Math and Logic

[DevilSolution]

Just a quickie, is all math logical? i.e if its not logical it just aint math

 

2+2=4 true math

2+4=6 false jibba jabba

 

The "rules" to the logic that defines math are man made? i.e BODMAS is just a set of rules we made up to describe the order of any equation.

 

Finally is math a product of logic? sort if inferred from q1

 

 

Regards.

[mathematic]

Is your question real or rhetorical?

[Vexen]

I think this question was articulated and answered more than 200 years ago by mathematicians and logicians.

 

If you were really interested in this topic you would pick up books on calculus, algebra and logic.

[ajb]

Finally is math a product of logic? sort if inferred from q1

 

No, you cannot reduce all of mathematics to pure logic.

 

Alfred North Whitehead and Bertrand Russell wrote a huge book trying to do this called Principia Mathematica. They wanted to reduce all mathematics to a set of axioms and some rules within the framework of symbolic logic. This idea is fundamentally flawed as shown by Gödel's incompleteness theorem; there must in fact be some truths of mathematics that cannot be derived from the axioms.

 

Now, if you were simply asking do mathematicians use logic, then the answer is yes all the time, but usually in an informal way.

I think this question was articulated and answered more than 200 years ago by mathematicians and logicians.

The drive to make all mathematics logic I think really took of in the 1900's with the book I mention above. Gödel's result is from 1931.

[DevilSolution]

If all math has to be logical then it has to be a product of logic? im aware of principia mathmatica but havent looked into godels theorem, i know its supposed to contradict the premise all math can be reduced to pure logic.

 

As a side question, does the above mentioned stop computer architecture being able to fully represnt reality?

[studiot]

 

The drive to make all mathematics logic I think really took of in the 1900's with the book I mention above. Gödel's result is from 1931.

 

 

 

David Hilbert and Erlangen?

[Vexen]

Are we below you?

I haven't studied the topic.

[ajb]

David Hilbert and Erlangen?

Hilbert is one of the founders of proof theory and mathematical logic.

 

If all math has to be logical then it has to be a product of logic?

It cannot be based on pure logic along is the loose statement. This is despite the use of informal logic all the time in mathematics.

 

 

As a side question, does the above mentioned stop computer architecture being able to fully represnt reality?

No idea.

[studiot]

 

Hilbert is one of the founders of proof theory and mathematical logic.

 

 

 

Yes indeed, which is why I mentioned him and Erlangen (which failed but was not fruitless in the end) as an addition to your post#4.

 

Erlangen was the Hilbert's program to codify mathematics.

It is also true that Russell, Whitehead and particularly Hilbert developed much deep logical thought of great subsequent value in the searches for their particular grails.

We owe much to each.

Edited December 14, 2014 by studiot

[Strange]

im aware of principia mathmatica but havent looked into godels theorem, i know its supposed to contradict the premise all math can be reduced to pure logic.

 

Not it just says that within any formal system (*) there are some things which are undecidable, i.e. which cannot be proved either true or false. Basically, this means you can develop your mathematics in two ways, depending on which you choose. (What, if anything, this says about reality is another question.) This is still all defined by the logic of mathematics.

 

(*) that is sufficiently ... blah blah blah

 

As a side question, does the above mentioned stop computer architecture being able to fully represnt reality?

 

I'm not sure what that means. You can write a program to model reality only as well as you understand it. Any computer architecture or language (*) is equally capable of this. If your model depends on some undecidable theorem, then you are free to choose either way (and perhaps see which models reality better).

 

(*) That is Turing complete

[studiot]

 

i know its supposed to contradict the premise all math can be reduced to pure logic.

 

There are two aspects to this.

 

Godel says that within any formal system there is at least one proposition, that can be made with the axioms and definitions of the system, but that cannot be proved within that system, yes.

 

But

Did I just mention axioms and definitions?

 

In order to have a formal system you must start with some axioms - no axioms no system!

 

Further again you require definitions to specify your axioms -These again have no 'proof', just plausibility or convenience.

 

Both the axioms and the definitions are separate from and additional to any Godel unproven propositions.

Edited December 15, 2014 by studiot

[DevilSolution]

axiomatic proofs are observational and not logical then? or a state of limbo given that we must presume an observational truth then apply logic to it? logic seems self defined in its essence, though we cant axiomatically prove it? and just because we cant axiomatically prove logic doesnt mean its not the pure form of math? i cant get my head around something mathematical that isnt logic, which therefor makes math a prouct of logic.

Edited December 17, 2014 by DevilSolution

[ZVBXRPL]

Just a quickie, is all math logical? i.e if its not logical it just aint math

 

2+2=4 true math

2+4=6 false jibba jabba

 

The "rules" to the logic that defines math are man made? i.e BODMAS is just a set of rules we made up to describe the order of any equation.

 

Finally is math a product of logic? sort if inferred from q1

 

 

Regards.

I don't understand why 2+4=6 is considered false, jibba jabba.

2 is 1 and 1. 4 is 1 and 1 and 1 and 1. 6 is 1 and 1 and 1 and 1 and 1 and 1.

It is just as logical as 2+2=4.

Edited January 3, 2015 by ZVBXRPL

[Sato]

I don't understand why 2+4=6 is considered false, jibba jabba.

2 is 1 and 1. 4 is 1 and 1 and 1 and 1. 6 is 1 and 1 and 1 and 1 and 1 and 1.

It is just as logical as 2+2=4.

 

The natural numbers with addition is an algebraic structure with a formal definition, the numbers defined as 0 and successors of 0 (so, 1 is the easy-to-read symbol for s(0), 2 is for s(s(0)), and so on). According to that system, the application of 2 and 2 to the addition function—that is, +(s(s(0)), s(s(0)))—will yield s(s(s(s(0)))) or 4. This is something that is simply understood as counting, but the point is that there is a rigorous, logical system that asserts 2 + 2 = 4, as well as 2 + 4 = 6.

 

Edit: Apologies, I misunderstood your post, which I skimmed initially, as saying that "2 is 1 and 4 is 6 and 6 is 1 and etc, is just as logical as 2+2=4".

Edited January 3, 2015 by Sato