Sunday, August 29, 2004


Amateur takes on Wiles's work

Now I'm not a professional mathematician. I do post about math on Usenet, but that's not an indication of expertise!

I'm not beholding to any mathematical interests though, so I feel no compulsion to protect a favored Golden Calf of the modern math world, which is an argument that supposedly proves something by one Andrew Wiles, which I fear doesn't, and I'll say exactly why I say it doesn't.

It'll be up to others to answer the charge, dismiss it, or consider that I might be right.

First off despite the assertions of great complexity to the area what mathematicians initially noticed isn't that complicated:

They had these things they called modular forms, and these things they called elliptic curves, which didn't seem at ALL related.

But there are these 4 numbers that you can get from elliptic curves, and find modular forms with the same 4 numbers. Those numbers are kind of like a description.

So there's some way that modular forms and elliptic curves could have the same description!

Mathematicians would check various elliptic curves and find they could always find some modular form to associate with it.

Taniyama and Shimura conjectured that there was a pattern here that held, as in fact modular forms and elliptic curves WERE related in some deep way, and that what mathematicians were noticing wasn't just one of those intriguing coincidences.

But you have the setup for a logical fallacy called Cum Hoc, Ergo Propter Hoc, where people see what looks like a pattern, and leap to a conclusion, though at this point mathematicians were ok, as it was only a conjecture.

It took Andrew Wiles coming in, with an attempt at proof by association for the logical fallacy to fully take hold.

The problem for many of you with such a charge is that it can seem esoteric. I've had two posters on sci.math where I've discussed this for a while actually come back to claim that Cum Hoc, Ergo Propter Hoc is about time, so it can't appy to mathematics!!!

But notice, it's actually about false implication, where you see a pattern, and your mind plays a trick on you and tells you that the pattern is proof of itself!!!

To date, while mathematicians now apparently mostly believe the Taniyama-Shimura Conjecture, they can't give you a reason why, or can they?

It turns out that if the charge of Cum Hoc, Ergo Propter Hoc is itself challenged, the next proper step is to ask for a null test.

What is a null test?

A null test is to go through the argument under challenge with the assumption that its conclusion is false, and find a contradiction with that assumption!

You see, math proofs begin with a truth and proceed by logical steps to a conclusion which then MUST BE TRUE.

But the conclusion follows from the previous steps in the proof, so any challenge to the conclusion must contradict a previous logical step, or the truth with which the proof begins.

Math proofs are perfectly logical.

There is no way for a math proof to fail a null test.

It is just not logically possible.

Therefore, any math proof can be challenged by assuming the opposite of its conclusion, and tracing through it until you reach the logical step where you end up with a contradiction.

The resolution to the contradiction, if you have a proof, is that your assumption is false and the conclusion IS true!

It's neat. It's beautiful. It's just cool.

Notice also that the null test, which can be requested whenever, and not just when you have a case of Cum Hoc, Ergo Propter Hoc, is a great way for someone who is not an expert in a particular feel to find a limited area to check.

For instance, with my challenge to Wiles's work, someone should find a single logical step where the assumption of a non-modular elliptic curve will cause a contradiction, and be able to give the exact section in his work where it occurs!!!

Then they can explain why it occurs and despite the entire work being hundreds of pages you have the ability to look at the crucial link without going through the entire thing.

You could call that logical step the keystone.

I'm asking for someone to produce the keystone in Wiles's work, which will ring out loud and clear if you assume the existence of a non-modular elliptic curve.

Let the full challenge—with witnesses now from alt.math.recreational and others throughout the world through the Internet—begin.

Saturday, August 28, 2004


My prime counting formula, other prime counting

For over two years I've talked about my prime counting discovery only to face a strange apparent lack of interest from mainstream mathematicians and outright hostility from mostly sci.math posters who tend to try and claim my work is not new.

Now I'm going to explain just a bit on how my work connects with what mathematicians found on their own.

For instance one particular sci.math poster has for years now charged that my work is just a copy of something called Legendre's Formula!

It turns out that Legendre's Formula uses a partial sieve function called phi(x,a) where phi(x,a) is the count of positive integers less than or equal to x which do not have the first 'a' primes as a factor.


The historical prime counting function is pi(x), which can be confusing to some as they see "pi" in front, but it's just the count of primes up to and including x.

Still pulling info from that page on Legendre's Formula:

phi(x,pi(sqrt(x))) = pi(x) - pi(sqrt(x)) + 1

So you can solve for pi(x), and if you figure out pi(sqrt(x)) and phi(x,pi(sqrt(x))) then you have a count of prime numbers.

Notice that phi(x,a) is a partial sieve function because you actually have to know what the a_th primes are. Like if a=2, it is fed the prime numbers 2 and 3.

Now I'll show how phi(x,a) works with a simple explanatory example.

For example, up to and including 10 you have

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

and pi(sqrt(10)) = 2, and those primes are 2 and 3.

So phi(10,2) = 3, and those numbers are

1, 5, 7

as those are the ones not divisible by 2 and 3, now since

phi(x,pi(sqrt(x))) = pi(x) - pi(sqrt(x)) + 1

you can solve for pi(x) to get

pi(x) = phi(x,pi(sqrt(x))) + pi(sqrt(x)) - 1


pi(10 = 3 + 2 - 1 = 4

which is the correct answer.

Notice you add back in to count 2 and 3 themselves, and subtract 1 for 1 which is counted in phi(x,a), and the 4 primes are

2, 3, 5, 7.

Now my prime counting function 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))],

S(x,1) = 0, p(x, y) = floor(x) - S(x, y) - 1, and S(x,y) is the sum of dS from dS(x,2) to dS(x,y)

where dS(x,y) is the count of composites up to and including x that have y as a factor which do not have any primes less than y as a factor, and the count of primes is given by p(x,sqrt(x)), so to connect with the traditional:

pi(x) = p(x,sqrt(x))

Now if you're counting primes you are just counting primes, so it's not surprising that various methods, though looking dramatically different MUST have underlying similarities because they're counting the same thing.

Now I explained my dS(x,y) function above but the S(x,y) function is the count of composites up to and including x which have the first primes up to and including y as factors.

Like S(10,3) = 5, and those composites are

4, 6, 8, 9, 10

so how do my dS(x,y) and S(x,y) relate back to Legendre's Formula?

The connecting relationship is

phi(x,a) = x - S(x,p_a) + pi(sqrt(x)) + 1

where I have to actually show the prime that is being referred to by the sieve in my S(x,y) function.

Now the physicists here can appreciate that, sure, you can have connecting formulas behind functions that ultimately do the same thing, but sci.math'ers have claimed for over two years, including recently in posts here, that the relationship "essentially" proves that my prime counting function is just Legendre's Formula.

It gets weirder though, as the phi(x,a) partial sieve function can be defined by a recurrence relationship:

phi(x,a) = phi(x, a-1) - phi(x/p_a, a-1)

which looks a lot like part of my dS(x,y) function, as consider

dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1, sqrt(y-1))],

and if you have that y is a prime, like let's say p_a, then

dS(x,p_a) = (p(x/p_a, p_a-1) - p(p_a-1, sqrt(p_a-1)))

where since

p(p_a-1, sqrt(p_a-1)) = pi(p_a-1)

so I have that

dS(x,p_a) = p(x/p_a, p_a-1) - pi(p_a-1)

and multiplying both sides by -1 and reordering a bit I finally have

-dS(x,p_a) = pi(p_a-1) - p(x/p_a, p_a-1)

and pulling down the recurrence relationship above so it's easier to directly compare

phi(x,a) = phi(x, a-1) - phi(x/p_a, a-1)

and yes they look similar!

One poster in particular has, for over two years now, repeated over and over again that my prime counting function is just Legendre's Formula based on just that similarity, but look closely, and it seems actually a bit odd that someone could make the case for over two years that those are exactly the same, and get away with it, while another sci.math'er has been more creative in claiming they are "essentially" the same.

You can imagine my consternation as a discoverer with having done the work to find my prime counting function only to see mathematicians not acting as expected while sci.math'ers would say the strangest things.

So, the short answer is, yes I can relate my work to a certain extent to what mathematicians already had, though notice I have to constrain my functions just a bit for them to connect with the sieve function.

That's easy enough though as instead of "y" I just put in "p_k" where k means the k_th prime.

But try to go the other way from what mathematicians have to my work.

The short of it is that there's no rational reason to accept the charge that my work is the same as Legendre's Formula but by now I'm sure many of you realize that the issue here isn't rationality.

It's about social stuff, academic politics, and a society of people—mathematicians—who clearly feel that they can make up their own rules.

Now I've given the derivation of my prime counting function, which you can read for yourself at my blog:

It is actually a nice introduction to the concepts of counting prime numbers, and all of the formulas make sense, and you don't need to be a top mathematician to understand it!

Now sci.math'ers have made this all personal, but it's knowledge.

Sure, many of you may just accept that mathematicians have the right to hide whatever they want, but it'd be like accepting that physicists should have the right to hide physics research.

Many of you may believe that it's my fault, and that I should have kissed butt or whatever it took to get mathematicians to like me enough to acknowledge my results, but you forget, I have the knowledge.

It's the world that is being blocked from it. Not me.

And to me, knowledge is a great thing, sure in and of itself it's not power, and I think my experiences show dramatically that knowledge is NOT power, as social forces allow people to dismiss just about anything.

Social forces are power. People can get together and simply tell themselves what they wish to believe and there's little you can do about it. There's no sense worrying about it too much as that's just reality.

Knowledge is NOT power.

But it's still immensely satisfying when you're standing on top of a mountain surveying a world. Sure it might be a little lonely, but I wouldn't have it any other way.

Hey, I found my own way to count prime numbers which advanced the literature, with one of the most beautiful derivations ever.

People act wacky about it? Sure! Of course they would!!! Of course mathematicians would behave screwy!!!

Of course. You see, they didn't make the find.

I did.

There's some guy I saw on TV who is trying to get in the record books for surfing every day for some number of years…

Someone has a world record for walking backwards…

Athletes currently are pushing themselves at the Olympics and some will receive medals.

People work to get attention, and society rewards them for standout accomplishments.

Mathematicians are just cheating with my work.

They are remaking the rules where a significant individual accomplishment can be ignored if the group charged with recognizing such accomplishments doesn't feel like it.

The are cheating in a world that knows the value of acknowledging accomplishment.

They have to know what they are doing.

The facts are on my side. I have my prime counting function itself, the derivation, and several different ways to show it to be important, unique, and even beautiful, but mathematicians are cheating.

Look at the Olympics and imagine the world the mathematicians would have, where if you don't like that runner, like if she makes you angry, you can refuse to give a medal.

If that guy throwing that javelin happens to not be nice to you or just because you don't really feel like it that day, who cares if his throw is a record?

You can just ignore him.

Mathematicians have been known for a while to be kind of out of it, and sort of in their own little world, but now it's clear that part of that world is deliberately ignoring an accomplishment like mine.

The proper punishment for them is to ignore what they claim are accomplishments.

Let them taste the pain they are willing to give to others.

Put the math people in the doghouse where they belong.

Wednesday, August 04, 2004


JSH: Where will you be in 5 years?

Obviously that's a rhetorical question though I'm sure some of you have plans for your future that extend out that far and beyond.

To many of you that may seem like a very long time, but it can be here before you know it, and if you do something today that will ruin your life in five years, what have you really got?

There are posters on sci.math who made a lot of posts a few years back about some of my work, and many who doggedly keep at it, making posts as if it matters if it takes a couple of years or even five years before the truth comes out.

Unless you think you'll be dead in five years, don't play that game.

I'm not in a hurry. I do research, think about lots of things, and submit papers to journals.

Journals are not Usenet, and they are not places where people care if I made you mad, or if it upsets you when I make grand claims.

Oh wait, so one journal did fall apart when pressured by sci.math posters, so what?

There are other journals.

To date Ioannis Argyros has not sent me a real reviewer's report about my paper Advanced Polynomial Factorization as the dirty secret (not secret any more) is that clearly not being able to handle the pressure from emails by sci.math posters Argyros sent me an email claiming my paper had not passed peer review, but included text from W. Dale Hall's post on sci.math the day before, and said that was the reviewer's report.

He lied. The chief editor for Southwest Journal of Pure and Applied Mathematics lied about a paper.

And yes, obviously, he doesn't read sci.math or he'd have known that W. Dale Hall posted what he'd emailed.

Yup, Southwest Journal of Pure and Applied Mathematics fell apart.

To date no one has found a critical error in my paper, though a while back someone did notice what were basically typos, and I acnowledged that when I realized they were correct.

You see, I do check into claims of error in my work.

I think some of you think that a month goes by, or six months, and you're home free, but that's not how it works.

Five years can go by, and then that day will come, when you are a pariah in the math world because the story will come out, and it won't matter what you did, or how much good you think you did, as your epitaph will be your shame.

That's why I don't get very excited about all of this any more.

Some of you are just destroying your lives, both what you've managed to accomplish and what you may accomplish until that day, when it all comes crashing down, and you are deservedly cast out of math society.

Some of you have dragged in one other person along with you—Ioannis Argyros—but I don't feel sorry for him because he made his choices.

He had the choice to do the right thing.

That paper is at another journal where it has been for the past two months.

That paper is not the only one I have at a journal, and as for journals, let's just say I learned that it was a mistake to use a small, electronic journal, where a chief editor could be so easily controlled.

So yeah, some of you may think it's over, but even if it takes five years, the truth will come out, and then, everything you've done, your whole life, will face a judgement of failure.

On that day, your life in the math world will be over.

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