Philosophical Inquiry

Note to all: this will be the most nerdy post I’ve ever written, so stand back and don’t read if you’re allergic to philosophy, mathematics, or both. Oh, and I finally found a LaTeX typer, so this post will be much more mathematically professional!

Utilitarianism, and all moral philosophy with the meta-ethic of consequentialism, is mathematical in nature. In general, consequentialists believe that the moral correctness of a thing is proportional to the effects that thing. Utilitarian philosophy emphasizes the effects of actions on people’s happiness or pleasure. So with that in mind let’s actually derive an equation that approximates Utilitarian thinking.

Let  be an operator that represents an action. It will map one level of happiness to another level of happiness. Mathematically speaking,

Where H(p,t) is the net happiness of a particular person p at a specific time t.

But now the question beckons, what is net happiness? It will be denoted as follows:

The raw happiness h is subtracted by the raw misery m at any time t, of any person p. It is important to make this distinction between raw and net happiness because, in accordance with the laws of life, the misery level of any person at any time is never zero. One can be happy but miserable, in pain, and not content at the same time; in such a state the net happiness would be in fact less than raw happiness. One can argue that the net happiness at any time is less than the raw happiness, due to unconscious pains that exist within each one of us. But I digress. More importantly, I claim that while net happiness can be negative (indicating some emotion that’s quite opposite from happiness), raw happiness and raw misery must be nonnegative – they are scalar measurements of raw emotion. (Also, an aside –  death means that H = 0, as no emotion can be felt after passing away.)

Jeremy Bentham attempted to formulate the misery and happiness functions when he was creating the theory. He argued in The Principles of Morals and Legislation that happiness (and, I will extend, for misery as well) is a linear combination of several components – intensity, duration, certainty, propinquity (“nearness”/closeness in space and time), fecundity (the ability for the emotion in one instance influence the emotion in a later one), purity (pure happiness or one diluted by misery?), and extent (how far reaching the emotion extends).

So this becomes a very difficult problem. First off what emotion is he referring to – raw or net? I will argue neither. The “raw” equations I have defined are valid at specific times with specific discrete persons, which do not factor in anything but intensity. Net emotion encompasses purity and certainty. But duration, propinquity, extent, and fecundity are still excluded. For the first three, we cannot look at the function at any one time; we must do more and take it through all times. In other words, we must take a summation, and from this I shall define total net happiness of a person:

where t0 is the time right now. This covers most things related to time, all the way to infinity (humans don’t live forever and they aren’t born “now”, but the reason for putting these limits will become evident later). It’s not a perfect model, but it will do for now.

Now, as for fecundity, that has everything to do with the change of net happiness. In other words, how does the net happiness H at this one instance change the net happiness H + dH? If fecundity is high then the change in net happiness will be great. At this time you’re probably thinking in terms of ∂H/∂t, and you’d be right. The key is to maximize that partial derivative. However, before you start freaking out about the explosion of terms, the integral in the total happiness function takes that into account – a function that, on average, decreases less/increases more than another function will have a larger area under the curve given same initial condition.

Believe it or not we’re almost done now! Let’s go back to the tenet of Utilitarianism: how does an action produce the greatest happiness for the most people? Well we have a formula for the happiness for one person, so lets start with an action’s effects for that one person. For an action to create the most happiness, it must optimally increase the existing happiness. We maximize the change in total happiness due to an action. Referring to old definitions and substituting:

See why now the limits make sense? An action at a current time will have an effect through the rest of time. Time before t0 is irrelevant.

Finally now we acknowledge that our actions have effects on more than one person. Since people are discreet beings (we don’t have infinitesimal fractions of human beings) we will take a discreet summation rather than an integral for this final part. The result will be defined as U(Â), the Utilitarian moral worth of an action:

where p0 to pn are all persons affected by the action.



This post is getting long so I’ll put this as a type of addendum. The major criticisms of Utilitarianism are of a) minority rights, and b) difficulty in predicting consequences. These are valid criticisms. Mathematically, a) is the neglecting of negative consequences for any one person because of overwhelmingly positive consequences for most others leading to a positive value for U. In other words the final summation through all persons drowns out the possible plights in minority persons. B) is the fact that the Utilitarian function is quite difficult to calculate using known integration techniques. 😉 A quick mental estimation could lead to faulty results.


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