Conceptual Math

December 30, 2011


by Paul R. Martin

Everyone has a concept of the world they live in. This concept, which first formed at birth, if not before, is ever-changing for each person as they go through life. The concept is used to make decisions about how to act and what to do during the day. It is sort of like a game where the objective is to get one's needs met and the challenge is to figure out the rules of the game.

Maslow outlined a hierarchy of needs beginning with the most basic and vital. These basic needs demand nearly all of one's attention until they are met. Once the basic needs are met, then needs at a higher level can be addressed. As you work your way up the hierarchy, the needs become less vital, or less urgent, and they transform into wants and from there into luxuries or indulgences. The happy situation is where most people are working at a high level on this hierarchy. As it is, there are some of us lucky enough to be working up there, while there are still some people working on their very survival down at the bottom of Maslow's hierarchy.

Returning to my first sentence, even though everyone has a concept of the world, those concepts are not all the same. In fact, there probably aren't two that are the same among all 7 billion people. For example, to some people, everyone is a benevolent friend and all you have to do is be nice and all your needs get met. To others, everyone is a threat and a danger to be confronted or avoided. To people in between those extremes, there are ways of distinguishing friend from foe and needs get met by acting differently based on those distinctions.

Each person's concept of the world evolves over time as they go through various experiences and as they see and experience the various consequences of their actions. People have different experiences, they react in different ways, and they interpret their observations differently. That is what leads to the uniqueness of each individual's concept of the world.

With that background painted, let me describe my particular unique concept of the world I live in. I developed a concept of the world early enough so that it proved to meet my needs fairly well, at least in the last several decades. I wasn't operating much above the subsistence level for the first four decades of my life, but things improved after that. Now I have the luxury of spending lavish amounts of time trying to extend my concept of the world even further. I really want to know how this world actually came to be, and I want to know how it works in all its splendor and detail. I don't need to know that in order to survive, but I simply want to know.

In my academic and avocational careers, I have developed some concepts of how mathematics relates to the world we live in, and it is those concepts I shall attempt to explain, in ordinary language, in this essay. It is my aim to show how those concepts might shed some light on the questions of how the universe came to be and how it works.

First I'll describe my concept of mathematics as a subject, and then I'll try to describe how some mathematical concepts relate to the world we live in.

Mathematics, believe it or not, is really all about language; mathematics is the attempt to answer the question, "What can you say for sure?"

Of course, you can't really "say" anything without language, so that is why I say mathematics is all about language. But it also seems that with so many people talking all the time, and now texting to boot, there is an awful lot of "saying" going on that is not mathematics. That is where the little kicker, "for sure", comes in. Almost everything everyone says in ordinary conversation is not "for sure".

Almost everything anyone says is based on opinions. These might be well-founded, and they may even be true, but almost all of them can be doubted, so we can't say they hold "for sure". In my particular concept of the world, there is only one statement that I claim is true "for sure", and that is, "Thought happens". But let's leave that thought aside for the moment and get back to mathematics.

What really goes on in mathematics is nothing more than a group of people who have agreed to be very careful in their use of language. They accumulate a collection of language statements (what other kind is there?) which they designate as "The body of mathematics". They have very strict rules for accepting any statement into this "body", and they sort of vote amongst themselves to decide which new candidates to accept.

To non-mathematicians, this body of mathematics is the interesting and useful part. The statements included in this body turn out to be useful to people for many reasons, from balancing their checkbooks all the way to people calculating energies in some hypothetical quantum chemistry problem. Just think of the many uses for calculators and computers.

But let's step back and look at mathematics as a whole and try to see how it fits into our society. Think of the subject of mathematics as having depth and breadth.

The depth of mathematics is how "hard" it is, or how "sophisticated" it is. At the shallowest depth we have the "easy" math of grade-school arithmetic: adding, subtracting, fractions, etc. At a greater depth we have the subjects of Geometry, Algebra, Calculus, and so on. Deeper yet we find subjects like Topology, Differential Geometry, and Foundations. I want to plumb down to those depths in this essay.

In contrast to the depth, the breadth of mathematics can be thought of as a horizontal spectrum that ranges from the practical to the theoretical. This range is sometimes characterized by dividing the field into "Applied Mathematics" and "Pure Mathematics". (You can bet that the label 'Pure' was coined by the mathematicians working at that end of the spectrum, just as it was the "Civilized" ancient Greeks who coined the term 'Barbarian'.)

This spectrum of breadth has a wide range. At the extreme left (let's say) are the applied mathematicians. These in the shallow water are people balancing their checkbooks and figuring their tips in the restaurant. As you go deeper there on the left, you encounter engineers who use fairly deep mathematics but they are using it to design bridges or computers.

As you move to the right a little, you run into the subject of Geometry at the High School level in the shallower water, and you run into research scientists in the deeper water. Moving right even further the shallow water gets less populated, and in the deeper water we see a shift from scientists with an interest in math to mathematicians with an interest in science. Further yet, we find the water getting deeper and the mathematicians losing interest in science and in anything else that is real for that matter. They are doing pure math. And at the extreme right, in the really deep and murky waters, we make a transition from pure mathematics to philosophy. And, hang on to your hat (or your SCUBA mask), it is in those deep murky waters that I want to give you a little taste and some of my thoughts.

So let's ignore the left end of the spectrum and all of its successful computations and computers, and look at "Pure Mathematics". Here's what goes on.

The aim of a professional mathematician is to come up with a statement which his peers will vote to accept into the accumulating body of mathematics. Such a successful statement is called a theorem.

The way the vote works, is that the mathematician writes down a series of statements, called a proof, that lead to the final one on the list, that one being the theorem itself, by way of applying rules of logic and deduction that the mathematicians have agreed upon. Other mathematicians then examine this proof looking for flaws or errors. If none is found, then after some due process, the new theorem is accepted.

The rules for the deductions used in the proof are the following, at least these are my concept of those rules.

1. Statements must parse in some language. That is, the statements must make sense to readers who understand that language.

2. Terms used in the statements must either be
a. clearly defined,
b. part of the grammar of the language that speakers of that language agree are understood the same by all speakers, or
c. what are called 'primitives', which I define below.

3. The designation 'true' is assigned to some statements in one of two ways:
a. The statement is accepted and designated as an axiom (or postulate—they are synonyms now).
b. The statement is the logical consequence, or implication, of several statements ahead of it in the list called the 'proof'.

4. The rules of inference applied in 3b are the rules of formal logic, which seem to have come from Aristotle a long time ago. (Why we accept these rules, and how we got them, has always puzzled me and I haven't figured it out yet. But I guess we don't need to know that for this essay.)

There is a specific set of terms, called 'primitives', mentioned in 2c. above, that are accepted by the mathematicians without definition. And, the mathematicians go out of their way not to define these terms, or in any way try to say exactly what they mean. Weird, I know, but that's how it is.

Now, to actually "do mathematics" there are two separate processes that go on. One is that the mathematician gets an inspiration or an idea and thinks about it and comes up with a rough strategy for convincing his fellows that he can make some true statement. This is called a 'conjecture' or 'hypothesis'. The second process is that the mathematician sits down and writes out the proof, line by line, roughly following his strategy. It is this proof which his fellows will examine to see whether they agree that the statement is indeed true. In reality, the two processes are iterative, intertwined, and they evolve as the convincing proof develops, if in fact it ever does. If a proof isn't discovered, then the proposition remains a conjecture. Some famous conjectures lasted for centuries before they were finally proved.

We are not going to "do any mathematics" here. I am no good at that. Instead I am going to try to describe some of the body of mathematics in plain old English using familiar examples from our world instead of the nearly unreadable mathematical notation that mathematicians use in their work. When you study formal pure mathematics in a classroom, they frown on drawing pictures on the board or using concrete examples from real life. In this essay we are going to ignore those frowns and feel free to use real examples and maybe even a few pictures. We won't be needing the formal notation.

So, without further delay, let's head over to those waters clear over to the right, on the boundary between mathematics and philosophy and jump in. We don't have to go very deep before we run into my definition of mathematics itself. It is the question of what can we say for sure.

Being on the boundary, we can consider this question from either the philosophical point of view or the mathematical point of view. Mathematicians answer this question with formalisms. That is, they have a strict form for expressing what they say and they insist on sticking to this form. They declare their primitives and their axioms, they define their terms, they infer true statements based on the axioms, newly defined terms, and previously demonstrated true statements. And they will claim that's all we can say for sure. The form, hence "formalism", of the true statements is always, "If some particular premise is true, then some specific consequence is also true." Statements of this form are frequently referred to as If...then statements. According to mathematicians, it is only statements of this form that we can say are true for sure.

The philosophers object by saying, "Wait a minute, what about all these sense impressions I experience that tell me that there is something like a world that exists out there and that is outside of your body of true statements. Can't I say anything for sure about this putative world? Like "Something exists"?"

And the mathematicians look up from their work on the other side of the boundary and say, "Well, you can say that as soon as you define 'something' and 'exists', or adopt those terms as primitives so you can even talk about them. Until you do, you can't really say anything about them." So unless the philosophers cross over the boundary and start doing mathematics, the mathematicians ignore them and get back to their work.

At this point it might be fun to look at what I think has been an interesting migration of philosophers over time. Back in Euclid's day, you couldn't tell a philosopher from a mathematician from a scientist. They were all the same people. In the intervening centuries, the ranks of philosophy have thinned and those who otherwise would have been philosophers became instead scientists and mathematicians. Let me explain what drove this migration.

Back when I listed the rules for doing mathematics, in rule 3a I said that we have the two words, 'axiom' and 'postulate' which are synonyms. The reason we have two words is because back in Euclid's time, they thought there was a difference.

Axioms were thought to be statements about our real world that were so obvious that we didn't need to prove them. They were just part of the world. Postulates, on the other hand were "starting point statements" that we don't know whether they are true in this world or not, but we agree to accept them as true anyway just to see what we can derive or deduce from them. If we deduce nonsense, or contradictions, we can always throw them away and start over.

Sometime during the nineteenth century, when non-Euclidean Geometry was developed, it dawned on philosophers and mathematicians that maybe those axioms weren't a sure part of the real world after all. They thought about re-designating the axioms of Geometry as postulates, but it was easier just to declare that axioms and postulates are the same thing and that neither one says anything for sure about the real world.

So now it was harder yet for philosophers to say anything for sure about the real world. Many, if not most of them, concentrated on the appearances of things in the world, called phenomena, and studied them. The ones that came up with anything useful at all, left the ranks of philosophy and formed a new group calling themselves scientists. That's a fairly recent happening. It resulted in the relatively new fields of science like chemistry, physics, biology, geology, and cosmology.

So philosophers migrated away from philosophy and became scientists, or in some cases mathematicians. This migration continued to erode the ranks of philosophy to the point where philosophers today are not much more than critics of science and mathematics claiming that they, the scientists and mathematicians, really have nothing to say for sure after all, either. (I know I am risking the ire of modern philosophers here, but remember, I am giving you my personal unique concept of the world.)

So far in this essay, I have described my concept of mathematics as a subject; now let's continue our dive into deeper waters but let's stay close to the boundary of philosophy and mathematics. That way we can venture across the boundary into philosophy and try to relate concepts to the real world, and we can also venture into the mathematical waters to see how those concepts are used in mathematics without regard to any world. We'll be moving back and forth across the boundary.

Let's start on the philosophical side with a question that has been asked by philosophers since before the time of Socrates, and that is still unanswered today. That question is whether the universe is made of things or made of stuff.

Those who claimed it was made of things said that if you took anything real and cut it into smaller and smaller pieces, you would eventually reach little chunks that were as small as you could go. There would be a limit. Democritus gave these small, indivisible, chunks the name "atom", which we still use today for some pretty small chunks of matter.

Those who thought the universe was made of stuff thought that there was no limit to how fine you could chop up matter, and that it was as smooth as butter all the way down. These philosophers guessed that at base, everything was made of water, or air, or some other substances or combinations that were ultimately smooth.

In modern times, this controversy is still alive but it has morphed into the question of particles versus waves. The particle proponents hang their hat on what they call The Standard Model, which is a classification scheme to describe all the various particles of the "Particle Zoo" mostly discovered in their accelerators during the '50s and '60s. The wave proponents, on the other hand, hang their hats on various field theories. They consider certain fields, like electric fields, or gravitational fields, to be smooth as butter all the way down.

The particle proponents jumped the gun and assigned Democritus' term 'atom' to a particle which later proved not to be indivisible. The name has stuck and therefore lost its original meaning. They are hesitant now to declare that they have found the ultimately smallest particle. Some of them have guessed that instead of small hard balls, those smallest particles might even look like tiny loops of string.

The proponents of stuff, have tried to unify the various fields they know about, and are trying to show that they are all parts of a most basic field they call the Higg's Field. There was also an idea called the aether which would have been the basic stuff of some of these fields. It was discredited over a century ago, but now seems to be having a resurgence.

The debate continues. We'll look at this controversy from a mathematical perspective later on.

Now, make sure your SCUBA regulator is in your mouth and your air supply is working, and let's go down pretty deep here on the mathematics side of the boundary with philosophy. I want to introduce you to the Foundations of Mathematics, but we won't go deep enough to drown.

Mathematicians laid their foundations on pure thought with no reference whatsoever to anything that might be real or part of any universe, whatever those things mean. They tried to pick, as primitives, the simplest concepts they could come up with and yet all agree that they understood what those concepts were without talking about them. They didn't even insist on or require any understanding. They simply agreed to adopt the terms without saying what they were or understanding them at all. They winked and looked aside, though, if you, perish the thought, came up with some concept in your mind that you associated with the primitive terms.

The primitive terms they settled on were 'Set', 'contains', 'element', and a few more.

With just those three terms, you can form some English sentences that parse. For example, you could say, "Set contains element". That's good, and you could say that it makes sense.

Or, you could say, "Contains element element". That one doesn't parse in English, so we won't allow it.

Or, you could say, "Element contains set". That one parses, and believe it or not, it is also acceptable in mathematics. It's a little tricky and might not seem to make sense, but you'll just have to take my word for it that it does.

One of the most powerful tools of mathematics is the use of symbols to stand for variables. For example, using the symbol 'x' to stand for some number. (I think this was invented by al-Khwarizmi, the Uzbeki inventor of Algebra.) All Algebra students are familiar with using 'x' as an unknown number in this way. Numbers are the most common types of variables in Algebra, but they are not the only types of variables to which we assign symbols in mathematics. For example, we can assign symbols to stand for elements, or even sets, or sets of sets.

So using that trick, and our three primitives, along with some English grammatical help, we can form sentences like, "X is a set". Here, 'X' is a symbol standing for some variable, 'set' is one of our primitives we don't have to define, and 'is a' is a construction of English that we assume every English speaker understands.

The sentence, "X is a set" means, "I am thinking of a particular set and I will refer to it as 'X', so when you see 'X", you should understand that I mean that particular set I was thinking about."

We could go on, but this kind of thing glazes the eyes over pretty quickly. So we won't.

Mathematicians who aren't glazed, or fazed, by the tedium of developing this kind of mathematics, have added to the body of mathematics a subject called Set Theory. Set Theory has been taken in turn to form the foundation of mathematics as a whole. Even the numbers themselves, are defined in terms of the concepts and theorems of Set Theory.

This will sound preposterous to anyone who hasn't actually tried to study mathematics at this depth this close to the boundary with philosophy, but let me give you two examples to give you an idea. First is my own experience.

When I studied undergraduate mathematics, my introduction to these deep waters came in a course taught by Prof. Grimm. We spent several weeks just to define the number '1'. Defining the numbers '0', '2', '3', and so on came quicker after that, but quite a few students dropped out of the course before we got to '1' just because their eyes had glazed over.

The second example is in Alfred North Whitehead and Bertrand Russell's monumental book titled, "Principia Mathematica". In this book, the authors attempted to develop the foundation for all mathematics starting with the philosophical basics that I sketched out above. If you crack this book open at virtually any random page, you will find it tightly packed, line after line, with the most cryptic mathematical notation you have ever seen. In this book, the number '1' is finally defined on page 345 (in my edition). As in my experience with Prof. Grimm, the number '2' came quicker. In the Principia the number '2' is defined on page 375.

The point is that pure mathematicians, working in the very deep water over to the right, do not take such seemingly simple concepts as the number '1' for granted. At least not any more.

That is a fairly modern development. From Pythagoras' time up until the turn of the 20th century, mathematicians, and philosophers accepted many notions as simply being given by nature. For example, Leopold Kronecker is famous for stating, in the 1880's, that, "God made the integers; all else is the work of man." But since then, mathematicians have tried to accept nothing as given, not even the integers.

Now, assuming you have enough air left in your SCUBA tank, let's dwell down here for a moment near the philosophy boundary and consider again the philosophers' question of things versus stuff at the base of reality. The mathematicians have been working around a similar dichotomy for a long time. This is the difference between counting and measuring.

We count things, but we measure stuff. We can count eggs, but we have to weigh, or otherwise measure, peanut butter. In mathematics, since ancient times, we have used arithmetic with its numbers to count things. And we have used geometry with its lines, planes and boxes, to measure things. These remained separate and distinct branches of mathematics until 1637 when René Descartes published his invention of Analytic Geometry in an appendix to an otherwise unimportant book. From then on, numbers could be used not only to count, but to measure things.

But that raised a new question: Can numbers be used to measure stuff down to arbitrarily small pieces? Well, "Of course", was the answer. You can use fractions to measure arbitrarily small pieces. And that works, for all practical purposes even to this day even for our most precise requirements.

It satisfied technologists way over to the left side of our pond, but not the mathematicians in the deep water we are hovering in. The question was, are there points on the geometric line that you can't hit using a fraction? It wasn't a real question, though, because Pythagoras had shown millennia prior that you couldn't find the square root of two that way. So, undeterred, the mathematicians invented a way of defining these "irrational" numbers. Then, together with the fractions, which were called the "rational" numbers, they formed the "real" numbers which people now speculate fill up the line with nothing left out. At least people couldn't find any more "holes" in the line that didn't have a real number assigned to it.

But there was still one more problem that didn't relate to the arithmetic-geometry merger. That was that the real numbers weren't enough to solve certain algebraic equations (the polynomial equations). For that, it required the definition of yet newer numbers called "imaginary" numbers. These are real number multiples of a single new number, called i which is the square root of -1. Then by adding a real number to an imaginary number we get what is called a "complex" number, and we have a whole new set of numbers which does allow us to solve those pesky polynomial equations.

Now bite down on your regulator and take in a nice big breath of air. We are going to stop with our description of the development of numbers. That is because that is all we will need. It's way more than we needed to do calculation, as I said earlier, but with the introduction of complex numbers, mathematicians are also satisfied that those are the only numbers they will need. (Actually that's not true. They still extend the numbers in the same way but they just quit calling them numbers. they have names like quaternions, vectors, and tensors, among other even more esoteric objects.)

We are now ready to ask some questions about the relationships between some mathematical concepts and some observations of the real world that seem to match up. We already have a long history of many mathematical concepts lining up almost exactly, or at least to within any tolerance we can measure, with features and structures in the real world. For example, the concepts of geometry apply directly to measuring farmers' fields, layouts of buildings, designs of machines, etc. But in this essay, we want to push the limits of our concept of reality to the extreme and ask whether mathematical concepts at these extremes apply or not.

What we are aiming at is the question of how mathematical concepts relate to reality. We know the laws of physics, which are stated in mathematical terms and which conform to the the body of mathematics, have been very successful predictors of reality. But we want to push the boundary and ask some questions at the edges that are still controversial. Then, in order to try to convey my personal concept of the world, which is the purpose of this essay, I will give you my views on the answers and try to explain my reasons for holding those views. Hopefully I have given you enough background for you to grasp my reasons for my views.

Here are the questions we are going to examine:

1. Is space or time infinite?
2. Are space and time smooth or discrete?
3. Is space constant?
4. Is time steady?
5. What is the number of spatial dimensions?
6. Are the extra dimensions "big" or "small"?
7. Is time a dimension like space?

These questions are all either controversial or ignored among scientists and philosophers. In my personal concept of the world, I have formed some opinions which answer these questions. As a fool treading where angels won't, I'll tell you my answers and why I think they hold. I welcome any logical refutation of my positions by anyone who cares to send me their comments. I will gladly change my world-view if I encounter a strong enough counter argument. So let's start down the list.

1. Is space or time infinite? I think neither one is infinite with one caveat.

I leave open the possibility that the front end of time, i.e. the past, might be infinite, although I don't think the term 'infinite' can be clearly defined. But the back end of time, the future, is definitely not infinite. If there is even such a thing as time, reality continues to move into the future. There may be no end to that movement, but that does not mean it is infinite. At each and every moment during any future, reality will have existed only for the time up to that point which is of finite length from any arbitrarily defined starting point. That length will always remain finite. I'm reminded of Woody Allen's comment that "Eternity is a very long time...especially toward the end."

As for space, I think it is finite in all respects.

This opinion is based on my attitude toward the Foundations of mathematics, which I briefly sketched out earlier. Mathematicians are divided in exactly what are the proper axioms to include at the very beginning. I mentioned that the foundation starts with Set Theory, but there are a few flavors of this depending on the axioms that are chosen. I am in water way over my head here, but here's how I see it.

There is another division among mathematicians who work on Foundations. That is the Constructivists versus the rest of them. I align with the Constructivists. The Constructivist position is that nothing should be accepted into the body of mathematics unless it can be specifically demonstrated or constructed. This makes the definition of anything "infinite" very tricky. I claim that it can't be done, but I am no expert and a lot of smart people disagree with me.

The commonly accepted starting point for set theory is the set of axioms spelled out by two people named Zermelo and Fraenkel. Mathematicians refer to these starting axioms and the resulting theory as ZF, taking the letters from their names. But these axioms by themselves are not enough to develop everything they wanted, in particular infinite sets (at least the way I understand it) so they typically add what is called the Axiom of Choice, or AC. The augmented theory with AC added is called ZFC. This theory is the most commonly accepted foundation for almost all mathematicians. Set theorists, however, accept and explore other alternative axioms.

Trying to clear up my understanding of the issue, I argued in various Internet forums that AC was what permits the construction of infinite sets, like the integers (so we don't have to get them from God as Kronecker did). People on the forums, who were smarter than me, ridiculed me and dismissed me saying that AC had nothing to do with the construction of the notion of infinity. I left the discussions with my tail between my legs muttering to myself that yes, it does.

Just recently, the author of Handbook of Analysis and Its Foundations pointed out a joke in that book on page 145 which is delightfully relevant here. I'll quote the passage:

"6.21. We ask again: Is the Axiom of Choice "true?" According to Bona [1977],

The Axiom of Choice is obviously true; The Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

The joke is that the three principles are equivalent, as we have just seen. Still, there is a point to the jest: Our intuition isn't reliable here."

I wish I had been able to fire that joke back at my forum interlocutors when they dismissed my opinions on AC. But at least now, I am encouraged to hold on to my position that the AC is instrumental in allowing infinity into the body of mathematics, and I think it should be disallowed in order to develop a body of mathematics more consistent with reality. That's just one poorly equipped man's opinion which I would love to have changed or confirmed by someone who really knows this stuff.

2. Are space and time smooth or discrete? I think both are discrete.

That means that I expect Quantum Mechanics to hold up better than General Relativity in the resolution of the inconsistency between them.

The reason for this position is a direct consequence of my position on question number 1. If there are no infinite sets in mathematics, as I would prefer, then there must be a largest integer and that means there must be a smallest non-zero interval. Fractions don't exist all the way down. I think that is the only consistent possibility for the body of mathematics developed on strictly constructivist axioms. And since I expect reality to match mathematics in this instance, it says that reality is grainy and not smooth. The notion of continuity will need major rework in my preferred mathematical system. I wish someone would tackle that job.

3. Is space constant? I think it is, but I believe those few scientists who don't ignore this question would say that it isn't.

Here's what I mean by that question: Does an inch stay an inch over time? Does a mile? A light-year? Well, here's a way of looking at it. Scientists claim that the visible universe is expanding. They argued over this until fairly recently, but there now seems to be a scientific "consensus" on the question. They think it is expanding.

They explain it by using the analogy of a balloon with dots inked on it and which is in the process of being inflated. As the diameter of the balloon increases, the distance, measured along the surface of the balloon, between any two dots also increases.

But if you ask, what about the dots themselves, don't they increase in size also? they answer, it's only an analogy. Think of pennies glued onto the balloon instead of dots. Now leave me alone.

So they conclude that even though the visible universe is expanding, planets and atoms don't. I am not so sure.

I took some numbers from Brian Greene's The Elegant Universe and did some calculations, which I have since lost and can't remember, but the conclusion was something like this:

If the expansion of the universe is linear, so that the rate of increase in the distance between things is proportional to the distance, then even though those distant galaxies are rushing away from us at near light-speed, or even greater, the rate of increase in the length of a mile-long iron rod would be in the ratio of one mile to the distance to the far galaxy. I calculated the rate to be something like a fraction of the width of an atom over a year's time. I think that might be measurable today, with some very expensive apparatus, but I doubt that anyone has ever measured it, and so I don't think they really know if atoms are increasing dots or static pennies.

Now, let's suppose that everything increases proportionally. Then how would we define length? With our rulers increasing right along with anything we measured, how would we know that the universe is expanding at all?

On the other hand, suppose that atoms and planets are like pennies and that they don't expand. Then we still have the problem of defining a ruler. There is no problem for smaller things, because neither the ruler nor the object is going to expand. But on an astronomical scale, I think there is a problem. I think space is constant and that there may be some other explanation other than Doppler Shift for the apparent expansion of the visible universe. Could we be looking through a zoom lens? No...forget I even said that.

4. Is time steady? My answer is 'No'. Here I think I agree with all the scientists.

Special Relativity tells us that twins traveling at different velocities will age differently, and our experience with GPS implementation confirms that different clocks run at different rates. There doesn't seem to be any such thing as steady time.

5. What is the number of spatial dimensions? I can only guess, but I would say between 5 and 20.

Many scientists would agree with me. Classical scientists and philosophers like Kant would disagree with me and say there are only 3 spatial dimensions. But more modern physicists, in particular the string theorists would agree with me. Their numbers vary, but 10 and 11 dimensions are common favorites. I think it is interesting that Plato also speculated that there were 11 dimensions in which our 3D world is analogous to his cave shadows.

6. Are the extra dimensions "big" or "small"? I think they are big. This is a major disagreement between me and the scientists.

I have been careful to note, for quite a while, any scientist's opinion which I can find on the reasons for believing that extra dimensions beyond our 3 do not exist. And, for those scientists who believe extra dimensions do exist, their reasoning that if extra dimensions do exist, they must be curled up so tiny that they are undetectable.

They all seem to assume that if large extra dimensions exist, we would be able to "see" them. I claim that is false. I claim that the reason we can't see them is that everything involved in "seeing" is three-dimensional. The objects we see are 3D, the eyeballs we use to form the image are 3D, and in fact the image formed is 2D, every piece of apparatus, or machinery is a 3D structure, the aspects of the light beams that transmit the visible information operate in 3D space, so I think it is unreasonable to expect that we should be able to see anything 4D or higher.

It would be as if Flatlanders, living in their 2D world, should expect to be able to see outside of their flat 2D world and "see" into the 3D space we live in.

This is another question I have been unable to successfully argue on the Internet forums. No one has given me a counter argument for my position, nor have they given me a convincing argument for the expectation to be able to "see" higher dimensions. The well-worn, and woefully erroneous and misleading, analogy of the hose-pipe, which is used by virtually every scientist who writes about the question, can be debunked easily just by considering it carefully. I plan to write a separate essay on that question alone because I feel it is so important, so I won't dwell on it here. Just know that the analogy is fallacious.

7. Is time a dimension like space? I think the answer is no. Most scientists would disagree.

Einstein introduced, or made use of, the notion of integrating the 3 spatial dimensions and the 1 temporal dimension to yield his 4-dimensional space-time continuum. Because of my opinion on question 4 above, I hold that time can't be considered as a dimension because it doesn't hold steady. At least not the familiar time we measure with clocks and calendars. It is a puzzle to me why scientists who agree with me on question 4 don't come to the same conclusion.

Here again, this is a subject that deserves a lot more discussion that would go well beyond the scope of this essay, which is plenty long the way it is.

I intend to write a series of follow-on essays which will pursue some of these ideas further. For now, I hope I have given you a glimpse of some of my concepts which lie somewhat outside the accepted boxes of science and have given you some food for thought. I also hope that if you take issue with anything I have said, that you will contact me with suggestions. My sincere thanks go out to anyone and everyone who has read this far.

Please send me an email with your comments.

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