“Mathematics and Scientific Representation” – a non-review

by Neil Rickert

In this post, I will discuss part of the forthcoming book “Mathematics and Scientific Representation” by Christopher Pincock.  The book is scheduled to be published on Jan 13, 2012.

I am calling this a “non-review” for several reasons.  Most obviously, the book is not out yet and I have not seen it.  I am limited to what is available for previewing online (see the link above).  I have access up through page 26 of what is listed as a 352 page book.  I can see the table of contents and the introductory chapter, but little else.  It would not be fair to claim to review the book based on so little of it.

A second reason this is not a review, is that the book is written primarily for philosophers.  In spite of the name of this blog, I am not a philosopher.  It is very hard for me to judge of how much interest this will be to philosophers.  I am a mathematician, but I can find little here that is of interest to me as a mathematician.  Perhaps I would find something of interest in the later chapters on philosophy of mathematics, were they available to me.  However, I’m skeptical of even that.  In my experience, there is little of interest to mathematicians in philosophy of mathematics that is written by professional philosophers, though philosophy written by mathematicians can be of value.

This non-review will be critical (i.e. negative).  I want to make clear that this should not be taken as a personal criticism of Chris Pincock, the book’s author.  As best I can tell, what he presents is well within the mainstream of philosophy.  It would be better to take this post as a criticism of philosophy, most particularly of philosophy of science.  If anything, I want to thank Chris for the clarity of his writing which makes it easier to see failings in philosophy of science.

The beginning

Let’s start with the first paragraph of chapter 1.  That paragraph reads:

The success of science is undeniable, but the nature of that success remains opaque.  To make a start on characterizing the success of science, I will stipulate that science is in the business of producing representations of the physical world.  These representations will be accurate just in case some region of the world is a certain way.  When approached from this angle, it seems obvious that the success of science should be characterized by saying that our scientific representations are, for the most part, accurate.  That is, they present this or that region to be a certain way, and in fact the world is that way.

The first sentence of that paragraph is perhaps the most honest statement that I have ever seen from a philosopher of science.  For it is an open admission that philosophers of science do not actually understand science.  Would that all philosophy of science were so brutally honest.

Pincock then stipulates that science is in the business of forming representations of the world.  Well, of course, that is true.  But isn’t everybody in that business?  If I note that “the cat is on the mat” then I have formed a representation.  Am I thereby doing science?  And if the cat actually is on the mat, then my representation is accurate.  Does that explain that my observation skills should be every bit as successful as science?  Clearly, something is missing in that picture of science.  Pincock is correct when he admits that philosophers of science do not understand the science about which they have appointed themselves expert.

The second paragraph begins: “This natural form of scientific realism is opposed on many fronts.”  And then it goes on to describe some of the alternative accounts that have been offered for the role of science.  Personally, I consider myself to be a scientific realist, yet I have dared to criticize Pincock’s first paragraph because I see it as far too simplistic.  Presumably, I will be seen as an opponent of “this natural form of scientific realism.”  Unfortunately, this seems to be a common reaction of philosophers.  If there is any criticism, they circle the wagons and declare the criticism to be an attack on realism.  But that way, bad philosophy will stay with us forever, because that sort of reaction shields philosophy from what could be healthy criticism of its flaws.

The second paragraph ends with a comment on the philosophy of Nancy Cartwright, who is presumably considered to be one of the opponents of scientific realism.  Personally, I view Cartwright as one of the better philosophers of science.  From a web page on her philosophy we read “Her ontological starting point is that the world is really dappled and not submitted under rigid structures of order and lawfulness (Cartwright 1999).”  That actually sounds about right to me.  It has been my impression that Cartwright does spend some time talking to real scientists (mostly physicists), and she has probably discussed that “ontological starting point” with them.  I suspect that many scientists would agree with the view that the world is dappled, as Cartwright suggests.

The role of mathematics

Toward the bottom of page 3, Pincock asks three questions in an attempt to pinpoint the role of mathematics in science:

(1) what does mathematics contribute to the representation, (2) how does it make this contribution, and (3) what must be in place for this contribution to occur?

My own answer to (1), is that mathematics contributes nothing to the representation.  And that makes the other questions inapplicable.  Of course, I am not denying that mathematics is important in science.  It is.  But to look for what mathematics contributes to the representation is to look in the wrong place.

Moving to page 5, we read:

Suppose, for example, we have a representation of the dynamics of a particular physical system like a cannonball.  Here the intrinsic mathematics of the representation includes the differential equations with which the trajectory of the cannonball is specified.

Hey, wait a minute.  Where did those differential equations come from?

How do I get a differential equation for “the cat is on the mat?”  That question probably sounds absurd.  And, indeed, it is absurd.  However, if science is merely what Pincock depicts it as in his first paragraph of the introductory chapter, then it is hard to see how differential equations would get in there for anything at all.  And that’s why I consider the approach to be simplistic.

Referring briefly back to page 4, Pincock mentions the use of mathematical idealization, which he takes as indicating that the mathematics is not part of reality in that case.  Here’s the story on differential equations – whenever scientists are using differential equations, they are engaging in mathematical idealization.  I’m not sure why Pincock fails to see that in his cannonball illustration.


I think I have already made my point.  Pincock and, more generally, philosophy of science does not understand science and does not understand where mathematics fits in.  I could continue to examine more of the available chapter, but it is mostly based on that initial misunderstanding.

Perhaps I shall try to come up with a post, in the next few days, explaining how mathematics really fits in.  If I do, then I expect that traditional philosophers will reject it as obviously wrong.  But then, I am a heretic.

3 Responses to ““Mathematics and Scientific Representation” – a non-review”

  1. May I ask what your academic background is? I am asking because you are accusing philosophers of not understanding science.

    Also, concerning the point you make toward the end (“My own answer to (1), is that mathematics contributes nothing to the representation.”), I was wondering what your reason is for this claim? As someone with a dual training in philosophy and in applied maths, I would tend to disagree with you. This prompts me to ask you why you belief this.



    • I have a Ph.D. in mathematics from Yale, which I’m pretty sure you could have discovered with a google search.

      A blog post is an expression of opinion by the author. It is not an assertion of authority. My credentials should not matter. If you don’t agree, then either ignore me or provide a reason that we can discuss.

      As someone with a dual training in philosophy and in applied maths, I would tend to disagree with you.

      That’s no surprise. I expected disagreement (and said so). There are plenty of scientists who disagree with me, too.

      This prompts me to ask you why you belief this.

      If you are looking for specifics, then I have some related posts. Check the “mathematics” category. Post comments on specific points of disagreement, if you want discussion.

      In general terms, I began studying human cognition around 25 years ago. And that led me to a new understanding of the nature of human knowledge. I had to revise a lot of my thinking. The way that you are looking at science and mathematics is probably a lot closer to how I looked at it before I took my current path.



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