Friday, August 12, 2005

Supposed purity, paradox

One of the weirder assertions in modern intellectual thought is of the "logical paradox" which makes no sense. Consider a supposed example where here's what Wikipedia says:

The Barber paradox is a paradox that relates to mathematical logic and set theory. The paradox considers a town with a male barber who shaves every man who does not shave himself, and no one else. Such a town cannot exist:

* If the barber does not shave himself, he must abide by the rule and shave himself.
* If he does shave himself, according to the rule he will not shave himself.

Thus the rule results in an impossible situation.


Whoever wrote that Wikipedia article or edited it to add that such a town cannot exist, deferred to practical reality, which does not allow those conditions, yet supposedly there is a paradox in logic there!!!

What gives?

The conditions as given reduce to the assertion that a set includes and excludes itself, so there is a direct contradiction--not a paradox--but mathematicians and logicians teach it as a paradox, so they are not telling the truth.

To understand the reduction, just read the description that's in the Wikipedia article! The requirement that the barber shaves every man who does not shave himself in the town directly contradicts with the barber shaving himself or not shaving himself, so you get that reduction to a statement that the barber shaves and does not shave himself.

You might think it a trivial issue, but let's resolve it anyway by relying on the real world, where it makes sense if in the town the barber shaves every man who does not shave himself, and no one else, except himself, as that barber can exist!!!

So you have the idea of an exception--the barber.

Seem trivial?

Well now we get to consider THE set that only has sets that do not include themselves and has all such sets--except itself.

Necessarily, the set made up of all and only sets that do not include themselves, except itself, cannot include itself, as then you'd have a contradiction as it would both include and exclude itself.

There must, logically then, be a set of all sets that exclude themselves, except itself.

I call that set a superset, as there is only one set that only has sets that exclude themselves and has them all, except itself.

Let that sink in--there can be only one.

Super sets of that type are unknown in "modern" set theory as it is primitive.

You need the real world to understand logic.

If mathematicians and logicians focused in the real world versus being "pure" they could have resolved the supposed paradox by seeing the contradiction and hypothesized the existence of the superset with the easy proof of its existence.

Concentrate your mind on a set that includes all sets that exclude themselves, except itself.

To me that is an example of super-information as in any reality any sentient beings who came across this issue would, if smart enough, eventually hypothesize the exact same set. There can be only one across all realities, all means of existence.

That's kind of cool.

James Harris

Tuesday, August 09, 2005

Logical theory versus illogical logicians

There's this guy named Russell who put forward what he thought was a logical paradox:



One variant of it is called the "Barber's Paradox" and I use it for laypeople as it might be easier to understand than starting with the full abstraction:

Consider a small village where all men are shaved, the barber is a man, and the barber shaves all and only those men who do not shave themselves.

But who shaves the barber?

The answer is that it's impossible for the barber to shave himself and not shave himself, which is what those conditions would require.

That is, the supposed paradox is with assertions that would require that the barber shave himself and not shave himself, which is a direct contradiction.

More abstractly, consider a set made up only of sets that exclude themselves that includes all such sets, but stop, before you run around with that idea, as it reduces to: consider a set that includes and excludes itself.

So why is this a big deal in logic?

Because, I guess, people a while back didn't deal much with exceptions.

But we're modern, well-educated people so let's toss in exceptions:

Consider a small village where all men are shaved, the barber is a man, and the barber shaves all and only those men who do not shave themselves, except himself, as the barber shaves himself


So let's handle the abstraction: consider a set that includes all and only sets that exclude themselves, except itself.

And just like that I've handled a big enough issue in logical theory that you can find quite a bit on it in books and on the web, and look at them to see that they do not provide the correct solution.

What? Yup. Look at them and they do not provide the correct solution.

Why don't they?

Best guess is that we have the benefit of something modern: computers.

So our perspective is broader.

James Harris

Sunday, July 03, 2005

Definition of mathematical proof

mathematical proof (noun): a mathematical argument that begins with a truth and proceeds by logical steps to a conclusion which then must be true.

Updating the page to add a traditionally formatted definition at the top of the page. After is the original post from 2005 without edits.  James Harris, February 6th, 2012.

A mathematical proof begins with a truth and proceeds by logical steps to a conclusion which then must be true.

I call that the functional definition of a mathematical proof, which I came up with a couple of years ago and posted on Usenet, as part of extreme mathematics as I practice it, is open research meaning you publicize your work.

An absolute definition would be:

A mathematical proof is a mathematical argument that begins with a truth and proceeds by logical steps to a conclusion which then must be true.

I like "logical steps" while others might prefer "valid mathematical steps".

By convention, the conclusion of a math proof is often called a theorem, and students are taught to state it up front, basically just saying the conclusion at the outset.

That convention is just that, a convention, of saying the conclusion up front, and then going through the steps to prove that conclusion.

So, a theorem is actually the conclusion of a math proof, which includes the theorem itself, as well as the argument proving it.

For some reason (surprise, surprise) you will not find my definitions of mathematical proof in the mainstream literature, as in my experience, mathematicians willfully ignore information that they don't like.

So why are my definitions worth mentioning?

Well, for some reason, the idea has become part of the mainstream thinking on proofs that they are creations of mathematicians that are "fragile" or delicate, with the idea of broken proofs, and failed proofs, etc., which comes about when failed arguments are called proofs.

But, consider the word "proof", as, for an example from another domain, if prosecutors claim to have proof that a suspect committed a crime, but it turns out their "proof" is not proof at all--but maybe a flight of fancy or reach by the prosecutors--it's not called failed proof!!!

They just didn't have proof.

There is a direct contradiction in declaring a flawed argument to be proof or a proof, and what I don't like about that being routine is it gives the wrong impression to the public where people don't realize that proofs are solid in a way not known in other areas--they are absolutely true.

Failed arguments are failed arguments and not proofs, no matter what anybody calls them.


Thursday, June 23, 2005

Partial differential, prime counting

So a few days ago I went through the basic derivation of my prime counting function, and noted that it uses a partial difference equation.

That partial difference equation is

dS(x,y) = (P(x/y,y-1) - P(y-1, sqrt(y-1)))(P(y, sqrt(y)) - P(y-1, sqrt(y-1)))


P(x,y) = [x] - S(x,y) - 1

with the constraint that if y>sqrt(x), then P(x,y) = P(x,sqrt(x)), which is necessary to handle the fact that [x/p] - 1 is the count of composites that have a prime p as a factor for a positive integer x, as long as p<=sqrt(x).

And finally, S(x,y) is the sum of dS from 2 to y or sqrt(x) if y>sqrt(x), as it has the same constraint, and S(x,1) = 0, by definition.

It's easy enough to move from the partial difference equation dS(x,y) to a partial differential equation:

ΔS(x,y) = (P(x/y,y-Δy) - P(y-Δy, sqrt(y-Δy)))(P(y, sqrt(y)) - P(y-Δy, sqrt(y-Δy)))

and divide by Δy to get

ΔS(x,y)/Δy = (P(x/y,y-Δy) - P(y-Δy, sqrt(y-Δy)))(P(y, sqrt(y)) - P(y-Δy, sqrt(y-Δy)))/Δy

and let Δy approach 0 to get

S'y(x,y) = (P(x/y,y) - P(y, sqrt(y))) P'(y, sqrt(y))

and now I also have

P(x,y) = x - S(x,y) - 1

and differentiating with respect to y, gives

P'y(x,y) = -S'y

so I have

P'y(x,y) = -(P(x/y,y) - P(y, sqrt(y))) P'(y, sqrt(y))

and a continuous function.

Now you can numerically integrate that and I'm not really up on numerical integration so I made a really basic program and did so, and found that I got values closer to pi(x) than li(x) but a bit further than R(x), but I did just a rough go at it.

To date, I've yet to find anyone to help me with the numerical integration!

If that partial differential integrates to a close value to pi(x) then that could be rather important news.


Thursday, June 16, 2005

Counting primes

One of my earlier results comes from a pure thinking exercise on how to count prime numbers as the start is a simple idea: consider a function that gives the count of composites for a particular prime that have that prime as a factor, but no lesser primes as factors.

For instance, up to 10, for 3 that function would give a count of 1, as 9 is the first composite that has 3 as a factor and no lesser primes, while 6 would be excluded as it has 2 as a factor.

Now call that function dS(x,pj) where x is where you're counting up to, and pj is the j_th prime, so my earlier result is dS(10,3) = 1.

Now generalizing a bit, to get the count of composites that have pj as a factor up to and including some x, you can use

floor(x/pj) - 1

where from now on, I'll use [x] = floor(x), so I have

[x/pj] - 1

for the count of composites with pj as a factor.

From that count I need to subtract the number of primes less than pj, which is j-1, so I have

[x/pj] - 1 - (j-1).

Now I need the count of composites up to and including x that have primes less than pj as a factor, which is another function, that is, I now need a function that is the count of composites up to and including some x for all primes less than pj, and I'll use S(x,pj-1), as the count of composites up to and including x that have primes less than pj as factors.

Here's where things are a little tricky as I now have my full dS(x,pj) function as

dS(x,pj) = [x/pj] - 1 - (j-1) - S(x/pj, pj-1)

where S(x/pj, pj-1) is the count of composites that multiply times pj to give a product less than or equal to x, where notice that pj must be less than or equal to sqrt(x) or the composite count given by [x/pj] - 1 will not be correct.

Now I know that in general the count of primes up to and including x is equal to x minus the count of composites minus 1, and that gives an idea for one more function, which I'll call P(x,pj), where

P(x,pj) = [x] - S(x,pj) - 1

which allows me to simplify dS(x, pj) to

dS(x,pj) = P(x/pj,pj-1) - (j-1)

where again pj has to be less than or equal to sqrt(x).

But notice that P(x,pj) is the full count of primes up to and including x, if pj is greater than or equal to the last prime less than sqrt(x), so I can remove all mention of primes with the following

dS(x,y) = (P(x/y,y-1) - P(y-1, sqrt(y-1)))(P(y, sqrt(y)) - P(y-1, sqrt(y-1)))

as if y is not prime then

P(y, sqrt(y)) - P(y-1, sqrt(y-1)) = 0

with the constraint that if y>sqrt(x), then P(x,y) = P(x,sqrt(x)) to keep the correct count.

I also now have

P(x,y) = [x] - S(x,y) - 1

and notice that dS(x,y) is the count of composites that have y as a factor that do not have primes less than y as a factor, which means that if y is composite dS(x,y) = 0, which is how I can remove mention of specific primes.

Now the count of composites for each prime less than or equal to the square root of x when added together just gives the full count of composites, so I have that the S function equals the sum of the dS functions for each prime less than or equal to sqrt(x), where I define S(x,1) = 0, and my prime counting function is complete.

Programming it is rather easy, and yes, it does work!

Though for speed the form where I have specific primes is much faster as otherwise you get recursion that determines that numbers are prime, over and over again, which slows it down.

As an exercise in pure thought, I doubt there are many derivations like it.

There is a surprise in the result in that I get a multi-dimensional prime counting function, which is P(x,y) which gives the same value as the traditional pi(x), if y=sqrt(x).

The other key fact is that dS(x,y) is a partial difference equation.

I came up with the gist of that derivation over three years ago.


Monday, June 13, 2005

3 Logic More Basics

In an earlier post I went over the necessity of three logical states, where you have 1 for true, -1 for false, but negatably true, and 0 for statements not negatably true.

To understand that system, consider the following.

One plus one equals two.

That sentence is true, but consider.

One plus one equals three.

That sentence is false, but negatably true, where by negation, I mean to negate one side of the equals:

One plus one does not equal three.


One plus one is not three.

To show the last logical state is a little harder, as I need a nonsense statement like:

The sjdlfy is green.

Here I've created the word "sjdlfy" by randomly typing, so it's not meaningful to give it a color!

Under the rules of 3 Logic its logical value can be rigorously determined to be 0 by trying to negate it, which gives:

The sjdlfy is not green.

As that is still nonsense, it has a logical value of 0.

I'll talk again about the Liar Paradox, as it's a well-known example, where 3 Logic does not give a contradiction, but the resolution can be confusing I think, as consider the sentence:

This sentence is false.

It has a logical value of -1, as it is negatable:

This sentence is not false.

Notice then the logical result that a sentence cannot truthfully declare itself to be false, even if some other sentence can declare it to be false, as true does not equal false, T does not equal F.

So while I can say, that sentence is false, a sentence cannot truthfully claim that it is false, as it could only do so, if truth and falsity were equals, but they are not.

You may say to yourself, "The sentence claims it is false, and I can say, yes, that sentence is false, so then the sentence is true!"

But at the start you note that the sentence is false!

The contradiction is in your own behavior. If the sentence is false, then THAT sentence is false, so it's not true, no matter what. In claiming it is false, it is attempting to equate truth with falsity, which is just false.

One way to look at it is as if a clever con-man were trying to convince you to part with your life's wages. This con-man needs to convince you that true is false, so that you will give up your money, so he just tells you that, true is false, right?

Nope. That won't work. He tries something more clever, where you assume truth, like with sentences, as given a sentence there is an assumption of truth.

So in claiming itself to be false, a sentence is relying on your own inner assumptions about sentences, which you can consider.

Now consider this example:

A sentence normally declares a truth as the purpose of sentences is to communicate information, but THIS sentence is going to tell you that it is false, so it is telling you that it is not communicating truth to you, but is in fact, not true, as I go ahead and just now say, this sentence is false.

Now negate it:

A sentence normally declares a truth as the purpose of sentences is to communicate information, and THIS sentence is not going to tell you that it is false, so it is telling you that it IS communicating truth to you, which is in fact, true, as I go ahead and just now say, this sentence is true.

And I did have to negate in several places as it is a compound sentence, but I think you should have some sense of what is happening with the original statement:

This sentence is false.

It's like a con-man saying: "Trust me, truth is falsehood, give me your money."

But, truth is not falsehood. Invest wisely.


Wednesday, June 01, 2005

Three Valued Logic

In logic there is the usual bi-valued "logic" where everything is supposed to be either true or false, and there are three valued logics, where I did a quick search this morning and saw stuff like true, false, or unknown, among other possibilities, like using numbers where 1 is true, 1/2 is in-between, or unknown and 0 is false.

I'll make the case that three valued logic is a necessity.

I suggest a three valued logic, where a statement has a logical value of 1, 0, or -1, where 1 is true, -1 is false, and negatably true, while 0 is unresolvable.

For instance

1+1 equals 2

is a logical statement with a truth value of 1.

1+1 equals 3

is an illogical statement with a truth value of -1.

Notice it is negatable, to

1+1 does not equal 3

with a truth value of 1.

So the negative of a false statement i.e. a statement with a truth value of -1, is a true statement with a truth value of 1.

So what about 0?

Consider the statement:

If a=b and c=d then gooses lay eggs.

That statement cannot be said to be false and negatable, as the negative is, maybe,

If a does not equal b and c does not equal d then gooses do not lay eggs.

It remains nonsensical, despite negations where there are different ways you can negate.

So it has a truth value of 0.

Notice that in bi-valued systems the nonsensical statement can give you problems. Possibly some would just say it's false, but then again, it just doesn't make sense.

As "doesn't make sense" is not very rigorous, I like the term "malformed".

The statement is illogical as it is malformed.

The correct form in abstract is,

If a=b and b=c then a = c.

That has a truth value of 1.

Now let's consider a "paradox".

"This sentence is false."

Is an example of what I've seen called the "Liar's Paradox".

It has a truth value in the system I've outlined of -1.

So why?

"This sentence is not false."

is the negation, with a truth value of 1.

The sentence is negatable, so it has a truth value of -1.

No paradox.

That may have happened so fast that you may not realize how quickly this system just processed a "paradox" that has challenged logicians for quite some time, so here's a link:

Now I emphasize three values in a system that can handle the "Liar Paradox" with only two, but that's because of malformed statements, which force you to have a formedness value.

So I see logic as about formedness, where a logical statement connects two truths, and it has a truth value of 1.

A false statement has a truth value of -1, and is negatable to a well-formed statement.

A statement that is not true nor can it be negated to a true statement is malformed with a truth value of 0.

If you disagree and believe that a bi-valued system can work, then you need to handle bizarre statements like:

"If the world were made of chocolate, and you were my friend, then robots would bleep like sheep."

Now then, in a two valued system, make sense of that statement!

James Harris

Friday, May 27, 2005

Logical Formedness Axioms

1. Identical sets are identical.

2. Different sets are different.

3. Statements contradicting axioms 1 or 2 are false or malformed.

4. A malformed statement is one for which a conclusion does not follow given its structure.

5. A false statement is one that while structurally correct is not true.

The "structure" I mean has to do with how the sentence is put together.

For instance, the following is a badly structured syllogism:

If x=1, and y=2, then x=y.

The basic structure for the syllogism--the well-formed structure--is,

if a=b, and b=c, then a=c,

and variations on that structure are to be considered malformed.

Notice that with the first given that x does not equal y, the conclusion is false, but the entire statement is malformed so that is secondary.

With the basic axioms established--and notice how simple they are--it's trivial to handle supposed paradoxes which reduce to attacking one of the first two axioms.

For instance, the so-called Russell Paradox reduces to the assertion that a set includes and excludes itself.

Let A be a set that includes itself, and let B be a set that excludes itself.

B is different from A.

Therefore, by axiom 2 any statement that B is A is malformed or false.

Stating that B is A and B is different from A is structurally wrong, so the full statement is malformed.

Notice also that the resolution to the supposed paradox is the well-formed statement:

Consider a set A that includes all and only sets, except itself, that exclude themselves.

Notice that the axioms prevent that set from including itself, as then you reduce to a set that both includes and excludes itself, against axiom 2 as I explained.

James Harris

Tuesday, April 12, 2005

My Mathematical Journey

Ten years ago this month for various reasons I thought to myself it might be fun to try and find simple solutions to famously hard math problems, so I went at it.

Believe it or not, I assumed that I could use modern problem solving techniques to quickly find what I figured people weren't really looking for, as from what I gathered mathematicians believe that all the easy stuff was found years ago, and they pride themselves on building only on what's known.

Like with the spherical packing problem or Fermat's Last Theorem, I thought maybe there were simple answers, but because today's mathematicians start by looking at what other mathematicians did, and trying to build on it, maybe if I just started from scratch, using modern problem solving techniques, I could find something that was missed.

After a couple of weeks I actually thought I was done. I was very, very wrong, and here now about a decade later, I wonder at how naive I was.

Still if I'd known what was ahead, would I have tried?

Who knows, but I did try and now I have mathematical discoveries that span number theory.

Possibly I can go into the journey itself in this blog, as it's been a remarkable adventure, with lots of bumps along the way, and some strange characters and misadventures.

But through it all has been the beauty of discovery and the power of logic, and thought, as I worked very hard for some years to get some answers.

And I found some.

James Harris

Sunday, March 20, 2005

Object Ring

The object ring is defined by two conditions, and includes all numbers such that these conditions are true:

1. 1 and -1 are the only rationals that are units in the ring.

2. Given a member m of the ring there must exist a non-zero member n such that mn is an integer, and if mn is not a factor of m, then n cannot be a unit in the ring.