The Empiricist Strikes Back: A Platonic Dialogue by MC Complete

Daniel: Don’t get me wrong, Stewart, I’m not a hard core empiricist.  But they empiricists were right about some things.

Stewart: What do you have in mind?

Daniel: Math, for example.

Stewart: Really?  You think math is the crown jewel of empricism?  Very well then, I’ll play your game.  Tell me, Daniel, what are we to make of the Euclidean postulate that between any two points one can draw a straight line?

Daniel: What’s the problem?  Take out a sheet of paper and a pencil, make two dots anywhere, then connect them.  Bang!  Straight line.

Stewart:  But it’s not a breadthless line.

Daniel: Breadthless?  Who said anything about breadthless?

Stewart: John Stuart Mill.  He said that lines are “limit concepts”.

Daniel: Yeah, that whole “limit concepts” strategy was a big mistake.  Lines aren’t limit concepts.  They’re just very thin.

Stewart: And what about the postulate that all lines can be bisected by a perpendicular line?

Daniel: That’s empirical too -- as long as you interpret it correctly.  What it really means is that all sufficiently long lines can be bisected that way.  Between you and me, if a line is really too short to be bisected, it’s not much of a line, is it?  It’s more of a dot, wouldn’t you say?

Stewart: And what about heartbeats?  How can you empirically justify the proposition that 2 heartbeats plus 3 heartbeats equals 5 heartbeats?

Daniel: That’s easy.  Take an MP3 file with 5 heartbeats.  I can play that file and count 1, 2, 3, 4, 5 -- and then I can play the file again and count 1, 2, 1, 2, 3.

Stewart: And what is 500 red marbles plus 500 blue marbles?

Daniel: Oh, about 1000 marbles, I’d say.

Stewart: “About” 1000?  You seem to have anticipated my line of argument.

Daniel: You were going to say that if I counted the marbles, I probably wouldn’t come up with exactly 1000.

Stewart: Yes, that’s exactly what I was going to say.  How did you know?

Daniel: First of all, I read your book.  Second of all, I’m the one writing the dialogue.  Anyway, here’s my point.  Arithmetic with big numbers, like geometry and arithmetic with continuous quantities, is only approximate.  I mean you have to reinterpret the propositions to be approximate propositions.  So we usually say 500 + 500 = 1000, but what we really mean is 500 + 500 is about equal to 1000.

Stewart: And what about infinity?

Daniel: Infinity?  What’s the big deal with infinity?

Stewart: Well, imagine a universe with only 100 objects.

Daniel: OK.

Stewart: In such a universe, is the proposition “75 * 2 = 150” true or false?

Daniel: Good question.  First of all, let’s agree that the proposition “75 * 2 = 150” is the outcome of applying the arithmetic algorithms.

Stewart: OK.

Daniel: And the arithmetic algorithms themselves work in the Hundredverse -- for all actual quantities that we can test.

Stewart: OK.

Daniel: So the expression “75 * 2 = 150” is kind of an “if then” statement -- it means that if we had a set of 75 objects that we could double, we’d expect the result to be 150.  Since there is no such set in the Hundredverse, the “if” part is not satisfiable.

Stewart: That sounds kind of like deductivism.

Daniel: I think it is a kind of deductivism, but traditional deductivism is agnostic about the justification for the axioms.  I think the justification is empirical.

Stewart: And what is that justification?  Enumerative induction?  The web of knowledge?

Daniel: Neither.  The justification for math is more or less the same as the justification for physics: experiment.

Stewart: But mathematicians don’t do experiments.

Daniel: They don’t need to.  The physicists do the experiments for them.  In order to predict the outcome of a physics experiment you need physics *and* math -- so the success of the experiment justifies both kinds of inputs.

Stewart: And what about mathematical induction?  And the completeness axiom?

Daniel: I’m not smart enough to know if these mathematical tools are used in physics or not, but I would have to go with Quine here.  If they are used in physics, they are justified by the experiments that test them.  If not, they are toys and curiosities.  Of course, sometimes a caterpillar of pleasure can become a butterfly of knowledge.  This happens in physics too, for example in the case of relativity.

Stewart: And you would also go with Quine in rejecting the certainty and necessity of mathematics?

Daniel: Yes.  As you say in your account of Quineinism, “We cannot conceve of 7 + 5 being anything other than 12.  This, however, is a psychological feature of human beings”.