Tag Archives: arguments

Axioms, definitions and agreement

A while ago I posted A Problem of Language, a response to an article claiming that Scala was not a functional language. This isn’t an attempt to revive that argument (and please don’t respond to it with such attempts. I’m likely to ignore or delete comments on the question of whether Scala is a functional language). It’s a post which is barely about programming, except by example. Really it’s a post about the philosophy of arguments.

My point was basically that without a definition of “functional language” (which no one had provided) it was a meaningless assertion to make.

Unfortunately this point isn’t really true. I think I knew that at the time of writing but glossed over it to avoid muddying the waters, as it’s false in a way that doesn’t detract from the basic point of the article, but it’s been bugging me slightly so I thought I’d elaborate on the point and the basic ideas.

Let’s start with what’s hopefully an unambiguous statement:

Brainfuck is not a functional language

Hopefully no one wants to argue the point. :-)

Well, why is brainfuck not a functional language? It doesn’t have functions!

So, we’re making the following claim:

A functional language must have a notion of function

(in order to make this fully formal you’d probably have to assert some more properties functions have to satisfy. I can’t be bothered to do that).

Hopefully this claim is uncontroversial.

But what have we done here? We’ve, based on commonly agreed statements, proved that Brainfuck is not functional without having defined “functional programming language”. i.e. my claim that you need a definition in order to meaningfully claim that a language is not functional is false.

What you need in order to make this claim is a necessary condition for the language to be functional. Then on showing that condition does not hold you have demonstrated the dysfunctionality of the language.

But how do we arrive at necessary conditions without a definition? Well, we simply assert them to be true and hope that people agree. If they do agree, we’ve achieved a basis on which we can conduct an argument. If they don’t agree, we need to try harder.

A lot of moral arguments come down to this sort of thing. Without wanting to get into details, things like arguments over abortion or homosexuality frequently come down to arguments over a basic tenet: Do you consider a fetus to be of equal value to a human life, do you consider homosexuality to be inherently wrong, etc. (what I said about arguments RE Scala holds in spades for arguments on these subjects). It’s very rare for one side to convince the other of anything by reasoned argument, because in order to construct a reasoned argument you have to find a point of agreement from which to argue and that point of agreement just isn’t there.

Mathematically speaking, what we’re talking about is an Axiom. Wikipedia says:

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.

I consider this definition to be true, but perhaps a bit obfuscated. I’d like to propose the following definition. It’s overly informal, but I find it’s a better way to think about it:

An axiom is a point which we agree to consider true without further discussion as a basis for arriving at an agreement.

(This may give the hardcore formalists a bit of a fit. If so, I apologise. :-) It is intended to be formalist more in spirit than letter )

The most important part of this is that axioms are social tools. They don’t have any sort of deeper truth or meaning, they’re just there to form a basis for the discussion.

This entry was posted in Numbers are hard, rambling nonsense and tagged , on by .