Posts tagged: Mathematics

Thought and the Environment

This is the first post I’m doing through an add-on to FireFox called ScribeFire.  Let’s hope it works as planned.  Again, I apologize for the last of blog activity.  I am going to try hard to play a larger part in Web 2.0 starting right…now!

So, in my last post (ages ago) I talked about how a child’s interaction with their environment actually helps them formulate the symbol system that they use for throught.

Interestingly, interaction with the environment not only helps us to develop the ability to think, but it continues to support thought.  This often leads to an aspect of memory known as context specificity.  This means that you recall things easier in the environment in which you learned them.  For instance, people can remember recipes easier in their kitchen, where you use them.  There is a good chance that when you cook you probably take out everything you need and arrange it in such a fashion that it supports you in both reminding you how to cook the food and helping you keep track of where you are in the process and what you need to do next.  Atleast I hope that’s what happens.  I can make about 4 things, so this probably isn’t an ideal example for me to use.

The most interesting experience with this for me was when I started taking more advanced mathematics.  This was after spending a lot of time studying the above phenomena.  Mathematics is a complex truth-preserving symbol system that can be visually represented on paper.  So, when you look at a math problem being worked on, you are looking at thought.  Mathematics is basically a formalized discipline of thought with a standardized symbol system.  I can look at how someone reasoned about a math problem simply by looking at how they worked it out on paper, as long as I know the symbol system they are using.

So, this system system helps someone think through a problem, by supporting them in keeping track of where they are and where they can go with it.  It helps them to make sure that the steps they are taking are valid.  And, allows them to communicate their thought process with others, provided they share a common understanding of the system.

That’s the power of a truth-preserving symbol system and the means by which to represent it visually.  We can think through and communicate more complex problems.  This is a major step.  What kind of ways can and do computer assist as with this?

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The Origin of Thought

In my previous post, I explained how people have mental models of the world around them.  These models have “moving parts” that can be manipulated such that rules are followed and truth is preserved.  This manipulation can be called reasoning.  Where do these models and their symbol-systems come from?

Simply put, they come from experience.  In developmental psychology there is something called the naive sciences.  Young children have not been formally introduced to science, but that doesn’t stop them from being scientists.  Indeed, a toddler has an understanding of physics and mathematics even though they have not been formally taught these.  Children’s experiences in the world consist of tiny informal experiments.  They learn the laws of gravity, friction, and momentum simply by living in a world where these things act upon both them and the objects that they come into contact with.  Naive science is the science that we learn from experience before we are formally taught it.  This is where our mental models and symbol systems begin to form.  We internalize the things we externally experience.  We then use this to act in a way that achieves desired outcomes that our mental models predict.

This is the beginning of abstract thought and reasoning.  Next I’ll discuss how we offset thought onto our environment in order to assist with constraints put on our working memory.  Trust me, I’m going somewhere.  :-)

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The Components of Thought

It’s been a while since I formally studied cognition, and I haven’t kept up the reading as well as I should have.  It doesn’t help, of course, that my profession requires a lot of investment in studying within it.  However, I want to try to help you understand why software development is so meaningful to me…a lover of the study of human cognition.

How do you think about things?  How do you percieve the world around you?  There is a theory out there that says that you model the world around you mentally.  These models are usually called mental models.  They don’t have to manifest themselves as physical models do.  They are not constrained by space or time.  And, only you, can truly understand how you model the world around you.  That doesn’t, however, prevent Cognitive Science from trying to take a glimpse.

Most biological and cognitive theoretical frameworks come down to evolution.  The question is asked, “How does this trait either keep you from getting killed or assist you in procreating?”  The reason for this is that random mutations that lead to either not being killed or procreating a lot before you die are most likely the ones being selected for as humans evolved.  This is because those that have an advantageous mutation are more likely to live longer and procreate more.  Their progeny are also more likely to have the same mutation which will give them the same advantages.  This, by the way, is of course theory and not shared by everyone.  I’m mentioning it here for context.

Let’s get back to mental models.  What are they made out of?  Another theory is that they are based on abstract symbols systems.  A symbol system is a system made up of symbols along with the rules for manipulating those symbols.  These rules amount to affordances and constraints.  Affordance are what you can do, and constraints are what you can’t.  This might be difficult to visualize.  So, let’s concretize it a bit.  Mathematics uses a symbol system.  Think about algebra.

1 + x = 3

In order to find out what x is you have to manipulate the symbols.

1 + x – 1 = 3 -1

Doing this leads to the following.

x = 2

As long as the left and right side of the equals sign are balanced you are within the constraints of the symbol system.

What about this symbol system makes it useful?  As long as each manipulation of it is within its constraints, it preserves truth.  In other words, each manipulation of it gives us an accurate picture of the situation.  And, mathematics models reality.

Now let’s go back to the mind.  If we have an accurate picture of the world around us what advantage does that give us?  Think of it this way, what could an accurate understanding of physics give us if we are pondering walking in front of a moving automobile or off the side of a building?  If the symbol-system guiding our mental model of the situation preserves truth, it preserves our life.  I can reason that the automobile will hit me and kill me or that I will hit the ground and be killed.  Reasoning, it can be said, is simply a manipulation of a mental model based on a symbol-system.  If that symbol-system preserves truth then the reasoning is accurate.

So, what I have tried to establish above is that people have mental models of the world around them.  These mental models are made up of symbol systems whose manipulations must be guided by rules that preserve truth in order for them to live (and procreate).  They use these models to solve problems and guide their behavior.

Next I’ll discuss how these models are formed.

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