shhhhhhhhhhhhhhh... cmere..

Wow James! I was expecting a short reply and I got a very interesting story.

Yes if only I can get the numbers together then I would understand half of it. I would first have to study each mathematician and their discoveries to get more out of this story in which I will. Starting with Euclid, Cantor, and Archimedes.

It is very interesting and it seems like math will continue to come with new discoveries that will help technology evolve.

If a man such as Cantor can start with nothing and end up inventing new mathematics in its own category then there is much to be yet discovered.

What you have discovered will help future mathematicians discover something useful in the future.

Great write!

I am sure many who stumble upon this thread will have something to think about.

Well, there's actually quite a bit more to it. But I'm not about to post it all here.

Other mathematicians that took part in paving the way for Cantor's craziness include the following:

Empodecles (ancient Greek)
Plato
Aristotle
Joseph Liouville
Dedekind (famous for the "Dedekind cut")

Actually the "Dedekind-cut" is an abstract method that uses ideas that began with Empodecles and relies on a method of exhaustion which is a precursor to calculus.

The Dedekind cut gives a superficial realitiy to the concept of irrational numbers. I say that this is a superficial concept because it's actually an approach that uses a very crude idea of calculus to define these numbers.

However, the modern formal definition of the calculus limit as introduced by Karl Weierstrass actually shows why Dedekind cuts are not defining anything other than a calculus limit. Which does not suggest that these numbers exist at all. It merely shows that they can be defined as a 'limit'. But limits, as Weierstrass has formally defined them are nothing more than proving trends.

In fact, anyone who has passed a course on calculus knows that to prove a limit exist all that is required is that trends be proven.

But proving trends does not proof the existence of the thing they are heading toward. In fact, we know that we can prove the existence of limits that don't exist. This is a standard exercies in undergraduate calculus.

By proving the existence of limits that don't exist, I simply mean that we can prove that limit do exist, for points in the range of a function where the point is undefined by the function.

In othe words, say we have a rational function whose denominator passes through zero. The function is undefined at that point. Yet the calculus limit for that point may very well exist. So we say that the limit exists.

But we know that the actual point does not!

Calculus limits do not equal points! We see this all the time in practice. This is why calculus will survive. Calculus never truly claimed that limits actually exist as points. They simply exist as limits. They may or may not exist as a point depending on the circumstances. There's a difference, and even modern mathematics recognizes this difference formally. Karl Weierstrass clearly understood this very well when he wrote the defintion for the calculus limit.

In fact, I'm a little surprised that Weierstrass didn't jump on this in terms of the definition of number. He probably just wasn't thinking along those lines. He was thinking solely in terms of clariflying the concepts of infinitesimals that had been introduced by Leibniz and refined by Cauchy.

Some other players that were important at the time of Cantor, were Gottlob Frege and Giuseppe Peano.

Both of these men were actually on the RIGHT TRACK!

In fact Giuseppe Peano actually had it CORRECT!

But his idea was rejected in favor of Cantors for reason that I won't go into at this time. But sadly they were basically religious in nature.

Another thing that modern mathematicians probably don't think about a lot is the power of religious thought in the days of Cantor's works.

There was a very profound interest in making the defintion of number "Pure". Peano's idea, although the correct idea, seemed to be impure.

What do I mean by impure?

Well, it simply tied the idea of number to the idea of the quantitative property of a thing!

Duh?

Isn't that what number SHOULD BE?

But this was viewed as tainting the idea of number with the unholiness of the real world.

Cantor's idea of defining number based on nothing seemed pristine and divine.

Unfortuantely, it's a totally erroneous concept that has nothing to do with what the concept of number. Number is a concept that is supposed to reflect the quantitative property of the universe.

But Cantor's "Empty Set Theory" removes the idea from what it was originally supposed to represent and has taken it into a totally unrelated realm of nothingness.

We really need to go back and reexam the orginal proposal of Peano.

Peano had it right. Unfortunate he didn't understand the details and therefore he wasn't able to present the idea convincingly.

I now have the full explantion.

So we need to rewind the history of mathematics a coupld hundred years and redefine Set Theory correctly.

This will have a HUGE impact on modern day mathematics.

At least in theoretical physics.

Don't worry about engineering. Engineers don't often work with infinities.

So planes will still fly and bridge will still stand.

This will only affect the more abstract mathematics used in things like String Theory and Quantum Mechanics.

It's not going to affect your bank account. Unless of course you have balances that are infinite.

Very interesting indeed.

If Peano had it right then how come other modern scientists don't see it today? Or do they?

In terms of modern scientist then those who are not influenced by religion and have the freedom of expressing their views.

If Peano's proposal is valid then I am amazed that no one sees it today.

If you are the only one then I would truly take advantage and write this book as soon as possible.

Very educational. Thank you for sheding your knowledge on this. I learn alot already

Well, it's actually wrong to say that Peano had the right idea. Perhaps it's better to just say that he was on the right track.

He didn't know how to formalize it and therefore he couldn't truly propose it as a complete idea. He just had an intuitive notion. The notion was correct, but he didn't know how to build it into an actual formalism.

It would be like he said, "Hey I have an idea!"

And they said, "Ok show us how your idea works"

And Peano replied, "Well, I don't now how to make it work, I just have this idea"

In the meantime Cantor said, "Well I have a different idea and I can make my idea WORK!"

Cantor was no dummy. In fact, he was an extremely brilliant man. His ideas weren't stupid. They just don't properly reflect the quantitative nature of the universe. Instead they represent a totally abstract notion that actually fits in more with the Greek Notion that number should be defined as the answer to equations.

Cantor was looking at it entirely from this point of view. Which was the accepted view.

Peano was actually looking at the idea of number ontologically. He had the right idea. In other words, he knew that the idea of number should correctly represent the quantitative nature of the physical world. But he didn't know how to put this into an abstract formalism.

Cantor may very well have been far more brilliant than Peano. They were just thinking differently.

If Cantor had truly understood where Peano was coming from, Cantor himself might have seen the truth of Peano's insight. Unfortunately Cantor had already been sucked into the mindset of the algebraic approach to number as had many other prominent mathematicians. After all, this idea had been accepted since the ancient Greeks. Just look at how long it took before the mathematical community actually put this into a formalism. Almost two millennia! Prior to that the idea of number was entirely intuitive with no sound mathematical formal definition at all. Number was simply accepted to be anything that satisfies an equation!

In any case, Cantor has his formalism all worked out. So he was quite impressive. He could do all sorts of mathematical tricks with his established Set Theory, whilst Peano couldn't even get his Set Theory off the ground.

This would be like two people showing up at NASCAR. Peano can't get his car running. In the meantime Cantor is running victory laps to impress the crowd!

Now don't be fooled by all of this. Cantor's car did NOT run perfectly. It was spitting fire and sputtering smoke and leaking oil all over the track, and even backfiring at times!. In fact, there have been many band aids added to Cantor's Set Theory since that initial time. So Cantor didn't have all the bugs worked out by far. But his Set Theory was "up and running".

Also, Cantor's Set Theory predicts an infinity of difference sized infinities! A lot of mathematicians were not pleased with that at all. However, Cantor was very persuasive and even had proof that this has to be the case! His proof still stands to this very day! However, it's a false proof. And the reason becomes crystal clear once we get Peano's Set Theory up and running!

Peano never did get his race car started. So he couldn't even say how many infinities his Set Theory would predict! He never got it running well enough to even see that far.

Well, I have Peano's Set Theory running perfectly. It runs as smooth as a Swiss Watch. It's a NATURAL.

It predicts exactly ONE infinity. No more. In fact, it even denies the possibility of any more.

So what about Cantor's proof of why there absolutely must be more than one size of infinity?

Well, Peano's Set Theory shows why Cantor's proof is not valid.

Also, all of the band aids that had to be placed onto Cantor's set theory fall away. And underneath them the wounds are healed in amazing and insightful ways.

Maybe I shouldn't call this "Peano's Set Theory" since he didn't truly complete it. Maybe I should just refer to it as the "Natural Set Theory". The Set Theory that comes directly from NATURE.

It's not man made at all. I didn't invent this theory. I discovered it. It's not mine. It belongs to Mother Nature. Eventually mankind will discover this theory because it's the truth. It's not just a made-up guess. It's physics.

Cantor's Set Theory is entirely a manmade whim. It was Cantor's idea and it was based on the idea that the concept of number is the concept of solutions to algebraic equations. But that's not the true meaning of number. In other words, that's not Mother Nature's number.

Cantor dialed an empty number
and got a number if infinities
Mother Nature peed her pants
and laughed at the obscenities

For those who wonder what the mathematician Giusippe Peano discovered:

((Note: Please correct me if this information provided is incorrect. I have copied it from a website out of interest of the mathematician.))

In 1886 Peano proved that if f (x, y) is continuous then the first order differential equation dy/dx = f (x, y) has a solution.

The existence of solutions with stronger hypothesis on f had been given earlier by Cauchy and then Lipschitz.

Four years later Peano showed that the solutions were not unique, giving as an example the differential equation dy/dx = 3y2/3 , with y(0) = 0.

In addition to his teaching at the University of Turin, Peano began lecturing at the Military Academy in Turin in 1886.

The following year he discovered, and published, a method for solving systems of linear differential equations using successive approximations.

However Emile Picard had independently discovered this method and had credited Schwarz with discovering the method first.

In 1888 Peano published the book Geometrical Calculus which begins with a chapter on mathematical logic.

This was his first work on the topic that would play a major role in his research over the next few years and it was based on the work of Schröder, Boole and Charles Peirce.

A more significant feature of the book is that in it Peano sets out with great clarity the ideas of Grassmann which certainly were set out in a rather obscure way by Grassmann himself.

This book contains the first definition of a vector space given with a remarkably modern notation and style and, although it was not appreciated by many at the time, this is surely a quite remarkable achievement by Peano.

In 1889 Peano published his famous axioms, called Peano axioms, which defined the natural numbers in terms of sets.

These were published in a pamphlet Arithmetices principia, nova methodo exposita which, according to [5] were:-

... at once a landmark in the history of mathematical logic and of the foundations of mathematics.

The pamphlet was written in Latin and nobody has been able to give a good reason for this, other than [5]:-

... it appears to be an act of sheer romanticism, perhaps the unique romantic act in his scientific career.

Genocchi died in 1889 and Peano expected to be appointed to fill his chair. He wrote to Casorati, who he believed to be part of the appointing committee, for information only to discover that there was a delay due to the difficulty of finding enough members to act on the committee. Casorati had been approached but his health was not up to the task.

Before the appointment could be made Peano published another stunning result.

He invented 'space-filling' curves in 1890, these are continuous surjective mappings from [0,1] onto the unit square.

Hilbert, in 1891, described similar space-filling curves.

It had been thought that such curves could not exist. Cantor had shown that there is a bijection between the interval [0,1] and the unit square but, shortly after, Netto had proved that such a bijection cannot be continuous.

Peano's continuous space-filling curves cannot be 1-1 of course, otherwise Netto's theorem would be contradicted. Hausdorff wrote of Peano's result in Grundzüge der Mengenlehre in 1914:-

This is one of the most remarkable facts of set theory.

In December 1890 Peano's wait to be appointed to Genocchi's chair was over when, after the usual competition, Peano was offered the post. In 1891 Peano founded Rivista di matematica, a journal devoted mainly to logic and the foundations of mathematics.

The first paper in the first part is a ten page article by Peano summarising his work on mathematical logic up to that time.

Peano had a great skill in seeing that theorems were incorrect by spotting exceptions. Others were not so happy to have these errors pointed out and one such was his colleague Corrado Segre.

When Corrado Segre submitted an article to Rivista di matematica Peano pointed out that some of the theorems in the article had exceptions. Segre was not prepared to just correct the theorems by adding conditions that ruled out the exceptions but defended his work saying that the moment of discovery was more important than a rigorous formulation. Of course this was so against Peano's rigorous approach to mathematics that he argued strongly:-

I believe it new in the history of mathematics that authors knowingly use in their research propositions for which exceptions are known, or for which they have no proof...

It was not only Corrado Segre who suffered from Peano's outstanding ability to spot lack of rigour.

Of course it was the precision of his thinking, using the exactness of his mathematical logic, that gave Peano this clarity of thought. Peano pointed out an error in a proof by Hermann Laurent in 1892 and, in the same year, reviewed a book by Veronese ending the review with the comment:-

We could continue at length enumerating the absurdities that the author has piled up. But these errors, the lack of precision and rigour throughout the book take all value away from it.

From around 1892, Peano embarked on a new and extremely ambitious project, namely the Formulario Mathematico. He explained in the March 1892 part of Rivista di matematica his thinking:-

Of the greatest usefulness would be the publication of collections of all the theorems now known that refer to given branches of the mathematical sciences ... Such a collection, which would be long and difficult in ordinary language, is made noticeably easier by using the notation of mathematical logic ...

In many ways this grand idea marks the end of Peano's extraordinary creative work.

It was a project that was greeted with enthusiasm by a few and with little interest by most.

Peano began trying to convert all those around him to believe in the importance of this project and this had the effect of annoying them. However Peano and his close associates, including his assistants, Vailati, Burali-Forti, Pieri and Fano soon became deeply involved with the work.

When describing a new edition of the Formulario Mathematico in 1896 Peano writes:-

Each professor will be able to adopt this Formulario as a textbook, for it ought to contain all theorems and all methods.

His teaching will be reduced to showing how to read the formulas, and to indicating to the students the theorems that he wishes to explain in his course.

When the calculus volume of the Formulario was published Peano, as he had indicated, began to use it for his teaching. This was the disaster that one would expect.

Peano, who was a good teacher when he began his lecturing career, became unacceptable to both his students and his colleagues by the style of his teaching.

One of his students, who was actually a great admirer of Peano, wrote:-

But we students knew that this instruction was above our heads. We understood that such a subtle analysis of concepts, such a minute criticism of the definitions used by other authors, was not adapted for beginners, and especially was not useful for engineering students.

We disliked having to give time and effort to the "symbols" that in later years we might never use.

The Military Academy ended his contract to teach there in 1901 and although many of his colleagues at the university would have also liked to stop his teaching there, nothing was possible under the way that the university was set up.

The professor was a law unto himself in his own subject and Peano was not prepared to listen to his colleagues when they tried to encourage him to return to his old style of teaching.

The Formulario Mathematico project was completed in 1908 and one has to admire what Peano achieved but although the work contained a mine of information it was little used.

However, perhaps Peano's greatest triumph came in 1900. In that year there were two congresses held in Paris.

The first was the International Congress of Philosophy which opened in Paris on 1 August. It was a triumph for Peano and Russell, who attended the Congress, wrote in his autobiography:-

The Congress was the turning point of my intellectual life, because there I met Peano.

I already knew him by name and had seen some of his work, but had not taken the trouble to master his notation. In discussions at the Congress I observed that he was always more precise than anyone else, and that he invariably got the better of any argument on which he embarked.

As the days went by, I decided that this must be owing to his mathematical logic. ... It became clear to me that his notation afforded an instrument of logical analysis such as I had been seeking for years ...

The day after the Philosophy Congress ended the Second International Congress of Mathematicians began. Peano remained in Paris for this Congress and listened to Hilbert's talk setting out ten of the 23 problems which appeared in his paper aimed at giving the agenda for the next century.

Peano was particularly interested in the second problem which asked if the axioms of arithmetic could be proved consistent.

Even before the Formulario Mathematico project was completed Peano was putting in place the next major project of his life.

In 1903 Peano expressed interest in finding a universal, or international, language and proposed an artificial language "Latino sine flexione" based on Latin but stripped of all grammar.

He compiled the vocabulary by taking words from English, French, German and Latin.

In fact the final edition of the Formulario Mathematico was written in Latino sine flexione which is another reason the work was so little used.

Peano's career was therefore rather strangely divided into two periods.

The period up to 1900 is one where he showed great originality and a remarkable feel for topics which would be important in the development of mathematics.

His achievements were outstanding and he had a modern style quite out of place in his own time. However this feel for what was important seemed to leave him and after 1900 he worked with great enthusiasm on two projects of great difficulty which were enormous undertakings but proved quite unimportant in the development of mathematics.

Of his personality Kennedy writes in [5]:-

... I am fascinated by his gentle personality, his ability to attract lifelong disciples, his tolerance of human weakness, his perennial optimism. ... Peano may not only be classified as a 19th century mathematician and logician, but because of his originality and influence, must be judged one of the great scientists of that century.

Although Peano is a founder of mathematical logic, the German mathematical philosopher Gottlob Frege is today considered the father of mathematical logic.

Next up, who was Gottlob Frege?

All of these mathematicians did a lot of work in different fields. Gottlob Frege also offered ideas of how to define the natural numbers.

In fact, many mathematicians will argue today that all of these ideas are 'valid'. They will simply say that have a myriad of ways of defining what number 'means'. And in a very real sense this is actually true of modern mathematics.

In fact, I just took a two refresher courses, on the history of of mathematics, and both of those mathematicians have defined number in different ways.

One of them sees the idea of number in a very abstract way in terms of patterns and transformations. The other offers many different definitions for number, and in fact, proclaims that the concept of number within mathematics is an incomplete definition. He calls it a 'moving target' that mathematicians are constantly following.

This is truly hilarious to me. Because I actually know precisely what they are doing. They are like a dog chasing their own tail and they don't even realize that they are doing it.

Have you ever gone to the bank to deposit $100 and have the teller ask you, "What definition of number are you using today?"

I doubt it very seriously.

The bottom line is that their truly is no need to have multiple defintions of number. One defintion will do just fine.

Now, some people might say, "Oh but there are Whole numbers, and Prime numbers, and Rational Numbers, and Irrational numbers, and Imaginary numbers, and so on and so forth.

But there's no need to have different definitions for all those numbers. In fact all those number concepts fit perfectly well into Natural Set Theory. Only one definition is required.

Also this isn't precisely what mathematicians mean when they speak of different number defintions. They aren't really all that concerned about the precise defintions of numbers because from their point of view that more of an Applied Mathematics question. They aren't really concerned about that so much. Use whatever definition you like when you apply your numbers, they'll say, and MORE POWER TO YOU!

But that's totally missing the point.

Moreover, no matter what they claim, it's Cantor's definition of number that they actually use as the foundation for Set Theory without a doubt. And Set Theory is what drive much of Abstract Algebra and Group Theory.

But then again. I don't even know why I care what they do to be honest about it. If they could get by for almost 2000 years without even bothering to formally define the concept of number at all then I guess it isn't very important to them.

I would think it would be important to physicsts though.

I mean, if physicsts are going to use pure mathematics to describe the universe in terms of String Theory, for example, you'd think they'd want to use a definition of number that actually matches the quantitative nature of the universe, and not some crazy idea of some pure mathematicians who coudln't care less whether their crazy ideas match up with phsyical reality or not.

All I'm saying is that such a "Natural Mathematics" actually exists, and Modern mathmematical formalism isn't it.

Mathematics is natural

When I observe Mathematics, I always start to wonder what kind of superhumans make the mathematics up, for the mathematics was full of bizarre definitions, and unreadably long proofs. There was also some devilishly clever short proof, but they did not make it easier, for I knew that one had to posses divine intelligence to come up with them. For example, I did not understand how a mortal could read, much less invent, a statement of a theorem spanning three pages or its proof and most of all why?

However I later come to realize that behind every definition, every theorem, every proof, there is an idea.

Often the simplest implementation of the idea does not work, and the original approach has to be modified or supplemented with technical conditions.

When the correct piece of mathematics is found, the original idea is buried deeply. Hence, explaining the idea involves far more than writing down logically correct mathematical statements. A good explanation requires description of the whole process of the discovery. So this takes some considerable time to invent I can imagine!

I can imagine the process of discovery is always messy, and it is tempting to reveal as little of it as possible. An honest description of the convoluted twists of mind that culminate in a discovery is both long and embarrassing. The discoverer is thus tempted to impress the reader with the shortest argument possible. Since it is rarely possible, monstrosities, dense with equations, are born.

Now natural mathematical are thus extremely rare as I don't find much on it searching on the internet. Yet I can't help but appreciate the thought that mathematics was natural from the beginning and those who do it actually enjoy mathematics as a passion.

With that I will finish with a quote that I like personally for I always appreciate those teachers who make a subject dummy proof for an average mind such as myself to understand. In otherwords a natural process that will allow the student to learn at a moderate rate to understand something that could be beneficial to them in the future.

The quote is:

The ultimate goal of mathematics is to eliminate any need for intelligent thought. - Alfred N. Whitehead

To me this means that mathematics should be a natural process in the end. If it where only true, but I do admire those teachers who try to break it down for one to understand. Now that in itself is a vast intelligent person who can do that.

Which leads me to another question.

Is 2 + 2 really 4?

I mean yes the world and probally everyone will agree that it is 4.

but is it really 4?

I mean what if one said it is 3?

The world will say he is wrong of course.

Yet let us say he provides proof that it is 3?

Would the world adjust to it or would they say no it remains 4?

I know this sounds like a stupid question for most people, but think about it for a moment when mathematics was discovered and the person came up with the question what is two plus two?

and he or she then said well it is 4.

Then everyone agreed for it is obvious if 2 things are standing on one side and two things are standing on the other side then you have four.

But it amazes me nevertheless that many mathmatical problems come up with their answers because of a discovery of a supernatural mind that somehow figured it out on his or her own.

I could imagine that math is always being tested continously with different ways of getting from one point to another.

It is also amazing that perhaps we have only discovered a ounce of what mathematics can probably do in the future concerning newer methods or ideas in finding them.

Yes, you could imagine that. But this doesn't mean that what you imagine is true.

Mathematicians do not even ask questions like, "Are these mathematical results ontologically correct with respect to the physical reality that we live in.

They just don't ask those questions at all anymore. They have totally accepted that mathematical formalism is completely divorced from the physical world.

They have completely accepted the Platonic picture that Jeremy posted.



They accept this as a GIVEN!

And what does this picture truly state!

It states that pure mathematics give rise to the world.

The world give rise to mind.

Then mind tries to comprehend pure mathematics that somehow encompasses far more than the real world.

Yet, most mathematicians hold to the believe that mathematics is real and NOT invented by the whim of man.

In other words, when we discover a mathematical relationship it's because it actually exists and not because we just arbitrarily made it up.

In fact, this is why mathematics is based on proofs. Once something has been proven mathematically it is accepted to be a genuine eternal truth. This mathematicial truth exists somwhere outside of BOTH the physical universe and the minds of men!

It has to!

If we accept that mathematics actually has some ulimately truth, and exists BEFORE we discover it, yet does NOT ARISE from the physical universe, then what other conclusion can there be other than to recognize that it must pre-exist somewhere else?

This model demands a 'mind of God' that exists seperate from both the physical universe AND the minds of men!

But it's a wrong model.

And this comes right back to what you just said,...



This is absolutely true. But not in an ontological sense!

It's being tested ONLY against it's very own AXIOMS!

And what I'm saying is that those basica axioms are FALSE with respect to the true quantitative nature of the unvierse.

Mathematicians are constantly testing their mathematicas against their axioms. But they never go beyond that to question their axioms because the axioms have been accepted as absolutes.

I'm saying that these fundamental axioms that they test everything against are ultimately wrong.

Of course, only in a subtle way.

It's not going to change the Pythagorean theorm, or Eulcid's foundations of Geometry. It's not going to change engineering mathematics of how we build airplanes or bridges, etc. And it's not going to change the way we do accounting.

What it will change are concepts that are on the cutting-edge of mathematics. Mainly concepts that deal in infintities!

As long as we remain in the realm of the finite our current mathematical formalism works just fine. The problems that we have introduced into mathematical formalism only affect INFINITY and how we deal with it.

This may seem totally irrelavent to a layperson, but all of the mathematics of Quantum Physics, String Theory, Differential Geometries, and many topics in Abstract Algebra and Group Theory DO deal with infinities routinely!

In fact, the concept of INFINITY is precisely the PROBLEM that is preventing General Relativity from being melded together with Quantum Mechanics. Currently when we try to meld them together they don't work because they yeild results of INFINITIES which make NO SENSE in modern mathematics!

But in the corrected form of mathematics, these conceps of INFINITY are described and defined in a totally different WAY.

This is why modern mathematics can NEVER discover their problem because they have already lifted themsleves up by their own bootsraps and they refuse to let go of those bootstraps.

Yes, they TEST their ideas, but they test them against their own bootsrapts (i.e. their accepted axioms!).

And it is these fundamental axioms that are wrong. So as long as they keep using these fundamental axioms their conclusions will always be MOOT with respect to the actual quantitative nature of the universe.

This is what I mean about them needing to come down to the basement of the Cathedral of Mathematics and LOOK into the abyss of the crack that lies at the foundation of their entire discipline!

You mention: In other words, when we discover a mathematical relationship it's because it actually exists and not because we just arbitrarily made it up.

So it has existed all along? So this means that human minds have just discovered something that already is and just presented what they discovered to the public which was further elaborated by other mathematicians. Is this correct?


Also you mention about how most mathematicians use axioms as a solid foundation and ultimate truth in resolving many mathematical propositions.

So this means if they didn't use this (using something else) they would open up a whole new world actually of understanding the possibility of more mathematical solutions in hand.

What could a mathematician use if not the axiom would interest me?

I find it compelling that I can even conduct a conversation in mathematics although I have no degrees in it

but nevertheless, (actually laughing so hard where tears are coming out), it is interesting to know the different possiblities you present in the subjects.

You really need to chat with some scientists who truly understand everything you are mentioning. You would probably get alot more out of it and maybe even come up with new conclusions or ideas

Hey at least I am honest with my educational level. Many I fear are not honest and claim they know it all.

Yet yes I think it is interesting that ones mind can discover something and use it later to construct the many things we take for granted on this world.

Math is truly everywhere you look. It reminds me of that movie with Russel Crowe acting like the mathematic genius who sees the world differently then most. He sees everyhing in mathematics. It is truly like a magical world unheard of.

What a beautiful mind this mathematician truly has and for that matter all mathematicians who do math with a passion and even discovered new things such as yourself James.

Not most. ALL mathematicians do this. This is how mathematics must be done. I'm not suggesting to do away with 'axioms'. I'm just suggesting that some axioms are true (meaning they are in agreement with the quantitative nature of the universe), and some are not true (meaning that they have no meaningful realtionship to the quantitative nature of the universe.

Physicsts use "postulates". Mathematicians use "axioms"

But these all reduce to "premises".

Physicists try to choose "postulates" that they believe correctly reflect the true nature of physical reality. They test their theories against nature, and if the theories don't hold up, they go back and reexam their postulates.

Mathematicians have gone beyond the point of even considering whether or not their "axioms" match up with anything physical. They have jumped off the edge of the cliff of total abandonment of physical reality into the realm of meaningless abstraction.

Abstraction removed completely from any physical reality. And they are comfortable with that. Why? I have no clue! It seems to me like a totally ludicous thing to do.

Would we expect aliens to jump off this same cliff? And if they did, then why would we expect their mathematics to agree with ours?

We could only expect it to agree if we BELIEVE that there was some universal reality in the abyss that we dove into.



ABSOLUTELY!

In fact, they have done this themselves on many occassions.

For example, for the longest time it was assumed that Euclid's axiom was true, that through any given point that is not on a line, only one parallel line can pass through that point.

But then they thought, "What if we change that axiom, what would happen?"

Then they discovered positively curved space (spherical geometry) and negatively curved space (hyperbolic geometry). These are now known as the "non-Euclidian" geometries. Euclidean geometry is considered Zero-curved space (or Flat space).

So yes, by simply changing their axiom they discovered new and wonderful things! Things that are TRUE by the way!

All of these geometries actually exist in the physical universe depending on the curvature of spacetime in the particular region of the universe that is being considered.

So they do question some axioms.

The axiom they refuse to question is Cantor's axiom of the existence of an empty set. And the reason they refuse to question it is because from their point of view it's obviously TRUE. Zero exists!

But that's a deep story I won't go into here.



It's not that they should do away with axioms. It's simply that some of the axioms they are using are wrong and need to be revisited and corrected.

The axiom of the existence of the empty set is a false axiom that must be discarded and replaced with a new one.

I have that new axiom, but I'm hoarding it for now. I'm not sure if I want to give away the gold just yet.

There's a lot to this. This is something I've known since I was in my 20's. I actually came to this insight throught a strange path.

I was studying the Machian Principle of General Relativity that gives a plausible explanation for what gives rise to inertia.

The study of inertia was one of my pet projects. The fact that anything can have inertia is amazing because it demands MEMORY!

The can be seen in one of Zeno's paradoxes. If time is discrete then each point in time is frozen. So think of an arrow in flight. Imagine it at any POINT in time. It's STILL at that POINT.

So how does the arrow know at the next point in time that it was traveling? What gives it inertia? It must somehow remember his history to know that it was flying along horizontally, and not just dropping straight down from rest.

Well, I have since resolved this question. However, at the time the concept of how things can 'remember' their inertia was elusive to me.

Earnest Mach, had an idea called the Machian Principle that relied on the spacetime equations of General Relativity. However, a PURE mathematican by the name of Kurt Godel came along and proved the Mach's Principle was false.

Godel's proof was a purely mathematical one with no physics involved. This drew me into the work of Godel. And that lead me to his famous "Incompleteness Theorem". This is a theorem that says that mathematics is necessarily imcomplete, and that any statements that reference the natural numbers are necessarily inconsistent.

This actually was a huge discovery in pure mathematics and rocked the math world.

In any case, this lead me to looking into why this is true. And that lead me to Cantor's set theory, and that lead me to recognize the flaw in the formal definition of number.

Once I saw what the flaw was I corrected it. And that is how I came up with the actual true mathematical formalism of the universe.

On an interesting side note, when mathematics is defined correctly Kurt Godel's proof that mathematics is inconsistent falls away.

It's not that Godel was wrong, it's simply that with the new correct defintion of number, Godel's theorem no longer applies.

Thus mathematics regains consistency when using the correct definition of number.

Also another very interesting thing as you had previously mentioned:



Yes, one of the things that comes from this correct definition of the concept of number is a very profond and enlightening understanding of the nature of irrational "numbers". Which aren't really numbers at all in this new formalism, but instead are recognized as irrational "realtionships", and we can see why they arrise and what causes thier property irrationality.

This is something that modern mathematics can never discover because they are locked into viewing irrational relationships as actual 'numbers'. So they have hidden their true nature forever.

Well with that I must say that it is highly time for you to write that book. Even if it is only a page a day. You will have a book done in one year unless it is a encyclopedia or something

I also think the mathematic and physic community would love to see a different approach for a change.

Who knows it might help invoke younger generations with new ideas that will one day create a spaceship that will finally fly to other galaxies?

Something I wish I could of experienced.

Thank you for a partial history lesson on math. It shows me that math has come a long ways and at the same time it looks like it still has a long ways to go.

You're right John.

I truly have a responsiblity to share this information with humanity. And it's certainly not that I'm trying to hoard it. I've just been too lazy to write it out. And that's truly disgusting.

Although, I confess there is more to it than that.

Like you say, this information does have potential to open up entirely new things in physics. Things that could potentially end up in resulting in space travel, or building bombs that could totally blast away the entire planet Earth.

This is another concern I have. I saw what humanity did with the information that Einstein came up with. They built atomic BOMBS with it!

What will they do with this information? Blow up the whole planet?

There's also something else I'm currently working on right now. Something that would totally shock the math and physics world entirely and instantly force them to recognize the truth of my words.

Not that I'm seeking fame, but I truly don't feel up to arguing about it.

Anyway, I've recently come to realize that they may very well be a way to calculate the value of planck's constant using nothing more than "pure mathematics"!

Of course, I'd be uing the REAL math, and not the modern artifical math.

But this would be a jaw-dropper. It would prove beyond any shadow of a doubt that I hold the key to truth. Being able to calculate a physical property of the physical universe using pure mathematical arugments with no need to actually measure anything physical would be a clincher that I'm onto something that modern mathematics can't even dream of doing.

I actually have ideas on how this might be accomplished. Actually if I complete this calculation all I would need to do is publish that one result and I would instantly be a household name around the world. The headlines will read, "Abracadabra pulls Planck's constant out of a pure mathematical hat!" No tricks involved! It's pure magick!

That would be a clincher that I'm onto something profound.

Sadly I'm not there yet, and it may prove to be more illusive than I think. But it's a fun calculation to work on anyway.

Abracadabra maneuvers his wand
o'er an abstract mathematical hat
pulling out numbers from beyond the beyond
before taking a sabbatical nap

Yes that is something one would have to really think about. You are right.

I can understand Einstein's concern at the time because he didn't want the Nazi regime to come up with such a weapon first and use it.

I wish governments would actually spend and concentrate more money and efforts in space projects then defense projects.

With what you have discovered may help trigger ideas in creating inventions that allow more space flight outside the rim of the universe we are in now.

I can't help but wonder when the Europeans have mentioned there is a planet out there that may have the same properties as Earth.

If we can only get our ships to fly faster.

Well I would really spend the time to get the book done. Sacrifice 6 months of writing! I mean just wake up and write. Go to bed writing. When you go somewhere write. Just write, write, write until you have that book done.

Sacrifice that time! You will be thankful if you did.

The scientific community will thankyou for the contribution to your discovery.

Otherwise they will never know and it was nothing more then a idea that no one has heard of.

A mathematician, a biologist and a physicist are sitting in a street cafe watching people going in and coming out of the house on the other side of the street.
First, they see two people going into the house. Time passes. After a while, they notice three persons coming out of the house.
The physicist: "The measurement wasn't accurate."
The biologists: "They have reproduced".
The mathematician: "If now exactly one person enters the house then it will be empty again."

The religious person, "It's a miracle! And it proves that God exists!"

Yes that is what they would say wouldn't they too funny

Well, that would truly be horrible actually.

Humans can't even get along with each other, and all they do is rape their own planet with absolutely no respect for it whatsoever.

And you want to send these idiots out into the universe to corrupt other worlds?

Shame on you John!

All the more reason to keep it hush hush!