Are Numbers Reality?

 

Numerology is a discredited pseudo-science, but in its updated form, it has attracted the serious attention of a few premier scientists.

 

Ancient philosophers, such as Pythagoras, were deeply impressed by the orderliness and mystery of numbers.  On the one hand, there was a very practical side, while on the other, numbers seemed to speak out from the unplumbed depths of reality.

 

This should be no surprise.  We all know that numbers are critical in very many areas of life, from accounting to zoology, from rocket science to sports.

 

But numbers, while being the heart of the most exacting of disciplines, are also, the most abstract of concepts.  After all, what, really, is a number?  Consider the number, 7, for instance.  One can count seven days in a week, seven steps on a stairway, or seven stars in a constellation.  But while one can have seven of something, one cannot have just “seven,” all by itself.

 

Numbers obey rules.  The rules are known as mathematics.  Where do these rules come from?  Are they universal?  Can they ever change?  Two plus three equals five, and it matters not one’s opinion on the matter.  Ignore the rules, and disaster strikes, whether it be in the form of a rocket exploding on the launch pad, or a tax audit.

 

This is where the mystery comes into play.  Are numbers simply a product of our mind?  Are they simply something we construct?  Or, are they a fundamental reality, no less so than space and time, no less so than quarks and leptons, no less so than our conscious minds?

 

When cosmologists deign to explain the universe, they do so in terms of numbers.  At the fundamental level, the physical universe is defined by its mathematical constants.  According to the Many Universes Hypothesis (MUH), these constants are arbitrarily assigned to various universes by random chance.  But the numbers underlying the constants may be so deeply embedded into reality that they are, in fact, reality itself.

 

Well-known physicist Dr Max Tegmark proposes that the universe is not only described by mathematics, but it actually is mathematics.  He supports this proposal by pointing out that everything that is observed in nature obeys mathematical principles—and that the obedience is so strict that one cannot separate physical reality from the underlying math.

 

Here is an excerpt from Wiki[edia

 

Tegmark's MUH [mathematical universe hypothesis] is: Our external physical reality is a mathematical structure. That is, the physical universe is not merely described by mathematics, but is mathematics (specifically, a mathematical structure). Mathematical existence equals physical existence, and all structures that exist mathematically exist physically as well. Observers, including humans, are "self-aware substructures (SASs)". In any mathematical structure complex enough to contain such substructures, they "will subjectively perceive themselves as existing in a physically 'real' world".

The theory can be considered a form of Pythagoreanism or Platonism in that it proposes the existence of mathematical entities; a form of mathematical monism in that it denies that anything exists except mathematical objects; and a formal expression of ontic structural realism.

 

That which we are seeking to explain . . .

In another discussion website,
Dana pointed out that, when we are seeking
to explain consciousness,
that which we are seeking to explain
is that which is doing the seeking.

Ultimates and Absolutes

The center of a circle has no center.
It IS the center.
It is not the center of itself, because then the center would not be the center.

The beginning has no beginning.

The origin has no origin.

God was not created.
He did not create Himself.
God is the Creator.

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How Many Finite Integers are There?

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An integer is a whole number, like 1, 3, 99 and so forth.  Non-integers are fractions, such as 1/3, 5/4, and 99.5

No matter how large a finite integer is, you can always add one to it, without reaching infinity. 

This fact can be expressed as “Let X = X+1,” in repeating form, so that X continues to increase endlessly.

Simply stated, it is, in effect, counting from one, toward infinity, but never getting there.

As a consequence of this recursion, it is stated by experts that there must be an infinite number of finite integers.

But this makes no sense.  It sounds like a self-contradiction.  How can we make sense of it?

Let’s start with the title of George Gamow’s seminal (and short) book, 1,2,3. . .Infinity.  The title itself brings up the question, how high do we have to count to reach infinity?

The answer, of course, is that no matter how high we count, by ones, we never reach infinity.  If we count by twos, we get the same result, never getting there.  If we count by millions or trillions, the same.  If we multiply by the square of the previous number—if we raise any finite number X to the finite power of Y—no matter what we do with finite numbers, we never, ever, reach infinity.

But the paradox remains.  How can there be an infinite number of finite integers, without ever reaching infinity?  It seems nonsensical.

Let’s look at the problem from a different angle.  Let’s consider a line segment of finite length.  You can easily draw one on a sheet of paper with a pencil and a straight-edge ruler.  Any convenient length will do.

Next, let us begin repeatedly dividing that line.  First, divide it in half, and then divide the half into halves, then one of those halves into half, and so forth.  What you get are line segments of ½, ¼, 1/8, 1/16 and so forth.  How many times can the line be divided?  You can quickly see that (in principle) there is no end to it.  You never get a line segment so small that to divide it again would result in a length of zero.

But wait.  If you could (but you can’t) divide the line an infinite number of times, you would get a length of zero.  In fact, according to geometry, any finite line segment consists of an infinite number of geometric points, each of which has a length of zero.  Confused?  Yeah, me too.

Perhaps this will allow us, then, to (at least in principle) count from one, two, three, . . . all the way to the end of the line, to infinity.  Will it?

No.  Here is why.  To count, point by point, from one end of the line to the other, one would have to bear in mind that the length of each point is zero.  Therefore, any finite number of zeros adds up to zero.  Five times zero is zero.  Five kazillion times zero is still zero.  Since zero is the length of the first (endmost) point, one never gets past the first point.  Traveling a length of one point, two points, ten, a million or any finite number of points, never gets you past point one.  Never.

However, you can get to point infinity.  You do it by moving a finite length, either part way along the line, or all at once.  Each finite length of the line contains an infinite number of points, but never a finite number of points (which as we said, is zero length).

If you have trouble picturing this, you are in good company.  Nobody can fully understand it.  Some people think they can, but they are just too embarrassed to admit that we are as smart as they are, because they have degrees and titles, and what good are those if they are no smarter than you and me?

In my humble (haha) opinion, there is a way out of the paradox.  It is to make a distinction between the words, “infinite” and “endless.”  In the dictionary, they are the same, but in math and geometry, they have different implications, subtle but important.  The word, “infinite,” carries the implication that you got there, you got to infinity, simply by moving any finite distance along a line.  Any such movement travels an infinite number of points.  You got there.  But “endless” means you are never done, you always have further to go, as in “Let X = X + 1.” 

Is there any practical point to all this?  Scientists think so, and so do philosophers.  You see, there is a question as to whether space is continuous, or grainy.  Is space analog, or digital?

The ancient Greeks gave this some thought, and they came up with an ingenious answer to the question of space.  It’s grainy, that is, there is a unit of space that is of finite size, but so unimaginably small that, because of the very nature of space, no smaller size exists. It is called, the Planck Length.

How did they figure this out?  They took the example we used, of a line segment with an infinite number of zero-length points, and realized that, if space were like that, we could never move.  No matter how many points of length we might move, the distance moved would add up to zero.  Even if, somehow, we could move an infinite number of points at a time, then the finite distance we moved could be any distance at all, since an infinite number of points includes a line of one inch, one meter, or a kazillion miles.

Only by moving a finite number of Planck Lengths can we ever move at all.

So, how many finite integers are there?  We never get to the answer.

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Why is the Universe as it Is?

 “Explain, as you would a child.”

--General Sarris, brutal commander of a space-battle-ship crewed by Fatu-Krey soldiers, in the movie, Galaxy Quest.

http://extraterrestrials.wikia.com/wiki/Sarris

 

= = = = =

 

The world is not flat.  It is a sphere.  Nobody knows why.

 

Oh wait, you say.  Of course we know why.   Scientists have explained it to us, as would they, to a child.  But they are wrong.  They have no idea why the world is round.  Their explanation is based on after-the-fact discovery that the world is indeed round.  Once you know that what you are explaining is certainly a fact, it is relatively simple to gather more facts, and then claim that these “more” facts explain the first fact.

 

But then, how do you explain the “more” facts, except with even more, “more” facts?

 

Now all this may seem silly, at first, but it is not.  It is simply a child-like way of explaining to adults how to answer a child’s simple questions.  Why is the sky blue?  Why is grass green?  Why do we fall down instead of up?  The more we explain, the more the child questions those explanations.

 

We never get to the final answer.

 

When scientists explain to us why the world is round, they explain to us that gravity attracts atoms to each other, and these atoms clump together into a ball shape.  The ball gets bigger and bigger, until it forms into a star or a planet or some other body.

 

Of course, all this is brilliant as far as it goes, but then, how far does it really go?  Why does gravity attract instead of repel?  Why is gravity as strong as it is, and neither stronger nor weaker?  And then there is the even bigger question, why is there gravity at all?  Why can’t there just be no gravity?

 

Why is there a universe?  Why is the universe the way it is?  Why is it exactly, precisely the way it is?  Could it have been any different?   A little?  A lot?

 

Scientists tell us that the universe is the way it is because it is governed (or shaped) by something called physical constants.  These physical constants seem to be nothing more than numbers.  There is a number for the strength of gravity.  It is written as 6.674×10⁻¹¹ N·kg^–2·m².

 

These physical constants have to be within exceedingly narrow limits in order for there to be life, civilization and technology.  Even the tiniest variation in the cosmological constant would make it impossible for atoms to exist, and therefore of course, humans.

 

There are two possible explanations for why the universe appears to be fine tuned to support us.  One explanation is Intelligent Design (or God).  Another is random chance.

 

Neither of these explanations is acceptable to physicalist scientists.  The “God” theory, or anything similar, is rejected for reasons that seem unclear to us, unless it is simply an insurmountable bias.  The random chance theory is, by its very nature, too unlikely to be plausible.

 

So then, what is left?

 

Scientists have found a way to make the random chance theory work—or, so they think.  Instead of depending on chance to make our one universe as it is, they have supposed that there are vast numbers of universes—so many, in fact, that no matter how small the chance of a universe like ours, there are so many chances that at least one of them is almost sure to happen.  Roll the dice enough times, and all combinations will eventually appear.

 

But wait.  Why does the universe have constants at all?  Why does it have twenty-seven of them, instead of three, or three thousand?  Why are the values of the constants what they are?  Could each of them be any value at all?  Is there no limit?

 

A die roll may land any number from one to six.  No, not really.  If the die is four-sided, it can never land a six.  If it has hundreds of sides, it will rarely land a six.  Dice are not randomly manufactured with random numbers of sides.  The point is, randomness can operate only within nonrandom parameters.  Those nonrandom parameters must be established by—designed by—by whom?  Cosmic intent?  God?

 

If scientists must improvise to explain away Fine Tuning of our one universe, then how do they explain the constants of the multi-universe which they suppose gives rise to all other universes plus ours?  The many universes hypothesis explains nothing.  It simply adds to the problem.

 

So the challenge to science is, “Explain, as you would a child.”