[EDIT: My question can be refined to, how does Heron's account of the behavior of light fit into a classical causal account of nature? Especially, is his account a kind of natural locomotion in which light "striving" for the shortest path qualifies as a kind of final cause? Or if not, what kind of explanation is it? Or does it find no home at all in an Aristotelian physical framework (which would be hard for me to believe)? I fear that mentioning modern physics in my question below was a distraction -- it was the context in which the question arose for me, but is not really essential.]

Much of modern physics can be written in terms of so-called variational principles. One common example is Fermat's principle of least time: out of all the ways that light could get from A to B, it chooses the quickest route. And so on, to Lagrangian and Hamiltonian formulations of mechanics etc.

I came across a physics text that claimed, casually and briefly, that such principles are final causes in the Aristotelian sense.

I am looking for a second+ opinion/elaboration on that claim. To what extent would the classical tradition (Aristotle, Aquinas) really accept such a variational principle as satisfying the definition of a final cause?

I have found numerous quotes in secondary literature of ancient or medieval authors who ascribe an economical character to nature's workings. A significant example is Heron of Alexandria who not only claimed (in a predecessor to Fermat's principle) that light "strives to move over the shortest possible distance, since it has not time for slower motion" but drew quantitative conclusions from this principle. Another is Grosseteste, also in the context of optics, "nature always acts in the mathematically shortest and best possible way."

Did these authors understand principles like this to be expressions of final causality, or something else? Ideally, is there a passage where a classical author explicitly considers whether such a principle -- likely Heron's -- is a final cause, or clearly treats it as a final cause or as not a final cause but something else?

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    I've never heard the term "variational principle" before (can you provide a citation?), but Aristotle's concept of final cause is the belief that all objects tend towards their final end.
    – virmaior
    Sep 5, 2015 at 3:05
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    @virmaior: I mean the term "variational principle" as used in modern physics; some ref's are W'pedia, Cornelius Lanczos' Variational principles of mechanics. They involve some quantitative maximum or minimum property that nature "strives" for (per the Heron quote). Sep 5, 2015 at 4:10
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    idea makes sense. I'd just never heard the term. It's a good and interesting question.
    – virmaior
    Sep 5, 2015 at 4:13

3 Answers 3


Aristotles discerns four kinds of causes:

  • material (subject to change)

  • efficient (that which changes)

  • formal (the shape of change)

  • final (that for which change occurs)

and he considers them as explanatory principles of an 'inquiry into nature'; an adequate explanation should deploy all four causes; but in terms of explanatory power it is the final cause which stands pre-eminent; and in particular over the efficient (which, in a sense, and in contrast to the final cause, can be seen as the initial cause).

The 'variational principle', is as nothing without something to be a principle of; this something is that which is 'subject to change' - ie a system of particles, a pendulum or that bit of physical reality I see before me: this is the material cause.

Further, the variational principle, is a principle; and as a principle, it is a principle of change; this suggests that we see it as the efficient cause.

Moreover, when Aristotle asks for the efficient cause of a bronze statue, he doesn't identify this as the man who pours the bronze, and shapes it - the artisan - he says it is the art of bronze-casting; thus, and by analogy; and thinking of 'the art of' as the 'law of'; then, in this bit of reality before me, where particles come, and go, and collide; it isn't the collisions that are the efficient cause of their motion on impact, but that which regulates their motion - the aforementioned principle - again.

It is by identifying Final and Formal causes as genuine causes of change, that Aristotle in his own opinion, differs from his predecessors, who he claims were content with the material and efficient causes; in his account the Final and Formal often coincide.

The question is, whether we can identify such causes here; for Aristotle admits not all natural processes have such: his example is the eclipse of the moon; and it appears that this is the case here.

What's been shown is that the variational principle is an efficient cause; but not a final cause (which isn't to deny that such a cause exists at a different ontological level).


I think there is probably quite a bit more that can be said;especially when takes the long view of variational principles; the account above uses modern notions of variational principles; but there are notions too like the conatus of Spinoza (ie 'striving'); and also the maxim of Liebniz: 'best of all possible worlds' especially when Aristotle suggests that ends are to be considered not by what is last but by what is best.

  • I was intrigued by your interpretation that it is not the sculptor but the sculptor's art (his practical knowledge, I take it) that is the efficient cause. Is this a common reading of Phys. II 3? A little earlier (c.194b30) the examples of efficient cause are the man who gives advice and the father of a child, rather than an art of advice-giving or of fatherhood. Unfortunately (or interestingly?) both translations I can find are ambiguous on this point, HG,WC. Sep 12, 2015 at 1:37
  • @gnarledroot: it's not my interpretation but one I read in the SEP on the Aristotles thinking on causality; how common a reading it is I'm not sure. That's definitely an interesting point, though I don't recall reading it; thanks for bringing it up. Sep 12, 2015 at 3:31
  • In the same entry there is an interesting aside on how they take Aristotle to defend his thinking on final causes; he argues it is neccessary to explain the regularities we seen in nature; it is suggestively close to Humes attack on cause - which would suggest that Hume was influenced by this; but his solution was a kind of psychologism which was then thoroughly formalised by Kant. Sep 12, 2015 at 3:35

Variational principles play a main role in todays physics under the name Lagrange principle. That is a very general principle which allows to derive the fundamental equations of the physical domain in question.

Lagrange principle states that the real path of the system is distinguished from all possible paths by the fact that a certain integral is minimal.

The latter fact should not be taken as a kind of teleology. Because the formulation by the integral is mathematically equivalent to a formulation by a system of differential equation (Euler equations of the Lagrange principle). And differential equations are considered a means for causally determined developement.

Hence mathematics dose not discriminate between causa finalis and causa efficiens.

  • I would agree in general that differential equations are ripe for causal interpretation. But the gap in my understanding that remains is, what is the connection to classical-medieval definitions of final causality, and-or natural locomotion? Sep 5, 2015 at 20:01
  • @gnarledRoot Causa finalis first looks to the goal which is to be reached in the future and then decides about the actual path to choose. Some people see an analogue to Lagrange's principle: The goal is to find a path such that the integration of a certain physical quantity along the path, taken from present time to the choosen time in future, is minimal. The goal is to minimalize that integral as if this goal were a causa finalis. - I wanted to stress that the analogue is an illusion: From a mathematical point of view Lagrange's principle is equivalent to a system of differential equations.
    – Jo Wehler
    Sep 5, 2015 at 21:24
  • This is becoming a different question but is irresistibly connected, and interesting to pursue. The mere mathematical duality of the formulations by itself does not tell us which interpretation is an illusion, if any. So something else must be entering into the argument that I am missing? I am more persuaded by your earlier statement that mathematics does not discriminate among the causes. Sep 6, 2015 at 0:15

Not directly. Aristotle's final causes are forms that an artifact or a living organism are supposed to achieve when fully developed. Variational principle in optics, on the other hand, applies not to the light's destination, but to the path it takes to get there. To fit it into Aristotle's scheme one needs an extra "time" dimension in which light's path "evolves" into the optimal one. Aristotle entertained no such fancy. And since variational principles prescribe the exact path light (or other system) must take they cipher Aristotle's efficient causes, not his final ones. However, in a way medieval scholasts did have an extra dimension in the logical progression from God to the creation, albeit collapsed into a single step. So God's design could be seen as the final cause explaining the economy of nature, just like features of an artifact are explained by its final cause, the creator's design.

But with a single created world connection to optimality is tenuous, optimal is only such compared to non-optimal. Heron's observation was essentially overlooked by scholasts, and only resurfaced when Fermat generalized it. Variational principles of physics stem not from Aristotle's teleological intuitions, but from Leibniz's modal ones, whose motivation came not from economy of nature but from theodicy. Leibniz did have medieval predecessors, explicitly cited in his Theodicy. Duns Scotus asserted that God had many alternative chains of events to choose from, and Molina even inserted an intermediate step between God's essence and the act of creation, the so-called middle knowledge. In it God saw "what each such will would do... were it to be placed in this or that or indeed in infinitely many orders of things". In orders of things one easily recognizes possible worlds, but Leibniz added a missing key ingredient. God did not just choose this particular world for inscrutable divine reasons, he chose the best of possible worlds, and our reason is capable of discerning the signs of this bestness. Maupertuis' original formulation of the least action principle bears the hallmarks of trying to fulfill this task.

Alas, it was not to be. We now know that light, and other systems, need not minimize action even locally, they might also maximize it, or just follow a stationary path that neither minimizes nor maximizes. In quantum theory light follows all possible paths, and interferes with itself to produce a probability distribution clustering around stationary paths. So while variational functionals proved to be a lasting presence in fundamental physics, optimality did not.

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