April 28, 2013

Why I don’t like philosophy of mathematics

I recently posted a link to an explanation of the philosophy of mathematics.  While I thought that Balaguer’s explanation was very good, I also remarked that I don’t find the philosophy of mathematics to be useful.  In this post, I’ll say why I don’t find it useful.

Toward the end of his explanation, Balaguer presents the following argument for platonism:

1. Semantic platonism is true–i.e., ordinary mathematical sentences like ‘2 + 2 = 4’ and ‘3 is prime’ are straightforward claims about abstract objects (or at any rate, they purport to be about abstract objects). Therefore,
2. Mathematical sentences like ‘2 + 2 = 4’ and ‘3 is prime’ could be true only if platonism were true–i.e., only if abstract objects existed. But
3. Mathematical sentences like ‘2 + 2 = 4’ and ‘3 is prime’ are true. Therefore,
4. Platonism is true.

Balaguer, who says he is a fictionalist and not a platonist, questions step 3 in that argument.  However, it seems to me that step 2 is already mistaken.  People simply do not use “true” in the way that step 2 supposes.

April 24, 2013

What philosophy of mathematics is about

Found via a reference at the M-Phi blog, here a pretty clear statement on what the philosophy of mathematics is all about:

This does, indeed, seem to capture much of what philosophers of mathematics are studying.  However, it fails to persuade me that such study is useful to mathematicians.

Tags:
January 28, 2013

HSW – Kepler’s laws are false

While my title line might seem dramatic, I want to be clear that this post is not intended as a criticism of Kepler, or of Kepler’s laws.  Rather, it is critical of the view that scientific laws are true descriptions of the world.  This post is intended as part of my series on how science works.  My aim is to describe my own understanding of Kepler’s laws.

The basis of Kepler’s laws

In case some of my readers are not familiar with them, Kepler’s laws are an attempt to account for the motion of the planets in our solar system.  Kepler’s laws were preceded by the Ptolemaic idea that the planets moved in cycles and epicycles.  Galileo argued, instead for the idea of Copernicus, that the planets traveled in circular paths around the sun.  I presume that Kepler was looking for something a little more precise than the Copernican circles.

December 25, 2012

Mathematics and logic

Yesterday, Massimo Pigliucci posted on the relation between mathematics 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.

December 2, 2012

HSW2 – How I see Newton’s mechanics

This continues my discussion of how science works, a topic that I introduced in a recent post.  The “HSW” in the title of this post is intended to indicate that.  My plan, for this post, is to describe how I look at Newton’s laws.  I won’t be discussing his law of gravity here, mostly to keep this post reasonably short.  I might post on that at a future time.

A note on history

I am not an historian.  My primary concern is with how the science works, rather than with how it was discovered.  If you think that I have said something about history, then you have misunderstood.  Some of what I am discussing here might actually be due to Galileo or to other scientists.

October 18, 2012

On David Deutsch on AI

Physicist David Deutsch has an interesting article on AI in aeon magazine.  I thank Ant for bringing it to my attention in a comment on another blog.  My view of AI is rather different from that of Deutsch, though I agree with some of what he has to say.

I started this blog in order to discuss some of what I have learned about human intelligence, as a result of my own study of AI.  It turns out that I have not actually posted much that is directly on the topic of AI.  So I am using this post mainly as a vehicle to present my own views, though I will present them in the form of commentary on Deutsch’s article.  I’ll note that Deutsch uses the acronym AGI for Artificial General Intelligence, by which he means something like the intelligence of humans to be created artificially.

October 12, 2012

On science and scientism

Coel Hellier has a new post on his blog, on the subject of scientism:

The tagline of coelsblog is “Defending Scientism” so it is no surprise that Coel is a proponent of scientism.  However, his post also brings out some points on the nature of science, and that’s what I want to discuss here.

September 14, 2012

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.

August 30, 2012

Over at the Uncommon Descent blog, poster vjtorley has posed “Ten Questions for Professor Coyne.”  I am not a spokesman for Jerry Coyne, and I disagree with some of what he writes.  But I thought I would try giving my own answers to those questions.  I’m pretty sure that Jerry Coyne would disagree with me on some of the answers.

Question 1 – Is science the only road to knowledge?

I’ll note that there is some ambiguity on what is meant by “knowledge.”  For myself, I would never claim that science is the only way to all knowledge, though it is an excellent way to knowledge about the natural world.  In any case, vjtorley breaks this question into several parts.