But it's not a stupid question at all. The "Trinity" diagram of Plato's Pure Mathematical World, does indeed suggest to many people that "Pure Mathematics" exists "Somewhere out there" which is neither in the physical world NOR in the minds of men!

This is precisely what the THIRD GLOBE represents! It suggests the existence of a pure mathematics that exists totally seperate from BOTH the physical universe AND from the minds of men!



Exactly, and that mathematics exist in precisely the same place that Pink Elephants reside! In the minds of men!




Well, I didn't truly mean to jump on you personally. I confess that I have a bone to pick with this "Trinity" of Plato.

You may not think of it as being the "mind of God" but many people DO!

And this model just perpetuates this false idea that some knowledge actually exists beyond the physical universe AND the minds of men.

Thus leading to the conclusion that there MUST BE a THIRD place in which this Pure Mathematics resides (i.e. Outside of the physical world, AND outside of the minds of men!)

And that's a false delusion that is totally unnecessary, IMHO.

Sorry, but the whole Platonic Model is a touchy thing for me because so many people have accepted it almost as a "GIVEN". But there's nothing to support it truly.

In fact, I've seen this in so many math books, and even more so in philosophy books. It's such a popular model that it's almost taken for granted as a basic truth of reality. It implies (or at the very least suggests) that there is an intelligence that clearly exists outside of both the physical world, and the minds of men.

This is actually quite misleading.

YES! YES! and NO! NO! to BOTH of you!

You're both simultaneously totally correct, and totally wrong!

But it would require that I write my entire book to explain WHY this is true!

I guess I better get tuit!

I've been putting off writing it for DECADES!

I'm just a lazy BUM!

At least I am partially right. That has to count for something

Well, in terms of current modern mathematics you're actually 100% right.

They haven't even begun to scratch the surface. Mathematics will NEVER be complete the way they currently define it.

They question then becomes, "Have they even defined mathematics correctly?"

And that's a very serious question in and of itself.

EPSPECIALLY if we're going to asks question like, "Would aliens come to the same conclusions we're coming to?"

If we wish to claim that mathematics is the language of nature, perhaps that's a hint to how we MUST NECESSARILY define it?

In fact, this is the topic of the very first page of my book.

I've actually written quite a bit of it. I just need to get it back out and edit it. Mainly for clarity.

Just out of curiosity, does anyone here understand the proof of why any three p-adic numbers must necessarily form an isoceles triangle?

Also, do equalateral triangles count as isoceles in this case, or are equalateral triangles ruled out of p-adic space?

Just curious?

Too funny!

This is what I read:

Does anyone understand the proof of al;kjfd;ajofja jaodjfoajfd slash joajfdjj double ;adn;dj?

Who knows I am sure one might keep up with your mathematics genius? Probably Jeremy understands what you are talking about

Who knows you might just be having a best seller in all of the college classes across the country and end up teaching again

These particular questions are actually quite elementary. Anyone familiar with p-adic numbers should know this. I'm just too lazy to look it up and I can't recall it off the top of my head because it's been a while since I've worked with them.

The other question I was pondering just now is whether all n-dimensional spaces can be euclidean? (i.e. exhibit zero curvature)

I'm thinking they can. It seems obvious at first thought, but my mind has been failing me lately. Sometimes things that seem obvious aren't so obvious after further contemplation.

You mentioned that I should go back to teaching, but I've actually been thinking about going back as student!

I'd like to study ellipic curves and modular functions along with p-adic numbers and n-dimension spacetime geometries. I actually have specific physics problems in quantum mechanics that I would like to apply these to.

It's a bit depressing to have an interest in learning more about these things whilst simultaneously recognizing that my mind is out to lunch.

I think I better stick with giving up.

Er,... I mean,... accepting realitistic limitations.

I should have done this when I was a kid. The problem is that when I was a kid I was too busy learning algebra and calculus.

I'm a S-L-O-W learner. To this I confess.

What I really need is a bunch of students working under me doing all the thinking! Owl just tell them what to think about.

Like little human computers.

Then owl take the data they spit out and put it together in meaningful ways.

Well here is something that might be useful to you James.

It’s well-known that the ring of integers admits infinitely many structures of a metrized topological ring. The standard archimedean metric is the one we learn in grade school, and for each prime , there is also a -adic metric , where is the largest such that if it exists, and if . The -adic metrics (unique up to normalization) are called ultrametrics, because they satisfy the following strengthening of the triangle inequality: . It implies that all triangles are isosceles, and for any two balls with nonempty intersection, one contains the other. Each of the above metrics can be uniquely extended to , and completing yields the field of real numbers in the archimedean case, and the -adic fields for each prime . These fields are useful in number theory for a variety of reasons, e.g., they have much simpler arithmetic structure (Diophantine equations, and more generally, first order sentences are decidable, and the Galois groups of these fields are pro-solvable), and the complete topology allows questions to be answered using analytic techniques, such as Newton’s method (aka Hensel’s lemma).

has a unique maximal compact subring, called , the ring of -adic integers, which is both the completion of under the -adic metric and the subset of elements of with non-negative valuation. in turn has a unique maximal ideal, , which generates the topology, and the quotient, called the residue field, is the field with elements. The topology on is fractal in nature, and in fact, is homeomorphic to the Cantor set. Elements of can be conveniently written as Laurent series in with coefficients in the residue field, but addition and multiplication are more complicated than in the Laurent series field because of nontrivial carries. You will typically see this presentation explained as writing rationals in base , but letting the expansion extend infinitely far to the left. Standard examples include and . We cannot let the expansion extend infinitely far in both directions, because multiplication becomes hard to define.

Now that we understand , I will spend the rest of the post discussing some distinguished extensions. We have the following strict inclusions:



The last three fields are ring-theoretically isomorphic to , assuming the axiom of choice, and the existence of an embedding is employed in Deligne’s theory of weights. There are additional extensions of interest, such as , a maximal tamely ramified extension, and , a maximal abelian extension, but they are beyond the scope of this post. I’ll add some references at the end.

is known as a maximal unramified extension of . It is given by adjoining all th roots of unity, for . The elements of this field can be represented by Laurent series in , with coefficients in , although this is an imprecise way to describe it. There are two standard ways to remedy this: First, one can say that the coefficients are Teichmüller lifts of elements in , in other words, prime-to- roots of unity or zero. Alternatively, one can work with Witt vectors, representing the coefficients as . Witt vectors become very messy if you try to do anything explicit (Lenstra once remarked, “the formulas do not fit in the head of a civilized mathematician of the twenty-first century.”).

is an algebraic closure of . Typical elements not in are roots of , -power roots of unity, and more exotic examples like roots of (discovered by Chevalley). Their expansions in powers of do not necessarily have exponents that form a discrete subset of the rationals. Bjorn Poonen’s undergraduate thesis has partially expanded in powers of two, and the set of exponents with nonzero coefficients seems to have order type , although I don’t know if anyone has bothered to prove it. Kedlaya’s third paper on the arXiv describes which series lie in this field, using twist-recurrence relations, and shows that the aforementioned order type is the largest possible for algebraic elements.

is the -adic completion of . One can generate transcendental elements in by summing a rapidly decaying sequence of elements of increasing degree such as , and the proof that they are transcendental is based on the fact that the action of the Galois group preserves distances. Gouvea’s undergraduate textbook has a detailed construction of a Cauchy sequence in that doesn’t converge in . is one of the most commonly used base fields for -adic analysis and geometry, since it is complete and algebraically closed while still having a countable dense subset. A lot of classical Banach space theory works well here, e.g., open mapping theorem, closed graph theorem.

is the spherical completion of , although I don’t think the notation is completely standardized. A spherically complete metric space is one for which every sequence of nested balls of finite radius has nonempty intersection. This implies completeness, since that is the special case in which the radii of the balls approach zero, and it is necessary for the Hahn-Banach theorem to hold. Rather surprisingly, is not spherically complete, and in fact, one can reasonably say that most nested sequences of balls in have empty intersection. One example (pointed out to me by Bjorn) is given by taking the balls of the form , where the exponents are reciprocals of primes. The radii decrease to 1, but there is no Cauchy sequence of algebraic elements that begin with this sort of expansion.

How do we write elements of ? The answer quite simple, and is found in Poonen’s undergraduate thesis. The elements are exactly those power series with coefficients given by Teichmüller representatives of , such that the set of exponents with nonzero coefficients forms a well-ordered subset of the rationals. Apparently, the hardest part of the proof was showing that this set was closed under addition and multiplication.

One might ask if there are natural extensions of that don’t add geometry (e.g., taking rational functions shouldn’t count). The answer is no. There is a notion of maximally complete nonarchimedean field, because if we fix the data of the characteristic, the residue field, and the value group, then there is a maximal field with respect to those properties (assuming the data can actually come from a field), and it is unique up to nonunique isomorphism. However, there is a theorem asserting that maximally complete fields and spherically complete fields are the same thing. In particular, given any proper containment of fields such that they both yield the same data, the smaller field cannot be spherically complete.

The whole framework above more or less extends to the fraction field of any complete discrete valuation ring, such as . Since the field of complex numbers is algebraically closed, the Laurent series field has no unramified extensions. The algebraic closure is the field of Newton-Puiseaux series given by , which is not -adically complete, and its completion is not spherically complete, closely mirroring the local field case. The absolute Galois group of is , isomorphic to that of , but instead of a Frobenius, there is the monodromy operator . You can think of the spectra of both fields as really small circles, so the Zariski topology only sees a point, while the étale topology is a strong enough magnifying glass to see the finer structure. This picture is loosely connected to the arithmetic topology view of Spec as a three-ball with the primes forming embedded knots.

Further reading:

Schneider has a relatively new book on nonarchimedean functional analysis. I had expected it to be light reading, but I was wrong.

Bosch, Güntzer, and Remmert have a book that introduces nonarchimedean analytic geometry. You should skip the first six chapters, unless you really like Japanese rings.

The Arizona Winter School held a -adic geometry workshop this March, and Conrad has excellent notes on the foundations of rigid analytic spaces. Most of the speakers have excellent notes.

I don’t recommend Tate’s Inventiones article on rigid analytic spaces, although it has a cute joke about following Grothendieck fully and faithfully.

For information on other interesting algebraic extensions of , look in Serre’s Local Fields, and Serre’s chapter VI in Cassels and Frohlich. In fact, you might as well get everything you can find by Serre, because he writes beautifully.

http://sbseminar.wordpress.com/2007/08/21/p-adic-fields-for-beginners/

also more on scientific programs on field equations

http://www.fields.utoronto.ca/programs/scientific/03-04/p-adic/abstracts.html

Thank's John.

That's a very interesting information. That actually covers the Group Theory of p-adic numbers in general.

I didn't realize that intersecting spheres in p-adic space necessarily contain each other. That's another interesting property of p-adic space that I'd like to understand better.

I already have some clue of these things in the pure abstract sense of the algebraic rules that define them.

Where my specific interest lies is in the intuitive understanding of these concept geometrically in terms n-dimensional spaces.

This is a path that most pure mathematicians avoid. But my interest is more physical than mathematical. Pure mathematicians are happy with axiom, rules, and the pure abstract conclusions that follow from those abstract rules and axioms.

My quest is to gain a more intuitive understanding of what the physics is behind these things.

And YES, I believe that there necessarily must be physics behind them. In fact, it's my convinction that if I can show that the physics fails, then I will necessarily and simultaneously be able to show why certain axioms of pure mahtematics are false with respect to the actual physics of the universe.

If this can be shown (which I believe it can be in many cases), especially with respect to the abstract algebra of Group Theory, then it necessarily follows that the axioms that have lead to those conclusions are necessarily inventions of man, and have no intrinsic relationship the actual physical universe in which we live.

This is paramount because we're talking about axioms here, and not merely conclusions.

In other words, we will have proven that the Pink Elephant is necessarily a totally man-made illusion, and not an intrinsic property of the universe.

In fact, I think I already know how to do this.

Thank's for the info John.

You've been far more help than you realize, I'm sure.

Just when you publish that book mention a thank you from John(smiless) for your professional scientific help on the matter.

Just joking

I wish there were more minglers interested in the topic to really have a deep discussion on it. Usually such discussions can lead to new ideas that help advance further in a particular topic.

If anything I hope this thread lasts a long time allowing those who are interested to learn something perhaps new each time they visit it.

"If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics." - Roger Bacon

Ugh!

Metalwings theory of Applied Mathmatic and how to use it go get Chicks.

There is usually a lot of misunderstanding about what math actually is and how it applies the the real world. This is called "Applied Mathematics". Theoretical math deals more with ways the formuli work in ways that are interesting but may or may not have any practical use.

Many animal's brains (all to some degree) are computers. Many do internal math much better than humans in order to solve physics problems. The animal does not know or care about physics, it simply wants the course trajectory calculated to miss the branches, slow enough to catch prey, and land safely at zero velocity, for example.

Man has the (far as we know) unique ability to convert this math-physics conversion problem into symbolization. He created the concept of math and dimensions to describe to himself ways to predict math/physics problems. None of the multidimensionals we use in modern mathematics actually exist. What happens in reality is real and the math is our creation to visualize how they work. Much of what we think of as reality can be described mathematically and much of what can be described mathematically is just an exercise in mathematics.

Early man had to throw the spear
in a fraction of a second
to reach the point
in the future
through the wind
to hit the deer
as he jumped
as he ran
away at angles
predicting the turns
the pull of gravity
the force of impact
the location of vital organs
in time for dinner
in a way man cannot fully describe mathematically today

so he could buy her a steak.

Do not confuse math with reality. Math is a human construct simply to describe reality, and in some cases to describe itself. No mathematical construct is "real", but it can still put meat on the table ... or mankind into a far away star system.

Absolutely!

I couldn't agree with you more Metalwing.

You are absolutely 100% correct! This is indeed the state of modern mathematical formalism!

This is precisely why I put my statment into the form of a conditional statement.

I simply say,

"IF mathematics is supposed to properly discribe the quantitative properties of the physical universe, THEN modern mathematical formalism is incorrect"

You are one of the mathematicians who would disagree with the hypothesis of this conditional statement, and therefore accept the statement overall as being true, because you see the hypothesis as being false.

However, once you do this it would seem to me that you would be the first to recognize then, that aliens would not agree with our mathematics.

You would also agree that mathematics is not a universal language of nature since you clearly recognize it as being nothing more than the whimisical imaginagion of men.

Therefore you should readily see why a pure mathematical theory cannot be depended upon to reveal the truth of nature (i.e. String Theory is merely a mathematical guess based on the pure construct of man-made whims).

I disagree with this whole view.

In other words, I disagree that this is what mathematics should be. I believe that mathematics can indeed exist as a pure language of nature!

And if we treated mathematics as a science instead of a whim then we'd learn the true language of nature.

You said:


Again I agree with you 100%.

This is indeed the view of modern mathematicians. They view "Applied Mathematics" as merely a subset of a more encompassing "Theoretical Mathematics".

What I'm saying is that this is false.

It's not false with respect to modern mathematical formalism.

It's false with respect to the language of nature. (i.e. what mathematics should be)

That's my stance.

I agree with what you are saying because you are speaking about the current state of the actual formalism.

What I am saying that this formalism is wrong.

Do you see what I'm saying?

It's wrong based on the conditional statement I gave above.

and below,..

"IF mathematics is supposed to properly discribe the quantitative properties of the physical universe, THEN it's wrong!"

That's where I'm coming from.

And the concept of "Applied Mathematics" doesn't cut it.

Sorry.

That's a totally false notion.

This is what I am claiming.

In other words, real mathematics (i.e. the quantitative language of the universe, is not properly described by our current modern "Theorectical Mathemaics", therefore the delusion that real mathematics is simply a subset of "Theorectical Mathemaics" is false.

The true quantitative language of nature is entirely different!

Well, I shouldn't say "entirely" different. It's simply subtly different in very important ways.

In ways that are totally rejected and not accounted for in either "Theoretical Mathematics" or any subset of Theoretical Mathematics that we might refer to as Applied Mathematics.

That's a very wrong notion.

But unfortunately, it is very highly held as the correct way to view these things.

The current mathematical community would totally agree with you!

In fact, they are the ones that taught you that this is the case.

So you would get an A and move to the head of the class, and I would be expelled for being a rough rebel.

Yep. Fer sure!

You mention: I believe that mathematics can indeed exist as a pure language of nature!

And if we treated mathematics as a science instead of a whim then we'd learn the true language of nature.

Would you be so kind as to elaborate more on this James?

2 + 2 = somewhere between 4, and 5.

Well this is why I need to write a book.

However, I can offer you a crude synopsis. (I had to break this into the following four posts which I have enumerated for your convenience.)

Part I

If you enjoy fairytales, the history of mathematics is a whooper!

Our current modern mathematical formalism was truly spawned by the mathematical thinking of the ancient Greeks and we followed their lead perhaps a little too closely.

Now I realize there was actually quite a bit of "mathematics" that had existed prior to the Greeks. Primarily the work of the Egyptians and Babylonians, although this earlier work was more along pragmatic lines of actual arithmetic. Although they also began to explore some concepts of abstract algebra. However, their algebra was created specifically for solving pragmatic problems of calculations. So the "mathematics" of the Egyptians and Babylonians was fairy pragmatic in design. Even though they too often associated the concept of number with mystical power of the gods.

However, the Greeks were also strongly influenced by the Jains. The Jains were from the far east, mainly India, who were also thinking about concepts of mathematics and number. The Jains were highly spiritual and viewed mathematics and number in a very mystical way. These would have been the same people who viewed the world spiritually in terms of Buddhism, Taoism or variations on those themes.

The Greeks where strongly influenced by the Jains and so they too viewed number as being mystical and spiritual. Moreover the Greeks themselves were highly philosophical and surprisingly unscientific. Like Einstein, they tended to do thought experiments, rather than perform actual physical experiments. Of course this isn't entirely true. The Pythagoreans did indeed work with physical geometric examples, as well as using vibrating strings to investigate the mathematical nature of number.

Although at this point I truly believe that they felt that number was giving rise to the musical notes, rather than the other way around. Now this may seem like an odd statement, but in truth, it's a quite profound realization. This is because inherent in their initial choice of fundamental string length rests the hidden fact that they can begin with a string of arbitrary length, yet the same interger relationships will always hold. This should have been a clue to reveal to them that notes give rise to number, and not the other way around. But for some reason they never even asked that question. They were totally in a spiritual mindset believing that number gives rise to manifestation, and not the other way around.

Ironically they address many other very deep and profound questions. One of which was the question of whether the universe is a continuum, or whether it is made of discrete individual points. Wellm if we think of this in terms of pure thought it can be a very perplexing concept. To believe that the universe is a collection of individual points seems utterly absurd. What would be between the points? Obviously there would need to exist more points between any existing points!

So the overwhelming consensus was that the universe must necessarily be continuous. It's obvious! It can be no other way!

Well actually a Greek philosopher by the name of Zeno wrestled with this very question and decided that the universe must necessarily be discrete. He argued that in order to get from one point to the next we must first move to the point midway between. And therein lies a paradox. In order to get to the midway point we must first move to the point midway between the us and the midway point and so on, ad infinitum. Thus Zeno's conclusion was that in order to move we would need to complete an infinite number of tasks which would be impossible!

Zeno was so vehement about this he actually proposed the very same question in a dozen or so ways trying to convey the absurdity of the notion of a continuum. He argued that the universe necessarily must be discrete if motion is to be possible. Zeno has to be my favorite Greek Philosopher. I believe he was perfectly correct. Motion would be impossible in a universe that is a continuum. In fact the very notion of points that are not the same point would be an impossible notion in a universe that is a continuum.

So Zeno was on the right track to discovering the true nature of number. In fact, I personally give him credit for being one of the very first to have recognized that the universe must necessarily be Quantum in nature.

Response to Smiless - Part II

Despite Zeno's genius he was basically dismissed as being a kook. We can obviously move, and the universe is obviously a continuum. Next!

The concept of a continuous universe prevailed. And this erroneous notion still prevails to our present day, in spite of the fact that Max Planck has ultimately showed us that we do indeed live in a discrete and discontinuous universe. And Quantum Mechanics has confirmed it.

The Pythagoreans were a strange bunch of philosophers. Actually the Pythagorean 'school' was more like a 'church'. There were a very strong feelings that number had something to do with the mind of God. In fact, numbers were believed to be 'perfect'. Well, actually they had even defined specific numbers to be mathematically perfect, but just the same, the very concept of number in general was considered to be divine. In particular the whole numbers, or integers. In fact, that’s all that existed at the time, or so it seemed.

Well, as time when on the Pythagoreans discovered that the square root of two could not be expressed as a ratio of whole numbers. They were actually able to prove this result which is not difficult to do. Thus the square root of two was clearly an irrational number. (i.e. it is a quantity that cannot be expressed as the ration of integers!)

This was paramount! This meant to the Greeks that their divine Integers were contaminated! There existed numbers that could not be expressed as a ratio of the Integers! This was totally devastating to their spiritual view of number. In fact, stories have it that the Pythagoreans actually took the man who made this discovery out to sea and drowned him as a sacrifice to the gods for his malevolent discovery!

The Pythagoreans found this particularly disturbing because, as you well know, the Pythagorean theorem requires the finding square roots for its solutions (i.e. a² + b² = c²). Thus to find the value of c we must necessarily find its square root.

Now this is very interesting because this was a major turning point in mathematics. In fact, I would personally say that this was the first mistake made in mankind’s formalization of mathematics.

Rather than appealing to nature for the answer (i.e. taking a scientific point of perspective to view the problem ontologically), instead the Pythagoreans chose to travel down the path of pure abstract thought. They actually allowed for a change in their very definition of number. Although in as sense they had never truly defined the concept of number prior to this. They just assumed that number was some mysterious spiritual property of the mind of God of which they stood in awe.

However, what they had actually done here was to formally define the meaning of number without even truly realizing what they had done! Their new definition of number was to simply say that Number is the result of any algebraic expression or mathematical equation. Now this definition was truly an invention of the whim of man. Mother Nature probably peed herself with laughter when she saw humans creating this totally erroneous definition for the meaning of her quantitative nature.

It's a totally erroneous concept that isn't supported by nature. It’s ontological false. It’s a definition created by the whim of man.

Response to Smiless - Part III

Now this is where philosophers and mathematicians would raise their voices and object. They would say, "But we're talking about the LENGTH of a leg of a triangle here! That LENGTH is IRRATIONAL! And it EXISTS in NATURE! Therefore this is NOT just a WHIM of MAN you IDIOT!!!"

Well, maybe all philosophers would not be so uncouth as this, but I'm sure that they would still raise this objection in a refined and polite manner.

However, this objection stems from the totally erroneous idea that distances are indeed continuous and can be broken down into infinitely many smaller distances between infinitely many points in a continuum.

Now this idea is not supported by Zeno, and has actually been shown to be false by modern day Quantum Mechanics. We actually live in a Quantum universe, not a Continuous universe.

Just because we have a cute little algebraic formula that relates the lengths of sides of triangles doesn't mean that the “numbers” we come up with are a genuine quantitative properties of those distances. In fact, is it truly even meaningful to say that a distance can even have a quantitative property?

Who's leg are will pulling here?

Mother Nature has got to be peeing herself with laughter over the whole ordeal.

This whole formalism has gone awry already.

It only gets worse as the history of mathematics unfolds.

There were many people who had an opportunity to catch this mistake, but instead of catching it, they actually contributed to embedding this erroneous concept even more concretely into the formalism of mathematics.

I could name many mathematicians who contributed to formalizing this idea over the millennia with Georg Cantor epitomizing this idea by inventing the notion of an "empty set". He introduced this notion into Set Theory formally as an unproven axiom which has become a foundational pillar of much modern work in mathematics. This is especially true in the field of Abstract Algebra and Group Theory.

In truth, all mathematicians did not agree with this approach to defining number. (note: Modern Set Theory ultimately formally defines our modern formal concept of number)

Henri Poincaré, a very influential and prominent mathematician of Cantor’s day had this to say about Cantor's set theory, "Cantor's Set Theory is a disease from which future generation must recover"

I'm in total agreement with these words of Poincaré.

Leopold Kronecker, also a prominent mathematician of the time had this to say, "God created the Integers, all the rest is the work of man."

Again, I'm in total agreement.

Does this, mean that irrational numbers cannot exist? Well, yes and no.

Yes, if we redefine the meaning of number then these irrational relationships can still exist as solutions to pure abstract equations, but they would be recognized to be just that - relationships, and truly not satisfy the actual formal definition of number.

So, no they would no longer be numbers. They would mathematically lose their status of being mumbers, but like Pluto who recently lost his status of being a planet did yet did not instantly vanish from the universe, neither would irrational relationship vanish. They would simply be recognized for what they are; quantitative relationships, and not valid numbers.

This may seem rather esoteric, but I assure you that it’s not. Changing this formal definition of number will have a profound affect on mathematical formalism in many areas, specifically all areas that are heavily dependent upon Abstract Algebra and Group Theory.

Response to Smiless - Part IV

Now this concept of correcting the formal definition of number may seem trivial, but I assure you that it's not.

Cantor's Empty Set Theory leads to a multitude of infinities. That's right! According to modern day mathematics there are an infinity of infinities! Each one being a different cardinal size than the other!

In other words, according to Cantor's "Empty Set Theory" infinity comes in sizes! And there are as many different-sized infinites as there are real numbers! (i.e. infinitely many sizes of infinity!)

Most laypeople have no clue how crazy mathematicians truly are!

Georg Cantor is the only man in history to start out with nothing (i.e. and empty set) and end up with more than everything! (i.e. infinities larger than infinity!)

Now you may ask, "Do you have as solution to this dilemma?"

My answer is, "Yes I do".

There is another way to define number which avoids this problem. In fact, all I did to find the answer was ask Mother Nature. She smiled and gave it to me along with a pat on the back.

So now I have the correct true definition of number. But so far I've just been standing here holding onto it. I'm too lazy to write a book and share it with the rest of the world.

People might say, "What do you mean by the true definition of number?"

Well, I mean the correct ontological idea of the quantitative nature of the universe. There is only one such correct idea. All other ideas are whimsical inventions of men that have nothing to do with the true quantitative nature of the universe.

In other words, I have the definition that aliens would most likely agree with. Especially if they also arrived at their definition of number scientifically (i.e. ontologically rather than via pure whimsical imagination driven by spiritual awe and superstition.)

Actually Euclid almost had it! He was so close!

As you know Euclid formally defined many concepts in Geometry. However, he cheated a bit and used axioms instead of appealing to Mother Nature for a precise ontological definition.

He actually chose good axioms and if he had thought about them more deeply he might have recognized that he had the solution to Zeno's paradoxes right at his fingertips. But instead of exploring them more deeply he just accepted what he knew to be true and stated them as axioms. That's a shame, he was so close to the real definition of number but couldn't see it because he was thinking purely in terms of moving forward with his geometry. He wasn't concerned with Zeno's paradoxes evidently.

Archimedes also had a great opportunity to discover the true nature of number but instead of making that discovery, he too, took a more abstract approach and basically invented the early forms of calculus instead. He could have gone two ways; back to the definition of number, or forward to calculus. He chose to move forward. Which was cool. But he missed a great opportunity to correct this error.

By the way, modern calculus is not wrong. In fact, Karl Weiestrass beautifully formalized the definition of the calculus limit in a way that is totally compatible with the true nature of number. So modern calculus is safe from this error of the poorly defined concept of number. Calculus itself will not be affected by correcting the definition of number. We have Karl Weiestrass to thank for that.

In any case, I'm think I'm the only person on planet Earth who knows the answer to this riddle. I know what the problem is, how it was introduced, where the errors were formally supported, why they were supported, and how this erroneous definition of number affects Abstract Algebra, and Group Theory, and why it doesn’t affect Calculus. I've even discovered something truly beautiful concerning the nature of irrational relationships, when, how, and why they arise.

I really need to write the book!

Gee, I almost wrote half of it here, but not really, there is far more to tell. It's a real saga, and you can see why! This error was introduced at the very dawn of mathematical formalism, and formally supported the whole way though history, so the whole formalism is riddled with the affects of this erroneous notion.