The great thing about a conversation like this one is that since no-one knows the answers we can have a lot of fun defending different viewpoints! I must admit, however, that I don't really have a strong opinion on this matter one way or the other, so it is wonderful for me to find myself attracted to what you are saying - and then to try to play devil's advocate with your ideas.

You say that there is a lot of applied math in biology. With my devil's advocate (tail?) on, I would say that really you are only allowed to say that that which we call (applied) math can be found in biology. I mean to say, that if you are saying that echolocation and the like actually are mathematics, then you already have a platonic view, to some extent. You're saying that there is an embodiment of an abstract
principle, and that the universe (or the bit of it identified with the bat) is somehow solving equations to work out the source of an echo. This may be true, or it may be true that our mathematics is (as I think you want to say) a language, honed by evolution, for describing patterns, some of which can be found in nature. So I think that there may be a contradiction in your advocacy of "finding math in nature" and "math being an evolutionary product". If it is not a contradiction, then it may instead be a circular argument, in that you say (a) math evolved, because (b) math is in nature, because (a) math evolved. The problem with this is the same as when I pointed out that to say that the bat is "doing" math (unconsciously, presumably) is to assume that that which we call math is identical to how the universe works. And this is unknown.

You also point to what are known as Darwinian preadaptations: incidental features of an organism which were not directly selected, but which later turned out to be advantageous in another, unforeseen way. Your example of feathers, which are hypothesised to have evolved for warmth and possibly plumage display, but which later enabled controlled flight, is a good one. You then seem to want to argue that math was selected
for its utility in the (energy? length? speed? mass?) range of our bodies, and then had a Darwinian preadaptation utility in more advanced forms of math. First, I'm not convinced that being able to do math consciously has a utility of any form. We may certainly need it subconsciously, like the bat, or like a recent study of a certain Brazilian tribe which showed that they have excellent spherical geometry intuition but not even natural numbers, arithmetic, or sense of time or object persistence. Secondly, even if the first point were true, we still come back to the question of why we should be able to extend math beyond the range of our bodies and still find it has utility. Just because Darwinian preadaptations exist doesn't help to explain our Big Question - in fact, it tells us almost nothing about it. Why should math extend this way? Why should the universe be comprehensible at all?

As a footnote of sorts, Stuart Kauffman has written a lot about the mathematical structure of evolution, and John Barrow has been one of the brave few to raise the question of the mystery of the universe's comprehensibility - and potential limits on our own understanding of it.

## A devil's advocate responds . . .

Hi Daniel.

The great thing about a conversation like this one is that since no-one knows the answers we can have a lot of fun defending different viewpoints! I must admit, however, that I don't really have a strong opinion on this matter one way or the other, so it is wonderful for me to find myself attracted to what you are saying - and then to try to play devil's advocate with your ideas.

You say that there is a lot of applied math in biology. With my devil's advocate (tail?) on, I would say that really you are only allowed to say that that which we call (applied) math can be found in biology. I mean to say, that if you are saying that echolocation and the like actually

aremathematics, then you already have a platonic view, to some extent. You're saying that there is an embodiment of an abstractprinciple, and that the universe (or the bit of it identified with the bat) is somehow solving equations to work out the source of an echo. This may be true, or it may be true that our mathematics is (as I think you want to say) a language, honed by evolution, for describing patterns, some of which can be found in nature. So I think that there may be a contradiction in your advocacy of "finding math in nature" and "math being an evolutionary product". If it is not a contradiction, then it may instead be a circular argument, in that you say (a) math evolved, because (b) math is in nature, because (a) math evolved. The problem with this is the same as when I pointed out that to say that the bat is "doing" math (unconsciously, presumably) is to assume that that which we call math is identical to how the universe works. And this is unknown.

You also point to what are known as Darwinian preadaptations: incidental features of an organism which were not directly selected, but which later turned out to be advantageous in another, unforeseen way. Your example of feathers, which are hypothesised to have evolved for warmth and possibly plumage display, but which later enabled controlled flight, is a good one. You then seem to want to argue that math was selected

for its utility in the (energy? length? speed? mass?) range of our bodies, and then had a Darwinian preadaptation utility in more advanced forms of math. First, I'm not convinced that being able to do math consciously has a utility of any form. We may certainly need it subconsciously, like the bat, or like a recent study of a certain Brazilian tribe which showed that they have excellent spherical geometry intuition but not even natural numbers, arithmetic, or sense of time or object persistence. Secondly, even if the first point were true, we still come back to the question of

whywe should be able to extend math beyond the range of our bodies and still find it has utility. Just because Darwinian preadaptations exist doesn't help to explain our Big Question - in fact, it tells us almost nothing about it. Why should math extend this way? Why should the universe be comprehensible at all?As a footnote of sorts, Stuart Kauffman has written a lot about the mathematical structure of evolution, and John Barrow has been one of the brave few to raise the question of the mystery of the universe's comprehensibility - and potential limits on our own understanding of it.