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Introduction: 

In this article I describe concept sharing and take a look at Fermat’s Last Theorem. While it’s a common belief that math is cumulative, so that for example, to do calculus you need to know how to do the math at the lower grades, it might be that there is some basic math missing from our understanding of the overall mathematical structure. Here I present a different concept which subsets existing mathematics and has many applications.

Overlapping Shadows:

An interesting notion can come about from something observed in nature, the overlapping of shadows. Consider a cup of tea placed on a table and lit with two lights from above, one from the left and another from the right, as illustrated below:

Three areas of shadows are formed. The one in the center is the overlapping of two shadows. 

Consider two points intersecting from the areas of the overlapping shadows. We may make an analogy of two points being placed together with the overlapping of the shadows at the centre. Think of items you see around you, some touching other items, some like a book on a table, by the force of gravity. Others like books placed together on a shelf. Here is where two points of the different items, one from each item, can be seen as being together.

Now consider a point on one of the shadows which doesn’t intersect another shadow. This too has no extent. But as we’ve seen, something of no extent( two points placed together) can be expressed as the combination of two items which also have no extent( two separate points). Can this one point be seen as multiple as well as singular? Think of a hand of bananas. Each banana is connected to the others at the top of the hand.

Duplication also exists in nature, we can think of cell division.

Seeing this duplication with the shadows is possible. Consider two teacups and four lights. Creating sets of shadows as shown above.

The shaded area is where two shadows from teacup 1 and two shadows from teacup 2 intersect. Labeling points with “p”, I can associate p(1)(1) and p(1)(2) two duplicates and p(2)(1) and p(2)(2) the two other duplicates with this area. Then the original points p(1) and p(2) can be groups of these two. Then we may think p(1) and p(2) as two points which may be separated as in the analogy of the items touching each other but not p(1)(1) and p(1)(2) or p(2)(1) and p(2)(2).

In the case of a mathematical point, a duplication would have to be located in the same position.Since points are defined as having position only, a duplicate would have to take the same position. But I already have a concept of no extent without duplicates. If two duplicates are together it should lead to a single point. The usual case of a point is singular. So this is not a duplication in the usual sense.

Duplication exists in the form of concept sharing. The idea that concepts can be multiple. This duplicates the concept. Yet since the duplicate is identical we must remove the initial concept so there is room for this new concept.

The case of a point being only singular could be removed. Then I have room for this concept sharing idea. The concept sharing of point.

we must be able to then remove the concept which was there originally and replace it with the shared concept. It must be possible for shared concepts to exist somehow, as we can imagine it.. 

A point and its partner must be multiple so must take up the space of one point only. Yet we already have this notion of no extent without extra points.

All other possibilities for the number of shared concepts are removed.

There must be a “container” of concepts to take the concept out of. A concept of concept space. Here the two shared concepts can be seen as different. That is, I can separate them. I have another dimension of the concept, in concept space. So for points or places I have “places of places”

So I need the concept of a “concept container”

It is the same concept as the concept in question, yet it contains the concept in question. Meaning that it exists at a lower level so that the concept in question could be removed by two or more sharing concepts (equivalent to the concept in question). 

We cannot have both the initial concept  and sharing concepts at the same time. We need to remove the initial concept which means we need the lower level. Since we can always have the same concept as the concept in question and another dimension of the concept is possible since I can have the sharing of two or more of the initial concept and the concept can be extended using the extender “of” as in concept of concept. You can think of this as a nesting of concepts.

We can imagine that concepts can be multiple. Since it must be somehow possible to have sharing concepts, this lower level must also exist.

Akin to the overlapping shadows, concepts are not concrete-as they are abstract ideas, two or more of the same concept can co-exist. They share and fit into the concept which was there originally.

A symbol may represent two different concepts, such as “8” representing an amount, a measurement or a label.

In the case of points or places, the lower concept level is a place of places or location of locations. The two shared points may move off to two different places of places and we can then see how they can be made different (they can be defined at two different places of places).

When they separate, they do not leave a void, there must be a place of places. They may be separated as a place can take on a new place of places. See the diagram below. I am using a jagged line to show the opening into the new place of places space. This can be made two dimensional, leading to a new plane.

The Concept Sharing of a Number:

Numbers are exact concepts. In the above case, we can think of them as the number of shadows at the center. They have exact boundaries and some way of showing we have two there or three there, ect.

Then borrowing from the notion of overlapping shadows we should be able to hide numbers together and they would be “two hidden as one” as well. (concept sharing) if the mathematical objects represented by the numbers had the same boundaries, like the shadows at the center. 

Other than the further darkness of the overlapping shadows, we cannot see or imagine that there are two separate shadows there. Two or more numbers can be hidden as one since natural numbers represent exact positions.  Similarly with two numbers hidden as one we can not see or imagine them together. Yet our logic tells us this can be so.

Then to this end let us create another number dimension, a dimension of number of numbers. This must already exist since it should be possible to put two or more numbers together at a beginning. Let the usual case be that the number of numbers is only 1. But now let us expand into the next dimension and allow the number of numbers to be 2.

So for example with the number 1, let us take away the original number 1 (since we have another underlying dimension, we can do this) and replace it with two new numbers 1(1) and1(2). These are together like the two shadows but do not form one number.

The two numbers together can be notated ((1(1)((1(2)). 1(1) is “peeking out” from behind 1(2). Shown by the use of a half parentheses. Seen more clearly here: (a(a).

Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects, also concept sharing, hidden as one.

The objects are concept sharing as well and are somehow different from each other. We give the two hidden objects two new numbers 1(1) and 1(2). 

In the case of mathematical objects there is no external way of telling how many objects there are, previously it was assumed it was only one. We can state how many we wish at the onset thus fixing a certain mathematical system. Then we need the concept sharing of a number to indicate how many objects we wish to be there.

A new plane:

Points are also exact concepts. In the Euclidean plane they are places, with the notion of no extent (zero-dimensional), in the plane. We should be able to have two together using two new numbers 0(1) and 0(2) identifying that we have two points. (0 is indicating an origin)

An object of no extent existing together with another object of no extent, would still have no extent- but there could be two objects here, under another mathematical system. 

The two new points 0(1) and 0(2) can be different by first uncovering a new place dimension, a place of places. This must already exist because there must be some way to have two points exist together and still be two points.

In a similar way as we uncovered the new number dimension (the number of numbers) we can uncover the new place dimension. 

Take the original point out (we can do this since we have a new underlying dimension of place, a place of places) and replace it with the two new points. This can be done for the whole plane of points.

That is, there is nothing special about the origin, so any point of the usual plane can be removed and we can replace it with a “sharing” of two points. So that we have a set of doubled points. 

The New Plane of Numbers:

We can build another set of numbers by creating a number plane rather than a number line.

Start with place(2) space (places of places). This can be begun by considering two points which have an infinite uncountable number of parts. They can be extended outward perpendicular to each other. Then let this be the beginning. The places of places space so created contains points which are equivalent but not identical. They are equivalent in that they all start defined together at some place of place of places but not identical in that they all have a different place of place. Then let us place an arbitrary number of points(2)(these are the points which can move in the more primitive underlying place of places space) placed at the same origin. These can consist of a countably infinite number of points.

Now we can build a new number line by moving two points(2) out to some unit length of 1 unit. Do this for both axes. So that in each direction we have a distance of 1 unit marked, now also I have 1 point(2) at 1 and mark it with 1 This is a location (0,0) moved from the origin, so represents some distance. If I want to I can move these numbers in the place(2)a axis as they are associated with points(2). 

In the first quadrant we can make a number (1(1) by moving a part at 1 on one axis and the other part at 1 on the other axis to combine at the place(2)  (1,1). To do this we have to move two parts out to each axis, but this is okay because we can have an arbitrary number of parts there.

Now we can make (2(2) by the same process. Move two parts from the origin out 2 units and move one part up 2. units from the x axis and one part from the y axis to come together at the number (2(2).

We can go on making a midline of new numbers (1(1), (2(2), (3(3),…

Right now we are only interested in the numbers in this number plane that makes sense to us. That is, the numbers on the midline. 1 and 3 could also share but there would be no higher number here at the first level.

Similar to negative numbers, these could have some meaning, but I won’t explore this here.

We can have operations on these numbers such as addition (1(1) +(1(1)= (2(2) = 2*(1(1)- if we combine the new number system with the system of numbers point(2) numbers-and multiplication (1(1)* (1(‘)= (1(1) or (1(1) *(2(2) = (2(2)= 2*(1(1) as well.

But here is something interesting, since we can combine the two point(2) number systems (1(1) =sqrt(2), if I consider the midline as continuous and the axis to be discreet values of natural number on the axis and multiples of sqrt(2). Now we have to make this space different from that of which we are already aware. Let any distance be in the new dimension and let it be negative, as we are moving away from zero, but not in the first dimension.

Then since (1(1)^n=(1(1) then sqrt(2) =sqrt(2)*sqrt(2) =sqrt(2)*sqrt(2)*sqrt(2)… And we have an equivalence of these numbers. This is also seen intuitively as I am moving parts out to the different place(2)’s and these parts are equivalent for a certain set of numbers in this space.

Fermat’s Last Theorem:

Now I want to take a look at the general statement of Fermat’s Last Theorem, seeing if I can find a way to show its truth. It is said that it is unlikely that Fermat had a proof but my ideas could have been known at his time( since in a sense they are basic), so maybe with this new understanding he worked out what I did.

Many people have tried to prove the theorem in the past with elementary methods that would have been available at Fermat’s time. But here I am introducing something new and powerful which didn’t exist before, but might have been known and then subsequently lost.

Now consider that I might have three numbers a, b, c with c related to a and b. I am thinking specifically of a+b=c or a^2+b^2=c^2 or a^3+b^3=c^3…..

Let these be extended to the second form of numbers so that I have (a^n, a^n), (b^n, b^n), (c^n, c^n) or a^n(1(1), b^n(1(1), c^n(1(1)=a^n(1(1)+b^n(1(1), n is some natural number all these (1(1)-type numbers are on the midline. Here are two examples:

So for example with n=3, 1^3+2^3=(9^(1/3))^3 but is 9^(1/3) an integer?

b(1(1) is between c(1(1) and a(1(1). C’ is the largest number.

I can move the midline so it is congruent with the horizontal axis, as shown previously.

Then since (1(1)^n=(1(1) and that means sqrt(2)=sqrt(2)*sqrt(2)..

Then a and b are integers, c^n(1(1)=a^n(1(1)+b^n(1(1) and we assume c is an integer too.

If I take the nth root of (c^(1/n))^n I must have an integer times (1(1) on the midline.

I am left with c^(1/n)*(1(1). but (1(1)=(1(1)^q. This expression has to be a multiple of sqrt(2).

Does c^(1/n)*(1(1)^q=m*sqrt(2) for m a natural number or m a multiple of sqrt(2)? or c^(1/n)*sqrt(2)^q=m*sqrt(2). M can be a natural number to give the multiples of the sqrt(2) or m can be a multiple of the sqrt(2) to give the natural numbers that might occur as we expand sqrt(2)^q.

So an equation we can come up with is m=c^(1/n)*(2)^(1/2)(q-1). I wish the horizontal axis to only consist of integer values or multiples of sqrt(2) of places but any value of places of places.

Since a^n and b^n are at least 1 each (n=1) and a and b are at least 1, then c is at least 2. so m is somewhere between 2 and c.

But c and 2 are not points but points(2) which can move(at the other level). Let c and 2 and vary on the places(2) axis. These are fixed at their values in place(1) so the first equation in place(1) still holds.

These places of places (place(2)’s)can be numbered. Let them have numbers associated with them which mirror the number system given to the first level.

Before the moving numbers move they are associated with numbers on the place(2) number system. Since these associated numbers in the place(2) system have the same relative values, the equation holds for them. We can have a sharing of two concepts at two different levels. The equation still holds true for these sharings as well as the individual numbers are not moving at the beginning. Then as the places move, the placements change at different rates so that the equation still holds in the place(2) number system.

Then let c and 2 move together on the horizontal place(2) axis so that we have them all together at m. We can move at two different rates as the underlying placements change in such a way to always keep the equation true in placement space as well as place space. As in algebra when we give a letter to a number, here we are giving a letter ‘p’ to a specific number we come to which is fixed at m while the places move. In this number system it is not the numbers but the number of the numbers which count.

Then p = (p)^(1/n)*(p)^(1/2)*(q-1).  We are not interested in the number but the number of the numbers. This general number is a number so it obeys the exponent rule. So looking at the exponents then, we can ask is there any n and q for which 1=(1/n)+(1/2)*(q-1)?. In the cubic case, i.e. let n=3. For which the answer is no. For the squared case 1=1/2+1/2 (n=2, q=2) and the linear case 1=1+0 (n=1, q=1). For any case of n>2, this does not work.

So for the extended number system we are restricted when we examine (c(c)^n, (a(a)^n and (b(b)^n given they are linked by Fermat’s last Theorem . This number system was found by starting with the co-ordinate axis and forming a midline. I started this way to give a point of reference.

But suppose we look at it as the midline going forward to the usual number system. The concept sharing of a number is a more basic understanding of numbers. With (4(4) for example the sum of the underlying points leads to single item, what we know of already as a point. The so too the numbers which represent it follow. So that (4(4) creates 4. It makes sense that this result is a more basic feature of numbers.

Once the discovery is made for n, we can see how this is true in the usual number system. The underlying point and number systems can be considered to be previously hidden from us.