I believe that even in a sincere pursuit of Kṛṣṇa consciousness the border between reality and imagination is hard to determine. We also have another often used term, speculation, and we use it mostly negatively because it means lack of knowledge. What is the reality then, as opposed to speculations about reality? And what if our speculations are true? Would it make any practical difference?

“Reality” is hard to define in itself, even in the strictly material sense of things. We can count things, for example, things are real and so we can say that numbers that we count must also be real. And yet there are negative numbers, too, and you can’t count negative number of things. This means that our definition of what is real need to be expanded. Are negative numbers unreal just because we can’t represent a negative number with real objects?

Well, no, they are not unreal if we redefine what a number is. If, instead of a result of a count, we talk about difference between two counts, negative numbers become real. We can say that 2 is bigger than 3 by -1. Of course we can also say that 2 is smaller than 3 by 1 but the point is that if we accept the concepts of bigger and smaller and introduce them into our number system we’ll get negative numbers very quickly.

So, imagine we start with the sequence of “real” numbers, 1,2,3.. etc. “Negative numbers” means that now the sequence looks like ..,-3,-2,-1,0,1,2,3,.. The left side of this sequence does not really exists but it’s a helpful concept when dealing with the right side of the sequence in a real, tangible world.

When we represent this sequence geometrically, as an infinite line stretching both left and right from 0 we discover the need for fractions – points on the line between 1 and 2, for example. We can say fractions are real because we know what half a pizza is even though you can’t pick up more than 1 and less than 2 pebbles on the beach. Negative fractions are also easy as they complement positive fractions, even though no one can eat a minus half pizza.

Is minus half pizza real? You can’t touch it, you can’t eat it, but it’s a helpful concept nevertheless in cases where you count how many pizzas you need to order if you know how many people will eat how many slices, and then you discover that three pizzas won’t be enough, you will be half pizza short, or you’ll have a negative half pizza balance. That’s how people and countries count debt, too, and I just wrote three posts about Greece. These things are pretty real.

Then there’s a little diversion with irrational numbers, which might not be a very apt name. Irrational people are those with crazy, unjustifiable ideas, irrational numbers are not. They are just points on that same line that can’t be expressed as ratios, or fractions. π is one of such points. Ironically, we know it as a ratio of a circle’s circumference to its diameter but, in order to be “rational”, the number must be a ratio of two integers (integers are “real” numbers and their negative counterparts). Turns out it’s impossible to have both the circumference and the diameter to be either integers or their fractions at the same time.

That’s another class of numbers that we know must exist because they are points on the same line somewhere between every “real” numbers we know. Of course we can question existence of the line itself. 1,2,3 is not a line, and 1.1, 1.2, 1.3 is not a line either, they are separate points. They might get so close as to look like a line but to actually complete the line we need to fill the space between them with ink, and it’s this space that represents irrational numbers. Do they really exists or are they just a helpful concept? π is a very helpful number in a real life because it helps us approximate things like the size of a rope needed to tie up the Earth, for example. It doesn’t matter that we can never say exactly how big a circumference of a coin is, or that of a car tire, we just accept approximate values for convenience. We can’t measure the tire’s diameter or radius exactly either. 17″ is good enough for most practical uses. Not to mention that circumference of a circle is a real thing, the rope that goes around it exists in real life and has a real length, we just can’t say what it is using our “rational” numbers.

Are these numbers still “real”? Hard to say with certainty. Is “good enough approximation” a valid definition of reality.

Then we can look at our line stretching left and right, having all the rational and irrational numbers on it, and say – who does it have be one-dimensional? Why can’t we add a Y dimension to it and so describe every point on the same paper we drew our line on as a pair of X and Y coordinates? All the numbers on our existing line will tell us X and all the numbers on the perpendicular axis will tell us Y. God has given us a two dimensional surfaces to draw on, why don’t we use them fully?

Theoretically, it’s a very simple concept, we use two-dimensional diagrams or drawings of functions all the time. Will these new, two-dimensional numbers be “real”, though? In a sense that 1,2 and 3 were real they most certainly won’t but, since we’ve already added non-existing -1,-2,-3 and then π, we might just as well give up on this original idea of what “real” means and accept these new numbers into our family.

In mathematics these new numbers are called complex because they are “pairs”. Our original 1 becomes (1,0) pair, -2 become (-2,0), π become (π,0) and so on. What would the second coordinate signify in “real” world, though? Units of what? The way history happened was that it was called “imaginary” first and then the name stuck and this name is even less apt than “irrational”. The truth is, we can’t find a “real” equivalent of this unit but mathematically it’s simple – it’s a square root of minus 1. i=√(-1), can’t type any better symbol than this √ for square roots, sorry. Anyway, we can only imagine if it existed and that’s where the name came from.

In reality, though, all we did was to expand our definition of what “number” is. First from actual counts of things to relationships between counts, then to fractions, then to numbers between fractions, now we just added a second, God given dimension, that’s all.

How do we know that they are still “numbers”? Mathematicians found some common properties, like the definition of what it means to be equal, rules for addition and multiplication that come with their own properties and formulas, and so if these rules and properties still hold, and they do, then these imaginary things must be “numbers”. They can easily rewrite the entire math which started with 1,2,3 into a math with (1,0), (2,0),(3,0) and it would still work even for (1,1),(2,1),(3,1) and so on. They ARE numbers in a mathematical sense of the world.

Do they describe “reality” in any sense? Rarely, but they still can be very helpful. In engineering they describe certain functions that take “real” input and produce “real” output but are impossible to calculate without using “imaginary” numbers. In this sense our real world can’t exist or can’t work without these imaginary numbers.

Perhaps we’d be better of redefining what reality is instead and accept that what we can count is just a small subset of reality with a much larger part forever hidden from our view. This is where we come to atheism and Kṛṣṇa consciousness but, unfortunately, I’ve run out of space and time. Continue tomorrow.