Ethics is not mathematics. For, unlike mathematics, ethics cannot function solely based on a set of axioms, or ‘absolutely true staring points for reasoning,’ like a + b = b + a. Based on axioms, we can build an entire world  (‘mathematics’) in which we can be sure that, only by following these rules of inference, we will always end up with the truth, the truth and nothing but the truth. Hence it’s understandable that philosophers have thought to themselves: ‘Damn, how cool would it be if we could apply the same trick to ethics; that we, confronted with any action, could decide whether the action would be right or wrong?’ Surely: society has tried to build its very own rule-based system, the system of law. But is this a truly axiomatic system? Are there truly fundamental rights from which the rules of justice can be inferred? Let’s take a look at that.

Immanuel Kant made the distinction between hypothetical imperatives and categorical imperatives. These are two ‘kinds of rules’, with the first ‘being applicable to someone dependent upon him having certain ends‘; for example, if I wish to acquire knowledge, I must learn. Thus we’ve got: desired end (‘knowledge’) + action (‘learning’) = rule. Categorical imperatives, on the other hand, denote ‘an absolute, unconditional requirement that asserts its authority in all circumstances, both required and justified as an end in itself.’ We can see that there is no desired end present in this kind of rule; only the ‘action = rule‘-part.

But how could a categorical imperative be applied in practice? A belief leading up to a categorical imperative could for example be: Gay marriage is okay. Period. That would imply that, you believe that, irrespective of the conditions present in a particular environment – thus no matter whether there is a republic or democratic regime, whether the economy is going great or not – gay marriage is okay. However, as it stands, it is not yet a categorical imperative, since this claim doesn’t urge you (not) to do something (such as ‘You shall not kill’, which is a categorical imperative). The rightful categorical imperative would be something like (G): ‘You should accept gay marriage.’ This is an unconditional requirement that asserts its authority in all circumstances and is justified as an end in itself

Now: let’s assume that you’re talking to someone who doesn’t agree with (G). Because now it gets interesting, for now you have to make a decision: you either stick to (G) or you reformulate (G) into a hypothetical imperative. The first option is clear: you just say ‘I believe that gay marriage should be allowed always and everywhere. Period.’ Seems fair, right? But what if the person you’re talking to would respond by saying, ‘Okay…so even when citizens would democratically decide that gay marriage is unacceptable?’

Now you have got a problem, for this might be situation in which two of your categorical imperatives are contradictory, such as (G) and (D): ‘Decisions coming about through a democratic process should be accepted.’ Both (G) and (D) are unconditional rules: they should be acted on irrespective of the situation you’re in. But this is clearly impossible, for (G) forces you to accept gay marriage, while (D) forces you to do the opposite.

You could of course say that (G) is merely your belief (you believe that gay marriage should be accepted, not that this particular democratic society should find this too), but then you seem to fall into a form of moral relativism. Given that you don’t want that to happen, you have to decide which one is the true categorical imperative: (G) or (D)? And which one can be turned into ‘merely’ a hypothetical imperative?

You could of course decide to turn (D) into (D.a): ‘Only if you believe that a decision has come about through a democratic process and is a good decision, you should accept the decision.’ Or you could turn (G) into (G.a): ‘Only if the decision has come about through a democratic process, gay marriage should be accepted.’ But is this really how we form our moral judgements? Is (D.a) truly a rule you believe to be ‘fair’? And (G.a): do you truly believe that gay marriage is okay only if it is accepted by society? That is: do you make the moral value of gay marriage dependent upon the norms prevalent within a society? I doubt it.

So we are stuck; stuck into a paradox, a situation in which two absolute rules are contradictory, and the only way out is through turning at least one of them into an unintuitive and seemingly inadequate hypothetical imperative. So what to conclude? We’ve seen that categorical imperatives look powerful; as if they can truly guide our lives for once and for all; no more need to search for conditions that might be relevant to our judgements. But we’ve also seen that when two categorical imperatives are contradictory – that is, when two rules cannot be followed at the same time – changes have to be made: at least one of them has to be turned into a hypothetical imperative. In order to do so, a certain ‘value hierarchy’ is required, based upon which these categorization decisions can be made. Hence it seems that even Kant’s absolute ethics – with its absolute categorical imperatives – seems to be relative: relative to (the value of) other imperatives, that is. Therefore mathematical ethics, as presented above, seems to be impossible.

But what do you think?