Posts tagged ‘mathematics’

May 27, 2015

Searle on direct realism

by Neil Rickert

As I hinted in my previous post, I want to discuss some aspects of Searle’s theory of perception.

Searle makes a good start with:

I believe the worst mistake of all is the cluster of views known as Dualism, Materialism, Monism, Functionalism, Behaviorism, Idealism, the Identity Theory, etc. The idea these theories all have in common is that there is some special problem about the relation of the mind to the body, consciousness to the brain, and in their fixation on the illusion that there is a problem, philosophers have fastened onto different solutions to the problem. (page 10).

I agree that those are mostly mistakes.  Searle continues with:

A mistake of nearly as great a magnitude overwhelmed our tradition in the seventeenth century and after, and it is the mistake of supposing that we never directly perceive objects and states of affairs in the world, but directly perceive only our subjective experiences.

That is Searle’s statement about his direct realism.  I do support the view that perception is direct, but I avoid the term “direct realism” because the word “realism” seems to carry some unnecessary metaphysical baggage.

August 12, 2014

Constrained invention

by Neil Rickert

This will mostly be a copy of what I recently posted in a Yahoo groups discussion.  And, incidentally, Yahoo badly mangled that post (stripped out most of the formatting).

As background, I’ll note that in an earlier Yahoo groups post, I had indicated that I was opposed to the view that perception is passive.  This seemed to puzzle some participants in the discussion.  So my post — the one I am quoting — was intended to explain what I mean when I say that perception is active.

The quoted post

You guys need to get out more. You are trapped in a world of logic, and unable to think outside that box.

You both seem committed to God’s eye view thinking, though you may be in denial over that. So you see perception as a system to report to you what is seen by the hypothetical God. But how could that ever work?

May 4, 2014

Coffee cups and donuts

by Neil Rickert

There’s a saying among mathematicians, that a topologist is someone who cannot tell the difference between a coffee cup and a donut.  I’ll discuss that in this post, and I’ll suggest implications beyond mathematics.

Usually, when we say this, we are thinking of the donut and the coffee cup as two-dimensional surfaces.  Once we go to the three-dimensional objects, nobody denies that the donut has a soft and spongy texture which makes it clearly different from a coffee cup.

Topology

Let’s start with a brief rundown on what is topology.  It is a branch of mathematics where we discuss ideas such as continuity, convergence, etc.  A classic example of convergence is with the sequence 0.9, 0.99, 0.999, …  We can see that the sequence gets closer and closer to 1, and we say that it converges to 1.  So topology has something to do with the geometric ideas of getting closer.  But it does so without needing a notion of metric (or distance).

November 3, 2013

Convention (2) – word usage vs. behavior

by Neil Rickert

This continues the series that I started at

Today’s post distinguishes between conventions of word usage, and conventions of other kinds of behavior.  Word use is, of course, a kind of behavior.

I’ll give an example of each.

A word use convention

In his “Truth by convention,” Quine writes:

A contextual definition sets up indefinitely many mutually analogous pairs of definienda and definientia according to some general scheme;  an example is the definition whereby expressions of the form ‘sin —/cos —‘ are abbreviated as ‘tan —‘.

May 3, 2013

Truth and axioms in mathematics

by Neil Rickert

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.

April 28, 2013

Why I don’t like philosophy of mathematics

by Neil Rickert

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

by Neil Rickert

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.

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January 28, 2013

HSW – Kepler’s laws are false

by Neil Rickert

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

by Neil Rickert

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

by Neil Rickert

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.