William Briggs is giving free epistemology courses on Substack and in a recent one (Minute 18:00 in the video here), there is mention of whether an inductive process (one that proceeds from the specific to the general) can prove something. It’s a light-hearted topic for a change.
Most philosophers say induction can’t prove things, but — due to the way that philosophic knowledge transmits vertically, rather than horizontally like most other knowledge does —most philosophers are wrong. The vertical transmission of beliefs can incorporate and retain errors, even if the horizontal transmission self-corrects.
Why no one-pound fleas?
Fleas are tiny, and the image below shows a 500 micron scale at bottom right which is equivalent to the thickness of a fingernail:
To say that “no fleas weigh one pound” is to engage in inductive inference (from specific to general) because, in order to make it a deduction, then one would have had to have observed 100% of all of the fleas which have ever existed in the world.
But merely observing a small fraction of fleas, and then claiming something to be true for all of them, requires making an inductive inference. Fleas jump far — 200 times their own body length — which is farther than would be physically possible using muscle contractions (no living organism, using just muscle alone, can jump that far):
They accomplish it by storing up tension in their legs and then releasing it like a compacted spring. But fleas have skeletons on the outside (exoskeletons) and their legs can be likened to cylindrical beams, where we can apply standard beam theory.
The G-forces that some fleas undergo during a jump are phenomenal: over 200 G’s. This means that, for a split-second during take-off, they “weigh” over 200 times what they would weigh when placed on a weight scale.
But when this G-force gets scaled up to the new and larger force that would be required if a flea weighed one pound, and standard beam theory mechanics is applied, then their scaled-up legs buckle by too much and break. The leg of a flea is like a cantilever, a beam fixed on one end, as shown on the right side of the image below:
Notice how it doesn’t take that many Newtons (1 kilogram-force = 9.8 N) to buckle a cantilever beam when it is only fixed at one end — compared to the beam at left which is fixed at both ends. The leg of the flea is even less sturdy, because the force is not applied vertically (straight down the shaft) like it is with standard beam-buckling.
Here is an excerpt (from the first reference cited below) showing what happens when you double the size of a hollow cyclinder which represents a flea’s leg:
The load that a leg must bear (F) and the cost of producing and moving the exoskeleton increase 8-fold, while the force produced by a muscle only increases 4-fold. Relative to the F's that must be resisted, the maximum force exerted by the appendage per muscle force exerted is only half of that at the smaller size.
Resistance to bending and to breaking while bearing body weight or locomoting are also reduced by 50% if size doubles, whereas resistance to bowing is reduced by 75% and to kinking by 87.5%. This suggests that leg failure by kinking may determine the maximum allowable size, S(m) in Equation (20), for a given insect body plan.
With every doubling of size you lose another 87.5% in resistance to kinking. But if the leg gets a permanent kink in it, then you get leg failure. A heavy (recently-fed) flea begins at about 0.001 grams in weight, but to reach one pound, it must be able to reach 454 grams. The first doubling only gets the flea up to 0.002 grams though.
Reaching just the first 1.0 gram of body weight requires 10 doublings!
This means that it cannot ever be the case that an actual flea weighs one pound — it is “too much to ask” from their body structures. Other limitations, such as passive diffusion of absorbed oxygen, rather than using circulation to carry oxygen, make it so that fleas can never ever get very big — because they’d “suffocate.”
But this then means that we can say something that is true of all fleas, after only examining some of them — that we can move from the particular to the general (forming an inductive inference to all fleas).
The reverse is true for elephants
Elephants cannot ever exist at a body weight of one pound because being an elephant requires being a living organism made up of certain tissues and body structures. If you do not have the tissues and the body structures that an elephant has, then it proves that you are not an elephant.
Living tissue can only exist when it is made up of the required concentration of cells: somewhere around 200 million cells per cubic centimeter of tissue. The density of cells inside of tissues is tightly-controlled. In fact, it is 100 times more tightly-controlled than the variability in the size of individual cells to begin with!
It turns out that the cells only “work” (to maintain life and function) when they are found to be inside of a certain density range. The size of individual cells also has a lower limit, like one based on the fixed size of transporters which must be found in the cell membrane to let the cell engulf the necessary molecules for metabolism.
The lower limit of the radius of a functioning (living) cell is about 2 nanometers — a radius which would lead to cell surface area with just enough space for a single transporter being fixed into the membrane of that cell.
It isn’t physically possible for a cell to ever go below that size.
But if there is an absolute lower limit on cell size, and an absolute lower limit on the cell density required for functioning tissue, then there is an absolute lower limit on the possible size of a given organism, such as an elephant.
Because both elephants and fleas cannot ever be found to have a mass leading to a weight of one pound, it means that it can be known — even without looking (without empirical verification) — that all fleas have less mass than all elephants. It can be proven to be true — it just has been — without empirical verification of the matter.
Easier example with casino dice
Casino dice rolls run from a result of 2 (“snake eyes”) up to a result of 12 (“box cars”) but you only need to examine one die and make one calculation in order to generalize or infer that all dices rolls — past, present, and future — have, do, and will come out to less than 13.
You don’t even have to witness a single roll of the dice to formulate a generalization which is true for all dice rolls. After you become aware of “crucial details” — such as a 6 being the highest result of a single die, and the fact that two dice get summed up — you can apply those crucial details to all dice rolls.
There is no need to record successive dice rolls, a rudimentary process unfortunately called “induction by enumeration.” That’s because you can successfully generalize without enumerating anything (without empirically verifying anything). This method of “induction by crucial details” was also performed on the fleas and elephants above.
Reference
The Scales That Limit: The Physical Boundaries of Evolution. Front. Ecol. Evol., 07 August 2019. Sec. Biogeography and Macroecology. Volume 7 - 2019 | https://doi.org/10.3389/fevo.2019.00242
Polilov, A., Makarova, A. The scaling and allometry of organ size associated with miniaturization in insects: A case study for Coleoptera and Hymenoptera. Sci Rep 7, 43095 (2017). https://doi.org/10.1038/srep43095
Sutton GP, Burrows M. Biomechanics of jumping in the flea. J Exp Biol. 2011 Mar 1;214(Pt 5):836-47. doi: 10.1242/jeb.052399. PMID: 21307071. https://journals.biologists.com/jeb/article/214/5/836/33598/Biomechanics-of-jumping-in-the-flea
Ruan Y, Konstantinov AS, Shi G, Tao Y, Li Y, Johnson AJ, Luo X, Zhang X, Zhang M, Wu J, Li W, Ge S, Yang X. The jumping mechanism of flea beetles (Coleoptera, Chrysomelidae, Alticini), its application to bionics and preliminary design for a robotic jumping leg. Zookeys. 2020 Feb 24;915:87-105. doi: 10.3897/zookeys.915.38348. PMID: 32148424; PMCID: PMC7052025. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7052025/
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