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Mathematical Explanations in Saatsi

In Discussion on March 15, 2010 by bcnjake

Something has been bothering me about Saatsi’s handling of the cicada case, and I’m curious what people think.  His account of a “non-mathematical” explanation for cicada periods doesn’t seem like a non-mathematical explanation to me.  Using sticks of various lengths to demonstrate why cicadas have life cycles of 13 or 17 years (or n years in any n-x – n+y period) still seems to depend on mathematics for explanation.  Simply because he’s using sticks instead of a pencil and paper doesn’t mean he’s not using mathematics to explain.

Even if you grant the stick explanation, no one would say that the sticks explain the cicada periods; we would still have the underlying question why do the sticks explain?  The only answer to that question I can think of involves number theory.  So, one of a few things is going on here.  (1) Saatsi is wrong and mathematics is explanatory in the cicada case.  (2) There’s some argument that I’m missing, which I grant is possible.  (3) Saatsi’s argument is wrong, but there’s a different argument that can be made in favor of non-mathematical explanation.

Thoughts?  I have my suspicions, but I fear they’re tainted by my inclination to endorse mathematical explanation.

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11 Responses to “Mathematical Explanations in Saatsi”

  1. I think that one thing that makes it difficult is that math represents so well, and we’re so used to it doing our representing for us, that it’s hard to wrap the mind around other representations. This is evidenced by the fact that Saatsi continues to use numbers even as he’s talking about the sticks he wants to replace them with.

    I find it easier to imagine God doing this (though why he needs to go through the stick intermediate I don’t know 🙂 ). God is able to pick sticks that are proportional to each other in the exact same way that the lengths of time are proportional. While we *can* put numbers to those proportions, God doesn’t need to and doesn’t bother. Laying out the proportional sticks shows that the ones that correspond to actual cycles overlap the least. This leads God to believe that the reason the actual cycles correspond is that when they are that length they will overlap the least.

    I’ve been thinking of the challenge for users of the indispensability argument in these terms generally. I.e. The Platonists challenge is to provide an example of scientific explanation in which God would not be able to avoid math in understanding it. Quite frankly, I’m skeptical that there is one.

  2. Staatsi’s argument regarding magnitudes being prime to one another is essentially a geometric argument. Euclid conducted a great deal of number theory geometricly in Book 5 to 10 of the Elements.

    Check out the definitions around numbers 10 and 11 of book 7. Euclid has a wicked kewl geomentric proof of the infinitude of prime numbers in book 10 (not directly on topic).

    http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/bookVII.html

    Pincock’s argument is stronger. We already believe mathematics is true, so all these belief inferences do not apply. I also take issue with the First proposition of EIA, but that is another matter entirely.

  3. Josh,

    In dealing with proportion and ratios (I assume that the matching between the sticks and lengths of time works because the ratios are equivalent), aren’t you still using math as an explanation?

    I see the argument as:
    [1] Sticks are proportional in the same way that time is proportional.
    [2] Therefore, the length of sticks maps directly onto time.
    [3] Sticks can be used to represent life cycles, since the length of a stick is analogous to a life cycle.
    [4] Sticks whose ends correspond with other sticks’ ends will have overlapping life cycles.
    [5] Sticks with the fewest corresponding ends will have the greatest evolutionary advantage.
    [6] Sticks that are proportional to cicada life cycles will be evolutionarily advantaged in this system, since they will have the fewest corresponding ends.
    [7] Therefore, sticks can be used to demonstrate the evolutionary advantage in cicada life cycles.

    But this argument depends on proportions, which are mathematical properties. Using Pincock’s replacement test, if we take out the proportion premise [1], the argument isn’t explanatory at all. You need [1] to make sense of [2] and [3], and you can’t get the rest of the argument without [2] and [3].

    Put differently, just because God wouldn’t need to put numbers to the proportions doesn’t mean that the proportions lack mathematical properties that explain; if the proportions lacked said properties, how would it still be true that cicada life cycles are evolutionarily advantaged? It seems that you would simply be asserting that this is the case, rather than explaining.

    Thoughts?

  4. I’m happy to see the debates have already started in advance of Tuesday’s meeting! My own take on the God perspective is that we lack complete knowledge of the natural world, and this is one big reason why mathematics is so useful in science.

    On Saatsi’s explanation in terms of sticks, I think that even if the explanation avoids the full mathematics of our theory of natural numbers, it can still be seen to be equivalent to a weaker mathematical theory. This is because a weak theory of numbers 1 through n (for some small n like 100) can be interpreted in these concrete terms. So, in terms of my draft, Saatsi is raising the problem of weaker alternatives.

    For those in the blogosphere, you might like the exchange from last summer here: http://hnsttl.blogspot.com/2009/07/colyvan-blocks-easy-road-to-nominalism.html

  5. I tried to post this on my own thread many hours ago, but the thing is only letting me submit new lines ‘for approval’ but seems to let me post at will in reply. Here is my only mostly related position:

    The Universe is contained by nothing (definition of universe).

    Therefore logic is contained within the universe and the universe is not contained by logic.

    The truths of mathematics are those things that logically follow from certain axioms, postulates, and definitions. There can be more than one mathematics as there are different choices one may make in foundational assumptions. Mathematics is an artifact of logic; it is the logical consequences of framing assumptions and definitions. See Saatsi’s footnotes,numbers 4&6, especially living in a non-Euclidean reality.

    If logic cannot contain the universe, then mathematics cannot contain the universe. Platonic forms of mathematical objects cannot contain or constrain the universe. For such things to contain or constrain the universe, they would exist outside the universe, but nothing is outside of everything.

    The universe has a logical and mathematical like appearance, but these things cannot contain or constrain existence. Something much stranger than platonic forms is going on here.

  6. Jake is challenging Sattsi’s conception of non-mathematical explanation. I think Sattsi’s main point is that we don’t need to appeal to prime numbers to explain the phenomenon. Jake is saying that Sattsi’s non-mathematical explanation still uses natural numbers.

    I am thinking that there might be two ways for an explanation to be mathematical. The first way is to use some terms in mathematics (e.g., natural numbers) but do not need the mathematical features to explain the phenomenon. If this explanation is regarded mathematical, then it is merely mathematical in a trivial sense. I take Sattsi’s non-mathematical explanation to be of this kind. The second way is to use mathematical terms (e.g., prime numbers) and the mathematical features of the terms to explain the phenomenon. The mathematical explanation Sattsi rejects is of this kind.

  7. Jake,

    [Replying as though I hadn’t read Chris’ reply yet.]

    I think it looks like this. We normally use abstracta, numbers, to talk about proportions, but that is not necessary. Saatsi’s case uses concreta, sticks, to do so instead. The numbers can certainly be used to represent the sticks as well as the cicada pattern. But someone with no knowledge of numbers (or who could consciously not use them, like God) could reason from the sticks directly to the patterns. (Of course, it would be difficult to get the proportions right without using a numbered measuring system, but that’s why I find it helpful to think about God doing it.)

    Basically I’m just not convinced that proportions are *inherently* mathematical, which is what I take you to be getting at when you say “Put differently, just because God wouldn’t need to put numbers to the proportions doesn’t mean that the proportions lack mathematical properties that explain…” Or perhaps you’re saying that just because we can explain in another way doesn’t mean that the math isn’t still explanatory. My response to that line would be that it may be true, but doesn’t seem like it’s helpful to the Platonist vis-a-vis the indispensability argument.

    Chris,

    Thanks for the commenting. Let me throw something out there that might apply to both of your comments, and see what you think about it:

    The Platonist is making a metaphysical claim. As such, any argument involving explanation has to be about *possible* explanation; otherwise it would only buy an epistemic conclusion. This is (really this time!) why I like thinking about it in God-terms. It’s also why I wouldn’t be worried about the possibility of reinterpreting the sticks as a weaker math.

    It’s *also* why I have a slight quibble with your replacement test as given in the paper. I’m fine with a test where there’s no longer explanation when the maths are removed. But it seems you also want to include a weakened explanation. This doesn’t seem right to me. After all, it seems that one could give a stronger explanation of the bending of light if one posited evil demons that deflected things a bit here and there. It would explain why relativistic predictions are close, but not exact. Still, we don’t take it as anything like strong evidence of the demons’ existence that the explanation is weaker when they are removed from it.

    I’m looking forward to talking with you about all this stuff tomorrow!

  8. Just a quick two-cents: I keep thinking about the difference between “how possibly” vs. “how actually” explanations when I see these posts. In my crude mind, mathematical descriptions sometimes (not always) provide a broad “how possibly” explanation. I say “broad” because it does more than provide counter-factual information, you don’t need to entertain possible worlds to understand how mathematical descriptions work.

    Notice, mathematical descriptions are multiply realizable in a very broad way. Again, I say “broad” because you don’t have to imagine any actual substance substituting for the parameters in the theory to understand the mathematical description. Second, they provide alternatives. Is this what you mean, Chris?

  9. My couple of coins on this subject matter:
    I agree with Jake and Chris that the stick system is still a mathematical claim, albeit a weaker one (the properties instantiated by the sticks are mathematical, just as the mathematical claim 1+1 = 2 can be instantiated with the manipulation of stones on a table). Therefore, Sattsi is not giving a non-mathematical explanation, but an equally explanatory weaker mathematical claim.

    However, I have a problem with this as being an “alternative” and “weaker” explanation, which is related to the issue brought up by Michael today. If the original explanandum is “why is the life-cycle period in prime numbers?” Then the original explanation is the best explanation, one that explains why the periods are “prime numbers.” The Sattsi explanation explains the question “why is the life-cycle period 13 and 17?” The prime number theory and the weaker stick-proportion theory are both explanations. Only then would the discussion of the inability to pick between the weaker and stronger explanation be meaningful.

    To Andre:
    When I think of multiply realizable, I think of laws. Otherwise, I see no reason of why we need to concern ourselves with unified mathematical descriptions given that they are merely coincidental. Today Chris mentioned that finding such regular patterns does not commit us to laws, however, the conclusion in the cicada, etc., cases are describing mixed biological/mathematical laws. How different is this from natural laws, or why would it be non-related to natural laws at all?

    I am very worried that I may have misunderstood something. Please correct me if I did.

  10. To everyone, thanks for a great discussion. I really got a lot out of it, and will try to improve my draft over the next month or so and let you know when it is available online.

    Andre: If we focus just on my type-(ii) case of isolating recurring features, then I agree that these mathematical explanations will usually involve abstract features which are multiply realizable across kinds of systems. So, right away, it is possible to see how these explanations unify by factoring out irrelevant details. For me this is a large part of the reason for the mathematics increasing the explanatory power when compared to a microphysical description.

    Lynn: As I said in the seminar I don’t have a clear view on laws, so I want to allow that some of these generalizations are not laws, but that they still play a role in explanation. In biology it seems like we often want to explain contingent events, or even patterns of events which are contingent on features of the particular history of the Earth. So, if what we are trying to explain is not a law, then I think we should be allowed to appeal to similar general claims when we give the explanation.

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