Sunday, March 3, 2013

Riddles in the dark

I stated earlier a position that mathematical concepts are just our way of making sense of our internal logical reality, not so much making sense of reality "out there". I repeated this view in a later post, stating:
Mathematical truths are merely descriptions of the internal laws governing our processing of the world outside. Specific laws and notations of mathematics are our ways of making sense of the external world. To say that there are 10 bottles in a pack merely means to say that we have grouped the matter in the pack into 10 objects (per our definition of "ten" and "objects") in our heads. There is nothing more to it. 
The reason why Math is useful is because our brain can model the reality pretty well (at least to a certain extent), having been created/evolved for that purpose, and the brains of people are quite similar in this capacity, such that these models can be shared and mutually recognized as either true or false (or, rather, good predictors of reality or bad ones). 
 As a matter of support for this view, consider that mathematicians and physicists choose different kinds of Mathematics to describe the world. Eucledean mathematics can allow you to build a pyramid, but not circumnavigate the world. And Newton had to invent of a whole new set of mathematical concepts in order to prove his theory of gravitation. Same for Einstein: he had to adopt a radically new set of mathematical models to describe his view of the physical reality. 
 So, while the physical reality is "external" and objective, our ways of understanding and modelling it are internal and subjective, pure products of our minds.
Let me illustrate this point of view with a few examples:

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What is 400+200? Most of you will say: 600.

But imagine we are on a sphere that is 1000 miles in circumference. Each of us can be at most 500 miles away from each other (half the circumference). Imagine we are standing on the equator of the sphere, and you start walking away from me along the equator. You are 200 miles away. Then you walk 200 more miles. How far are you? 400 miles.

Now, pay attention: you walk 200 miles more. How far are you? You may be tempted to say: 600 miles, but remember that the sphere allows us to be at most 500 miles apart from each other, along the surface of the sphere. What happened is that you reached the opposite end of the sphere and then started circumnavigating it back towards me, along the other side. So, you're still 400 miles away.

On such a sphere, 400 mi +200 mi = 400 mi, when measuring distance between two objects.

The same is actually true for Einstein's Theory of Relativity. Once you start moving really fast, the speeds don't add up as they do in our everyday reality and they can never add up above the speed of light, c. So, what is 0.6 c + 0.5 c? It's some number smaller than c.

And according to Einstein's General Relativity, sometimes 3-dimensional space curves.

* * *

Another example would be of a wall clock. According to wall-clock arithmetic, 11 o'clock + 3 hours = 2 o'clock. 11 + 3 = 2. One way to state this is that the number line of the clock is curved onto itself, such that 12 = 0.

* * *

Imagine a string of numbers:

N = 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + ... to infinity.

Can we figure out what N is equal to?

Well, we can write N as:

N = (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + ... = 0 + 0 + 0 + 0 + ... = 0

But, another, equally valid way of writing this is:

N = 1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + … = 1 + 0 + 0 + 0 + … = 1

So, N is equal to both 0 and 1.

But, we could also say that:
N = 1 − 1 + 1 − 1 + …, so:
1 − N = 1 − (1 − 1 + 1 − 1 + …) = 1 − 1 + 1 − 1 + … = N,
1 − N = N; thus N = 0.5

So, we can get three different answers for N just based on the simple concepts of addition of integers and infinity.

* * *
Then there are statements like: "This statement is not true". Is the statement in quotation marks true?

There is the well-known case of Xeno's paradoxes which was solved through invention of Calculus.

And so on...

* * *

All these examples show that specific systems of notation and logic get us only so far. A specific method of "keeping track" of our internal world's perceptions of the outside works for only certain kinds of perception; other methods may be devised for different kinds, and even within the methods we already know well, there may be limits of their applicability.

I think it may be true that the reality "outside" has certain logic to it, and we are mapping out different aspects of this "grand-logic" in our head using many different kinds of "mini-logic". This doesn't just apply to Math, but to any kind of logic, symbolic and formal or not.

The point is that we must be prepared for the idea that there are "creases" in the outside's grand-logic where our internal mini-logics flow into each other (or transition sharply). There may also be areas of the grand-logic where "here be dragons".

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