Banner
    Why Do We Have Ten Fingers?
    By Mark Changizi | May 17th 2010 03:07 PM | 19 comments | Print | E-mail | Track Comments
    About Mark

    Mark Changizi is Director of Human Cognition at 2AI, and the author of The Vision Revolution (Benbella 2009) and Harnessed: How...

    View Mark's Profile
    In How Many Limbs Should Humans Have? I described my Limb Law, an empirical law I discovered which relates how long an animal’s limbs are to how many limbs it has. This law is explained by virtue of animals having evolved a limb design that minimizes the amount of needed materials to reach out into the world (see links to my academic work in the previous piece).

    To see the Limb Law in action, go to my web site  where you can play with an animal’s limb length and watch how the optimal number of limbs changes. Roughly speaking, the animal designs you can create in this program are the ones we find on Earth (…among radially-directed-limbed animals).

    The Limb Law applies to more than just animal legs. By “limbs” I refer to any appendages that reach out, and so the hypothesis applies to hands as well, but where a hand’s “limbs” are its digits. 



    The only thing we must keep in mind in order to apply the Limb Law to hands is that hands are not free-range animals, but are, rather, connected to an animal. Hands have digits pointing away from the arm that connects to the hand, and so have only about half of the digits one would expect if the hand were roaming the world on its own.

    In light of this fingers-are-the-hand’s-limbs observation, in this piece I’d like to ask…

    Why do we have ten fingers?
     

    In addition to being fundamentally interesting, this question also has deep implications for why we use a base-10 number system (rather than a base-2 or base-8 system, each which would arguably be better).

    How can the Limb Law tell us how many fingers we should have, given that it only tells us the relationship between limb length and number of limbs? 

    Because hands like ours have plausible constraints on how long their fingers should be. Hands must close, i.e., their fingers must be able to reach back over the palm and cover it up. And that simple requirement is enough to enable us to predict roughly how many fingers we should have. 
    Recall that the Limb Law was that the number of limbs, N≈2π/k, where k was the “limb ratio,” k = L/(L+R), where L is limb length and R the radius of the body. 

    The demand that finger length be approximately the diameter of the palm means that the finger length should be about twice the palm’s “radius”. So, L≈2R. It follows that k ≈L/[L + (L/2)] = 2/3. And, plugging in k=2/3 into the equation for the number of limbs, N, we have N≈2π/ (2/3) = 3*π ≈ 9.42. 

    That is, given that fingers must be roughly as long as the diameter of one’s palm, then there should be about 9.42 fingers poking out from the circumference of the palm.

    But remember that palms aren’t animals living freely on their own, but are attached to arms, and thus we expect palms to have digits on only about one half of their circumference. So, 9.42 is twice what we should expect for the number of fingers. Divide 9.42 by 2 and we have 4.78 fingers per hand. Or, about five.

    Could it be that your run-of-the-mill alien would also have ten fingers, and thus get saddled with base-10?

    Comments

    logicman
    Why do we have ten fingers?

    The simplest gripping device has two opposed jaws.  Any misalignment causes a gripped object to emphasise the misalignment as the pressure of grip increases.  This may be cured by using three jaws as two jaws or fingers opposed by one.

    The three jaw grip cannot readily be used as a manipulator.  A four-finger device is not much better.  If there are two fingers opposed by two fingers then grip is improved at the expense of manipulation.

    Four fingers and an opposable thumb represent a good compromise.  The four fingers can move independently to a great degree while the thumb steadies an object in opposition to one, two or three fingers.  The human hand is something like a quadruped with an almost prehensile tail.

    It is most important in comparing humans to animals to note that we humans have only eight fingers and two opposable thumbs.  It is only in mathematics that it is accurate to refer to all ten digits as fingers.

    On counting up my fingers
     I discover I have ten.
    I'm not much good at maths
     so I must count them all again.

    This miracle of nature
     I can't help but contemplate -
    However could I count to ten
     If I had only eight?


    Mark Changizi

    Nice.

    For any animal in the plot in my previous piece, there may be a hypothesis specific to that animal and its habitat explaining why it has that many limbs and why limb length is what it is. And yet none of those individual hypotheses would be able to explain why the more general Limb Law holds. By my way of thinking, my Limb Law explains the expected baseline number of limbs expected for any limb length, but to explain the exact number of limbs (which may be a little above or below the predicted line) one needs to get a more specific hypothesis.
    so, wouldn't any aliens number of fingers depend on the overall size of the alien and it's palms?

    what if the aliens can do telekinesis - would they have vestigal hands?

    jtwitten
    "Law" may be coming it a bit high. Relativity and evolution are "only" theories after all.

    I've been looking at my own hand and it seems that my wrist only occupies of my hand radius. So, where are my extra two fingers? I feel so sub-optimal.
    Mark Changizi

    I'd say that my hypothesis is a theory, but that the empirical relationship itself (shown in the previous piece) between number of limbs and limb length is a "law" (but just as happy to say it's an empirical regularity).

    As for how much of the hand has been selected to have "limbs", it seems plausible that roughly the distal half would want fingers, not any parts of the proximal half.


    Your rule gives N an upper limit of 2π.

    Do octopi exceed this because their limbs are flexible in pretty much any direction?

    Mark Changizi

    In my treatment (e.g., see the third section of this chapter of my first book http://bit.ly/b2EgWr ), I treat cases like octopus as fitting a "planar" design, in the sense that the limbs all emanate from a perimeter and can point along the plane. ...even though they can go any which way.  As you'll see in the chapter, the model itself is very simple, in the spirit of treating cows as spheres.
    Scotch my last post. Brain farting in action.

    Cool stuff.

    We have a maximum of 5 digits per limb, because ancient ancestors of all subsequent quadrupeds settled on 5 digits per limb, after initially starting out with a higher number (7 or 8 digits per limb - see for example, http://www.dinosaurjungle.com/prehistoric_animals_acanthostega.php ).

    These ancient ancestors of terrestrial vertebrates weren't planning ahead what mammals, or crocodiles, or dinosaurs, or frogs, might need million of years later on their limbs. Evolution only adopts for the current environment. So the fact that 5 digits per limb suited the life style of these ancient tetrapods hundreds of millions of years ago, set the maximum number of digits per limb at 5 in all subsequent tetrapods (amphibians, reptiles, birds, and mammals).

    (Note: Evolving additional digits seems to be extremely tough. I don't know of any tetrapod lineage that did so - I don't think there are any - the closest is the Giant Panda that uses 5 forward digits, and has a pseudo-digit made from its extended wrist bone).

    Anyway, the maximum digits per limb in tetrapod was set at 5, hundreds of millions of years ago.

    Some lineages of tetrapods have lost digits (e.g. hooved animals, or birds having fused fingers in their wings), but of course many haven't. Once a digit is lost, it doesn't come back. For example, you might argue that it would be very useful for some flightless birds to be able to grip their food, but none re-evolved hands.

    Now look at our evolutionary past. Our ancestors were very different from us. At various stages they were amphibian-like, then reptile-like, then probably a bit dog-like, then rat-like, then squirrel-like, then monkey-like, then ape-like, and finally human-like, and then human. None of these evolutionary stages were planning ahead for the number of fingers than humans would need in millions of years time. And if it had suited them to lose a finger, we wouldn't have regained it (remember, no animal has). So it's a happenstance of history that we have 5 fingers.

    Mark Changizi

    The fact that number of digits has tended to only fall among tetrapods (from polydactylous to pentadactylous and lower in many cases) could mean there is some kind of (genetic or developmental) difficulty in adding digits, as you suggest. But abnormal polydactyly is fairly common in vertebrates (including humans), and often has a hereditary component. On this basis it would seem that adding a digit is possible. And, evolution can take more creative approaches as well, like the Giant Panda extra pseudo-digit you mentioned.

    Rather than supposing that there is some kind of difficulty in adding digits, or some kind of upper limit of five, an alternative hypothesis is that the original polydactylous tetrapod had simply "too many" digits for most hand designs relevant for terrestrial environments, and that tetrapods ever since have been disproportionately losing digits to fill in vacant spots in design space, with only the occasional added digit. That is, the tendency for digit loss over time may be due to adaptive selection pressures, not a no-adding-digits constraint.

    So, I'm not convinced that developmental / genetic constraints force a five digit maximum.
     
    Also, I'm of course not suggesting that prior "evolutionary stages are planning ahead for the number of fingers humans would need in millions of years time."

    And, at any rate, all this is beside the point. Let's suppose that some kind of historical accident were to force exactly five digits on all progeny of an animal, and that some of those progeny became primates with our hand design. Is it true that it is a historical accident that we have five fingers, in this thought experiment? Not quite. Being stuck with five digits would have constrained the kinds of hand (and body) designs possible for this animal's progeny. Some hand designs, and animal designs, would then be out of reach to this lineage. The hand designs within reach for such a lineage would be ones which work really well with five digits. ...and one such hand design is the "grasper" one where the digit length is of similar length to the palm diameter, very roughly our hand. The question is whether our hand/digit design is optimal in some hypothesized sense. My suggestion is that our 5 digits and our digit-length-to-palm ratio are "designed for one another" (because that relationship is consistent with cheap reaching-out wiring costs). My suggestion is an engineering hypothesis, not a historical hypothesis. If five-ness was historically fixed, then what historically evolved was the length of the digits to become a proper grasping hand. To put it another way, if our long ago ancestors had, say, three fingers, and could not add new ones, then primates would probably not have evolved in the first place, because the hand would have led the lineage down new design paths.
    There is also a recessive human trait for having 6 digits on each hand, which is dominant in most monkey and the great apes. How do you explain that? I don't think design can be explained so brutishly, as there are more factors to consider, mainy not easily quantifiable, such as sexual selection.

    Mark Changizi

    My empirical "Limb Law" showing how number of limbs varies with (relative) limb length is fairly strong, something you can read about in the previous piece I wrote for SB (http://www.scientificblogging.com/mark_changizi/how_many_limbs_should_hu... ).

    The argument for the fiveness for our hands, though, is very approximate. The expectation would be that hand-graspers (who therefore want their fingers approximately similar in length to their palm diameter) should tend to have roughly five fingers. But this is a kind zeroth-order hypothesis about what to expect, and more specific design criteria may shift the design. Maybe four, maybe six. About five. But not two fingers on a hand, and not 10 or 50.
    jtwitten
    Flippant semantics aside, it seems that this does not resolve the problem that this approximate number of digits is quite prevalent throughout the mammalian clade in species that have no evolutionary history of "grasping" hands.
    Mark Changizi

    The claim is not that grasping hands are necessary for five-ishness; only sufficient. 
    [Grasping hands => digit length similar to hand diameter],
    and because (as a special instance of the limb law)
    [digit length similar to hand diameter<=> around five digits],
    we have... [Grasping hands => around five digits]
    But not the other way around.

    BTW, on semantics, "law" is regularly used in biology to refer to wide-ranging empirical generalizations, like allometric scaling laws. This limb law is essentially a variety of allometric scaling law (although in this case, the proportionality constant, and not just the exponent, is particularly interesting).
    Hfarmer
    Interesting.  I think the reason we have a base 10 number system has more to do with various accidents of history.  Based 10 is so common now that it's hard to remember that people have used other number systems in the past.   Those civilizations that did fell or were conquered for reasons that had nothing to do with numbers. 
    The Roman Numeral system was not base 10.  It also wasn't very good for anything other than simple counting.  However it did rule the world for over a millennium. 

    The Mayan numeral system was interesting because it was a base 20 system which only used three basic symbols.  A dot for multiples of one. A bar for multiples of five.  A sea shell for zero.  Which is interesting since they kind of have the advantage of a base 3 system by using only three primitives to build up their symbols.   I have heard archaeologist speculate that base 20 appealed to the Mayans because they generally wore sandals and therefore had 20 convenient counting tools instead of 10.  

    By accident of history the Visigoths of Spain had used Roman numerals and were conquered by the Umayyad Caliphate and the Moors who used base 10. Then the Spanish imposed base 10 on the Mayans who had collapsed under their own weight shortly before the Spanish showed up.   

    Had the Mayans had the wherewithal to travel to Europe and conquer perhaps we would all use base 20. 

    Or if the world was warmer and we all walked around barefoot we would use base 20. 

    One more thing. It could be argued that by way of our computer electronics we use base two way more than we use base 10 these days.   


    Good post. 
    Science advances as much by mistakes as by plans.
    Mark Changizi
    Nice.  One might argue that the only reason base-10 is even ever considered is because of our fingers.
    logicman
    Mark: I don't know if you are already familiar with this paper on polydactyly in fossils found in Greenland - in letters to Nature:

    http://pondside.uchicago.edu/oba/faculty/coates/23.CoatesClack1990polyda...

    The human hand is a wonderful piece of engineering.  It is a compromise design for shape and texture sensing in three dimensions and for grip, manipulation, impact and signalling.  Very handy.  ( Sorry, couldn't resist. )
    Mark Changizi
    Yes, I know that paper. (And although it wasn't part of my data set, because I'd need to know where the fingers actually separate, in terms of skin, their digit length relative to hand width is smaller than for humans, as the empirical limb expects.) Very footy, indeed. Mark