Archive for ‘mathematics’

April 5, 2012

How I see logic

by Neil Rickert

I think it was in 4th grade elementary school, that I first heard of logic.  The teacher introduced it, with some examples to illustrate the use of logic.  I recall that it seemed rather easy to do logic.  All that I had to do was use clear thinking.  So I never did actually try to learn the rules of logic.  I just went by clear thinking, as appropriate.

I am posting this as one of the posts on my own views of philosophical topics.  No doubt my view of logic is partly shaped by the fact that I am a mathematician.  And, as a mathematician, I of course use a lot of logic.  But my basic view is still much the same as in 4th grade.  That is, I see logic as clear thinking.  It is a formalized or formalizable view of clear thinking, but it is nevertheless something that comes from clear thinking.

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December 5, 2011

Why mathematics is useful to science

by Neil Rickert

Philosophers, including philosophers of science, talk about representations.  For example, the statement “the cat is on the mat” might be a representation of one aspect of the world.  When mathematicians talk of representations, they think of a representation as a mapping of one space into another.  For the way that philosophers talk of representations, a mathematician might think of that as a mapping from reality to the space of linguistic expressions.

One way of having representations of the world is to come up with ad hoc methods of representing little bit of the world, and then tossing them together, willy nilly, to provide a more comprehensive representation of the world as a whole.  For want of a better term, I’ll call that way of representing the “willy nilly method” and I will refer to a representation result from such a method as a willy nilly representation.  The naming of houses by the first builder in my post “The parable of the three builders” could be considered a willy nilly representation.

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October 25, 2011

Mathematical duality (2)

by Neil Rickert

In an earlier post  I discussed the idea of mathematical duality, and I used the duality between the time domain and the frequency domain, as seen with Fourier transforms, to illustrate the idea.

Today, I will discuss the simpler duality that we see in linear algebra (the study of vector spaces).  This will only be an overview.  There are many fine textbooks on linear algebra if you are looking for a more detailed discussion.

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October 13, 2011

Mathematical duality

by Neil Rickert

There are many dualities that mathematicians study.  In this post, I shall discuss two of them, in order to illustrate some of the ideas involved.

Line, point duality

As a simple example of mathematical duality, consider the duality of points and lines in Euclidean geometry.  And recall from your school days, that when we talk of a line in Euclidean geometry, we are talking of an infinite straight line extended in both directions.  The short “lines” that we actually draw can be called “line segments” so as to distinguish them from the extended lines.

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August 15, 2011

The improbability of ID

by Neil Rickert

ID proponents are frequently appealing to probabilistic arguments as evidence for their ID claims.  Unfortunately, most of the presented arguments are wrong.  There’s a particularly egregious example posted today at the Uncommon Descent blog.  The author of that post, JonathanM, apparently managed to get into a debate with Massimo Pigliucci.  He quotes Pigliucci as saying:

No evolutionary biologist I know…actually attaches probabilities to specific evolutionary events of the type you are talking about. There is no way to do that.

JonathanM then goes on to cite places where evolutionists have used probabilistic reasoning.  Apparently, JonathanM has no understanding of the difference between probabilities of specific events, and the use of probabilities over populations.

Here’s an illustration of the problem.  If I shuffle a deck of cards, and then deal out a bridge hand, I will have produced a highly improbable event.  If you were to list a particular hand before I had shuffled and dealt the card, then the probability calculation would show that the hand you listed was very unlikely.  If I had then dealt that actual hand, you would have reason to question whether I had been cheating.  However, once a hand has already been dealt, it makes no sense to compute the probability for that hand.  It does not tell us anything useful.

If you really wanted to look at a few hands that I had dealt, to find evidence of cheating, there is a way to do that.  You would need an alternative explanation as to how those hands were dealt.  And then you could calculate the conditional probability:  given that this hand was dealt, what is the conditional probability that it was dealt by method X (say, standard shuffling) rather than by method Y (your alternative).

It isn’t the direct probability of the hand that matters, it is that conditional probability.  And we can only use that method if we have sufficient data to realistically estimate the condition probability.

Unfortunately, the ID proponents don’t seem to understand this.  They do not use conditional probabilities in their arguments.  Perhaps this is because an estimate based on conditional probabilities would show that natural causes are far more probable than supernatural causes.

It is not just JonathanM who is confused about this.  His blog post has been made into a “sticky” and thus highlighted on the Uncommon Descent blog.  So whoever makes the decisions about such highlighting is presumably just as confused.

After citing his examples of statistics applied to population genetics, JonathanM comments “To this, I received no response.”  That, I can understand.  By this time, Pigliucci must have recognized that Jonathan was driven by ideology, and unwilling to learn anything.

July 4, 2011

Geometry and logic (3)

by Neil Rickert

In my previous post in this series I talked about the idea of partition, or dividing up the world.

Let’s suppose that we start with the world, and then divide it into two parts.  Call those parts A and B.  Perhaps it is the division into night and day, something that newborn infants have problems with but begin to get after a while.  Next we divide those parts, based on some other characteristics.  So A is further partitioned into parts A1 and A2, while B is further partitioned into parts B1 and B2.

If we continue partitioning in this way, we will have a way of organizing the world into the partitions.  And that organization will look a bit like a tree, something similar to a family tree.  If our partition was into two parts at each stage, we will finish up with a particular type of tree diagram that is known as a binary tree.

There is a natural way of finding items on a binary tree.  And that is to use logic.  At each branch point you encounter, you make the decision “if what I am seeking has characteristic 1, then go down branch 1; otherwise go down branch 2.”

My suggestion here is that the reason we find logic to be useful, is that as part of our cognition and perception of the world, we have been applying what I have called “geometric method” to organize our world into a nested series of partitions.  We use this geometric or partitioning method as a way of identifying objects, and the usefulness of logic is a consequence of our basing object recognition on such a partitioning scheme.

June 23, 2011

Geometry and logic (2)

by Neil Rickert

In my first post for this series, I mentioned “geometric method,” but I did not explain what I mean when I use that expression.  That is what I want to discuss today.  In doing so, I widen our horizons beyond mathematics, and look at how we use geometric methods in the real world.  In particular, I shall discuss its epistemic significance.

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June 22, 2011

Geometry and logic

by Neil Rickert

While preparing to write this post, I did a google search for “geometric method.”  Many of the results of the search were about logic rather than about geometry as, for example, in Spinoza’s geometric method.  In a way that makes sense, for Euclid’s geometry is famous for its use of self-evident axioms.  However, that is not at all what I think of when I think about geometry and geometric method.

The way logic is used in geometry is very different from the way it is used in ordinary propositional logic.  And that is what I mainly want to discuss here.

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May 21, 2011

Mathematics and science

by Neil Rickert

What is the role of mathematics in science?

This question seems to puzzle some people.  Personally, I never found it at all puzzling.  Much of the development of mathematics was done by scientists, because they needed that mathematics in their science.

Science is not simply a matter of describing or representing nature, as some seem to believe.  Describing is the job of journalists, not of scientists.  Of course, science does make descriptions.  But descriptions do not come for free.  Facts do not just pop into our heads.  Science has had to develop many methodologies, in order to make it possible to describe various aspects of reality.  And that development of suitable methodologies is a large part of science, and a large part of what distinguishes science from other areas of human endeavor.

In a recent post, I suggested that mathematical knowledge was mainly knowledge of the consequences of following methodologies.  I illustrated that by suggesting that arithmetic is a study of the consequences of following the methodology of counting.  I could equally have pointed to traditional geometry (or ruler-compasses geometry) as a study of the consequences of using a portable measuring rod.  The initiative to study these methodologies came from the importance of those methodologies.  And the mathematics thus developed has proved very useful.

More generally, scientists attempt to be as systematic as possible, in their development and use of methodologies appropriate to the area that they are studying.  One could almost say that “Be systematic, young man” is one of the important principles of science.  Mathematics, with its interest in patterns and regularities, is very much the abstract study of systematicity.  So it should be no surprise that mathematics is useful to scientists.  And it should be no surprise that scientists sometimes look to systematic methods studied in abstract mathematics, so as to see if those systems can be adapted for use by science.

May 13, 2011

Mathematical knowledge

by Neil Rickert

Philosophers seem to be a bit puzzled as to how there can be such a thing as mathematical knowledge.  And what puzzles them is the question of what mathematical entities (such as numbers) really are.  It’s a bit strange that they are puzzled, but then philosophy is a bit strange anyway.

The philosophical account of knowledge typically begins with ontology, the question of what exists.  And ontology is taken to be part of metaphysics.  So if mathematical entities, such as numbers, are fictions (as a fictionalist claims), then they don’t really exist.  Or if numbers exist only in a Platonic world of ideal forms (as the mathematical Platonist asserts), then their existence seems a tad dubious.  Once having established what exists, the philosopher typically says that knowledge is in the form of true beliefs (or believed facts).  However, according to philosophers, those beliefs have to be about something that exists.  If a belief is about fictional entities, then it isn’t about anything in the world, so it could not possibly be a fact.

We begin to see the difficulty.  Philosophers want statements to be propositions, descriptive statements about the world, before they can be considered to be knowledge.  But since mathematical statements purportedly describe only fictional entities or entities in an imagined Platonic world, they do not seem to fit the requirement to be descriptions.

The idea that mathematical truths are not facts seems rather strange to most mathematicians.  For those mathematical statements are among our most certain facts.  Moreover, many of those mathematical facts have proven enormously useful in the sciences.

The mistake is to treat mathematical statements as if they were descriptions.  Mathematics is not descriptive; it is methodological.  Mathematical statements are statements about important methods that are used.  And, among other things, those methods are often used to form factual statements (descriptions).

As an illustration, consider simple integer arithmetic (sometimes referred to as number theory).  The idea of arithmetic arises from counting, and counting is a method that is used to acquire factual information about the world.  The addition operation in arithmetic comes from the idea of counting in groups, and then combining the counts of the groups to yield a total count.  And multiplication comes from the idea of counting in groups in a very systematic way, so that all of the groups have the same count.

Arithmetical statements about numbers are really statements about counts.  We use numbers as place holders for the counts, so that our theoretical analysis of the consequences of counting does not depend on whether we are counting sheep or counting beans or counting people.  Arithmetic is so broadly useful, because it is applicable to any kind of counting.  The fact that numbers are abstract, are really fictional counts used as place holders to stand for real counts, is what makes arithmetic so general and so broadly useful.  And yet it is this very feature, this aspect of mathematics that makes it so general and so useful, that causes philosophers to question the epistemic status of mathematics.