Ontroduction

As a programmer I always have to take great care that nowhere in the code something is divided by zero (x/0). I consider the fact that our mathematical system does not allow such a basic operation a major flaw.

The root of the problem is, as I see it, the claim that
1 = 1 and therfor
1 + 1 = 2 or that
1 – 1 = 0.
Behind this claim, which goes through all mathematical thinking, is the idea of constancy and/or equality. That one thing should be equal to itself from one instant to the next or that there indeed should be two things that have at least one perfectly equal property.

Mathematics aims to reduce all the properties of an object to only the ones that matter for a certain calculation and tries to make calculations based on these abstractions before putting them back to the object or objects in question claiming that the results are in some sense “true”.

I’d like to give an example that illustrates the problem:

Imagine you need a bicycle, but you have no money. You look around and soon you find a job were you are paid 8 MU/h (read monetary units per hour and fill in which ever currency you prefer).

The price of the bicycle you want is 35 MU. A quick calculation tells you that you need to work for 35 / 8 = 4.375 hours. Great! Next morning at 8 am you are there, you skip all the breaks and at 12:22:30 you go to the boss to get paid.

The boss, an otherwise reasonable man, gives you 32 MU, telling you that he only pays full hours, no exception made.

Grinding your teeth you take the 32 MU and go to the bicycle shop. There you find your bike. You ask for a discount explaining to the clerk your problem, but he is unyielding, so for the moment the value of your 32 MU is, in terms of how many bikes you can buy, zero.

But luck is on your side, you discover that one of the bikes has a little scratch to its frame and the clerk gives in. You get the bike for 32 MU. The clerk, who all the time thought he had 2.0 bikes realizes he only had 1.9143. You have in terms of functionality 1.0 bikes and not only 0.9143. So while you bought 1.0 bike only 0.9143 were sold.

You may now (correctly) protest, that the boss only counts the property “hour” not the properties “minutes and seconds”, that you count the number of functional bikes while the clerk counts some monetary equivalent to the products he sells, so everybody in this story calculates different properties of certain objects and therefor comes to different results. But as I will show, it is not as easy as that.

Counting stuff

If you move an apple from table A to table B and then move it back again to table A the mathematical equation to calculate the number of apples on the table B would be 1 – 1 = 0.

Every time you take the apple from tabel A, move it to B and back to A you get 1 – 1 + 1 = 1

Most human 7-year old children are able to calculate that. However, most mice and dogs will disagree. They can tell that the apple has left something behind. A smell.

Every time you add and then subtract an apple to and from table B a trace of the apple remains on the table. Some molecules, some cells, some specks of apple skin.

Even if each apple only leaves a millionth of a billionth of its weight behind each time you add and then subtract one apple, if you repeat this operation a million billion times the equivalent of one apple would finally be created on table B and while still left on table A, because you did put it back. With other words

0 + 1000 000 000 000 000 000 * (+1 – 1) = 1 (for table B) while

1 + 1000 000 000 000 000 000 * ( -1 + 1) = 1 (for table A).

This is somewhat surprising since we all learned at school that 0 + 1 – 1 = 0 and 1 – 1 + 1 =1 and 0 + 1 = 1 and not 2 and that any number multiplied by 0 equals 0, (x * 0 = 0). Even if x is a very large number.

But apparently not if you count apples. This is sad since math-teachers are particularly fond of counting apples. Not so many, though.

You could of course say that after many moves there is really the smashed equivalent of a halv apple on each tabel, minus a significant left on your hands and all that has fallen on the floor or evaporated.

There is a world that I would like to call “the mathematical dreamworld” (TMD) in which you can add and subtract apples a million and a billion and a trillion times and the apple remains fresh and red and juicy. Problems arise when people from TMD move to a world one might call “reality”.

Reality is harsh and dirty. Every society on earth has created criteria to distinguish between humans and non-humans. Race, color, sex, age. Number, shape and attachment of limbs, number of cells, IQ, wealth, name, mother tongue, religion, position in relation to a womb or a test tube, growth of hair on certain parts of the body, shape of brain waves and so on. You cannot count a thousand humans without some overlooked, unloved, forgotten and ignored. So if one society couns thouse humans they will come up with let’s say 987 (not couning the dead), another society will say 1008 (counting pregnant woman as 2. Others again will count zero, because those 1000 happen to be of the wrong race.

One problem is time. Everything changes over time. Values, ideas, number of living cells, number of molecules, weight, size, temperature. In TMD there is no time and therefor no change. Many TMDrians go as far as to claim that in reality time is an illusion. An interesting claim. I will come back to that later when discussing the cosmological implications of my reasoning.

As an owner of a grocery store I know, that if I have 100 apples for sale there will always be some that my customers will refuse to buy. They simply do not qualify as apples in their eyes. The apples will remain on the shelf, overlooked, unloved, forgotten and ignored. Ignored, at least, until they start to turn half liquid and find a new way off the shelf and onto the floor. Do thees apples offer a convincing argument for the non-illusional property of time to TMDrians?

The other problem is the claim that there indeed exists integer numbers (whole numbers as 1, 5 or 5 496 872) outside TMD. A TMDrian will tell you that you will get a correct mathematical result if you include this millionth of a billionth of the apples weight in the equation. True. But the example above illustrates that when dealing with reality integers will lead you into trouble. Either you treat things you count as integers and accept that 1 + 1 mostly but not always equals 2, or you treat them as units that have to be measured, which will give you a different kind of headache.

Measurements

Infinity is a nasty thing to deal with. If you want to measure the length of a stick, you might do that by comparing its length to an other stick of “known” or defined length, a yardstick for instance. Using a looking glass you might measure the length of your stick with an accuracy of some micrometers. 1 micrometer = 0,000 001 m. Using a microscope or an electron microscope you might get down to some fractions of a nanometer. 1 nanometer = 0,000 000 001 m.

Compared to an infinitesimal accurate measurement this is pretty rough.

The theory of relativity tells us that space and time are deformed by the presents of a mass. The influence of this mass over space decreases with increased distance. But it never becomes truly zero. This means that any mass anywhere in space has a tiny, tiny influence on the space in which we are doing our measurements. And if only one little particle of dust in some unknown dust cloud billions of light years away changes its position by the length of an electron in relation to our stick, the fabric of time and space around our stick is affected. Maybe only at some trillionth decimal, but an inaccuracy at some trillionth decimal is still pretty rough compared to infinite accuracy. So if you would attempt to measure the stick by comparing it with you yardstick, even if you cover the distance from the yardstick to the stick with the speed of light, and even if you had some magic device that actually could compare the length of two object with unbelievable accuracy by the time the device moves from the yardstick to the stick and back, the length of both will have changed in relation to each other just because of this one tiny particle of dust. If you now add all the suns nearby, not to mention flies, dust and bacteria only a few meters away, cosmic radiation actually passing through our objects and all the electrons, neutrons and protons within the sticks constantly changing position and speed, any dream of an infinitely accurate measurement turns out to be a night mare.

Only if a number has an infinitely determinable value it qualifies as an integer or rational number, like 5,0… or 2,66…. Otherwise it is an irrational number of which only a finite number of decimals can be known, all other decimals remain unknown and/or indeterminable.

It is in my view a strange phenomenon, that the mathematics we use everyday treats our reality as if it was the mathematical dreamworld in which, no doubt, such wonderful things as equal numbers, integers or even zero exists.

It is convenient to count all the apples in the basket and make calculations of an expected future income. Although it never comes. It is fun to calculate how many planks we need to build a house. But, honestly, do we not all buy always some more “just to make sure”.

Math and reality never add up. The square of which we calculate the area is never really square. The trousers size L do sometimes fit, sometimes not. Some mornings we can hardly recognize ourselves in the mirror. There is no such thing as an Euclidean plane, since space and time are constantly under change. One object changes its properties from one instant to the next and can therefor never be considered equal to itself. Two different object can not be equal to each other.

I’m not against the use of math or physics or science, quite the contrary. I simply feel that one has to take great care when using math to make statements about reality.

If, as I tried to show, integers and real numbers indeed do not exist in reality (or their existence can at least not be taken for granted or proven), this has the following consequences on, what one might call, math of reality:

• x does not exactly equal x
• x/x is not exactly 1
• x – x is not exactly 0

Zero

I’d like to give the number zero some more attention.

The greatest mystery of all is not this wonderful phenomenon we call life, not the unbelievable balance of the universal constants that make suns and supernovas and different atoms and all the rest possible, but the apparent fact that the universe at all exists. How can that be?

Before and beyond time, space, matter, energy, and physical laws there must have been nothing. The complete absents of anything. Not emptiness, but nothingness. No time, no space. Nothing that had the capability of having any properties. It is for me inconceivable that everything should always has been. Even if time now so to say has the property of being infinite, this property must have come somehow into existence.

Once and somewhere Nothing changed and turned into Something. Or to put it into mathematical terms, the sole existence of the value zero turned into the complete absents of the value zero. The mathematical dreamworld that consistent of only this sole value turned into the multitude of reality. And in reality there is always something. In this reality there is no zero. Cannot be. The complete Nothingness of non-existence has been banished from reality. There is always time and space. And time and space are affected by the presents of matter and energy.

Even in the emptiest emptiness of the most absolute vacuum that is possible in our universe a pair of particles can spontaneously pop into existence. They might annihilate each other again a fraction of a second later but they have been there. If they had mass, they produce a gravity wave that will forever travel through our universe and have an effect on every particle it hits.

In this universe of existence there is no place for non-existence, for true zero. And if zero does not exist, this affects mathematics:

1 minus 1 cannot be zero and therefor 1 cannot equal 1.

If 1 does not equal 1, 1 times 1 and 1 divided by 1 cannot equal 1. And so on.

There is no equality, no constancy and no zero in this universe and we have to take this into account when we make calculations, when we try to describe the world with equations, or make predictions using math.

We must be aware of the fact, that the math we use to describe the world is at present fundamentally wrong. It is convenient to “equal out” terms when constructing equations, but we must be aware that every time we do that, there is a tiny rest of something we simply ignore. Applied to everyday problems the problem might not turn up, but when dealing with chaos, very large, very small things or cosmological problems the accumulated error might be significant.

Call for a new Mathematics

To be continued…