First Order Predicate Logic(FOPL)) are all tautologies. It follows that their theorems will be tautologies also. Hence logic is a tautology.

But math(arithmetic) can shown to be part of logic(cf Russell & Whitehead). It therefore follows that math is a tautology. This means math statements do not say anything about the world, they are statements about other statements. 1+1=2 is neither true nor false statement; it is a meta-statement. Note 1+1 and 2 are well formed formula (wff) in FOPL. 1+1=2 simply states that a wff 1+1 can be sustituted by another wff 2 in any FOPL wff.

This is what I conclude. What do you think.



Regards,
Happy Philosophizing.

PS: Not sure sure if axioms of Predicate Logic are tautologies. But I guess since ZF set theory <=> predicate logic, at least finitist math is a tautology.

And by the way, the predicate logic I refer to above is the original version developed by Frege and Russell which was axiomatic. Modern predicate calculus seems more descriptive than deductive. It is more like a language rather than a deductive system. Modern equivalent of the concept will probably be some deductive system like Hilbert Deductive system.
Tayluh
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Posted Feb 27, 2013 - 7:20 PM:

"Math is truth."

This is the only reason Computers are higher then Humans on the Scales of Evolution.
Unlike us, They cannot lie unless told to lie.
prothero
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Posted Feb 27, 2013 - 8:25 PM:

Except for the unusual efficacy of mathematical systems in modeling the physical world and allowing for prediction of some types of physical events (engineering, medicine, physics, chemistry, etc), I might agree with you. Mathematics would seem to be much more useful than simple tautology. smiling face
fishfry
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Posted Feb 27, 2013 - 11:06 PM:

prothero wrote:
Except for the unusual efficacy of mathematical systems in modeling the physical world and allowing for prediction of some types of physical events (engineering, medicine, physics, chemistry, etc), I might agree with you. Mathematics would seem to be much more useful than simple tautology. smiling face


All mathematical truths are theorems derivable (in principle) mechanically from a handful of axioms.

But!! Not every valid mathematical derivation is interesting. If you started with the usual axioms and fed them into a "valid argument generator," you'd get a list of valid theorems. But most of them would be pointless and trivial.

It takes human mathematicians to put together related groups of theorems that are interesting and that bear meaningfully on other mathematical or physical problems.

A computer could prove a theorem in group theory. But could a computer recognize the importance of making the definition of a group?
the.yangist
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Posted Feb 27, 2013 - 11:20 PM:

Yeah, Wittgenstein said that, but did he prove it?

johnson.mafoko
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Posted Feb 28, 2013 - 9:27 AM:

I am currently reading this so that I make a more informed contribution to OP. In the meanwhile I will like to know whether the axiom of choice is the same as (or equivalent or implies) axiom of reducibility. How are they related? My initial impression is that Zermelo,von-newman and David Hilbert seems to have illiminated the need for axiom of reducibility in their axiomization of arithmentic, but Godel seems to suggest that it is similar to axiom of choice:

"But he observes that "this procedure[axiom of choice] seems to presuppose arithmetic in some form or other" (p. 134), and he states in the next paragraph that "the question of whether (or to what extent) the theory of integers can be obtained on the basis of the ramified hierarchy must be considered as unsolved." (p. 135)"(from wikipedia article)

he seems to suggest that introduction of axiom of choice pressupposes arithmetic(that is russell's logical axiomization with axiom of reducibility), while others(zermelo,von-newman,hibert etc) claim to have rid math of problematic axiom of reducibility (from ramified theory of types)

Ps. I need a book on these things instead of relying on wikipedia. Any recommendations? I need one which will sum up all Mathematical Logic(proof theory,model theory,recursion theory and set theory), especially from computation theory view.

regards,

johnson.

Edited by johnson.mafoko on Mar 2, 2013 - 12:48 AM
sealight
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Posted Mar 1, 2013 - 8:14 AM:

johnson.mafoko wrote:
Tautologies, says Wittgenstein, are senseless propositions, which means, they don't say anything about world. Axioms of logical theories(Propositional logic and First Order Predicate Logic(FOPL)) are all tautologies. It follows that their theorems will be tautologies also. Hence logic is a tautology.

But math(arithmetic) can shown to be part of logic(cf Russell & Whitehead). It therefore follows that math is a tautology. This means math statements do not say anything about the world, they are statements about other statements. 1+1=2 is neither true nor false statement; it is a meta-statement. Note 1+1 and 2 are well formed formula (wff) in FOPL. 1+1=2 simply states that a wff 1+1 can be sustituted by another wff 2 in any FOPL wff.

This is what I conclude. What do you think.




To understand the problem better may I ask you to give an example of a proposition that is not a tautology?

Also, logic, if I am not wrong, is not responsible for truthfulness of propositions that it uses. In contrast math operates with truthful statements, like 1+1=2. This statement is truthful in math although it is nonreal (in the sense that there is nothing that is equivalent to the statement in reality). So I do not see why math(arithmetic) is a part of logic.

In my opinion math is a dynamic system (for simplicity) with objects and actions. Logic is responsible for actions only. It shows a way how to create a new object from old ones. It does not tell us if the new or old objects are true.

Edited by sealight on Mar 1, 2013 - 8:22 AM
Mystermenace
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Posted Mar 1, 2013 - 10:26 AM:

I have two dogs, both named "One".

When I make them share a kennel I call that "Two".

The process of calling them and catching them and getting them in the kennel and getting them to quiet down is called "and".

One and One is Two. Is this a tautology?

Athis
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Posted Mar 5, 2013 - 5:25 PM:

Tayluh wrote:
"Math is truth."

This is the only reason Computers are higher then Humans on the Scales of Evolution.
Unlike us, They cannot lie unless told to lie.



Unlike us they have to be told to lie; or tell the truth; or do anything at all.


sealight
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Posted Mar 5, 2013 - 6:36 PM:

Mystermenace wrote:
One and One is Two. Is this a tautology?


Looks like neither tautology nor math. Why is this example in this topic?
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