We are going to take a slight moment now, and try to help put your mind at ease. Approaching such concepts as the ones mentioned previously and the one that will soon be forthcoming can be an intimidating experience. Hence, it is hoped that by pointing out how this intimidation is 'natural', we can limit the damage it will cause. The following is from this article, which is an excellent introduction to one of the concepts that is going to be half of our axiom. If you feel like you need a 'warm-up' article, that would be a good one.
The word after "infinity" in my dictionary is "infirm," a definition of which is "weak of mind." This is how many of us who are not mathematically inclined feel upon contemplating infinity.(To see how mathematicians regard infinity, see Working With Infinity: A Mathematical Perspective, which is also an excellent warm-up article.)
To continue with the quote.
We feel weak because our finite minds can only go so far with the concept, and because every time we think we're on the verge of securing even a shadowy understanding, we're tripped up by something. A friend of mine once told me that trying to hold her hyperactive toddler was like trying to hold a live salmon. Infinity is like that for us "infirm ones": slippery as a salmon, forever eluding our grasp.Note, many of the excerpts in this work are available without interruption here. Again, since this work is partially at attempt to bring the two perspectives (philosophical and mathematical) together, it would probably be worth your time to read both of the introductory texts.
Before we dive in, let's take a moment to remember to breathe and then take a plunge into deep thought.
For many of us uncomfortable with infinity, the word number can be defined as "that which makes numb."Hence, one can with a nod to Marx, call the the rest of this work "Opiates for the math's."
As this is not intentionally a work of humor, that will be my last attempt at such endeavors. To continue from the philosophical perspective...
We may think, for starters, that we're well on our way to getting a sense of infinity with the notion of no biggest number. There's always an ever larger number, right? Well, no and yes. Mathematicians tell us that any infinite set--anything with an infinite number of things in it--is defined as something that we can add to without increasing its size. The same holds true for subtraction, multiplication, or division. Infinity minus 25 is still infinity; infinity times infinity is--you got it--infinity. And yet, there is always an even larger number: infinity plus 1 is not larger than infinity, but 2infinity is.The concept of the 'exponent' is going to be crucial to understand before we continue. If you have forgotten from your schooling, or have yet to be introduced to the concept, a short refresher is here. [we'll wait....o.k.]
'Infinity' is a strange concept in mathematics because while one can imagine a single infinity and how subtracting or multiplying it doesn't change it, it is more difficult to see how ∞∞ is quite a bit more. From the mathematical perspective linked to earlier.
Also, as I've mentioned, there are essentially two kinds of mathematics. One of them is known as intuitionism, which is the less standard one. Intuitionists, unlike other mathematicians, don't want to essentially go beyond the very first orders of infinity. They don't allow infinities of higher order. And they show that within intuitionism, you can develop practically all the mathematics that you need. Of those two different mathematical theories, one allows in a whole bunch of infinities, the other doesn't. Yet they derive pretty much the same kind of results.So now that we know the problem we are trying to deal with, and how to illustrate with a simple graphed axiom how multiple infinities can exist and be used in conjunction with one another.Once again you see that the philosophical question--What kind of infinities exist?--is not really answered. It's left as a philosophical puzzle. I think it's fair to say that mathematics shows you what's coherent, but what's actually true about the universe is left a philosophical question. In fact, mathematics has shown us during the 20th century that you can have all sorts of mathematical theories out there that are compatible with all sorts of possible universes. I think the philosophical puzzle remains where it was 2,500 years ago and is likely to remain that way.[Dr. Reviel Netz]
First we are going to start with a simple axiom. This axiom, I can only say, came to me a few years back as I was thinking about stuff. Essentially it is derived from the question, "What is halfway between infinity and zero?"
The answer I came up with is 1 (one). This the first step in 'graphing' the axiom. It just takes a bit of non-linear thinking.
When graphed this looks like so.
As shown in the drawing, "Infinity" is placed as one axis. "Zero" is placed on the other, and basic N-squared curve shows that a line with a slope of 1 precisely divides each end of the spectrum of all possible numbers.
Now, there are two assumptions inherent in this first step. Those are that both 0 and ∞ are not numbers themselves, but the limits of numbers. This is something of a change from traditional number systems where 'zero' is considered a real number. As is evident from the above picture, the curved line, which can be drawn simply by taking any number (N) and multiplying it by itself (N=N2 and repeating this process ad infinitum, will never reach the limit. There is no number that can be multiplied by itself to get zero as the result. Even under the current rules of multiplication, there is no way to get zero as a result unless one starts with it in the equation. Hence, zero is NOT a real number in the number system we are currently exploring.
Inherent in the graph is also the basic idea that numbers, given the 'power law' will trend towards the limits of ∞ or 0 depending on which side of '1' they are currently found. Hence, one can draw the conclusion that the number 1 is precisely halfway between the limits of infinity and zero. Additionally, this gives rise to the equation, ∞ * 0 = 1 (infinity times zero = 1)*.
As mentioned, this is a major change in the conception of numbers. It is easy to see how a concept like 'zero' was originally introduced. After all, one can hold up 3 fingers. 5 fingers and 2 toes, or even 10 fingers and NO toes. We have always called that 'no toes' zero, and then later continued to build upon the mistake. "No toes" is completely indisquishable from "no fingers", "no kittens", "no infinities", and even "no people". As such, it is a bit outside the realm of rational thought, and should be treated as such. Hence it, like infinity, becomes a limit for a number system rather than an integral "trump card", where anything multiplied by nothing equals nothing. Multiplying by nothing assumes that one can start with nothing, and since the philosophy of the writer is antagonistic to the concept of 'ex nihilo', zero is not considered a real number for our purposes.
Note: zero has been confusing people for a good long while.
We now come to considering the first appearance of zero as a number. Let us first note that it is not in any sense a natural candidate for a number. From early times numbers are words which refer to collections of objects. Certainly the idea of number became more and more abstract and this abstraction then makes possible the consideration of zero and negative numbers which do not arise as properties of collections of objects. Of course the problem which arises when one tries to consider zero and negatives as numbers is how they interact in regard to the operations of arithmetic, addition, subtraction, multiplication and division. In three important books the Indian mathematicians Brahmagupta, Mahavira and Bhaskara tried to answer these questions.[source]
As most are well aware, division by zero has long been one of math's Achilles heels. In this particular version of it's history, we skip over 500 years before making anything close to progress.
Bhaskara wrote over 500 years after Brahmagupta. Despite the passage of time he is still struggling to explain division by zero. He writes:-Note: The reference article then goes on a tangent about Mayan mathematics, which was base 20 rather than 10, which is to say, they used all their digits, instead of just their hands. Neither system used general relativity to define the numbers themselves (as the concept wouldn't show up for a number of centuries). By 'relative' numbers, I am describing a situation where numbers do not exist in a vacuum. They cannot be created 'ex nihilo'. You must start with a number and work from there. Hence we have N2 as a more natural number system than simply N (i.e. A non-linear process where that which exists first effects itself, rather than a linear progression of digits numbering things that exist in and of themselves...fingers and toes, for example). This also removes the problem of situation like that of the diagonal of a square, and points to a resolution of some latent issues involving 'meta-physics'.A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.So Bhaskara tried to solve the problem by writing n/0 = ∞. At first sight we might be tempted to believe that Bhaskara has it correct, but of course he does not. If this were true then 0 times ∞ must be equal to every number n, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero. Bhaskara did correctly state other properties of zero, however, such as 02 = 0, and √0 = 0.
More on 'meta-physics' a bit later, but the general argument of "Quantum Philsophy" is that they are no longer needed. We have enough of a grasp of physics itself that we can do away with most of the 'meta' parts [such as those mentioned here]. Such concepts have become extraneous and can be removed post haste by applying Occam's razor and losing nothing that is necessary for understanding.
NEXT SECTION: Extending the Axiom to Higher Dimensions
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