# Integral Calculus and Utilitarianism

I came across this image relating calculus to happiness.

It's obviously not meant to be particularly serious, but I was nevertheless inclined to give it some thought. The integral concept in mathematics essentially defines the net accumulation of a changing quantity, and (act) utilitarianism in ethics holds that the action that maximizes the net utility is the (relatively) moral one. It seems natural to me therefore to relate integral calculus to utilitarianism.

For example, define a set of functions to be the "happiness" of different living sentient beings (with each function corresponding to each being) which can be directly (or indirectly[ 1 ]--I don't know how these things are dealt with in utilitarian ethics) attributed to some action at some time t as a function of time. The sign distinguishes pleasure from suffering. The "bounds" of the integral for each function can be from zero--which we can define to be when the action starts--to the death of the sentient being. Sum up the integrals for each sentient being, and you have essentially quantified the "morality" of a particular action. One action can be said to be "relatively moral" to another if it has the greater integral. In fact, that set can include all sentient beings--unaffected ones would simply have an integral of zero.

Does any of this make sense? Has it been studied? Of course, the niceties of my argument may be incorrect, but I was just curious in general whether or not mathematics, in particular integral calculus, can be used to model utilitarianism theoretically. Again, it seems natural, even though it can't really be applied.

[ 1 ]: As a side note, I would say that the "direct" and "indirect" affect issue is related to the Year and a Day rule in law. There is also the butterfly effect and "randomness". These, however, are more general issues I would reckon.

• Integral calculus is nothing special and a very broad topic. Utilitarianism is per definition connected to calculus and thus also to integral calculus. It is like asking whether the square root is connected to some part of physics.
– user2953
Commented Jul 6, 2015 at 10:01
• Yeah, economists use integrals to capture things like this all the time. People integrate utility functions and expenditures to compare effectiveness of policies over time, and otherwise treat market forces as differential equations. Since classical economics and utilitarianism are basically isomorphic, a whole lot of people have looked into this.
– user9166
Commented Jul 7, 2015 at 22:13
• There is something similar in ethics: normative rational choice theory. Rational choice theory tells you that you should do whatever has the best expected utility. Expected utility is a function of two factors: the first one is the utility of each of the different possible outcomes of action A, the second one is the probability of these outcomes. See this for the juicy details.
– rds
Commented Aug 10, 2015 at 19:37
• Am I the first person to notice that the 1/t here is incorrect? This should be happiness(t)*dt so happiness*dt is close enough, but why the 1/t? Does this mean the same amount of happiness is worth less as you get older and t increases?
– user935
Commented Mar 17, 2016 at 3:36

Why does it seem natural? Simply because of its phrasing as 'maximising the sum of human happiness'?

Consider, that a cup is the sum of all the molecules that make it up; this tells me nothing about cups.

Possibly the influence was the other way round given the influence of Newtons physics on the mechanical philosophy of the 18C.

A modern defence of utilitarian ethics is given in Rawls A theory of Justice; his 'foundational axiom' is the veil of ignorance.

In his brief outline of the main classical arguments of utilitarianism there is nothing there that can be construed as the integral calculus; but a more definite influence of Playos Republic is discernible.

And in the main body of the text there is nothing there either; nor any other kind of mathematics.

The implication with this model is that it's possible to know one's happiness at each point in time along the way. Consider one possible counter-argument offered by Herodotus: when Croesus asked Solon who was the happiest person in the world, Solon told the story of others who were happier than Croesus. When Croesus asked Solon why Solon did not consider Croesus the happiest, Solon responded that it was impossible to judge the happiness of one's life until it was at an end.

So, there is at least one different approach.

You are assuming that the utility of (say) a consumption path is the sum (or the integral) of instantaneous utilities. That is, you've assumed that utility is time-separable in the sense that there all cross-partial derivatives (with respect to different points in time) vanish. Economists frequently make this assumption, because it makes their models tractable, but it's hard to justify from first principles.