# What is the tradition of reasoning behind Archimedes's “a priori” derivation of the law of the lever?

Today we get the law of the lever by empirical experiment or see it as a special case of the principle of least action.

Assuming only gravity acts, with weights of masses m₁, m₂ and corresponding distances of the weights r₁, r₂ from the mounting of the lever, we can state the law of the lever as:

m₁ / m₂ = r₂ / r₁

Archimedes tried to prove the law of the lever. The reasoning behind his proof is something like this (the full proof is too long to present here): if we look at a lever, for example with masses 2 kg and 3 kg and corresponding distances of 3 meters and 2 meters from the distance:

we see that if we split the weights into masses of 0.5 kg and place them so that the center of gravity of their corresponding group does not differ from the center of gravity of the original weights, like this

we recognize that there must be an equilibrium.

Archimedes' proof was criticized by Ernst Mach, stating that it can at most show that the law of the lever has the form m₁ / m₂ = f(r₂) / f(r₁) with some function f(x) (x being the distance from the mounting). There are others who argue that Mach's interpretation was unfair. But this is not the point of this question.

Reasoning as in Archimedes' proof is quite alien to modern physics. Mach judged it to be a result from the “mania for demonstration” of the ancient Greeks. But if Archimedes' proof is not (proto-)modern, what are its influences? Where do we place it in the different strains of ancient Greek thought?

Is there no tradition it fits into (is there a truly “Archimedean philosophy of science”?)? Or is it influenced by Pythagoreanism? Or is it an example of Aristotelian science, maybe arguing with the help of formal causation?

• arguing from symmetry isn't alien in the physical tradition; Galileo did it too. – Mozibur Ullah Mar 2 '17 at 4:41
• Unless you think Galileo isn't part of modern physics... – Mozibur Ullah Mar 2 '17 at 4:50
• @MoziburUllah yes, Galileo is modern physics – or rather the founder of it. Don't be so snarky ;-) And I know of symmetry arguments in physics, of course. But they are just a way to reduce the numbers of coordinates relevant to solve a problem. This does not seem to be the case here. Or do you think there is an a priori reason to believe why the yellow 0.5 kg weight that ends up on the other side of the mounting point gives not rise to a very different physical situation? Though Archimedes' proof seems intuitively convincing, it seems he injects an additional premise somewhere. – wolf-revo-cats Mar 2 '17 at 5:35
• Main influence : (pseudo-)Aristotle, Mechanica; see also: Joyce van Leeuwen, The Aristotelian Mechanics: Text and Diagrams, Springer (2016) – Mauro ALLEGRANZA Mar 2 '17 at 6:56
• See also Paolo Palmieri, The empirical basis of equilibrium: Mach, Vailati, and the lever (2008). – Mauro ALLEGRANZA Mar 2 '17 at 8:12

## 2 Answers

I believe the tradition in question is that of Gedankenexperiments, or thought experiments, and they are certainly used in modern science to motivate and/or justify physical principles, just recall Newton's bucket spinning for absolute space, or Einstein's elevator falling for the equivalence principle. Ironically, it is Mach who coined the term Gedankenexperiment. Thought experiments attracted attention of a number of philosophers recently, Brown even wrote a whole book analyzing their role in science, freely available online. See also Thought Experiments in Biology by Schlaepfer et al. and Irvine's Thought Experiments in Scientific Reasoning.

Already at the dawn of modern science Galileo gave an argument (based on a thought experiment), similar in spirit to the Archimedes's, and with the same "magical" quality to it of establishing "a priori" something that should be empirical. The claim was that the falling bodies fall at rates independent of their weights. It goes something like this. Tie a heavier weight to a lighter one. When they fall together the lighter weight will slow down the heavier one, so the composite will fall slower than the heavier weight. On the other hand, since the weights are added the composite should fall faster than it. The only way to reconcile the two conclusions is to assume that the rate does not depend on the weight. Discussion of this thought experiment, and references, can be found under Is Galileo's argument about falling bodies logically flawed?

In short, there is no magic. Galileo's reasoning relies on a hidden empirical assumption, that bodies tied together would fall the same way as the the bodies just touching each other. This seems "right", but why does it seem "right"? It is the "physical intuition", of course, the product of accumulated experience from dealing with many special cases. It can be "folk" or acquired through working for a long time in a particular field. The accumulated knowledge remains tacit, but can be elicited by a clever set-up and draw broad consent from other practitioners. This is the role and the use of constructing thought experiments.

Archimedes's argument is not magical either, it relies on the assumption that equilibrium is not disturbed by replacing many weights with a single combined weight placed at their center of mass. The consensus of scholars seems to be that Archimedes considers this to be a postulate, and with it his derivation of the law of the lever is valid, see e.g. Palmieri's The Empirical Basis of Equilibrium. But where does this assumption come from if not from empirical experience condensed into intuition? Archimedes surely understood this as he explicitly characterizes his center of mass based arguments in the Method of Mechanical Theorems to be only heuristic, and reproves them by the method of exhaustion.

Brown drew perhaps the most extreme conclusions from the import of thought experiments, he is a self-identified modern Platonist, with capital P:

"...we really do have some a priori knowledge of nature. Of course, the great bulk of our knowledge must be accounted for along empiricist lines; but there is, I contend, the odd bit that is a priori and it comes from thought experiments. Not all thought experiments generate a priori knowledge. Only a very select class is capable of doing so. This a priori knowledge is gained by a kind of perception of the relevant laws of nature which are, it is argued, interpreted realistically."

Few take this position seriously, see e.g. Hacking's critique in Do Thought Experiments Have a Life of Their Own? ("We have been inoculated against Brown's variety of platonism for too long. Most of us see at once that his idea, of immediate acquaintance with universals, abstract objects, just does not do any explanatory work for us"). To most the "a priori knowledge of nature" is the empirically accumulated intuitions, unrealized until elicited by a thought experiment, but compelling once realized:

"What, after all, is applied mathematics, but the deduction of what we don't know from what we know? There is of course an ancient tradition that says deduction merely makes us aware of what was contained in our premises. Very well, but it still leads us to new awareness, new understanding of the world... Thought experiments are valuable when they bring out, in a succinct way, a conceptual tension between two ways of thinking, and force us to come to grips with it."

Thought experiments are common in modern philosophy too. Kripke and Putnam are famous for invoking "linguistic intuitions" to argue for their causal theory of reference and linguistic externalism. But in philosophy thought experiments are often treated as compelling arguments in their own right (as there is no experimental check on them afterwards), and that is highly controversial. Here is Cummins in Reflections on Reflective Equilibrium:

"Theorists regularly claim support from disputed intuitions when there is no resolution in sight. Indeed, disputed intuitions are often the linchpin on which everything turns. Consider the role of Twin-Earth cases in current theories of content. It is commonplace for researchers in the Theory of Content to proceed as if the relevant intuitions were undisputed. Nor is the reason for this practice far to seek. The Putnamian take (Putnam 1975) on these cases is widely enough shared to allow for a range of thriving intramural sports among believers. Those who do not share the intuition are simply not invited to the games."

Today we get the law of the lever by empirical experiment or see it as a special case of the principle of least action.

The principle of least action was founded on the principle of the law of the lever. For example, Lagrange in his Analytical Mechanics (cf. its introduction), decided to base his entire physics off the principle. Least action and virtual velocities/forces were consequences of it.

Mach judged it to be a result from the “mania for demonstration” of the ancient Greeks.

Capecchi's 2012 History of Virtual Work Laws §3.1.2.1 "Proof of the law of the lever" (p. 48-50) concludes on p. 50 by mentioning ancient and modern criticisms of Archimedes's proof. Mach considered Archimedes's proof circular (The Science of Mechanics p. 13-14):

But surprising as the achievement of Archimedes and his successors may at the first glance appear to us, doubts as to the correctness of it, on further reflection, nevertheless spring up. From the mere assumption of the equilibrium of equal weights at equal distances is derived the inverse proportionality of weight and lever-arm! How is that possible? If we were unable philosophically and a priori to excogitate the simple fact of the dependence of equilibrium on weight and distance, but were obliged to go for that result to experience, in how much less a degree shall we be able, by speculative methods, to discover the form of this dependence, the proportionality!

Mach was a positivist, so in a way it is strange that right before what I quoted, he mentioned Lagrange's "concise disposal of the problem" being "only possible to the practised mathematical perception." Yet, Lagrange's argument seems no different from Archimedes's.

if Archimedes' proof is not (proto-)modern, what are its influences? Where do we place it in the different strains of ancient Greek thought?

For a good comparison between Aristotle and Archimedes, see pp. 11-15 of: