Archive for May, 2011

May 30, 2011

Heretical Scientific Realism

by Neil Rickert

I have disagreed with parts of traditional epistemology in some of my earlier posts.  So it will surely be no surprise that I have disagreements with scientific epistemology.  In this post, I will discuss some of those disagreements in the context of scientific realism.

For a quick review of the traditional view on scientific realism, I suggest the Wikipedia entry and the Stanford Encyclopedia entry.  As the Stanford Encyclopedia says, “Debates about scientific realism are centrally connected to almost everything else in the philosophy of science, for they concern the very nature of scientific knowledge.”  I shall be contrasting my view (which I am describing as heretical) with some of the positions expressed in a more traditional view.

The Wikipedia entry opens with:

Scientific realism is, at the most general level, the view that the world described by science is the real world, as it is, independent of what we might take it to be.

I agree with that.  Of course, the actual descriptions provided by science might be imperfect, as most philosophers of science would agree.  The important point is that, imperfect as they may be, it is the real world that is being described.  The Wikipedia entry continues with:

Within philosophy of science, it is often framed as an answer to the question “how is the success of science to be explained?”

I also agree there, that accounting for the success of science is an important part of philosophy of science.  Beyond that point, I find myself disagreeing with much of the traditional view.

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.

May 9, 2011

Thoughts on philosophy of mathematics

by Neil Rickert

It is usually said that most mathematicians are mathematical platonists or mathematical realists (those two are often considered to be the same).  It should be pointed out, however, that most mathematicians have not studied a lot of philosophy so what they say about their mathematical philosophy is not always well thought out.  Perhaps that same comment applies to what I am posting here.

There are good accounts of the philosophy of mathematics in both Wikipedia and the Stanford Encyclopedia.  A major issue in the philosophy of mathematics is the question of what are mathematical entities such as numbers.  Most mathematicians agree that mathematical entities are abstract, so are not physical.  The platonist view is supposedly that numbers exist in a platonic world of ideal forms.  Personally, I take numbers and other mathematical entities to be useful fictions.  I suppose that makes me a fictionalist, though others might consider me a platonist.  But then the platonic world of forms seems like a fictional place.  My main disagreement with platonists is over the Continuum Hypothesis.  As Gödel and Cohen have shown, the continuum hypothesis is independent of the other axioms of set theory.  I see that as settling the issue, while many platonists believe that there is a fact of the matter which we have yet to settle.

Some philosophers seem to think that the status of numbers and other entities has important implications for mathematical knowledge, or for the question of whether there is such a thing as mathematical knowledge.  And some think that this also raises questions about why mathematics has turned out to be so useful to science.  Yet, to me, it is quite clear that there is mathematical knowledge, and it seems quite apparent as to why mathematics is useful in the sciences.  And whether or not numbers are fictions, platonic entities, or anything else has little relevance to those issues.

I intend to discuss mathematical knowledge, and the relation of mathematics to the sciences in future posts.  The current post is a warm up exercise.

I tend to see mathematical formalism as a kind of fallback position.  That is, mathematics can be described as if it consisted of playing formal games with symbols.  But I doubt that many mathematicians actually think about mathematics in that way.  Similarly, I suspect that most mathematicians would find logicism (the view that mathematics is a part of logic) to be implausible.

I see intuitionism and the constructivism of Errett Bishop as interesting, and as a bit like doing mathematics with one’s hands tied.  I’m sure that Bishop didn’t see it that way.  However, I am not at all persuaded by the philosophical arguments often used to support intuitionism and constructivism.

That’s a casual overview of how I see mathematics.  I shall be more detailed in some upcoming posts.

May 8, 2011

Answering those questions on knowledge

by Neil Rickert

Last week, I posted some questions.  Unsurprisingly, there was only one commenter who tried to answer.  In this post, I will provide my own answers.

1.  If epistemology provides a useful account of knowledge, why is it that many scientists find it useless?

The account of knowledge provided by epistemology has very little to do with how scientists see knowledge.  In his book “The Concept of Mind,” Gilbert Ryle already argued that knowing how is more basic than “knowing that” and he criticized the intellectualist tradition of defining knowledge in terms of propositions.  Most scientists would be more comfortable with Ryle’s view of knowledge.

2.  Why do scientists find mathematics to be of great value, while epistemology has difficulty accounting for mathematical knowledge?

In my opinion, mathematics is already a better theory of knowledge than epistemology.  Parts of mathematics were developed by scientists to serve their needs.  (I plan to expand on this idea in some upcoming posts about mathematics).

3.  Why are television programs such as “Sesame Street” considered by many to be of educational value, when they are entirely fictional and therefore have no true beliefs?

This is also consistent with Ryle’s view on “knowing how”.  Sesame street is about acquiring skills and enhancing one’s abilities.

May 1, 2011

Questions on knowledge

by Neil Rickert

In this post, I pose some questions for those who take knowledge to be justified true belief.  I am treating them as rhetorical questions, so that I don’t actually expect any answers.  I doubt that there are any good answers that are consistent with the JTB characterization of knowledge.  However, I welcome comments on the questions.

  1. If epistemology provides a useful account of knowledge, why is it that many scientists find it useless?
  2. Why do scientists find mathematics to be of great value, while epistemology has difficulty accounting for mathematical knowledge?
  3. Why are television programs such as “Sesame Street” considered by many to be of educational value, when they are entirely fictional and therefore have no true beliefs?