Truth and axioms in mathematics

There’s been some discussion of truth in mathematics in the comments to my previous post.  Here, I want to expand a little on my view and express puzzlement at the idea that axioms are themselves true or false.

In response to a question, said “Actually, I take axioms to be neither true nor false, and I take the truth of mathematical theorems to be relative to the assumed axioms.”  Let me restate that in terms of the Peano axioms for ordinary arithmetic.

1. The Peano axioms are neither true nor false.  Rather, they are definitional statements.  They define that part of mathematics known as Peano Arithmetic (or PA, or simply arithmetic).
2. Theorems proved in PA are true in a relative sense.  Their truth is relative to the PA axioms.  They are true as used within PA, but perhaps not even meaningful outside of PA.

Mathematics and logic

Yesterday, Massimo Pigliucci posted on the relation between mathematics and logic:

• The (complicated) relationship between math and logic

so I though I would offer my opinion on that topic.  I see things differently from Massimo, but that’s probably just the different perspective as see by a mathematician (me) and a philosopher.

Massimo cites Peter Cameron (a mathematician) and Sharon Berry (a philosopher – actually a student of philosophy of mathematics).  Check Massimo’s post for the links.

Mathematical Fictionalism

Massimo Pigliucci has a post on mathematical platonism, so I thought it appropriate to discuss that in conjunction with my own version of mathematical fictionalism.

Pigliucci begins with three principle of platonism, which he takes from the SEP entry:

1. Existence: There are mathematical objects;
2. Abstractness: Mathematical objects are abstract;
3. Independence: Mathematical objects are independent of intelligent agents and their language, thought, and practices.

Here’s the parallel principles for my version of fictionalism:

1. Mathematical objects are useful fictions.  They have no actual existence, but it is useful to talk about them as if they existed.
2. Mathematical objects are abstract.  I take this as a consequence of their being fictions.
3. Mathematical objects are mental constructs, so are not strictly independent of the intelligent agents who talk about them.  However, if some alien intelligence exists — let’s call them Martians, to have a name — were to construct their own mathematics for reasons analogous to why we construct mathematics, then many of their mathematical fictions would have truth conditions analogous to those of our mathematics.

My fictionalist version of independence is weaker than the platonist version, though it seems adequate for mathematics.

How I see logic

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.

Why mathematics is useful to science

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.

Mathematical duality (2)

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.

Mathematical duality

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.

The improbability of ID

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.

Geometry and logic (3)

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.

Geometry and logic (2)

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.