Saturday, October 20, 2007

Mathematical consistency

My prior thinking on logic and then the connection between logic and mathematics through equality finally lead me to realizing that consistency in mathematics is about identity.

So any valid mathematical statement must when numbers are used reduce to an identity, and mathematics is distinguished from the rest of logic only in that its valid statements reduce to numerical tautologies which are identities, like 1=1. That is, mathematics is just a subset of logic.

And mathematical truth can be defined as any mathematical statement where introducing any numbers valid under the conditions given will lead to an identity.

For example, with x=3, y=4 and z=5, it is true that x2 + y2 = z2 as introducing the numbers gives

9 + 16 = 25

which is

25 = 25.

Interestingly, by definition then any valid mathematical statement will give a tautology when numbers are introduced, so any statement that does not, is not a valid mathematical statement.

So by definition, relying merely on the equals being equal, every valid mathematical statement will lead to a tautology, and correct mathematics is consistent.

Extending to logic is straightforward but slightly outside of the scope of this post, though, easily enough I can note that in logic any valid logical statement will lead to a tautology.

And then truth is about tautology, and truth itself can be defined easily enough.

Truth is any unchanging object, concept or thing.

So truth is only about lack of change. Truths are absolutes.

James Harris

Saturday, October 13, 2007

Logic and equality

If you think globally about it EVERY mathematical statement reduces to an identity, like

x2 + y2 = z2, if x=3, y=4, and z=5, you have

9 + 16 = 25

25 = 25

and if that does not occur you do not have a valid mathematical statement.

Another way to say it is that if you plug in all the numbers and do all the mathematical operations then at the end you just get an identity, which in logic is a tautology as it is always true.

So every mathematical statement that is valid can be reduced to an identity.

And now it makes sense to move to logic and find a surprisingly easy way to connect mathematics and logic--through equality.

A few years ago I was pondering something or other (not sure what) and I realized that every logical statement just maps a truth to itself, or as I said to myself then, every logical statement connects a truth to itself.

Like with mathematics, for a logical statement, you must reduce to a tautology if it is a valid logical statement.

You must. There is no logical statement in existence that does not map a truth to itself.

When you have equals, you have equality, in logic, like in mathematics.

That may seem like a strange thing to emphasize, as if you have "=" then doesn't it just mean equal? Well, if it does then you can clear up some things considered big issues in modern logic so that must mean logicians do not necessarily know that equals means equal.

Once I realized that then I realized that anything else could NOT be a logical statement, just like 1=0 is not mathematically valid, as it breaks the equals I like to say.

So if you make sure the equals, as in the equals sign or a declaration of equality, actually mean equal, you must have a logical statement, and intriguingly you can then clean up ALL the supposed logical paradoxes.

I like the consideration of the set of all sets that exclude themselves, except itself, as that one is the easiest.

(Um, in case you didn't notice I resolved the supposed paradox with that "except".)

Harder I think for people to understand is considering a sentence that declares itself to be false.

e.g. This sentence is false.

It is trying to say that it is truthfully false, mapping truth to false, so it is an illogical statement like 1=0 is not a valid mathematical statement. So it doesn't make sense to then say, but if it is false, and is saying it's false, isn't it true? Because to be true it'd have to map a truth to itself, but no truth is false, so it cannot be true!

So it's like the sentence is saying, True = False. And that breaks the equals.

In what I call 3 logic I declare that sentence to be false as it's negatably true and then I continue as if it's not a big deal, but one way to look at that sentence is as a direct declaration that true equals false, which is not letting the equals be equal.

Now in allowing in a system of logic false statements I relied on the idea that a false statement is negatably true, like if you say 1 is 3 three, yeah that's false, but if you say 1 is NOT 3, that is true, so the previous is negatably true.

But I had to have a third type--something other than true or false--as what if you have sjdfkj jumps ships?

But what is a "sjdfkj"?

So if you negate that to get sjdfkj does not jump ships, you still have gobbledygook, so I call that the 0 type, neither true nor false, but still illogical, as the only logical statements are true ones.

Oh I'd like to emphasize that the set of all sets that exclude themselves, except itself, is a complete solution and gives you a single set, as there can be only one in all of infinity.

To see what logicians say on this see: "Russell's Paradox"

The use of the exception class also gives you the way to resolve the related so-called "Barber's Paradox", simply by using an exception.

The Barber shaves every man in the village who doesn't shave himself, except himself.

Now to me using exceptions is natural and rather neat as I have a background in computer programming.

Now then if a logical statement connects a truth to itself then a false statement is NOT a logical statement, nor is one with a truth value of 0, where I say false statement for the false one and for the 0 one I say malformed.

Just like if you have a supposedly valid mathematical equation and it reduces to


then you know it isn't valid!

I think that it's easier to get this in mathematics because mathematics has to be right to work practically, while the discipline of logic is often just about talking, so if people say one thing but do another--as exceptions are, I'm sure, not new to you--then you can say paradox when you simply ignore using equals in a consistent manner.

So no logical statement can contradict itself; therefore, there can be no logical paradox.

Notice that is true because a logical statement connects a truth to itself, like a mathematical statement connects a number to itself.

So an illogical statement cannot connect a truth to itself, and no logical statement can contradict itself, as it is like saying that 1=1 cannot contradict itself.

So fundamentally mathematics and logic are connected in this necessity that equals means equal!

And I think they should teach that in colleges and universities, so I suggest you go see if they do.

James Harris